Studies in Fuzziness and Soft Computing

Ali EmrouznejadMadjid Tavana Editors

Performance Measurement with Fuzzy Data Envelopment Analysis

Studies in Fuzziness and Soft Computing

Volume 309

Series editor

Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Polande-mail: kacprzyk@ibspan.waw.pl

For further volumes:http://www.springer.com/series/2941

About this Series

The series ‘‘Studies in Fuzziness and Soft Computing’’ contains publications onvarious topics in the area of soft computing, which include fuzzy sets, rough sets,neural networks, evolutionary computation, probabilistic and evidential reasoning,multi-valued logic, and related fields. The publications within ‘‘Studies inFuzziness and Soft Computing’’ are primarily monographs and edited volumes.They cover significant recent developments in the field, both of a foundational andapplicable character. An important feature of the series is its short publication timeand world-wide distribution. This permits a rapid and broad dissemination ofresearch results.

Ali Emrouznejad • Madjid TavanaEditors

Performance Measurementwith Fuzzy DataEnvelopment Analysis

123

EditorsAli EmrouznejadOperations and Information

Management GroupAston Business SchoolAston UniversityBirminghamUK

Madjid TavanaBusiness Systems and Analytics DepartmentLa Salle UniversityPhiladelphiaUSA

ISSN 1434-9922 ISSN 1860-0808 (electronic)ISBN 978-3-642-41371-1 ISBN 978-3-642-41372-8 (eBook)DOI 10.1007/978-3-642-41372-8Springer Heidelberg New York Dordrecht London

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To our motherswho inspired us to teach

Preface

Since its introduction in 1978, Data Envelopment Analysis (DEA) has become oneof the preeminent non-parametric methods for measuring efficiency and produc-tivity of decision making units. DEA models are now employed routinely in areasthat range from assessment of public sectors such as hospitals and healthcaresystems, schools and universities to private sectors such as banks and financialinstitutions. The advantage of DEA is to accommodate multiple inputs and mul-tiple outputs for measuring the relative efficiencies of a set of homogeneousdecision making units (DMUs).

One limitation of the conventional DEA models is that they can only handlecrisp input and output data. However, the observed values of the input and outputdata in real-world problems are sometimes imprecise or vague. The aim of thisbook is to study various fuzzy methods for dealing with the imprecise andambiguous data in DEA. This monograph is the first in fuzzy DEA (FDEA).It contains both the authors’ research work on fuzzy DEA and other developments,especially in the last 10 years, and it is a good indication of the outgrowth of thefield of fuzzy data envelopment analysis.

With the exception of some basic notions in DEA and fuzzy theory, the book iscompletely self-contained. Important concepts in fuzziness and measuring effi-ciency are carefully motivated and introduced. Specifically, we have excluded anytechnical material that does not contribute directly to the understanding of fuzzy orDEA. Many other excellent textbooks are available today that discuss DEA inmuch more technical detail than is provided here. This book is aimed at upper-level undergraduate as well as beginning graduate students who want to learn moreabout fuzziness in DEA or who are pursuing research in fuzzy DEA and relatedareas.

The main objective of this book is to provide the necessary background to workwith existing fuzzy DEA models. Once the material in this book has been mas-tered, the reader will be able to apply fuzzy DEA models to his or her problems formeasuring comparative efficiency of decision making units with imprecise data.

To facilitate this goal, the Chap. 1 provides a literature review and summary ofthe current research in fuzzy DEA. In this introductory chapter we present aclassification scheme with six primary categories, namely, the tolerance approach,the a-level-based approach, the fuzzy ranking approach, the possibility approach,fuzzy arithmetic, and the fuzzy random/type-2 fuzzy set. We discuss each

vii

classification scheme and group the FDEA papers published in the literature overthe past three decades.

This is followed by Chap. 2, where K. Sam Park discusses the development ofimprecise DEA (IDEA) that broadens the scope of applications to efficiencyevaluations involving imprecise information such as various forms of ordinal andbounded data often occurring in practice. The primary purpose of Chap. 2 is toreview what has been developed so far, including the body of concepts andmethods that go by the name of IDEA. This chapter is a review of IDEA thatcomprises (a) why one may look at imprecise data and how to elicit impreciseinformation, (b) how to calculate the efficiency measures, and (c) how one caninterpret the resulting efficiency.

In Chap. 3, Pei-Huang Lin provides a framework for dealing with qualitativedata in DEA using fuzzy numbers. A two-level mathematical programming hasbeen developed to implement a fuzzy extension principle to a crisp DEA model tofind a-cuts of leveled fuzzy efficiency based on crisp observations and a-cuts offuzzy factors. This chapter, further, provides reliable fuzzy numbers representingqualitative data, using DEA models as experts to integrate objective productiondata and subjective information to generate possible values for qualitative data. Anapplication to university performance evaluation is used to demonstrate thisframework.

Y. Shen, E. Hermans, T. Brijs, and G. Wets, in Chap. 4, present the extension ofthe DEA-based composite indicators (CI) model by incorporating a fuzzy rankingapproach for modeling qualitative data. This model has been applied to construct acomposite alcohol performance index for road safety evaluation of a set ofEuropean countries. As discussed in this chapter, comparisons of the results withthe ones from the imprecise DEA-based CI model show the effectiveness of theproposed model in capturing the uncertainties associated with human thinking, andfurther imply the reliability of using this approach for modeling both quantitativeand qualitative data in the context of CI construction.

Further theoretical development in the area of fuzzy DEA is discussed inChap. 5, where I. Sirvent and T. León criticized the DEA method when used forranking the fuzzy efficiencies obtained from FDEA models. They further devel-oped the concept of fuzziness to cross-efficiencies evaluation. Hence their modelrelies on the dual multiplier formulation of the CCR model and the fuzzy effi-ciency of a given DMU is defined in ratio form in terms of the input and outputweights obtained. The advantage of this model is that it allows us to define thecross-efficiencies in a manner analogous to that of the fuzzy efficiency.

While most fuzzy DEA models are developed for radial efficiency, that dealsonly with proportional changes of inputs/outputs and neglects the input/outputslacks, Chap. 6 extends the concept of slack-based measure (SBM) model, whichis one of the commonly used non-radial models to fuzzy environment. In thischapter, J. Puri and S. Prasad Yadav extended the idea of mix-efficiency to fuzzyenvironments and developed the concept of fuzzy mix-efficiency in fuzzy DEA.This chapter provides both the input and output orientations of fuzzy mix-efficiency. Further, the a–cut approach is used to evaluate the fuzzy input as well

viii Preface

as fuzzy output mix-efficiencies for each DMU. Moreover, a new method isprovided for ranking the DMUs on the basis of fuzzy input and output mix-efficiencies. To ensure the validity of the proposed methodology, an application inbanking is used to illustrate the usefulness of the proposed methodology to thebanking sector in India.

An alternative ranking model for fuzzy efficiency measures through a formal-ized fuzzy DEA model has been developed in Chap. 7 by M. R. Ghasemi,J. Ignatius, and S. M. Davoodi. This chapter first identifies some drawbacks of thecurrent ranking models, ranging from the inability to provide satisfactory dis-crimination power to simplistic numerical examples that handle only symmetricalfuzzy numbers. As a result, a fuzzy DEA-CCR model using a linear rankingfunction is proposed to incorporate fuzzy inputs and fuzzy outputs that areasymmetrical in nature.

On another issue, DEA with inexact discretionary inputs are discussed inChap. 8 by M. Zerafat Angiz L. This chapter first explores the relationshipbetween fuzzy concepts and the efficiency score in DEA. Accordingly, a new DEAmodel for handling crisp data using the fuzzy concept is proposed. In addition, therelationship between possibility sets and the efficiency score in the traditional crispCCR model is presented which provides an alternative perspective of interpretingefficiency scores. Furthermore, based upon the proposed approaches, two non-discretionary models are introduced in which some inputs or outputs, in a fuzzysense, are inexact discretionary variables.

Chapter 9 explains how conventional DEA treats a system as a whole unit whenmeasuring efficiency, ignoring the operations of the component processes. Net-work DEA, on the other hand, takes the component processes into consideration,with results that are more representative and can be used to identify inefficientcomponents. In this chapter, C. Kao discusses network DEA for fuzzy observa-tions and has developed two approaches, the membership grade and the a-cut,which are proposed for measuring the system and process efficiencies via two-level mathematical programming. The model associated with the latter approach istransformed into a conventional one-level program so that the existing solutionmethods can be applied. Since the data are fuzzy, the measured efficiencies arealso fuzzy.

Along the same line of network DEA, Chap. 10 provides a general approach tohandle fuzzy data when units under assessments are formed by a network ofprocesses. S. Lozan and P. Moreno first discussed conventional network DEAapproaches with crisp data and then presented further development of networkDEA with uncertainty that used fuzzy data. Then they explained the need to dealwith general networks of processes which can have fuzzy input or output data.In this chapter, several fuzzy DEA approaches are extended to network DEA.The resulting models are illustrated using a dataset from the literature.

The remainder of this book is devoted to some applications of fuzzy DEAstarting with Chap. 11, an application of fuzzy DEA approach for occupationalsafety. Esra Bas proposed an integrated fuzzy DEA for evaluating differentdepartments of a company with respect to the occupational safety investment.

Preface ix

In another application, Chap. 12 proposed integrating fuzzy intermediatefactors in supply chain efficiency evaluation. Q. Xia, L. Liang and F. Yangdiscussed that the effective supply chain management (SCM) depends on thereasonable performance evaluation to the entire supply chain. This chapter inte-grates fuzzy intermediate factors in supply chain efficiency evaluation and pro-poses a fuzzy supply chain data envelopment analysis (FSCDEA) model based onprevious supply chain DEA models. Finally, the chapter uses the FSCDEA modelto assess the operational efficiency of a group of bank branches.

As another application of DEA in supplier evaluation, A. Amindoust andA. Saghafinia, in Chap. 13, proposed an a-cut approach fuzzy DEA model withlinguistic ratings for the assessment of candidate suppliers. A hypothetical appli-cation is provided to demonstrate the applicability and feasibility of the method.

With a different perspective but in the same area of supply change, A. Awasthi,K. Noshad, and S. Singh Chauhan, in Chap. 14 present a model for supplierperformance evaluation using a hybrid fuzzy DEA approach. Hence, this chapterpresents a multi-stage approach based on fuzzy DEA for supplier quality evalu-ation. The criteria for supplier performance evaluation are obtained using theDelphi technique. The hierarchy for the criteria and preferential relations betweenthem are developed using the principles of the Analytic Hierarchy Process (AHP).Supplier performance evaluation is performed using fuzzy data envelopmentanalysis.

November 2013 Ali EmrouznejadMadjid Tavana

x Preface

Acknowledgments

This book would not have been possible without the help of a number of people.We are grateful to the series editor, Professor Janusz Kacprzyk, for his support. Wewould also like to extend our appreciation to the contributors and reviewers fortheir critical review of the chapters and the insightful comments and suggestionsprovided. The editors would like to thank Dr. Holger Schaepe (Springer Editor,Studies in Fuzziness and Soft Computing), Mr Shine David, and Miss Ramya-krishnan Murugesan for their editorial assistance in producing this volume. Wehope the readers will share our excitement with this important scientific contri-bution to the body of knowledge in Fuzzy Data Envelopment Analysis.

xi

Contents

1 The State of the Art in Fuzzy Data Envelopment Analysis . . . . . . . 1Ali Emrouznejad, Madjid Tavana and Adel Hatami-Marbini

2 Imprecise Data Envelopment Analysis: Concepts, Methods,and Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47K. Sam Park

3 A General Framework of Dealing with Qualitative Datain DEA: A Fuzzy Number Approach . . . . . . . . . . . . . . . . . . . . . . 61Pei Huang Lin

4 Fuzzy Data Envelopment Analysis in CompositeIndicator Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Yongjun Shen, Elke Hermans, Tom Brijs and Geert Wets

5 Cross-Efficiency in Fuzzy Data EnvelopmentAnalysis (FDEA): Some Proposals . . . . . . . . . . . . . . . . . . . . . . . . 101Inmaculada Sirvent and Teresa León

6 Fuzzy Mix-efficiency in Fuzzy Data EnvelopmentAnalysis and Its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Jolly Puri and Shiv Prasad Yadav

7 Ranking of Fuzzy Efficiency Measures via Satisfaction Degree. . . . 157M.-R. Ghasemi, Joshua Ignatius and S. M. Davoodi

8 Inexact Discretionary Inputs in Data Envelopment Analysis . . . . . 167Majid Zerafat Angiz Langroudi

9 Network Data Envelopment Analysis with Fuzzy Data . . . . . . . . . 191Chiang Kao

10 Network Fuzzy Data Envelopment Analysis . . . . . . . . . . . . . . . . . 207Sebastián Lozano and Plácido Moreno

xiii

11 An Application of Fuzzy Data Envelopment AnalysisApproach for Occupational Safety . . . . . . . . . . . . . . . . . . . . . . . . 231Esra Bas

12 Integrating Fuzzy Intermediate Factors in SupplyChain Efficiency Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243Qiong Xia, Liang Liang and Feng Yang

13 Supplier Evaluation and Selection Using a FDEA Model . . . . . . . . 255Atefeh Amindoust and Ali Saghafinia

14 Supplier Performance Evaluation Using a HybridFuzzy Data Envelopment Analysis Approach . . . . . . . . . . . . . . . . 271Anjali Awasthi, Khoshrow Noshad and Satyaveer Singh Chauhan

Biography of Editors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

xiv Contents

Chapter 1The State of the Art in Fuzzy DataEnvelopment Analysis

Ali Emrouznejad, Madjid Tavana and Adel Hatami-Marbini

Abstract Data envelopment analysis (DEA) is a methodology for measuring therelative efficiencies of a set of decision making units (DMUs) that use multipleinputs to produce multiple outputs. Crisp input and output data are fundamentallyindispensable in conventional DEA. However, the observed values of the input andoutput data in real-world problems are sometimes imprecise or vague. Manyresearchers have proposed various fuzzy methods for dealing with the impreciseand ambiguous data in DEA. This chapter provides a taxonomy and review ofthe fuzzy DEA (FDEA) methods. We present a classification scheme with sixcategories, namely, the tolerance approach, the a-level based approach, the fuzzyranking approach, the possibility approach, the fuzzy arithmetic, and the fuzzyrandom/type-2 fuzzy set. We discuss each classification scheme and group theFDEA papers published in the literature over the past 30 years.

An earlier version of this chapter was published as Hatami-Marbini et al. [1].

A. Emrouznejad (&)Aston Business School, Aston University, Birmingham, UKe-mail: a.emrouznejad@aston.ac.uk

M. TavanaLindback Distinguished Chair of Information Systems and Decision Sciences, BusinessSystems and Analytics Department, La Salle University, Philadelphia, PA 19141, USAe-mail: tavana@lasalle.eduURL: http://tavana.us

M. TavanaBusiness Information Systems Department, Faculty of Business Administration andEconomics, University of Paderborn, 33098 Paderborn, Germany

A. Hatami-MarbiniLouvain School of Management, Center of Operations Research and Econometrics (CORE),Université catholique de Louvain, L1.03.01 1348 Louvain-la-Neuve, Belgiume-mail: adel.hatamimarbini@uclouvain.be

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_1, � Springer-Verlag Berlin Heidelberg 2014

1

Keywords Data envelopment analysis � Fuzzy sets � Tolerance approach � a-levelbased approach �Fuzzy ranking approach �Possibility approach �Fuzzy arithmetic �Fuzzy random � Type-2 fuzzy set

1 Introduction

Data envelopment analysis (DEA) was first proposed by Charnes et al. [2], and is anon-parametric method of efficiency analysis for comparing units relative to theirbest peers (efficient frontier). Mathematically, DEA is a linear programming(LP)-based methodology for evaluating the relative efficiency of a set of decisionmaking units (DMUs) with multi-inputs and multi-outputs. DEA evaluates theefficiency of each DMU relative to an estimated production possibility frontierdetermined by all the DMUs. The advantage of using DEA is that it does notrequire any assumption on the shape of the frontier surface and it makes noassumptions concerning the internal operations of a DMU. Since the original DEAstudy by Charnes et al. [2], there has been a continuous growth in the field. As aresult, a considerable amount of published research papers and bibliographies haveappeared in the DEA literature, including those of Seiford [3], Gattoufi et al. [4],Emrouznejad et al. [5], and Cook and Seiford [6].

The conventional DEA methods require accurate measurement of both theinputs and outputs. However, the observed values of the input and output data inreal-world problems are sometimes imprecise or vague. Imprecise evaluations maybe the result of unquantifiable, incomplete and non-obtainable information. Someresearchers have proposed various fuzzy methods for dealing with this impre-ciseness and ambiguity in DEA. Since the original study by Sengupta [7, 8] therehas been a continuous interest and increased development in fuzzy DEA (FDEA)literature. In this study, we review the FDEA methods and present a taxonomyby classifying the FDEA papers published over the past two decades into sixprimary categories, namely, the tolerance approach, the a-level based approach,the fuzzy ranking approach, the possibility approach, the fuzzy arithmetic, and thefuzzy random/type-2 fuzzy set as well as a secondary category to group the pio-neering papers that do not fall into the six primary classifications. This studyupdates the previous review of Hatami-Marbini et al. [1] on FDEA and it providesthe complete source of references on FDEA since its inception two decades ago.This chapter is organized into five sections. In Sect. 2, we present the funda-mentals of DEA. In Sect. 3, we review the FDEA principles. In Sect. 4, we presenta summary development of the FDEA followed by a detailed description of theFDEA methods in the literature. Conclusions and future research directions arepresented in Sect. 5.

2 A. Emrouznejad et al.

2 The Fundamentals of DEA

There are basically two main types of DEA models: a constant returns-to-scale(CRS) or CCR model that was initially introduced by Charnes et al. [2] and avariable returns-to-scale (VRS) or BCC model that was later developed by Bankeret al. [9]. The BCC model is one of the extensions of the CCR model where theefficient frontiers set is represented by a convex curve passing through all efficientDMUs.

DEA can be either input- or output-orientated. In the first case, the DEA methoddefines the frontier by seeking the maximum possible proportional reduction ininput usage, with output levels held constant, for each DMU. However, for theoutput-orientated case, the DEA method seeks the maximum proportional increasein output production, with input levels held fixed.

Figure 1 illustrates a simple VRS output-oriented DEA problem with twooutputs, Y and Z, and one input, X. The isoquant L1L2 represents the technicalefficient frontier comprising P1, P2, and P3 which are technically efficient DMUsand hence on the frontier. If a given DMU uses one unit of input and producesoutputs defined by point P, the technical inefficiency of that DMU is represented asthe distance PP0, which is the amount by which all outputs could be proportionallyincreased without increasing the input. In percentage terms, it is expressed by theratio OP/OP0, which is the ratio by which all the outputs could be increased.

An input oriented DEA model with m input variables (x1; . . .; xm), s outputvariables (y1; . . .; ys) and n decision making units (j ¼ 1; 2; . . .; n) is presented inModel 1a (for CCR model) and Model 1b (for BCC model). The only differencebetween these two models is the inclusion of the convexity constraints ofPn

j¼1 kj ¼ 1 in the BCC model.

Model 1a: A basic CCR model Model 1b: A basic BCC model

min hp

s:t:Xn

j¼1kjxij� hpxip; 8i;

Xn

j¼1kjyrj� yrp; 8r;

kj� 0; 8j:

min hp

s:t:Xn

j¼1kjxij� hpxip; 8i;

Xn

j¼1kjyrj� yrp; 8r;

Xn

j¼1kj ¼ 1; kj� 0; 8j:

DEA applications are numerous in many applications such as financial services,agricultural, health care services, education, manufacturing, telecommunication,supply chain management, and many more. For a recent comprehensive bibliog-raphy of DEA see Emrouznejad et al. [5]. Recently fuzzy logic has been intro-duced to DEA for measuring efficiency of DMUs under uncertainty mainly whenthe precise data is not available. The rest of this chapter focuses on the use of fuzzysets in DEA.

1 The State of the Art in Fuzzy Data Envelopment Analysis 3

3 The FDEA Principles

The observed values in real-world problems are often imprecise or vague.Imprecise or vague data may be the result of unquantifiable, incomplete and non-obtainable information. Imprecise or vague data is often expressed with boundedintervals, ordinal (rank order) data or fuzzy numbers. In recent years, manyresearchers have formulated FDEA models to deal with situations where some ofthe input and output data are imprecise or vague.

3.1 Fuzzy Set Theory

The theory of fuzzy set has been developed to deal with the concept of partial truthvalues ranging from absolutely true to absolutely false. Fuzzy set theory hasbecome the prominent tool for handling imprecision or vagueness aiming attractability, robustness and low-cost solutions for real-world problems. Accordingto Zadeh [10], it is very difficult for conventional quantification to reasonablyexpress complex situations and it is necessary to use linguistic variables whosevalues are words or sentences in a natural or artificial language. The potential ofworking with linguistic variables, low computational cost and easiness of under-standing are characteristics that have contributed to the popularity of thisapproach. Fuzzy set algebra developed by Zadeh [11] is the formal body of theorythat allows the treatment of imprecise and vague estimates in uncertainenvironments.

Zadeh [11], p. 339 states ‘‘The notion of a fuzzy set provides a convenient point ofdeparture for the construction of a conceptual frame-work which parallels in manyrespects the framework used in the case of ordinary sets, but is more general thatthe latter and, potentially, may prove to have a much wider scope of applicability.’’

Fig. 1 An output-orientedDEA with two outputs andone input

4 A. Emrouznejad et al.

The application of fuzzy set theory in multi-attribute decision-making (MADM)became possible when Bellman and Zadeh [12] and Zimmermann [13] introducedfuzzy set into the field of MADM. They cleared the way for a new family of methodsto deal with problems that had been unapproachable and unsolvable with standardtechniques [see Chen and Hwang [14] for a numerical comparison of fuzzy andclassical MADM models]. Bellman and Zadeh’s [12] framework was based on themaximin principle and the simple additive weighing model of Yager and Basson[15] and Bass and Kwakernaak [16]. Bass and Kwakernaak’s [16] method is widelyknown as the classic work of fuzzy MADM methods.

In 1992, Chen and Hwang [14] proposed an easy-to-use and easy-to-understandapproach to reduce some of the cumbersome computations in the previous MADMmethods. Their approach includes two steps: (1) converting fuzzy data into crispscores; and (2) introducing some comprehensible and easy methods. In additionChen and Hwang [14] made distinctions between fuzzy ranking methods and fuzzyMADM methods. Their first group contained a number of methods for finding aranking: degree of optimality, Hamming distance, comparison function, fuzzymean and spread, proportion to the ideal, left and right scores, area measurement,and linguistic ranking methods. Their second group was built around methods forassessing the relative importance of multiple attributes: fuzzy simple additiveweighting methods, fuzzy analytic hierarchy process, fuzzy conjunctive/disjunctivemethods, fuzzy outranking methods, and fuzzy maximin methods. The group withthe most frequent contributions is fuzzy mathematical programming. Inuiguchiet al. [17] have provided a useful survey of fuzzy mathematical programmingapplications including: flexible programming, possibilistic programming, possibi-listic programming with fuzzy preference relations, possibilistic LP using fuzzymax, possibilistic LP with fuzzy goals, and robust programming.

Recently, fuzzy set theory has been applied to a wide range of fields such asmanagement science, decision theory, artificial intelligence, computer science,expert systems, logic, control theory and statistics, among others [18–30].

3.2 Fuzzy Set Theory and DEA

The data in the conventional CCR and BCC models assume the form of specificnumerical values. However, the observed value of the input and output data aresometimes imprecise or vague. Sengupta [7, 8] was the first to introduce a fuzzymathematical programming approach in which fuzziness was incorporated into theDEA model by defining tolerance levels on both the objective function and con-straint violations.

Let us assume that n DMUs consume varying amounts of m different inputs toproduce s different outputs. Assume that ~xij (i ¼ 1; 2; . . .;m) and ~yrj

(r ¼ 1; 2; . . .; s) represent, respectively, the fuzzy input and fuzzy output of the jth

1 The State of the Art in Fuzzy Data Envelopment Analysis 5

DMUj (j ¼ 1; 2; . . .; n). The primal and its dual fuzzy CCR models in input-orientedversion can be formulated as:

Primal CCR model (input-oriented)

min hp

s:t:Xn

j¼1

kj~xij� hp~xip; 8i;

Xn

j¼1

kj~yrj�~yrp; 8r;

kj� 0; 8j:

ð1Þ

Dual CCR model (input-oriented)

max hp ¼Ps

r¼1ur~yrp

s:t:Pm

i¼1vi~xip ¼ 1;

Ps

r¼1ur~yrj �

Pm

i¼1vi~xij� 0; 8j;

ur; vi� 0; 8r; i:

ð2Þ

where vi and ur in model (2) are the input and output weights assigned to the ithinput and rth output. If the constraint

Pnj¼1 kj ¼ 1 is adjoined to (1), a fuzzy BCC

model is obtained and this added constraint introduces an additional variable, ~u0,into the dual model where these models are respectively shown as follows:

min hp

s:t:Pn

j¼1kj~xij� hp~xip; 8i;

Pn

j¼1kj~yrj�~yrp; 8r;

Pn

j¼1kj ¼ 1;

kj� 0; 8j:

ð3Þ

max wp ¼Ps

r¼1ur~yrp þ u0

s:t:Pm

i¼1vi~xip ¼ 1;

Ps

r¼1ur~yrj �

Pm

i¼1vi~xij þ u0� 0; 8j;

ur; vi� 0; 8r; i:

ð4Þ

6 A. Emrouznejad et al.

4 The FDEA Methods

The applications of fuzzy set theory in DEA are usually categorized into fourgroups Lertworasirikul et al. [31, 32], Lertworasirikul [33], Karsak [34]: (1) Thetolerance approach, (2) The a-level based approach, (3) The fuzzy rankingapproach, (4) The possibility approach. In this study, we expand this classificationand add two new groups: the fuzzy arithmetic and the fuzzy random/type-2 fuzzyset.

In this section, we provide a mathematical description of each approach fol-lowed by a brief review of the most widely cited literature relevant to each of thesix approaches. In addition to the six above mentioned approaches, we introduce anew category to group the pioneering papers that do not fall into any of the aboveclassifications. A summary development of the FDEA is listed in Table 1.

4.1 The Tolerance Approach

The tolerance approach was one of the first FDEA models that was developed bySengupta [7] and further improved by Kahraman and Tolga [35]. In this approachthe main idea is to incorporate uncertainty into the DEA models by definingtolerance levels on constraint violations. This approach fuzzifies the inequality orequality signs but it does not treat fuzzy coefficients directly. The intricate limi-tation of the tolerance approach proposed by Sengupta [7] is related to the designof a DEA model with a fuzzy objective function and fuzzy constraints which mayor may not be satisfied (Triantis and Girod [36]. Although in most productionprocesses fuzziness is present both in terms of not meeting specific objectives andin terms of the imprecision of the data, the tolerance approach provides flexibilityby relaxing the DEA relationships while the input and output coefficients aretreated as crisp.

4.2 The a-Level Based Approach

The a-level approach is perhaps the most popular FDEA model. This is evident bythe number of a-level based papers published in the FDEA literature. In thisapproach the main idea is to convert the FDEA model into a pair of parametricprograms in order to find the lower and upper bounds of the a-level of themembership functions of the efficiency scores. Girod [37] used the approachproposed by Carlsson and Korhonen [38] to formulate the fuzzy BCC and freedisposal hull (FDH) models which were radial measures of efficiency. In thismodel, the inputs could fluctuate between risk-free (upper) and impossible (lower)bounds and the outputs could fluctuate between risk-free (lower) and impossible

1 The State of the Art in Fuzzy Data Envelopment Analysis 7

Tab

le1

Fuz

zyD

EA

refe

renc

ecl

assi

fica

tion

from

1999

to20

13(1

73P

aper

s)

The

tole

ranc

eap

proa

ch(2

Pap

ers)

Sen

gupt

a[7

]S

engu

pta

[8]

The

a-le

vel

base

dap

proa

ch(7

7P

aper

s)L

iu[6

4]K

hali

li-D

amgh

ani

and

Tav

ana

[65]

Mug

era

[66]

Kha

lili

-Dam

ghan

ian

dT

agha

vifa

rd[1

28]

Che

net

al.

[67]

Fat

hian

dIz

adik

hah

[79]

Sri

niva

saR

aju

and

Nag

esh

Kum

ar[8

5]H

atam

i-M

arbi

niet

al.

[86]

Saa

tiet

al.

[87]

Pur

ian

dY

adav

[59]

Aza

deh

etal

.[8

2]W

ang

and

Yan

[127

]K

aoan

dL

in[6

2]A

zade

het

al.

[81]

Zho

uet

al.

[91]

Zho

uet

al.

[93]

Hat

ami-

Mar

bini

etal

.[8

0]K

hali

li-D

amgh

ani

and

Hos

sein

zade

hL

otfi

[125

]Z

eraf

atA

ngiz

etal

.[2

31]

Abt

ahi

and

Kha

lili

-Dam

ghan

i[1

21]

Kha

lili

-Dam

ghan

ian

dT

agha

vifa

rd[1

24]

Gha

panc

hiet

al.

[83]

Rez

aie

etal

.[8

4]K

hali

li-D

amgh

ani

etal

.[1

26]

Mos

tafa

ee[1

18]

Kho

shfe

trat

and

Dan

eshv

ar[1

20]

Kha

lili

-Dam

ghan

ian

dA

btah

i[2

35]

Kao

and

Liu

[60]

Kao

and

Lin

[54]

Nou

raet

al.

[115

]Z

hou

etal

.[9

0]A

zade

het

al.

[77]

Man

sour

irad

etal

.[1

17]

Aza

deh

etal

.[7

7]A

zade

han

dA

lem

[107

]Z

eraf

atA

ngiz

etal

.[1

14]

Hat

ami-

Mar

bini

etal

.[7

4]C

hian

gan

dC

he[5

8]S

aati

and

Mem

aria

ni[7

5]L

iuan

dC

huan

g[9

2]W

ang

etal

.[1

10]

Nou

raan

dS

aljo

oghi

[109

]T

lig

and

Reb

ai[1

13]

Hat

ami-

Mar

bini

and

Saa

ti[7

3]H

osse

inza

deh

Lot

fiet

al.

[111

]Ja

hans

hahl

ooet

al.

[89]

Li

and

Yan

g[5

6]L

iu[8

8]G

hapa

nchi

etal

.[7

2]A

zade

het

al.

[106

]K

arsa

k[3

4]Ja

hans

hahl

ooet

al.

[100

]K

aoan

dL

iu[5

3]S

anei

fard

etal

.[9

8]A

zade

het

al.

[71]

Liu

etal

.[9

7]A

llah

vira

nloo

etal

.[1

02]

Hos

sein

zade

hL

otfi

etal

.[1

03]

Saa

tian

dM

emar

iani

[69]

Hsu

[96]

Wu

etal

.[7

0]K

aoan

dL

iu[5

1]Z

hang

etal

.[5

2]K

uoan

dW

ang

[55]

Ent

ani

etal

.[9

5]T

rian

tis

[40]

Kao

and

Liu

[49]

Saa

tiet

al.

[68]

Guh

[48]

Kao

[47]

Che

n[1

8]K

aoan

dL

iu[4

6]

(con

tinu

ed)

8 A. Emrouznejad et al.

Tab

le1

(con

tinu

ed)

The

tole

ranc

eap

proa

ch(2

Pap

ers)

Kao

and

Liu

[42]

Gir

odan

dT

rian

tis

[39]

Tri

anti

san

dG

irod

[36]

Mae

daet

al.

[41]

Gir

od[3

7]T

hefu

zzy

rank

ing

appr

oach

(43

Pap

ers)

Bei

ranv

and

etal

.[1

41]

Aza

deh

etal

.[1

45]

Ahm

ady

etal

.[1

71]

Am

indo

ust

etal

.[1

80]

Sef

eedp

ari

etal

.[1

39]

Cha

ngan

dL

ee[1

34]

Moh

eb-A

liza

deh

etal

.[1

79]

Hat

ami-

Mar

bini

etal

.[1

]E

mro

uzne

jad

etal

.[1

70]

Aza

deh

etal

.[1

43]

Aza

deh

etal

.[1

69]

Hat

ami-

Mar

bini

etal

.[1

68]

Aza

deh

etal

.[1

44]

Hat

ami-

Mar

bini

etal

.[1

40]

San

eiet

al.

[132

]H

osse

inza

deh

Lot

fiet

al.

[178

]Ja

hans

hahl

ooet

al.

[165

]G

uo[1

31]

Sol

eim

ani-

dam

aneh

[162

]H

atam

i-M

arbi

niet

al.

[163

]Ju

an[1

75]

Bag

herz

adeh

vala

mi

[177

]H

osse

inza

deh

Lot

fiet

al.

[167

]H

osse

inza

deh

Lot

fian

dM

anso

uri

[155

]Z

hou

etal

.[1

56]

Guo

and

Tan

aka

[130

]Ja

hans

hahl

ooet

al.

[159

]N

oora

and

Kar

ami

[157

]S

olei

man

i-da

man

eh[1

61]

Hos

sein

zade

hL

otfi

etal

.[1

50]

Hos

sein

zade

hL

otfi

etal

.[1

49]

Pal

etal

.[1

53]

Jaha

nsha

hloo

etal

.[1

52]

Saa

tian

dM

emar

iani

[142

]S

olei

man

i-da

man

ehet

al.

[147

]L

eeet

al.

[174

]M

olav

iet

al.

[146

]Ja

hans

hahl

ooet

al.

[76]

Lee

[173

]D

ia[1

72]

Leó

net

al.

[136

]L

ertw

oras

irik

ul[3

3]G

uoan

dT

anak

a[1

29]

The

poss

ibil

ity

appr

oach

(21

Pap

ers)

Pay

anan

dS

hari

fi[2

02]

Ned

eljk

ovic

and

Dre

nova

c[1

90]

Zha

oan

dY

ue[1

89]

Wen

etal

.[1

98]

Wan

gan

dC

hin

[201

]H

ossa

inza

deh

etal

.[2

00]

Lin

[188

]K

hoda

bakh

shi

etal

.[1

94]

Wen

etal

.[1

96]

Wen

and

Li

[195

]Ji

ang

and

Yan

g[1

93]

Wu

etal

.[1

87]

Ram

ezan

zade

het

al.

[191

]G

arci

aet

al.

[186

]L

ertw

oras

irik

ulet

al.

[31]

Ler

twor

asir

ikul

etal

.[3

2]L

ertw

oras

irik

ulet

al.

[185

]L

ertw

oras

irik

ul[3

3]L

ertw

oras

irik

ulet

al.

[183

]L

ertw

oras

irik

ulet

al.

[184

]G

uoet

al.

[182

]

(con

tinu

ed)

1 The State of the Art in Fuzzy Data Envelopment Analysis 9

Tab

le1

(con

tinu

ed)

The

tole

ranc

eap

proa

ch(2

Pap

ers)

The

fuzz

yar

ithm

etic

(11

Pap

ers)

Aza

diet

al.

[212

]R

azav

iH

ajia

gha

etal

.[2

11]

Ale

met

al.

[210

]M

irhe

daya

tian

etal

.[2

36]

Mir

heda

yati

anet

al.

[237

]M

irhe

daya

tian

etal

.[2

09]

Jafa

rian

-Mog

hadd

aman

dG

hose

iri

[206

]R

aei

Noj

ehde

hiet

al.

[208

]A

bdol

iet

al.

[205

]W

ang

etal

.[2

04]

Wan

get

al.

[203

]T

hefu

zzy

rand

om/t

ype-

2fu

zzy

set

(7P

aper

s)Z

eraf

atA

ngiz

etal

.[2

20]

Tav

ana

etal

.[2

22]

Tav

ana

etal

.[2

19]

Qin

etal

.[2

17]

Qin

and

Liu

[215

]Q

inan

dL

iu[2

16]

Qin

etal

.[2

14]

Oth

erde

velo

pmen

tsin

fuzz

yD

EA

(12

Pap

ers)

Zer

afat

Ang

izan

dM

usta

fa[2

32]

Bag

herz

adeh

Val

ami

etal

.[2

33]

Zer

afat

Ang

izet

al.

[231

]Z

eraf

atA

ngiz

etal

.(2

010)

Zer

afat

Ang

izet

al.

[230

]Z

eraf

atA

ngiz

etal

.[2

28]

Qin

and

Liu

[214

]L

uban

[227

]U

emur

a[2

26]

Hou

gaar

d[2

25]

She

than

dT

rian

tis

[224

]H

ouga

ard

[223

]

10 A. Emrouznejad et al.

(upper) bounds. Triantis and Girod [36] followed up by introducing the fuzzy LPapproach to measure technical efficiency based on Carlsson and Korhonen’s [38]framework. Their approach involved three stages: First, the imprecise inputs andoutputs were determined by the decision maker in terms of their risk-free andimpossible bounds. Second, three fuzzy CCR, BCC and FDH models wereformulated in terms of their risk-free and impossible bounds as well as theirmembership function for different values of a. Third, they illustrated the imple-mentation of their fuzzy BCC model in the context of a preprint and packaging linewhich inserts commercial pamphlets into newspapers. Furthermore, their paperwas clarified in detail using the implementation road map by Girod and Triantis[39]. Triantis [40] extended his earlier work on FDEA [36] to fuzzy non-radialDEA measures of technical efficiency in support of an integrated performancemeasurement system. He also compared his method to the radial technical effi-ciency of the same manufacturing production line which was described in detail byGirod [37] and Girod and Triantis [39]. Meada et al. [41] used the a-level basedapproach to obtain the fuzzy interval efficiency of DMUs.

Kao and Liu [42] followed up on the basic idea of transforming a FDEA modelto a family of conventional crisp DEA models and developed a solution procedureto measure the efficiencies of the DMUs with fuzzy observations in the BCCmodel. Their method found approximately the membership functions of the fuzzyefficiency measures by applying the a-level approach and Zadeh’s extensionprinciple Zadeh [43], Zimmermann [44]. They transformed the FDEA model to apair of parametric mathematical programs and used the ranking fuzzy numbersmethod proposed by Chen and Klein [45] to obtain the performance measure of theDMU. Solving this model at the given level of a-level produced the intervalefficiency for the DMU under consideration. A number of such intervals could beused to construct the corresponding fuzzy efficiency. Assume that there aren DMUs under consideration. Each DMU consumes varying amounts of m dif-ferent fuzzy inputs to produce s different fuzzy outputs. Specifically, DMUj con-sumes amounts ~xij of inputs to produce amounts ~yrj of outputs. In the modelformulation, ~xip and ~yrp denote, respectively, the input and output values for theDMUp. In order to solve the fuzzy BCC model (4), Kao and Liu [42] proposed apair of two-level mathematical models to calculate the lower bound ðwpÞLa and

upper bound ðwpÞUa of the fuzzy efficiency score for a specific a-level as follows:

ðwpÞLa ¼ minðXijÞLa � xij �ðXijÞUaðYrjÞLa � yrj �ðYrjÞUa

8r;i;j

~wp ¼ maxXs

r¼1

uryrp þ u0

s:t:Xm

i¼1

vixip ¼ 1;

Xs

r¼1

uryrj �Xm

i¼1

vixij þ u0� 0; 8j;ur ;vi � 0; 8r;i:

8>>>>>>>>>><

>>>>>>>>>>:

ð5Þ

1 The State of the Art in Fuzzy Data Envelopment Analysis 11

ðwpÞUa ¼ maxðXijÞLa � xij �ðXijÞUaðYrjÞLa � yrj �ðYrjÞUa

8r;i;j

~wp ¼ maxXs

r¼1

uryrp þ u0

s:t:Xm

i¼1

vixip ¼ 1;

Xs

r¼1

uryrj �Xm

i¼1

vixij þ u0� 0; 8j;

ur; vi� 0; 8r; i:

8>>>>>>>>>><

>>>>>>>>>>:

ð6Þ

where ðXijÞLa ; ðXijÞUa� �

and ðYrjÞLa ; ðYrjÞUa� �

are a-level form of the fuzzy inputsand the fuzzy outputs respectively. This two-level mathematical model can besimplified to the conventional one-level model as follows:

ðwpÞLa ¼ maxXs

r¼1

urðYrpÞLa þ u0

s:t:Xs

r¼1

urðYrpÞLa �Xm

i¼1

viðXipÞUa þ u0� 0;

Xs

r¼1

urðYrjÞUa �Xm

i¼1

viðXijÞLa þ u0� 0; 8j;j 6¼ p;

Xm

i¼1

viðXipÞUa ¼ 1; ur; vi� 0; 8r; i:

ð7Þ

ðwpÞUa ¼ maxXs

r¼1

urðYrpÞUa þ u0

s:t:Xs

r¼1

urðYrpÞUa �Xm

i¼1

viðXipÞLa þ u0� 0;

Xs

r¼1

urðYrjÞLa �Xm

i¼1

viðXijÞUa þ u0� 0; 8j;j 6¼ p;

Xm

i¼1

viðXipÞLa ¼ 1; ur; vi� 0; 8r; i:

ð8Þ

Next, a membership function is built by solving the lower and upper boundsðwpÞLa ; ðwpÞUa� �

of the a-levels for each DMU using models (7) and (8). Kao andLiu [42] have used the ranking fuzzy numbers method of Chen and Klein [45]to rank the obtained fuzzy efficiencies. Kao and Liu [46] also used the method ofKao and Liu [42] to calculate the efficiency scores by considering the missingvalues in the FDEA based on the concept of the membership function in fuzzy settheory. In their approach, the smallest possible, most possible, and largest possiblevalues of the missing data are derived from the observed data to construct atriangular membership function. They demonstrated the applicability of theirapproach by considered the efficiency scores of 24 university libraries in Taiwan

12 A. Emrouznejad et al.

with three missing values out of 144 observations. Kao [47] further introduced amethod for ranking the fuzzy efficiency scores without knowing the exact form oftheir membership function. In this method, the efficiency rankings were deter-mined by solving a pair of non-linear programs for each DMU. This approach wasapplied to the ranking of the twenty-four university libraries in Taiwan with fuzzyobservations. Guh [48] used a FDEA model similar to Kao and Liu [42] toapproximate the fuzzy efficiency measures. However, Kao and Liu [42] developedtheir model under the VRS assumption and Guh [48]’s model was developed underthe CRS assumption.

Kao and Liu [49] integrated the maximum set–minimum set method of Chen[50] into the FDEA model proposed by Kao and Liu [42] and built pairs of non-linear programs and ranked the DMUs with fuzzy data. In their approach, therewas no need for calculating the membership function of the fuzzy efficiency scoresbut the input and output membership functions must be known. Kao and Liu [51]applied their earlier method Kao and Liu [42] to determine the fuzzy efficiencyscores of fifteen sampled machinery firms in Taiwan. Zhang et al. [52] proposed amacro model and a micro model for the efficiency evaluation of data warehousesby applying DEA and FDEA models. They used the FDEA solution proposed byKao and Liu [42], which transformed FDEA models to bi-conventional crisp DEAmodels by a set of a-level values. Kao and Liu [53] proposed a modification to theKao and Liu’s [46] method to handle missing values. In their method, they used aFDEA approach and obtained the efficiency scores of a set of DMUs by using thea-level approach proposed by Kao and Liu [42].

Kao and Lin [54] first created the corresponding fuzzy numbers for ordinal datausing the DEA multipliers and then adopted a pair of two-level mathematicalprograms of Kao and Liu [42] for measuring the fuzzy efficiencies for distincta-cuts. Kuo and Wang [55] applied a FDEA method to evaluate the performanceof multinational corporations in the face of volatile exposure to exchange rate risk.They employed the FDEA model suggested by Kao and Liu [42] to the infor-mation technology industry in Taiwan. Li and Yang [56] proposed a FDEA-discriminant analysis methodology for classifying fuzzy observations into twogroups based on the work of Sueyoshi [57]. They used the Kao and Liu’s [42]method and replaced the fuzzy LP models by a pair of parametric models todetermine the lower and upper bounds of the efficiency scores. By applying theKao and Liu’s [42] method and the fuzzy analytical hierarchy procedure, Chiangand Che [58] proposed a new weight-restricted FDEA methodology for rankingnew product development projects at an electronic company in Taiwan. Puri andYadav [59] proposed fuzzy CCR and fuzzy slack-based measurement (SBM)models and defined the fuzzy input mix-efficiency model (FIME) based on thea-cut method developed by Kao and Liu [42]. To ensure the validity of theirproposed FDEA model, they proposed a fuzzy correlation coefficient method byusing the expected value approach for computing the expected interval and theexpected value of the fuzzy correlation coefficient between the fuzzy inputs andfuzzy outputs. They then introduced a defuzzification method for ranking theDMUs using the FIME model. Kao and Liu [60] extended the fuzzy version of the

1 The State of the Art in Fuzzy Data Envelopment Analysis 13

two-stage DEA approach where its deterministic model was initially introduced byKao and Hwang [61]. Their model was inspired by the work of Kao and Liu [42] inobtaining the lower and upper bounds of the efficiency of each DMU and its sub-DMUs for different a-cuts. Kao and Lin [62] explored a method for measuring thefuzzy efficiency of parallel production systems which involved a number ofindependent processes where the input/output data are fuzzy numbers. Theincorporated parallel model with deterministic data can be found in Kao [63].Based on the work of Kao and Liu [42], the two-level programming model wasproposed to calculate the lower and upper bounds of efficiency for distinct a-cuts.

To expand the fuzzy two-stage DEA model proposed by Kao and Liu [60, 64]presented a ranking method for fuzzy overall efficiency scores using total utilitieswhen the precise membership functions of the overall efficiencies obtained fromfuzzy two-stage model are anonymous. He also took into account the possibility ofimposing multiplier bounds to derive the efficiency rankings. Kao and Liu [60]’sstudy inspired Khalili-Damghani and Tavana [65] to propose a fuzzy networkDEA model for measuring the performance of agility in supply chains. Mugera[66] exploited the FDEA method of Kao and Liu [42] to measure the technicalefficiency of dairy farms. Chen et al. [67] incorporated Kao and Liu [42]’s tech-nique into the SBM model to evaluate risk characteristics and estimate efficienciesin the banking sector.

Saati et al. [68] suggested a fuzzy CCR model as a possibilistic programmingproblem and transformed it into an interval programming problem using the a-levelbased approach. The resulting interval programming problem could be solved as acrisp LP model for a given a with some variable substitutions. Model (9) proposedby Saati et al. [68] is derived for a particular case where the inputs and outputs aretriangular fuzzy numbers:

max wp ¼Ps

r¼1y0rp

s:t:Ps

r¼1y0rj �

Pm

i¼1x0ij� 0; 8j;

viðaxmij þ ð1� aÞxl

ijÞ� x0ij� viðaxmij þ ð1� aÞxu

ijÞ; 8i; j;urðaym

rj þ ð1� aÞylrjÞ� y0rj� urðaym

rj þ ð1� aÞyurjÞ; 8r; j;

Pm

i¼1x0ip ¼ 1; ur; vi� 0; 8r; j:

ð9Þ

where ~xij ¼ ðxlij; x

mij ; x

uijÞ and ~yrj ¼ ðyl

rj; ymrj ; y

urjÞ are the triangular fuzzy inputs and

the triangular fuzzy outputs, and x0ij and y0rj are the decision variables obtained fromvariable substitutions used to transform the original fuzzy model into a parametricLP model with a 2 0; 1½ �. Saati and Memariani [69] suggested a procedure fordetermining a common set of weights in FDEA based on the a-level methodproposed by Saati et al. [68] with triangular fuzzy data. In this method, the upperbounds of the input and output weights were determined by solving some fuzzy LPmodels and then a common set of weights were obtained by solving another fuzzyLP model. Wu et al. [70] developed a buyer-seller game model for selecting

14 A. Emrouznejad et al.

purchasing bids using fuzzy values. They adopted the FDEA model proposed bySaati and Memariani [69] to obtain a common set of weights in FDEA. Azadehet al. [71] proposed an integrated model of FDEA and simulation to select theoptimal solution between some scenarios which were obtained from a simulationmodel and determined the optimum operators’ allocation in cellular manufacturingsystems. They used a FDEA model to rank a set of DMUs based on Saati et al.[68]’s method. In addition, they clustered the FDEA ranking of the DMUs by thefuzzy C-Means method to show a degree of desirability for operator allocation.Ghapanchi et al. [72] employed FDEA to evaluate the performance of enterpriseresource planning (ERP) packages. In their approach, inputs and outputs indiceswere first determined by experts’ opinions which were evaluated using linguisticvariables characterized by triangular fuzzy numbers and then a set of potentialERP systems was considered as DMUs. They applied a possibilistic-programmingapproach proposed by Saati et al. [68] and obtained the efficiency scores of theERP systems at different a values.

Hatami-Marbini and Saati [73] developed a fuzzy BCC model which consideredfuzziness in the input and output data as well as the u0 variable. Consequently, theyobtained the stability of the fuzzy u0 as an interval by means of the method pro-posed by Saati et al. [68]. Hatami-Marbini et al. [74] used the method of Saati et al.[68] and proposed a four-phase FDEA framework based on the theory of thedisplaced ideal. Two hypothetical DMUs called the ideal and nadir DMUs areconstructed and used as reference points to evaluate a set of information technologyinvestment strategies based on their Euclidean distance from these reference points.Chen [18] modified the a-level approach and proposed an alternative FDEA tohandle both the crisp and fuzzy data. Saati and Memariani [75] developed a fuzzySBM based on the a-level approach. They transformed their fuzzy SBM model intoa LP problem by using the approach proposed by Saati et al. [68].

Azadeh et al. [77] applied FDEA, fuzzy C-means and computer simulation todetermine the optimal scenario selection in cellular manufacturing. They developeda computer simulation model to determine distinct operator layouts. The output oftheir simulation was converted into fuzzy numbers to preserve information andsubsequently the FDEA proposed by Saati et al. [68] was used to evaluate thesimulation alternatives at different levels of uncertainty. A degree of desirability forthe operator allocation was ultimately identified by the clusters obtained from thefuzzy C-means method. Saati et al. [78] used the idea of Saati et al. [68] to present aFDEA model with discretionary and non-discretionary factors in both the input andoutput-oriented CCR models. Fathi and Izadikhah [79] duplicated Saati et al. [78]’smethod in an alternative method. Hatami-Marbini et al. [80] proposed a fuzzyadditive DEA model for evaluating the efficiency of peer DMUs with fuzzy data byutilizing Saati et al. [68]’s a-level approach.

Azadeh et al. [81] explored an integrated approach for performance evaluationof health safety environment divisions, involving DEA and FDEA, to lessen thehuman error and the data imprecision. They applied the method proposed by Saatiet al. [68] to deal with the FDEA model. Azadeh et al. [82] presented an integrateddecision support system, called AutoAssess, to measure performance and analyze

1 The State of the Art in Fuzzy Data Envelopment Analysis 15

the continuous improvement of the DMUs. AutoAssess utilizes FDEA, principlecomponent analysis, DEA, numerical taxonomy, and the Spearman correlationexperiment. They used the method proposed Saati et al. [68] when inputs andoutputs are characterized by triangular fuzzy numbers. Ghapanchi et al. [83]proposed a four-step method for project portfolio selection involving (i) modelingthe problem; (ii) assessing the projects and selecting potential candidate projectsusing FDEA based on Saati et al. [68]’s method; (iii) generating portfolios anddetermining the maximal portfolios; and (iv) assessing the maximal portfoliosusing FDEA based on Saati at al. [68]’s method. Rezaie et al. [84] employed theFDEA method of Saati et al. [68] to evaluate and rank 50 companies in the TehranStock Exchange. Srinivasa Raju and Nagesh Kumar [85] explored the performanceevaluation of an irrigation system in the fuzzy environments. The FDEA methodproposed by Saati et al. [68] was adopted to deal with the impreciseness in theirrigation systems. Hatami-Marbini et al. [86] presented a FDEA model to providethe positive-normative use of fuzzy logic in a NATO enlargement application byusing the a-level technique developed by Saati et al. [68]. Saati et al. [87] usedSaati et al. [68]’s method to present a FDEA method for clustering operating unitsin a fuzzy environment by considering the priority between the clusters and thepriority between the operating units in each cluster simultaneously.

Liu [88] developed a FDEA method to find the efficiency measures embeddedwith the assurance region (AR) concept when some observations were fuzzynumbers. He applied an a-level approach and Zadeh’s extension principle [43, 44]to transform the FDEA/AR model into a pair of parametric mathematical programsand worked out the lower and upper bounds of the efficiency scores of the DMUs.The membership function of the efficiency was approximated by using differentpossibility levels. Thereby, he used the Chen and Klein’s [45] method for rankingthe fuzzy numbers and calculating the crisp values. Let us consider the relative

importance of the inputs and outputs asLIdUIq� vd

vq� UId

LIq, d\q ¼ 2; . . .;m; and

LOdUOq� ud

uq� UOd

LOq, d\q ¼ 2; . . .; s; respectively.

The two parametric mathematical programs proposed by Liu [88] are asfollows:

ðWpÞLa ¼ maxXs

r¼1

urðyrpÞLa

s:t:Xs

r¼1

urðyrjÞUa �Xm

i¼1

viðxijÞLa � 0; 8j; j 6¼ p;

�vd þ ILdqvq� 0; vd � IU

dqvq� 0; 8d\q;

�ud þ OLdquq� 0; ud � OU

dquq� 0; 8d\q;

Xm

i¼1

viðxipÞUa ¼ 1; ur; vi� 0; 8r; j:

ð10Þ

16 A. Emrouznejad et al.

ðWpÞUa ¼ maxXs

r¼1

urðyrpÞUa

s:t:Xs

r¼1

urðyrjÞLa �Xm

i¼1

viðxijÞUa � 0; 8j; j 6¼ p;

�vd þ ILdqvq� 0; vd � IU

dqvq� 0; 8d\q;

�ud þ OLdquq� 0; ud � OUuq� 0; 8d\q;

Xm

i¼1

viðxipÞLa ¼ 1; ur; vi� 0; 8r; j:

ð11Þ

where ILdq ¼

LIdUIq; IU

dq ¼UIdLIq; OL

dq ¼LOdUOq

and OUdq ¼

UOdLOq

. Jahanshahloo et al. [89],Zhou et al. [90] and Zhou et al. [91] proposed some corrections to Liu’s [88]model. Liu and Chuang [92] applied the FDEA/AR model suggested by Liu [88]and evaluated the performance of 24 university libraries in Taiwan based on themethod proposed by Kao and Liu [46]. Zhou et al. [93] developed a generalizedFDEA model with assurance regions based on the generalized precise DEA modelof Yu et al. [94]. They used the a-cut based approach to calculate the upper andlower bounds of the efficiency score for a given a.

Entani et al. [95] proposed a DEA model with an interval efficiency consistingof the efficiencies obtained from the pessimistic and the optimistic viewpoints.They also developed this approach for fuzzy input and output data by using a-levelsets. Hsu [96] applied a simple FDEA model to a balanced scorecard with anapplication to multi-national research and development projects. The FDEAmethod included both crisp and linguistic variables processed by a four-stepframework. Liu et al. [97] developed a modified FDEA model to handle fuzzy andincomplete information on weight indices in product design evaluation. Theytransformed fuzzy information into trapezoidal fuzzy numbers and consideredincomplete information on indices weights as constraints. They used an a-levelapproach to convert their FDEA model into a family of conventional crisp DEAmodels. Saneifard et al. [98] developed a model to evaluate the relative perfor-mance of DMUs with crisp data based on the l2 � norm. They used the rankingfuzzy numbers method of Jiménez [99] to determine a crisp a-parametric modeland solve the fuzzy l2 � norm model.

Jahanshahloo et al. [100] developed a fuzzy l1 � norm model with trapezoidalfuzzy inputs/outputs that was initially suggested by Jahanshahloo et al. [238] forsolving the crisp data in DEA. They applied the ranking fuzzy numbers method ofJiménez [99] to the fuzzy l1 � normmodel and obtained a crisp a-parametricmodel. Allahviranloo et al. [102] introduced the notion of fuzziness to deal withimprecise data in DEA. They proposed a fuzzy production possibility set withconstant returns to scale to calculate the upper and lower relative efficiency scoresof the DMUs by using the a-level approach. Hosseinzadeh Lotfi et al. [103]applied the method of DEA-discriminant analysis proposed by Sueyoshi [104] tothe imprecise environment. They first modified Sueyoshi’s model with crisp data

1 The State of the Art in Fuzzy Data Envelopment Analysis 17

and then developed it using fuzzy inputs and outputs based on the concept of thea-level approach. Karsak [34] proposed an extension of Cook et al. [105]’s modelto evaluate crisp, ordinal and fuzzy inputs and outputs in flexible manufacturingsystems by determining the optimistic (the upper bound) and pessimistic (thelower bound) of the a-level of the membership function of the efficiency scores.Azadeh et al. [106] used a triangular form of fuzzy inputs and outputs instead ofthe crisp data and proposed a FDEA model for calculating the efficiency scores ofthe DMUs under uncertainty with application to the power generation sector. Theytransformed the fuzzy CCR model into a pair of parametric programs using thea-level approach and found the lower and upper bounds of the efficiency fordifferent a-values. Their contribution to the FDEA literature is in the developmentof the membership functions and not the crisp measure of the efficiencies. Theyused the a-level to transform the FDEA model into a series of conventional crispDEA models. Azadeh and Alem [107] also used this FDEA method [106] for thevendor selection problem which was taken from Wu and Olson [108].

Noura and Saljooghi [109] proposed an extension of a definite class of weightfunction in FDEA based on the principle of maximum entropy in order to providecircumstances for the compatibility and stability in ranking of interval efficiencyscores of DMUs at various a values. Wang et al. [110] proposed a FDEA–Neuralapproach with a self-organizing map for classification in their neural network. Theyused the upper and lower bounds of efficiency scores at different possibilistic levelsin their model. Hosseinzadeh Lotfi et al. [111] developed two methods for solvingthe fuzzy CCR model with respect to fuzzy, ordinal and exact data. They used ananalogue function to transform the fuzzy data into exact values in the first method. Inthe second approach, they applied an a-level approach based on the Kao [112]’smethod to obtain the interval efficiency scores for DMUs. Tlig and Rebai [113]proposed an approach based on the ordering relations between LR-fuzzy numbers tosolve the primal and the dual of FCCR. They suggested a procedure based on theresolution of a goal programming problem to transform the fuzzy normalisationequality in the primal of FCCR. Zerafat Angiz et al. [114] show the advantages andshortcomings of the fuzzy ranking approach, the defuzzification approach, the tol-erance approach and the a-level based approach. They proposed an a-level approachto retain fuzziness of the model by maximizing the membership functions of inputsand outputs. They also compared their results with the results from Saati et al. [68].

Noura et al. [115] developed a fuzzy version of the DEA and the fuzzypreference relation introduced by Wu [116] using the a-cut based approach.Mansourirad et al. [117] proposed a DEA model in favor of efficiency measure-ment where the output weights are characterized by fuzzy numbers. The concept ofthe a-cut based approach was taken into consideration to calculate the componentsof the fuzzy output weights. Mostafaee [118] extended the economic efficiencymodels in non-convex technologies to both the interval approach and fuzzy set.Mostafaee [118] applied the idea of Mostafaee and Saljooghi [119] to deal withuncertainty. In the fuzzy case, he transformed the model to an interval model usingthe a-cut approach in order to calculate the interval revenue efficiency for a givennumber of a levels. Khoshfetrat and Daneshvar [120] demonstrated that a unique

18 A. Emrouznejad et al.

non-Archimedean infinitesimal, called Epsilon, as a lower bound of all multipliersof fuzzy inputs and fuzzy outputs may not precisely measure the efficiency scoresof weak efficient DMUs. In response, they proposed a method for identifying anadequate lower-bound for each weight in FDEA based on the a-cut approach.

Abtahi and Khalili-Damghani [121] proposed a FDEA approach to measure theefficiency of just-in-time implementation and supply chains, respectively, based onthe idea of Despotis and Smirlis [122] and the a-cut method. Zerafat Angiz et al.[123] extended the a-cut based approach for solving FDEA models by defining theconcept of the ‘‘local a-level’’ and considering uncertainty. Their contribution led toa multi-objective LP method which was transformed to a LP model usingArchimedean goal programming. Khalili-Damghani and Taghavifard [124] devel-oped a fuzzy three-stage DEA method for measuring the efficiency of serial pro-cesses and sub-processes for just-in-time practices. The relational two-stage DEAmodel of Kao and Hwang [61] was extended to a three-stage DEA model with fuzzyobservations. They first obtained the interval data using the a-cut based method andthen used Despotis and Smirlis [122]’s method to solve the FDEA model.

Khalili-Damghani and Hosseinzadeh Lotfi [125] developed a method formeasuring the productivity of Iranian traffic centers where the input and the outputdata were characterized by fuzzy numbers. The basic idea was proposed byDespotis and Smirlis [122] for interval data. They first used the concept of thea-cuts method to obtain the interval inputs and outputs for a given a level. Theythen mathematically re-formed the proposed model by removing the a variablefrom the proposed model. Khalili-Damghani et al. [126] extended the two-stageDEA model of Kao and Hwang [61] to a fuzzy programming model and calculatedthe efficiency of the process and its sub-processes. They used the method ofDespotis and Smirlis [122] and the a-cuts method to obtain the LP models withoptimistic and pessimistic viewpoints. Wang and Yan [127] presented FDEA/ARevaluation method to select the most appropriate manufacturing mode using thea-level approach. Khalili-Damghani and Taghavifard [128] considered a specialcase of network DEA consisting of two sub-processes for each DMU with fuzzyparameters. They converted the precise form of the model proposed by Kao andHwang [61] to the fuzzy form in order to calculate the interval efficiency score of aDMU and its two sub-DMUs using the a-cut approach. To discriminate betweenthe DMUs (and sub-DMUs), Khalili-Damghani and Taghavifard [128] proposedthree distinct categories based on the interval efficiency bounds. They also appliedthe sensitivity analysis of Jahanshahloo et al. [76] to specify the radius of stabilityof the optimistic and pessimistic situations for the DMUs and sub-DMUs.

4.3 The Fuzzy Ranking Approach

The fuzzy ranking approach is also another popular technique that has attracted agreat deal of attention in the FDEA literature. In this approach the main idea is tofind the fuzzy efficiency scores of the DMUs using fuzzy linear programs which

1 The State of the Art in Fuzzy Data Envelopment Analysis 19

require ranking the fuzzy set. The fuzzy ranking approach of efficiency mea-surement was initially developed by Guo and Tanaka [129]. They proposed afuzzy CCR model in which fuzzy constraints (including fuzzy equalities and fuzzyinequalities) were converted into crisp constraints by predefining a possibility leveland using the comparison rule for fuzzy numbers. Assuming there are n DMUsunder evaluation, the efficiency of the DMUj with m symmetrical triangular fuzzyinputs and s symmetrical triangular fuzzy outputs is denoted by ~xij ¼ ðxij; cijÞ and~yrj ¼ ðyrj; drjÞ, respectively, where xij and yrj are the center, and cij and drj are thespread of fuzzy numbers. Guo and Tanaka [129] proposed the following LP modelwith two objective functions:

maxu;v

hp ¼Xs

r¼1

ðuryrp � ð1� aÞurdrpÞ

s:t: maxv

Pm

i¼1vicip

s:t:Pm

i¼1ðvixip � ð1� aÞvicipÞ ¼ 1� ð1� aÞe;

Pm

i¼1ðvixip þ ð1� aÞvicipÞ� 1þ ð1� aÞe;

vi� 0; 8i:

9>>>>>>>>>=

>>>>>>>>>;

! Modelð12� 1Þ

Xs

r¼1

ðuryrj þ ð1� aÞurdrjÞ�Xm

i¼1

ðvixij þ ð1� aÞvicijÞ; 8j;

Xs

r¼1

ðuryrj � ð1� aÞurdrjÞ�Xm

i¼1

ðvixij � ð1� aÞvicijÞ; 8j;

ur � 0; 8r:

ð12Þ

where a 2 ½0; 1� is a predetermined possibility level by decision-makers and theunity number in the right hand side of the first constraint of model (2) is supposedlya symmetrical triangular fuzzy number 1 ¼ ð1; eÞ. Note that if cij ¼ drj ¼ 0, then,the traditional CCR is obtained and if max½cp1

�xp1; . . .; cp1

�xps� � e in (12-1), there

exists an optimal solution in (12).The fuzzy efficiency of each DMU under evaluation with the symmetrical

triangular fuzzy inputs fxip and outputs fyrp is obtained for each a possibility level as

a non-symmetrical triangular fuzzy number ehp ¼ elp; e

mp ; e

up

� �as follows:

emp ¼

u�r yrp

v�i xip; el

p ¼ emp �

u�r yrp � drp 1� að Þ� �

v�i xip þ cip 1� að Þ� � ; eu

p ¼u�r yrp þ drp 1� að Þ� �

v�i xip � cip 1� að Þ� � � em

p

where u�r and v�i are obtained from (12), and, elp; e

up and em

p are the left, right spreads

and the center of the fuzzy efficiency ehp , respectively. Because of using apredefined a 2 0; 1½ � Guo and Tanaka [129]’s method can also be classified withina-level approaches.

20 A. Emrouznejad et al.

Guo and Tanaka [130] extended their earlier work [129] and introduced a fuzzyaggregation model to objectively rank a set of DMUs by integrating multipleattribute fuzzy values. Guo [131] further applied a novel FDEA model in a casestudy for a restaurant location problem in China by integrating the FDEA modelproposed by Guo and Tanaka [129] with the fuzzy aggregation model proposed byGuo and Tanaka [130]. Sanei et al. [132] used the sensitivity analysis model ofCooper et al. [133] with fuzzy data, and they applied the approach of Guo andTanaka [129] to build their fuzzy model for determining the stability radius fordifferent a values. Chang and Lee [134] extended the integrated DEA and knap-sack models proposed by Cook and Green [135] to select an optimal group ofprojects in the fuzzy environment. The proposed fuzzy programming model wasconverted into a non-linear programming model based on the method proposed byGuo and Tanaka [129]. Similar to the approach proposed by Guo and Tanaka[129], León et al. [136] developed a fuzzy BCC model (3). However, in Guo andTanaka [129]’s method, a fuzzy efficiency score is obtained for each possibilitylevel a while in León et al. [136]’s method, a crisp efficiency score is obtained foreither all or each of the possibility levels. León et al. [136] proposed two differentFDEA models depending on the ranking method used to interpret the fuzzyinequalities. The first model uses the ranking method of Ramík and Rímánek [137]to obtain a crisp efficiency score of DMUp in which all the possible values of thevarious variables for all the DMUs at all the possibility levels are considered. Thismodel can be expressed as follows:

min hp

s:t:Pn

j¼1kjxL

ij� hpxLip; 8i;

Pn

j¼1kjyL

rj� yLrp; 8r;

Pn

j¼1kjxR

ij � hpxRip; 8i;

Pn

j¼1kjyR

rj� yRrp; 8r;

Pn

j¼1kjxL

ij �Pn

j¼1kjcL

ij� hpxLip � hpcL

ip; 8i;Pn

j¼1kjyL

rj �Pn

j¼1kjdL

rj� yLrp � dL

rp; 8r;

Pn

j¼1kjxR

ij þPn

j¼1kjcR

ij � hpxRip þ hpcR

ip; 8i;Pn

j¼1kjyR

rj þPn

j¼1kjdR

rj� yRrp þ dR

rp; 8r;

Pn

j¼1kj ¼ 1; kj� 0; 8j:

ð13Þ

In model (13), the fuzzy inputs and the fuzzy outputs, respectively, are ~xij ¼

xLij; x

Rij ; c

Lij; c

Rij

� �and ~yrj ¼ yL

rj; yRrj; d

Lrj; d

Rrj

� �in which xL

ij and yLrj are the left centers,

yRrj and xR

ij are the right centers of the inputs and outputs, respectively, while cLij and

dLrj are the left spreads, and cR

ij and dRrj are the right spreads of the inputs and

outputs, respectively. The second model of León et al. [136] uses the ranking

1 The State of the Art in Fuzzy Data Envelopment Analysis 21

method of Tanaka et al. [138] to calculate the efficiency score of DMUp for eachpossibility level a 2 ½0; 1�. This model can be formulated as follows:

min hp

s:t:Pn

j¼1kjxL

ij� hpxLip; 8i;

Pn

j¼1kjxR

ij � hpxRip; 8i;

Pn

j¼1kjxL

ij � L�i ðaÞPn

j¼1kjcL

ij� hpxLip � L�i ðaÞhpcL

ip; 8i;

Pn

j¼1kjxR

ij þ R�i ðaÞPn

j¼1kjcR

ij � hpxRip þ R�i ðaÞhpcR

ip; 8i;

Pn

j¼1kjyL

rj� yLrp; 8r;

Pn

j¼1kjyR

rj� yRrp; 8r;

Pn

j¼1kjyL

rj � L0�i ðaÞPn

j¼1kjdL

rj� yLrp � L0�i ðaÞdL

rp; 8r

Pn

j¼1kjyR

rj þ R0�i ðaÞPn

j¼1kjdR

rj� yRrp þ R0�i ðaÞdR

rp; 8r;

Pn

j¼1kj ¼ 1; kj� 0; 8j:

ð14Þ

A fuzzy set of efficient DMUs can be defined based on the optimal solution formodel (14) so that the decision maker is able to identify sensitive DMUs and toselect the appropriate possibility level. When the data are assumed to be sym-metrical triangular fuzzy numbers that are denoted by ~xij ¼ ðxij; cijÞ and~yrj ¼ ðyrj; drjÞ, respectively, where xij and yrj are the center, and cij and drj are thespread of fuzzy numbers, model (14) can be written as:

min hp

s:t:Pn

j¼1kjxij � ð1� aÞ

Pn

j¼1kjcij� hpxip � ð1� aÞhpcip; 8i;

Pn

j¼1kjxij þ ð1� aÞ

Pn

j¼1kjcR

ij � hpxip þ ð1� aÞhpcip; 8i;

Pn

j¼1kjyrj � ð1� aÞ

Pn

j¼1kjdrj� yrp � ð1� aÞdrp; 8r;

Pn

j¼1kjyrj þ ð1� aÞ

Pn

j¼1kjdrj� yrp þ ð1� aÞdrp; 8r;

Pn

j¼1kj ¼ 1; kj� 0; 8j:

ð15Þ

Note that L�i ðaÞ ¼ R�i ðaÞ ¼ L0�i ðaÞ ¼ R0�i ðaÞ ¼ 1� a, a 2 ½0; 1�. We can alsocategorize León et al. [136]’s method as an a-level approach since they useda 2 ½0; 1� in their model. Sefeedpari et al. [139] evaluated the technical efficiencyof poultry eggs producers using the FDEA model proposed by León et al. [136].

Hatami-Marbini et al. [140] extended a fuzzy CCR model for evaluating theDMUs from the perspective of the best and the worst possible relative efficiency

22 A. Emrouznejad et al.

by utilizing León et al. [136]’s approach. Beiranvand et al. [141] applied a geneticalgorithm to identify the optimal a possibility level of León et al [136]’s FDEAmethod. Then, in order to rank all of the DMUs, a closeness coefficient index wasobtained by combining the two various efficiencies. Jahanshahloo et al. [76]proposed a fuzzy ranking method for solving the SBM model in DEA when theinput-output data are triangular fuzzy numbers. Saati and Memariani [142]addressed some shortcomings of the FDEA proposed by Jahanshahloo et al. [76]and suggested several corrections to their method. Azadeh et al. [143] developedan integrated algorithm containing DEA and FDEA for measuring the efficiency ofwireless communication sectors for 42 countries with uncertain data. They usedprincipal component analysis and the Spearman correlation technique to verify andvalidate the DEA models. The FDEA method proposed by Jahanshahloo et al. [76]was implemented to study the effects of government subsidies and interventions (ifany) through regions. Azadeh et al. [144] proposed a hybrid method for thelocation optimization of solar plants using an artificial neural network and FDEA.Their study took account of the FDEA model explored by Jahanshahloo et al. [76].For validation of the results of the FDEA, they used DEA when a=1. They thendetermined the best a-cut based on the normality test. Azadeh et al. [145] proposedan adaptive-network-based fuzzy inference system-FDEA algorithm for improvingthe long-term natural gas consumption forecasting and analysis. They used theFDEA method proposed by Jahanshahloo et al. [76] to study the behavior of gasconsumption.

Molavi et al. [146] introduced two FDEA models in which the objectivefunction and fuzzy constraints of the fuzzy CCR model were transformed intocrisp conditions by using LR-fuzzy numbers and the ranking method of Ramík andRímánek [137]. Soleimani-damaneh et al. [147] addressed some computationaland theoretical shortcomings of several FDEA models including Kao and Liu [42],León et al. [136], Lertworasirikul et al. [31], Guo and Tanaka [129] andJahanshahloo et al. [76]. Furthermore, they proposed a fuzzy BCC model using thefuzzy number ranking method proposed by Yao and Wu [148] for trapezoidalfuzzy data in DEA.

Hosseinzadeh Lotfi et al. [149] applied trapezoidal fuzzy data to Jahanshahlooet al.’s [101] DEA method, in which a fuzzy fixed cost was equitably assigned toall DMUs in such a way that the efficiency scores were not changed. They used afuzzy ranking method to solve the fuzzy model in which each fuzzy constrainttransformed to three crisp constraints. Hosseinzadeh Lotfi et al. [150] adopted thelinear ranking function proposed by Maleki [151] to present fuzzy CCR modelswith triangular fuzzy data. Jahanshahloo et al. [152] suggested a method to dealwith the DEA-based Malmquist productivity index for all DMUs with triangularfuzzy inputs and outputs. They applied a linear ranking function proposed byMaleki [151] to transform their fuzzy LP model into a group of the conventionalcrisp DEA models. Pal et al. [153] used a FDEA approach and a-parametricinequalities in quality function deployment. They used a fuzzy CCR model basedon the method proposed by Lai and Hwang [154].

1 The State of the Art in Fuzzy Data Envelopment Analysis 23

Hosseinzadeh Lotfi and Mansouri [155] considered the extended DEA-discriminant analysis method proposed by Sueyoshi [57] as fuzzy data and changedtheir fuzzy model into a crisp model using the linear ranking function proposed byMaleki [151]. Zhou et al. [156] developed a FDEA method to evaluate the effi-ciency performance of real estate investment programs. They applied the rankingfuzzy numbers to solve their model and designed a ‘‘relatively effective controller’’which considered controlling the diversity of the method. Noora and Karami [157]adopted triangular fuzzy data to establish a fuzzy non-radial DEA model andapplied a ranking function proposed by Maleki et al. [158] to transform the fuzzyLP into the crisp DEA models. Jahanshahloo et al. [159] applied the linear rankingfunction proposed by Mahdavi-Amiri and Nasseri [160] to change the fuzzy costefficiency model into a conventional LP problem. Soleimani-damaneh [161] usedthe fuzzy signed distance and the fuzzy upper bound concepts to formulate a fuzzyadditive model in DEA with fuzzy input-output data. Soleimani-damaneh [162] putforward a theorem on the FDEA model which was proposed by Soleimani-damaneh[161] in order to show the existence of distance-based upper bound for the objectivefunction of the model.

Hatami-Marbini et al. [163] proposed a FDEA model to assess the efficiencyscores in the fuzzy environment. They used the proposed ranking method in Asadyand Zendehnam [164] and obtained precise efficiency scores for the overallrankings of the DMUs. They compared their method with the FDEA methodsproposed by Soleimani et al. [147] and León et al. [136]. They also applied theirmodel to sixteen bank’s branches. Jahanshahloo et al. [165] further introduced analternative approach for solving the fuzzy l1 � norm method in DEA with fuzzydata based on the comparison of fuzzy numbers proposed by Tran and Duckstein[166] to change fuzzy LP to a crisp model form. Hosseinzadeh Lotfi et al. [167]generalized a multi-activity network DEA to fuzzy inputs and outputs which wereformed by triangular membership functions. They used a ranking function toconvert the multi-activity network FDEA into a multi-activity network crisp DEAmodel. Hatami-Marbini et al. [168] proposed an interactive evaluation process formeasuring the efficiency of peer DMUs in FDEA with consideration of the deci-sion makers’ preferences. By applying the fuzzy LP of Jiménez et al. (2007) basedon the ranking method introduced by Jiménez [99], they constructed a DEA modelwith fuzzy parameters to calculate the fuzzy efficiency of the DMUs for different alevels. Then, the decision maker identifies his/her most preferred fuzzy goal foreach DMU under evaluation and a ranking order of the DMUs can be obtained by amodified Yager index.

Azadeh et al. [169] presented a three-step algorithm for tackling a special caseof single-row facility layout problems. In the first step, discrete-event-simulationwith deterministic and fuzzy data was used to model the process. In the secondstep, the verification and validation of the previous results were studied using crispsimulation. In the final step, the range-adjusted measure as a measure in the non-radial DEA model was utilized with fuzzy parameters for ranking the simulationresults and finding the optimal layout design. Similar to Hatami-Marbini et al.[168], the authors used a possibilistic programming method proposed by Jiménez

24 A. Emrouznejad et al.

et al. (2007) to convert the FDEA model to an equivalent DEA model for differenta- cuts. Emrouznejad et al. [170] developed two methods for measuring the overallprofit Malmquist productivity index (MPI) when the inputs, outputs, and pricevectors are fuzzy or vary in intervals. They first extended a fuzzy version of theconventional MPI model using a ranking method and then defined an interval forthe overall profit MPI of each DMU. They classified the DMUs into six groupsaccording to the intervals obtained for their overall profit efficiency and MPIs.Ahmady et al. [171] generalized DEA with double frontiers (Wang and Chin 2009)from precise mathematical DEA modelling to fuzzy formulations in order to copewith ambiguity and fuzziness in supplier selection problems. They employed theranking method defined by Zimmermann [44] to provide the optimal solutions forthe FDEA models.

In this section, we also review a related method, called ‘‘defuzzificationapproach’’, proposed by Lertworasirikul [33]. In this approach, which is essentiallya fuzzy ranking method, fuzzy inputs and fuzzy outputs are first defuzzified intocrisp values. These crisp values are then used in a conventional crisp DEA modelwhich can be solved by an LP solver. Dia [172] developed an alternative FDEAmodel based on fuzzy arithmetic operations and ranking of fuzzy numbers. Afuzzy aspiration level was used to change the model into a crisp DEA model andthe fuzzy results outlined the practical and robustness aspects of the fuzzymethodology. Lee [173] and Lee et al. [174] have also proposed FDEA models forCCR and BCC by defuzzifying fuzzy inputs and outputs into crisp values andusing them in conventional DEA models. Juan [175] proposed a two-stage deci-sion support model by using a hybrid DEA and case-based reasoning model. In thisapproach, the center of gravity method (CGM) suggested in [176] was used totransform the fuzzy data into crisp data and build a conventional CCR model.Bagherzadeh valami [177] introduced a cost efficiency model with triangular fuzzyinput prices and proposed a method for comparing the production cost of the targetDMU with the minimum cost fuzzy set. Hosseinzadeh Lotfi et al. [178] proposed aFDEA model to evaluate a set of DMUs where all parameters and decisionvariables were fuzzy numbers. They changed their fuzzy model into a multipleobjective LP model and solved the LP model by using a lexicography method. Thedefuzzification approach is simple but the uncertainty in input and output variables(i.e., possible range of values at different a-levels) is not effectively considered[114]. Hatami-Marbini et al. [1] developed a fully fuzzified CCR model to obtainthe fuzzy efficiency of the DMUs by utilizing a fully fuzzified LP model, in whichall of the input-output data and variables (including their weights) were fuzzynumbers. In spite of its simplicity, the defuzzification approach has not been usedby DEA researchers and practitioners. The lack of interest in the defuzzificationapproach might be due to the fact that with the defuzzification approach thefuzziness in the inputs and outputs is effectively ignored [113].

Moheb-Alizadeh et al. [179] adopted a multi-criteria DEA for location–allocation problems with fuzzy input and output quantities. They determined thelocation of facilities as well as the volume of assigned demands for each locatedfacility such that not only the total cost involving transportation and fixed location

1 The State of the Art in Fuzzy Data Envelopment Analysis 25

costs is minimized but also the efficiency scores of the facilities are maximized.They presented a two-step approach to solve their fuzzy multi-objective non-linearprogramming problem. In the first step, the problem was transformed into a crispmulti-objective programming using the defuzzification method of Carlsson andKorhonen [38] and then an efficient solution of such a multi-objective programmingproblem was computed using the minimum deviation method. To quantify theimpreciseness in the environmental factors, Amindoust et al. [180] incorporatedfuzzy logic into DEA to assess the green suppliers. They defuzzified the FDEAmodel with the center of area method.

4.4 The Possibility Approach

The fundamental principles of the possibility theory are entrenched in Zadeh’s [43]fuzzy set theory. Zadeh [43] suggests that a fuzzy variable is associated with apossibility distribution in the same manner that a random variable is associatedwith a probability distribution. In fuzzy LP models, fuzzy coefficients can beviewed as fuzzy variables and the constraints can be considered to be fuzzy events.Hence, the possibilities of fuzzy events (i.e., fuzzy constraints) can be determinedusing possibility theory. Dubois and Prade [181] provide a comprehensive over-view of possibility theory.

Guo et al. [182] initially built FDEA models based on possibility and necessitymeasures and then Lertworasirikul [33] and Lertworasirikul et al. [183], [184] haveproposed two approaches for solving the ranking problem in FDEA models calledthe ‘‘possibility approach’’ and the ‘‘credibility approach’’. They introduced thepossibility approach from both optimistic and pessimistic view points by consid-ering the uncertainty in fuzzy objectives and fuzzy constraints with possibilitymeasures. In their credibility approach, the FDEA model was transformed into acredibility programming-DEA model and fuzzy variables were replaced by‘expected credits’, which were obtained by using credibility measures. The math-ematical details of the credibility model can be found in Lertworasirikul et al. [32].

Lertworasirikul et al. [31], [185] proposed a possibility approach for solving afuzzy CCR model in which fuzzy constraints were treated as fuzzy events. Theytransformed the FDEA model into a possibility LP problem by using the possi-bility measures of the fuzzy events. In the special case, if the fuzzy data wereassumed to be trapezoidal fuzzy numbers, the possibility DEA model becomes aLP model. The proposed possibility CCR model of Lertworasirikul et al. [31]where they applied the concept of chance-constrained programming (CCP) andpossibility of fuzzy events are represented by the following:

26 A. Emrouznejad et al.

max hp ¼ �f

s:t:�Xs

r¼1

ur~yrp

�U

b ��f ;

ðXm

i¼1

vi~xip

�U

a0� 1;

�Xm

i¼1

vi~xip

�L

a0� 1;

�Xs

r¼1

ur~yrj �Xm

i¼1

vi~xij

�L

a � 0; 8j;

ur; vi� 0; 8r; i:

ð16Þ

where b 2 ½0; 1�, a0 2 ½0; 1� and a 2 ½0; 1� are predetermined admissible levels ofpossibility. The purpose of (16) is to maximize �f so that

Psr¼1 ur~yrpof the first

constraint can achieve with a ‘‘possibility’’ level b or higher, subject to the possi-bility levels being at least a0 and a in other constraints. In other words, at the optimalsolution, the value of

Psr¼1 ur~yrp is obtained at least equal to �f with the possibility

level b; while at the same time all constraints are satisfied at the predeterminedpossibility levels.

Lertworasirikul et al. [32] further developed possibility and credibilityapproaches for solving fuzzy BCC models. They applied the concept of chance-constrained programming (CCP) and possibility of fuzzy events (fuzzy constraints)to the primal and dual of the fuzzy BCC models in order to obtain possibility BCCmodels. This approach also studied the relationship between the primal and dualmodels of the fuzzy BCC. The efficiency obtained through their possibilityapproach to the primal and dual models provided the upper bound and the lowerbound for each DMU for a given possibility level. Next, in the credibilityapproach, they replaced the ‘‘expected credits’’ with fuzzy variables to cope withthe uncertainty in fuzzy objectives and fuzzy constraints. Hence, their fuzzy BCCmodel was transformed into a credibility programming-DEA model. An efficiencyscore for each DMU was obtained from the credibility approach as a representativeof its possible range. Unlike the possibility approach, the decision makers did nothave to determine any parameters or rank fuzzy efficiency values in the credibilityapproach. According to the possibility BCC approach proposed in [32], the primalproposed model can be represented by the following model:

1 The State of the Art in Fuzzy Data Envelopment Analysis 27

max hp ¼�Xs

r¼1

ur~yrp

�U

b � u0

s:t:�Xm

i¼1

vi~xip

�U

a0� 1;

�Xm

i¼1

vi~xip

�L

a0� 1;

�Xs

r¼1

ur~yrj �Xm

i¼1

vi~xij

�L

a � u0� 0; 8j;

ur; vi� 0; 8r; i:

ð17Þ

where b 2 ½0; 1�, a0 2 ½0; 1� and a 2 ½0; 1� are the predetermined admissible levelsof possibility.

Similarly, according to the possibility approach [32], the dual proposed modelcan be represented by the following model:

min hB

s:t:�hB~xip �

Pn

j¼1kj~xij

�U�a1� 0; 8i;

�Pn

j¼1kj~yrj � ~yrp

�U�a2� 0; 8r;

Pn

j¼1kj ¼ 1;

kj� 0; 8j:

ð18Þ

where �a1 2 ½0; 1� and �a2 2 ½0; 1� are predetermined admissible levels of possi-bility. Garcia et al. [186] introduced a FDEA approach to rank failure modesidentified by means of the occurrence, severity and detection indices. Theirmethod allowed the experts to use linguistic variables in assigning more importantvalues to the considered indices. They utilized the possibility approach proposedby Lertworasirikul et al. [31] in their model to solve their FDEA problem. Sim-ilarly, Wu et al. [187] used the formulation of Lertworasirikul et al. [31] in theirFDEA model to cope with the quantitative and linguistic variables in the efficiencyanalysis of cross-region bank branches in Canada.

Lin [188] proposed a three-phase method as a decision support tool using anintegrated analytic network process and FDEA approach to tackle the personnelselection problem. In phase 1, a fuzzy scheme was employed to appraise theapplicants using linguistic variables. In phase 2, the analytic network processtechnique was employed to obtain the global criteria weights with regards to thedecision–makers’ preferences. In phase 3, a possibility DEA-CCR model based onLertworasirikul et al. [31] was used and the global criteria weights from theprevious phase were considered as weight restrictions to measure the relativeeffectiveness of the applicants at different possibility levels. Zhao and Yue [189]

28 A. Emrouznejad et al.

evaluated the mutual funds management companies in China, in which theirinternal structure involved investment research competence and marketing servicecompetence. They extended the two-subsystem FDEA model based on the FDEAmodel proposed by Lertworasirikul et al. [31]. Nedeljkovic and Drenovac [190]used the possibility approach of Lertworasirikul et al. [31] to measure the effi-ciency of delivery post offices.

Ramezanzadeh et al. [191] proposed a CCR model with a chance-constrainedprogramming approach and used the a-level method and a fuzzy probabilitymeasure to rectify the randomness by the classical mean-variance method ofCooper et al. [192]. Jiang and Yang [193] proposed a fuzzy chance-constrainedcreditability programming DEA model. Khodabakhshi et al. [194] formulated twoalternative fuzzy and stochastic additive models to determine returns to scale inDEA. They developed the fuzzy and stochastic DEA models based on the possi-bility approach and the concept of chance constraint programming, respectively.Wen and Li [195] proposed a hybrid algorithm integrating fuzzy simulation and agenetic algorithm to solve a FDEA model based on the credibility measure.Recently, Wen et al. [196] extended the CCR model to a FDEA model based onthe credibility measure presented by Liu [197]. They designed a hybrid algorithmcombining a fuzzy simulation and a genetic algorithm to rank all the DMUs withfuzzy inputs and outputs.

Wen et al. [198] investigated the sensitivity and stability analysis of the FDEAmodel developed by Wen and You [199] with respect to the concept of thecredibility measure. Hossainzadeh et al. [200] presented a FDEA method usingCCP. The FDEA model was first converted into a multi-objective programmingmodel by considering optimistic, pessimistic and expected values. A goal pro-gramming technique was applied to obtain the LP model. Wang and Chin [201]introduced a FDEA method to measure the optimistic and the pessimistic effi-ciencies of DMUs using a fuzzy expected value approach with either fuzzy or crispmultipliers. They used the geometric average to aggregate the two extreme effi-ciencies and rank the DMUs. Payan and Sharifi [202] extended a method formeasuring the fuzzy MPI using credibility theory.

4.5 The Fuzzy Arithmetic

The papers published in this group focus on the fact that decision makers are notallowed to convert a fuzzy fractional programming to a LP model using con-ventional methods. That is to say,

Psr¼1 ur~yrj

�Pmi¼1 vi~xij cannot be transformed

intoPs

r¼1 ur~yrj by settingPm

i¼1 vi~xij ¼ ~1. Though ~1 is presumed to be crisp (i.e.,Dual CCR model (2)) there are a variety of methods for measuring efficiency in afuzzy environment that mostly involve extensive computational efforts. As aresult, methods in this category focus heavily on fuzzy arithmetic to cope with thefuzziness of the input and output data in the DEA models.

1 The State of the Art in Fuzzy Data Envelopment Analysis 29

Wang et al. [203] briefly argued how to treat fuzzy data using fuzzy arithmetic.Along this line, Wang et al. [204] proposed two FDEA models with fuzzy inputsand outputs by means of fuzzy arithmetic. They converted each proposed fuzzyCCR model into three LP models in order to calculate the efficiencies of the DMUsas fuzzy numbers. In addition they developed a fuzzy ranking approach to rank thefuzzy efficiencies of the DMUs. Abdoli et al. [205] studied the productivity of a setof knowledge workers using a FDEA-based approach in which the fuzzy efficien-cies were computed using Wang et al. [204]’s method. Jafarian-Moghaddam andGhoseiri [206] extended a multi-objective static DEA model of Chiang and Tzeng[207] to the fuzzy dynamic multi-objective DEA model. They transformed thefuzzy multiple objectives programming problem to one objective programmingproblem using the method introduced by Zimmermann [44]. Furthermore, Jafarian-Moghaddam and Ghoseiri [206] incorporated missing values, as triangular fuzzynumbers, into their model. After obtaining the LP model using the previous method,they solved the model based on the FDEA method developed by Wang et al. [204].Raei Nojehdehi et al. [208] studied the production frontier and production possi-bility set in FDEA. They defined the production possibility set as a fuzzy set whereall production plans are treated as its member with different degrees of membership.Mirhedayatian et al. [236] used FDEA to obtain weights in the fuzzy analytichierarchy process (AHP). They considered the fuzzy arithmetic approach to solvethe FDEA model. Their proposed approach was utilized for determining thesuperior tunnel ventilation system. Mirhedayatian et al. [237] proposed a networkDEA model for evaluating the firms in green supply chain management. Theirmethod encompasses dual-role factors, undesirable outputs and fuzzy data and theyused the principal fuzzy calculation to solve the FDEA model. Similar to Hatami-Marbini et al. [74], Mirhedayatian et al. [209] used FDEA in the technique for orderperformance by similarity to the ideal solution (TOPSIS) by means of the fuzzyarithmetic method proposed in Wang et al. [204] to examine the relative weldingprocess selection factors and evaluate different welding processes. Alem et al. [210]proposed an efficiency analysis method using FDEA and fuzzy AHP. They firstcalculated the fuzzy efficiency via the FDEA method of Wang et al. [204] and theapplied the fuzzy AHP to rank the fuzzy efficiency scores. Razavi Hajiagha et al.[211] extended a FDEA model by using intuitionistic fuzzy data consisting of amembership and non-membership functions. They applied an arithmetical operatorto obtain the LP problem. Azadi et al. [212] applied fuzzy input-output data topresent a goal directed benchmarking model for benchmarking and selecting thebest suppliers. To resolve their fuzzy mathematical DEA model, Azadi et al. [212]used fuzzy arithmetic to obtain the performance and benchmarks of each supplier.Khalili-Damghani and Taghavifard [124] and Khalili-Damghani and Taghavifard[128] used fuzzy arithmetic to remove the a from the a-cut model and achieve a LPmodel.

30 A. Emrouznejad et al.

4.6 The Fuzzy Random/Type-2 Fuzzy Set

Zadeh [10] initially introduced the type-2 fuzzy set as an extension of the prevalentfuzzy set in which uncertainty is incorporated into the membership function of afuzzy set. In addition, many complex systems often involve randomness andfuzziness simultaneously. In response, Kwakernaak [213] proposed fuzzy randomvariables to tackle performance measurement in such systems. In this section, wereview a few recent studies on the application of type-2 fuzzy set and fuzzyrandom variables in DEA.

Qin et al. [214] developed a DEA model with type-2 fuzzy inputs and outputs todeal with linguistic uncertainties as well as numerical uncertainties with respect tofuzzy membership functions. Based on the expected value of a fuzzy variable, theyused a reduction method for type-2 fuzzy variables and built a FDEA model usingthe generalized credibility measure. Qin and Liu [215] proposed a class of fuzzyrandom DEA (FRDEA) models with fuzzy random inputs and outputs whererandomness and fuzziness coexisted in an evaluation system and the fuzzy randomdata were characterized with known possibility and probability distributions. Theyalso proposed a hybrid genetic algorithm and stochastic simulation approach toassess the objective function of the proposed DEA. Qin and Liu [216] also pro-posed another approach similar to the method proposed in [215]. They includedthe chance functions in the objective and constraint functions which were subse-quently converted into the equivalent stochastic programming forms and solvedwith a hybrid genetic algorithm and Monte Carlo simulation method. Qin et al.[217] incorporated the fuzzy possibility theory of Liu and Liu [218] into DEAmodels in which the inputs and outputs were type-2 fuzzy with known possibilitydistributions. Based on mean reduction methods, the fuzzy generalized expectationDEA models were developed where the inputs and outputs were independent type-2 triangular fuzzy variables.

Qin et al. [217] presented the equivalent parametric forms of the constraints andthe generalized expectation objective. Tavana et al. [219] developed three FDEAmodels with respect to probability-possibility, probability-necessity and proba-bility-credibility constraints in which fuzziness and randomness coexisted in theevaluation problems. In addition, they illustrated the proposed model by using acase study for the base realignment and closure (BRAC) decision process at theU.S. department of defense (DoD). Zerafat Angiz et al. [220] used a fuzzy non-radial DEA method to determine the ideal solution and distance function of type-2attributes using a TOPSIS methodology. The principal idea of FDEA was tomaximize all the membership functions of the fuzzy parameters. The proposedFDEA model was thus transformed into the multi-objective program that wassolved using a method proposed by Zerafat Angiz et al. [221].

Tavana et al. [222] first developed three DEA models for estimating the radialefficiency of DMUs in the presence of random-fuzzy (Ra-Fu) variables withPoisson, uniform and normal distributions. They then advanced the formulation ofthe possibility-probability and the necessity-probability DEA models with Ra-Fu

1 The State of the Art in Fuzzy Data Envelopment Analysis 31

parameters for where the Ra-Fu data contained normal distributions with fuzzymeans and variances. Tavana et al. [222] lastly presented the general possibility-probability and necessity-probability DEA models with fuzzy thresholds.

4.7 Other Developments in Fuzzy DEA

In this section, we review several FDEA models that do not fall into the fuzzyranking approach, the tolerance approach, the a-level based approach, the possi-bility approach the fuzzy arithmetic, and the fuzzy random/type-2 fuzzy setcategories. Hougaard [223] extended scores of technical efficiency used in DEA tofuzzy intervals and showed how the fuzzy scores allow the decision maker to usescores of technical efficiency in combination with other sources of availableperformance information such as expert opinions, key figures, etc. Sheth andTriantis [224] introduced a fuzzy goal DEA framework to measure and evaluatethe goals of efficiency and effectiveness in a fuzzy environment. They defined amembership function for each fuzzy constraint associated with the efficiency andeffectiveness goals and represented the degree of achievement of that constraint.Hougaard [225] introduced a simple approximation for the assessment of effi-ciency scores with regards to fuzzy production plans. This approach did notrequire the use of fuzzy LP techniques and had a clear economic interpretationwhere all the necessary calculations could be performed in a spreadsheet making ithighly operational. Uemura [226] introduced a fuzzy goal based on the evaluationratings of individual outputs obtained from the fuzzy loglinear analysis and thenproposed a fuzzy goal into the DEA. Luban [227] proposed a method inspired bySheth and Triantis’s [224] work and used the fuzzy dimension of the DEA modelsto select the membership function, the bound on the inputs and outputs, the globaltargets, and the bound of the global targets.

Zerafat Angiz et al. [228] proposed an alternative ranking approach based onDEA in the fuzzy environment to aggregate preference rankings of a group ofdecision makers. They applied their method to a preferential voting system withfour stages. Although they considered data as ordinal relations, stage 1 defined afuzzy membership function for ranking a set of alternatives to find the idealalternative. In the second stage they used the FDEA model proposed in ZerafatAngiz et al. [229] to obtain the ideal solution. In the last two stages, they proposeda method to aggregate the results to a single score using subjective weightsobtained from comparative judgments for ranking the alternatives. Zerafat Angizet al. [230] proposed a multi-objective mathematical model using the fuzzyconcept on the multipliers for ranking the efficient units.

Zerafat Angiz et al. [231] also presented a ranking method in the preferentialvoting system using DEA and the concept of a fuzzy set. Their method first con-structed fuzzy numbers based on the number of votes for the first ranked DMU.Second, the nearest point to a fuzzy number concept was used to define a dummyideal alternative. Finally, the efficiency of the alternatives in a pairwise comparison

32 A. Emrouznejad et al.

with the dummy ideal alternative was calculated using the DEA method. ZerafatAngiz and Mustafa [232] used fuzzy concepts to deal with non-discretionary dataembedded in DEA models. Bagherzadeh Valami et al. [233] considered theproduction possibility set as a fuzzy set where the input and output data vary inthe interval.

5 Conclusions, Limitations and Directions for FutureResearch

Fuzzy set theory has been used widely to model uncertainty in DEA. Althoughother models such as probabilistic/stochastic DEA and statistical preference (e.g.bootstrapping) are also used to model uncertainty in DEA, in this chapter we focuson the fuzzy set DEA papers published in the English-language academic journals.We present a classification scheme with six primary categories, namely, the tol-erance approach, the a-level based approach, the fuzzy ranking approach, thepossibility approach, the fuzzy arithmetic, and the fuzzy random/type-2 fuzzy set.While most of these approaches are powerful, they usually have some theoreticaland/or computational limitations and sometimes applicable to a very specific sit-uation (e.g., Soleimani-damaneh et al. [147]). For example, the tolerance approachuses fuzzy inequalities and equalities instead of fuzzy inputs and fuzzy outputs.The most popular FDEA group, a-level based approach, often provides a fuzzyefficiency score whose membership function is constructed from a-level eventhough models related to this approach are not computationally efficient becausethis group mostly requires a large number of LP models according to variousa-levels (e.g., Soleimani-damaneh et al. [147]). In this study, the importance offuzzy ranking approach in the literature is ranked second (see Table 1) while aconsiderable limitation of this group is that different fuzzy ranking methods mayresult in different efficiency scores. In the possibility approach, the proposedmodels may not be adapted to other DEA models (e.g., Soleimani-damaneh et al.[147]), and we believe that this approach requires complicated numerical com-putations compared to other approaches.

In summary, FDEA is best known for its distinct treatment of the imprecise orvague input and output data in the real-world problems. As shown in Fig. 2, FDEAis a growing field with many practical and theoretical developments. Nevertheless,we believe that FDEA is still in its early stages of development.

A wide variety of applications and proliferation of models have demonstratedthat FDEA is an effective approach for performance measurement in problemswith imprecise and vague data. Nevertheless, there are a number of challengesinvolved in the FDEA research that provide a great deal of fruitful scope for futureresearch:

• A unified process: It is imperative to provide a unified FDEA approach forpracticing managers and novice users. This need is clearly illustrated by the

1 The State of the Art in Fuzzy Data Envelopment Analysis 33

large number of models and the proliferation of frameworks, at times confusingor even contradictory. A unified process similar to the COOPER-framework[234] can provide the novice users with a clear-cut procedure for solving FDEAproblems. Experienced users can use the unified process for modeling depth andbreadth.

• User-friendly software: Although there are several DEA software packages inthe market, none of them are capable of handling fuzzy data and FDEAmodeling.

• Real-life applications: Most of the papers published in the literature have usedsimple examples or small sets of hypothetical data to illustrate the applicabilityof the models. We encourage researchers to use real-world case studies indemonstrating the applicability of their models and exhibit the efficacy of theirprocedures and algorithms.

• Sensitivity analysis: There is a need for comprehensive studies focusing onsensitivity analysis strategies in FDEA. Fuzzy data by definition are not fixed.As a result, the results from fuzzy models are less robust and more likely tochange over a period of time or even during the model-building phase. Con-sequently, there is a need for elaborate and comprehensive sensitivity analysismethods and procedures to deal with the changing nature of FDEA models.We hope that our research will benefit a wide range of users who desire to solve

real-life DEA problems with vague or imprecise data. The taxonomy and thecomprehensive review of the literature provided here should lead to a betterunderstanding of FDEA and its applications.

21

2 23

4

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4

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11

22

1817

20

24

3

0

5

10

15

20

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19

92

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In P

ress

Nu

mb

er o

f pap

ers

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Fig. 2 Three decades of fuzzy DEA development (1992–2013)

34 A. Emrouznejad et al.

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1 The State of the Art in Fuzzy Data Envelopment Analysis 45

Chapter 2Imprecise Data Envelopment Analysis:Concepts, Methods, and Interpretations

K. Sam Park

Abstract DEA has proven to be a useful tool for assessing efficiency or produc-tivity of organizations. While DEA assumes exact input and output data, thedevelopment of imprecise DEA (IDEA) broadens the scope of applications to effi-ciency evaluations involving imprecise information which implies various forms ofordinal and bounded data often occurring in practice. The primary purpose of thisarticle is to review what has been developed so far, including the body of conceptsand methods that go by the name of IDEA. This review comprises (a) why we look atimprecise data and how to elicit imprecise information, (b) how to calculate theefficiency measures, and (c) how we can interpret the resulting efficiency. Specialemphasis will be placed on how to deal with strict inequality types of imprecise data,such as strict orders and strict bounds, rather than weak inequalities. A generalapproach to these strict imprecise data is presented, in order to arrive at efficiencyscores. This approach first constructs a nonlinear program, transform it into a linearprogramming equivalent, and finally solve it via a two-stage method.

Keywords Imprecise data envelopment analysis � Fuzzy sets � Fuzzy DEA

1 Introduction

This article is concerned with the use of imprecise data in data envelopmentanalysis (DEA). Imprecise data implies that some data are known only to theextent that the true values lie within prescribed bounds while other data are knownonly in terms of ordinal relations. [1, 2] showed how DEA could be extended totreat ordinal data. To deal with all aspects of imprecise data in DEA, [3] proposed

K. S. Park (&)Business School, Korea University, Anam-5 Ga, Seongbuk, Seoul 136-701, Koreae-mail: sampark@korea.ac.kr

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_2, � Springer-Verlag Berlin Heidelberg 2014

47

a body of concepts and methods that go by the name of imprecise data envelop-ment analysis (IDEA). There have since been a number of refinements, extensions,and applications [4–11]. These studies have also developed different methods forsolving a nonlinear IDEA problem because some inputs and outputs are unknowndecision variables with values to be determined in the model. Although thecomputational algorithms are different, they result in the same efficiency scoresand, hence, the same efficiency classifications into efficient and inefficient groups.

Still further extensions of the IDEA approach have been made. [12] treatedinterval or bounded data in DEA and showed how the upper and lower bounds ofefficiency could be achieved in order to accomplish more detailed classifications ofefficiency performance, a three-group efficiency classification rather than thecustomary two-group partition. Later, [13] developed an extended method tohandle ordinal as well as interval data and obtain an upper and lower bound onefficiency, upon which the three-group partition was based. [14] considered asimilar partition but in a broader frame of imprecise data in that the three-groupclassification was made for arbitrary imprecise data, including any combinationsof bounded and ordinal data.

Meanwhile, [15, 16] dealt with a slightly different situation in which fuzzy datawere involved in DEA, referred to as fuzzy DEA. See also [17] and [18], for moreinformation on fuzzy DEA. Despite the different kinds of data, such fuzzy DEAapproaches usually generate bounded data from the given fuzzy data during theefficiency evaluations and efficiency classifications and therefore closely relatefuzzy DEA to IDEA. In fact, such bounded data can be viewed as the simplestform of fuzzy data and an example of imprecise data as well, so the fuzzy DEA orIDEA approach can be used to deal with these bounded data. However, a morecustomary structure of fuzzy data is based on the concept of possibility distribu-tions, specified usually by pessimistic, optimistic, and most likely estimatessolicited from expert judgments. Given the customary fuzzy data in DEA, thefuzzy approach would be more appropriate than IDEA approach. In contrast, asmentioned previously, imprecise data encompass the bounded and ordinal forms ofinput and output data, and can hence be expressed by a system of linearinequalities on data. The IDEA approach has been developed to deal primarilywith these imprecise data for efficiency evaluations. Note, however, that commonto the fuzzy and imprecise data is that some of the important factors are qualitativein nature and hence those exact values cannot readily be determined in advance.

In this article, we emphasize that imprecise data frequently takes the form ofstrict inequalities, such as strict orders and strict bounds, for the following reasons.First, ordinal data frequently arise in practical decision making, and these data canbe viewed as strict rather than weak inequality relations. Second, attributing evenbounded data to strict relations is more appealing in that the decision variablevalues need not be anchored at the extreme upper and lower bounds that mightgenerally evidence little desirability. Why is the treatment of strict informationdifficult? Strictly speaking, the standard algorithms for solving linear programmingproblems cannot be used directly, because the feasible region for the permissibledecision variable values becomes an open set.

48 K. S. Park

En route to evaluating efficiency, the majority of previous studies have replacedthese strict inequalities with weak inequalities, by employing a small positivenumber. As this replacement closes the feasible region of decision variable values,it circumvents certain troubling questions that arise when utilizing a mathematicalprogramming approach to the efficiency evaluations. However, there are no hardand fast rules for selecting the factual small value and, even if the choice ispossible, the resultant assessments depend profoundly on that choice.

In this article, we therefore demonstrate (a) how to elicit strict imprecise data,(b) how to compute the efficiency scores with these strict data, and (c) how we caninterpret the resulting efficiency. The computational method presented herein isgeneral enough to deal with strict data as well as weak and exact data all together,without selecting any small value for the strict information. This general approachfirst constructs a nonlinear program, transform it into a linear programmingequivalent, and finally solve it via a two-stage method, in order to arrive at effi-ciency ratings. How to interpret the resulting efficiency is also important. We notethat the interpretations in the ordinary DEA may alter when imprecise data areincluded in DEA, and they depend heavily on the type or nature of imprecise data.

The article is organized as follows. First, the IDEA model is shown briefly. Wethen demonstrate how to elicit strict information in great detail. This is followed bythe computational method to obtain efficiency ratings and a numerical example.We then show how to interpret the efficiency solutions. Finally, we conclude thearticle with a summary and a sketch of further research opportunities.

2 IDEA Model

The IDEA model proposed by [3] can be represented by

z�o ¼ maxPs

r¼1lryro

s:t:Ps

r¼1lryr �

Pm

i¼1xixi� 0

Pm

i¼1xixio ¼ 1

l ¼ ðlrÞ� e; x ¼ ðxiÞ� e

9>>>>>>>>=

>>>>>>>>;

ð1Þ

yr 2 Dþr ; r ¼ 1; . . .; sxi 2 D�i ; i ¼ 1; . . .;m

�

ð2Þ

Here, yr ¼ yr1; . . .; yrnð ÞT and xr ¼ xi1; . . .; xinð ÞT respectively represent thecolumn vectors of outputs produced and inputs consumed by n decision makingunits (DMUs) under consideration. The yro, xio data represent the outputs and inputsfor DMUo, as the DMUj to be evaluated, j = 1,…, n. The sets of variables l and xare multipliers associated with outputs and inputs and e[ 0 is a non-Archimedean

2 Imprecise Data Envelopment Analysis 49

element. The sets Dr+, Di

- in (2) represent the constraint sets of imprecise data for yr

and xi, respectively. As mentioned before, imprecise data encompass both ordinaland bounded data. More generally, each of the Dr

+, Di- sets can be assumed to be a

convex polyhedron in n-dimensional space, formed by a system of linearinequalities representing arbitrary linear-type imprecise data. Throughout weassume yr C 0, yr = 0 for all r and xi C 0, xi = 0 for all i.

Model (1) is a nonlinear program since the input and output data are knownimprecisely and their values are to be determined. However, several methods havebeen developed to transform the nonlinear model into a linear program [4, 5, 7, 8,11]. The optimal solution zo

* to (1) then represents the efficiency score of the DMUo

to be evaluated. A two-group efficiency classification can then be made in whichDMUo is classified as efficient if zo

* = 1, otherwise it is classified as inefficient.Regarding the upper bound, as mentioned in the introduction, many algorithms

are already available to solve the IDEA model in (1). However, we find that thesealgorithms have some drawbacks or limitations especially in dealing with strictordinal data and partial orders which may often appear in practice. We thereforepresent a different method to achieve the upper bound. We first define the strictordinal data, present the method, and show a numerical example.

3 Strict Imprecise Information

3.1 A Synopsis

Let there be a set of DMUs, U = {Uj : j = 1, …, n}, each of which is evaluatedbased on several inputs and outputs. Denote their values by vr(Uj) for output factorr, but assume these values are not known exactly. Various ways may be possible toestimate these values.

The foundation of measurement can be traced back to [19], who described thatquestions based on binary and quaternary relationships could be used to establish avalue function like vr. A consequence of using the binary relationship is that U1 ispreferred to U2 if and only if vr(U1) [ vr(U2). For the quaternary relationship, thepreference difference between U1 and U2 is greater than that between U3 and U4 ifand only if vr(U1) – vr(U2) [ vr(U3) – vr(U4). In addition, [20] extended thequaternary preference system to order the differences in the strength of preferencebetween pairs of alternatives and to relate it to the multi-attribute value theory[21].

Our purpose is not to specify such a value function to arrive at a precise valueof each DMU on a factor. The complete value judgment may be difficult and timeconsuming to implement in practice. Rather it is more realistic to assume thatthe marginal value function is unknown. Denoting the unknown marginal value of

50 K. S. Park

Uj, vr(Uj), by yrj, we have imprecise information such that yr1 [ yr2 and yr1 –yr2 [ yr3 – yr4. These relations can alternatively be represented by

yr1 � yr2� e ð3Þ

yr1 � yr2 � yr3 þ yr4� e ð4Þ

where e[ 0 is an unknown constant and can be assumed to be a non-Archimedeanvalue.

Type (3) is called strict order and, if e = 0, it becomes a weak order. Type (4) ismore complicated but can be regarded as a strict difference order. Regardingoutput r, (3) means that DMU1 outperforms (or is preferred to) DMU2, and (4)means that the difference in the production performance between DMU1 andDMU2 is greater than that between DMU3 and DMU4.

3.2 Measurement Scheme

Basic to the techniques of eliciting types (3) and (4) are the concepts of order andrelation. Common to those techniques is the concept of paired comparisons in a setof DMUs. These concepts and techniques serve as a foundation of measurementand are widely utilized in the multi-criteria decision analysis (MCDA) literature todescribe a measurable value function [20] and to elicit a precise value function[21] or imprecise values [22–24]. It is also noted that such concepts and methodsdeveloped for MCDA could be used for DEA because of the structural similaritybetween MCDA and DEA [25–27]. Indeed the DEA approach has been utilized forranking alternatives and examining multi-criteria decision situations [1, 2].

Besides the theoretical touchstone, there is an appealing practical usefulness oftype (4) as well as (3). We can think of type (3) as one-level strength of preferenceand type (4) as a two-level. Type (3) results from the fact that U1 is preferred toU2. We can effectively realize type (4) by employing a strong and weak preferencein the same framework of paired comparisons. Given U1 is strongly preferred to U2

and U3 is weakly preferred to U4, it gives rise to type (4). The quaternary rela-tionship reduces to binary and, hence, one does not need to respond to preferencedifference questions which may place a cognitive burden on the decision maker.[28] investigated this and showed that the two levels of strength of preferencecould be extended to more levels such as a five-level: For a fixed factor,A(i, j): Ui is very strongly preferred to Uj.B(i, j): Ui is strongly preferred to Uj.C(i, j): Ui is moderately preferred to Uj.D(i, j): Ui is weakly preferred to Uj.E(i, j): Ui is very weakly preferred to Uj.

2 Imprecise Data Envelopment Analysis 51

Using this five-level system, we can construct many constraints of type (4) aswell as (3). For example, on output factor r, given A(1, 2) and B(3, 4), then wehave constraint (4) in addition to (3) and yr3 [ yr4. If the same judgments occur oninput factor i, we then have xi2 – xi1 [ xi4 – xi3, xi2 [ xi1 and xi4 [ xi3. Note that adifferent number of levels such as three or four may be selected appropriately.

Such a measurement system is easily understood by management and can bewidely utilized to elicit imprecise values, in particular, for qualitative factors.Many practical problems of efficiency analysis or decision making using DEA mayoften involve one or more qualitative factors. Possible areas of application arenumerous and include project selection, location and policy analysis, managerialand operational performance evaluation, among others.

The nature of imprecise data will be problem specific and will depend uponparticulars of the problem in question such as prior knowledge or experiences withthe factors involved. Generally, there are various types of imprecise data includingthose mentioned above. The MCDA literature contains a more extensive discus-sion on the use of imprecise data (e.g., [22, 28, 24]), where various forms ofimprecise data that occur possibly in practice are described. The general methodwe will present in the next section can deal with all aspects of such imprecise data.

In addition, [2] showed another 5-point scale system for evaluating DMUs onqualitative factors, which was illustrated with an application to a R&D projectselection. Common to the above rating scheme and this evaluation system is theconcepts of order and relation. The difference is that the former uses the concept ofpaired comparisons but the latter does not. The concept of paired comparisons hasbeen a common practice in decision analysis and may give rise to a richer col-lection of preference information (or imprecise data). Specifically, Cook et al.attempted to gather information like type (3), while we could gather types (3) and(4) simultaneously. Addition of type (4) reduces the feasible region of unknownvariables considerably [23]. However, this does not imply that the former schemeis more appropriate or convenient than the latter one in terms of practical use.

4 Efficiency Computations

Let there be i = 1, …, m inputs, r = 1, …, s outputs, and j = 1, …, n DMUs.Recall the vector of outputs produced yr = (yr1, …, yrn)T and the vector of inputsconsumed xi = (xi1, …, xin)T by n DMUs. Although we can treat arbitrary linearimprecise data, we focus on handling strict inequalities as in (3) and (4). We canexpress these inequalities in the matrix form such that Aryr B -e and Bixi B -efor each output and input. The number of rows of the Ar, Bi matrices correspondsto the number of constraints on the yr, xi variables, respectively. The vector e is(e, …, e)T in appropriate sizes.

52 K. S. Park

Incorporating these imprecise data into DEA, we have the following IDEAmodel:

maxPs

r¼1lryro

s:t:Ps

r¼1lryr �

Pm

i¼1xixi� 0

Xm

i¼1

xixio ¼ 1

lr;xi� e; 8r; iAryr � � e; 8rBixi� � e; 8i

ð5Þ

As shown in [6, 7], consider the following transformation technique:

Yr ¼ Yr1; . . .; Yrnð ÞT¼ lryr; 8rXi ¼ Xi1; . . .;Xinð ÞT¼ xixi; 8i

ð6Þ

We can then reduce model (5) to the following linear program:

maxPs

r¼1Yro

s:t:Ps

r¼1Yr �

Pm

i¼1Xi� 0

Xm

i¼1

Xio ¼ 1

lr;xi� e; 8r; iArYr � � lre; 8rBiXi� � xie; 8i

ð7Þ

with the variables all constrained to be nonnegative.Looking at (7), we find that the concrete value of epsilons needs to be specified

en route to computing the efficiency score. This implies the score depends on theepsilon value selected. To address this problem, we now modify model (7) to

2 Imprecise Data Envelopment Analysis 53

maxPs

r¼1eoYr

s:t:Ps

r¼1IYr�

Pm

i¼1IXi� 0

Xm

i¼1

eoXi ¼ 1

ArYrþlre� 0; 8rBiXiþxie� 0; 8i�lr ��e; 8r�xi��e; 8i

ð8Þ

The eo [ <n is the row unit vector with one in the oth place. If DMUo=1 isevaluated, then eo = (1, 0, …, 0). If DMUo=2 is evaluated, eo = (0, 1, 0, …, 0).The I is the n 9 n identity matrix.

Dual to linear program (8) becomes

min h� e

�Ps

r¼1sþr þ

Pm

i¼1s�i

�

s:t: kIþprAr � eTo ; 8r

pre�sþr ¼ 0; 8rkI�qiBi� heT

o ; 8i�qieþs�i ¼ 0; 8i

ð9Þ

Here, k = (k1, …, kj, …, kn) C 0 is a row vector in <n. We then havesþr ¼ pr e and s�i ¼ qiefor all r and i. Model (9) is thus reduced to

min h� eðPs

r¼11pr þ

Pm

i¼11qiÞ

s:t: kIþ prAr � eTo ; 8r

kI� qiBi� heTo ; 8i

ð10Þ

where 1 = (1, …, 1) in appropriate sizes and all variables except for h are non-negative. Note that the square term e2 involved between (9) and (10) can beharmlessly replaced by another non-Archimedean constant.

We can now utilize the two-stage DEA method of [29] for solving problem(10). Therefore, without specifying the epsilon values, we can achieve the effi-ciency score h *, slacks, and a set of the kj

* values, which implies the peer group.

54 K. S. Park

5 Illustrative Example

Table 1 provides an example involving three DMUs which each produce twooutputs using two inputs. Two quantitative factors are given exact data but theother two qualitative factors are evaluated using the five-level system described inSect. 3. Then, y1 = (20, 20, 20)T and x1 = (15, 15, 20)T for the exact data. Therelations given for the unknowns y2 are {y21 [ y22 [ y23 C 0, y21-y23 [ y21-

y22 [ y22-y23 C 0}. Thus,

Ay2 ¼�1 2 �1�1 1 00 �1 1

0

@

1

Ay2� � e; y2� 0

Applying similar reasoning to the second input yields

Bx2 ¼�1 2 �11 �1 00 1 �1

0

@

1

Ax2� � e; x2� 0

Model (10) is then specified as

min h� eðsþ1 þ s�1 þ 1pþ 1qÞs:t: ky1�sþ1 ¼ y1o

kIþpA� eTo

kx1þs�1 ¼ hx1o

kI�qB� heTo

ð11Þ

Note that the objective function and all constraints in (11) remain unchangedduring the evaluation of different DMUs. However, like the ordinary DEA com-putation, we need to change the data in the right-hand side only according to thechange of DMUo under evaluation. When evaluating DMUo=1, we use eo = (1, 0,0) together with y1o = 20 and x1o = 15. When evaluating DMUo=3, we useeo = (0, 0, 1), y1o = 20, and x1o = 20.

Using (11), we have the efficiency results in Table 2. Only DMU1 appears to beefficient. DMU2 is compared to DMU1 and has positive slacks for the two qual-itative factors. This is a result of the fact that DMU1 outperforms DMU2 on the twoqualitative factors while these DMUs have the same exact data for the other two

Table 1 Mixtures of exact and imprecise data

DMUj Outputs Inputs

Revenue(y1)

Satisfaction(y2)

Cost(x1)

Judgment(x2)

1 20 A(1, 3) 15 A(1, 3)2 20 B(1, 2) 15 B(2, 3)3 20 C(2, 3) 20 C(1, 2)

2 Imprecise Data Envelopment Analysis 55

factors. DMU3 has a positive slack of 5 for the first input in comparison withDMU1, and also has positive slacks for the two qualitative factors that are largerthan those of DMU2. We do not know the exact values of those slacks for theordinal data, but we clearly identify the existence of positive slacks and theirrelative intensities. Note that, for the ordinal data, the amount of output producedand the amount of input used for each DMU rely on the value of epsilon.

6 Interpretations

In this section, we provide how to interpret the efficiency results from the effi-ciency model, which includes a radial measure of efficiency (h*), slacks, and thereturns-to-scale classification. Their interpretations are clear in the ordinary DEAwith exact data. For instance, the radial measure refers to the proportion of inef-ficiency present in all inputs so a percentage reduction is needed for all the inputsto improve efficiency. A positive slack for a particular input stands for a furtherinefficiency in that input. The returns-to-scale classification has to do with therelative scale sizes of DMUs in terms of input usages and output productions.However, such interpretations may alter when imprecise data are incorporated intoDEA, and they depend on the type or nature of imprecise data.

The nature of imprecise data is problem specific and will depend upon par-ticulars of the problem in question such as prior knowledge or experiences with thefactors involved. For example, on a qualitative factor such as ease of use, skilllevel of labors or managers, may take the form of only ordinal relations as when anexpert may say DMU1 is best, DMU2 is second, and so on. These DMUs mightalternatively be categorized into several groups with respect to performances thatare regarded as good, moderate, or poor. If such ordinal data are imposed in DEA,then one can only regard the values of radial efficiency as relations of ‘‘better’’ and‘‘worse.’’ This implies a preference relation between DMUs in terms of theirperformances, since the ordinal data can be viewed as a preference relation. Thecase of positive slacks, however, leads to somewhat more discriminating treatmentin that one can treat these values in a cardinal manner as for the factors with exactdata, as in the ordinary DEA, but not for the ordinal factor. The returns-to-scaleclassification has nothing to do with the scale sizes of DMUs since we cannot treatthe given ordinal data in a cardinal manner. More strictly, this classification ismeaningless when ordinal data are imposed in DEA.

Table 2 Efficiency results for the DMUs in Table 1

DMUj Efficiency(h*)

ReferentDMUs

Slacks

1 1 k1* = 1 No

2 1 k1* = 1 s2

+ = 2e, s2- = 1e

3 1 k1* = 1 s1

- = 5, s2+ = s2

- = 3e

56 K. S. Park

The other forms of imprecise data occur in practice. For example, on a quali-tative factor, an expert may say that the performance of DMU1 is more than twotimes and less than three times that of DMU2. Alternatively for the same factor theexpert may say that the performance of DMU2 is between 70 and 80 % of theperformance level of DMU1. Like ordinal data, this kind of bounded data comesfrom subjective judgments and represents preference values rather than objectivedata. Thus the interpretations similar to those for ordinal data can apply to thesituation with these bounded data. Note that, unlike the case of ordinal data, onemay treat the radial measure and slacks in a cardinal manner: For example, basedon the radial measure, a 10 % increase in the given 80 % performance level isrequired and, based on the positive slack, a further increase of 2 % in this factor’slevel needs to be made in order to become 90 % in the end. This interpretation ispossible but it is not directly operational to improve from 80 to 90 %. Namely theproblem of how to improve it still remains. It is also hard to relate such percentagedata to the scale sizes of DMUs. The only one clear trait underlying these per-centage data is more-the-better in terms of output.

Even some quantitative factors may involve imprecise data. For example, themanpower employed in a DMU could fail to be constant and may be so volatilethat they can only be said to fluctuate within upper and lower bounds. Althoughthis is an example of bounded data, the nature is quite different from the abovebounded data. The current bounded data are based primarily upon objectiveinformation so we can apply the interpretations like those in the ordinary DEA tothe case of these bounded data.

7 Conclusions

We have here provided how to elicit strict imprecise information, how to computeefficiency with such information, and how to interpret the resulting efficiency. Ageneral method for obtaining the upper bound on efficiency is demonstrated. Thismakes the treatment of partial orders possible and deals effectively with generalforms of strict orders in DEA. The general method uses a unique instrument, linearprogram, no matter what types of imprecise data are given, and it has no additionalconcern about the process of determining exact data. In the interpretation stage, weemphasize that the interpretations depend on the type or nature of imprecise data.For instance, when ordinal data are involved in DEA, the returns-to-scale classi-fication is meaningless.

Finally, revisiting the work of [3–5] we can identify some paths for furtherresearch. Besides imprecise data in DEA they also dealt with assurance region(AR) conditions on the multiplier variables, as in [30, 31] and the combinedvariable-data transformations employed in the cone-ratio envelopment of [32].Hence they developed one unified approach referred to as AR-IDEA. In contrast,

2 Imprecise Data Envelopment Analysis 57

we do not deal with AR bounds in the present article because we want to focus onanalyzing technical efficiency. Thus, a series of extensions is warranted in that ARbounds could be incorporated into our developments and the efficiency classifi-cations could then be extended to further prioritize the efficiency status of DMUs.

References

1. Cook, W.D., Kress, M., Seiford, L.M.: On the use of ordinal data in data envelopmentanalysis. J. Oper. Res. Soc. 44, 133–140 (1993)

2. Cook, W.D., Kress, M., Seiford, L.M.: Data envelopment analysis in the presence of bothquantitative and qualitative factors. J. Oper. Res. Soc. 47, 945–953 (1996)

3. Cooper, W.W., Park, K.S., Yu, G.: IDEA and AR-IDEA: models for dealing with imprecisedata in DEA. Manag. Sci. 45, 597–607 (1999)

4. Cooper, W.W., Park, K.S., Yu, G.: An illustrative application of IDEA (imprecise dataenvelopment analysis) to a Korean mobile telecommunication company. Opns. Res. 49,807–820 (2001)

5. Cooper, W.W., Park, K.S., Yu, G.: IDEA (imprecise data envelopment analysis) with CMDs(column maximum decision making units). J. Oper. Res. Soc. 52, 176–181 (2001)

6. Kim, S.H., Park, C.K., Park, K.S.: An application of data envelopment analysis in telephoneoffices evaluation with partial data. Comp. Oper. Res. 26, 59–72 (1999)

7. Park, K.S.: Simplification of the transformations and redundancy of assurance regions inIDEA (imprecise DEA). J. Oper. Res. Soc. 55, 1363–1366 (2004)

8. Zhu, J.: Imprecise data envelopment analysis (IDEA): a review and improvement with anapplication. Eur. J. Oper. Res. 144, 513–529 (2003)

9. Zhu, J.: Efficiency evaluation with strong ordinal input and output measures. Eur. J. Oper.Res. 146, 477–485 (2003)

10. Zhu, J.: Imprecise DEA via standard linear DEA models with a revisit to a Korean mobiletelecommunication company. Opns. Res. 52, 323–329 (2004)

11. Cook, W.D., Zhu, J.: Rank order data in DEA: a general framework. Eur. J. Oper. Res. 174,1021–1038 (2005)

12. Despotis, D.K., Smirlis, Y.G.: Data envelopment analysis with imprecise data. Eur. J. Oper.Res. 140, 24–36 (2002)

13. Kao, C.: Interval efficiency measures in data envelopment analysis with imprecise data. Eur.J. Oper. Res. 174, 1087–1099 (2006)

14. Park, K.S.: Efficiency bounds and efficiency classifications in DEA with imprecise data.J. Oper. Res. Soc. 58, 533–540 (2007)

15. Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets andSys 113, 427–437 (2000)

16. Kao, C., Liu, S.T.: Data envelopment analysis with missing data: an application to universitylibraries in Taiwan. J. Oper. Res. Soc. 51, 897–905 (2000)

17. Hougaard, J.L.: Fuzzy scores of technical efficiency. Eur. J. Oper. Res. 115, 529–541 (1999)18. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets and Sys 119,

149–160 (2001)19. Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement. Academic

Press, New York (1971)20. Dyer, J.S., Sarin, R.K.: Measurable multiattribute value functions. Opns. Res. 27, 810–822

(1979)21. Keeney, R., Raiffa, H.: Decisions with Multiple Objectives. John Wiley and Sons, New York

(1976)

58 K. S. Park

22. Sage, A.P., White, C.C.: ARIADNE: a knowledge-based interactive system for planning anddecision support. IEEE Trans. Sys. Man Cybern. 14, 35–47 (1984)

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24. Park, K.S.: Mathematical programming models for characterizing dominance and potentialoptimality when multicriteria alternative values and weights are simultaneously incomplete.IEEE Trans. Sys. Man Cybern. Part A 34, 601–614 (2004)

25. Belton, V., Vickers, S.P.: Demystifying DEA – a visual interactive approach based onmultiple criteria analysis. J. Oper. Res. Soc. 44, 883–896 (1993)

26. Stewart, T.J.: Relationships between data envelopment analysis and multi-criteria decisionanalysis. J. Oper. Res. Soc. 47, 654–665 (1996)

27. Bouyssou, D.: Using DEA as a tool for MCDM: some remarks. J. Oper. Res. Soc. 50,974–978 (1999)

28. Malakooti, B.: Ranking and screening multiple criteria alternatives with partial informationand use of ordinal and cardinal strength of preferences. IEEE Trans. Sys. Man Cybern. Part A30, 355–368 (2000)

29. Arnold, V., Bardhan, I., Cooper, W.W., Gallegos, A.: Primal and dual optimality in computercodes using two-stage solution procedures in DEA. In: Aranson, J., Zionts, S. (eds.),Operations Research: Methods, Models and Applications. Quorum Books, Westport, CT,pp. 57–96 (1998)

30. Thompson, R.G., Dharmapala, P.S., Thrall, R.M.: Linked-cone DEA profit ratios andtechnical efficiency with application to Illinois coal mines. Int. J. Prod. Econ. 39, 99–115(1995)

31. Thompson, R.G., Langemeier, L.N., Lee, C.T., Lee, E., Thrall, R.M.: The role of multiplierbounds in efficiency analysis with applications to Kansas farming. J E’conom. 46, 93–108(1990)

32. Charnes, A., Cooper, W.W., Huang, Z.M., Sun, D.B: Polyhedral cone-ratio DEA models withan illustrative application to large commercial banks. J. E’conom. 46: 73–91 (1990)

2 Imprecise Data Envelopment Analysis 59

Chapter 3A General Framework of Dealingwith Qualitative Data in DEA: A FuzzyNumber Approach

Pei Huang Lin

Abstract The original DEA models only deal with quantitative data because thealgebraic operations on qualitative data are meaningless. This chapter provides theframework of dealing with qualitative data in DEA through fuzzy numbers. Atfirst, use fuzzy numbers representing qualitative data. Then apply sets of two-levelmathematical programing to implement fuzzy extension principle to crisp DEAmodel to find a-cuts of leveled fuzzy efficiency based on crisp observations anda-cuts of fuzzy factors. Adequate number of a-cuts determines the fuzzy efficiency.Furthermore, to provide persuadable fuzzy numbers representing qualitative data,use DEA models as experts to integrate objective production data and subjectiveinformation to generate possible values of qualitative data. Based on possiblevalues of qualitative data, the shapes of fuzzy numbers are determined. To increasereadability of fuzzy efficiency for most decision-makers, apply K-medoids clus-tering method along with Hausdorff distances to convert these efficiencies intoqualitative efficiencies. Finally, a case of university performance evaluationdemonstrates the framework.

Keywords Qualitative data � Membership functions � DEA � K-medoids �Clustering analysis

1 Introduction

In competitive business environments, increasing efficiency continuously isessential for most managers. Efficiency is total weighted outputs per one unit oftotal weighted inputs. Decision-making units (DMUs) which produce more outputs

P. H. Lin (&)Department of Management and Information Technology, Southern Taiwan University ofScience and Technology, Tainan 71005, Taiwane-mail: phlin@mail.stust.edu.tw

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_3, � Springer-Verlag Berlin Heidelberg 2014

61

with less quantity of inputs have higher efficiency. Calculating efficiencies isstraightforward for single-input single-output systems, but not for multiple inputsand multiple outputs systems due to the difficulty of designating weights (relativeimportance). Many studies have endeavored to measure efficiencies for multipleinputs and multiple outputs system.

Farrell [1] was one of the earliest researches, using efficiency frontiers withoutapplying weights explicitly to measure relative efficiencies of DMUs. Under theassumption of constant returns to scale, he determined production functions (orfrontiers) first, based on the characteristics that frontier’s slops are negative and nounit is between frontiers and origin. Then the relative efficiencies of enveloped unitswere calculated. The concept is attractive, but the frontier searching processes, withthe number of factors and units increasing, are cumbersome. The technique did notprevail until Charne, Cooper, and Rhodes [2] (denoted CCR) invented the dataenvelopment analysis (DEA) technique which determines the relative efficienciesthrough mathematical programming systematically. DEA reduces computationdramatically and provides much management information simultaneously.

DEA is data oriented and a non-parametric method, exploring the piecewiselinear production functions. Unlike CCR model for constant returns to scale,Banker, Charnes and Cooper [3] (denoted BCC) further proposed the model forvariables returns to scale. After that, many derived models have been applied invaried situations, and DEA techniques have become a popular tool for manyapplication fields, such as agriculture, management, engineering, science, etc. formeasuring efficiency and allocating resources [4–6].

In general, observations are either quantitative or qualitative. Quantitative dataare precise value, and qualitative data (or levels) are imprecise value. Many scalesuse qualitative data to measure preference attitudes. For example, a telecommuni-cation company uses 1 to 5 rating scale, measuring branch service performances.Score 1 is ‘‘very good’’, score 2 ‘‘good’’, score 3 ‘‘normal’’, score 4 ‘‘unsatisfactory’’,and score 5 ‘‘poor’’. Both scores and linguistic terms are symbols, representingordinal levels of service quality.

Qualitative data possess both categorical and ordinal properties. Categoricalproperty implies that observations in the same category have the same preference todecision makers, while ordinal property suggests that observations in different cat-egories have different preference. The distances among qualitative data are unclear,and often depend on situations at that time. When observations are unclear, missing,or hard to be measured numerically timely, qualitative data is useful to present theirstatus. Although qualitative data may includes categorical data in some textbooks, inthe following discussion, qualitative data is for ordinal data (or levels) exclusively.

DEA is implemented by a set of algebraic operations, which work on precisevalues. To simplify analysis, analysts often quantify qualitative data to exactvalues and apply DEA to them directly. For example, designate numerical value 5to score 1 or ‘‘very good’’, value 4 to score 2 or ‘‘good’’, value 3 for score 3 or‘‘normal’’, and so on. That is, the value of very good is 1.25 times of the value ofgood, 1.67 times of the value of normal, 2.5 times of the value of unsatisfactory,and 5 times of the value of poor.

62 P. H. Lin

Such dealing simplifies analysis, but leads to some drawbacks: (1) assigningappropriate values for qualitative data is difficult or impossible; (2) reasonably,efficiencies derived from qualitative observations are imprecise. The resultedprecise efficiencies make decision makers ignore uncertainty and take risks. Insummary, dealing DEA with qualitative data should concern (1) quantifyingqualitative data appropriately to reflect their imprecise properties, (2) building andsolving models appropriately to deal with imprecise ordinal information, and(3) providing resulted efficiency that is, besides being consistent with qualitativeobservations, easy to read.

Many literatures have discussed DEA dealing with qualitative data. Cook et al.[7, 8] used variables, binary variables, and mathematical formulas to representdifferent levels of qualitative data. Cook et al. [9, 10] proposed upper boundsolution approaches for DEA with quantitative data and qualitative data, andnamed such problems as imprecise DEA. Some authors [11–13] provided alter-native approaches for upper bounds. Some authors [14–16] provided lower boundapproaches. Cook and Zhu [17] further provided a general framework dealing withqualitative data. In general, most methods converted, extremely, qualitative datainto optimistic or pessimistic quantitative data without violating ordinal relationsfor the upper bound or lower bound efficiencies. Kao [15] pointed out such dealinghas significant drawbacks. If ordinal relations are concerned only, quantifiedvalues of qualitative levels are intervals, which overlap one another indistin-guishably. That is all level ranges are very loose, and almost overlap one anothercompletely. The implication of qualitative data is that observers cannot distinguishsmall differences among interesting objects but can judge different levels on ascale. Otherwise, they could not assign an appropriate level to an interestingobject. If all quantified levels overlap indistinguishably, it is impossible forobservers to designate an appropriate level to an object. Therefore, applyingordinal relations only cannot describe qualitative data properly.

One may ask what the proper values for a level are. Obviously, different peoplehave different answers because of the imprecision of qualitative data. However,there is an interval, which most people may accept, and moving away from theinterval more, a value will be agreed on by less people, and no one would believethat value when it is far away from the interval. From this point of view, fuzzynumbers are an appropriate tool to represent qualitative data. Fuzzy numbers arean extension of real numbers, a special case of standardized convex fuzzy sets.Any fuzzy set ~A is defined as a set of order pairs:

x; l~AðxÞ� �

jx 2 X� �

where l~A xð Þ : X ! 0; 1½ � is the membership function of ~A, showing the possibilityof each x in discourse X belonging to ~A [18]. Unlike traditional sets, they restrict anelement whether yes/no belongs to the sets; fuzzy sets use soft constraints to defineelements belonging to the sets. Each element uses membership grade to show theintensity that it belongs to a set. Since fuzzy sets are able to describe impreciseconcept, so it is suitable to use fuzzy numbers representing qualitative data.

3 A General Framework of Dealing with Qualitative Data in DEA 63

Several authors [19–21] have developed DEA models embedded with fuzzynumbers; such model is called fuzzy DEA.

Although fuzzy numbers is a useful tool to represent qualitative data, it isdifferent from qualitative data in some aspects. When calculating efficiencies infuzzy DEA, each unit chooses possible values independently from correspondingfuzzy numbers without regarding others units. Contrarily, when fuzzy notationsrepresent qualitative data, values from the same fuzzy notion should always havethe same value in different DMUs and values from different fuzzy notations shouldkeep their ordinal relations as qualitative data. Therefore, it is inappropriate toapply fuzzy DEA to deal with qualitative data represented by fuzzy numbersdirectly. In the following discussion, to distinguish fuzzy numbers and fuzzynumbers representing qualitative data in DEA models, the latter is named asleveled fuzzy numbers.

Kao and Lin [22] introduced binary variables to synchronize qualitative data infuzzy DEA, and used two-level mathematical programing to determine the fuzzyefficiencies. Their approaches extended fuzzy DEA to qualitative data, and couldintegrate subjective information of qualitative data into efficiency models easily.

A convincing result of fuzzy DEA counts heavily on the degree that fuzzynumbers truly represent qualitative data. That is to assign appropriate membershipfunctions for fuzzy numbers. Many methods can construct membership functions[23–25], such as direct polling, direct rating, reverse rating, interval estimating,paired comparisons, experts interacting, etc. Most of them based either onobjective information or on subjective opinions. Kao and Lin [22] proposed analternative approach, which used DEA technique as a tool to integrate objectiveproduction data and subjective information to determine fuzzy numbers forqualitative data. It does not need field experts; rather, it is an objective, timesaving,and effective method to constructing membership functions.

Efficiencies derived from fuzzy numbers are fuzzy. Fuzzy efficiencies areinformative for some fuzzy experts, but it is hard to read for most readers.Therefore, it is better to show qualitative efficiencies. Lin [26] used K-medoidsclustering method along with Hausdorff distances to convert fuzzy efficiencies toqualitative efficiencies. Qualitative efficiencies also increase usability. Many sta-tistical methods can work with qualitative data to explore crucial factors related toefficiencies.

According to the above introduction, the following sections discuss (1) how touse two-level mathematical programming to solve fuzzy efficiencies of fuzzy DEAmodel embedded with leveled fuzzy numbers, (2) how to use DEA as a tool todetermined fuzzy numbers representing qualitative data, (3) how to use K-medoidsmethod converting fuzzy efficiencies to qualitative efficiencies. Finally, a casedemonstrates all procedures mentioned above, and then conclusions followed.

64 P. H. Lin

2 Solving Fuzzy DEA Model with Leveled Fuzzy Numbers

This section discusses the procedure solving DEA models embedded with leveledfuzzy numbers [22], which extends from the work of Kao and Liu [19] for fuzzyDEA.

To enhance the following discussion, the popular BCC model is introducedfirst. BCC model calculates efficiencies on variable returns to scale, and CCRmodel is a special case of BBC model on constant returns scale. Suppose there aren production decision-making units. The unit j (j = 1, 2, …, n) uses m inputfactors Xj = (X1j, X2j, …, Xmj) to produce s output factors Yj = (Y1j, Y2j, …, Ysj),then the input oriented BCC model is

Ek ¼ max: u0 þXs

r¼1

urYrk

!,Xm

i¼1

viXik

s:t: u0 þXs

r¼1

urYrj

!

�Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

ð1Þ

where Ek is the efficiency of the kth unit; u0, ur and vi are decision variables. ur andvi are weights to input and output factors, respectively. Parameter e is a tinypositive value (named as non-Archimedean value). To distinguish variables andobservations, the former is in lower case and the latter in upper case. Objectivefunction is efficiency. The first set of constraints ensures that efficiencies of allunits based on the weights of the kth unit are less than or equal to 1. It follows thetraditional concept that it is impossible to create more energy during energytransformation. Weights ur, vi� e means no factor can be ignored. Variable u0

allows production functions to deviate from origin. If u0 = 0, then the productionsystem is constant returns to scale, CRS, and if u0 [ 0, it is increasing returns toscale, IRS, and if u0 \ 0, it is decreasing returns to scale, DRS.

Model (1) is linear-fractional programming. Let the denominator as 1, andconvert it to linear model.

Ek ¼ max: u0 þXs

r¼1

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þXs

r¼1

urYrj �Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

ð2Þ

3 A General Framework of Dealing with Qualitative Data in DEA 65

The dual of Model (2) is

Ek ¼ min: h� eXm

i¼1

s�i þXs

r¼1

sþr

!

s:t:Xn

j¼1

kjXij þ s�i ¼ hXik; i ¼ 1; . . .;m

Xn

j¼1

kjYrj � sþr ¼ Yrk; r ¼ 1; . . .; s

Xn

j¼1

kj ¼ 1

s�i ; sþr ; kj� 0; i ¼ 1; . . .;m; r ¼ 1; . . .; s; j ¼ 1; . . .; n

h unrestricted in sign

ð3Þ

where h, kj are decision variables; s�i , sþr are slake and surplus variables,respectively. From the economical point of view, the dual problem is to find atheoretical competing system that can produce the same outputs with the leastquantity of inputs, and then measures the efficiency of the assessed unit accord-ingly. The first set of constraints asks a total input from all decision-making unitsis not more than that of the kth unit. The second set of constraints asks a totaloutputs from all decision-making units is not less than the kth unit. The thirdconstraint restricts the competing production system to a convex combination ofall units. The h in the dual problem is corresponding to a constraint of equality inprimary problem, so it is unsigned. k ¼ k1; k2; . . .; knð Þ is an intensity vector ofthe competing system. Model (2) and (3) are dual problem each other, they havethe same optimum solution Ek.

Now, it is time to discuss how to solve DEA models with fuzzy numbers.Suppose there are n decision-making units. To simplify explanation without losinggenerality, further suppose there is only one output factor Y1 which is fuzzyobservations, that is ~Y1 ¼ ~Y11; ~Y12; . . .; ~Y1n

� �. Solving fuzzy DEA models is to

determine all possible efficiencies and corresponding membership grades, based on

all possible inputs and outputs, (x, y), where y1 ¼ y11; y12;; . . .; y1n

� �, y1 2

S1 � S2 � . . .� Sn: Sj ¼ y1jjl~Y1jy1j

� �[ 0

n ois the support (denoted sup) of ~Y1j.

Model (2) determines efficiencies. To avoid confusion between ~Ek and Ek, denoteefficiencies derived from Model (2) as e. Then apply extension principle to deter-mine the membership grade of each e with respect to ~Ek [18, 27]. Calculating allpossible efficiencies and corresponding membership grades is cumbersome. Kaoand Liu [19] turned to a-cuts to implement extension principle. The a-cut (or alevel) of any fuzzy number ~A, denoted (A)a, is defined as xjl~AðxÞ� a; a 2 0; 1ð �

� �.

An a-cut is a crisp set. Kao and Liu determined (Ek)a of fuzzy efficiencies based on

66 P. H. Lin

crisp observations and (Y1)a of the fuzzy output factor through two-level mathe-matical programming. Normally, adequate numbers of Ekð Þai

(i = 1,…, n) deter-

mine ~Ek [23, 28].Further, suppose output factor Y1 is qualitative observations on 1 - L level

scale. The first level yields the highest output and the Lth level the lowest output.

Let leveled fuzzy number ~Zl represent level l. Then ~Y1j ¼PL

l¼1~ZlBðlÞj where BðlÞj

are binary variables to signify whether the jth decision-making units is level l. If its

level is l, BðlÞj ¼ 1, otherwise BðlÞj ¼ 0. Thus the possible level value vector z = (z1,

z2, …, zL) at a-level are z 2 C ¼ zjz 2 Z1ð Þ � Z2ð Þ � � � � � ZLð Þ; zl� zðlþ1Þ� �

.Substitute all possible x, y to Model (2) yields (Ek)a. The work seems cumbersome;actually the upper bound and the lower bound are enough to determine (Ek)a. Theupper bound and the lower bound are

Ekð ÞUa ¼ maxz2C

Ek x; yð Þ ð4Þ

Ekð ÞLa¼ minz2C

Ek x; yð Þ ð5Þ

Two sets of two-level mathematical programming can implement Model (4)and Model (5), respectively.

Ekð ÞUa ¼ max:Zlð ÞLa � zl� Zlð ÞUa

l ¼ 1; . . .; L

zl� zlþ1

l ¼ 1; . . .; L� 1

max: u0 þ u1

XL

l¼1

BðlÞk zl þXs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þ u1

XL

l¼1

BðlÞj zl þXs

r¼2

urYrj �Xm

i¼1

viXij� 0;

j ¼ 1; . . .; n

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

8>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>:

ð6Þ

The outer level of Model (6) generates all possible values of z, and passes themto the inner level that determines corresponding efficiencies and returns efficienciesto the outer level. Finally, the outer level chooses the maximum value as the upper

bound, Ekð ÞUa .

Similarly, Model (7) implements Model (5) for Ekð ÞLa .

3 A General Framework of Dealing with Qualitative Data in DEA 67

Ekð ÞLa¼ min:Zlð ÞLa � zl� Zlð ÞUa

l ¼ 1; . . .; L

zl� zlþ1

l ¼ 1; . . .; L� 1

max: u0 þ u1

XL

l¼1

BðlÞk zl þXs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þ u1

XL

l¼1

BðlÞj zl þXs

r¼2

urYrj �Xm

i¼1

viXij� 0;

j ¼ 1; . . .; n

u0 unrestricted in sign

8>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>:

ð7Þ

Since both levels in Model (6) have the same maximum operation, they can beintegrated as a maximizing mathematical programming.

Ekð ÞUa ¼ max: u0 þ u1

XL

l¼1

zlBðlÞk þ

Xs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þ u1

XL

l¼1

zlBðlÞj þ

Xs

r¼2

urYrj �Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

Zlð ÞLa � zl� Zlð ÞUa ; l ¼ 1; . . .; L

zl� zlþ1; l ¼ 1; . . .; L� 1

zL� 0

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

ð8Þ

Model (8) is nonlinear. Nonlinear term u1zl can be linearized by changingvariable techniques. Let wl = u1zl, then the model becomes

68 P. H. Lin

Ekð ÞUa ¼ max: u0 þXL

l¼1

wlBðlÞk þ

Xs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þXL

l¼1

wlBðlÞj þ

Xs

r¼2

urYrj �Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

u1 Zlð ÞLa �wl� u1 Zlð ÞUa l ¼ 1; . . .; L

wl�wlþ1; l ¼ 1; . . .; L� 1

wL� 0

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

ð9Þ

Model (9) is linear model. If optimum solutions of wl and u1 are wl* and u1

*, thenthe optimum of zl is zl

* = wl*/u1

*.Similarly, Model (7) can be simplified as a minimizing mathematical pro-

gramming, but it needs some tricks. Since Model (7) is a min–max program,convert the maximum program in inner level to a minimum program first throughits dual form, see Model (3), and then integrate the min–min program to a mini-mum program.

Ekð ÞLa¼ min: h� eXm

i¼1

s�i þXs

r¼1

sþr

!

s:t:Xn

j¼1

kjXij þ s�i ¼ hXik; i ¼ 1; . . .;m

Xn

j¼1

kj

XL

l¼1

zlBðlÞj � sþ1 ¼

XL

l¼1

zlBðlÞk

Xn

j¼1

kjYrj � sþr ¼ Yrk; r ¼ 2; . . .; s

Xn

j¼1

kj ¼ 1

Zlð ÞLa � zl� Zlð ÞUa ; l ¼ 1; . . .; L

zl� zlþ1; l ¼ 1; . . .; L� 1

zL� 0

s�i ; sþr ; kj� 0;

i ¼ 1; . . .;m; r ¼ 1; . . .; s; j ¼ 1; . . .; n

h unrestricted in sign:

ð10Þ

3 A General Framework of Dealing with Qualitative Data in DEA 69

Model (10) is nonlinear, and its variable relationship is complex, comparedwith Model (8). Converting it to a liner form is difficult. This does not cause aproblem because many mathematical packages solve nonlinear programs, such ascommercial package Lingo.

In summary, Model (9) and (10) determine the upper bound and the lowerbound of (Ek)a. Enumerating adequate numbers of Ekð Þai

for different ai simulates~Ek. Theoretically, more a-cuts make the shape of emulated fuzzy efficiency moresmoothly; the tradeoff is the computation cost. In general, ten a-cuts are enough. Ifthe shape is nonlinear, it needs more cuts; mostly, 25 cuts are enough empirically.

3 Determine Fuzzy Numbers for Qualitative Data

This section discusses the method proposed by Kao and Lin [22], which uses DEA asa tool to determine the fuzzy numbers representing qualitative data. To determiningfuzzy numbers is to explore possible values and their grades for the presentedqualitative data.

DEA models give any unit k the most preferable values to weights and decisionvariables. These values are similar to opinions of the expert k to these variables.Thus one can use variables representing qualitative data and solve DEA models toexplore their possible quantified values. That is, DEA models can act as expertsto integrate objective and subjective information and then to express his/herquantifying opinions to qualitative data.

Suppose there are n decision-making units. A production system has m inputfactors and s output factors. All factors are quantitative data except output Y1 that isqualitative data on 1–L level scale. The first level has the highest beneficial outputvalue. If level l is quantified as zl� 0; Y1 of the jth unit is Y1j 2

zljl ¼ 1; 2; . . .; L; zl [ zðlþ1Þ� �

and Y1j ¼PL

l¼1 zlBðlÞj where BðlÞj are binary variable.

If its level is l then BðlÞj ¼ 1, otherwise BðlÞj ¼ 0. That is Y1j ¼ 0; . . .; 0; zl; 0; . . .; 0½ �.The strong inequalities zl � zlþ1� 0 can be converted into zl � zlþ1� d where d is atiny positive value. Substitute Y1, and insert ordinal relations and assurance regionsof decision variables into Model (2), gaining the following efficiency model.

70 P. H. Lin

Ek ¼ max: u0 þ u1

XL

l¼1

zlBðlÞk þ

Xs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þ u1

XL

l¼1

zlBðlÞj þ

Xs

r¼2

urYrj �Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

zl � zlþ1� d; l ¼ 1; . . .; L� 1

zL� 0

zl 2 C; l ¼ 1; . . .; L

ur 2 X1; r ¼ 1; . . .; s

vi 2 X2; i ¼ 1; . . .;m

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

ð11Þ

where u0, ur, vi and zl are decision variables, C is assurance regions of all zl, X1 andX2 are assurance regions of output and input weights, respectively. The modelchooses the most favorable values of decision variables for the assessed unit k. Theterms u1zl are nonlinear. Let wl = u1zl and convert Model (11) into linear modelby changing variable technique.

Ek ¼ max: u0 þXL

l¼1

wlBðlÞk þ

Xs

r¼2

urYrk

s:t:Xm

i¼1

viXik ¼ 1

u0 þXL

l¼1

wlBðlÞj þ

Xs

r¼2

urYrj �Xm

i¼1

viXij� 0; j ¼ 1; . . .; n

wl � wlþ1� u1d; l ¼ 1; . . .; L� 1

wL� 0

wl 2 _C; l ¼ 1; . . .; L

ur 2 X1; r ¼ 1; . . .; s

vi 2 X2; i ¼ 1; . . .;m

ur; vi� e; r ¼ 1; . . .; s; i ¼ 1; . . .;m

u0 unrestricted in sign

ð12Þ

where _C is assurance region of wl. If the optimum solutions of wl and u1 are w�l andu�1, then the optimum solutions of zl are z�l ¼ w�l

�u�1. The wl can be thought as the

weight (or importance) of level l. The model is consistent with the characteristics

3 A General Framework of Dealing with Qualitative Data in DEA 71

of qualitative data—the same level with the same quantified value and the differentlevels governed with their ordinal relations.

Model (11) and (12) determine efficiencies based on the favorable quantifiedvalues of qualitative data. It’s worth noting that assurance regions must impose onzl and wl in order to gain meaningful quantified values. For instance, Beasley [29]rated research performance on scale of Star, A+, A and A-, when appraising 52physical and chemistry departments in UK. He determined subjectively that theimportance of Star is at least two times greater than A+, and A+ at least two timesgreater than A, and A at least two times greater than A-, and Star at most 12 timesgreater than A-.

Calculating efficiency for each unit gains a set of quantified values for a qual-itative scale, and there are n set of quantified values after calculating efficiencies forn different units. Thus, each qualitative level has n possible quantitative values.Then, we can use these possible values to determine the corresponding fuzzynumbers. However, it is known that variables (wl, ur, vi) of DEA models havemultiple solutions, based on Model (1). That is (wl*, ur*, vi*) and (cwl*, cur*, cvi*)can arrive at the same optimum solution where c is a positive value. Thus, in orderto integrate opinions from different units, optimum values of their decision vari-ables need to be standardized beforehand. The standardized procedure is to dividethe weight of each level by the average weight of qualitative data in the same scalein each run of Model (12). That is, standardized wl

*(l = 1,…, L) is zl, zl = wl*/

((w1* þ w2

* þ …þ wL*)/L). Another way is to divide each level value by the average

value of all qualitative data in the same scale in each run of Model (11).

If level l has n different quantized values zðjÞl (j = 1,…, n), then the minimum,25 and 75th percentile, and the maximum can be chosen as the four vertexes oftrapezoidal membership function l~Zl

xð Þ [22, 30, 31]. Equation (13) is trapezoidalmembership function, denoted [a1, a2, a3, a4]

l~ZlðxÞ ¼

x� a1ð Þ= a2 � a1ð Þ; a1� x� a2

1; a2� x� a3

x� a4ð Þ= a3 � a4ð Þ; a3� x� a4

0; otherwise

8>><

>>:ð13Þ

One may prefer using triangular membership functions to doing fuzzy analysis,

then the minimum, medium, and maximum of zðjÞl (j = 1,…, n) is the threevertexes of triangular membership functions.

In summary, DEA analysis lets every unit select the most favorable values toquantify qualitative data. Then standardize each value by the average values in thesame scale in each run. Collect all standardized quantified values from differentunits and choose the minimum, 25th and 75th percentile, and the maximum as fourvertexes of a trapezoidal membership function. Providing decision variablesassurance regions is important to gain reasonable quantifying values of qualitativedata.

72 P. H. Lin

4 Converting Fuzzy Efficiencies to Qualitative Data

This section discusses how to apply clustering analysis to convert fuzzy efficienciesto qualitative data. Clustering analysis is an explorative technique, which separatesa pool of objects into groups based on some common characteristics of objects.Therefore, apply clustering analysis to converts fuzzy efficiencies into qualitativedata.

Punj and Stewart [32] used 12 real cases to study Pros and Cons of differentclustering methods. At first, they used different methods to measure the commoncharacteristics in each group. After that, they applied different clustering methodsto pools of objects and compared the clustered results with the original groups.They concluded that K-means (clustering), a non-hierarchical method, along withcentroids assigned non-randomly has the best performance. The following isminimum variance method (or Ward’s method), a hierarchical method. The third isaverage linkage method, a hierarchical method.

K-means and its variety apply iterated procedures to search centroids and theirgroup members until reaching some satisfactory conditions. At first, it locatescentroids by some ways, then allocates all objects to the nearest centroids, andcalculates the new centroids according to the positions of group members. Then,the procedures of allocating members and locating centroids iterate until someterminated conditions are satisfied [33]. The terminated condition could be thecentroid position changing within a small range. K-means is good for crispobservations, but with some shortcomings. Gullo et al. [34] indicated that there aretwo shortcomings when K-means deals with observations formulated as proba-bility density functions. First, centroids based on the expected values are crispvalues, which is not precise to represent uncertain observations. Second, thecomputation cost of calculating the distances between centroids and uncertainobservations. K-means dealing with fuzzy observations has similar problems. Itneeds to calculate the distances between the centroids and fuzzy observations. Thecomputation cost is high. Furthermore, representing a group of fuzzy observationsby a crisp centroid is imprecise. Hence, Lin [26] applied k partition aroundmedoids (PAM, denoted K-medoids) to do cluster analysis for fuzzy observations.K-medoids (clustering) are a variety of K-means [35].

K-medoids chooses a number, say k, of objects as medoids, initially, from thepool of observations. Medoids are group centers. Then allocate the rest observa-tions to existing groups and sum all distances between medoids and its groupmembers as the total distance. The searching strategy of K-medoids is to changemedoids and theirs members continuously to reduce the total distance until itreaches a terminated conditions. In contrast to K-means, which calculates thedistances between centroids and observations every iteration, K-medoids onlyneeds to calculate the distances between objects once at beginning. Therefore, thedistances calculation is simple. If observations are fuzzy, medoids and groupcenters are fuzzy too, so K-medoids reflects precisely the fact that centers of fuzzygroup members are fuzzy. Furthermore, K-medoids is more robust than K-means

3 A General Framework of Dealing with Qualitative Data in DEA 73

for outliers, because it tries to minimize total distances rather than to allocateobservations to the nearest center [35]. Many cluster analyses have appliedvarieties of K-medoids for a large amount of data with uncertainty [34, 36]. It isoften that the number of units in DEA is less than 150, so integer mathematicalprogramming can implement K-medoids easily.

Implementing K-medoids needs to know distances between observations. Dis-tance is a similarity measure or dissimilarity measure. Similarity measure is tojudge the degree of similarity between objects on some interesting attributes.Many applications such as decision analysis, pattern recognition, machine leaning,marketing forecasting are often based on it. Many authors have proposed manysimilarity measures for fuzzy observations [37–41]. These fuzzy similarity mea-sures can be categorized based on properties and applications, including geometricdistance model, set theoretic approach, pattern recognition approach, Hausdorffdistance, etc. [42]. Choosing similarity measures are often problem oriented.Categorizing body motion and detecting subtle difference in facial expressionoften use different measures with different sensitivity. The studies of Pappis andKaracapilidis [43] and Chen et al. [44] about fuzzy approximate equality are alsovaluable for choosing an appropriate similarity measure. Tversky [45] and Santiniand Jain [46] concluded from psychological literature that the difference betweensimilarity measures based on human perceptions and Euclidean distances arestatistically significant and cannot be ignored. That indicates the importance ofbehavioral studies to selecting similarity measures.

Zwick et al. [42] studied the correlations between 19 common fuzzy similaritymeasures and human judgment systematically. Their experiments applied differentapproaches to six fuzzy linguistic terms, such as dubious, possible, certain, etc. fordichotomous classification and differentiation judgment between linguistic terms.Experiments showed all different similarity measures and experimental subjectsreached the same correct results for coarse binary classification. For differentiationjudgment, three types of Hausdorff measures and subjects had the highest corre-lations. Another measure also had high correlation with subjects, but it is based onlogical operations. Logical operations give two fuzzy sets, without intersection,similarity index as 0 without regarding how far between two fuzzy observations[43]. Therefore, such a measure is not concerned here. They also pointed out thatHausdorff distance between two fuzzy observations based on an a-cut at a specificlevel even have higher correlation with subjects than multiple a-cuts. Manyapplications have applied Hausdorff type distances and their variety to measuresimilarity among fuzzy observations [47–50]. In summary, it is a good choice toapply Hausdorff distance measures in clustering analysis for fuzzy observations,because they are highly correlated to human judgments and the clustering resultswill be more acceptable for most stakeholders.

Intuitively, Hausdorff distance is the longest distance of members of twosubsets in a metric space moving to another set through the shortest distance. Letinft2T

d(s, t) is the shortest distance of a point s in a set S moving to another set

T. Then the Hausdorff distance between S, T is

74 P. H. Lin

dH S; Tð Þ ¼ max: sup :s2S

inf :t2T

d s; tð Þ; sup :t2T

inf :s2S

d s; tð Þ� �

ð14Þ

The first term in brace of Eq. (14) is the longest distance from S to T, and thesecond term is the longest distance from T to S. According to Eq. (14), theHausdorff distance of two intervals [l1, u1] and [l2, u2] is [49]

max: l1 � l2j j; u1 � u2j jf g ð15Þ

There is not a universal definition of Hausdorff distance between two fuzzynumbers. Usually, it is determined based on some Hausdorff distances betweentheir a-cuts. Equation (16a), (16b), (16c) lists three common Hausdorff distancesbetween two fuzzy numbers ~S and ~T , see Fig. 1.

d� ¼ max:ai¼1

jSLai� TL

aij; jSU

ai� TU

aij

n oð16aÞ

d1 ¼ max:ai2ð0; 1�

SLai� TL

ai

; SU

ai� TU

ai

n oð16bÞ

dg ¼1n

Xn

i¼1

dH ½SLai; SU

ai�; ½TL

ai; TU

ai�

�ð16cÞ

where [SLai

, SUai

] and [TLai

, TUai

] are a-cuts of ~S and ~T , respectively, at level ai. d* isHausdorff distance of cut at ai = 1. d? is the largest Hausdorff distance of ai [ (0,1], and dg is the average Hausdorff distance of a set of ai.

Given Hausdorff distances among fuzzy observations, one can apply K-medoidsmethods to implement cluster analysis by integer programming [35]. Assume thereare n fuzzy observations ~E ¼ ~Eiji ¼ 1; 2; . . .; n

� �to be partitioned into k groups.

Distances between the ith and jth observations are d(~Ei ~Ej). Let bi be binary var-iable. If the ith observation is a medoid then bi = 1, otherwise bi = 0. Let pij also abinary variable. If the jth observation is in the same group as the ith observationwhich is a medoid, then pij = 1, otherwise pij = 0. Thus model of K-medoids is

0.0

0.2

0.4

0.6

0.8

1.0

μ

X

S T

iLSα i

USα iUTαi

LTαi

Fig. 1 The Hausdorffdistance between two fuzzynumbers

3 A General Framework of Dealing with Qualitative Data in DEA 75

TD ¼ min:Xn

i¼1

Xn

j¼1

d ~Ei; ~Ej

� �pij

s:t:Xn

i¼1

pij ¼ 1; j ¼ 1; . . .; n

pij� bi; i; j ¼ 1; . . .; n

Xn

i¼1

bi ¼ k

bi; pij 2 0; 1f g; i; j ¼ 1; . . .; n

ð17Þ

where the objective function is the total distance of observations and their cor-responding medoids. The first set of constraints restricts each observation to onecorresponding medoid. The second set of constraints prevents each observationbinding to an observation, which is not a medoid. The third constraint declares thenumber of partitioned groups.

Assigning a proper number of groups, k, is important. It ensures that obser-vations in a group are similar and in different groups are different significantly.Scree plot, see Fig. 2, is a popular tool to determine the number of groups. It showsthe relationship between total distances and partitioned group number, and itsshape is like a curved arm. The cut-off partitioned number is at the elbow. That is,when the partitioned number is less than the number; increase the partitionnumber, decrease total distance significantly, and increase the similarity withingroups. In contrast, with more of that, increasing partitioned numbers has littleeffect on increasing similarity within groups, and observations in the same groupintrinsically may be separated with high probability. Therefore, the partitionednumber at elbow is often recommended [33]. However, decision makers maychoose the number subjectively, such as 3 or 5, based on management perspective.

0 1 2 3 4 5 6

Dis

tanc

e

Groups

Elbow

Fig. 2 Scree plot

76 P. H. Lin

In short, to convert fuzzy numbers into qualitative data, let Formula(16) determine the Hausdorff distances between observations, and run Model(17) along with different partition numbers. Then choose the partition number atelbow on scree plot or subjectively. Thus, medoids and its members are deter-mined. Normally, the ordinal relation among medoids is obvious. If not, applyfuzzy ranking method, such as Chen and Klein [51], to determine their ranks.Finally, each fuzzy number has the same rank as its medoid.

5 Case Study

This section demonstrates how the methods discussed in former sections deal withqualitative data in DEA.

Beasley [29] used three input factors X1–X3 and five output factors Y1–Y5

measuring efficiencies of 52 chemical departments in UK. Inputs are generalexpenditure (X1), equipment expenditure (X2), research income (X3), and outputsare number of undergraduates (UGs or Y1), number of postgraduates taught (PGs-Tor Y2), number of research postgraduates (PGs-R, Y3), research outcome (Y4), andresearch quality(Y5). Since there was no suitable data to evaluate research outcomeat that time, research income was used as a proxy of it for their high correlation.All factors are quantitative except research quality, which is ranked into four levels(Star, A+ , A, and A–) and expressed by four binary variables B5–B8. Table 1 listsperformance data of universities. Then department efficiencies are determined byModel (18), which is based on CCR model along with some weight constraints andrelevant ordinal information. CCR model is a special case of BCC model withoutu0, see Model (2).

Ek ¼ max:X4

r¼1

urYrk þX8

r¼5

wrBrk ð18aÞ

s:t:X3

i¼1

viXik ¼ 1 ð18bÞ

X4

r¼1

urYrj þX8

r¼5

wrBrj �X3

i¼1

viXij� 0; j ¼ 1; . . .; 52 ð18cÞ

ur � 0; r ¼ 1; . . .; 4; wr � 0; r ¼ 5; . . .; 8; vi� 0; i ¼ 1; 2; 3 ð18dÞ

u3� 1:25u2� 1:252u1; u3� 2u1 ð18eÞ

0:51 �X3

r¼1

urYrk

,X4

r¼1

urYrk þX8

r¼5

wrBrk

!

� 0:765 ð18fÞ

3 A General Framework of Dealing with Qualitative Data in DEA 77

Table 1 Data for the chemistry departments of 52 UK universities [29]

No General exp(£000s)X1

Equip exp(£000s)X2

Research income(£000s)X3, Y4

UGs PGs PGs Research rating

T R Star A+ A A-

Y1 Y2 Y3 B5 B6 B7 B8

1 446 21 183 62 0 37 0 0 0 12 670 53 288 137 0 43 0 0 1 03 1,459 69 288 225 3 63 0 0 1 04 613 95 73 92 0 12 0 0 0 15 2,043 256 1,050 253 27 118 1 0 0 06 686 46 436 137 18 27 0 0 0 17 2,227 620 981 305 0 159 1 0 0 08 696 93 354 81 0 31 0 0 0 19 1,027 148 578 187 0 42 0 0 1 010 1,155 113 545 126 31 90 0 1 0 011 620 115 565 76 5 49 0 0 0 112 984 138 198 166 0 32 0 0 1 013 880 78 488 119 9 29 0 0 1 014 440 51 217 50 0 20 0 0 0 115 667 281 111 116 0 29 0 0 0 116 685 50 191 92 11 15 0 0 0 117 2,545 210 763 320 9 82 0 0 1 018 919 61 419 173 0 49 0 0 1 019 1,259 82 496 195 0 56 0 1 0 020 734 33 142 46 26 48 0 0 0 121 1,760 742 1,061 167 0 141 1 0 0 022 1,487 479 521 240 3 42 0 0 1 023 1,106 170 430 164 3 37 0 1 0 024 962 131 152 122 0 33 0 0 0 125 1,238 67 490 157 4 60 0 1 0 026 1,208 89 397 158 26 49 0 0 0 127 1,920 191 544 268 0 81 0 1 0 028 1,758 196 1,162 237 9 105 0 0 1 029 1,211 79 540 157 0 52 0 0 1 030 1,409 122 527 263 0 94 0 1 0 031 3,337 654 1,780 707 0 211 1 0 0 032 908 120 336 162 5 36 0 0 0 133 1,492 127 613 152 18 102 0 0 0 134 1,346 78 250 223 0 64 0 1 0 035 1,620 420 1,224 199 2 124 1 0 0 036 691 65 407 122 2 27 0 0 0 137 1,324 144 565 189 13 104 0 1 0 038 927 148 359 147 0 43 0 0 1 039 947 146 724 236 7 54 0 0 1 040 370 32 102 58 1 10 0 0 0 141 360 73 122 89 0 7 0 0 0 142 849 32 258 158 3 53 0 1 0 043 764 89 317 132 0 31 0 0 1 0

(continued)

78 P. H. Lin

0:51 �X52

j¼1

X3

r¼1

urYrj

,X52

j¼1

X4

r¼1

urYrj þX8

r¼5

wrBrj

!

� 0:765 ð18gÞ

0:664 �X8

r¼5

wrBrk

,

u4Y4k þX8

r¼5

wrBrk

!

� 0:997 ð18hÞ

0:664 �X52

j¼1

X8

r¼5

wrBrj

,X52

j¼1

u4Y4j þX8

r¼5

wrBrj

!

� 0:997 ð18iÞ

w5� 2w6� 22w7� 23w8;w5� 20w8 ð18jÞ

0:8F �X52

j¼1

v2X2j

,X52

j¼1

X3

i¼1

viXij � 1:2F ð18kÞ

F ¼X52

j¼1

X2j

,X52

j¼1

X3

i¼1

Xij ð18lÞ

0:4v1 � v3 � 0:6v1 ð18mÞ

The first three sets of constraints are fundamental of CCR model, and otherconstraints are assurance regions of weights and ordinal relations of researchranks. Constraint (18e) declares the relative importance regarding research post-graduates, postgraduates taught, and undergraduates. Research postgraduates are1.25 times more important than postgraduates taught are, postgraduates taught are1.25 times more important than undergraduates are, and research postgraduates areequal to or less than 2 times importance of undergraduates. Constraint (18f)restricts weighted sum of teaching over weighted sum of research to 0.51–0.765for the kth university. Constraint (18h) restricts weighted research quality over

Table 1 (continued)

No General exp(£000s) X1

Equip exp(£000s) X2

Research income(£000s) X3, Y4

UGs PGs PGs Research rating

T R Star A+ A A-

Y1 Y2 Y3 B5 B6 B7 B8

44 560 99 196 100 0 24 0 0 0 145 1,029 126 391 164 2 39 0 0 0 146 619 21 136 73 0 13 0 0 0 147 1,381 254 812 292 0 71 0 1 0 048 2,253 131 360 354 7 94 0 0 1 049 768 38 324 142 0 25 0 0 0 150 696 73 408 121 0 29 0 0 0 151 421 18 105 57 0 15 0 0 0 152 1,714 112 945 269 15 77 0 1 0 0

3 A General Framework of Dealing with Qualitative Data in DEA 79

weighted sum of research output to 0.664–0.997. Constraint (18g) and (18i) aresimilar to constraint (18f) and (18h), but a total value of all universities is con-cerned. Constraint (18j) shows the relative importance of every adjacent level isequal to or less than value 2, and highest rank Star is less than 20 times moreimportant than the lowest rank A–. Constraint (18k) and (18l) are to restrict theratio of total weighted equipment expenditure over total weighted expenditure ofall universities to somewhere between 0.8–1.2 times of original correspondingratio. The last constraint (18m) is to restrict the weight of research income to0.4–0.6 times of general expenditure’s. Assurance regions are drawn up to makeweighting variables and level values consistent with experience or polices. In thisexample, relations between different types of expenditure are objective experience,and relations between teaching outcome and research outcome would governdeveloping directions of universities.

Running Model (18) for all universities yields 52 (point) efficiencies, see Col 2in Table 2, and 52 sets of level weights w5–w8 for research quality. Differentuniversities have different favorable weights, so these weights reflect opinions ofdifferent universities to level values, which can be used to construct fuzzy numbersrepresenting qualitative data. Before combining them together, these weights arestandardized by their average in each run because Model (18) has multiple solu-tions. Figure 3 shows Box plots of four level weights. The left and the right handside boundaries of each box are 25 and 75th percentile, respectively, the verticalbar inside a box is a median, and short vertical bars at both end of whiskers are 5and 95th percentiles, respectively. Use the minimum, 25 and 75th percentile, andthe maximum as the four vertexes to determine the shape of trapezoidal mem-bership functions. Hence, Star = [2.133, 2.526, 2.932, 2.963], A+ = [0.593,0.610, 0.842, 1.067], A = [0.295, 0.302, 0.421, 0.533] and A- = [0.150, 0.151,0.211, 0.270], see Fig. 4. These fuzzy numbers have some traits: (1) 0+-cutintervals increase fast, (2) there is no overlapping boundary, (3) the distancebetween star and A+ is large, which implies star is more important than A+ .Those traits are mainly influenced by constraints from (18e) to (18m). If reducemultiplies 2 and 20 in constraint (18j) into 1.25 and 12, respectively, these resultedfuzzy numbers overlap one another and the distances among them shrink. Thus,providing correct and sufficient information is crucial to gaining reasonable fuzzynumbers representing qualitative data.

Given fuzzy numbers Star, A+, A, and A- as ~Zl, l = 1,…, 4, combine them

with four binary variables B5–B8 into a leveled fuzzy variable ~Y5j ¼P4

l¼1~ZlBðlÞj .

The revised model of university efficiencies is to substituteP8

r¼5 wrBrj ¼u5P4

l¼1 zlBðlÞj , and add a-cuts constraints Zlð ÞLa � zl� Zlð ÞUa and ordinal constraints

zl � zlþ1Pd into Model (18). Since leveled fuzzy numbers already replace qual-itative data, constraint (18j) is not necessary. To gain the efficiency of the kthuniversity, ~Ek, is to determine a set of Ekð Þa through their lowers bound and upperbounds by the revised model. The procedures are similar to applying Model(8) and Model (10). After that, a set of Ekð Þa determine ~Ek.

80 P. H. Lin

Table 2 Point, interval, and qualitative efficiencies of 52 UK universities

No Pointefficiency

Interval efficiency Level

a = 0+ a = 0.5 a = 1

1 0.81 [0.67, 0.93] [0.70, 0.91] [0.73, 0.88] 22 0.91 [0.77, 1.00] [0.80, 1.00] [0.83, 0.98] 13 0.64 [0.51, 0.74] [0.54, 0.72] [0.57, 0.70] 34 0.54 [0.45, 0.64] [0.47, 0.62] [0.49, 0.60] 4*

5 1.00 [0.78, 1.00] [0.84, 1.00] [0.90, 1.00] 1*

6 0.68 [0.52, 0.83] [0.56, 0.80] [0.59, 0.76] 37 0.96 [0.74, 0.96] [0.79, 0.96] [0.86, 0.96] 18 0.51 [0.42, 0.60] [0.44, 0.58] [0.46, 0.56] 49 0.66 [0.54, 0.74] [0.56, 0.74] [0.69, 0.72] 3

10 0.87 [0.85, 0.87] [0.86, 0.87] [0.86, 0.87] 211 0.59 [0.46, 0.70] [0.49, 0.68] [0.51, 0.65] 412 0.66 [0.55, 0.76] [0.57, 0.74] [0.60, 0.71] 313 0.61 [0.52, 0.72] [0.54, 0.70] [0.56, 0.66] 314 0.58 [0.50, 0.71] [0.52, 0.67] [0.54, 0.63] 415 0.56 [0.45, 0.66] [0.47, 0.64] [0.49, 0.61] 416 0.56 [0.46, 0.65] [0.48, 0.63] [0.51, 0.61] 417 0.44 [0.33, 0.53] [0.35, 0.52] [0.37, 0.49] 418 0.78 [0.63, 0.87] [0.67, 0.87] [0.70, 0.84] 319 0.75 [0.74, 0.75] [0.74, 0.75] [0.74, 0.75] 220 0.64 [0.51, 0.76] [0.54, 0.73] [0.57, 0.71] 321 0.89 [0.70, 0.90] [0.75, 0.90] [0.80, 0.90] 222 0.51 [0.40, 0.59] [0.42, 0.58] [0.45, 0.56] 423 0.71 [0.68, 0.71] [0.68, 0.71] [0.69, 0.71] 224 0.49 [0.39, 0.57] [0.41, 0.56] [0.43, 0.54] 425 0.72 [0.71, 0.73] [0.71, 0.73] [0.71, 0.72] 226 0.50 [0.00, 0.63] [0.00, 0.60] [0.00, 0.56] 5*

27 0.65 [0.63, 0.65] [0.64, 0.65] [0.64, 0.65] 328 0.53 [0.40, 0.64] [0.43, 0.63] [0.45, 0.59] 429 0.58 [0.47, 0.66] [0.50, 0.65] [0.52, 0.63] 430 0.88 [0.86, 0.88] [0.87, 0.88] [0.87, 0.88] 231 0.98 [0.81, 0.99] [0.84, 0.98] [0.88, 0.98] 132 0.57 [0.43, 0.70] [0.46, 0.68] [0.48, 0.64] 433 0.45 [0.00, 0.58] [0.00, 0.54] [0.00, 0.48] 534 0.81 [0.80, 0.81] [0.81, 0.81] [0.81, 0.81] 235 1.00 [0.80, 1.00] [0.85, 1.00] [0.90, 1.00] 136 0.60 [0.48, 0.72] [0.50, 0.69] [0.53, 0.67] 337 0.87 [0.84, 0.87] [0.85, 0.87] [0.85, 0.87] 238 0.67 [0.56, 0.75] [0.58, 0.74] [0.61, 0.72] 338 0.80 [0.63, 0.93] [0.67, 0.92] [0.70, 0.89] 240 0.68 [0.59, 0.83] [0.61, 0.78] [0.63, 0.73] 341 0.80 [0.68, 0.91] [0.70, 0.88] [0.73, 0.85] 242 1.00 [1.00, 1.00] [1.00, 1.00] [1.00, 1.00] 1

(continued)

3 A General Framework of Dealing with Qualitative Data in DEA 81

Table 2 (continued)

No Pointefficiency

Interval efficiency Level

a = 0+ a = 0.5 a = 1

43 0.73 [0.62, 0.82] [0.64, 0.81] [0.67, 0.78] 344 0.66 [0.54, 0.77] [0.56, 0.75] [0.59, 0.72] 3*

45 0.51 [0.39, 0.63] [0.41, 0.61] [0.43, 0.57] 446 0.52 [0.44, 0.60] [0.46, 0.58] [0.47, 0.55] 447 0.81 [0.78, 0.81] [0.79, 0.81] [0.79, 0.81] 2*

48 0.54 [0.00, 0.65] [0.00, 0.64] [0.00, 0.60] 549 0.61 [0.48, 0.73] [0.51, 0.71] [0.54, 0.68] 350 0.60 [0.47, 0.71] [0.50, 0.69] [0.52, 0.66] 451 0.65 [0.56, 0.79] [0.58, 0.74] [0.60, 0.70] 352 0.71 [0.70, 0.71] [0.70, 0.71] [0.70, 0.71] 2

0.5 1.0 1.5 2.0 2.5 3.0

u5

u7

u8

u6

Weight

Fig. 3 Box plots of fourresearch quality level weights

0 0.5 1 1.5 2 2.5 3

0.2

0.4

0.6

0.8

1 A– StarA+AµFig. 4 Four fuzzy numbers

representing research quality

82 P. H. Lin

Table 2 lists three a-cuts, a = 0+, 0.5, 1 of efficiency from Col 3 to Col 5.Efficiencies only happen within cuts at a = 0+. The highest possibility is cuts ata = 1. Cuts at a = 0+ enclose point efficiencies, which shows the resulted fuzzyefficiencies are consistent with point efficiencies.

Fuzzy analysis reveals efficiencies being vague due to qualitative observation.There are three efficient units (No. 5, 35, 42) with efficiency 1, based on pointefficiency, while fuzzy efficiencies point out that only No. 42 is truly efficient. SeeFig. 5. Besides, No. 2 is another efficient unit at possibility level of a = 0.5. Fuzzyefficiencies uncover which one is truly efficient, and find a unit is efficient in aloosened possibility level. Interested readers can refer to Kao and Lin [22], whichprovides detailed discussions about the benefit of fuzzy efficiency compared withpoint efficiency.

Furthermore, partition 52 universities into groups based on their efficiencies toconvert them into qualitative data. First, apply formula (16c) to determine theirpairwise Hausdorff distances dg on a –cuts at a = 0, 0.1,…, 1. Then apply Model(17) to do cluster analysis for different number of groups. The results, in each run,are total distance, a set of medoids, and their members. Medoids have optimalsolution b�i ¼ 1 and their members are z�ij ¼ 1.

Figure 6 is the scree plot, showing the relationship between total distances andthe number of groups. The elbow is between four and five groups. We choosesubjectively five as partition groups. Therefore, separate universities into fivegroups, and rank them into 1–5 level accordingly. Level 1 is the best, and level 5 isthe worst. Five medoids with efficiencies from high to low are No. 5, 47, 44, 4, and26. Each group has 6, 13, 15, 15, and 3 members, respectively. Members at eachlevel are (2, 5*, 7, 31, 35, 42), (1, 10, 19, 21, 23, 25, 30, 34, 37, 39, 41, 47*, 52),(3, 6, 9, 12, 13, 18, 20, 27, 36, 38, 40, 43, 44*, 49, 51), (4*, 8, 11, 14, 15, 16, 17, 22,24, 28, 29, 32, 45, 46, 50), and (26*, 33, 48). A superscript ‘‘*’’denotes a medoid.Col 6 in Table 2 shows ranks of universities.

The above analysis is on Hausdorff distance dg. Now it is conducted here againby other Hausdorff distance d� and d? based on formula (16a) and (16b),respectively, with similar results by different distance measures. Only somemedoid shifts within groups and one member drifts across groups. Although many

0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

µ

Efficiency

No 42No 35No 5No 2Fig. 5 Fuzzy efficiencies of

university No. 2, 5, 35 and 42

3 A General Framework of Dealing with Qualitative Data in DEA 83

analysts believe that analysis based on average would reach results more stably,Zwick et al. [42] suggested using d* to simplify computation and this case confirmstheir suggestions. Therefore, readers can use d* in clustering analysis withoutlosing accuracy and releasing computation burden significantly.

6 Conclusions

This chapter presents a general framework of dealing with DEA model containingqualitative factor through fuzzy numbers. To solve a fuzzy efficiency ~E in a fuzzyDEA model is to determine a set of its a-cuts, Ea. Two sets of two-level mathe-matical programming can implement extension principle to determine the lowerbound and the upper bound of Ea in crisp DEA models, where qualitative data arerepresented by binary variable and variables. Fuzzy efficiencies are more infor-mative than ordinary point efficiencies.

To provide convincing fuzzy numbers representing qualitative data, DEAmodels can act as experts to provide the opinions about the weight of qualitativedata. Weights of qualitative data in each run are standardized beforehand by theiraverage weight. Then weights of each qualitative level from all runs determine theshape of corresponding fuzzy number. For a trapezoidal-shaped membership, fourvertexes are minimum, 25 and 75th percentile, and maximum. In addition, readingqualitative efficiencies is easier than reading fuzzy efficiencies. K-medoids clus-tering method can convert fuzzy efficiencies into qualitative data properly. Theclustering method is robust to outlier observations. It’s medoids are fuzzy, so usingthem to represent the center of a set of fuzzy observations is very convincing.The recommended distance measure in clustering analysis is Hausdorff distance.Different types of Hausdorff distance, which based on 0+-cuts, 1-cuts, or multiplelevel cuts, yield the similar results in the demonstration case. These results are

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

Groups

Dis

tanc

e

Fig. 6 Total distance andpartitioned groups

84 P. H. Lin

similar to those of some former researchers on behavior experiments. Hence,0+-cuts and 1-cuts are recommended to determine Hausdorff distances in clusteringanalysis for converting fuzzy efficiencies to qualitative data.

This chapter has discussed the general framework of solving DEA modelcontaining qualitative factors. There are still interesting issues remaining in thistopic. Providing directions of improving and efficient targets are often interested inDEA. If DEA contains qualitative data, how to provide qualitative improvingsuggestions from quantitative analysis is worthwhile to study. In addition, clus-tering analysis implies that ranks are in a complete sequence. This is not alwaystrue, and it raises issues regarding how to convert fuzzy efficiencies into ranksproperly in different situations. Finally, the guidelines of collecting subjectiveopinions to yield reasonable fuzzy numbers representing qualitative data areinteresting for most people who want to apple the framework.

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3 A General Framework of Dealing with Qualitative Data in DEA 87

Chapter 4Fuzzy Data Envelopment Analysisin Composite Indicator Construction

Yongjun Shen, Elke Hermans, Tom Brijs and Geert Wets

Abstract Data envelopment analysis (DEA) as a performance evaluation meth-odology has lately received considerable attention in the construction of compositeindicators (CIs) due to its prominent advantages over other traditional methods. Inthis chapter, we present the extension of the basic DEA-based CI model byincorporating fuzzy ranking approach for modeling qualitative data. By interpretingthe qualitative indicator data as fuzzy numerical values, a fuzzy DEA-based CImodel is developed, and it is applied to construct a composite alcohol performanceindicator for road safety evaluation of a set of European countries. Comparisons ofthe results with the ones from the imprecise DEA-based CI model show theeffectiveness of the proposed model in capturing the uncertainties associated withhuman thinking, and further imply the reliability of using this approach formodeling both quantitative and qualitative data in the context of CI construction.

Keywords Alcohol performance index � Composite indicators � Data envelop-ment analysis � Fuzzy ranking approach � Qualitative data � Road safety

Y. Shen (&) � E. Hermans � T. Brijs � G. WetsTransportation Research Institute (IMOB), Hasselt University, Wetenschapspark 5 bus 63590 Diepenbeek, Belgiume-mail: yongjun.shen@uhasselt.be

E. Hermanse-mail: elke.hermans@uhasselt.be

T. Brijse-mail: tom.brijs@uhasselt.be

G. Wetse-mail: geert.wets@uhasselt.be

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_4, � Springer-Verlag Berlin Heidelberg 2014

89

1 Introduction

As a performance evaluation methodology, data envelopment analysis (DEA) istraditionally used to measure the so-called relative efficiency of a homogeneous setof decision making units (DMUs) by allowing direct peer comparisons on the basisof multiple inputs and multiple outputs [1]. However, as noted by Adolphson et al.[2], it is possible to adopt a broader perspective, in which DEA is also appropriatefor comparing any set of homogeneous units on multiple dimensions. Based on thisperspective, DEA has been introduced to the field of composite indicators (CIs),which is to aggregate a set of individual indicators that measure multi-dimensionalconcepts but usually have no common units of measurement [3]. The mostattractive feature of DEA, relative to the other methods in developing a CI, such asregression analysis (RA), principal components analysis (PCA), factor analysis(FA), analytic hierarchy process (AHP), and the technique for order preference bysimilarity to ideal solution (TOPSIS) (see also Saisana and Tarantola [4], OECD[3], and Bao et al. [5]) is that, each DMU obtains its own best possible indicatorweights, i.e., the weights resulting in the highest index score for a DMU. Thisimplies that dimensions on which the DMU performs relatively well get a higherweight. It is thereby also called ‘benefit of the doubt’ (BOD) approach [6]. In thisway, policymakers could not complain about unfair weighting, because each DMUis put in the most favorable light, and any other weighting scheme would generate alower composite score. In other words, if a country turns out to be underperformingbased on the most favorable set of weights, its poor performance cannot be tracedback to an inappropriate evaluation process [7]. Due to the aforementioned char-acteristic, the DEA-based CI construction has been widely explored in severalrecent studies such as environmental performance index [8], human developmentindex [9], macro-economic performance index [10], sustainable energy index [11],technology achievement index [12], and road safety performance index [13, 14].

However, as a ‘data-oriented’ technique, the applicability of DEA in the con-struction of CIs relies mostly on the quality of information about the indicators. Inother words, obtainment of measurable and quantitative indicators is commonlythe prerequisite of the evaluation. Under many conditions, however, quantitativedata are inadequate or inappropriate to model real world situations due to thecomplexity and uncertainty of the reality. Therefore, it is essential to take intoaccount the presence of qualitative indicators when making a decision on theperformance of a DMU. Very often it is the case that an indicator can, at most, bespecified with either ordinal measures, from best to worst, or with the help ofexperts’ subjective judgments, such as ‘high’, ‘medium’ and ‘low’. Under thesecircumstances, the basic DEA models are not capable of deriving a satisfactorysolution. Generally, two strategies have appeared in the literature to the treatmentof qualitative data within the DEA framework. One is to reflect the rank position ofeach DMU with respect to each ordinal indicator by setting corresponding con-straints, which results in the so-called imprecise DEA (IDEA) (see e.g., Cooperet al. [15]; Cook and Zhu [16]). The other is to deal with the natural uncertainty

90 Y. Shen et al.

inherent to some production processes by means of fuzzy mathematical pro-gramming, such as the tolerance approach developed by Sengupta [17] andKahraman and Talgo [18], the a-level based approach introduced by Meada et al.[19], the defuzzification and the possibility approach proposed by Lertworasirikul[20] and Lertworasirikul et al. [21], and the fuzzy ranking approach developed byGuo and Tanaka [22]. All of them are collectively named as fuzzy DEA (FDEA).

In this Chapter, we investigate FDEA, and more specifically, the fuzzy rankingapproach, to model qualitative data in the construction of CIs. Based on a briefreview of the basic DEA model and the DEA-based CI model in Sect. 2, we elaboratethe development of a FDEA-based CI model in Sect. 3. In Sect. 4, the proposedmodel is illustrated by constructing a composite alcohol performance index for roadsafety evaluation of a set of European countries, and the results are compared withthe ones from the IDEA model. The chapter ends with conclusions in Sect. 5.

2 DEA-based CI Model

Data envelopment analysis initially developed by Charnes et al. [1] is a non-parametric optimization technique which employs linear programming tools toobtain the empirical estimates of multiple inputs and multiple outputs related to aset of DMUs. During the last decades, a number of different formulations have beenproposed in the DEA context, the best-known of which is probably the Charnes–Cooper–Rhodes (CCR) model, and its multiplier form is presented as follows.

E0 ¼ maxXs

r¼1

uryr0

s:t:Xm

i¼1

vixi0 ¼ 1;

Xs

r¼1

uryrj �Xm

i¼1

vixij� 0; j ¼ 1; � � � ; n

ur; vi� e; r ¼ 1; � � � ; s; i ¼ 1; � � � ;m

ð1Þ

The above linear program is computed separately for each DMU, and thesubscript, 0, refers to the DMU whose relative efficiency is to be evaluated. yrj andxij are the rth output and ith input respectively of the jth DMU. ur is the weightgiven to the rth output, vi is the weight given to the ith input, and e is a small non-Archimedean number [23] for preventing the model to assign a weight of zero tounfavorable factors.

To use DEA for CI construction, i.e., aggregating a set of individual indicatorsinto one overall index, however, only inputs or outputs of the DMUs will be takeninto account in the model. Mathematically, the DEA-based CI model (DEA-CI) canbe realized by converting the DEA model in (1) into the following constrainedoptimization problem, which is also known as the CCR model with constant inputs.

4 Fuzzy Data Envelopment Analysis in Composite Indicator Construction 91

CI0 ¼ maxXs

r¼1

uryr0

s:t:Xs

r¼1

uryrj� 1; j ¼ 1; � � � ; n

ur � e; r ¼ 1; � � � ; s

ð2Þ

The n DMUs are now to be evaluated by combining s different outputs (orindicators) with higher values indicating better performance, while the inputs ofeach DMU in model (1) are all assigned with a value of unity. This linear programis run n times to identify the optimal index score for all DMUs by selecting theirbest possible indicator weights separately. In other words, the weights in theobjective function are chosen automatically with the purpose of maximizing thevalue of DMU0’s index score and also respect the less than unity constraint for allthe DMUs. Meanwhile, all the weights are required to be positive. In general, aDMU is considered to be best-performing if it obtains an index score of one in (2),whereas a score less than one implies that it is underperforming.

3 Fuzzy DEA-CI Model

In model (2), the performance evaluation is generally assumed to be based upon aset of quantitative data. However, in situations where some indicators might betterbe represented in either ordinal measures or the help of expert subjective judg-ments, the standard DEA-CI model cannot be used directly, because ordinal (orqualitative) data cannot be simply treated as numerical ones for which a score of 2is twice as large as a score of 1. The most that can be judged is that the former oneis preferred to or more important than the latter in a maximization context. Inrecent years, fuzzy set theory [24] has been proposed as a valuable way to quantifyimprecision and vagueness in DEA framework, and a number of different fuzzyDEA models has been developed (see e.g., Hatami-Marbini et al. [25]). In CIconstruction, by interpreting the qualitative indicator data as fuzzy numericalvalues which can be represented by means of fuzzy numbers or fuzzy intervals, thebasic DEA-CI model (2) can also be naturally extended to the following fuzzy one:

CI0 ¼ maxXs

r¼1

ur~yr0

s:t:Xs

r¼1

ur~yrj.1; j ¼ 1; � � � ; n

ur � e; r ¼ 1; � � � ; s

ð3Þ

where ~yrj denotes the rth fuzzy indicator value of the jth DMU.

92 Y. Shen et al.

The resulted fuzzy DEA-based CI model (FDEA-CI) takes the form of a fuzzylinear programming problem with fuzzy coefficients in the objective function andalso the constraints. Therefore, to compute the final index score for each DMU,some fuzzy operations including ‘maximizing a fuzzy variable’ and ‘fuzzyinequality’ are required. In what follows, we simply recall how to perform thebasic operations of arithmetics and the comparison of fuzzy intervals for rankingpurposes. To be more precise, we deal with LR-fuzzy numbers whose definition isas follows.

Definition 1 [26] A fuzzy number ~M is an LR-fuzzy number,~M ¼ ðmL;mR; aL; aRÞL;R, if its membership function has the following form:

l ~MðrÞ ¼L mL�r

aL

� �; r�mL

1; mL� r�mR

R r�mR

aR

� �; r�mR

8>><

>>:ð4Þ

where the subset ½mL;mR� consists of the real numbers with the highest chance ofrealization, aL is the left spread, aR is the right spread, and L and R are referencefunctions defining the left and the right shapes of the fuzzy number, respectively,which should satisfy the following conditions:

L;R : 0;1! 0; 1;

LðxÞ ¼ Lð�xÞ;RðxÞ ¼ Rð�xÞ;Lð0Þ ¼ 1;Rð0Þ ¼ 1; and

LðxÞ and RðxÞ are strictly decreasing and upper semi-continuous on supp( ~M) ¼ r : l ~MðrÞ[ 0

� �.

In addition, an LR fuzzy number becomes an LL fuzzy number whenLðxÞ ¼ RðxÞ, an LL fuzzy number with LðxÞ ¼ max 0; 1� xj jð Þ is known as atriangular fuzzy number, and a symmetrical LL fuzzy number is for the case ofaL ¼ aR.

Let us now recall the definition of the maximum of two fuzzy numbers.

Definition 2 [27] Let ~M and ~N be two fuzzy numbers and h a real number,h 2 0; 1½ �. Then ~MJh ~N if and only if, 8k 2 h; 1½ �, the following two statementshold:

inf s : l ~MðsÞ� k� �

� inf t : l~NðtÞ� k� �

sup s : l ~MðsÞ� k� �

� sup t : l~NðtÞ� k� � ð5Þ

where inf stands for infimum (lower bound or minimum), and sup stands forsupremum (upper bound or maximum).

4 Fuzzy Data Envelopment Analysis in Composite Indicator Construction 93

Hence, for LR-fuzzy numbers with bounded support, and using this rankingmethod, at a given possibility level h, expression (5) becomes

mL � L�ðkÞaL� nL � L0�ðkÞbL 8k 2 h; 1½ �mR þ R�ðkÞaR� nR þ R0�ðkÞbR 8k 2 h; 1½ �

ð6Þ

Therefore, using LR fuzzy numbers in the FDEA-CI model (3), i.e.,

~yrj ¼ ðylrj; yurj; arj; brjÞ, the constraintPs

r¼1ur~yrj.1 can then be considered as

inequalities between an LR fuzzy number and a real number, and the use of anordering relation in (6) allows us to convert this fuzzy constraint into a crisp

inequality as:Ps

r¼1ur yurj þ brjR�ðhÞ� �

� 1.1

Concerning ‘maximizing a fuzzy variable’, i.e., maxPs

r¼1ur~yr0, still using the

ordering relation in (6), this objective function can then be decomposed into two

crisp relations as: maxPs

r¼1ur ylr0 � ar0L�r0ðhÞ� �

and maxPs

r¼1ur yur0 þ br0R�r0ðhÞ� �

,

h 2 0; 1½ �, which should be maximized simultaneously. To this end, a weighted

function k1Ps

r¼1ur ylr0 � ar0L�r0ðhÞ� �

þ k2Ps

r¼1ur yur0 þ br0R�r0ðhÞ� �

with k1� 0,

k2� 0, and k1 þ k2 ¼ 1 is used to obtain the compromise solution. Three situa-tions are usually considered, which are optimistic if k2 ¼ 1, pessimistic if k1 ¼ 1,and indifferent if k1 ¼ k2.

Thus, the FDEA-CI model (3) can now be transformed in the following crisplinear programming problem:

CI0 ¼ max k1

Xs

r¼1

ur ylr0 � ar0L�r0ðhÞ� �

þ k2

Xs

r¼1

ur yur0 þ br0R�r0ðhÞ� �

s:t:Xs

r¼1

ur yurj þ brjR�rjðhÞ

� �� 1; j ¼ 1; � � � ; n

ur � e; r ¼ 1; � � � ; s

ð7Þ

Definition 3 DMU0 is called fuzzy best performing if and only if it obtains afuzzy index score of one at least at one possibility level h. Otherwise, it is fuzzyunderperforming.

Definition 4 DMU0 is called fuzzy non-dominated best performing if and only ifit obtains a fuzzy index score of one at all possibility levels h.

In particular, if indicators ~yrj are assumed to be symmetrical triangular fuzzynumbers, which are often used to represent the uncertainty of information for

1 Ps

r¼1ur ylrj � arjL�ðhÞ� �

� 1 is always satisfied whenPs

r¼1ur yurj þ brjR�ðhÞ� �

� 1.

94 Y. Shen et al.

simplification, they can then be denoted by the pairs consisting of the corre-sponding centers and spreads, ~yrj ¼ ðyrj; arjÞ, r ¼ 1; � � � ; s, j ¼ 1; � � � ; n, and themodel (7) can be substantially simplified as follows:

CI0 ¼ max k1

Xs

r¼1

ur yr0 � ð1� hÞar0ð Þ þ k2

Xs

r¼1

ur yr0 þ ð1� hÞar0ð Þ

s:t:Xs

r¼1

ur yrj þ ð1� hÞarj

� �� 1; j ¼ 1; � � � ; n

ur � e; r ¼ 1; � � � ; s

ð8Þ

Note that for triangular fuzzy numbers, L�rjðhÞ ¼ R�rjðhÞ ¼ 1� h;

0� h� 1; r ¼ 1; � � � s. The fuzzy index score of DMU0 can then be defined as

{Ps

r¼1u�r yr0 � ð1� hÞar0ð Þ,

Ps

r¼1u�r yr0,

Ps

r¼1u�r yr0 þ ð1� hÞar0ð Þ}, which represents the

pessimistic, indifferent, and optimistic situation, respectively.

4 Application and Discussion

To illustrate the use of the proposed FDEA-CI model, we apply it to construct analcohol performance index for a set of European countries based on both quan-titative and qualitative indicators. In road safety context, driving under the influ-ence of alcohol is believed to increase the risk and severity of road crashes morethan most other traffic law violations [28]. Therefore, it is valuable to compare thesituation of drinking and driving between countries for the sake of better under-standing of this risk factor in each country. In doing so, several relevant indicatorscan be considered. First, the percentage of road fatalities attributed to alcohol,which represents the consequence of drinking and driving from the view of thefinal outcome level, is commonly used as a representative alcohol indicator forcross-country comparison. Moreover, at the intermediate outcome level, analcohol performance indicator is also developed, which is the percentage ofdrivers above the legal blood alcohol concentration (BAC) limit in roadsidechecks. In addition to the above two quantitative indicators, one more indicatorrelated to policy output, i.e., the effectiveness of overall enforcement againstdrinking and driving, is also suggested to supplement the alcohol performance ofa country. Such a policy performance indicator, derived from the Global StatusReport on Road Safety prepared by the World Health Organization [29], in whichthe respondents were asked to reach a consensus on their assessment of theenforcement in the country, is qualitative in nature, and can only take the form ofordered classes rated on a 0–10 scale (with 0 represents the worst drink drivingenforcement while 10 the best) rather than numerical values for the purpose ofdescription, comparison and evaluation of this risk factor for various countries.

4 Fuzzy Data Envelopment Analysis in Composite Indicator Construction 95

Data on these three indicators for the 28 European countries2 are presented inTable 1, in which the first two quantitative indicators are normalized using thedistance to a reference approach [3] so as to ensure that they are expressed in thesame direction with respect to their expected road safety impact, i.e., a highindicator value should always correspond to a low crash/injury risk. Taking thepercentage of alcohol-related fatalities as an example, the Netherlands performsthe best (1.000) while Slovenia worst (0.078), and all other countries’ values liewithin this interval.

Table 1 Normalized numerical data and ordinal data on three alcohol indicators for 28 Europeancountries

Alcohol indicators

% of alcohol-relatedfatalities

% of driversabove legalalcohol limitin roadside checks

Effectivenessof overallenforcementon drinkingand driving

AT 0.463 0.116 9BE 0.654 0.068 3BG 0.855 0.123 7CY 0.182 0.137 4CZ 0.675 0.145 9DK 0.143 0.301 8EE 0.080 0.860 8FI 0.136 0.593 8FR 0.123 0.263 4DE 0.306 0.093 4EL 0.432 0.273 7HU 0.283 0.279 5IE 0.119 0.237 5IT 0.992 0.098 7LV 0.175 0.218 7LT 0.321 0.555 6LU 0.248 0.102 5NL 1.000 0.081 9NO 0.159 0.142 4PL 0.438 0.091 7PT 0.610 0.137 8RO 0.423 0.070 8SK 0.607 0.067 9SI 0.078 0.122 6ES 0.402 0.398 7SE 0.357 1.000 6CH 0.230 0.141 6UK 0.228 0.051 5

2 Missing data are imputed by using Multiple Imputation in SPSS 20.0 [30].

96 Y. Shen et al.

To combine these three alcohol indicators into one index score, symmetricaltriangular fuzzy numbers are first used for the ordinal data in this study, which aredefined as in Table 2.

In addition, to guarantee that all the three indicators will be used to some extentby the models, the share of each of these three indicators in the final index score isrestricted to lie within the interval [0.1, 0.5], yet is rather broad to allow a highlevel of flexibility, and the e value is chosen as 0.0001.

The alcohol performance index score of the 28 European countries can now becomputed by applying the FDEA-CI model (8). The results are shown in Table 3,together with the ones from the IDEA-CI model. For more information on thismodel, we refer to Shen et al. [31].

By using the FDEA-CI model, fuzzy index scores are obtained based on dif-ferent possibility levels of h. In practice, the given possibility degree by decisionmakers reflects their attitude on uncertainty. When h = 1, the ordinal data areactually treated as numerical ones and the same index scores are obtained for eachcountry, no matter whether the decision makers are in a pessimistic, indifferent, oroptimistic consideration. When the given value of h becomes lower, it means thedecision makers are more cautious. As a consequence, a wider range of indexscores will be derived. In such a way, the uncertainties associated with humanthinking are effectively interpreted. Taking Belgium as an example, which wasassigned the lowest value of 3 for this ordinal indicator among all the 28 Europeancountries, it obtains an index score of 0.392 when h = 1. That is, decision makershave no doubt about this value in representing the true performance of Belgiumwith respect to this indicator, which is half of the value of 6 and one third of 9.When h decreases to 0.5, this implies that decision makers are no longer fully sureabout the relation between 3 and 6, and the other numbers. In other words, thevalue of 6 could be more (or less) than twice as large as the value of 3, and themost that can be judged is that the former one is preferred to or more importantthan the latter. As a result, an interval index score is obtained for Belgium, whichis between 0.359 (pessimistic) and 0.401 (optimistic), with a medium value of0.382 (indifferent). The widest interval is derived when h = 0, which is {0.318,0.373, 0.409}. Among all the 28 European countries, Sweden is the only non-dominated best-performing country since it obtains the fuzzy index score of one at

Table 2 Representation of symmetrical triangular fuzzy numbers for the ordinal indicator values

Ordinaldata (~yrj)

Symmetrical triangularfuzzy numbers ðyrj; arjÞ

Ordinaldata (~yrj)

Symmetrical triangularfuzzy numbers ðyrj; arjÞ

0 0; 110

� �1 1

10 ;110

� �

2 210 ;

110

� �3 3

10 ;110

� �

4 410 ;

110

� �5 5

10 ;110

� �

6 610 ;

110

� �7 7

10 ;110

� �

8 810 ;

110

� �9 9

10 ;110

� �

10 1; 110

� �

4 Fuzzy Data Envelopment Analysis in Composite Indicator Construction 97

all possibility levels h. Whereas for other countries, their ranking could be slightlychanged when different possibility level and consideration are taken into account.

Moreover, by comparing the alcohol performance index scores of the 28European countries with the ones from the IDEA-CI model, in which a crisp indexscore is achieved, we find that the FDEA-CI score is lower than the one from theIDEA-CI model, even in the optimistic situation with the lowest possibility levelof h. This can be partly explained by the fact that a relatively small and constantvalue of e is used in the IDEA-CI model to reflect the minimum allowable gapbetween the two ranking positions in terms of the indicator value, which results inan extreme index score for each country. In other words, based on the same evalue, the index score from the FDEA-CI model would not exceed the one fromthe IDEA-CI model. Nevertheless, a high correlation coefficient (0.989) is deducedbetween the IDEA-CI score and the FDEA-CI score (taking h = 0.5 and the

Table 3 Composite alcohol performance index scores of 28 European countries based on theFDEA-CI model and the IDEA-CI model

FDEA-CI IDEA-CI

h = 0 h = 0.5 h = 1

SE {0.872, 0.947, 1.000} {0.940, 0.973, 1.000} {1.000, 1.000, 1.000} SE 1.000CZ {0.768, 0.792, 0.812} {0.795, 0.806, 0.816} {0.820, 0.820, 0.820} CZ 0.880ES {0.684, 0.733, 0.775} {0.729, 0.752, 0.774} {0.773, 0.773, 0.773} ES 0.847LT {0.670, 0.727, 0.778} {0.721, 0.750, 0.776} {0.774, 0.774, 0.774} FI 0.833PT {0.694, 0.727, 0.749} {0.726, 0.740, 0.752} {0.755, 0.755, 0.755} PT 0.826FI {0.686, 0.720, 0.750} {0.719, 0.735, 0.751} {0.752, 0.752, 0.752} LT 0.803BG {0.672, 0.703, 0.729} {0.703, 0.717, 0.730} {0.732, 0.732, 0.732} EL 0.780EL {0.634, 0.679, 0.717} {0.674, 0.696, 0.715} {0.713, 0.713, 0.713} BG 0.776AT {0.624, 0.642, 0.658} {0.645, 0.654, 0.662} {0.666, 0.666, 0.666} AT 0.711IT {0.598, 0.623, 0.643} {0.623, 0.634, 0.644} {0.646, 0.646, 0.646} IT 0.679NL {0.566, 0.579, 0.590} {0.581, 0.587, 0.592} {0.594, 0.594, 0.594} NL 0.678EE {0.535, 0.556, 0.574} {0.554, 0.564, 0.573} {0.572, 0.572, 0.572} DK 0.626HU {0.468, 0.518, 0.558} {0.509, 0.532, 0.553} {0.547, 0.547, 0.547} HU 0.623PL {0.505, 0.523, 0.537} {0.523, 0.531, 0.538} {0.539, 0.539, 0.539} EE 0.589DK {0.496, 0.513, 0.526} {0.513, 0.521, 0.528} {0.530, 0.530, 0.530} PL 0.567SK {0.467, 0.475, 0.482} {0.476, 0.480, 0.484} {0.486, 0.486, 0.486} SK 0.563LV {0.440, 0.459, 0.474} {0.458, 0.466, 0.474} {0.474, 0.474, 0.474} RO 0.562RO {0.446, 0.456, 0.466} {0.457, 0.462, 0.467} {0.469, 0.469, 0.469} LV 0.500CH {0.402, 0.427, 0.448} {0.424, 0.435, 0.446} {0.443, 0.443, 0.443} DE 0.488LU {0.357, 0.389, 0.414} {0.382, 0.397, 0.410} {0.405, 0.405, 0.405} CH 0.474DE {0.339, 0.384, 0.423} {0.371, 0.394, 0.414} {0.404, 0.404, 0.404} BE 0.466IE {0.360, 0.386, 0.405} {0.382, 0.393, 0.403} {0.401, 0.401, 0.401} LU 0.464FR {0.340, 0.380, 0.408} {0.371, 0.389, 0.404} {0.399, 0.399, 0.399} FR 0.450BE {0.318, 0.373, 0.409} {0.359, 0.382, 0.401} {0.392, 0.392, 0.392} IE 0.425CY {0.300, 0.336, 0.362} {0.327, 0.343, 0.357} {0.351, 0.351, 0.351} CY 0.415NO {0.291, 0.324, 0.347} {0.316, 0.330, 0.343} {0.337, 0.337, 0.337} NO 0.393UK {0.290, 0.304, 0.315} {0.302, 0.309, 0.314} {0.314, 0.314, 0.314} UK 0.324SI {0.250, 0.258, 0.264} {0.257, 0.261, 0.264} {0.264, 0.264, 0.264} SI 0.268

98 Y. Shen et al.

indifferent situation as an example). This not only demonstrates the robustness oftheir ranking results, but also implies the reliability of using fuzzy rankingapproach for modeling qualitative data.

5 Conclusions

In this chapter, we investigated the usage of fuzzy ranking approach in the DEAframework for modeling both quantitative and qualitative data in the context ofcomposite indicator construction. By interpreting the qualitative indicator data asfuzzy numerical values, a fuzzy DEA-based CI model was developed, and it wasfurther transformed into a crisp linear programming problem. The model wasdemonstrated by combining three alcohol indicators (two quantitative and onequalitative) into an alcohol performance index score for the 28 European countries.The analysis of the results showed that fuzzy index scores obtained based ondifferent possibility levels were powerful in capturing the uncertainties associatedwith human thinking, which was therefore superior over the imprecise DEA-basedCI model that only resulted in a crisp index score. However, the high similarity ofthe ranking result based on these two models verified its robustness and alsoimplied the reliability of using the fuzzy ranking approach for modeling qualitativedata. In the future, exploration on the dual envelopment formulation of this modeland on the usage of other fuzzy techniques such as the a-level based approach andthe possibility approach, are worthwhile.

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30. IBM Corp. IBM SPSS missing values 20. http://www.ibm.com/spss (2011)31. Shen, Y., Ruan, D., Hermans, E., Brijs, T., Wets, G., Vanhoof, K.: Modeling qualitative data

in data envelopment analysis for composite indicators. Int. J. Syst. Assur Eng Manag 2(1),21–30 (2011)

100 Y. Shen et al.

Chapter 5Cross-Efficiency in Fuzzy DataEnvelopment Analysis (FDEA):Some Proposals

Inmaculada Sirvent and Teresa León

Abstract Different techniques have been proposed in the literature to rankdecision making units (DMUs) in the context of Fuzzy Data Envelopment Anal-ysis. In our opinion, those that result from using a ranking method to order thefuzzy efficiencies obtained are susceptible to a serious criticism: they are not basedon objective criteria. Cross-efficiency evaluation was introduced as an extension ofDEA aimed at ranking the DMUs. This methodology has found a significantnumber of applications and has been extensively investigated. In this chapter, wediscuss some difficulties that arise with the definition of fuzzy cross-efficienciesand we propose a fuzzy cross-efficiency evaluation based on the FDEA model byGuo and Tanaka. Such model relies on the dual multiplier formulation of the CCRmodel and the fuzzy efficiency of a given DMU is defined in a ratio form in termsof the input and output weights obtained. This allows us to define the cross-efficiencies in an analogous manner to that of the fuzzy efficiency. The resultingcross-efficiencies are consistent in the sense that the cross-efficiency of a givenDMU, calculated with its own input and output weights, is equal to the relativeefficiency of this unit. We illustrate our methodology with an example.

Keywords Data envelopment analysis � Fuzzy mathematical programming �Cross-efficiency � Ordering

I. SirventCentro de Investigación Operativa, Universidad Miguel Hernández, Alicante, Spain

T. León (&)Departament de Estadística e Investigación Operativa, Universitat de Valencia,Valencia, Spaine-mail: Teresa.Leon@uv.es

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_5, � Springer-Verlag Berlin Heidelberg 2014

101

1 Introduction

Data Envelopment Analysis (DEA), as introduced in Charnes et al. [1], is amethodology for the assessment of relative efficiency of a set of decision makingunits (DMUs) that use several inputs to produce several outputs. For each DMU, itprovides efficiency scores in the form of a ratio of a weighted sum of the outputs to aweighted sum of the inputs. DEA results classify DMUs into two groups, those thatare efficient and define the Pareto frontier and those that are inefficient. However, inmany practical applications decision makers are interested in a ranking beyond thisclassification. Many authors claim that we should not derive an ordering of the unitsbased on the efficiency scores since these scores are not comparable as a result ofthe fact that the different DMUs attach different weights to the inputs and outputswhen being evaluated. Thus, different techniques have been proposed in the liter-ature to rank DMUs in the context of DEA (see Adler et al. [2] for a review of thesemethods). One such method that has found a significant number of applications andhas been extensively investigated theoretically is cross-efficiency evaluation.

The idea of cross-efficiency evaluation (Sexton et al. [3], and Doyle and Green[4]) is to assess each unit with the DEA weights of all the DMUs instead of withonly its own weights. To be specific, the cross-efficiency score of a given unit isusually calculated as the average of the efficiency scores (the cross-efficiencies) ofsuch unit obtained with the profiles of weights of all the DMUs. Unlike the DEAself-evaluation, this provides a peer evaluation of the different units, which makesit possible to derive an ordering.

In fuzzy data envelopment analysis (FDEA) literature we can find differentapproaches to define DMU rankings. In our opinion, those that result from usingsome ranking fuzzy number method to rank the obtained fuzzy efficiencies aresusceptible to the same criticism described before regarding ranking units based onthe ordering of their crisp DEA efficiency scores. Mainly that they are not based onobjective criteria. Thus, it should be desirable to have available a methodologysuch as cross-efficiency evaluation in the FDEA context in order to providerankings of DMUs that can be accepted by a broader number of practitioners.

We do not find possible to give a unique definition of cross-efficiency evalu-ation in FDEA by the same reasons that many different definitions of efficiencyhave been given (see for instance Kao and Liu [5], Guo and Tanaka [6], León et al.[7] and Lertworasirikul et al. [8]). On the other hand, as we will comment on inthis chapter, it is not possible either to propose a general route to straightforwardlyuse every existing FDEA model to perform the corresponding cross-efficiencyanalysis. For example, FDEA models based on the primal envelopment formula-tion of the CCR model do not directly provide the input and output weights so it isnot clear how the cross-efficiencies associated to the relative efficiency assess-ments provided by these models can be obtained. Thus, after exploring differentpossibilities, we have decided to make some proposals of fuzzy cross-efficiencyevaluation based on the model of Guo and Tanaka [6]. We have selected thisFDEA model because it is based on the dual multiplier formulation of the CCR

102 I. Sirvent and T. León

model and the fuzzy efficiency of a given DMU is defined in a ratio form in termsof the obtained input and output weights. This allows us to define the cross-efficiencies in an analogous manner to that of the fuzzy efficiency so that theresulting cross-efficiencies are consistent in the sense that the cross-efficiency of agiven DMU calculated with its own input and output weights is equal to therelative efficiency of this unit.

The chapter is organized as follows: In Sect. 2 we briefly describe the standard(crisp) cross-efficiency evaluation methodology. In Sect. 3 we discuss some dif-ficulties that arise when defining the cross-efficiencies in the fuzzy context andpropose a fuzzy cross-efficiency evaluation based in the fuzzy DEA model of Guoand Tanaka. We illustrate our methodology with an example in Sect. 4. Lastsection concludes.

2 Cross-Efficiency Evaluation in Data EnvelopmentAnalysis

Let us assume that we have n DMUs which use m inputs to produce s outputs.These can be described by means of the vectors ðXj; YjÞ, j = 1,…,n. The standardcross-efficiency evaluation is based on the CCR model (Charnes et al. [1]), whichprovides a measure of the relative efficiency of a given DMU0 by solving thefollowing problem

Max h0 ¼l0Y0

v0X0

s:t: :l0Yj

v0Xj� 1 j ¼ 1; . . .; n

v� 0m; l� 0s

ð1Þ

This is the ratio form of the CCR model, which can be converted into thefollowing linear problem, called the dual multiplier formulation, by using theresults on linear fractional problems in Charnes and Cooper [9]

Max h0 ¼ l0Y0

s:t: v0X0 ¼ 1l0Yj � v0Xj� 0 j ¼ 1; . . .; n

v� 0m; l� 0s

ð2Þ

In the standard cross-efficiency evaluation the optimal solution of (2) for eachDMUd, ðvd; ldÞ provides the DEA weights that are used to calculate the cross-efficiency of a given DMUj, j = 1,…,n, as follows

Edj ¼l0dYj

v0dXjð3Þ

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 103

and, therefore, the cross-efficiency score of DMUj is defined as the average of thethese cross-efficiencies

�Ej ¼1n

Xn

d¼1

Edj; j ¼ 1; . . .; n ð4Þ

which measures the average efficiency according to all DMUs.The DMUs can then be ranked according to the values of these cross-efficiency

scores.However, as it has been frequently noted in the DEA literature, the possible

existence of alternate optima for the weights when solving the CCR model is adifficulty which may reduce the usefulness of cross-efficiency evaluation since thismay lead to different cross-efficiency scores and rankings of DMUs depending onthe particular optimal solution that is used to compute the cross-efficiencies. Theuse of alternative secondary goals to the choice of weights among the alternativeoptimal solutions has been suggested as a potential remedy to the possible influ-ence of this difficulty. The idea in most of the existing proposals that use thisapproach is to implicitly provide for each DMU a set of optimal weights obtainedafter imposing some condition on the resulting cross-efficiencies for all the DMUs.This is the case of the two best known procedures to select DEA weights in cross-efficiency evaluation: the benevolent and aggressive formulations (Sexton et al.[3], and Doyle and Green [4]). The benevolent formulation selects weights thatmaintain the self-efficiency score of the unit under assessment while enhancing thecross-efficiencies of the other DMUs as much as possible, whereas the aggressiveformulation also maintains the self-efficiency score while diminishing the rest ofcross-efficiencies. (See also Liang et al. [10], for extensions of these models andWang and Chin [11], or Ramón et al. [12], for other alternative secondary goals).

3 Fuzzy Cross-Efficiency Evaluation: Some ProposalsBased on the FDEA Model Due to Guo and Tanaka

The key issue of any cross-efficiency evaluation approach in FDEA is how todefine the cross-efficiencies of the different DMUs. Once these are defined, dif-ferent alternative proposals could be given depending on the manner in which thecross-efficiencies are aggregated in order to provide the cross-efficiency scoresand/or the ordering procedure selected to rank these scores in case of these beingfuzzy numbers. Further variations of the particular resulting approach couldeventually be given by proposing different criteria to select among the optimalinput and output weights, if any, as it happens in the crisp context.

Therefore, we should start our development by providing a definition of thecross-efficiency of a given DMU with the weights of any other unit similar to thatin (3) in the fuzzy context. The main difficulty we face when addressing this task is

104 I. Sirvent and T. León

that different fuzzy DEA methods have been proposed to assess the relative effi-ciency of a set of DMUs with the CCR model when some of the input and/oroutput data are imprecise (see Hatami-Marbini et al. [13], for a taxonomy andreview of FDEA methods) so that different proposals of the cross-efficiency of agiven DMUj with the weights of DMUd in FDEA can be given depending on theparticular definition of relative efficiency considered. We could initially believethat any of the efficiency scores provided by the different existing FDEA modelscould be naturally extended to define cross-efficiencies, but it is not really so.

It is the case of the FDEA models based on the envelopment formulation of theCCR model as that in León et al. [7], since they do not directly provide the inputand output weights that are needed to obtain the cross-efficiencies of the DMUs.While when assessing the relative efficiency of a set of DMUs with the envel-opment formulation of the CCR model in DEA these weights can be obtained as aby-product via the duality theory in linear programming allowing so to computethe cross-efficiencies of all the DMUs, for the case of Fuzzy Linear Programming,different definitions of duality have been given (see for instance Rodder andZimmermann [14], Verdegay [15], Bector and Chandra [16], Wu [17], and Ramik[18]) so there is not a unique via to obtain, if possible, the optimal dual variables.Thus, depending on the definition that we accept, we could construct differentprimal–dual pairs verifying different conditions but, in our opinion, it seems dif-ficult that some of them can lead to a consistent definition of cross-efficiencies.

On the other hand, those FDEA models based on the CCR multiplier formu-lation that do not allow defining the corresponding efficiency score as a ratio of aweighted sum of the outputs to a weighted sum of the inputs cannot be eitherstraightforwardly used to define cross-efficiencies consistent with the definition ofrelative efficiency associated with the corresponding FDEA model. It is the case,for example, of Lertworasirikul et al. [8], where the possibilistic efficiency score �fof a DMU at a given possibility level a is defined as the optimal value of apossibility programming model and equals a weighted sum of the upper limits ofthe outputs’ a-cuts. However, the two limits of the inputs’ a-cuts are involved inthe normalization constraint so that it is not clear which denominator could be usedto define the cross-efficiencies.

Finally, even FDEA models based on the ratio formulation of the CCR modelalso exhibit important difficulties when trying to define consistent cross-efficiencies.It is the case, for example, of the model of Kao and Liu [5], because it uses differentfrontiers to compute each of the two limits of the a-cuts of the fuzzy efficiencyscores. Thus, different combinations of weights and limits of the a-cuts of the inputsand outputs can be used to compute the limits of the cross-efficiencies’ a-cuts,although none of them seems providing a satisfactory result.

Given that we do not find then possible to suggest a general approach to extendthe use of any FDEA model to compute cross-efficiencies, we have selected aparticular model, that of Guo and Tanaka [6], and make some proposals to performa cross-efficiency analysis based on the cross-efficiencies we have defined from theinformation provided by this model. Our intention has been to respect the basic

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 105

ideas of the authors trying to be consistent with their approach. We have chosenGuo and Tanaka’s FDEA model for two main reasons: that their model is based onthe multiplier formulation and that the efficiency score of each DMU is explicitlydefined in terms of a ratio of a weighted sum of the outputs to a weighted sum ofthe inputs. Moreover, since our definition of cross-efficiency has been raised byfollowing Guo and Tanaka’s scheme for assessing relative efficiency, this meansthat: (1) it has been defined only for the case in which the fuzzy inputs and outputsare symmetrical triangular numbers and (2) the analysis is referred to a particularpossibility level h between 0 and 1 predefined by the decision-maker.

In order to make this chapter self-contained, let us briefly summarize Guo andTanaka’s FDEA model. (We recommend reading the original paper for a betterknowledge of this model).

3.1 The Fuzzy DEA Model of Guo and Tanaka

A fuzzy number A is said to be a triangular fuzzy number and denoted by A ¼ðcl; a; crÞ if its membership function has the following form:

lAðxÞ ¼1� a�x

cl; a� cl� x� a

1� x�acr; a� x� aþ cr

0; otherwise

8<

:

where a 2 R is the center and cl; cr [ 0 are the left and right spreads of A,respectively.

When the two spreads of A are the same, that is, cl ¼ cr, then A is said to be asymmetrical triangular fuzzy number and is simply denoted by A ¼ ða; cÞ.

Considering a set of n DMUs described by m symmetrical triangular fuzzyinputs and s symmetrical triangular fuzzy outputs, Guo and Tanaka extend theCCR model (2) to be the following fuzzy DEA model:

max l0Y0

s:t: v0X0 � ~1

l0Yj.v0Xj j ¼ 1; . . .; n

v� 0m; l� 0s

ð5Þ

where Xj ¼ ðxj; cjÞ is the m-dimensional fuzzy input vector and Yj ¼ ðyj; djÞ is thes-dimensional fuzzy output vector of the jth DMU, j = 1,…, n, and ~1 denotes thefuzzy number 1 ¼ ð1; eÞ, where e\1 is a predefined spread. Moreover, xj � cj [ 0and yj � dj [ 0, 8j, are assumed because only positive inputs and outputs areconsidered.

Next we explain what these authors understand by a fuzzy inequality and bymaximizing a fuzzy variable as well as how they interpret the problem of finding v

106 I. Sirvent and T. León

such that v0X0 � ~1 in order to model (5) can be solved. In what follows, we willdenote by h 2 ½0; 1� a possibility level predefined by the decision maker.

Given two symmetric triangular fuzzy numbers Z1 ¼ ðz1;w1Þ and Z2 ¼ ðz2;w2Þ,the relation Z1.Z2 is defined in [6], by the following inequalities:

z1 � ð1� hÞw1� z2 � ð1� hÞw2;

z1 þ ð1� hÞw1� z2 þ ð1� hÞw2:

In addition, maximizing a symmetrical triangular fuzzy variable Z ¼ ðz;wÞ isunderstood as maximizing z1 � ð1� hÞw1, that is, maximizing the lower limit ofthe h-cut of Z.

Finally, the problem of finding v such that v0X0 � ~1 is converted into the fol-lowing optimization problem:

max v0c0

s:t: v0x0 � ð1� hÞv0c0 ¼ 1� ð1� hÞev0x0 þ ð1� hÞv0c0� 1þ ð1� hÞev� 0m

ð6Þ

Thus, problem (5) is rewritten as the following LP problem:

max l0y0 � ð1� hÞl0d0

s:t: v0c0� g0

v0x0 � ð1� hÞv0c0 ¼ 1� ð1� hÞev0x0 þ ð1� hÞv0c0� 1þ ð1� hÞel0yj � ð1� hÞl0dj� v0xj � ð1� hÞv0cj j ¼ 1; . . .; n

l0yj þ ð1� hÞl0dj� v0xj þ ð1� hÞv0cj j ¼ 1; . . .; n

v� 0m; l� 0s

ð7Þ

where e ¼ maxj¼1;...;nðmaxi¼1;...;m cij=xijÞ and g0 is the optimal value of (6).The efficiency score of a given DMU0 with the symmetrical triangular fuzzy

input vector X0 ¼ ðx0; c0Þ and fuzzy output vector Y0 ¼ ðy0; d0Þ is then defined as anon-symmetrical triangular fuzzy number E ¼ ðxl; g;xrÞ as follows:

g ¼ l�0y0

v�0x0; xl ¼ g� l�0ðy0 � d0ð1� hÞÞ

v�0ðx0 þ c0ð1� hÞÞ ; xr ¼l�0ðy0 þ d0ð1� hÞÞv�0ðx0 � c0ð1� hÞÞ � g;

ð8Þ

where v� and l� are the optimal solutions of (7), and xl;xr and g are the left andright spreads and the center of the fuzzy efficiency E, respectively. In addition, theauthors show that the center g satisfies g� 1.

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 107

3.2 Some Fuzzy Cross-Efficiency Evaluation ProposalsBased on the FDEA Model of Guo and Tanaka

Taking into account the definition of the fuzzy relative efficiency of a given DMUprovided by (8), we can similarly define the cross-efficiencies of the different units.

Definition 1 If ðvd ; ldÞ is an optimal solution of (7) for a given DMUd, then thefuzzy cross-efficiency of DMUj, j = 1,…,n, obtained with the weights of DMUd isdefined as a non-symmetrical triangular fuzzy number Edj ¼ ðxl

dj; gdj;xrdjÞ as

follows:

gdj ¼l0dyj

v0dxj; xl

dj ¼ gdj �l0dðyj � djð1� hÞÞv0dðxj þ cjð1� hÞÞ ; xr

dj ¼l0dðyj þ djð1� hÞÞv0dðxj � cjð1� hÞÞ � gdj

As it is usually done, we next define the fuzzy cross-efficiency score of DMUj,j = 1,…,n, as the average of its fuzzy cross-efficiencies obtained with the weightsof all the DMUs. Making use of the arithmetic with triangular fuzzy numbers (see[19], for instance) then the fuzzy cross-efficiency score of DMUj is given by thenon-symmetrical triangular fuzzy number �Ej ¼ ð�xl

j; �gj; �xrj Þ, j = 1,…,n, where:

�gj ¼ 1n

Pn

d¼1gdj; �xl

j ¼ 1n

Pn

d¼1xl

dj; �xrj ¼ 1

n

Pn

d¼1xr

dj

Remark: It is straightforward to check that gdj� 1; 8d; j, and also

�gj ¼ 1n

Pnd¼1 gdj� 1. It suffices to sum the last two inequalities in model (7).

As mentioned in Sect. 2, the possible existence of alternate optima whensolving the CCR model is a difficulty of cross-efficiency evaluations since this maylead to different cross-efficiency scores depending on the choice of weights that ismade. Guo and Tanaka’s model may have also alternative optimal solutions, as wewill show in the next section with an example. Therefore, our proposal of fuzzycross-efficiency evaluation may exhibit the same difficulty as the one mentioned inthe DEA context. Similarly to the crisp cross-efficiency evaluation, in which somealternative secondary goal is used to choose among the alternate optimal weightsprovided by the CCR model, we propose a solution in our context. To be precise,we propose a fuzzy benevolent and a fuzzy aggressive formulation which willallow us to choose among the alternate optimal weights of a given DMUd whenassessed with the FDEA model of Guo and Tanaka.

As explained in the previous section, the choice of weights which not onlymaximizes the efficiency of a particular DMU under evaluation as a primary goalbut, as a secondary goal, also maximizes the other DMUs’ cross-efficiencies, isknown as the benevolent formulation.

In their efficiency assessment approach, Guo and Tanaka consider that maxi-mizing a symmetrical triangular fuzzy variable is maximizing its lower limit.Therefore, for keeping the consistency with such definition we are interpreting the

108 I. Sirvent and T. León

idea of ‘‘maximizing the fuzzy cross-efficiency of the DMUj using the weights ofDMUd’’ as maximizing the lower limit of Edj ¼ ðxl

dj; gdj;xrdjÞ, i.e. maximizing

gdj � xldj ¼

l0dðyj � djð1� hÞÞv0dðxj þ cjð1� hÞÞ : ð9Þ

Maximizing globally the fuzzy cross-efficiencies of all the DMUs would lead toa non-linear model having in its objective a sum of ratios, which might be difficultto solve. In the (crisp) DEA literature, it has been proposed the use of differentsurrogates (see Doyle and Green [4], and Liang et al. [10],) to deal with thisdifficulty. Thus, we are going to proceed in the same manner by adapting to ourcontext one of the formulations proposed in Liang et al. [10], which is frequentlyused in practical applications. This formulation uses as surrogate the differencesbetween the denominator and the numerator of the cross-efficiencies.

Liang et al. define the ideal point as that weight vector ðv; lÞ for which every

DMU is efficient, i.e., l0Yj=v0Xj ¼ 1 or l0Yj � v0Xj ¼ 0 for all j = 1,…,n. In theabsence of such an ideal point, they argue that a reasonable objective is to treataj ¼ v0Xj � l0Yj, aj� 0, as goal achievement variables, and for each DMUd derivea set of input and output weights ðvd; ldÞ that is an optimal solution of the CCRmodel, and at the same time minimizes the total deviation from the ‘‘ideal point’’Pn

j¼1 aj.In our case, the input and output weights of DMUd that should be chosen to

compute the fuzzy cross-efficiencies should be analogously obtained by selectingamong the optimal solutions of (7) that minimizing the corresponding goalachievement variables aj ¼ v0dxj þ ð1� hÞv0dcj � l0dyj þ ð1� hÞl0ddj, which meansto maximize somehow the sum of the lower limits of the cross-efficiencies of allthe DMUs. Thus, we propose the following benevolent formulation:

minXn

j¼1

j6¼d

aj

s:t: l0dyd � ð1� hÞl0ddd ¼ bd

v0dcdd

v0dxd � ð1� hÞv0dcd ¼ 1� ð1� hÞev0dxd þ ð1� hÞv0dcd � 1þ ð1� hÞel0dyj � ð1� hÞl0ddj� v0dxj � ð1� hÞv0dcj j ¼ 1; . . .; n

l0dyj þ ð1� hÞl0ddj� v0dxj þ ð1� hÞv0dcj j ¼ 1; . . .; n

l0dyj � ð1� hÞl0ddj � v0dxj � ð1� hÞv0dcj þ aj ¼ 0 j ¼ 1; . . .; n; j 6¼ d

vd � 0m; ld � 0s; aj� 0 j ¼ 1; . . .; n; j 6¼ d

ð10Þ

where bd is the optimal value of problem (7).

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 109

We should remark that in the same way that ad ¼ v0dXd � l0dYd is constant forany optimal solution of the CCR model when assessing DMUd, the goal variablead ¼ v0dxd þ ð1� hÞv0dcd � l0dyd þ ð1� hÞl0ddd is also constant for every optimalsolution of (7) when assessing DMUd. In fact, it is easy to check thatad ¼ 1þ ð1� hÞð2g0 � eÞ � bd.

Following a similar reasoning, we can also define a fuzzy aggressive formu-lation to select input and output weights when performing the fuzzy cross-efficiency evaluation we have proposed based on the FDEA model of Guo andTanaka. In this case, we should seek weights that minimize the cross-efficienciesof the other units as a secondary goal. Therefore, a reasonable way to obtain thedesired weights is through the solution of a linear problem whose objectivefunction is maximizing

Pnj¼1j6¼d

aj with the same constraints as those in model (10).

Once we have obtained the fuzzy cross-efficiency scores of all the DMUs, wehave to rank them. Unlike what happens with real numbers which have theirnatural order, there is no consensus on how to sort fuzzy numbers. A usualapproach is to convert a fuzzy quantity into a real number and base the comparisonof fuzzy quantities on that of real numbers. Many different indices have beenproposed in the literature for the comparison of fuzzy quantities and none of themis commonly accepted. Each method suffers from defects but also presents someintuitive advantages over its competitors. As said in Wang and Kerre [20], ‘‘If onetries to develop a new method aiming at the improvement of an establishedordering procedure, one normally designs some examples in which the newlydeveloped method derives a more reasonable resulting ranking than the known oneby his intuition’’. These authors establish then some reasonable properties for theordering of fuzzy quantities and check several ordering indices to see whether theysatisfy them or not.

Given that the fuzzy cross-efficiency scores we intend to rank are triangularfuzzy numbers, many of the indices found to be relatively reasonable for theordering of fuzzy numbers in Wang and Kerre [20] are equivalent. We are going touse one such index denoted as Y2 in that paper.1 This index was defined by YagerYager [25] and it is very easy to obtain in the particular case we address. For agiven triangular fuzzy quantity A ¼ ðcl; a; crÞ it is computed as

Y2 ¼ aþ 14 ðcr � clÞ

Therefore, our fuzzy cross-efficiency evaluation finishes by computing thisindex Y2 associated to each fuzzy cross-efficiency score �Ej ¼ ð�xl

j; �gj; �xrj Þ,

j = 1,…,n, and providing the corresponding ranking of DMUs.

1 A triangular fuzzy number is, in particular, convex and its height is equal to 1. Therefore,according to Remarks 2.1 and 2.2 in Wang and Kerre [20], Y2 is equal both to an index proposedby Campos and Muñoz [21] and the index of Liou and Wang [22] in the particular case ofconsidering an optimism index equal to 1/2. Besides, it is straightforward to check that Y2 alsocoincides with Fortemps and Roubens’ index [23]. Finally, Proposition 2.1 states also anequivalence of Y2 with a third index proposed in Choobineh [24].

110 I. Sirvent and T. León

4 Numerical Example

In this section, we present a numerical example in order to illustrate the use of themethodology proposed here. The data are taken from Guo and Tanaka [6], andrecorded in Table 1. They consist of 5 DMUs (A, B, C, D and E) with two fuzzyinputs and two fuzzy outputs.

Table 2 shows the fuzzy relative efficiencies provided by model (7) of Guo andTanaka for different possibility levels h, where e ¼ maxj2 A;B;C;D;Ef gðmaxi¼1;2 cij=xijÞ ¼ 0:1707. The (crisp) relative efficiencies for the case h = 1have been obtained from the CCR model.

Prior to perform the proposed cross-efficiency analysis we are going to showthat model (7) can have alternate optimal solutions so that it made actually sense topropose some procedure to select among the optimal weights in order to computethe cross-efficiencies.

For example, it is easy to check that both ðv1D ; v2D; l1D; l2DÞ ¼ ð0:243902;0; 0; 0:160853Þ and ðv1D ; v2D; l1D;l2DÞ ¼ ð0:243902; 0; 0:228242; 0:052050Þ areoptimal solutions of model (7) when evaluating the relative efficiency of DMU D atlevel h = 0.4 (and so are, obviously, all the convex combinations of these twosolutions). The same happens with DMU B with the weights ðv1B ; v2B; l1B; l2BÞ ¼ð0; 0:623306; 0:114942; 0:190736Þ and ðv1B ; v2B; l1B; l2BÞ ¼ ð0; 0:623306;0:407982; 0Þ for this h-value. Thus, different cross-efficiencies can be obtained whenperforming the cross-efficiency analysis at this level of possibility depending on theparticular optimal weights selected.

Table 2 Efficiency scores for different h-values. Model of Guo and Tanaka

h A B C D E

0 (0.136, 0.812,0.187)

(0.075, 0.975,0.115)

(0.215, 0.820,0.301)

(0.159, 0.933,0.317)

(0.124, 0.792,0.234)

0.2 (0.120, 0.820,0.146)

(0.081, 0.980,0.090)

(0.182, 0.823,0.233)

(0.186, 0.945,0.245)

(0.152, 0.803,0.184)

0.4 (0.093, 0.827,0.108)

(0.061, 0.984,0.067)

(0.141, 0.827,0.170)

(0.144, 0.959,0.177)

(0.118, 0.814,0.137)

0.6 (0.064, 0.835,0.071)

(0.041, 0.989,0.044)

(0.097, 0.831,0.110)

(0.100, 0.972,0.115)

(0.082, 0.825,0.090)

0.8 (0.033, 0.842,0.035)

(0.021, 0.995,0.022)

(0.050, 0.836,0.054)

(0.052, 0.986,0.056)

(0.042, 0.837,0.044)

1 (0, 0.855, 0) (0, 1, 0) (0, 0.861, 0) (0, 1, 0) (0, 1, 0)

Table 1 Data (Source: [6])

Variable A B C D E

x1 (4.0, 0.5) (2.9, 0.0) (4.9, 0.5) (4.1, 0.7) (6.5, 0.6)x2 (2.1, 0.2) (1.5, 0.1) (2.6, 0.4) (2.3, 0.1) (4.1, 0.5)y1 (2.6, 0.2) (2.2, 0.0) (3.2, 0.5) (2.9, 0.4) (5.1, 0.7)y2 (4.1, 0.3) (3.5, 0.2) (5.1, 0.8) (5.7, 0.2) (7.4, 0.9)

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 111

Table 3 shows the cross-efficiency scores of the five DMUs computed with theoptimal weights provided by the benevolent formulation (10) for the possibilitylevels h considered in Table 2. These cross-efficiency scores are depicted inFigs. 1, 2, 3, 4, 5. Again, the (crisp) cross-efficiency scores for the case h = 1 havebeen obtained from the corresponding benevolent formulation of Liang et al. [10].It can be seen that, as expected, the center of the cross-efficiency scores are largerand the spreads are smaller as the level of possibility h increases.

Finally, Table 4 records the value of the index Y2 associated to the fuzzy cross-efficiency scores of the five DMUS for the considered possibility levels as well asthe corresponding rankings of DMUs. We can see that, although it has not to bealways the case, for this example the obtained ranking of DMUs does not changefor the different h-values lower than 1. DMU B is the best unit, followed by DMUsD, E, C and A, respectively. When no uncertainty is considered, that is, whenh = 1, the value of Y2, recorded in the second last row of Table 3 simply equalsthe (crisp) cross-efficiency score provided by the conventional cross-efficiencyevaluation of the centers of the fuzzy triangular inputs and outputs. In this case, a

Table 3 Cross-efficiency scores for different h-values. Benevolent formulation

h A B C D E

0 (0.133, 0.810,0.166)

(0.057, 0.957,0.064)

(0.207, 0.812,0.272)

(0.155, 0.929,0.198)

(0.185, 0.853,0.230)

0.2 (0.110, 0.816,0.131)

(0.046, 0.965,0.051)

(0.171, 0.818,0.213)

(0.128, 0.937,0.156)

(0.152, 0.859,0.181)

0.4 (0.085, 0.823,0.096)

(0.035, 0.973,0.038)

(0.133, 0.825,0.156)

(0.099, 0.945,0.114)

(0.117, 0.866,0.134)

0.6 (0.058, 0.830,0.063)

(0.024, 0.981,0.025)

(0.091, 0.832,0.102)

(0.068, 0.954,0.075)

(0.080, 0.874,0.088)

0.8 (0.030, 0.837,0.031)

(0.012, 0.990,0.013)

(0.047, 0.840,0.050)

(0.035, 0.963,0.037)

(0.041, 0.881,0.043)

1 (0, 0.854, 0) (0, 1, 0) (0, 0.860, 0) (0, 0.984, 0) (0, 1, 0)

Cross-efficiency scores (h=0)

0

0,2

0,4

0,6

0,8

1

1,2

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

DMU A

DMU B

DMU C

DMU D

DMU E

Fig. 1 Cross-efficiencyscores h ¼ 0 Benevolentformulation

112 I. Sirvent and T. León

Cross-efficiency scores (h=0.2)

0

0,2

0,4

0,6

0,8

1

1,2

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

DMU ADMU BDMU CDMU DDMU E

Fig. 2 2. Cross-efficiencyscores h ¼ 0:2. Benevolentformulation

Cross-efficiency scores (h=0.4)

0

0,2

0,4

0,6

0,8

1

1,2

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

DMU A

DMU B

DMU C

DMU D

DMU E

Fig. 3 Cross-efficiencyscores h ¼ 0:4. Benevolentformulation

Cross-efficiency scores (h=0.6)

0

0,2

0,4

0,6

0,8

1

1,2

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

DMU A

DMU B

DMU C

DMU D

DMU E

Fig. 4 Cross-efficiencyscores h ¼ 0:6. Benevolentformulation

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 113

change is observed in the resulting ranking of DMUs. DMU E rises at the firstposition of the ranking, tied with DMU B. However, as we have just mentioned,for all other h-values considered, this DMU ranked third. This supports the ideathat it is necessary to take into account the uncertainty of the data in any decisionmaking process.

5 Conclusions

Both DEA and FDEA results classify DMUs into two groups: efficient and inef-ficient. However, in many practical applications decision makers are interested in aranking beyond this classification. In FDEA literature we can find differentapproaches to define DMU rankings. In our opinion, those that result from usingsome ranking fuzzy number method to order the obtained fuzzy efficiencies aresusceptible to a serious criticism: they are not based on objective criteria. Thus, it

Cross-efficiency scores (h=0.8)

0

0,2

0,4

0,6

0,8

1

1,2

0,5 0,6 0,7 0,8 0,9 1 1,1 1,2

DMU A

DMU B

DMU C

DMU D

DMU E

Fig. 5 Cross-efficiencyscores h ¼ 0:8. Benevolentformulation

Table 4 Ranking of units for different h-values

h A B C D E

0 Y2 0.818 0.959 0.828 0.939 0.864rank 5 1 4 2 3

0.2 Y2 0.822 0.966 0.829 0.943 0.867rank 5 1 4 2 3

0.4 Y2 0.826 0.973 0.831 0.949 0.871rank 5 1 4 2 3

0.6 Y2 0.832 0.981 0.835 0.955 0.875rank 5 1 4 2 3

0.8 Y2 0.838 0.990 0.840 0.963 0.881rank 5 1 4 2 3

1 Y2 0.854 1 0.860 0.984 1rank 4 1 3 2 1

114 I. Sirvent and T. León

should be desirable to have available a methodology, such as classic cross-effi-ciency evaluation, in order to provide rankings of DMUs that can be accepted by abroader number of practitioners.

In this chapter, we discuss some difficulties that arise with the definition offuzzy cross-efficiencies and accordingly we propose a fuzzy cross-efficiencyevaluation based in the fuzzy DEA model by Guo and Tanaka. We choose thismodel because it is based on the dual multiplier formulation of the CCR model andthe fuzzy efficiency of a given DMU is defined in a ratio form in terms of theobtained input and output weights.

As it happens in the CCR model, Guo and Tanaka’s model may have alternativeoptimal solutions which may lead to different cross-efficiency scores depending onthe choice of weights that is made. Therefore we propose some alternativesecondary goals to choose among the alternate optimal weights.

We would like to remark that our main interest has been the choice of theprofiles of FDEA weights that the different DMUs use in the calculation ofthe cross-efficiencies rather than how to deal with such cross-efficiencies once theyare obtained. We simply aggregate them with an arithmetic mean. Then the cross-efficiencies provided by the FDEA weights of the different DMUs are all attachedthe same aggregation weight and, consequently, the same importance. Obviously,the assumption of equal aggregation weights for all the profiles of FDEA weightscan be relaxed, so that we may attach different importance to the cross-efficienciesprovided by different DMUs. The use of a weighted average of cross-efficiencies,with aggregation weights that are not necessarily equal, allows the decision maker(DM) to introduce some flexibility in the analysis by incorporating his/her pref-erences into the evaluation to be made.

Once we have obtained the fuzzy cross-efficiency scores of all the DMUs, werank them by using a ranking index defined by Yager. This index satisfies manyreasonable properties for the ordering of fuzzy quantities and is very easy to obtainin the particular case we address of triangular cross-efficiency scores. However,other ranking methods could also be reasonable.

Acknowledgments This work has been partially supported by the Ministerio de Ciencia eInnovación of Spain, Research Projects TIN2009-14392-C02-01 and MTM2009-10479.

References.

1. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units.Eur. J. Oper. Res. 2(6), 429–444 (1978)

2. Adler, N., Friedman, L., Sinuany-Stern, Z.: Review of ranking methods in the dataenvelopment analysis context. Eur. J. Oper. Res. 140(2), 249–265 (2002)

3. Sexton, T.R., Silkman, R.H., Hogan, A.J.: Data Envelopment Analysis: Critique andExtensions. In: Silkman, R.H. (ed.) Measuring Efficiency: An Assessment of DataEnvelopment Analysis, pp. 73–105. Jossey-Bass, San Francisco (1986)

5 Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA) 115

4. Doyle, J.R., Green, R.H.: Efficiency and cross-efficiency in DEA: derivations, meanings anduses. J. Oper. Res. Soc. 45(5), 567–578 (1994)

5. Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst.113, 427–437 (2000)

6. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119,149–160 (2001)

7. León, T., Liern, V., Ruiz, J.L., Sirvent, I.: A fuzzy mathematical programming approach tothe assessment of efficiency with DEA models. Fuzzy Sets Syst. 139(2), 407–419 (2003)

8. Lertworasirikul, S., Fang, S.C., Joines, J.A., Nuttle, H.L.W.: Fuzzy data envelopmentanalysis (DEA): a possibility approach. Fuzzy Sets Syst. 139, 379–394 (2003)

9. Charnes, A., Cooper, W.W.: Programming with linear fractional functionals. Naval Res.Logistics Quart. 9, 181–186 (1962)

10. Liang, L., Wu, J., Cook, W.D., Zhu, J.: Alternative secondary goals in DEA cross-efficiencyevaluation. Int. J. Prod. Econ. 113(2), 1025–1030 (2008)

11. Wang, Y.M., Chin, K.S.: A neutral DEA model for cross-efficiency evaluation and itsextension. Expert Syst. Appl. 37(5), 3666–3675 (2010)

12. Ramón, N., Ruiz, J.L., Sirvent, I.: On the choice of weights profiles in cross-efficiencyevaluations. Eur. J. Oper. Res. 207, 1564–1572 (2009)

13. Hatami-Marbini, A., Emrouznejad, A., Tavana, M.: A taxonomy and review of the fuzzy dataenvelopment analysis literature: two decades in the making. Eur. J. Oper. Res. 214, 457–472(2011)

14. Rodder, W., Zimmermann, H.J.: Duality in fuzzy linear programming, In: Fiacco, A.V.,Kortanek, K.O. (eds.) External Methods and System Analysis, Berlin, New York, 415–429(1980)

15. Verdegay, J.L.: A dual approach to solve the fuzzy linear programming problem. Fuzzy SetsSyst. 14(2), 131–141 (1984)

16. Bector, C.R., Chandra, S.: On duality in linear programming under fuzzy environment. FuzzySets Syst. 125, 317–325 (2002)

17. Wu, H.C.: Duality theory in fuzzy linear programming problems with fuzzy coefficients.Fuzzy Optim. Decis. Making 2, 61–73 (2003)

18. Ramik, J.V.: Duality in fuzzy linear programming: some new concepts and results. FuzzyOptim. Decis. Making 4, 25–39 (2005)

19. Pedrycz, W., Gomide, F.(eds): An introduction to Fuzzy Sets: Analysis and Design. The MITPress, Cambridge (1998)

20. Wang, X., Kerre, E.E.: Reasonable properties for the ordering of fuzzy quantities (I). FuzzySets Syst. 118, 375–385 (2001)

21. Campos, L., Muñoz, A.: A subjective approach for ranking fuzzy numbers. Fuzzy Sets Syst.29, 145–153 (1989)

22. Liou, T., Wang, J.: Ranking fuzzy numbers with integral value. Fuzzy Sets Syst. 50, 247–255(1992)

23. Fortemps, P., Roubens, M.: Ranking and defuzzification methods based on areacompensation. Fuzzy Sets Syst. 82, 319–330 (1996)

24. Choobineh, F., Li, H.: An index for ordering fuzzy numbers. Fuzzy Sets Syst. 54, 287–294(1993)

25. Yager, R.R.: A procedure for ordering fuzzy sets of the unit interval. Inf. Sci. 24, 143–161(1981)

116 I. Sirvent and T. León

Chapter 6Fuzzy Mix-efficiency in Fuzzy DataEnvelopment Analysis and Its Application

Jolly Puri and Shiv Prasad Yadav

Abstract Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiencies of a homogeneous setof decision making units (DMUs) which utilize multiple inputs to produce multipleoutputs. It consists of two types of DEA models: radial models and non-radialmodels. A radial model deals only with proportional changes of inputs/outputs andneglects the input/output slacks whereas a non-radial model deals directly with theinput/output slacks. The slack based measure (SBM) model is a non-radial modelthat results into the SBM efficiency which can be further decomposed into radial,scale and mix-efficiency. The mix-efficiency is a measure to estimate how well theset of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure is limited to crisp input and output data which may not alwaysbe available in real life applications. However, in real life problems, data may beimprecise or fuzzy. In this chapter, we extend the idea of mix-efficiency to fuzzyenvironments and develop a concept of fuzzy mix-efficiency in fuzzy DEA. Weprovide both the input and output orientations of fuzzy mix-efficiency. The a-cutapproach is used to evaluate the fuzzy input as well as fuzzy output mix-efficiencies of each DMU. Further, a new method is provided for ranking theDMUs on the basis of fuzzy input and output mix-efficiencies. Moreover, to ensurethe validity of the proposed methodology, we illustrate a numerical example andapplied the proposed methodology to the banking sector in India.

Keywords Fuzzy DEA � Fuzzy mix-efficiency � Fuzzy ranking approach �Banking performance evaluation

J. Puri � S. P. Yadav (&)Department of Mathematics, Indian Institute of Technology Roorkee,Roorkee 247667, Indiae-mail: spyorfma@gmail.com

J. Purie-mail: puri.jolly@gmail.com

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_6, � Springer-Verlag Berlin Heidelberg 2014

117

1 Introduction

Data envelopment analysis (DEA), initially developed by Charnes, Cooper andRhodes [1], is basically a generalization of Farrell’s technical efficiency measureto multiple inputs and multiple outputs case [2]. It is a linear programming basednon-parametric technique for evaluating the relative efficiencies of homogeneousdecision making units (DMUs) which utilize multiple inputs to produce multipleoutputs. It constructs a non-parametric piecewise frontier (surface) over the dataand using this frontier it computes a maximal performance measure for each DMUrelative to that of all other DMUs with the restriction that each DMU lies on theefficient frontier or is enveloped by the frontier. The DMUs which lie on thefrontier are called the efficient DMUs. The efficiency value of an efficient DMU isequal to 1. On the other hand, the DMUs which are enveloped by the frontier arecalled the inefficient DMUs. The efficiency value of an inefficient DMU liesbetween 0 and 1. Since 1978 DEA has got comprehensive attention both in theoryand applications. Based on the original DEA model [1], various theoreticalextensions have been developed which can be seen in Banker et al. [3], Charneset al. [4], Peterson [5], Tone [6], and Cooper et al. [2]. More detailed reviews ofthe DEA methodology can be seen in Seiford and Thrall [7], Seiford [8], Zhu [9],and Ray [10]. The models in DEA are categorized as radial models andnon-radial models. A radial model deals with proportional changes of inputs oroutputs. The CCR model [1], the first DEA model, is a radial model and evaluatesthe CCR efficiency of a DMU which reflects the proportional maximum input(output) reduction (augmentation) rate that is common to all inputs (outputs).There are two major drawbacks of this model. First one is that in real life situa-tions, not all inputs (outputs) act in a proportional way and second one is that whilereporting the efficiency score it neglects the input–output slacks. Therefore, theradial models may mislead the decisions of those decision makers who select theefficiency score as the only index for measuring the performance of their DMUs.On the other hand, a non-radial model puts aside the assumption of proportionatechanges in inputs and outputs, and deals directly with the input/output slacks. Theslack-based measure (SBM) model [6] is a non-radial model and possesses threeorientations, namely, input-oriented SBM model, output-oriented SBM model andnon-oriented SBM model. It results into the SBM efficiency which can be furtherdecomposed into radial, scale and mix-efficiencies. The results of both the CCRand the SBM models are used to evaluate the mix-efficiency [11]. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputsare produced) together [12, 13].

The conventional DEA models are limited to crisp inputs and outputs. How-ever, in real life applications, uncertainty often exists in inputs and outputs whichcan be represented by fuzzy sets or fuzzy numbers. In order to deal with uncertainand inexact input–output data in DEA, the notion of fuzziness has been introducedin DEA which resulted in its extension to fuzzy DEA (FDEA). The literature onFDEA can be seen in [14–19]. Several approaches have been proposed and several

118 J. Puri and S. P. Yadav

new are coming to introduce fuzzy data in DEA and to solve the resulted FDEAmodels. Sengupta [20] was the first to introduce fuzziness in the conventional DEAmodel by applying the principle of fuzzy set theory [21] and proposed a meth-odology to solve FDEA model. Hatami-Marbini et al. [16] presented a taxonomyand review of FDEA literature, and classified the approaches for solving FDEAmodels into four major categories, namely, tolerance approach [20], a-cutapproach [17, 22], fuzzy ranking approach [15] and possibility approach [23]. Inthe tolerance approach, the uncertainty is incorporated into the DEA modelsby defining tolerance levels on constraint violations. In this approach, theinequality or equality signs, present in the constraints, are fuzzified withouttreating the fuzzy coefficients directly. In a-cut approach, the main idea is totransform the fuzzy DEA model into a pair of parametric programs in order to findthe lower and upper bounds of the a-cuts of the fuzzy efficiencies. The fuzzyranking approach, initially proposed by Guo and Tanaka [15], is also anotherpopular approach in the FDEA literature. The main idea in this approach is totransform a FDEA model into crisp linear program in which fuzzy constraints areconverted into crisp constraints by predefining a possibility level and using thecomparison rule, also known as ranking function, for fuzzy numbers. In possibilityapproach, fuzzy constraints of the FDEA model are treated as fuzzy events and theFDEA model is transformed into a possibility linear programming problem byusing the possibility measures of the fuzzy events.

The traditional mix-efficiency measure is also limited to crisp input and outputdata which may not always be available in real life applications. However, due touncertainty in the availability of the crisp data in real life problems, inputs andoutputs are often imprecise or fuzzy. Therefore, in order to calculate mix-efficiency with imprecise or fuzzy data, there is a need to extend the traditionalmix-efficiency measure to fuzzy environments and develop a concept of fuzzymix-efficiency in FDEA. We provide both the input and output orientations offuzzy mix-efficiency. Tone [11] presented the input and output orientations of mix-efficiency by using the input and output oriented CCR and SBM models. Theconcept of fuzzy input mix-efficiency (FIME) is proposed by Puri and Yadav [18].For measuring FIME in [18], the input-oriented fuzzy CCR model (FCCRI) andthe input-oriented fuzzy SBM model (FSBMI) with fuzzy input and fuzzy outputdata have been proposed. In this chapter, the concept of FIME proposed by Puriand Yadav [18] is presented and fuzzy output mix-efficiency (FOME) is defined byproposing output-oriented fuzzy CCR model (FCCRO) and output-oriented fuzzySBM model (FSBMO) with fuzzy input and fuzzy output data.

The chapter is arranged as follows: Sect. 2 presents an overview of DEA withinput and output orientations of CCR model, SBM model and mix-efficiencies.Section 3 presents the description of FDEA with FCCRI, FCCRO, FSBMI andFSBMO models. Section 4 presents the methodology for solving FCCRI, FCCRO,FSBMI and FSBMO models. Section 5 gives the definitions of FIME and FOME.Section 6 presents a numerical illustration. Section 7 describes a new method forranking the DMUs. Section 8 presents an application of the proposed methodologyto the banking sector. The last Sect. 9 concludes the findings of our study.

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 119

2 Data Envelopment Analysis (DEA)

DEA is a linear programming based non-parametric method for evaluating therelative efficiencies of the DMUs which utilize multiple inputs to produce multipleoutputs. DEA models are classified as radial models and non-radial models. Thefirst DEA model proposed by Charnes et al. [1] is a radial model, popularly knownas the CCR model. The SBM model proposed by Tone [6] is a non-radial model.Both CCR and SBM models have input and output orientations. Assume that theperformance of a homogeneous set of n DMUs (DMUj; j = 1,…, n) is to bemeasured in terms of m inputs (xij; i = 1,…, m) and s outputs (yrj; r = 1,…, s). Letxik be the amount of the ith input used by the kth DMU to produce the rth outputyrk of the kth DMU. Assume that each input and output are positive real numbers.

2.1 Input and Output Orientations of CCR Model

Charnes et al. [1] proposed the CCR model with both input and output orientations.The input-oriented CCR (CCRI) model and output-oriented CCR (CCRO) modelevaluate the CCR input efficiency and CCR output efficiency of every DMUrespectively, and are presented in the envelopment form in Table 1. The CCRinput efficiency and the CCR output efficiency of the kth DMU are respectivelydenoted by hk

I and hkO.

Here sþrk and s0þrk are slacks in the rth output of the kth DMU in CCRI and CCRO

models respectively; s�ik and s0�ik are slacks in the ith input of the kth DMU in CCRI

and CCRO models respectively; gjk’s and g0jk ’s are non negative variables forj ¼ 1; 2; . . .; n ; e is the non-Archimedean infinitesimal. Due to nonzeroassumption of the data, we have 0\hk�

I � 1 for k ¼ 1; 2; . . .; n [2]. However,

hk�O � 1 for k ¼ 1; 2; . . .; n [2].

Definition 1 The kth DMU is said to be CCR input efficient if

(i) hk�I ¼ 1;

(ii) all slacks are zero, i.e., s�ik ¼ 0 for i ¼ 1; 2; . . .; m and sþrk ¼ 0 forr ¼ 1; 2; . . .; s:

The optimal solution of the CCRO model is related to that of CCRI model via:

hk�O ¼

1

hk�I

and g0�jk ¼g�jkhk�

I

j ¼ 1; 2; . . .; n:

120 J. Puri and S. P. Yadav

Tab

le1

CC

RI

mod

elan

dC

CR

Om

odel

Inpu

t-or

ient

edC

CR

(CC

RI)

mod

elO

utpu

t-or

ient

edC

CR

(CC

RO

)m

odel

hk I¼

min

h�

eXm i¼

1

s� ikþX

s

r¼1

sþ rk

!

subj

ectt

oh

x ik¼X

n j¼1

x ijg j

kþ

s� ik8i;

y rk¼X

n j¼1

y rjg j

k�

sþ rk8r;

e[

0;g j

k�

0;s� ik�

0;sþ rk�

08

i;r;

j;

hun

rest

rict

edin

sign

hk O¼

max

hþ

eXm i¼

1

s0� ikþX

s

r¼1

s0þ rk

!

subj

ectt

ox i

k¼X

n j¼1

x ijg0 jkþ

s0� ik8i;

hyrk¼X

n j¼1

y rjg0 jk�

s0þ rk8r;

e[

0;g0 jk�

0;s0� ik�

0;s0þ rk�

08

i;r;

j;

hun

rest

rict

edin

sign

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 121

2.2 Input and Output Orientations of SBM Model

Tone [6] presented the SBM model in both input and output orientations. Theinput-oriented SBM (SBMI) model and output-oriented SBM (SBMO) modelevaluate the SBM input efficiency and SBM output efficiency of every DMUrespectively, and are presented in Table 2. The SBM input efficiency and the SBMoutput efficiency of the kth DMU are respectively denoted by qk

I and qkO:

Here Sþrk and S0þrk are slacks in the rth output of the kth DMU in SBMI andSBMO models respectively; S�ik and S0�ik are slacks in the ith input of the kth DMUin SBMI and SBMO models respectively; kjk’s and k0jk’s are non negative variables

for j ¼ 1; 2; . . .; n: Also 0\qk�I ; qk�

O � 1 for k ¼ 1; 2; . . .; n:

Definition 2 The kth DMU is said to be SBM input efficient if

(i) qk�I ¼ 1;

(ii) all input slacks are zero, i.e., S�ik ¼ 0 for i ¼ 1; 2; . . .; m:

Definition 3 The kth DMU is said to be SBM output efficient if

(i) qk�o ¼ 1

(ii) all output slacks are zero, i.e., Sþrk ¼ 0 for r ¼ 1; 2; . . .; s:

2.3 Input and Output Mix-Efficiencies

The input mix-efficiency (IME) of the kth DMU is denoted by wkI and is defined as

the ratio of SBM input efficiency of the kth DMU to CCR input efficiency of thekth DMU. Thus,

Table 2 SBMI model and SBMO modelInput-oriented SBM (SBMI) model Output-oriented SBM (SBMO) model

qkI ¼ min 1� 1

m

Xm

i¼1

S�ikxik

subject to xik ¼Xn

j¼1

xij kjk þ S�ik 8i;

yrk ¼Xn

j¼1

yrj kjk � Sþrk 8r;

kjk � 0; S�ik � 0; Sþrk � 0 8 i ; r; j

1

qkO

¼ max 1þ 1s

Xs

r¼1

S0þrk

yrk

subject to xik ¼Xn

j¼1

xij k0jk þ S

0�ik 8i;

yrk ¼Xn

j¼1

yrj k0jk � S

0þrk 8r;

k0jk � 0; S0�ik � 0; S

0þrk � 0 8 i ; r; j

122 J. Puri and S. P. Yadav

wk�I ¼

qk�I

hk�I

:

Due to assumption of the positive data, we have 0\qk�I � 1 and

0\hk�I � 1 for k ¼ 1; 2; . . .; n: Also qk�

I � hk�I [11]. This implies that 0\wk�

I � 1 ;

and wk�I ¼ 1 if and only if qk�

I ¼ hk�I holds. The kth DMU is said to have the most

efficient combination of inputs if wk�I ¼ 1 holds, even though it may be technically

inefficient.The output mix-efficiency (OME) of the kth DMU is denoted by wk

O and isdefined as the ratio of SBM output efficiency of the kth DMU to CCR outputefficiency of the kth DMU. Thus,

wk�O ¼

qk�O

hk�O

:

Due to assumption of the positive data, we have 0\qk�O � 1 and hk�

O � 1 for k ¼1; 2; . . .; n: Also qk�

O � hk�O [11]. This implies that 0\wk�

O � 1 ; and wk�O ¼ 1 if and

only if qk�O ¼ hk�

O holds. The kth DMU is said to have the most efficient combi-

nation of outputs if wk�O ¼ 1 holds.

3 Fuzzy Data Envelopment Analysis (FDEA)

Conventional DEA models are limited to crisp inputs and outputs which may notalways be available in real life applications. However, in real life problems, inputsand outputs are often imprecise or uncertain. In FDEA, the imprecision anduncertainty in inputs and outputs can be represented by fuzzy sets or fuzzynumbers. Several efforts have been made by many researchers to handle fuzzyinput and fuzzy output data in FDEA. Both CCR and SBM models were extendedto fuzzy CCR model [15, 24] and fuzzy SBM model [25, 26] in which fuzzy input–output data is represented by fuzzy numbers. Assume that the performance of ahomogeneous set of n DMUs (DMUj; j = 1,…, n) is to be measured. Each DMUj

(j = 1,…, n) utilizes m fuzzy inputs (~xij; i = 1,…, m) to produce s fuzzy outputs(~yrj; r = 1,…, s). Let ~xik be the amount of the ith fuzzy input used by the kth DMUto produce ~yrk amount of the rth fuzzy output of the kth DMU. Assume that thefuzzy inputs and fuzzy outputs are positive fuzzy numbers (FNs), in particular,triangular fuzzy numbers (TFN) [27].

Definition 4 A TFN ~A ¼ ða1; a2; a3Þ is defined by the membership function l~A

given by

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 123

l~AðxÞ ¼

x� a1

a2 � a1; a1\x� a2;

x� a3

a2 � a3; a2� x\a3;

0; otherwise:

8>>>>>><

>>>>>>:

3.1 Input and Output Orientations of Fuzzy CCR Model(FCCR Model)

The input-oriented FCCR (FCCRI) model [18] and output-oriented FCCR(FCCRO) model evaluate the FCCR input efficiency and FCCR output efficiency ofevery DMU respectively, and are presented in Table 3. The FCCR input efficiency

and the FCCR output efficiency of the kth DMU are denoted by ~hkI and ~hk

O

respectively.Here ~sþrk and ~s0þrk are fuzzy slacks in the rth fuzzy output of the kth DMU in

FCCRI and FCCRO models respectively; ~s�ik and ~s0�ik are fuzzy slacks in the ithfuzzy input of the kth DMU in FCCRI and FCCRO models respectively; gjk’s andg0jk’s are non-negative variables for j ¼ 1; 2; . . .; n ; e is the non-Archimedeaninfinitesimal.

3.2 Input and Output Orientations of Fuzzy SBM Model(FSBM Model)

The input-oriented FSBM (FSBMI) model [18] and output-oriented FSBM(FSBMO) model evaluate the FSBM input efficiency and FSBM output efficiencyof every DMU respectively, and are presented in Table 4. The FSBM input effi-ciency [18] and the FSBM output efficiency of the kth DMU are denoted by ~qk

I and~qk

O respectively.

Here ~Sþrk and ~S0þrk are fuzzy slacks in the rth fuzzy output of the kth DMU in

FSBMI and FSBMO models respectively; ~S�ik and ~S0�ik are the fuzzy slack in the ithfuzzy input of the kth DMU in FSBMI and FSBMO models respectively; kjk’s andk0jk’s are non-negative variables for j ¼ 1; 2; . . .; n:

124 J. Puri and S. P. Yadav

Tab

le3

FC

CR

Im

odel

and

FC

CR

Om

odel

Inpu

t-or

ient

edF

CC

R(F

CC

RI)

mod

elO

utpu

t-or

ient

edF

CC

R(F

CC

RO

)m

odel

~ hk I¼

min

~ h�

eXm i¼

1

~ s� ikþX

s

r¼1

~ sþ rk

!

subj

ectt

o~ h

~ x ik¼X

n j¼1

~ x ijg j

kþ

~ s� ik8i;

~ y rk¼X

n j¼1

~ y rjg j

k�

~ sþ rk8r;

e[

0;g j

k�

0;~ s� ik�

~ 0;~ sþ rk�

~ 08;

i;r;

j

~ hk 0¼

max

~ hþ

eXm i¼

1

~ s0� ikþX

s

r¼1

~ s0þ rk

!

subj

ectt

o~ x i

k¼X

n j¼1

~ x ijg0 jkþ

~ s0� ik

8i;

~ h~ y r

k¼X

n j¼1

~ y rjg0 jk�

~ s0þ rk

8r;

e[

0;g0 jk�

0;~ s0� ik�

~ 0;~ s0þ rk�

~ 08;

i;r;

j

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 125

Tab

le4

FS

BM

Im

odel

and

FS

BM

Om

odel

Inpu

t-or

ient

edF

SB

M(F

SB

MI)

mod

elO

utpu

t-or

ient

edF

SB

M(F

SB

MO

)m

odel

~ qk I¼

min

1�

1 m

Xm i¼1

~ S� i ~ x ik

subj

ectt

o~x i

k¼X

n j¼1

~x ijk j

kþ

~ S� ik8i;

~ y rk¼X

n j¼1

~ y rjk j

k�

~ Sþ rk8r;

k jk�

0;~ S� ik�

~ 0;~ Sþ rk�

~ 08

i;r;

j

1 ~ qk O

¼m

ax1þ

1 s

Xs

r¼1

~ S0þ rk

~ y rk

subj

ectt

o~ x i

k¼X

n j¼1

~ x ijk0 jkþ

~ S0� ik

8i;

~ y rk¼X

n j¼1

~ y rjk0 jk�

~ S0þ rk

8r;

k0 jk�

0;~ S0� ik�

~ 0;~ S0þ rk�

~ 08

i;r;

j

126 J. Puri and S. P. Yadav

4 Methodology for Solving FCCRI, FCCRO, FSBMI

and FSBMO Models

Various efforts have been made by eminent researchers to solve FDEA models.Many researchers have developed several approaches to handle fuzzy input andoutput data in FDEA models. One among such approaches is the a-cut approach[17, 22] which has been widely used in various studies of FDEA. The main idea ina-cut approach is to transform a FDEA model into a pair of parametric crispprograms in order to find the lower and upper bounds of the a-cuts of the fuzzyefficiencies. Kao and Liu [17] developed a procedure to solve FDEA model bytransforming it into a family of crisp DEA models using a-cuts and Zadeh’sextension principle [21]. The same procedure is used in Puri and Yadav [18] toconvert FCCRI and FSBMI models into crisp linear DEA models. In this chapter,the FCCRI, FCCRO, FSBMI and FSBMO models are converted into a family ofcrisp DEA models using the same procedure which is as follows:

Let Sð~xijÞ and Sð~yrjÞ be the supports of the ith fuzzy input ~xij ði ¼ 1; 2; . . .;mÞand rth fuzzy output ~yrj ðr ¼ 1; 2; . . .; sÞ of the jth DMU (j = 1,…, n) respectivelywhich are given by

Sð~xijÞ ¼ fxijj l~xijðxijÞ [ 0g and Sð~yrjÞ ¼ fyrjj l~yrj

ðyrjÞ [ 0g ð1Þ

The a-cuts of ~xij and ~yrj are denoted by ð~xijÞa and ð~yrjÞa respectively, and aredefined as

ð~xijÞa ¼ f xij 2 Sð~xijÞ j l~xijðxijÞ� a g ¼ ½ ðxijÞLa ; ðxijÞUa � 8 i; j ð2aÞ

¼ minxij

f xij 2 Sð~xijÞ j l~xijðxijÞ� a g ; max

xij

f xij 2 Sð~xijÞ j l~xijðxijÞ� a g

� �

8 i; j

ð2bÞ

and ð~yrjÞa ¼ f yrj 2 Sð~yrjÞ j l~yrjðyrjÞ� a g ¼ ½ ðyrjÞLa ; ðyrjÞUa � 8 r; j ð3aÞ

¼ minyrj

f yrj 2 Sð~yrjÞ j l~yrjðyrjÞ� a g ; max

yrj

f yrj 2 Sð~yrjÞ j l~yrjðyrjÞ� a g

� �

8 r; j

ð3bÞ

where 0\a� 1:Further, the FCCRI, FCCRO, FSBMI and FSBMO models can easily be trans-

formed into crisp models by using the above a-cuts. Since each input and outputare taken in terms of FNs, the efficiency scores should also be FNs. Let the fuzzy

efficiencies be represented by ~hkI ;

~hkO; ~qk

I and ~qkO with the membership functions

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 127

l~hkI; l~hk

O; l~qk

Iand l~qk

Orespectively. Let Sð~hk

I Þ; Sð~hkOÞ; Sð~qk

I Þ and Sð~qkOÞ be the

supports of the fuzzy efficiency scores ~hkI ;

~hkO; ~qk

I and ~qkO of the kth DMU

respectively and are defined by

Sð~hkI Þ ¼ fhk

I j l~hkIðhk

I Þ [ 0g; Sð~hkOÞ ¼ fhk

Oj l~hkOðhk

OÞ [ 0g; ð4aÞ

Sð~qkI Þ ¼ fqk

I j l~qkIðqk

I Þ [ 0g and Sð~qkOÞ ¼ fqk

Oj l~qkOðqk

OÞ [ 0g ð4bÞ

The a-cuts ða 2 ð0; 1�Þof ~hkI ;

~hkO; ~qk

I and ~qkO are respectively defined as

ð~hkI Þa ¼ fh

kI 2 Sð~hk

I Þ j l~hkIðhk

I Þ� a g ¼ ½ ðhkI Þ

La ; ðh

kI Þ

Ua � 8 i; j ð5aÞ

¼ minhk

I

fhkI 2 Sð~hk

I Þ j l~hkIðhk

I Þ� ag; maxhk

I

f hkI 2 Sð~hk

I Þ j l~hkIðhk

I Þ� ag" #

8 i; j

ð5bÞ

ð~hkOÞa ¼ fh

kO 2 Sð~hk

OÞ j l~hkOðhk

OÞ� a g ¼ ½ ðhkOÞ

La ; ðh

kOÞ

Ua � 8 i; j ð6aÞ

¼ minhk

O

fhkO 2 Sð~hk

OÞ j l~hkOðhk

O� ag; maxhk

O

fhkO 2 Sð~hk

OÞ j l~hkOðhk

O� ag" #

8 i; j

ð6bÞ

ð~qkI Þa ¼ f qk

I 2 Sð~qkI Þ j l~qk

Iðqk

I Þ� a g ¼ ½ ðqkI Þ

La ; ðqk

I ÞUa � 8 i; j ð7aÞ

¼ minqk

I

fqkI 2 Sð~qk

I Þ j l~qkIðqk

I Þ� ag; maxqk

I

fqkI 2 Sð~qk

I Þ j l~qkIðqk

I Þ� a g" #

8 i; j

ð7bÞ

and ð~qkOÞa ¼ f qk

O 2 Sð~qkOÞ j l~qk

Oðqk

OÞ� a g ¼ ½ ðqkOÞ

La ; ðqk

OÞUa � 8 i; j ð8aÞ

¼ minqk

O

f qkO 2 Sð~qk

OÞ j l~qkOðqk

O� a g ; maxqk

O

f qkO 2 Sð~qk

OÞ j l~qkOðqk

O� a g" #

8 i; j

ð8bÞ

128 J. Puri and S. P. Yadav

ðhkI Þ

La ¼ min

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

min h� ePm

i¼1s�ik þ

Ps

r¼1sþrk

� �

subject to h xik ¼Pn

j¼1

xij gjk þ s�ik 8i;

yrk ¼Pn

j¼1

yrj gjk � sþrk

8r;

e [ 0; gjk � 0;s�ik� 0; sþ

rk� 0 8 i; r; j

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð9aÞ

ðhkI Þ

Ua ¼ max

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

min h� ePm

i¼1s�ik þ

Ps

r¼1sþrk

� �

subject to h xik ¼Pn

j¼1

xij gjk þ s�ik 8i;

yrk ¼Pn

j¼1

yrj gjk � sþrk

8r;

e[ 0; gjk � 0;s�ik� 0; sþ

rk� 0 8i; r; j

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

; ð9bÞ

ðhkOÞ

La ¼ min

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

max hþ ePm

i¼1s0�ik þ

Ps

r¼1s0þrk

� �

subject to xik ¼Pn

j¼1

xij g0jkþ s0�ik 8i;

hyrk ¼Pn

j¼1

yrj g0jk� s0þ

rk8r;

e[ 0; g0jk� 0; s0�

ik� 0; s0þ

rk� 0 8 i ; r; j;

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð10aÞ

ðhkOÞ

Ua ¼ max

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

max hþ ePm

i¼1s0�ik þ

Ps

r¼1s0þrk

� �

subject to xik ¼Pn

j¼1

xij g0jkþ s0�ik 8i;

hyrk ¼Pn

j¼1

yrj g0jk� s0þ

rk8r;

e [ 0; g0jk� 0; s0�

ik� 0; s0þ

rk� 0 8 i ; r; j

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

; ð10bÞ

ðqkI Þ

La ¼ min

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

min 1� 1m

Pm

i¼1

S�ikxik

subject to xik ¼Pn

j¼1

xij kjk þ S�ik 8i;

yrk ¼Pn

j¼1

yrj kjk � Sþrk

8r;

kjk � 0;S�ik� 0; Sþ

rk� 0 8 i; r; j

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð11aÞ

ðqkI Þ

Ua ¼ max

ðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

min 1� 1m

Pm

i¼1

S�ikxik

subject to xik ¼Pn

j¼1

xij kjk þ S�ik 8i;

yrk ¼Pn

j¼1

yrj kjk � Sþrk

8r;

kjk � 0;S�ik� 0; Sþ

rk� 0 8 i; r; j

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð11bÞ

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 129

and

1

ðqkOÞ

La

¼ minðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

max 1þ 1s

Ps

r¼1

S0þrkyrk

subject to xik ¼Pn

j¼1

xij k0jk þ S0�ik 8i;

yrk ¼Pn

j¼1

yrj k0jk� S0þ

rk8r;

k0jk� 0; S0�

ik� 0; S0þ

rk� 0 8 i ; r; j;

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð12aÞ

1

ðqkOÞ

Ua

¼ maxðxijÞLa � xij � ðxijÞUaðyrjÞLa � yrj � ðyrjÞUa

8 i; r; j

max 1þ 1s

Ps

r¼1

S0þrkyrk

subject to xik ¼Pn

j¼1

xij k0jk þ S0�ik 8i;

yrk ¼Pn

j¼1

yrj k0jk� S0þ

rk8r;

k0jk� 0; S0�

ik� 0; S0þ

rk� 0 8 i ; r; j;

8>>>>>>><

>>>>>>>:

9>>>>>>>=

>>>>>>>;

ð12bÞ

Further, from (9a), (9b), (11a), (11b), (12a) and (12b), we can find the ‘mini-mum efficiency’ of the targeted DMU by taking lower bound outputs for thetargeted DMU and upper bound outputs for other DMUs, and upper bound inputsfor the targeted DMU and lower bound inputs for other DMUs. Similarly, we canfind the ‘maximum efficiency’ of the targeted DMU by taking upper bound outputsfor the targeted DMU and lower bound outputs for other DMUs, and lower boundinputs for the targeted DMU and upper bound inputs for other DMUs.

In crisp DEA, hk�O ¼ 1=hk�

I holds [2]. However, in FDEA

~hk�O ¼

1~hk�

I

, ð~hk�O Þa ¼

1

ð~hk�I Þa, ðhk�

O ÞLa ; ðh

k�O Þ

Ua

� �¼ 1

ðhk�I Þ

La ; ðh

k�I Þ

Ua

� �,

ðhk�O Þ

La ; ðh

k�O Þ

Ua

� �¼ 1

ðhk�I Þ

Ua

;1

ðhk�I Þ

La

" #

, ðhk�O Þ

La ¼

1

ðhk�I Þ

Ua

and ðhk�O Þ

Ua ¼

1

ðhk�I Þ

La

:

ð13Þ

Therefore, for solving output-oriented models (10a) and (10b), we will use thesimilar conditions which were used to solve input-oriented models (9b) and (9a)respectively. It means from (10a) and (10b), we can find the ‘minimum efficiency’of the targeted DMU by taking lower bound inputs for the targeted DMU andupper bound inputs for other DMUs, and upper bound outputs for the targetedDMU and lower bound outputs for other DMUs. Similarly, we can find the‘maximum efficiency’ of the targeted DMU by taking upper bound inputs for thetargeted DMU and lower bound inputs for other DMUs, and lower bound outputsfor the targeted DMU and upper bound outputs for other DMUs. Thus, models(9a), (9b), (10a), (10b), (11a), (11b), (12a) and (12b) reduce to the following crispmodels:

130 J. Puri and S. P. Yadav

ðhkI Þ

La ¼ min h� e

Xm

i¼1

ðs�ikÞU þ

Xs

r¼1

ðsþrkÞL

!

subject to h ðxikÞUa ¼Xn

j¼1; j 6¼k

ðxijÞLa gjk þ ðxikÞUa gjk þ ðs�ikÞU 8i;

ðyrkÞLa ¼Xn

j¼1; j 6¼k

ðyrjÞUa gjk þ ðyrkÞLa gjk � ðsþrkÞL 8r;

e [ 0; gjk� 0; ðs�ikÞU � 0; ðsþrkÞ

L� 0 8i; r; j

ð14aÞ

ðhkI Þ

Ua ¼ min h� e

Xm

i¼1

ðs�ikÞL þ

Xs

r¼1

ðsþrkÞU

!

subject to h ðxikÞLa ¼Xn

j¼1; j6¼k

ðxijÞUa gjk þ ðxikÞLa gjk þ ðs�ikÞL 8i;

ðyrkÞUa ¼Xn

j¼1; j 6¼k

ðyrjÞLa gjk þ ðyrkÞUa gjk � ðsþrkÞU 8r;

e [ 0; gjk� 0; ðs�ikÞL� 0; ðsþrkÞ

U � 0 8i; r; j:

ð14bÞ

ðhkOÞ

La ¼ min hþ e

Xm

i¼1

ðs0�ik ÞL þ

Xs

r¼1

ðs0þrk ÞU

!

subject to h ðxikÞLa ¼Xn

j¼1; j6¼k

ðxijÞUa g0jk þ ðxikÞLa g0jk þ ðs0�ik ÞL 8i;

ðyrkÞUa ¼Xn

j¼1; j 6¼k

ðyrjÞLa g0jk þ ðyrkÞUa g0jk � ðs0þrk ÞU 8r;

e [ 0; g0jk � 0; ðs0�ik ÞL� 0; ðs0þrk Þ

U � 0 8i; r; j

ð15aÞ

ðhkOÞ

Ua ¼ min hþ e

Xm

i¼1

ðs0�ik ÞU þ

Xs

r¼1

ðs0þrk ÞL

!

subject to h ðxikÞUa ¼Xn

j¼1; j 6¼k

ðxijÞLa g0jk þ ðxikÞUa g0jk þ ðs0�ik ÞU 8i;

ðyrkÞLa ¼Xn

j¼1; j 6¼k

ðyrjÞUa g0jk þ ðyrkÞLa g0jk � ðs0þrk ÞL 8r;

e [ 0; g0jk� 0; ðs0�ik ÞU � 0; ðs0þrk Þ

L� 0 8i; r; j:

ð15bÞ

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 131

ðqkI Þ

La ¼ min 1� 1

m

Xm

i¼1

ðS�ikÞU

ðxikÞUa

subject to ðxikÞUa ¼Xn

j¼1; j 6¼k

ðxijÞLa kjk þ ðxikÞUa kjk þ ðS�ikÞU 8i;

ðyrkÞLa ¼Xn

j¼1; j6¼k

ðyrjÞUa kjk þ ðyrkÞLa kjk � ðSþrkÞL 8r;

kjk� 0; ðS�ikÞU � 0; ðSþrkÞ

L� 0 8i; r; j

ð16aÞ

ðqkI Þ

Ua ¼ min 1� 1

m

Xm

i¼1

ðS�ikÞL

ðxikÞLa

subject to ðxikÞLa ¼Xn

j¼1; j6¼k

ðxijÞUa kjk þ ðxikÞLa kjk þ ðS�ikÞL 8i;

ðyrkÞU ¼Xn

j¼1; j 6¼k

ðyrjÞLa kjk þ ðyrkÞUa kjk � ðSþrkÞU 8r;

kjk � 0; ðS�ikÞL� 0; ðSþrkÞ

U � 0 8i; r; j:

ð16bÞ

and

1

ðqkOÞ

La

¼ max 1þ 1s

Xs

r¼1

ðS0þrk ÞL

ðyrkÞLa

subject to ðxikÞUa ¼Xn

j¼1; j 6¼k

ðxijÞLa k0jk þ ðxikÞUa kjk þ ðS0�ik ÞU 8i;

ðyrkÞLa ¼Xn

j¼1; j 6¼k

ðyrjÞUa k0jk þ ðyrkÞLa kjk � ðS0þrk ÞL 8r;

k0jk � 0; ðS0�ik ÞU � 0; ðS0þrk Þ

L� 0 8i; r; j:

ð17aÞ

1

ðqkOÞ

Ua

¼ max 1þ 1s

Xs

r¼1

ðS0þrk ÞL

ðyrkÞUa

subject to ðxikÞLa ¼Xn

j¼1; j6¼k

ðxijÞUa k0jk þ ðxikÞLa kjk þ ðS0�ik ÞL 8i;

ðyrkÞU ¼Xn

j¼1; j 6¼k

ðyrjÞLa k0jk þ ðyrkÞUa kjk � ðS0þrk ÞU 8r;

ð17bÞ

k0jk � 0; ðS0�ik ÞL� 0; ðS0þrk Þ

U � 0 8i; r; j:

132 J. Puri and S. P. Yadav

The exact form of the membership functions of the fuzzy efficiencies~hk

I ;~hk

O; ~qkI and ~qk

O are not known explicitly. However, the sets of intervals

ðhkI Þ

La ; ðh

kI Þ

Ua

� �a 2 ð0; 1�; k ¼ 1; 2; . . .; nj

� ; ðhk

OÞLa ; ðh

kOÞ

Ua

� �a 2 ð0; 1�; k ¼ 1;j

�

2; . . .; ng; ðqkI Þ

La ; ðqk

I ÞUa

� �a 2 ð0; 1�; k ¼ 1; 2; . . .; nj

� and ðqk

OÞLa ; ðqk

OÞUa

� �a 2j

�

ð0; 1�; k ¼ 1; 2; . . .; ng can reveal the shape of the membership functions l~hkI; l~hk

O;

l~qkI

and l~qkOrespectively. Since fuzzy inputs (~xij; i ¼ 1; 2; . . .; m) and fuzzy out-

puts (~yrj; r ¼ 1; 2; . . .; s) are TFNs, the membership functions l~hkI; l~hk

O; l~qk

Iand

l~qkOcan be approximated by the triangular membership functions whose a-cuts are

represented by the sets of intervals ðhkI Þ

La ; ðh

kI Þ

Ua

� �a 2 ð0; 1�; k ¼ 1;j

�

2; . . .; ng; ðhkOÞ

La ; ðh

kOÞ

Ua

� �a 2 ð0; 1�; k ¼ 1; 2; . . .; nj

� ; ðqk

I ÞLa ; ðqk

I ÞUa

� �a 2 ð0; 1�j

�

; k ¼ 1; 2; . . .; ng and ðqkOÞ

La ; ðqk

OÞUa

� �a 2 ð0; 1�; k ¼ 1; 2; . . .; nj

� respectively.

The fuzzy efficiencies ~hkI ;

~hkO; ~qk

I and ~qkO can be approximated by TFNs with

supports given by

Sð~hkI Þ ¼ fhk

I j l~hkIðhk

I Þ [ 0g ¼ ðhkI Þ

L0 ; ðh

kI Þ

U0

�;

Sð~hkOÞ ¼ fh

kOjl~hk

Oðhk

OÞ [ 0g ¼ ðhkOÞ

L0 ; ðh

kOÞ

U0

�;

Sð~qkI Þ ¼ fqk

I j l~qkIðqk

I Þ [ 0g ¼ ðqkI Þ

L0 ; ðqk

I ÞU0

�and Sð~qk

OÞ ¼ fqkOj l~qk

Oðqk

OÞ[ 0g ¼ ðqk

OÞL0 ; ðqk

OÞU0

�:

The FCCR input efficiency ~hkI of the kth DMU is a fuzzy number and it can be

expressed by its a-cuts ð~hkI Þa; a 2 ð0; 1� as ~hk

I ¼ [a að~hkI Þa; k ¼ 1; 2; . . .; n; where

ð~hkI Þa ¼ ðh

kI Þ

La ; ðh

kI Þ

Ua

� �; a 2 ð0; 1�; k ¼ 1; 2; . . .; n; and each ðhk

I ÞLa and ðhk

I ÞUa are

obtained from the optimal objective function values of (14a) and (14b)respectively.

We know that for any a1; a2 2 ð0; 1� and a1� a2;ð~hkI Þa2� ð~hk

I Þa1: On general-

izing, we get for 0\a1� a2� . . .� an ¼ 1 that ð~hkI Þan¼1 � ð~h

kI Þan�1

� . . . �ð~hk

I Þa2� ð~hk

I Þa1� ð~hk

I Þ0: Graphically, intervals ð~hkI Þa; a 2 ½0; 1� are represented in

Fig. 1.

Thus, the set of intervals ðhkI Þ

La ; ðh

kI Þ

Ua

� �a 2 ½0; 1�; k ¼ 1; 2; . . .; nj

� can reveal

the shape of the membership function l~hkI

of ~hkI : Then ~hk

I by using a-cuts

ð~hkI Þa; a 2 ð0; 1� and Sð~hk

I Þ can be approximated by a TFN ððhkI Þ

L0 ; ðh

kI Þ1; ðh

kI Þ

U0 Þ;

i.e., ~hkI ððh

kI Þ

L0 ; ðh

kI Þ1; ðh

kI Þ

U0 Þ:

In a similar way, the FCCR output, FSBM input and FSBM output efficienciescan also be approximated by TFNs.

The FCCR output efficiency ~hkO by using a�cuts ð~hk

OÞa; a 2 ð0; 1� and Sð~hkOÞ

can be approximated by a TFN ððhkOÞ

L0 ; ðh

kOÞ1; ðh

kOÞ

U0 Þ; i.e.,

~hkO ððhk

OÞL0 ; ðh

kOÞ1; ðh

kOÞ

U0 Þ:

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 133

The FSBM input efficiency ~qkI by using a�cuts ð~qk

I Þa; a 2 ð0; 1� and Sð~qkI Þ can

be approximated by a TFN ððqkI Þ

L0 ; ðqk

I Þ1; ðqkI Þ

U0 Þ; i.e., ~qk

I ððqkI Þ

L0 ; ðqk

I Þ1; ðqkI Þ

U0 Þ:

The FSBM output efficiency ~qkO by using a�cuts ð~qk

OÞa; a 2 ð0; 1� and Sð~qkOÞ

can be approximated by a TFN ððqkOÞ

L0 ; ðqk

OÞ1; ðqkOÞ

U0 Þ; i.e., ~qk

O ððqkOÞ

L0 ;

ðqkOÞ1; ðqk

OÞU0 Þ:

5 Fuzzy Input and Output Mix-Efficiencies

The fuzzy input mix-efficiency (FIME) [18] of the kth DMU, denoted by ~wkI ; is

defined as the ratio of FSBM input efficiency ~qkI of the kth DMU to the FCCR input

efficiency ~hkI of the kth DMU. Let ~qk

I ¼ ðqk1; qk

2; qk3Þ and ~hk

I ¼ ðhk1; hk

2; hk3Þ be the

fuzzy efficiencies approximated as TFNs. Then by using arithmetic operations on

TFNs, ~wkI can be defined as

~wkI ¼

~qkI

~hkI

; ~hkI 6¼ ~0 ð18Þ

¼ ðqk1; qk

2; qk3Þ

ðhk1; hk

2; hk3Þ¼ ðqk

1; qk2; qk

3Þ ðhk1; hk

2; hk3Þ�1

¼ ðqk1; qk

2; qk3Þ

1

hk3

;1

hk2

;1

hk1

!

; hk1 [ 0

4( )k L

I αθ

2( )k L

I αθ

( )k

I αθ

1( ) ( ) ( )n n

k L k U kI I Iα αθ θ θ= =

α

1nα =

1nα −

4α

3α

2α

1α

0

1( )

n

k LI αθ

− 1( )

n

k UI αθ

−

4( )k U

I αθ

3( )k L

I αθ 3( )k U

I αθ

2( )k U

I αθ

1( )k L

I αθ1

( )k UI αθ

0( )k LIθ 0( )k U

Iθ

Fig. 1 Shapes of themembership functions of ~hk

I

134 J. Puri and S. P. Yadav

qk1

hk3

;qk

2

hk2

;qk

3

hk1

!

; hk1 [ 0: ð19Þ

The fuzzy output mix-efficiency (FOME) of the kth DMU, denoted by ~wkO; is

defined as the ratio of FSBM output efficiency ~qkO of the kth DMU to the FCCR

output efficiency ~hkO of the kth DMU. Let q� k

O ¼ ðq0k1 ; q0k2 ; q0k3 Þ and ~hkO ¼

ðh0k1 ; h0k2 ; h0k3 Þ be the fuzzy efficiencies approximated as TFNs. Then by using

arithmetic operations on TFNs, ~wkO can be defined as

~wkO ¼

~qkO

~hkO

; ~hkO 6¼ ~0 ð20Þ

¼ ðq0k1 ; q0k2 ; q0k3 Þðh0k1 ; h0k2 ; h0k3 Þ

¼ ðq0k1 ; q0k2 ; q0k3 Þ ðh0k1 ; h0k2 ; h0k3 Þ�1

¼ ðq0k1 ; q0k2 ; q0k3 Þ 1

h0k3;

1

h0k2;

1

h0k1

!

; h0k1 [ 0

q0k1h0k3;q0k2h0k2;q0k3h0k1

!

; h0k1 [ 0: ð21Þ

6 Numerical Illustration

To ensure the validity of the proposed methodology, we consider the performanceevaluation problem of 18 DMUs in terms of 2 inputs and 2 outputs. Let all inputsand first output are in terms of TFNs, and second output is exact. The input andoutput data is shown in Table 5.

The a�cuts ð~hkI Þa; ð~h

kOÞa; ð~qk

I Þaand ð~qkOÞa of ~hk

I ;~hk

O; ~qkI and ~qk

O respectively for18 DMUs are evaluated by using models from (14a) to (17b) at different values ofa 2 ð0; 1� and are shown in Tables 6, 7, 8 and 9.

Further, the graphical representations of the fuzzy efficiencies ~hkI ;

~hkO; ~qk

I and~qk

O for 18 DMUs are shown in Figs. 2, 3, 4 and 5. The figures reveal that the shape

of the membership functions of ~hkI ;

~hkO; ~qk

I and ~qkO for every DMU can be

approximated by a TFN as shown in Tables 10 and 11. Furthermore, the fuzzy

mix-efficiencies ~wkI and ~wk

O are evaluated by using Eqs. (19) and (21) respectively.

The values of ~wkI and ~wk

O; shown in Tables 10 and 11 respectively, indicate that~0\~wk

O� ~1 ; while ~wkI may or may not lie between ~0 and ~1 because the right part of

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 135

~wkI may take the value greater than or equal to 1 due to the division operation of

TFNs. Here ~0 ¼ ð0; 0; 0Þ and ~1 ¼ ð1; 1; 1Þ:In FDEA, the relationship between ~hk

I and ~hkO can be seen from Figs. 2 and 3.

The figures reveal that the shapes of the membership functions of ~hkO in Fig. 3 are

the mirror images of the shapes of the membership functions of ~hkI in Fig. 2

respectively. From the relations derived in (13), we conclude that the shapes of the

membership functions of ~hkO are the mirror images of the shapes of the mem-

bership functions of ~hkI :

Table 5 Fuzzy input and fuzzy output data for 18 DMUs

DMUS Input 1 (I1) Input 2 (I2) Output 1 (O1) Output 2 (O2)

1 (250, 253 257) (4, 5, 6) (183, 187, 190) 902 (263, 268, 270) (8, 10, 12) (190, 194, 197) 1303 (253, 259, 262) (3, 3, 3) (220, 220, 220) 2004 (180, 180, 180) (4, 6, 7) (158, 160, 163) 1005 (254, 257, 260) (3, 4, 6) (202, 204, 205) 1736 (246, 248, 249) (1, 2, 3) (190, 192, 193) 1707 (270, 272, 272) (5, 8, 10) (190, 194, 194) 608 (327, 330, 331) (11, 11, 11) (195, 195, 195) 1459 (324, 327, 327) (8, 9, 10) (200, 200, 200) 15010 (327, 330, 331) (4, 7, 9) (170, 171, 172) 9011 (320, 321, 322) (13, 16, 19) (172, 174, 176) 10012 (325, 329, 332) (12, 14, 14) (209, 209, 209) 20013 (280, 281, 281) (13, 15, 17) (160, 165, 167) 16314 (306, 309, 312) (10, 13, 14) (195, 199, 201) 17015 (290, 291, 292) (10, 12, 12.5) (184, 188, 189) 18516 (330, 334, 337) (13, 17, 19) (165, 168, 170) 8517 (244, 249, 250) (1, 1, 1) (173, 177, 179) 13018 (213, 216, 219) (17, 18, 19) (165, 167, 169) 160

136 J. Puri and S. P. Yadav

Tab

le6

a�cu

tsð~ h

k IÞ a¼ðh

k IÞL a;ð

hk IÞU a

��

ofF

CC

Rin

put

effi

cien

cy~ hk I

for

diff

eren

tva

lues

ofa2ð0;1�

DM

Uðh

k IÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

7841

0.78

920.

7968

0.80

390.

8108

0.81

750.

8241

0.83

050.

8368

0.84

310.

8493

20.

7843

0.78

780.

7914

0.79

500.

7986

0.80

220.

8058

0.80

940.

8130

0.81

660.

8202

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

8869

0.89

070.

8946

0.89

840.

9023

0.90

610.

9100

0.91

390.

9178

0.92

170.

9265

60.

8767

0.88

380.

8941

0.90

590.

9234

0.94

160.

9604

0.98

011.

0000

1.00

001.

0000

70.

7654

0.76

850.

7715

0.77

460.

7777

0.78

070.

7838

0.78

690.

7904

0.79

620.

8018

80.

6621

0.66

360.

6651

0.66

670.

6682

0.66

980.

6713

0.67

280.

6744

0.67

590.

6775

90.

6880

0.68

940.

6908

0.69

220.

6936

0.69

500.

6964

0.69

780.

6992

0.70

060.

7020

100.

5656

0.56

720.

5688

0.57

050.

5743

0.57

780.

5812

0.58

440.

5875

0.59

050.

5934

110.

5886

0.59

060.

5926

0.59

450.

5965

0.59

850.

6005

0.60

250.

6046

0.60

660.

6086

120.

7602

0.76

270.

7652

0.76

770.

7702

0.77

270.

7752

0.77

770.

7803

0.78

280.

7853

130.

7309

0.73

270.

7345

0.73

630.

7381

0.73

990.

7417

0.74

350.

7453

0.74

710.

7489

140.

7144

0.71

810.

7217

0.72

530.

7290

0.73

270.

7363

0.74

000.

7437

0.74

740.

7511

150.

7988

0.80

100.

8032

0.80

540.

8077

0.80

990.

8121

0.81

430.

8166

0.81

880.

8210

160.

5384

0.54

090.

5434

0.54

580.

5483

0.55

080.

5533

0.55

590.

5584

0.56

100.

5635

170.

8938

0.98

391.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

9216

0.92

510.

9286

0.93

210.

9356

0.93

910.

9426

0.94

620.

9497

0.95

330.

9569

(con

tinu

ed)

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 137

Tab

le6

(con

tinu

ed)

DM

Uðh

k IÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

DM

Uðh

k IÞU a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

8910

0.88

690.

8827

0.87

860.

8745

0.87

030.

8661

0.86

200.

8577

0.85

350.

8493

20.

8638

0.85

850.

8532

0.84

790.

8425

0.83

850.

8348

0.83

110.

8275

0.82

380.

8202

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

9593

0.95

610.

9529

0.94

970.

9464

0.94

310.

9398

0.93

650.

9332

0.92

980.

9265

61.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

70.

8356

0.83

250.

8293

0.82

610.

8229

0.81

950.

8161

0.81

270.

8091

0.80

550.

8018

80.

6916

0.69

010.

6887

0.68

730.

6859

0.68

450.

6831

0.68

170.

6803

0.67

890.

6775

90.

7189

0.71

690.

7148

0.71

270.

7109

0.70

940.

7079

0.70

640.

7050

0.70

350.

7020

100.

6191

0.61

670.

6142

0.61

170.

6092

0.60

670.

6041

0.60

150.

5988

0.59

610.

5934

110.

6254

0.62

370.

6220

0.62

030.

6186

0.61

700.

6153

0.61

360.

6119

0.61

030.

6086

120.

8044

0.80

250.

8005

0.79

860.

7967

0.79

480.

7929

0.79

100.

7891

0.78

720.

7853

130.

7606

0.75

950.

7583

0.75

710.

7559

0.75

470.

7536

0.75

240.

7512

0.75

010.

7489

140.

7741

0.77

180.

7695

0.76

710.

7648

0.76

250.

7602

0.75

790.

7556

0.75

330.

7511

150.

8337

0.83

240.

8311

0.82

990.

8286

0.82

730.

8261

0.82

480.

8235

0.82

230.

8210

160.

5846

0.58

250.

5803

0.57

820.

5761

0.57

400.

5719

0.56

980.

5677

0.56

560.

5635

171.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

9819

0.97

940.

9768

0.97

430.

9718

0.96

930.

9668

0.96

430.

9618

0.95

930.

9569

138 J. Puri and S. P. Yadav

Tab

le7

a�cu

tsð~ h

k OÞ a¼ðh

k OÞL a;ð

hk OÞU a

�� of

FC

CR

outp

utef

fici

ency

~ hk Ofo

rdi

ffer

ent

valu

esof

a2ð0;1�

DM

Uðh

k OÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

11.

1214

1.12

651.

1318

1.13

711.

1424

1.14

791.

1534

1.15

901.

1646

1.17

041.

1762

21.

1574

1.16

461.

1719

1.17

931.

1869

1.19

261.

1979

1.20

321.

2085

1.21

381.

2192

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

51.

0423

1.04

581.

0494

1.05

301.

0566

1.06

031.

0640

1.06

771.

0716

1.07

541.

0794

61.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

71.

1944

1.19

881.

2034

1.20

811.

2129

1.21

791.

2230

1.22

821.

2336

1.23

931.

2451

81.

4459

1.44

891.

4519

1.45

491.

4579

1.46

091.

4639

1.46

691.

4699

1.47

291.

4760

91.

3907

1.39

481.

3989

1.40

311.

4067

1.40

971.

4126

1.41

551.

4185

1.42

141.

4243

101.

6088

1.61

531.

6219

1.62

861.

6355

1.64

251.

6497

1.65

701.

6645

1.67

231.

6802

111.

5978

1.60

221.

6065

1.61

091.

6153

1.61

971.

6241

1.62

851.

6329

1.63

741.

6418

121.

2427

1.24

561.

2486

1.25

161.

2546

1.25

761.

2606

1.26

361.

2666

1.26

971.

2727

131.

3139

1.31

591.

3179

1.32

001.

3220

1.32

401.

3261

1.32

811.

3302

1.33

221.

3343

141.

2916

1.29

551.

2995

1.30

341.

3073

1.31

121.

3152

1.31

921.

3232

1.32

721.

3312

151.

1990

1.20

081.

2027

1.20

451.

2063

1.20

821.

2100

1.21

181.

2137

1.21

551.

2174

161.

7078

1.71

411.

7203

1.72

661.

7330

1.73

931.

7457

1.75

201.

7584

1.76

491.

7713

171.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

181.

0184

1.02

101.

0237

1.02

631.

0289

1.03

161.

0343

1.03

691.

0396

1.04

231.

0450

(con

tinu

ed)

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 139

Tab

le7

(con

tinu

ed)

DM

Uðh

k OÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

DM

Uðh

k OÞU a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

11.

2746

1.26

621.

2541

1.24

281.

2322

1.22

201.

2122

1.20

281.

1937

1.18

491.

1762

21.

2749

1.26

911.

2634

1.25

781.

2521

1.24

651.

2410

1.23

551.

2300

1.22

461.

2192

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

51.

1275

1.12

261.

1178

1.11

301.

1083

1.10

361.

0989

1.09

421.

0896

1.08

501.

0794

61.

1406

1.13

141.

1184

1.10

381.

0829

1.06

201.

0412

1.02

031.

0000

1.00

001.

0000

71.

3041

1.29

901.

2939

1.28

881.

2837

1.27

871.

2737

1.26

871.

2631

1.25

391.

2451

81.

5101

1.50

671.

5032

1.49

971.

4963

1.49

291.

4895

1.48

611.

4827

1.47

931.

4760

91.

4532

1.45

031.

4474

1.44

451.

4416

1.43

871.

4358

1.43

291.

4300

1.42

721.

4243

101.

7659

1.76

101.

7562

1.75

071.

7386

1.72

741.

7169

1.70

711.

6977

1.68

881.

6802

111.

6974

1.69

181.

6862

1.68

061.

6750

1.66

941.

6639

1.65

831.

6528

1.64

731.

6418

121.

3147

1.31

041.

3061

1.30

191.

2977

1.29

341.

2893

1.28

511.

2809

1.27

681.

2727

131.

3668

1.36

351.

3602

1.35

691.

3536

1.35

041.

3471

1.34

391.

3407

1.33

751.

3343

141.

3993

1.39

221.

3852

1.37

831.

3714

1.36

461.

3578

1.35

101.

3444

1.33

771.

3312

151.

2511

1.24

761.

2442

1.24

081.

2374

1.23

401.

2307

1.22

731.

2240

1.22

071.

2174

161.

8537

1.84

531.

8369

1.82

861.

8203

1.81

211.

8038

1.79

571.

7875

1.77

941.

7713

171.

1185

1.01

621.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

181.

0848

1.08

081.

0767

1.07

271.

0687

1.06

471.

0607

1.05

671.

0528

1.04

891.

0450

140 J. Puri and S. P. Yadav

Tab

le8

a�cu

tsð~ q

k IÞ a¼ðq

k IÞL a;ð

qk IÞU a

�� of

FS

BM

inpu

tef

fici

ency

~ qk Ifo

rdi

ffer

ent

valu

esof

a2ð0;1�

DM

Uðq

k IÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

5328

0.54

470.

5570

0.56

980.

5829

0.59

650.

6036

0.61

210.

6222

0.63

170.

6407

20.

4895

0.49

640.

5034

0.51

070.

5182

0.52

590.

5339

0.54

210.

5484

0.55

330.

5584

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

5824

0.59

630.

6111

0.62

740.

6472

0.66

780.

6895

0.71

220.

7362

0.75

680.

7750

60.

8569

0.87

110.

8858

0.90

150.

9184

0.93

670.

9565

0.97

801.

0000

1.00

001.

0000

70.

4944

0.50

210.

5101

0.51

840.

5270

0.53

600.

5453

0.55

510.

5613

0.56

570.

5705

80.

4214

0.42

660.

4318

0.43

710.

4423

0.44

760.

4529

0.45

810.

4628

0.46

440.

4659

90.

4416

0.44

790.

4543

0.46

090.

4676

0.47

440.

4814

0.48

690.

4896

0.49

220.

4946

100.

3763

0.38

320.

3904

0.39

790.

4058

0.41

400.

4202

0.42

330.

4266

0.43

000.

4335

110.

3639

0.36

760.

3715

0.37

550.

3781

0.38

050.

3829

0.38

540.

3880

0.39

060.

3932

120.

4779

0.48

280.

4878

0.49

190.

4932

0.49

440.

4957

0.49

700.

4982

0.49

950.

5008

130.

4388

0.44

050.

4423

0.44

400.

4458

0.44

770.

4495

0.45

140.

4532

0.45

520.

4571

140.

4344

0.44

010.

4459

0.45

180.

4578

0.46

390.

4701

0.47

450.

4775

0.48

050.

4835

150.

5019

0.50

700.

5121

0.51

630.

5179

0.51

940.

5210

0.52

260.

5241

0.52

570.

5273

160.

3345

0.33

840.

3423

0.34

620.

3496

0.35

180.

3541

0.35

640.

3587

0.36

110.

3635

170.

8892

0.93

721.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

5253

0.52

730.

5294

0.53

150.

5336

0.53

570.

5378

0.53

990.

5420

0.54

420.

5463

(con

tinu

ed)

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 141

Tab

le8

(con

tinu

ed)

DM

Uðq

k IÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

DM

Uðq

k IÞU a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

7322

0.72

150.

7112

0.70

130.

6917

0.68

250.

6736

0.66

500.

6566

0.64

850.

6407

20.

6131

0.60

730.

6014

0.59

540.

5897

0.58

410.

5786

0.57

340.

5682

0.56

320.

5584

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

9151

0.89

740.

8807

0.86

490.

8500

0.83

580.

8224

0.80

960.

7973

0.78

570.

7750

61.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

70.

6441

0.63

320.

6235

0.61

470.

6067

0.59

940.

5927

0.58

650.

5808

0.57

540.

5705

80.

4760

0.47

520.

4745

0.47

380.

4730

0.47

230.

4716

0.47

060.

4690

0.46

740.

4659

90.

5183

0.51

560.

5132

0.51

080.

5084

0.50

600.

5037

0.50

140.

4991

0.49

690.

4946

100.

5043

0.49

370.

4843

0.47

590.

4682

0.46

130.

4548

0.44

890.

4434

0.43

830.

4335

110.

4198

0.41

680.

4138

0.41

100.

4082

0.40

560.

4030

0.40

040.

3980

0.39

560.

3932

120.

5281

0.52

510.

5221

0.51

930.

5165

0.51

370.

5110

0.50

840.

5058

0.50

320.

5008

130.

4753

0.47

330.

4714

0.46

950.

4676

0.46

580.

4640

0.46

220.

4605

0.45

880.

4571

140.

5282

0.52

280.

5177

0.51

280.

5082

0.50

360.

4993

0.49

510.

4911

0.48

720.

4835

150.

5566

0.55

330.

5500

0.54

690.

5438

0.54

090.

5380

0.53

520.

5325

0.52

980.

5273

160.

3959

0.39

210.

3884

0.38

490.

3815

0.37

820.

3750

0.37

200.

3691

0.36

620.

3635

171.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

5626

0.56

090.

5593

0.55

760.

5560

0.55

440.

5527

0.55

110.

5495

0.54

790.

5463

142 J. Puri and S. P. Yadav

Tab

le9

a-cu

tsð~ q

k OÞ a¼ðq

k OÞL a;ð

qk OÞU a

�� of

FS

BM

outp

utef

fici

ency

~ qk O

for

diff

eren

tva

lues

ofa2ð0;1�

DM

Uðq

k OÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

5750

0.57

770.

5804

0.58

310.

5858

0.58

860.

5913

0.59

410.

5968

0.59

960.

6024

20.

6950

0.69

780.

7006

0.70

340.

7063

0.70

910.

7119

0.71

470.

7176

0.72

040.

7232

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

8668

0.87

030.

8738

0.87

730.

8808

0.88

430.

8878

0.89

140.

8949

0.89

850.

9020

60.

8705

0.87

830.

8916

0.90

530.

9195

0.93

410.

9492

0.96

481.

0000

1.00

001.

0000

70.

4142

0.41

540.

4166

0.41

780.

4190

0.42

020.

4215

0.42

270.

4239

0.42

510.

4263

80.

6096

0.61

130.

6129

0.61

450.

6162

0.61

780.

6194

0.62

110.

6227

0.62

440.

6260

90.

6359

0.63

740.

6389

0.64

040.

6420

0.64

350.

6450

0.64

650.

6480

0.64

950.

6510

100.

4347

0.43

600.

4373

0.43

850.

4398

0.44

100.

4423

0.44

360.

4448

0.44

610.

4474

110.

4792

0.48

070.

4822

0.48

370.

4853

0.48

680.

4883

0.48

980.

4913

0.49

280.

4943

120.

7425

0.74

490.

7474

0.74

980.

7523

0.75

470.

7572

0.75

960.

7621

0.76

460.

7670

130.

6921

0.69

480.

6976

0.70

040.

7032

0.70

600.

7088

0.71

160.

7144

0.71

720.

7200

140.

7037

0.70

680.

7098

0.71

290.

7160

0.71

910.

7222

0.72

530.

7284

0.73

150.

7346

150.

7611

0.76

410.

7670

0.76

990.

7729

0.77

590.

7788

0.78

180.

7847

0.78

770.

7907

160.

4073

0.40

890.

4105

0.41

210.

4137

0.41

530.

4170

0.41

860.

4202

0.42

180.

4235

170.

8253

0.90

901.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

8944

0.89

830.

9022

0.90

620.

9101

0.91

410.

9181

0.92

200.

9260

0.93

010.

9341

(con

tinu

ed)

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 143

Tab

le9

(con

tinu

ed)

DM

Uðq

k OÞL a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

DM

Uðq

k OÞU a

a=

0a

=0.

1a

=0.

2a

=0.

3a

=0.

4a

=0.

5a

=0.

6a

=0.

7a

=0.

8a

=0.

9a

=1

10.

6201

0.61

830.

6165

0.61

470.

6130

0.61

120.

6094

0.60

770.

6059

0.60

420.

6024

20.

7504

0.74

760.

7449

0.74

210.

7394

0.73

670.

7340

0.73

130.

7286

0.72

590.

7232

31.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

41.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

50.

9254

0.92

310.

9207

0.91

830.

9160

0.91

360.

9113

0.90

900.

9067

0.90

430.

9020

61.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

70.

4344

0.43

360.

4328

0.43

200.

4312

0.43

030.

4295

0.42

870.

4279

0.42

710.

4263

80.

6391

0.63

770.

6364

0.63

510.

6338

0.63

250.

6312

0.62

990.

6286

0.62

730.

6260

90.

6646

0.66

330.

6619

0.66

050.

6592

0.65

780.

6564

0.65

510.

6537

0.65

240.

6510

100.

4577

0.45

660.

4556

0.45

460.

4535

0.45

250.

4515

0.45

040.

4494

0.44

840.

4474

110.

5038

0.50

290.

5019

0.50

100.

5000

0.49

910.

4981

0.49

720.

4962

0.49

530.

4943

120.

7855

0.78

360.

7818

0.77

990.

7781

0.77

620.

7744

0.77

250.

7707

0.76

890.

7670

130.

7355

0.73

400.

7324

0.73

080.

7293

0.72

770.

7262

0.72

460.

7231

0.72

150.

7200

140.

7540

0.75

210.

7501

0.74

820.

7462

0.74

430.

7423

0.74

040.

7385

0.73

650.

7346

150.

8048

0.80

340.

8020

0.80

060.

7991

0.79

770.

7963

0.79

490.

7935

0.79

210.

7907

160.

4354

0.43

420.

4330

0.43

180.

4306

0.42

940.

4282

0.42

700.

4258

0.42

460.

4235

171.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

180.

9641

0.96

100.

9580

0.95

500.

9520

0.94

900.

9460

0.94

300.

9400

0.93

700.

9341

144 J. Puri and S. P. Yadav

0.9 0.95 10

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

34561718

0.54 0.56 0.58 0.6 0.62 0.640

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha 101116

0.65 0.7 0.75 0.8 0.85 0.90

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha1278912131415

Fig. 2 Shapes of the membership functions of ~hkI

1 1.05 1.1 1.150

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

34561718

1.6 1.7 1.8 1.90

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

101116

1.1 1.2 1.3 1.4 1.50

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

1278912131415

Fig. 3 Shapes of membership functions of ~hkO

0.35 0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

89101112131416

0.6 0.8 10

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

1345617

0.5 0.55 0.6 0.650

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

271518

Fig. 4 Shapes of membership functions of ~qkI

0.6 0.65 0.7 0.75 0.80

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

128912131415

0.85 0.9 0.95 10

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

34561718

0.4 0.45 0.50

0.2

0.4

0.6

0.8

1

Efficiency

Alp

ha

7101116

Fig. 5 Shapes of membership functions of ~qkO

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 145

Table 10 The values of ~hkI , ~qk

I and ~wkI approximated as the TFNs

DMU ~hkI

~qkI

~wkI

1 (0.7841, 0.8493, 0.8910) (0.5328, 0.6407, 0.7322) (0.5980, 0.7544, 0.9338)2 (0.7843, 0.8202, 0.8638) (0.4895, 0.5584, 0.6131) (0.5667, 0.6808, 0.7817)3 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)4 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)5 (0.8869, 0.9265, 0.9593) (0.5824, 0.7750, 0.9151) (0.6070, 0.8365, 1.0318)6 (0.8767, 1.0000, 1.0000) (0.8569, 1.0000, 1.0000) (0.8569, 1.0000, 1.1407)7 (0.7654, 0.8018, 0.8356) (0.4944, 0.5705, 0.6441) (0.5917, 0.7115, 0.8415)8 (0.6621, 0.6775, 0.6916) (0.4214, 0.4659, 0.4760) (0.6093, 0.6877, 0.7189)9 (0.6880, 0.7020, 0.7189) (0.4416, 0.4946, 0.5183) (0.6143, 0.7046, 0.7533)10 (0.5656, 0.5934, 0.6191) (0.3763, 0.4335, 0.5043) (0.6077, 0.7305, 0.8916)11 (0.5886, 0.6086, 0.6254) (0.3639, 0.3932, 0.4198) (0.5818, 0.6461, 0.7132)12 (0.7602, 0.7853, 0.8044) (0.4779, 0.5008, 0.5281) (0.5941, 0.6376, 0.6947)13 (0.7309, 0.7489, 0.7606) (0.4388, 0.4571, 0.4753) (0.5769, 0.6104, 0.6504)14 (0.7144, 0.7511, 0.7741) (0.4344, 0.4835, 0.5282) (0.5612, 0.6437, 0.7393)15 (0.7988, 0.8210, 0.8337) (0.5019, 0.5273, 0.5566) (0.6021, 0.6422, 0.6968)16 (0.5384, 0.5635, 0.5846) (0.3345, 0.3635, 0.3959) (0.5722, 0.6450, 0.7353)17 (0.8938, 1.0000, 1.0000) (0.8892, 1.0000, 1.0000) (0.8892, 1.0000, 1.1188)18 (0.9216, 0.9569, 0.9819) (0.5253, 0.5463, 0.5626) (0.5349, 0.5709, 0.6105)

Table 11 The values of ~hkO, ~qk

O and ~wkO approximated as the TFNs

DMU ~hkO

~qkO

~wkO

1 (1.1214, 1.1762, 1.2746) (0.5750, 0.6024, 0.6201) (0.4511, 0.5122, 0.5530)2 (1.1574, 1.2192, 1.2749) (0.6950, 0.7232, 0.7504) (0.5452, 0.5932, 0.6483)3 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)4 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)5 (1.0423, 1.0794, 1.1275) (0.8668, 0.9020, 0.9254) (0.7688, 0.8357, 0.8878)6 (1.0000, 1.0000, 1.1406) (0.8705, 1.0000, 1.0000) (0.7632, 1.0000, 1.0000)7 (1.1944, 1.2451, 1.3041) (0.4142, 0.4263, 0.4344) (0.3176, 0.3424, 0.3637)8 (1.4459, 1.4760, 1.5101) (0.6096, 0.6260, 0.6391) (0.4037, 0.4241, 0.4420)9 (1.3907, 1.4243, 1.4532) (0.6359, 0.6510, 0.6646) (0.4376, 0.4571, 0.4779)10 (1.6088, 1.6802, 1.7659) (0.4347, 0.4474, 0.4577) (0.2462, 0.2663, 0.2845)11 (1.5978, 1.6418, 1.6974) (0.4792, 0.4943, 0.5038) (0.2823, 0.3011, 0.3153)12 (1.2427, 1.2727, 1.3147) (0.7425, 0.7670, 0.7855) (0.5648, 0.6027, 0.6321)13 (1.3139, 1.3343, 1.3668) (0.6921, 0.7200, 0.7355) (0.5063, 0.5396, 0.5598)14 (1.2916, 1.3312, 1.3993) (0.7037, 0.7346, 0.7540) (0.5029, 0.5519, 0.5838)15 (1.1990, 1.2174, 1.2511) (0.7611, 0.7907, 0.8048) (0.6084, 0.6495, 0.6712)16 (1.7078, 1.7713, 1.8537) (0.4073, 0.4235, 0.4354) (0.2197, 0.2391, 0.2549)17 (1.0000, 1.0000, 1.1185) (0.8253, 1.0000, 1.0000) (0.7379, 1.0000, 1.0000)18 (1.0184, 1.0450, 1.0848) (0.8944, 0.9341, 0.9641) (0.8244, 0.8939, 0.9467)

146 J. Puri and S. P. Yadav

7 Ranking of the DMUs on the Basis of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI

and ~wkO

A convenient method for comparing fuzzy numbers is by use of the rankingfunction [28]. Let FðRÞ be a set of all fuzzy numbers defined on R: Then a rankingfunction < : FðRÞ ! Ris a function which maps each fuzzy number in FðRÞ to areal number inR:

Let ~A and ~B be two fuzzy numbers in FðRÞ: Then

(i) ~A �<

~B if and only if <ð~AÞ �<<ð~BÞ

(ii) ~A [<

~B if and only if <ð~AÞ [<<ð~BÞ

(iii) ~A ¼<

~B if and only if <ð~AÞ ¼<<ð~BÞ

For a TFN ~A ¼ ða1; a2; a3Þ; ranking function value rankð Þ is given by<ð~AÞ

¼ 14ða1 þ 2a2 þ a3Þ: ð22Þ

For ranking of the DMUs on the basis of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O; we use

the above linear ranking function. Let <ð~hkI Þ; <ð~h

kOÞ; <ð~qk

I Þ; <ð~qkOÞ; <ð~wk

I Þ and

<ð~wkOÞ be the ranks of ~hk

I ;~hk

O; ~qkI ; ~qk

O;~wk

I and ~wkO respectively. The ranks of

~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O in Tables 10 and 11 are evaluated using (22) and areshown in Table 12.

Table 12 indicates that the ranking of DMUs on the basis of ~hkI ;

~hkO; ~qk

I ; ~qkOand

~wkO is possible in conventional way, as the values of <ð~hk

I Þ; <ð~hkOÞ; <ð~qk

I Þ; <ð~qkOÞ

and <ð~wkOÞ lie between 0 and 1. But there is a DMU17 whose <ð~w17

I Þ[ 1 and thus

the ranking of DMUs on the basis of <ð~wkI Þis still unattainable. Therefore, for

doing exact ranking of the DMUs on the basis of ~wkI ;we propose a new method of

ranking based on <ð~qkI Þ and <ð~hk

I Þ:Table 12 reveals that <ð~qkI Þ � <ð~hk

I Þ and

<ð~qkOÞ � <ð~hk

OÞ for every DMU. Therefore by using <ð~qkI Þ � <ð~hk

I Þ; we define

the following index for ranking the DMUs on the basis of ~wkI :

Ið~wkI Þ ¼

<ð~qkI Þ

<ð~hkI Þ� 1: ð23Þ

Now, Ið~wkI Þ ¼ 1 if <ð~qk

I Þ ¼ <ð~hkI Þ:

For the efficient DMUs, we have <ð~qkOÞ ¼ <ð~hk

OÞ ¼ 1: Therefore for the

efficient DMUs, we get Ið~wkI Þ ¼ 1 ; while for inefficient DMUs we get Ið~wk

I Þ \ 1:

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 147

7.1 Proposed Algorithm to Rank the DMUs on the Basisof ~wk

I :

Step 1. Evaluate <ð~wkI Þ for each DMU

Step 2. If <ð~wkI Þ 2 ð0; 1� for every DMU, STOP the algorithm and rank the

DMUs with decreasing values of <ð~wkI Þ: Otherwise go to step 3

Step 3. Evaluate Ið~wkI Þ for every DMU using (23) and rank the DMUs with

decreasing values of Ið~wkI Þ:

In Table 12, <ð~w17I Þ[ 1:Therefore we find the ranking of the DMUs on the

basis of ~wkI by using the above algorithm which is shown in Table 13.

Table 13 shows that the FIME determines the ranking of the DMUs in the orderof DMU3¼

IDMU4 [

IDMU17 [

IDMU6 [

IDMU5 [

IDMU1 [

IDMU10 [

I

DMU7 [I

DMU9 [I

DMU8 [I

DMU2 [I

DMU16 [I

DMU11 [I

DMU15 [I

DMU14 [I

DMU12 [I

DMU13 [I

DMU18and the FOME determines the ranking

of the DMUs in the order of DMU3¼<

DMU4 [<

DMU6 [<

DMU17 [<

DMU18 [<

DMU5 [<

DMU15 [<

DMU12 [<

DMU2 [<

DMU14 [<

DMU13 [<

DMU1 [<

DMU9 [<

DMU8 [<

DMU7 [<

DMU11 [<

DMU10 [<

DMU16: The DMUs with

Table 12 The ranks of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O and ranking of the DMUs

DMU <ð~hkI Þ Rank <ð~qk

I Þ Rank <ð~wkI Þ Rank <ð~hk

OÞ Rank <ð~qkOÞ Rank <ð~wk

OÞ Rank

1 0.8434 7 0.6366 6 0.7602 – 1.1871 7 0.6000 14 0.5071 122 0.8221 8 0.5548 8 0.6775 – 1.2177 8 0.7230 10 0.5950 93 1.0000 1 1.0000 1 1.0000 – 1.0000 1 1.0000 1 1.0000 14 1.0000 1 1.0000 1 1.0000 – 1.0000 1 1.0000 1 1.0000 15 0.9248 6 0.7619 5 0.8280 – 1.0821 6 0.8991 6 0.8320 66 0.9692 4 0.9642 4 0.9994 – 1.0352 4 0.9676 3 0.9408 37 0.8011 10 0.5699 7 0.7141 – 1.2471 10 0.4253 17 0.3415 158 0.6771 15 0.4573 14 0.6759 – 1.4770 15 0.6252 13 0.4235 149 0.7028 14 0.4873 12 0.6942 – 1.4232 14 0.6506 12 0.4574 1310 0.5929 17 0.4369 16 0.7401 – 1.6838 17 0.4468 16 0.2658 1711 0.6078 16 0.3925 17 0.6468 – 1.6447 16 0.4929 15 0.3000 1612 0.7838 11 0.5019 11 0.6410 – 1.2757 11 0.7655 8 0.6006 813 0.7473 13 0.4571 15 0.6120 – 1.3373 12 0.7169 11 0.5363 1114 0.7477 12 0.4824 13 0.6470 – 1.3383 13 0.7317 9 0.5476 1015 0.8186 9 0.5283 10 0.6458 – 1.2212 9 0.7868 7 0.6446 716 0.5625 18 0.3643 18 0.6494 – 1.7760 18 0.4224 18 0.2382 1817 0.9734 3 0.9723 3 1.0020 – 1.0296 3 0.9563 4 0.9345 418 0.9543 5 0.5451 9 0.5718 – 1.0483 5 0.9317 5 0.8897 5

148 J. Puri and S. P. Yadav

<ð~wkI Þ and <ð~wk

OÞ less than one are said to be input mix-inefficient and output mix-inefficient respectively. The input mix-inefficient and output mix-inefficient DMUsshould respectively decrease their input mix and augment their output mix in orderto become fully efficient.

8 Application to the Banking Sector

The performance of the nationalized banks (NBs) in India is measured in terms of2 inputs and 2 outputs. The DMUs in our study are 19 NBs and are listed inTable 14. The inputs used in our study are (i) Labour and (ii) Total deposits.Labour is the number of employees working in each NB. Total deposit is the sumtotal of demand deposits, saving banks deposits and term deposits. The outputsused in our study are (i) Total investments and (ii) Performing loans. Totalinvestments include the investments done by the banks in (a) government secu-rities, (b) other approved securities, (c) shares, (d) debentures and bonds,(e) subsidiaries and/or joint ventures, and others. Performing loans are the per-forming assets which are calculated by subtracting non-performing loans/assetsfrom total advances. The data is taken from RBI [29] which is in the form of crispquantities. In this study, we have fuzzified the data. The second input, i.e., totaldeposits and each output are taken as TFNs. The objective of our study is to

measure ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O of 19 NBs for the period 2010–2011 and toknow which NBs are the best performers and which are the worst performers.

Table 13 The ranks of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O and ranking of the DMUs

DMU <ð~hkI Þ Rank <ð~qk

I Þ Rank Ið~wkI Þ Rank <ð~hk

OÞ Rank <ð~qkOÞ Rank <ð~wk

OÞ Rank

1 0.8434 7 0.6366 6 0.7548 6 1.1871 7 0.6000 14 0.5071 122 0.8221 8 0.5548 8 0.6749 11 1.2177 8 0.7230 10 0.5950 93 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 14 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 15 0.9248 6 0.7619 5 0.8239 5 1.0821 6 0.8991 6 0.8320 66 0.9692 4 0.9642 4 0.9948 4 1.0352 4 0.9676 3 0.9408 37 0.8011 10 0.5699 7 0.7114 8 1.2471 10 0.4253 17 0.3415 158 0.6771 15 0.4573 14 0.6754 10 1.4770 15 0.6252 13 0.4235 149 0.7028 14 0.4873 12 0.6934 9 1.4232 14 0.6506 12 0.4574 1310 0.5929 17 0.4369 16 0.7369 7 1.6838 17 0.4468 16 0.2658 1711 0.6078 16 0.3925 17 0.6458 13 1.6447 16 0.4929 15 0.3000 1612 0.7838 11 0.5019 11 0.6403 16 1.2757 11 0.7655 8 0.6006 813 0.7473 13 0.4571 15 0.6117 17 1.3373 12 0.7169 11 0.5363 1114 0.7477 12 0.4824 13 0.6452 15 1.3383 13 0.7317 9 0.5476 1015 0.8186 9 0.5283 10 0.6454 14 1.2212 9 0.7868 7 0.6446 716 0.5625 18 0.3643 18 0.6476 12 1.7760 18 0.4224 18 0.2382 1817 0.9734 3 0.9723 3 0.9989 3 1.0296 3 0.9563 4 0.9345 418 0.9543 5 0.5451 9 0.5712 18 1.0483 5 0.9317 5 0.8897 5

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 149

The a-cuts ð~hkI Þa; ð~h

kOÞa; ð~qk

I Þaand ð~qkOÞa of ~hk

I ;~hk

O; ~qkI and ~qk

O respectively for19 NBs are evaluated by using models from (14a) to (17b) at different values ofa 2 ð0; 1� and their graphical representations are shown in Figs. 6, 7, 8 and 9. The

figures reveal that the shape of the membership functions of ~hkI ;

~hkO; ~qk

I and ~qkO for

every DMU can be approximated by a TFN as shown in Tables 15 and 16. The

fuzzy mix-efficiencies ~wkI and ~wk

O are evaluated using (19) and (21) respectively,and are shown in Tables 15 and 16.

Further, the ranks of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O in Tables 15 and 16 areevaluated using (22) and are shown in Table 17. The ranking of DMUs on the

basis of <ð~hkI Þ; <ð~h

kOÞ; <ð~qk

I Þ; <ð~qkOÞ and <ð~wk

OÞ is done in a conventional way.

However, the ranking of DMUs on the basis of <ð~wkI Þis done by applying the

proposed method of ranking given in Sect. 7.1. In this case the algorithm termi-

nates at the Step 2 since the value of <ð~wkI Þ 2 ð0; 1� for each NB.

Table 17 indicates that DMU2, DMU8, DMU14 and DMU15, i.e., Andhra Bank,Corporation Bank, Punjab National Bank and Syndicate Bank are the best per-

former banks in terms of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O: On the other hand, DMU5

Table 14 NBs during the period 2010–2011

NBs

BC Bank names BC Bank names BC Bank names

1 Allahabad Bank 8 Corporation Bank 15 Syndicate Bank2 Andhra Bank 9 Dena Bank 16 UCO Bank3 Bank of Baroda 10 Indian Bank 17 Union Bank of India4 Bank of India 11 Indian Overseas Bank 18 United Bank of India5 Bank of Maharashtra 12 Oriental Bank of Commerce 19 Vijaya Bank6 Canara Bank 13 Punjab & Sind Bank7 Central Bank of India 14 Punjab National Bank

Note BC stands for bank code

0.922 0.9250

0.5

1

Efficiency

Alp

ha

0.932 0.9325 0.9330

0.5

1

Efficiency

Alp

ha

0.9366 0.93720

0.5

1

Efficency

Alp

ha

0.9419 0.94210

0.5

1

Efficiency

Alp

ha

0.999 0.9995 10

0.5

1

Efficiency

Alp

ha

0.9444 0.9450

0.5

1

Efficency

Alp

ha

0.957 0.95760

0.5

1

Efficency

Alp

ha

0.9694 0.9696 0.96980

0.5

1

Efficency

Alp

ha

0.8813 0.8814 0.88150

0.5

1

Efficency

Alp

ha

0.908 0.9085 0.909 0.90950

0.5

1

Efficency

Alp

ha

5

194 1 6 7

2,3,8,14,15

11

12 17 101391816

Fig. 6 Shapes of membership functions of ~hkI

150 J. Puri and S. P. Yadav

0.8835 0.8840

0.5

1

Efficiency

Alp

ha

0.776 0.7780

0.5

1

Efficiency

Alp

ha

0.849 0.85 0.851 0.8520

0.5

1

Efficiency

Alp

ha

0.841 0.8415 0.8420

0.5

1

Efficiency

Alp

ha

0.868 0.8685 0.8690

0.5

1

Efficiency

Alp

ha

0.996 0.998 10

0.5

1

Efficiency

Alp

ha

0.894 0.9 0.9020

0.5

1

Efficiency

Alp

ha

0.915 0.916 0.918 0.9190

0.5

1

Efficiency

Alp

ha

0.907 0.9075 0.9080

0.5

1

Efficiency

Alp

ha

0.912 0.91250

0.5

1

Efficiency

Alp

ha

0.9365 0.9370

0.5

1

Efficiency

Alp

ha

13 4

9

6

7

16 17 510

13 1912 18

2,8,14,15

11

Fig. 9 Shapes of membership functions of ~qkO

0.7882 0.78860

0.5

1

Efficiency

Alp

ha

0.821 0.822 0.8230

0.5

1

Efficiency

Alp

ha

0.73 0.732 0.7340

0.5

1

Efficiency

Alp

ha

0.7825 0.783 0.78350

0.5

1

Efficiency

Alp

ha

0.8837 0.8838 0.88390

0.5

1

Efficiency

Alp

ha

0.89 0.9 0.910

0.5

1

Efficiency

Alp

ha

0.9112 0.91140

0.5

1

Efficiency

Alp

ha

0.9213 0.9214 0.92150

0.5

1

Efficiency

Alp

ha

0.944 0.945 0.9460

0.5

1

Efficiency

Alp

ha

10

0.5

1

Efficiency

Alp

ha

1199

13 11

5 18 10 7 16 12

4 6

2,3,8,14,15

17

Fig. 8 Shapes of membership functions of ~qkI

1.099 1.1 1.101 1.1020

0.5

1

Efficiency

Alp

ha

1.1348 1.135 1.13520

0.5

1

EfficencyA

lpha

1.0444 1.0446 1.04480

0.5

1

Efficiency

Alp

ha

1.0312 1.0313 1.03140

0.5

1

Efficiency

Alp

ha

1.058 1.0585 1.0590

0.5

1

Efficiency

Alp

ha

1 1.0005 1.0010

0.5

1

Efficiency

Alp

ha

1.082 1.084 1.0850

0.5

1

Efficiency

Alp

ha

1.0724 1.0725 1.07260

0.5

1

Efficiency

Alp

ha

1.067 1.0675 1.0680

0.5

1

Efficiency

Alp

ha

1.0614 1.0616 1.06180

0.5

1

Efficiency

Alp

ha

12

918

1617 10 13

2,3,8,14,15

115 19 4 6 1 7

Fig. 7 Shapes of membership functions of ~hkO

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 151

Table 15 The values of ~hkI , ~qk

I and ~wkI approximated as the TFNs

DMU ~hkI

~qkI

~wkI

1 (0.9366, 0.9367, 0.9368) (0.8219, 0.8220, 0.8221) (0.8773, 0.8775, 0.8777)2 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)3 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)4 (0.9323, 0.9324, 0.9325) (0.9113, 0.9113, 0.9114) (0.9773, 0.9774, 0.9776)5 (0.9240, 0.9242, 0.9244) (0.7293, 0.7294, 0.7295) (0.7889, 0.7892, 0.7894)6 (0.9369, 0.9370, 0.9371) (0.8838, 0.8838, 0.8839) (0.9431, 0.9433, 0.9434)7 (0.9419, 0.9420, 0.9421) (0.7829, 0.7830, 0.7831) (0.8311, 0.8312, 0.8313)8 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)9 (0.9088, 0.9090, 0.9091) (0.8204, 0.8205, 0.8207) (0.9025, 0.9027, 0.9030)10 (0.9449, 0.9450, 0.9451) (0.7883, 0.7884, 0.7885) (0.8341, 0.8343, 0.8345)11 (0.9989, 0.9990, 0.9991) (0.9027, 0.9039, 0.9051) (0.9035, 0.9047, 0.9061)12 (0.9571, 0.9573, 0.9575) (0.9443, 0.9445, 0.9446) (0.9863, 0.9866, 0.9869)13 (0.9444, 0.9446, 0.9448) (0.8986, 0.8988, 0.8989) (0.9511, 0.9515, 0.9518)14 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)15 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)16 (0.8813, 0.8814, 0.8815) (0.7832, 0.7833, 0.7834) (0.8885, 0.8887, 0.8889)17 (0.9696, 0.9696, 0.9697) (0.9213, 0.9214, 0.9215) (0.9501, 0.9503, 0.9504)18 (0.9082, 0.9084, 0.9085) (0.7331, 0.7332, 0.7333) (0.8069, 0.8071, 0.8074)19 (0.9218, 0.9221, 0.9224) (0.8228, 0.8229, 0.8232) (0.8920, 0.8925, 0.8929)

Table 16 The values of ~hkO, ~qk

O and ~wkO approximated as the TFNs

DMU ~hkO

~qkO

~wkO

1 (1.0674, 1.0676, 1.0677) (0.9122, 0.9124, 0.9125) (0.8544, 0.8546, 0.8549)2 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)3 (1.0000, 1.0000, 1.0000) (0.7759, 0.7762, 0.7765) (0.7759, 0.7762, 0.7765)4 (1.0724, 1.0725, 1.0726) (0.8498, 0.8499, 0.8500) (0.7923, 0.7924, 0.7926)5 (1.0818, 1.0820, 1.0822) (0.9155, 0.9157, 0.9159) (0.8459, 0.8463, 0.8466)6 (1.0672, 1.0672, 1.0673) (0.8512, 0.8513, 0.8514) (0.7975, 0.7976, 0.7978)7 (1.0615, 1.0615, 1.0616) (0.8836, 0.8837, 0.8839) (0.8323, 0.8325, 0.8327)8 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)9 (1.1000, 1.1002, 1.1003) (0.8511, 0.8513, 0.8515) (0.7735, 0.7738, 0.7741)10 (1.0581, 1.0582, 1.0584) (0.9177, 0.9179, 0.9180) (0.8671, 0.8674, 0.8676)11 (1.0009, 1.0010, 1.0011) (0.9959, 0.9963, 0.9967) (0.9949, 0.9953, 0.9959)12 (1.0444, 1.0446, 1.0448) (0.9365, 0.9367, 0.9368) (0.8964, 0.8967, 0.8970)13 (1.0585, 1.0586, 1.0589) (0.8945, 0.8948, 0.8950) (0.8448, 0.8452, 0.8456)14 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)15 (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000) (1.0000, 1.0000, 1.0000)16 (1.1349, 1.1350, 1.1351) (0.8416, 0.8417, 0.8419) (0.7414, 0.7416, 0.7418)17 (1.0312, 1.0313, 1.0314) (0.8683, 0.8684, 0.8685) (0.8419, 0.8420, 0.8422)18 (1.1007, 1.1008, 1.1011) (0.9073, 0.9075, 0.9077) (0.8241, 0.8244, 0.8247)19 (1.0842, 1.0845, 1.0848) (0.8998, 0.9000, 0.9002) (0.8295, 0.8299, 0.8303)

152 J. Puri and S. P. Yadav

and DMU16, i.e., Bank of Maharashtra and UCO Bank are the worst performer

banks in terms of ~wkI and ~wk

O respectively. The order of performance of the NBs in

terms of ~wkI is given by DMU2¼

<DMU3¼

<DMU8¼

<DMU14¼

<DMU15 [

<DMU12 [

<DMU4 [

<DMU13 [

<DMU17 [

<DMU6 [

<DMU11 [

<DMU9 [

<DMU19 [

<DMU16 [

<DMU1 [

<DMU10 [

<DMU7 [

<DMU18 [

<

DMU5 and in terms of ~wkO is given by DMU2¼

<DMU8¼

<DMU14¼

<DMU15 [

<DMU11 [

<DMU12 [

<DMU10 [

<DMU5 [

<DMU1 [

<DMU18 [

<DMU19 [

<DMU13 [

<DMU7 [

<DMU17 [

<DMU9 [

<DMU4 [

<DMU16 [

<DMU6 [

<DMU3:

9 Conclusions

In this chapter, we have extended the input and output orientations of CCR andSBM models to fuzzy environments as the precise input and output data are notalways available in real life applications. We have proposed the FCCRI, FCCRO,FSBMI and FSBMO models, and these models are further transformed into thecrisp DEA models by using a�cut approach which makes them easy to implementin real life problems. By using FCCRI and FSBMI models, and FCCRO andFSBMO models, we have defined FIME and FOME respectively. The FIME and

Table 17 The ranks of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O and ranking of the DMUs

DMU <ð~hkI Þ Rank <ð~qk

I Þ Rank <ð~wkI Þ Rank <ð~hk

OÞ Rank <ð~qkOÞ Rank <ð~wk

OÞ Rank

1 0.9367 13 0.8220 13 0.8775 15 1.0676 13 0.9124 9 0.8546 82 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 13 1.0000 1 1.0000 1 1.0000 1 1.0000 1 0.7762 19 0.7762 174 0.9324 14 0.9113 8 0.9774 7 1.0725 14 0.8499 16 0.7924 165 0.9242 15 0.7294 19 0.7892 19 1.0820 15 0.9157 8 0.8463 96 0.9370 12 0.8838 11 0.9433 10 1.0672 12 0.8513 18 0.7976 157 0.9420 11 0.7830 17 0.8312 17 1.0615 11 0.8837 13 0.8325 128 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 19 0.9090 17 0.8205 14 0.9027 12 1.1002 17 0.8513 15 0.7738 1810 0.9450 9 0.7884 15 0.8343 16 1.0582 9 0.9179 7 0.8674 711 0.9990 6 0.9039 9 0.9048 11 1.0010 6 0.9963 5 0.9954 512 0.9573 8 0.9445 6 0.9866 6 1.0446 8 0.9367 6 0.8967 613 0.9446 10 0.8988 10 0.9515 8 1.0586 10 0.8948 12 0.8452 1014 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 115 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 1 1.0000 116 0.8814 19 0.7833 16 0.8887 14 1.1350 19 0.8417 17 0.7416 1917 0.9697 7 0.9214 7 0.9503 9 1.0313 7 0.8684 14 0.8420 1118 0.9084 18 0.7332 18 0.8071 18 1.1009 18 0.9075 10 0.8244 1419 0.9221 16 0.8230 12 0.8925 13 1.0845 16 0.9000 11 0.8299 13

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 153

FOME are obtained as fuzzy numbers which can be approximated as TFNs.Further, ranking function is used to compare the fuzzy efficiencies of the DMUsand to rank the DMUs on the basis of the evaluated ranks. A new ranking algo-rithm based on the ranks of FCCR input efficiency and FSBM input efficiency hasbeen developed for comparing and ranking DMUs in terms of FIME. The rankingof the DMUs not only provide the ranks to the DMUs, but also provide us theinformation about the best and worst performer DMUs on the basis of~hk

I ;~hk

O; ~qkI ; ~qk

O;~wk

I and ~wkO: To illustrate the validation and effectiveness of the

proposed methodology, we have presented a numerical example. Moreover, theproposed methodology has been applied to the banking sector in India in which

the performance of 19 NBs is measured in terms of ~hkI ;

~hkO; ~qk

I ; ~qkO;

~wkI and ~wk

O:It is shown that Andhra Bank, Corporation Bank, Punjab National Bank andSyndicate Bank are the best performer banks whereas Bank of Maharashtra and

UCO Bank are the worst performer banks in terms of ~wkI and ~wk

O respectively.According to the findings of our study, the input and output mix-inefficient NBsshould keep in mind the following suggestions:

• All the input mix-inefficient banks are suggested to decrease their input mix inorder to become fully efficient.

• All the output mix-inefficient banks are suggested to increase their output mix inorder to become fully efficient.

References

1. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units.Eur. J. Oper. Res. 2, 429–444 (1978)

2. Cooper, W.W., Seiford, L.M., Tone, K.: Data envelopment analysis: a comprehensive textwith models, applications, references and DEA-solver software, 2nd edn. Springer, NewYork (2007)

3. Banker, R.D., Charnes, A., Cooper, W.W.: Some models for estimating technical and scaleinefficiencies in DEA. Manag. Sci. 30(9), 1078–1092 (1984)

4. Charnes, A., Cooper, W.W., Seiford, L., Stutz, J.: A multiplicative model for efficiencyanalysis. Socio-Econ. Plann. Sci. 6, 223–224 (1982)

5. Petersen, N.C.: Data envelopment analysis on a relaxed set of assumptions. Manag. Sci.36(3), 305–313 (1990)

6. Tone, K.: A slacks-based measure of efficiency in data envelopment analysis. Eur. J. Oper.Res. 130, 498–509 (2001)

7. Seiford, L.M., Thrall, R.M.: Recent developments in DEA: the mathematical programmingapproach to frontier analysis. J. Econ. 46, 7–38 (1990)

8. Seiford, L.M.: Data envelopment analysis: the evolution of a state of the art. J. Prod. Anal.7(2), 99–137 (1996)

9. Zhu, J.: Quantitative models for performance evaluation and benchmarking: dataenvelopment analysis with spreadsheets and DEA excel solver. Kluwer AcademicPublishers, USA (2003)

154 J. Puri and S. P. Yadav

10. Ray, S.C.: Data envelopment analysis: theory and techniques for economics and operationsresearch. Cambridge University Press, Cambridge (2004)

11. Tone, K.: On mix efficiency in DEA. The Operations Research Society of Japan, 14–15(1998). http://ci.nii.ac.jp/naid/110003478327/en

12. Asbullah, M.A.: A new approach to estimate the mix efficiency in data envelopment analysis.Appl. Math. Sci. 4(43), 2135–2143 (2010)

13. Herrero, I., Pascoe, S., Mardle, S.: Mix efficiency in a multi-species fishery. J. Prod. Anal. 25,231–241 (2006)

14. Angiz, L., Emrouznejad, M.Z., Mustafa, A.: Fuzzy data envelopment analysis: a discreteapproach. Expert Syst. Appl. 39, 2263–2269 (2012)

15. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119,149–160 (2001)

16. Hatami-Marbini, A., Emrouznejad, A., Tavana, M.: A taxonomy and review of the fuzzy dataenvelopment analysis literature: two decades in the making. Eur. J. Oper. Res. 214, 457–472(2011)

17. Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst.113, 427–437 (2000)

18. Puri, J., Yadav, S.P.: A concept of fuzzy input mix-efficiency in fuzzy DEA and itsapplication in banking sector. Expert Syst. Appl. 40, 1437–1450 (2013)

19. Wang, Y.M., Luo, Y., Liang, L.: Fuzzy data envelopment analysis based upon fuzzyarithmetic with an application to performance assessment of manufacturing enterprises.Expert Syst. Appl. 36(3), 5205–5211 (2009)

20. Sengupta, J.K.: A fuzzy systems approach in data envelopment analysis. Comput. Math.Appl. 24, 259–266 (1992)

21. Zimmermann, H.J.: Fuzzy Set Theory and its Applications, 2nd edn. Kluwer-Nijhoff, Boston(1991)

22. Triantis, K.P., Girod, O.: A mathematical programming approach for measuring technicalefficiency in a fuzzy environment. J. Prod. Anal. 10(1), 85–102 (1998)

23. Lertworasirikul, S., Fang, S.C., Jeffrey, A., Joines, J.A., Nuttle, H.L.W.: Fuzzy dataenvelopment analysis (DEA): a possibility approach. Fuzzy Sets Syst. 139(2), 379–394(2003)

24. Guh, Y.Y.: Data envelopment analysis in fuzzy environment. Inf. Manag. Sci. 12(2), 51–65(2001)

25. Jahanshahloo, G.R., Soleimani-damaneh, M., Nasrabadi, E.: Measure of efficiency in DEAwith fuzzy input–output levels: a methodology for assessing, ranking and imposing ofweights restrictions. Appl. Math. Comput. 156, 175–187 (2004)

26. Saati, S., Memariani, A.: SBM model with fuzzy input-output levels in DEA. Aust. J. BasicAppl. Sci. 3(2), 352–357 (2009)

27. Chen, S.M.: Fuzzy system reliability analysis using fuzzy number arithmetic operations.Fuzzy Sets Syst. 66, 31–38 (1994)

28. Mahdavi-Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of acertain linear ranking function. Appl. Math. Comput. 180, 206–216 (2006)

29. Reserve Bank of India: Statistical Tables Relating to Banks in India, 2010–2011. http://rbidocs.rbi.org.in/rdocs/Publications/PDFs/FSTR141111.pdf

6 Fuzzy Mix-efficiency in Fuzzy Data Envelopment Analysis and Its Application 155

Chapter 7Ranking of Fuzzy Efficiency Measuresvia Satisfaction Degree

M.-R. Ghasemi, Joshua Ignatius and S. M. Davoodi

Abstract Fuzzy DEA models emerge as another class of DEA models that sup-port subjectivity in eliciting inputs and outputs for decision making units (DMUs).Though several approaches for solving fuzzy DEA models exist, there are somedrawbacks, ranging from the inability to provide satisfactory discrimination powerto simplistic numerical examples that handle only symmetrical fuzzy numbers. Toaddress these drawbacks, a fuzzy DEA-CCR model using a linear ranking functionis proposed to incorporate fuzzy inputs and fuzzy outputs that are asymmetrical innature. This chapter proposes a ranking method for fuzzy efficiency measuresthrough a formalised fuzzy DEA model.

Keywords Crisp efficiency measure � Fuzzy efficiency measure � Linear rankingfunction � Fuzzy satisfaction degree

1 Introduction

Since fuzzy DEA models take the form of fuzzy linear programming (LP) prob-lems, optimization methods developed for LP problems often can be extended tofuzzy DEA models. Though numerous approaches for solving fuzzy LP problemsexist, a more popular method is based on the concept of comparison among fuzzynumbers via a ranking function (see i.e. [1–3]). The comparison through a ranking

M.-R. Ghasemi � J. Ignatius (&) � S. M. DavoodiSchool of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysiae-mail: joshua_ignatius@hotmail.com

M.-R. Ghasemie-mail: reza_gasemi_54@yahoo.com

S. M. DavoodiDepartment of Mathematics, Islamic Azad University, Abadeh branch, Iran

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_7, � Springer-Verlag Berlin Heidelberg 2014

157

function requires a model to be defined equivalent to the fuzzy LP problem, thusensuring that the optimal solution of this model can be used as the optimal solutionof the fuzzy LP problem (see i.e. [3, 4]). In the same manner, a linear rankingfunction for comparing fuzzy numbers can be proposed as an approach for solvingfuzzy DEA models [5].

Most of the existing fuzzy DEA models in the literature require significantcomputational efforts, i.e. the Guo and Tanaka’s [6] fuzzy ranking approach needstwo linear programming problems to obtain the efficiency value of a given DMU,in which the optimal value of the objective function of the primary linear pro-gramming problem is used in the secondary linear programing problem and thenby using the optimal solution in the secondary problem, a formula is proposed toachieve fuzzy efficiency scores. To obtain the efficiency value of an evaluatedDMU (decision maker unit) with less computational efforts, the concept of com-paring fuzzy numbers through a ranking function can be proposed as a usefulapproach for the fuzzy DEA model that is usable and practicable for real-worldproblems.

The rest of the current chapter is structured as follows. We first present theconcept and properties of linear ranking functions in Sect. 2. Section 3 outlines amore thorough interpretation of the fuzzy DEA-CCR model using the linearranking function and illustrates the model with an established numerical example.Section 4 provides a ranking method of fuzzy numbers. In Sect. 5, by using theproposed ranking method, we rank the fuzzy efficiency measures provided by theproposed fuzzy DEA model. Section 6 concludes the current chapter.

2 Ranking Functions

Several approaches have been developed to solve fuzzy LP problems in the lit-erature (see i.e. [4, 7, 8]). A more popular method is based on the concept ofcomparing fuzzy numbers through a ranking function ([2, 3, 9]).

Remark 1 Let F Rð Þ denote the set of all trapezoidal fuzzy numbers. Byassuming a ¼ aL ¼ aU for a trapezoidal fuzzy number ~A ¼ aL; aU ; aL; aUð Þ, weobtain a triangular fuzzy number as ~A ¼ a; aL; aUð Þ.

A ranking function R is a map from FðRÞ into the real line R, R : FðRÞ ! R.Orders on F Rð Þ can be defined as follows:

~a� R~b if and only if R ~að Þ�R ~b

� �; ð1Þ

~a [ R~b if and only if R ~að Þ[ R ~b

� �; ð2Þ

~a ¼R~b if and only if R ~að Þ ¼ R ~b

� �; ð3Þ

where ~a and ~b are fuzzy numbers in FðRÞ. It is obvious that we may write ~a� R~b if

and only if ~b� R~a.

158 M.-R. Ghasemi et al.

Suppose that R be any linear ranking function, then

(i) ~a� R~b if and only if ~a� ~b� R0 if and only if �~b� R � ~a,

(ii) if ~a� R~b and ~b� R~c, then ~a� R~c.

It should also be noted that for any fuzzy number ~a and ~b in FðRÞ, and any kbelonging to R and the ranking function R, we have

R k~aþ ~b� �

¼ kR ~að Þ þ R ~b� �

: ð4Þ

Let us continue by considering following definition:

Definition 1 If ~A ¼ ðaL; aU ; aL; aUÞ is a trapezoidal fuzzy number, then the sup-port of ~Aand the modal set of ~A are defined as aL � aL; aU þ aUð Þ and aL; aU½ �respectively. Also, the a-level set of ~A is defined as

�Aa ¼ x 2 R leA xð Þ� a���

n o¼ aL � aL þ aaL; aU þ aU � aaU� �

;

and it can be assumed that ~ALa ¼ aL � aL þ aaL and ~AU

a ¼ aU þ aU � aaU ,respectively.

In this study, we present in a linear ranking function which is similar to theranking function developed by [2], which is defined as follows:

R ~að Þ ¼ 12Z1

0

inf ~aa þ sup ~aað Þda; ð5Þ

Hence, the value of the ranking function for a trapezoidal fuzzy number ~a ¼aL; aU ; aL; aUð Þ is obtained as

R ~að Þ ¼ 12ðaL þ aUÞ þ 1

4ðaU � aLÞ: ð6Þ

3 Fuzzy CCR Model Using a Linear Ranking Function

Given that there are n DMUs to be evaluated, which use m inputs to produce soutputs with the data of inputs and outputs being uncertain a fuzzy trapezoidalwith bounded support can be expressed as

~xij ¼ xLij; x

Uij ; a

Lij; a

Uij

� �; i ¼ 1; . . .;m; j ¼ 1; . . .; n; ~yij ¼ yL

rj; yUrj ; b

Lrj;b

Urj

� �; r

¼ 1. . .; s; j ¼ 1; . . .; n:

7 Ranking of Fuzzy Efficiency Measures via Satisfaction Degree 159

We use the input-oriented CCR model [10] to evaluate the relative efficiency ofthis set of DMUs. The CCR model can be transformed into the following fuzzy LPform using a linear ranking function.

max ~ho ¼R

Xs

r¼1

ur~yro

subject to :Xm

r¼1

vi~xio ¼R 1;

Xm

r¼1

vi~xij� R

Xs

r¼1

ur~yrj; j ¼ 1; . . .; n;

ur � 0; vi� 0; i ¼ 1; . . .;m; j ¼ 1; . . .; n;

ð7Þ

where ‘¼R’ and ‘� R’ indicate equality and 1 with respect to the ranking functionR.

In DEA, we use the optimal value of the objective function to compute the

efficiency of DMUo. Similarly, in the proposed model (7) the value of ~ho is appliedto clarify the fuzzy efficiency measure of DMUo.

The above fuzzy LP problem can be transformed into its crisp equivalent form.Let us continue by considering the expressions (1), (3), (4), and (6). According tothese expressions, the above fuzzy DEA model (7) can be expressed as follows:

maxho ¼Xs

r¼1

ur12

yLro þ yU

ro

� �þ 1

4bU

ro � bLro

� �

subject to :Xm

r¼1

vi12

xLio þ xU

io

� �þ 1

4aU

io � aLio

� �

¼ 1;

Xm

r¼1

vi xLij þ xU

ij þ12

aUij � aL

ij

� �

�Xs

r¼1

ur yLrj þ yU

rj þ12

bUrj � bL

rj

� �

; j ¼ 1; . . .; n;

ur � 0; vi� 0; i ¼ 1; . . .;m; j ¼ 1; . . .; n;

ð8Þ

Similar to DEA models, the optimal value of the objective function, ho in theproposed model (8), is used to find the crisp efficiency measure of DMUo. Ifho ¼ 1; we consider DMUo as efficient; otherwise, it is considered to be inefficient.The fuzzy efficiency of an evaluated DMU with the trapezoidal fuzzy output~yro ¼ yL

ro; yUro; b

Lro; b

Uro

� �is defined as a trapezoidal fuzzy number as follows:

~h�o ¼Xs

r¼r

u�r yLro;Xs

r¼r

u�r yUro;Xs

r¼r

u�r bLro;Xs

r¼r

u�r bUro

!

: ð9Þ

160 M.-R. Ghasemi et al.

Consider a numerical example with 10 DMUs, where each DMU consists oftwo asymmetric trapezoidal fuzzy inputs and outputs. This example is taken from[11] and it is reproduced as Table 1.

Using the data in Table 1, the results of the crisp efficiencies and fuzzy effi-ciencies obtained by model (8) and expression (9) are provided in Table 2. Withthe exception of DMU 1, 5 and 6, all the DMUs are efficient. DMUs 1, 5 and 6,assumed crisp inefficient values of 0.293, 0.446 and 0.450, respectively. The fuzzyefficiency values of DMUs 1, 5, and 6 are (0.295, 0.305, 0.039, 0.010), (0.436,0.456, 0.021, 0.021), and (0.437, 0.469, 0.033, 0.023) respectively. The rest of thefuzzy efficiency measures are presented in Table 2. It is obvious that efficientDMUs have larger fuzzy efficiency values than inefficient DMUs (as seen inTable 2).

Here, both crisp and fuzzy efficiency measures are considered. However, thefuzzy efficiency measures can provide a clear indication on the uncertainty inhuman decision making, thus making fuzzy measures obtained from fuzzyobservation more informative than crisp measures where the measures areassumed to be precise. Most importantly, the measures should reflect the realsituation, and the decision maker (DM) must not be overly confident with the crispresults, without taking into account their fuzzy counterparts.

Table 1 DMUs with two fuzzy inputs and fuzzy outputs

DMUs Inputs Outputs

~X1 ~X2 ~Y1 ~Y2

1 (16, 17, 1, 2) (40, 42, 2, 2) (30, 31, 4, 1) (16, 18, 2, 14)

2 (8, 11, 4, 1) (14, 18, 2, 4) (40, 44, 2, 2) (75.5, 79.5, 3.5, 4)

3 (14, 16, 7, 1) (11, 13, 1, 1) (10, 12, 4, 1) (86, 88, 7, 1)

4 (11, 13.5, 5.5, 1) (12.25, 15.5, 1.25, 2.5) (25, 28, 3, 1.5) (81, 82, 3, 5.5)

5 (16, 18, 7, 1) (46, 48, 2, 4) (42, 44, 2, 2) (16, 16, 3, 4.5)

6 (13.5, 15.75, 7.25, 1) (29.25, 31.75, 1.75, 3.25) (33.5, 36, 2.5, 1.75) (47.6, 49.87, 3.87, 3.5)

7 (8, 18, 5, 1) (15, 16, 2, 2) (40, 42, 2, 4) (76, 79.5, 5, 3)

8 (8, 12, 3, 1) (16, 16, 2, 4) (40, 46, 4, 2) (78, 79.5, 6, 4)

9 (6, 9, 4, 1) (12.5, 20, 2, 5) (40, 44, 1, 1) (74, 79.5, 5.5, 4.5)

Table 2 Result of crispefficiency and fuzzyefficiency measures

DMUs Crisp efficiency measures Fuzzy efficiency measures

(the value of h�o) (the value of~h�o)

1 0.293 (0.295, 0.305, 0.039, 0.010)

2 1.000 (0.966, 1.032, 0.046, 0.050)

3 1.000 (1.006, 1.029, 0.082, 0.012)

4 1.000 (0.980, 1.016, 0.055, 0.065)

5 0.446 (0.436, 0.456, 0.021, 0.021)

6 0.450 (0.437, 0.469, 0.033, 0.023)

7 1.000 (0.976, 1.023, 0.058, 0.061)

8 1.000 (0.965, 1.054, 0.087, 0.049)

9 1.000 (0.963, 1.042, 0.057, 0.048)

7 Ranking of Fuzzy Efficiency Measures via Satisfaction Degree 161

4 Ranking of Fuzzy Numbers

Ranking of fuzzy numbers is an important component of a decision process. Sincefuzzy numbers do not form a natural linear order, like real numbers, a key point inthe application of fuzzy set theory is how to compare fuzzy numbers. Here, weshall propose the fuzzy satisfaction degree using the following definition:

Definition [12] 2 Suppose that ~a and ~bare two fuzzy numbers in F Rð Þ, and at least one of them is not a real number.

The fuzzy satisfaction degree of ~a� ~b is defined as

P ~a� ~b� �

¼R1

0½max 0; bU

a � aLa

� �� max 0; bL

a � aUa

� ��da

R1

0aU

a � aLa

� �daþ R1

0ðbU

a � bLaÞda

: ð10Þ

According to Definition 2, P ~a� ~b� �

obviously has the following properties:

(1) P ~a� ~b� �

þ P ~b� ~a� �

¼ 1;

(2) P ~a� ~b� �

¼ 1 if and only if bL0 � aU

0 :

(3) P ~a� ~b� �

¼ 1 if and only if aL0 � bU

0

(4) If ~a � ~b, then P ~a� ~b� �

¼ P ~b� ~a� �

¼ 12.

(5) 0�P ~a� ~b� �

� 1 and 0�P ~b� ~a� �

� 1:

For example applying (10) for the four fuzzy numbers ~a ¼ 6; 7; 1; 1ð Þ,~b ¼ 6:5; 7:5; 2:5; 1:5ð Þ, ~c ¼ 3; 4:5; 1; 1:5ð Þ, and ~d ¼ 8; 9; 1; 1ð Þ yields

P ~a� ~b� �

¼ P ~b� ~a� �

¼ 0:5;

P ~a�~cð Þ ¼ 0:042 and P ~c� ~að Þ ¼ 0:958;

P ~a� ~d� �

¼ 0:938 and P ~d� ~a� �

¼ 0:062;

P ~b�~c� �

¼ 0:095 and P ~c� ~b� �

¼ 0:905;

P ~b� ~d� �

¼ 0:926 and P ~d� ~b� �

¼ 0:074;

P ~c� ~d� �

¼ 1 and P ~d�~c� �

¼ 0:

162 M.-R. Ghasemi et al.

For n fuzzy numbers ~aj j ¼ 1; . . .; nð Þin F Rð Þ, it can be assumed that Pij ¼Pð~ai� ~ajÞ and these values can then form a matrix:

P ¼

P11 P12 . . . P1n

P21 P22 . . . P2n

..

.

Pn1

..

.

Pn2

. . .

. . ....

Pnn

2

6664

3

7775

ð11Þ

According to the properties of the fuzzy satisfaction degree,8ij i ¼ 1; . . .n; j ¼ 1; . . .nð Þ;Pij þ Pji ¼ 1 and Pii ¼ 1

2. If we assume thatPi ¼

Pnj¼1 Pij; i ¼ 1; . . .; n, then the ranking of Pi can provide the ranking of the

fuzzy numbers.By considering the values of the fuzzy satisfaction degree of the above fuzzy

numbers ~a ¼ 6; 7; 1; 1ð Þ, ~b ¼ 6:5; 7:5; 2:5; 1:5ð Þ, ~c ¼ 3; 4:5; 1; 1:5ð Þ, and~d ¼ 8; 9; 1; 1ð Þ, matrix P is obtained as

P ¼

0:5 0:5 0:958 0:0620:5 0:5 0:905 0:074

0:042 0:095 0:5 00:938 0:926 1 0:5

2

664

3

775

So P4 ¼ 3:364 [ P1 ¼ 2:02 [ P2 ¼ 1:979 [ P3 ¼ 0:637 therefore, theabove fuzzy numbers can be ranked as ~d [ ~a J ~b [~c.

The results can be interpreted in the following way. Since ~d [ ~a with degree of0.938 and ~b [~c with degree of 0.905 it can be concluded that ~d is almost strictlygreater than ~a and ~b is almost strictly greater than ~c. Whereas, P ~a� ~b

� �¼

P ~b� ~a� �

¼ 0:5, which means that these two fuzzy numbers have almost the same

valuation which is why it is expressed as ~a J ~b. However, using the aboveproposed ranking method it can be stated that one of these two fuzzy numbers canbe preferred over another. Thus, the ranking of these fuzzy numbers can be shownas follows:

~d [ 0:938 ~aJ0:5~b [ 0:905 ~c

5 Ranking the Fuzzy Efficiency Measures Using the FuzzySatisfaction Degree

In the standard DEA models, inefficient DMUs have scores less than one. How-ever, efficient DMUs are identified by an efficiency score equal to one; so theseDMUs cannot be ranked. This is a problem that has been frequently discussed in

7 Ranking of Fuzzy Efficiency Measures via Satisfaction Degree 163

the literature which attributes to the lack of discrimination in DEA applications. Toovercome these discrimination power problems, a procedure for ranking efficientunits, called the super-efficiency model was first proposed by Andersen andPetersen (AP) [13].

Similar to the standard DEA models, there is a need to rank the efficient andinefficient DMUs in the Fuzzy DEA model (7). The method of ranking of fuzzynumbers in Sect. 3 can be used to rank the fuzzy efficiency values in model (7).Therefore, by considering the fuzzy efficiency measures in Table 2, and theircorresponding fuzzy satisfaction degree, matrix P (11) and the value ofPi i ¼ 1; . . .; 9ð Þ for each row of matrix can be presented as follows:

P ¼

0:5 0 0 0 0 0 0 0 0

1 0:5 0:529 0:498 1 1 0:501 0:511 0:504

1 0:471 0:5 0:467 1 1 0:471 0:490 0:478

1 0:502 0:533 0:5 1 1 0:503 0:513 0:506

1 0 0 0 0:5 0:469 0 0 0

1 0 0 0 0:531 0:5 0 0 0

1 0:499 0:529 0:497 1 1 0:5 0:511 0:504

1 0:489 0:510 0:487 1 1 0:489 0:5 0:493

1 0:496 0:522 0:494 1 1 0:496 0:507 0:5

2

66666666666666664

3

77777777777777775

and

P1 ¼ 0:500

P2 ¼ 6:043

P3 ¼ 5:877

P4 ¼ 6:057

P5 ¼ 1:969

P6 ¼ 2:031

P7 ¼ 6:040

P8 ¼ 5:967

P9 ¼ 6:016

Based upon the above values of Pi i ¼ 1; . . .; 9ð Þ it can be concluded that,

P4 [ P2 [ P7 [ P9 [ P8 [ P3 [ P6 [ P5 [ P1:

Therefore, the fuzzy efficiency measures in Table 2 can be ranked as

~h�4J0:502~h�2J0:501

~h�7J0:504~h�9 [ 0:507

~h�8 [ 0:510~h�3 [ 1

~h�6 [ 0:531~h�5 [ 1

~h�1:

6 Conclusion

We have established a fuzzy DEA approach using the concept of comparing fuzzynumbers through a linear ranking function that could incorporate fuzzy inputs andfuzzy outputs that are asymmetrical in nature. The input-oriented CCR model wasemployed as a base for the fuzzy DEA-CCR model, which can be converted intoits crisp equivalent form. Since the fuzzy efficiency measures can provide a clearindication of the uncertainty in human decision making, fuzzy efficiency measuresare provided as a means to gauge the reliability of its crisp equivalent. However, inorder to rank the fuzzy efficiency values, one fuzzy efficiency measure needs to beevaluated and compared to others, which may not be straightforward. We haveshown a ranking method which could be used to rank the fuzzy efficiency mea-sures of DMUs.

164 M.-R. Ghasemi et al.

References

1. Chang, W.: Ranking of fuzzy utilities with triangular membership functions. In: Proceedingsof International Conference on Policy Analysis and Systems, pp. 263–272 (1981)

2. Fortemps, P., Roubens, M.: Ranking and defuzzification methods based on areacompensation. Fuzzy Sets Syst. 82, 319–330 (1996)

3. Maleki, H.R.: Ranking functions and their applications to fuzzy linear programming. Far EastJ. Math.Sci. 4, 283–301 (2002)

4. Mahdavi-Amiri, N., Nasseri, S.H.: Duality in fuzzy number linear programming by use of acertain linear ranking function. Appl. Math. Comput. 180, 206–216 (2006)

5. Noora, A.A., Karami, P.: Ranking Functions and its Application to Fuzzy DEA. Int MathForum 3, 1469–1480 (2008)

6. Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119,149–160 (2001)

7. Maleki, H.R., Tata, M., Mashinchi, M.: Linear programming with fuzzy variables. Fuzzy SetsSyst. 109, 21–33 (2000)

8. Tanaka, H., Ichihashi, H., Asai, K.: A Formulation of fuzzy linear programming problembased on comparison of fuzzy numbers. Control Cybern. 13, 185–194 (1984)

9. Liou, T.-S., Wang, M.-J.J.: Ranking fuzzy numbers with integral value. Fuzzy Sets Syst. 50,247–255 (1992)

10. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units.Eur. J. Oper. Res. 2, 429–444 (1978)

11. Soleimani-damaneh, M.: Fuzzy upper bounds and their applications. Chaos Soliton. Fract. 36,217–225 (2008)

12. Liu, X.: Measuring the satisfaction of constraints in fuzzy linear programming. Fuzzy SetsSyst. 122, 263–275 (2001)

13. Andersen, P., Petersen, N.C.: A procedure for ranking efficient units in data envelopmentanalysis. Manag. Sci. 39, 1261–1264 (1993)

7 Ranking of Fuzzy Efficiency Measures via Satisfaction Degree 165

Chapter 8Inexact Discretionary Inputs in DataEnvelopment Analysis

Majid Zerafat Angiz Langroudi

Abstract In this chapter, the relationship between fuzzy concepts and theefficiency score in Data envelopment analysis (DEA) is dealt with. A new DEAmodel for handling crisp data using fuzzy concept is proposed. In addition, therelationship between possibility sets and the efficiency score in the traditional crispCCR model is presented. The relationship provides an alternative perspective ofviewing efficiency. With the usage of the appropriate fuzzy and possibility sets torepresent certain characteristics of the input data, many DEA models involvinginput data with various characteristics could be studied. Furthermore, based uponthe proposed models, two nondiscretionary models are introduced in which someinputs or outputs, in a fuzzy sense, are inexact discretionary variables. For thispurpose, a two-stage algorithm will be presented to treat the DEA model in thepresence of an inexact discretionary variable. With this relationship, a newperspective of viewing and exploring other DEA models is now made possible.

Keywords Data envelopment analysis � Fuzzy � Possibility distribution �Efficiency � Non-discretionary variables

1 Introduction

Since its inception 48 years ago, the theory of fuzzy sets has advanced in a varietyof ways and in many disciplines. Applications of fuzzy technology can be found inartificial intelligence, computer sciences, control engineering, decision theory,expert systems, logic, management sciences, operations research, robotics andothers [1].

M. Z. A. Langroudi (&)School of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysiae-mail: mzerafat24@yahoo.com

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_8, � Springer-Verlag Berlin Heidelberg 2014

167

Sugeno [2] defined a fuzzy measure. Banon [3] shows that very many measureswith finite universe, such as probability measures, belief functions, plausibilitymeasures and so on, are fuzzy measures in the sense of Sugeno. In recent years,some specific interpretations of fuzzy set theory have been suggested. One of themis possibility theory. In the framework of fuzzy set theory, Zadeh [4] introduced thenotion of a possibility distribution and the concept of a possibility measure, which isa special type of fuzzy measure proposed by Sugeno [2]. Possibility theory focusesprimarily on imprecision. Possibility theory used to correspond, roughly speaking,to the min–max version of fuzzy set theory, that is, to fuzzy set theory in which theintersection is modeled by the min operator and the union by the max operator. Thisinterpretation of possibility theory, however, is no longer correct. Rather, it hasbeen developed into a well-founded and comprehensive theory.

DEA researchers have begun using fuzzy concept for measuring efficiency andproductivity of DMUs since 1992. Some existing approaches for solving fuzzyDEA are the tolerance approaches; the fuzzy ranking approaches; the a-level basedapproaches; the defuzzification approaches; and the possibility approach [5].

Guo et al. [6] were pioneers in using the fuzzy DEA models based on possibilityand necessity measures. Alp [7] further extended the concept of possibilistic DEAby considering problems in handling real data such as fuzziness, impreciseness andincompleteness. In such cases, the difficulty in model building can be overcome byusing fuzzy set theory and concepts. In his study, only situations of incompletedata were considered and a new method for possibilistic DEA was introduced.Saati et al. [8] introduced a mathematical programming approach for measuringtechnical efficiency in a possibilistic environment.

Lertworasirikul et al. [9] further expanded the possibility approach to the FuzzyDEA model from an optimistic viewpoint. Lertworasirikul et al. [10] furtherdeveloped a fuzzy BCC model where the possibility and credibility approaches areprovided and compared with a s!Z level based approach for solving the fuzzyDEA models. Using the possibility approach, they revealed the relationshipbetween the primal and dual models of fuzzy BCC. Using the credibility approachthey showed how the efficiency value for each DMU can be obtained as a repre-sentative of its possible range.

Liu and Chuang [11] developed a method to find the fuzzy efficiency measuresembedded with the assurance region (AR) concept when some observations arefuzzy numbers. They utilized Zadeh’s extension principle to transform a fuzzyDEA-AR model into a family of crisp DEA-AR models to calculate the lower andupper bounds of efficiency scores at a specific level. From different possibilitylevels, a membership function is derived accordingly.

This research, however, is not about fuzzy DEA. It does not deal with fuzzyinput or output data. It is about the use of the fuzzy concept to handle crisp data inDEA. The usage of fuzzy concepts in handling certain crisp mathematical modelingsituations has resulted in the formulation of creative and efficient procedures. Thiscan be seen for example in the work of Zerafat Angiz et al. [12] and Emrouznejadet al. [13]. Motivated by these results, in this research an alternative interpretation

168 M. Z. A. Langroudi

of the CCR model using the possibility set is presented. The interpretation opens awindow for viewing efficiency in a new perspective. This new perspective is veryuseful in certain applications of the DEA models. An example application is shownin the handling of non-discretionary data.

Banker and Morey [14] initially introduced a DEA model in which the inputsare divided into two non-discretionary and discretionary variables. A number ofapproaches have been proposed for handling non-discretionary DEA models.Among them are the works of Golany and Roll [15], Ruggiero [16, 17], Muniz[18], Muniz et al. [19] and Cordero-Ferrera [20].

The remainder of this research is organized as follows. Section. 2 providessome background information about the possibility sets. The relationship betweenefficiency in DEA and the possibility sets together with some interpretations ofefficiency are presented in Sect. 3. An alternative view of efficiency based on fuzzyconcepts is presented in Sect. 4. A case study is illustrated in Sect. 5. In Sect. 6, anew non-discretionary DEA model is introduced. Finally, the conclusion is pre-sented in Sect. 7.

2 Fuzzy Events via Possibility Measures

In the framework of fuzzy set theory, Zadeh [21] proposed possibility theory whichis a special type of fuzzy measure for modeling and characterizing situationsinvolving uncertainty. The following definitions have been extracted fromLertworasirikul et al. [9, 10]:

Let P Xið Þ i ¼ 1; 2; . . .; nð Þ be the power set of a set Xi i ¼ 1; 2; . . .; nð Þ. A pos-sibility measure is a function p : P Xið Þ ! 0; 1½ � with the properties

1: p ;ð Þ ¼ 0; p Xið Þ ¼ 1

2: A � B) p Að Þ� p Bð Þ

3: p[

i2I

Ai

!

¼ supi2I

p Aið Þ Ai � P Xið Þ with an index set I;

ð1Þ

Xi;P Xið Þ; pið Þ is called the possibility space.

Based on the possibility measure, Zadeh [21] defines a fuzzy variable ~n asfollows:

l~n sð Þ ¼ p xi 2 Xi~n xið Þ ¼ s��

n o� �

¼ supxi2Xi

p xif g ~n xið Þ ¼ s��

n o; 8s 2 R:

ð2Þ

The Cartesian product of the possibility space X;P Xð Þ; pð Þ in whichX ¼ X1 � X2 � � � � � Xn, is defined as follows:

p Að Þ ¼ mini¼1;2;...;n

pi Aið Þ A ¼ A1 � A2 � � � � � Anj ;Ai 2 P Xið Þf g:

8 Data Envelopment Analysis 169

Consider ~a and ~b as fuzzy variables on the possibility spaces X1;P X1ð Þ; p1ð Þ andX2;P X2ð Þ; p2ð Þ, respectively. On the product possibility space X ¼ X1 � X2;ð

P Xð Þ; pÞ, the fuzzy event ~a� ~b is defined as

p ~a� ~b� �

¼sup

x12X1x22X2

p x1; x2ð Þ ~a x1ð Þ� ~b x2ð Þ��

� �� �

¼

The possibility of fuzzy event ~a� ~b is obtained from Expression (2) as follows:

p ~a� ~b� �

¼ sups;t2R

min l~a sð Þ; l~b tð Þ� �

js� t� �

:

Furthermore, the possibility of the fuzzy event a� ~b in which a is a crisp valueis given as

p a� ~b� �

¼ supt2R

l~b tð Þ s� tj� �

:

Lertworasirikul et al. [10] proved the following Lemma using the possibility offuzzy events concept.

Lemma 1 Let ~a1; ~a2; . . .; ~an be fuzzy variables with normal and convex mem-bership functions. Then,

p ~a1 þ ~a2 þ � � � þ ~an� bð Þ� a if only if ~a1ð ÞUa þ ~a2ð ÞUa þ � � � þ ~anð ÞUa � b: ð3Þ

where the symbol :ð ÞUa denotes the upper bound of the a-level set of~ai i ¼ 1; 2; . . .n:

If the fuzzy number ~ri ¼ ~rið ÞL0 ; ~rið ÞL1 ; ~rið ÞU1 ; ~rið ÞU0� �

; i ¼ 1; 2; . . .; n is trapezoi-dal, for any level of a such that 0� a� 1, the following is true:

p ~r1 þ ~r2 þ � � � þ ~rn� bð Þ� a if only if

1� að Þ ~r1ð ÞU0 þ � � � þ ~rnð ÞU0� �

þ a ~r1ð ÞU1 þ � � � ~rnð ÞU1� �

� bð4Þ

Figure 1 shows the trapezoidal fuzzy number ~r ¼ ~rð ÞL0 ; ~rð ÞL1 ; ~rð ÞU1 ; ~rð ÞU0� �

.

3 Interpretation of the CCR ModelUsing the Possibility Set

In this section, we present an alternative interpretation of the efficiency concept indata envelopment analysis. To begin with, we start with the CCR model and after afew substitutions and changes a possibility form of the model is obtained.A graphical illustration and explanation is given next and followed by somediscussion.

170 M. Z. A. Langroudi

3.1 Derivation of the Possibility Model

Consider the following DEA Model introduced by Charnes et al. [22]:

min h

s:t: hxip �Xn

j¼1

kjxij� 0 i ¼ 1; 2; . . .;m

Xn

j¼1

kjyrj� yrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n:

ð5Þ

In Model (5), xij i ¼ 1; 2; . . .;m; j ¼ 1; 2; . . .; nð Þ represents the quantity of inputi consumed by DMUj and yrj r ¼ 1; 2; . . .; s; j ¼ 1; 2; . . .; nð Þ is the quantity ofoutput r produced by DMUj. The variable h measures the efficiency of DMUp, andkj j ¼ 1; 2; . . .; nð Þ are the raw weights assigned to the peer DMUs when solving theDEA model.

By adding hPn

j¼1kjxij � h

Pn

j¼1kjxij to inequality (5), the following mathematical

programming model is obtained:

min h

s:t: hxip � hPn

j¼1kjxij þ h

Pn

j¼1kjxij �

Pn

j¼1kjxij� 0 i ¼ 1; 2; . . .;m

Pn

j¼1kjyrj� yrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n:

ð6Þ

0( )Lr 1( )L1( )U

0( )U

r

( )r

sμ

s

Fig. 1 Trapezoidal fuzzynumber ~r ¼ ~rð ÞL0 ; ~rð ÞL1 ;

�

~rð ÞU0 ; ~rð ÞU1 Þ

8 Data Envelopment Analysis 171

After some rearrangement, Model (6) is written as follows:

min h

s:t: hðxip �Pn

j¼1kjxijÞ þ 1� hð Þð0�

Pn

j¼1kjxijÞ� 0 i ¼ 1; 2; . . .;m

Pn

j¼1kjyrj� yrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n

ð7Þ

Assume that xip

� �U

1 ¼ 0 and xip

� �U

0 ¼ xip. In addition, consider the followingtrapezoidal fuzzy number:

~xip ¼ ~xip

� �L

0 ; ~xip

� �L

1 ; ~xip

� �U

1 ; ~xip

� �U

0

� �¼ �xip;�xip; 0; xip

� �:

Figure 2 illustrates such a fuzzy number.Using the fuzzy number ~xip, Model (7) is written as follows:

min h

s:t: h xip

� �U

0 �� Xn

j¼1

kjxij

�þ 1� hð Þ xip

� �U

1 �� Xn

j¼1

kjxij

�� 0 i ¼ 1; 2; . . .;m

Xn

j¼1

kjyrjrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n:

ð8Þ

By assuming that h ¼ 1� b, Model (8) is converted to the following non-linearprogramming problem:

1�max b

s:t: 1� bð Þ xip

� �U

0

��Xn

j¼1

kjxij

�þ b xip

� �U

1 �� Xn

j¼1

kjxij

�� 0 i ¼ 1; 2; . . .;m

Xn

j¼1

kjyrjrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n:

ð9Þ

Based on Expression (4), the possibility programming formulation of the DEAmodel is thus obtained as follows:

172 M. Z. A. Langroudi

max b

s:t: ðpð~xip�Xn

j¼1

kjxijÞ� 0Þ� b i ¼ 1; 2; . . .;m

Xn

j¼1

kjyrj� yrp r ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n:

ð10Þ

Afterwards, the above model is called the Possibilistic CCR model. Obviously,if equation

Pnj¼1 kj ¼ 1 is added to Model (5), the BCC model is obtained. Since

the equation does not play any role in the inequalities associated with the possi-bility sets of Model (10), the above derivation is also valid for the input orientedBCC model.

4 A Traditional CCR Model Basis on the Fuzzy Concept

In this section an alternative interpretation of efficiency based on the fuzzy conceptis presented. For this end, a new DEA model for handling crisp data using thefuzzy concept is proposed. Assume that all postulates to construct the productionpossibility set corresponding to constant return to scale are satisfied. Therefore, theproduction possibility set corresponding to the CCR model and the proposedmodel are similar. To present the new model, we introduce a triangular fuzzynumber with the following membership function.

In Model (5) consider inputs xip and outputs yrp related to the DMU underevaluation, say DMUp. Then, the membership function of fuzzy number ~xip isconsidered as follows:

l~xipxip

� �¼ xip � xip

xipxip� xip i ¼ 1; 2; . . .;m ð11Þ

ipx

0 1( ) ( )L Lip ip ipx x x= = − 0( )U

ip ipx x= 0( )Uip ipx x=

( )ipx

sμ

s

Fig. 2 Membership functionof the fuzzy number ~xip

8 Data Envelopment Analysis 173

The following linear programming model is proposed:

max�min model

max min l~xipxip

� �n oi ¼ 1; 2; . . .;mð Þ

n o

s:t:

Xn

j¼1

kjxij��xip i ¼ 1; 2; . . .;m

Xn

j¼1

kjyrj� yrp j ¼ 1; 2; . . .; s

�xip� xip i ¼ 1; 2; . . .;m

kj� 0 j ¼ 1; 2; . . .; n

ð12Þ

where �xip indicates the inputs that are necessary for the DMUp to be efficient.Obviously, Model (12) is always feasible and the value of optimum objective

function in (12) is non-negative and less than or equal to 1.Since the objective functions are the membership functions, the optimal value

in (12) does not exceed the maximum value of the membership values, i.e. a valueof 1. The values kp ¼ 1; kj ¼ 0 j 6¼ pð Þ and �xip ¼ xip are the feasible solutions inModel (12); thus it is always feasible.

Assume that a ¼ min l~xip�xip

� �n o, then:

max a

s:t:

C1Xn

j¼1

kjxij� xip i ¼ 1; 2; . . .;m

C2Xn

j¼1

kjyrj� yrp

C3 a� xip � xip

xipi ¼ 1; 2; . . .;m

C4 xip� xip i ¼ 1; 2; . . .;m

kj� 0 j ¼ 1; 2; . . .; n

a� 0

ð13Þ

This problem is now solved for each DMU.The main idea of our proposal is that there exists a relationship between the

efficiency scores in Model (5) and the membership values of the fuzzy numbers ~xip.The lesser the membership values the higher the efficiency score of the DMUunder evaluation.

174 M. Z. A. Langroudi

The following theorem indicates equivalency between the efficiency scores ofModels (5) and (13).

Theorem 1 Assume that h ¼ minh and a ¼ maxa are the optimal values of (5)and (13), respectively. Then, h ¼ 1� a. In other words, the treatment of Model(13) and the CCR model are similar.

In Model (6), consider a� xip�xip

xip, implying that axip� xip � xip, and

xip� xip � axip ¼ 1� að Þxip ð14Þ

There is a one-to-one correspondence between the values �xip and 1� að Þxip, andtherefore we can consider Eq. (14). Replacing the last inequality (14) in C1 inModel (13), the following linear programming problem is obtained:

max a

s:t:

C01Xn

j¼1

kjxij� xip� 1� að Þxip i ¼ 1; 2; . . .;m

C02Xn

j¼1

kjyrj� yrp

C03 a� xip � xip

xipi ¼ 1; 2; . . .;m

C04 xip� xip i ¼ 1; 2; . . .;m

kj� 0 j ¼ 1; 2; . . .; n

a� 0

ð15Þ

Assume h ¼ 1� a. Then Model (15) is converted to the following:

max 1� h

s:t:

C001Xn

j¼1

kjxij��xip� hxip i ¼ 1; 2; . . .;m

C002Xn

j¼1

kjyrj� yrp

C003 1� h� xip � �xip

xipi ¼ 1; 2; . . .;m

C004 �xip� xip i ¼ 1; 2; . . .;m

kj� 0 j ¼ 1; 2; . . .; n

h� 0

ð16Þ

8 Data Envelopment Analysis 175

In C003; 1� h� xip��xip

xipimplies that hxip��xip, thus, C003 is implied by C001. Since

0� xip��xip

xip� 1, 0� h� 1 is obtained from C003. On the other hand, considering

max 1� hð Þ ¼ 1�minh, Model (16) is written as follows:

1�min h

s:t: C001Pn

j¼1kjxij� hxip i ¼ 1; 2; . . .;m

C002Pn

j¼1kjyrj� yrp

kj� 0 j ¼ 1; 2; . . .; n

h� 0

ð17Þ

This means that h ¼ 1� a.By some substitutions, Model (5) implies Model (13). Simply put, Model (13)

demonstrates a fuzzy interpretation of the CCR model. The efficiency score in thismodel is the membership value of the point located at the intersection of the fuzzyinterval and the efficiency frontier.

4.1 Graphical Illustration

For a better understanding of the relationship between h and 1� a and also therelationship between the efficiency in the CCR model and the possibility set, werefer to Fig. 3a and b which provide two two-dimensional diagrams of a simpleefficiency case study in which only a single input (x) is used to produce a singleoutput (y). There are a number of DMUs in this case study, however, only the data

2 1( , )B x y3 1( , )C x y

D

E

*θ

Input

OutputEfficient frontier

1 1( , )A x y

DMUA3 1

( , )C x y

1 1( , )A x y2 1( , )B x y

H

F

*α

Output

Input

Efficient frontier

DMUA

(a) (b)

Fig. 3 a A fuzzy number related to efficiency. b An alternative fuzzy number related toefficiency

176 M. Z. A. Langroudi

for DMUA is plotted on the diagrams. Triangles CAE and CAF are the membershipfunctions of the fuzzy numbers which correspond to the input of DMUA. InFig. 3a, the membership function is in the form of the possibility aspect, whereasin Fig. 3b, it is in the impossibility aspect. These membership functions aresuperimposed onto the graphs. Notice that, the shape of the membership functionsreflects the characteristic of the input variables in the DEA (i.e., the smaller thelevel of the input the more efficient is the DMU).

In referring to Fig. 3a, BD is parallel to AE; therefore the two triangles CEA andCDB are similar. So, we have

CB

CA¼ BD

AEð18Þ

It is well known that the value CBCA is equivalent to the technical efficiency of

DMUA. Thus, since AE = 1, BDAE = h is the technical efficiency of DMUA which is

also the membership value corresponding to point B, the reference point of DMUA

on the efficient frontier.Furthermore, we consider Fig. 3b in which triangle CAF is similar to triangle

BAH. Therefore, based on Thales theorem or the Basic Proportionality theoremwhich states that the line drawn parallel to one side of a triangle divides the othertwo sides in the same ratio, we have

BA

CA¼ BH

CFCF ¼ 1ð Þ ð19Þ

And thus,

1� BA

CA¼ CA� BA

CA¼ CB

CAð19:1Þ

Obviously, the following expressions are obtained from expressions (18), (19),and (20):

BD ¼ 1� BH or h ¼ 1� a ð20Þ

In fact, the above equations illustrate the relationship between the efficiency inDEA and the optimal a- level of the new fuzzy number.

4.2 Discussion

We refer to Fig. 3a and b again which provide two two-dimensional diagramsillustrating the possibility interpretation of the efficiency of DMUA. For this pur-pose, two antonym keywords, production possibility and production impossibility,

8 Data Envelopment Analysis 177

are utilized. It is clear that if we consider 0� b� 1 as a measure of the productionpossibility, then 1� b is the measure of the production impossibility. In Fig. 3b,since point A is an observed value for DMUA, the value of the productionimpossibility corresponding to point A is considered zero. In other words, outputy1 can certainly be produced by input x1. If the input level is decreased to x3 ¼ 0,producing y1 is impossible. Therefore, the production impossibility of C x3; y1ð Þ is1. The production impossibility for producing output y1 using inputs betweenx3 ¼ 0 and x1 is a fuzzy concept which is illustrated by a triangular fuzzy numberdemonstrated by the dotted lines. At point B, which is on the efficient frontier, theproduction impossibility value is BH ¼ a. This value is related to the efficiencyscore of DMUA.

In Fig. 3a an alternative interpretation is presented based on the productionpossibility aspect. By reducing the input from the observed value at point A, wereached point B, which is on the efficient frontier. The production possibility valueat this point is BD ¼ h, which indicates the efficiency score of DMUA.

5 An Illustration Example

The following example demonstrates the correspondence between the proposedfuzzy and the possibilistic methodologies and the CCR model.

Assume that a major organization consists of six branches called DMUs. Due tothe recession, adjusting the budget is the organization’s agenda. Since the man-agement is interested in maintaining the current level of production, DEA as apowerful tool is chosen to determine the decrease in inputs by maintaining thecurrent level of production. In this case, the budget allocated to each branch isdivided into sub budgets, the budget related to the employee’s salary and theallocated budget associated with other affairs. In Table 1 a list of 6 DMUs withtwo inputs and two outputs measurements is given. Inputs I1 (in $100,000) and I2

(in $1,000,000) are the allocated budget associated with other affairs and thebudget regarding the employee’s salary, respectively. Outputs O1 (in 10,000 tons)

Table 1 The list of DMUs with two inputs and two outputs

DMUs I1 I2 O1 O2

1 1.50 1.50 1.40 0.352 4.00 0.70 1.40 2.103 3.20 1.20 4.20 1.054 5.20 2.00 2.80 4.205 3.50 1.20 1.90 2.506 3.20 0.70 1.40 1.50

178 M. Z. A. Langroudi

and O2 (in 10,000 tons) indicate the amount of the two products produced in thebranches. A ranking of these DMUs based on their efficiency scores is necessary.In this section, the data listed in Table 1 are ranked in order so that the proposedmethodology can be compared with the CCR model.

The efficiency of CCR and the proposed models are given in Table 2.Since h ¼ 1� a ¼ 1� b the results of the new proposed methods presented

in Sects. 3 and 4, and the CCR model are the same. After evaluation, managementis suggested to decrease the inputs of inefficient DMUs 1, 5 and 4 as follows:

DMU1 should decrease I1 and I2 from 1.50 and 1.50 to 1.07 and 1.07, respectively.DMU5 should decrease I1 and I2 from 3.50 and 1.20 to 3.46 and 1.19, respectively.DMU6 should decrease I1 and I2 from 3.20 and 0.70 to 2.86 and 0.63, respectively.

6 Inexact Discretionary Variables

The uses of fuzzy concepts in handling certain crisp mathematical modellingsituations have resulted in the formulation of creative and efficient procedures.This section describes another use of the fuzzy concept and the possibility in acrisp situation. To this end, an alternative application of the new fuzzy approachesin handling non-discretionary data is presented. For the newly proposed non-discretionary models, the usage of membership function replaces the need todetermine the discretionary index of a non-discretionary variable. The discre-tionary index concepts are used in some of the existing non-discretionary models.In real life applications, discretionary indexes are usually not known and arearbitrarily determined by decision makers.

Table 2 Efficiency scores for CCR and proposed models

DMUs CCR The result of possibilistic andfuzzy models (9) and (15)

h a and b

1 0.711111 0.2888892 1 03 1 04 1 05 0.988488 0.0115126 0.893962 0.106038

8 Data Envelopment Analysis 179

6.1 A Non-discretionary DEA Model Using the FuzzyConcept

In this section, an application of the possibility set concept that has been discussedin the previous section is presented. The context of the application is in thehandling of non-discretionary variables.

One of the significant concepts in data envelopment analysis is the use of non-discretionary variables. An input or output is called a non-discretionary variable ifit cannot be varied at the discretion of management or other users. Banker andMorey [14] were pioneers in this study by including non-discretionary variables inthe input-oriented DEA model. The Banker and Morey model, considering con-stant return to scale, is given by the following mathematical programming model:

min u

s:t:Pn

j¼1kjxij þ s�i ¼ u xip i 2 D

Pn

j¼1kjxij þ s�i ¼ xip i 2 ND

Pn

j¼1kjyrj � sþi ¼ yrp j ¼ 1; 2; . . .; s

kj� 0 j ¼ 1; 2; . . .; n

ð21Þ

In Model (19.1), s�i and sþi are the slack variables for the i-th input and r-thoutput, respectively, and the symbols D and ND refer to the discretionary andnon-discretionary variables, respectively.

6.2 Inexact Discretionary Variables Using Possibility Sets

Golany and Roll [15] pointed out that in many real-life efficiency studies, a factoris neither fully controllable nor totally uncontrollable. For example, managers canmake marginal alterations in personnel scheduling. However they have to complywith general guidelines of their organisation in many other aspects involving theuse of their human resources. In other words, the factor is partially controllable.To incorporate this factor into a DEA model, an index taking on the valuesbetween 0 and 1 is used to represent the degree of discretion that the DMU haswith respect to the factor. In this research, such a factor will be called an inexactdiscretionary variable and since a membership function (a fuzzy number concept)will be used to describe the factor instead of the discretionary index, the term fuzzynon-discretionary (FND) variable will also be used.

180 M. Z. A. Langroudi

Figure 4a and b illustrate two separate DEA studies with a single input and asingle output, in which the input variable corresponding to A is discretionary andthe input variable corresponding to B is inexact discretionary. The appropriatemembership functions are superimposed onto the graphs. Notice that, for themembership function of the discretionary variable, more weight is given for thesmaller values, reflecting a characteristic of the input variable in a traditional DEAmodel, which is, that the smaller the level of input the more efficient is the DMU.However, for the membership function of the inexact discretionary variable, moreweight is given for the observed/current value to reflect the reluctance on the partof the DMU to reduce the value.

The value x2 is the input level at point C, which is the projection of point B ontothe efficient frontier. In fact, x2 ¼ hx2 where h is the optimal solution of the CCRmodel related to DMUB (i.e., Model (1) where all the inputs are treated as dis-cretionary variables). The inclusion of the inexact discretionary variables or thefuzzy non-discretionary (FND) variables into Model (19.1) resulted in the fol-lowing non-linear programming models:

1�maxu ð22Þ

s:t: 1� uð Þ xip

� �U

0

��Xn

j¼1

kjxij

�þ u xip

� �U

1

��Xn

j¼1

kjxij

�� 0 i 2 ID ð22:1Þ

Xn

j¼1

kjxij� xip i 2 IND ð22:2Þ

1� uð Þ xip

� �U

0

��Xn

j¼1

kjxij

�þ u xip

� �U

1

�Xn

j¼1

kjxij

�� 0 i 2 IFND ð22:3Þ

1 1( , )A x y 2 2

( , )B x y

*2 2( , )C x y

DMUADMUB

Output

Input

Efficient frontier Efficient frontier

Output

Input

(a) (b)

Fig. 4 a Discretionary input. b Inexact discretionary input

8 Data Envelopment Analysis 181

Xn

j¼1

kjyrj� yrp

kj� 0 j ¼ 1; 2; . . .n

u� 0

ð22:4Þ

min d ð23Þ

s:t:C1 d xip

� �U

0

��Xn

j¼1

kjxij

�þ 1� dð Þ xip

� �U

1

��Xn

j¼1

kjxij

�� 0 i 2 ID

ð23:1Þ

C2Xn

j¼1

kjxij� xip i 2 IND ð23:2Þ

C3 d xip

� �U

0

��Xn

j¼1

kjxij

�þ 1� dð Þ xip

� �U

1

�Xn

j¼1

kjxij

�� 0 i 2 IFND

ð23:3Þ

C4Xn

j¼1

kjyrj� yrp

kj� 0 j ¼ 1; 2; . . .; n

d� 0

ð23:4Þ

The above models are obtained from Models (8) and (9) and can be solved usinga two-stage method. In the first stage, all variables are treated as discretionaryvariables and the traditional CCR model (5) or the newly proposed possibilityprogramming model (10) is used to find the efficiency score of each DMU (i.e. h).

These efficiency scores are then used to determine variables xip

� �U

1of Equations

(22.3) and (23.3). Variables xip

� �U

1are the projections of the inexact discretionary

variables xip onto the efficient frontier. They are determined using

xip

� �U

1¼ hxip ¼ xip. An example of such a variable is the value x2 in Fig. 4b. In the

second stage, the non-linear programming model (23) is transformed into a linearprogramming model and solved.

Once again, consider the case study in Sect. 5. The result of the model rec-ommends the management to adjust the inputs to gain an efficient status. Aftergetting this feedback from the DEA evaluation, the decision maker realized thatsuch a decrease in the employees’ salary may make trouble for organization(period). The management is aware that reducing the employee’s salary (I2) canlead to dangerous consequences. For instance, such adjustment can cause the

182 M. Z. A. Langroudi

s25 1( )Ux 1.186=

0( )U25x 1.20=

( )25x

sμ

Fig. 5 Membership function of the fuzzy number ex25

employee’s dissatisfaction which in turn influences the efficiency of organization.So, there is an inverse relationship between salary and employee’s satisfactionwhich can be described by a fuzzy number. Thus, they decide to gradually adjustthe reduction in the employee’s salary.

So we assume that the second input is an inexact discretionary variable.In the first stage of the proposed method where all variables are treated as

discretionary variables and the traditional CCR model or the newly proposedpossibility set equivalent model is used to find the efficiency score of each DMU,the result has been shown in Table 2. From the result, the values for variables

xip

� �U

1of Equation (23.3) are determined. As an example, for DMU5, x25

� �U

1 ¼hx25 ¼ 0:988488 1:2 ¼ 1:186. Based on the result, the management is suggestedto decrease the salary budget from $1,200,000 to $1,186,000. But management isaware that in these circumstances, such a reduction is not possible. Thus, thefollowing fuzzy number, based on the relationship between salary deduction andstaff satisfaction, is designed (Fig. 5):

Then, the following linear programming model for finding the efficiency ofDMU5 in stage 2 is solved:

minu

s:t:

1:50k1 þ 4:00k2 þ 3:20k3 þ 5:20k4 þ 3:50k5 þ 3:20k6� 3:50u

1:50k1 þ 0:70k2 þ 1:20k3 þ 2:00k4 þ 1:20k5 þ 0:70k6� 1:20uþ 1:186185 1� uð Þ1:40k1 þ 1:40k2 þ 4:20k3 þ 2:80k4 þ 1:90k5 þ 1:40k6� 1:90

0:35k1 þ 2:10k2 þ 1:05k3 þ 4:20k4 þ 2:50k5 þ 1:50k6� 2:50

ð24Þ

The overall result of the efficiency analysis when input 2 is inexact discre-tionary is shown in Table 3. Note that there are some suggestions for improvement

8 Data Envelopment Analysis 183

of both the input variables. As an example, for DMU5, reducing the input variable1 from 3.50 to �x15 ¼ 3:397 and the input variable 2 from 1.20 to �x25 ¼ 1:199 issuggested. This means, the reduction in salary budget is adjusted to $1,199,000instead of $1,186,000. As is seen in Table 3, the most adjusted salary reduction isapplied to DMU1 that is, �x21 ¼ 1:375. This means that the salary budget associatedwith DMU1 will be reduced from $1,500,000 to $1,375,000.

Efficiency scores corresponding to DMU1 in Tables 2 and 3, have not changed,because the efficient DMU associated with it, lies on the weak efficiency frontier.We can see this by solving the following dual form:

Max Z ¼ 1:40u1 þ 0:35u2

s:t:

1:50v1þ 1:50v2 ¼ 1

1:40u1 þ 0:35u2 � 1:50v1 þ 1:50v2ð Þ� 0

140u1 þ 2:10u2 � 4:00v1 þ 0:70v2ð Þ� 0

4:20u1 þ 1:05u2 � 3:20v1 þ 1:20v2ð Þ� 0

2:80u1 þ 4:20u2 � 5:20v1 þ 2:00v2ð Þ� 0

1:90u1 þ 2:50u2 � 3:50v1 þ 1:20v2ð Þ� 0

1:40u1 þ 1:50u2 � 3:20v1 þ 0:70v2ð Þ� 0

v1; v2; u1; u2� 0

The optimal solution u1 ¼ 0:004; u2 ¼ 0:006; v1 ¼ 0:007; v2 ¼ 0:000ð Þ con-firms such a claim.

6.3 Inexact Discretionary Variables the Using FuzzyConcept

In this sub section another approach based on the fuzzy concept is presented toanalyze an inexact discretionary variable. The Fig. 6a and b illustrate two DMUsA and B with a single input and a single output, in which the input variable

Table 3 Result of the CCR model in the presence of inexact discretionary variable

DMUj Inexact non-discretionary�x1j �x2j u

1 1.067 1.375 0.7112 – – 13 – – 14 – – 15 3.397 1.199 0.9716 2.581 0.686 0.807

184 M. Z. A. Langroudi

Ouput

Input

2 2( , )B x y

*2 2( , )C x y

Ouput

Input

1 1( , )A x y

(a) (b)

Fig. 6 a Efficiency in Fuzzy view of CCR. b Inexact discretionary input

corresponding to A is discretionary and the input variable corresponding to B isinexact discretionary.

The membership functions concerned with the input variables of the discre-tionary A and the inexact discretionary B are defined as follows:

l~x1x1ð Þ ¼

x1 � x1

x1x1� x1 and l~x2

x2ð Þ ¼x2 � x2

x2 � x02x2� x2 ð25Þ

The value x2 shows the input associated with the input variable of DMUC.DMUC is the decision-making unit corresponding to DMUB on the efficiencyfrontier. In fact, x2 ¼ hx2 in which h is the optimal solution of the CCR modelrelated to DMUB. Furthermore, assume that FND is the set of inexact discre-tionary. By adding the constraints related to FDN in Model (23), the followingmulti-objective linear programming problem is proposed:

max bi i 2 IFND

max a

s:t:

C1Pn

j¼1kjxij��xip i 2 ID

C2Pn

j¼1kjxij� xip i 2 IND

C3Pn

j¼1kjxij� xip i 2 IFND

C4Pn

j¼1kjyrjrp

C5 a� xip�xip

xipi 2 ID

C6 bi�xu

ip�xip

xuip�xl

ipi 2 IFND

a� 0; bi� 0 i 2 IFND; kj� 0 j ¼ 1; 2; . . .; n

ð26Þ

8 Data Envelopment Analysis 185

The above model is solved in a two-stage algorithm. At first, to the findmembership function related to C6, the point corresponding with the inexactdiscretionary input in the efficiency frontier should be recognized. xl

ip, in C6,

shows such a point and is obtained as xlip ¼ hxip ¼ xip. In Fig. 6a, the input x2 is

representative of such a point over efficiency frontier that is corresponding to theinexact discretionary input x2. To convert the above multi-objective programmingto a linear programming problem, the following mathematical programmingproblem is proposed:

max q

s:t:

C1Pn

j¼1kjxij��xip i 2 ID

C2Pn

j¼1kjxij� xip i 2 IND

C3Pn

j¼1kjxij��xip i 2 IFND

C4Pn

j¼1kjyrj� yrp

C5 q� a� xip��xip

xipi 2 ID

C6 q� bi�xu

ip��xip

xuip�xl

ipi 2 IFND

kj� 0 j ¼ 1; 2; . . .; n

a� 0

q� 0

bi� 0 i 2 IFND

ð27Þ

We refer again to the case study given in Sect. 5. Assume again that the secondinput in the case study is an inexact discretionary variable. The fuzzy numberassociated with this variable is defined according to C6. The fuzzy numbersassociated with the second input is defined as l~x52

x52ð Þ ¼ 1:20�x521:20�1:186185 and the linear

programming problem related to DMU5 is written as follows:

186 M. Z. A. Langroudi

maxq

s:t:

1:50k1 þ 4:00k2 þ 3:20k3 þ 5:20k4 þ 3:50k5 þ 3:20k6� x51

1:50k1 þ 0:70k2 þ 1:20k3 þ 2:00k4 þ 1:20k5 þ 0:70k6� x52

1:40k1 þ 1:40k2 þ 4:20k3 þ 2:80k4 þ 1:90k5 þ 1:40k6� 1:90

0:35k1 þ 2:10k2 þ 1:05k3 þ 4:20k4 þ 2:50k5 þ 1:50k6� 2:50

q� 3:50� x51

3:50

q� 1:20� x52

1:20� 1:1861850� x51� 3:50

0� x52� 1:20

ð28Þ

In the above model, the amount 1.186185 comes from xl2 ¼ hx2 ¼ 0:988488

1:2 that indicates the point corresponding to the number 1.2 located in the effi-ciency frontier. Table 4 indicates the efficiency scores, considering the inputvariable 2, as an inexact discretionary variable. It is noteworthy that there aresuggestions for improvement for both the input variables. In this case, reducing theinput variable 1 from 3.50 to 3.397 and the input variable 2 from 1.20 to 1.1996 issuggested. The efficiency score in the above mentioned problem is 0.9706094 andit is lower than the efficiency score of the CCR model which is 0.9884877.

7 Conclusion

The relationship between possibility sets and efficiency score in the traditionalcrisp CCR model has thus been presented. The relationship provides a new per-spective of viewing efficiency. With the usage of the appropriate possibility sets torepresent certain characteristics of the input data, many DEA models involving

Table 4 Result of the CCR model in the presence of inexact discretionary variable

�x11 �x12 a 1� a

DMU Inexact non-discretionary1 1.066667 1.163793 0.2241378 0.77586222 – – 0 13 – – 0 14 – – 0 15 3.460165 1.186342 0.01138147 0.98861856 2.839548 0.632889 0.0958722 0.9041278

8 Data Envelopment Analysis 187

input data with various characteristics could be studied. This paper described acase involving non-discretionary input data. The usage of the possibility setsreplaces the need to determine the discretionary index of a non-discretionaryvariable. The discretionary index concepts are used in some of the existing non-discretionary models. In real life applications, discretionary indexes are usually notknown and are arbitrarily determined by decision makers.

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8 Data Envelopment Analysis 189

Chapter 9Network Data Envelopment Analysiswith Fuzzy Data

Chiang Kao

Abstract Conventional data envelopment analysis (DEA) treats a system as awhole unit when measuring efficiency, ignoring the operations of the componentprocesses. Network DEA, on the other hand, takes the component processes intoconsideration, with results that are more representative and can be used to identifyinefficient components. This paper discusses network DEA for fuzzy observations.Two approaches, the membership grade and the a-cut, are proposed for measuringthe system and process efficiencies via two-level mathematical programming. Themodel associated with the latter approach is transformed into a conventional one-level program so that the existing solution methods can be applied. Since the datais fuzzy, the measured efficiencies are also fuzzy. The property of the systemefficiency slack being the sum of the process efficiency slacks, which holds in thedeterministic case, was found to hold for the fuzzy case as well. A simple networksystem with three processes is used to illustrate the proposed idea.

Keywords Network data envelopment analysis � Fuzzy sets � Two-levelprogramming � Extension principle

1 Introduction

Data envelopment analysis (DEA) deals with the performance measurement ofproduction systems which apply multiple inputs to produce multiple outputs.Conventionally, only the inputs consumed and outputs produced by the system areconsidered when measuring efficiency. However, in a production system, theinputs usually go through several processes, producing a number of intermediate

C. Kao (&)Department of Industrial and Information Management, National Cheng Kung University,Taiwan, Tainan, Republic of Chinae-mail: ckao@mail.ncku.edu.tw

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_9, � Springer-Verlag Berlin Heidelberg 2014

191

products, before being converted into the final outputs. If the operations of thecomponent processes are ignored, the resulting efficiency can be misleading. Kaoand Hwang [13] provided an example showing that a system can be efficient evenif all the component processes are not. It is also possible that when the processefficiencies of one DMU are dominated by another, the system efficiency of theformer is still greater than that of the latter. The operations of component processesmust therefore be considered to obtain meaningful results.

The methodology that takes the operations of component processes into accounthas been called network DEA [7]. The basic idea is to embed the productiontechnology of individual processes into the conventional DEA model when cal-culating system efficiency. Several models have been developed under thisframework (see the review of Kao and Hwang, [13]). One of which is the relationalmodel [10], where two processes connected by an intermediate product are relatedusing the same multiplier. The rationale is that if this intermediate product istreated as an output and is sold on the market, then an income equal to the priceis earned. On the other hand, if this intermediate product is used as an input and isbought from the market, then a cost equal to the price is incurred. Therefore, onemultiplier, representing the imputed price, is used for the intermediate product.Note that in the independent model, the intermediate product may have differentmultipliers for the process producing it and the process consuming it.

In the relational model, the system efficiency is the product of the processefficiencies if the processes are connected in series [12], and this is a weightedaverage of the process efficiencies if the processes are connected in parallel [11].For general network systems, the efficiency slack of the system is the sum of theefficiency slacks of all component processes [13].

The development of DEA for deterministic cases is mature, with numerousmodels and applications reported in the literature (see the review of Cook andSeiford [5]). Although a lot of effort has been devoted to studying DEA underfuzzy environments [6, 8, 9, 15, 17, 18, 20], it remains less well developed than itsdeterministic counterpart.

Let Xij and Yrj denote the ith input, i = 1,…, m, and rth output, r = 1,…, s,respectively, of the jth decision making unit (DMU), j = 1,…, n. The CCR modelfor calculating the efficiency of DMU k is [3]:

Ek ¼ max:Xs

r¼1

urYrk

,Xm

i¼1

viXik

s:t:Xs

r¼1

urYrj

,Xm

i¼1

viXij� 1; j ¼ 1; . . .; n

ur; vi� e [ 0; r ¼ 1; . . .; s; i ¼ 1; . . .;m;

ð1Þ

where ur and vi are virtual multipliers, and e is a small non-Archimedean number[2, 4]. Suppose that the inputs Xij and outputs Yrj are approximately known, and arerepresented by fuzzy numbers ~Xij and ~Yrj, with membership functions l~Xij

and l~Yrj,

192 C. Kao

respectively. When observations are fuzzy numbers, the measured efficiencyshould also be a fuzzy number. Let ~Ek denote the fuzzy efficiency of DMU k andl~Ek

the membership function. Conceptually, ~Ek can be expressed by the followingfuzzy mathematical program:

~Ek ¼ max:Xs

r¼1

ur ~Yrk

,Xm

i¼1

vi ~Xik

s:t:Xs

r¼1

ur ~Yrj

,Xm

i¼1

vi ~Xij� 1; j ¼ 1; . . .; n

ur; vi� e [ 0; r ¼ 1; . . .; s; i ¼ 1; . . .;m:

ð2Þ

Kao and Liu [15] developed a two-level mathematical program to calculate thea-cut of l~Ek

. By enumerating various values for a, the membership function of ~Ek

can be constructed.The two-level mathematical program must be transformed into a conventional

one-level program so that existing methods can be applied to obtain a solution.Using the property of relative efficiency, Kao and Liu [15] set the a-cut of eachfuzzy observation, which is an interval value, to one of the bounds to make theone-level transformation possible. When intermediate products are involved, theproblem becomes so complicated that this idea is no longer applicable. In thispaper, two approaches, the membership grade and the a-cut, are applied to for-mulate two-level mathematical programs. Both approaches can be used to find themembership functions of the system efficiency and process efficiencies; however,only the latter can be transformed into the conventional one-level program. Byapplying the a-cut approach, Kao and Liu [16] found that the property that thesystem efficiency is equal to the product of the process efficiencies of a two-stagesystem, while Kao and Lin [14] found that the property that the system efficiencyis equal to a weighted average of the process efficiencies in a parallel system,which holds for the deterministic case, also holds for the fuzzy one. In addition todeveloping the solution method, the relationship between the system efficiency andprocess efficiencies under a fuzzy environment will also be explored in this paper.In this context, the property of the slack of the fuzzy system efficiency being thesum of the slacks of the fuzzy process efficiencies is expected to hold.

The rest of this paper is organized as follows. In the next section, the networkrelational DEA model developed by Kao [10] for the deterministic case isreviewed. Section 3 extends the deterministic case to a fuzzy environment wherethe observations are fuzzy numbers. Two approaches are discussed, the mem-bership grade and the a-cut, and a solution method is developed for the latter.Section 4 provides an example to explain how to model a fuzzy network systemand how to measure efficiency, while Sect. 5 represents the conclusion of thiswork.

9 Network Data Envelopment Analysis with Fuzzy Data 193

2 Deterministic Case

Network systems have two basic structures: series and parallel; both can be for-mulated using the concept of network DEA to calculate efficiency. However, dueto the various possible connections among the processes which constitute thenetwork system, it is difficult to develop a general network DEA model for allsystems. Therefore, a simple example is used to illustrate how to model the net-work system under the framework of DEA, and a conceptually general model isthen provided.

Most network systems discussed in the literature are series systems. The mostcomplicated non-series system is a simple five-process system discussed in Sextonand Lewis [19]. Consider a simplified version of the three processes depicted inFig. 1, where Process 1 uses input X1 to produce intermediate product Z13, Process2 uses input X2 to produce intermediate product Z23, and Process 3 uses inter-mediate products Z13 and Z23, produced by Processes 1 and 2, respectively, toproduce the final output Y. When the operations of individual processes areignored, inputs X1 and X2 are used to produce output Y.

The conventional network DEA model requires the aggregated output of eachprocess to be less than or equal to its aggregated input in addition to the usualrequirements for the system. The relational model of Kao [10] further requires themultipliers for an intermediate product associated with different processes to be thesame. For the system in Fig. 1, the relational model for calculating the efficiencyof DMU k is:

Ek ¼ max: uYk= v1X1k þ v2X2kð Þs:t: uYj

�v1X1j þ v2X2j

� �� 1; j ¼ 1; . . .; n ð3:0Þ

w13Z13j

.v1X1j� 1; j ¼ 1; . . .; n ð3:1Þ

w23Z23j

.v2X2j� 1; j ¼ 1; . . .; n ð3:2Þ

uYj

.w13Z13

j þ w23Z23j

� �� 1; j ¼ 1; . . .; n ð3:3Þ

u; v1; v2;w13;w23� e

System

X1

X2

Process l

Process 2

13Z

YProcess 3

23Z

Fig. 1 A simple three-component network system

194 C. Kao

Constraint set (3.0) corresponds to the system and constraint sets (3.1)–(3.3)correspond to Processes 1 to 3, respectively. Note that intermediate product Z hasmultiplier w, no matter which process it is associated with. For example, Z13

j in(3.1) is the output of Process 1, and is also the input of Process 3, as appeared in(3.3). It always has multiplier w13, no matter what role it plays. The conventionalDEA model does not consider the operations of the processes; in other words,constraint sets (3.1)–(3.3) are excluded, and this leads to an over-estimation of thesystem efficiency.

Model (3.0–3.3) is a linear fractional program, which can be transformed intothe following linear program:

Ek ¼ max: uYk

s:t: v1X1k þ v2X2k ¼ 1

uYj � v1X1j þ v2X2j

� �� 0; j ¼ 1; . . .; n ð4:0Þ

w13Z13j � v1X1j� 0; j ¼ 1; . . .; n ð4:1Þ

w23Z23j � v2X2j� 0; j ¼ 1; . . .; n ð4:2Þ

uYj � w13Z13j þ w23Z23

j

� �� 0; j ¼ 1; . . .; n ð4:3Þ

u; v1; v2;w13;w23� e

The sum of process constraints (4.1)–(4.3) is equal to the system constraint (4.0).Hence, the system is efficient only if all its component processes are efficient. Eachconstraint is the difference between the aggregated output and the aggregated input,and the ratio of the former to the latter is the efficiency. After the optimal solution(u�, v�1, v�2, w13�, w23�) is obtained, the efficiency of the system is Ek ¼ u�Yk, and theefficiencies of the three processes are E1

k ¼ w13�Z13k

�v�1X1k, E2

k ¼ w23�Z23k

�v�2X2k ,

and E3k ¼ u�Yk

�w13�Z13

k þ w23�Z23k

� �, respectively. Let sk, s1

k , s2k , and s3

k be theslacks associated with (4.0)–(4.3), respectively, for DMU k. This implies thatsk ¼ s1

k þ s2k þ s3

k . Note that while (1-sk) is the efficiency of the system, (1-s1k),

(1-s2k), and (1-s3

k) need not be equal to the process efficiencies E1k , E2

k , and E3k ,

respectively, simply because v�1X1k þ v�2X2k is equal to one while v�1X1k, v�2X2k, andw13�Z13

k þ w23�Z23k are not necessarily equal to 1. From the relationship of

sk ¼ s1k þ s2

k þ s3k , it is not difficult to find that (1-Ek) is a linear combination of

(1-E1k ), (1-E2

k ), and (1-E3k ), expressed as: ð1� EkÞ ¼ ð1� E1

kÞv�1X1kþð1� E2

kÞv�2X2k þ ð1� E3kÞðw13�Z13

k þ w23�Z23k Þ. Note that this is not a weighted

average, because the sum of v�1X1k, v�2X2k, and ðw13�Z13k þ w23�Z23

k Þ is greater thanone.

Model (4.0–4.3) might have alternative solutions, which would producedifferent process efficiencies, although the system efficiency remains the same. Inorder to provide a common basis for all DMUs to compare their process

9 Network Data Envelopment Analysis with Fuzzy Data 195

efficiencies, the process efficiency can be calculated from a separate mathematicalprogram. The constraints are the same as those of Model (4.0–4.3), and only theobjective function needs to be changed to the process of interest. For example, themodel for calculating the efficiency of Process 1 is:

E1k ¼ max: w13Z13

k

s:t: v1X1k ¼ 1

uYj � v1X1j þ v2X2j

� �� 0; j ¼ 1; . . .; n

ð5:0Þ

w13Z13j � v1X1j� 0; j ¼ 1; . . .; n ð5:1Þ

w23Z23j � v2X2j� 0 j ¼ 1; . . .; n ð5:2Þ

uYj � w13Z13j þ w23Z23

j

� �� 0; j ¼ 1; . . .; n ð5:3Þ

u; v1; v2;w13;w23� e

Similarly, if the efficiencies of Processes 2 or 3 are desired, then only the objectivefunction and the first constraint need to be replaced, and the other constraintsremain unchanged.

This concept of modeling can be generalized to any network system that consistsof q processes, as shown in Fig. 2, where each process p uses a set of inputs Xp and aset of intermediate products Zap, produced by Process a, a = 1,…, q, to produce aset of outputs Yp and a set of intermediate products Zpb, to be used by Process b,b = 1,…, q. Some Xp, Zap, Zpb, and Yp vectors can be zero. Let vp, wap, wpb, and up

denote the sets of multipliers associated with the inputs, intermediate products to beconsumed, intermediate products produced, and the outputs of Process p, respec-tively. The network DEA model for calculating the efficiency of DMU k can beformulated as:

System

pZ1

.

.

.qpZ

Xp

Yp

Process p

1pZ...

pqZ

Fig. 2 General networksystem

196 C. Kao

Ek ¼ max:Xq

p¼1

upYpk

s:t:Xq

p¼1

vpXpk ¼ 1

Xq

p¼1

upYpj �Xq

p¼1

vpXpj� 0; j ¼ 1; . . .; n

ð6:0Þ

ðupYpj þXq

b¼1

wpbZpbj Þ � ðvpXpj þ

Xq

a¼1

wapZapj Þ� 0; p ¼ 1; . . .; q; j ¼ 1; . . .; n

up; vp;wap;wpb� e; a; b; p ¼ 1; . . .; q:

ð6:1Þ

Constraint set (6.0) corresponds to the system and constraint set (6.1) corre-sponds to Process p, p = 1,…, q. The first part of constraint set (6.1) represents theoutputs and intermediate products that flow out of Process p, while the second partrepresents the inputs and intermediate products that flow into Process p. Since it isrequired that an intermediate product has one multiplier, the sum of all q processconstraints for DMU j represented by (6.1) is exactly the system constraint (6.0).Constraint set (6.0) is thus redundant, and can be omitted. The objective function,Pq

p¼1 upYpk, is the system efficiency for DMU k and (upYpk ?Pq

b¼1 wpbZpbk )/

(vpXpk ?Pq

a¼1 wapZapk ) is the efficiency for Process p of DMU k.

When there are alternative solutions, the efficiency of Process p, Epk , can be

calculated by replacing the objective function of Model (6.0, 6.1) with

upYpk ?Pq

b¼1 wpbZpbk , and the first constraint with (vpXpk ?

Pqa¼1 wapZap

k ) = 1,so that the efficiencies of different DMUs have a common basis for comparison.

3 Fuzzy Case

When at least one observation is a fuzzy number, the network DEA modeldiscussed in the preceding section becomes a fuzzy network DEA model. Togeneralize the discussion, all observations are assumed to be fuzzy numbers sincecrisp values can be represented by degenerated fuzzy numbers with a domain thathas only one value.

Model (6.0, 6.1) is a general model for measuring the efficiency of a networksystem. When the observations are fuzzy numbers, it is very difficult to derive thefuzzy efficiency directly. One approach is to rely on Zadeh’s extension principle[21, 22], which describes the relationship between the membership function of ~Ek

and those of ~Xij, ~Zabj , and ~Yrj:

9 Network Data Envelopment Analysis with Fuzzy Data 197

l~Ekeð Þ ¼ sup

x;y;zminfl~Xij

ðxijÞ;l~YrjðyrjÞ;l~Zab

jðzab

j Þ;

8 i; j; r; a; bje ¼ Ek x; y; zð Þgð7Þ

where Ek(x, y, z) is defined by Model (6.0, 6.1). In this case, xij, yrj, and zabj are the

variables to be determined. Based on this relationship, two approaches can bedeveloped for obtaining l~Ek

, the membership grade and the a-cut.

3.1 The Membership Grade Approach

The right-hand-side of expression (7) is a two-level mathematical programmingmodel. For each set of xij, yrj, and zab

j , which produces the efficiency value e, the

minimum of the membership grades of l~XijðxijÞ, l~Yrj

ðyrjÞ, and l~Zabjðzab

j Þ is deter-

mined at the second level. The set of xij, yrj, and zabj values which produce the

largest membership grade obtained at the second level is determined at the firstlevel. This membership grade then becomes that of ~Ek for value e.

The two-level mathematical expression cannot be solved directly, and must betransformed into the conventional one-level mathematical model so that a solutioncan be obtained. Let h be the minimum membership grade obtained at the secondlevel for a set of xij, yrj, and zab

j values which has the efficiency value e, i.e.,

h = min{l~XijðxijÞ, l~Yrj

ðyrjÞ, l~Zabjðzab

j Þ, 8i, j, r, a, b | e =Ek(x, y, z)}. Since h is the

minimum of l~XijðxijÞ, l~Yrj

ðyrjÞ, l~Zabjðzab

j Þ, 8i, j, r, a, b, it must satisfy the conditions

of h � l~XijðxijÞ, h � l~Yrj

ðyrjÞ, h � l~Zabjðzab

j Þ, 8i, j, r, a, b. Expression (7) can thus be

formulated as:

l~Ekeð Þ ¼ max

x;y;zh

s:t: h� l~XijðxijÞ 8 i; j

h� l~YrjðyrjÞ 8 r; j

h� l~Zabjðzab

j Þ; 8 j; a; b

e ¼ Ek x; y; zð Þ

ð8Þ

where e = Ek(x, y, z) is defined by Model (6.0, 6.1).Model (8) is a conceptual expression, and must be converted to a form which is

solvable using existing methods. One problem in conversion is the membershipfunction. Suppose the fuzzy numbers are triangular, described by:

198 C. Kao

l~XðxÞ ¼x� að Þ= b� að Þ a� x� bc� xð Þ= c� bð Þ b� x� c

0 otherwise

8<

:ð9Þ

which requires x to lie between a and c, and for one of the following two situationsto hold:

(a) if a � x � b, then h � (x-a)/(b-a)(b) if b � x � c, then h � (c-x)/(c-b).

To describe this either-or constraint mathematically, a binary variable d isintroduced. The formulation is:

(a) a� xþMd; x� bþMd; h�ðx� aÞ=ðb� aÞ þMd(b)

b� xþMð1� dÞ; x� cþMð1� dÞ; h�ðc� xÞ=ðc� bÞ þMð1� dÞ ð10Þ

a� x� c; d ¼ 0 or 1

where M is a very large number. When d is equal to 1, the associated constraintsare redundant. For example, the constraint a � x ? Md always holds if d = 1,because in this case, a is always less than M. The binary variable d requires eitherd or (1-d) to be equal to 0, and the corresponding situation to hold. Each trian-gular fuzzy number needs eight constraints and one binary variable in modeling.For a problem of m inputs, s outputs, t intermediate products, and n DMUs, oneneeds ðmþ sþ tÞ � n binary variables and ðmþ sþ tÞ � n� 8 extra constraintsto model.

In Model (8), e =Ek(x, y, z) is a mathematical program. For the three-componentexample discussed in the preceding section, Model (4.0–4.3) applies. In this case,Model (8) uses a set of x1, x2, y, z13, and z23 values to produce the efficiency valuee from Model (4.0–4.3). This set of values is then used to calculate the mem-bership grades, with the minimum denoted as h. Different sets of x1, x2, y, z13, andz23 values that produce the efficiency score e from Model (4.0–4.3) are searchedfor at the first level to produce the largest value of h, and this h is l~Ek

(e). By

enumerating various values of e, the membership function for ~Ek can be obtainednumerically.

Since the second-level program involves another mathematical program, Model(4.0–4.3), and its objective value is required to be equal to a pre-specified value e,its transformation into the conventional one-level program appears to be impos-sible. The next subsection introduces a possible solution method.

9 Network Data Envelopment Analysis with Fuzzy Data 199

3.2 The a-Cut Approach

In contrast to the membership grade approach which views the membershipfunction vertically, the a-cut approach views the membership functionhorizontally.

According to Expression (7), l~Ek(e) is the minimum of l~Xij

ðxijÞ, l~YrjðyrjÞ,

l~Zabjðzab

j Þ, 8i, j, r, a, b. To satisfy l~EkðeÞ ¼ a, one needs l~Xij

ðxijÞ� a, l~YrjðyrjÞ� a,

l~Zabjðzab

j Þ� a, and at least one of l~XijðxijÞ, l~Yrj

ðyrjÞ, and l~Zabjðzab

j Þ is equal to a, 8i, j,

r, a, b, such that e ¼ Ek(x, y, z). Suppose all fuzzy observations have a triangularmembership function such that l~Xij

ðxijÞ� a and l~XijðxijÞ ¼ a have the same

domain. This is also true for l~YrjðyrjÞ and l~Zab

jðzab

j Þ.Denote ðXijÞa ¼[ðXijÞLa , ðXijÞUa ], ðYrjÞa ¼[ðYrjÞLa , ðYrjÞUa ], ðZab

j Þa ¼ [ðZabj Þ

La ,

ðZabj Þ

Ua ], and ðEkÞa ¼[ðEkÞLa , ðEkÞUa ] as the a-cuts of ~Xij, ~Yrj, ~Zab

j , and ~Ek, respec-

tively. To find the membership function of ~Ek, l~Ek(e), it suffices to find the lower

and upper bounds of the a-cut of l~Ek(e). From the above discussion, the upper

bound ðEkÞUa and lower bound ðEkÞLa of the a-cut of l~Ek(e) can be obtained as:

ðEkÞUa ¼ maxðXijÞLa � xij�ðXijÞUaðYrjÞLa � yrj�ðYrjÞUaðZab

j ÞLa � zab

j �ðZabj Þ

Ua

8 i; r; a; b; j

Ek x; y; zð Þ

ð11aÞ

ðEkÞLa ¼ minðXijÞLa � xij�ðXijÞUaðYrjÞLa � yrj�ðYrjÞUaðZab

j ÞLa � zab

j �ðZabj Þ

Ua

8 i; r; a; b; j

Ek x; y; zð Þ

ð11bÞ

where Ek(x, y, z) is defined in Model (6.0, 6.1). Note that Ek(x, y, z) is a mathe-matical program with maximization for the objective function. Therefore, Models(11a) and (11b) are two-level programs. For each set of xij, yrj, and zab

j values lyingin the respective a-cuts, the efficiency is calculated at the second level. The sets ofvalues which produce the largest and smallest efficiencies are determined at thefirst level.

While two-level programs are useful for modeling problems, they cannot besolved directly, and must be transformed into conventional one-level programs sothat the existing solution methods can be applied. Model (11a) is relatively simple

200 C. Kao

because both the first and second levels have the same direction for optimization;i.e., maximization. They can thus be combined together, with the objectivefunction of the second level as the objective function, and the constraints ofthe two levels combined as the constraints. Model (11b), in contrast, cannot becombined directly, because the directions for optimization of the two levels aredifferent. In order to obtain the same direction for optimization, the dual forthe second-level model is formulated. In this case, the two levels have the samedirection for optimization, i.e., minimization, and can thus be combined into a one-level program. Note that the resulting one-level programs are nonlinear.

The one-level transformations of Models (11a) and (11b) produce the upper andlower bounds of the a-cut of the fuzzy efficiencies, respectively. By enumeratingvarious values of a, the membership functions of the fuzzy efficiencies areobtained numerically.

A general formulation for all network systems is difficult. We provide a three-component example in the next section to illustrate the solution process and thedifference between the deterministic and fuzzy cases.

4 Example Problem

Consider the three-component network system in Fig. 1. Suppose there are fourDMUs (A, B, C, and D) whose efficiencies are to be evaluated. The data is shownin the left part of Table 1. When the operations of the three components areignored, the conventional CCR model measures all four DMUs as efficient. Theefficiency scores are shown in the middle of Table 1, under the heading of ‘‘CCREfficiency’’.

When the operations of the component processes are taken into account andModel (4.0–4.3) is used, the results, as shown in the right half of Table 1, underthe heading of ‘‘Relational model efficiency (slack)’’, are different. The processefficiencies can be obtained in two ways. One is to substitute the objective functionin Model (3.0–3.3) by the output-input ratio of the process whose efficiency is tobe measured, in the form of Model (5.0–5.3). The other is to use the optimalmultipliers obtained from Model (4.0–4.3), and substitute them into the corre-sponding ratio constraint in Model (3.0–3.3) to obtain the efficiency values. Notethat the second method might yield alternative solutions. In this example, the two

Table 1 Data and the deterministic efficiency measures for the given example

DMU Data CCR Relational model efficiency (slack)

X1 X2 Z13 Z23 Y Efficiency System Process 1 Process 2 Process 3

A 1 5 2 4 1 1 1-e (e) 1 (0) 4/5 (e) 1 (0)B 4 6 8 6 2 1 6/7 (1/7) 1 (0) 1 (0) 6/7 (1/7)C 9 6 12 6 3 1 3/4 (1/4) 2/3(1/4) 1 (0) 1 (0)D 6 1 5 1 1 1 1-e (e) 5/12 (e) 1 (0) 1 (0)

9 Network Data Envelopment Analysis with Fuzzy Data 201

methods yield the same result, that none of the DMUs are efficient, although A andD are weakly efficient. This result is reasonable, because each DMU has at leastone component process which is inefficient.

Kao and Hwang [13] derived the following property for a network DEA, thatthe system efficiency slack is the sum of the efficiency slacks of all the componentprocesses. The numbers in parentheses in Table 1 show the efficiency slacksassociated with the efficiency values, and the above-mentioned property is clearlysatisfied. For example, DMU B has a system efficiency slack of 1/7, while its threecomponent processes have efficiency slacks of 0, 0, and 1/7, respectively, whichsum to 1/7.

Now, suppose that each observation is uncertain and can be represented by anisosceles triangular fuzzy number whose vertex has a value equal to that of thedeterministic case and the base has a length of 1. In other words, a deterministicvalue X is replaced with a triangular fuzzy number (X - 0.5, X, X ? 0.5). Fol-lowing the idea of the two-level programming discussed in Sect. 3.2 for calcu-lating the upper bound of the a-cut of the fuzzy efficiency of the system via Model(11a), the inner program is of the same as Model (4.0–4.3), except that X1j, X2j, Yj,Z13

j , and Z23j are now variables to be determined by the outer program. The

corresponding one-level program being transformed is:

ðEkÞUa ¼ max: uyk

s:t: v1x1k þ v2x2k ¼ 1

w13z13j � v1x1j� 0; j ¼ A;B;C;D

w23z23j � v2x2j� 0; j ¼ A;B;C;D

uyj � w13z13j þ w23z23

j

� �� 0; j ¼ A;B;C;D

ðXijÞLa � xij�ðXijÞUa ; i ¼ 1; 2; j ¼ A;B;C;D

ðYjÞLa � yj�ðYjÞUa ; j ¼ A;B;C;D

ðZabj Þ

La � zab

j �ðZabj Þ

Ua ; a ¼ 1; 2; b ¼ 3; j ¼ A;B;C;D

u; v1; v2; w13; w23� e

ð12Þ

Model (12) is used to calculate the upper bound of the a-cut of the fuzzyefficiencies. Specifically, the objective function uyk is the system efficiency, andw13z13

k /v1x1k, w23z23k /v2x2k, and uyk/(w13z13

k þ w23z23k ) are the efficiencies for Pro-

cesses 1, 2, and 3, respectively. The slacks from the three constraints are theefficiency slacks of the three processes, and the sum of these is the efficiency slackof the system.

To calculate the lower bound of the a-cut of the fuzzy efficiency via Model(11b) is a little complicated. First, the inner program must be represented by thedual of Model (4.0–4.3) to have the objective of minimization:

202 C. Kao

Ek ¼ min: h � e s�1 þ s�2 þ so1 þ so

2 þ sþ� �

s:t: hX1k �XD

j¼AajX1j � s�1 ¼ 0

hX2k �XD

j¼AbjX2j � s�2 ¼ 0

XD

j¼AajZ

13j �

XD

j¼AcjZ

13j � so

1 ¼ 0XD

j¼AbjZ

23j �

XD

j¼AcjZ

23j � so

2 ¼ 0XD

j¼AcjYj � sþ ¼ Yk

aj; bj; cj; s�1 ; s

�2 ; s

o1; s

o2; sþ � 0 j ¼ A;B;C;D

ð13Þ

Note that the redundant constraint set (4.0) has been deleted in formulating thedual. With the same objective of minimization, the inner and outer programs canthen be combined. The observations Xij, Yj, and Zab

j are now variables to bedetermined by the outer program. The resulting one-level program is:

ðEkÞLa ¼ min:h � e s�1 þ s�2 þ so1 þ so

2 þ sþ� �

s:t: hx1k �XD

j¼Aajx1j � s�1 ¼ 0

hx2k �XD

j¼Abjx2j � s�2 ¼ 0

XD

j¼Aajz

13j �

XD

j¼Acjz

13j � so

1 ¼ 0XD

j¼Abjz

23j �

XD

j¼Acjz

23j � so

2 ¼ 0XD

j¼Acjyj � sþ ¼ yk

ðXijÞLa � xij�ðXijÞUa ; i ¼ 1; 2; j ¼ A;B;C;D

ðYjÞLa � yj�ðYjÞUa ; j ¼ A;B;C;D

ðZabj Þ

La � zab

j �ðZabj Þ

Ua ; a ¼ 1; 2; b ¼ 3; j ¼ A;B;C;D

aj; bj; cj; s�1 ; s

�2 ; s

o1; s

o2; sþ � 0 j ¼ A;B;C;D

ð14Þ

Model (14) is used to calculate the lower bound of the a-cut of the fuzzyefficiencies. Since this model is conceptually the dual of Model (12), the associatedvariables in Model (12) can be obtained from the reduced cost and dual price ofModel (14). For example, the efficiency slacks of the three processes are thereduced costs of ak, bk, and ck, respectively, and the multipliers v1, v2, w13, w23,and u are the dual prices of the first five corresponding constraints. With thesevalues, the system and process efficiencies can be calculated. Table 2 shows theresults for DMU C. The efficiencies and slacks at a = 1 are exactly the same asthose of the deterministic case, because we set the value of the triangle vertex ofthe fuzzy number to the corresponding data of the deterministic case. At all avalues, the a-cut of the system efficiency slack is the sum of the a-cuts of the

9 Network Data Envelopment Analysis with Fuzzy Data 203

process efficiency slacks, indicating that the fuzzy efficiency slack of the system isthe sum of the fuzzy efficiency slacks of the component processes. This equalityalso indicates that (1-Ek) = (1-E1

k )v1X1k ? (1-E2k )v2X2k+ (1-E3

k )(w13Z13k þ

w23Z23k ). If we consider (1-Ek) as the inefficiency score, then the system ineffi-

ciency score is a linear combination of those of all the processes.To have a better idea of the shape of the fuzzy efficiencies, Fig. 3 depicts the

membership functions of the system efficiency, labeled as System, and three

Table 2 a-cuts of the fuzzy system and process efficiencies and slacks for DMU C of the givenexample

a System Process 1 Process 2 Process 3

1.0 Eff (0.7500, 0.7500) (0.6667, 0.6667) (1.0000, 1.0000) (1.0000, 1.0000)Slack (0.2500, 0.2500) (0.2500, 0.2500) (0.0000, 0.0000) (0.0000, 0.0000)

0.9 Eff (0.6430, 0.7969) (0.6119, 0.6859) (0.8898, 1.0000) (0.9417, 1.0000)Slack (0.2031, 0.3570) (0.2031, 0.2891) (0.0000, 0.0281) (0.0000, 0.0398)

0.8 Eff (0.5514, 0.8443) (0.5605, 0.7056) (0.7914, 1.0000) (0.8887, 1.0000)Slack (0.1557, 0.4486) (0.1557, 0.3253) (0.0000, 0.0542) (0.0000, 0.0691)

0.7 Eff (0.4726, 0.8914) (0.5119, 0.7258) (0.7030, 1.0000) (0.8401, 1.0000)Slack (0.1086, 0.5274) (0.1086, 0.3589) (0.0000, 0.0785) (0.0000, 0.0900)

0.6 Eff (0.4046, 0.9374) (0.4664, 0.7476) (0.6236, 1.0000) (0.7955, 1.0000)Slack (0.0626, 0.5954) (0.0626, 0.3902) (0.0000, 0.1012) (0.0000, 0.1040)

0.5 Eff (0.3459, 0.9812) (0.4234, 0.7677) (0.5520, 1.0000) (0.7543, 1.0000)Slack (0.0188, 0.6541) (0.0188, 0.4192) (0.0000, 0.1223) (0.0000, 0.1127)

0.4 Eff (0.2949, 0.9997) (0.3829, 0.7895) (0.4872, 1.0000) (0.7162, 1.0000)Slack (0.0003, 0.7051) (0.0003, 0.4462) (0.0000, 0.1420) (0.0000, 0.1169)

0.3 Eff (0.2506, 0.9997) (0.3446, 0.8119) (0.4284, 1.0000) (0.6808, 1.0000)Slack (0.0003, 0.7494) (0.0003, 0.4714) (0.0000, 0.1604) (0.0000, 0.1175)

0.2 Eff (0.2121, 0.9998) (0.3079, 0.8348) (0.3744, 1.0000) (0.6490, 1.0000)Slack (0.0002, 0.787 9) (0.0002, 0.4949) (0.0000, 0.1777) (0.0000, 0.1153)

0.1 Eff (0.1785, 0.9998) (0.2744, 0.8583) (0.3264, 1.0000) (0.6171, 1.0000)Slack (0.0002, 0.8215) (0.0002, 0.5168) (0.0000, 0.1938) (0.0000, 0.1108)

0.0 Eff (0.1493, 0.9998) (0.2421, 0.8824) (0.2820, 1.0000) (0.5882, 1.0000)Slack (0.0002, 0.8507) (0.0002, 0.5373) (0.0000, 0.2090) (0.0000, 0.1045)

0

0.5

1

0.2 0.4 0.6 0.8 1.0

P1 P2 P3

System

Fig. 3 Membershipfunctions of the system andthree process efficiencies ofDMU C

204 C. Kao

process efficiencies, labeled as P1, P2, and P3, respectively, enumerated for a = 0,0.1,…, 1.0. The graphs are fairly smooth, and more a values can be enumerated ifhigher precision is required.

5 Conclusion

Network data envelopment analysis captures the internal operations of a system,and thus it is more realistic and can be used to detect inefficient componentprocesses. This paper discusses how to calculate the system and process effi-ciencies when the observations are fuzzy numbers.

Two approaches are proposed in this work; one takes a vertical view and theother takes a horizontal approach using the membership function. The formerinvolves more constraints and is too complicated to be converted into a formwhich can be handled by existing solution methods, and finding a solution methodfor this approach is a direction for future research. The latter tackles the problembased on the a-cut of the membership function. The lower and upper bounds of thea-cut can be obtained from a pair of two-level mathematical programs. The two-level programs are transformed into conventional one-level programs so thatexisting solution methods for nonlinear programs can be applied to obtain asolution. In this paper, LINGO was used to find the solution for a simple networksystem with three processes.

For deterministic cases, Kao and Hwang [13] found that the system efficiencyslack is the sum of the process efficiency slacks. From the models developed in thispaper, it was found that this property also holds for fuzzy cases. Although the exactform of the membership function cannot be derived analytically, the solution methoddevised in this paper can be used to find the membership function numerically.

One deficiency of this paper is that the models developed herein are nonlinear.Although solving nonlinear programs is not a problem, NLP solvers are not aspopular as LP ones. It would therefore be more convenient if the models werelinear. Another topic that would be interesting to explore in future work is returnsto scale. The DEA models examined in this paper are CCR ones under theassumption of constant returns to scale, and if they can be extended to BCCones [1], then scale efficiencies can also be investigated.

References

1. Banker, R.D., Charnes, A., Cooper, W.W.: Some models for estimating technical and scaleefficiencies in data envelopment analysis. Manage. Sci. 30, 1078–1092 (1984)

2. Charnes, A., Cooper, W.W.: The non-Archimedean CCR ratio for efficiency analysis:A rejoinder to Boyd and Färe. Euro. J. Oper. Res. 15, 333–334 (1984)

3. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units.Euro. J. Oper. Res. 2, 429–444 (1978)

9 Network Data Envelopment Analysis with Fuzzy Data 205

4. Charnes, A., Cooper, W.W., Rhodes, E.: Short communication: Measuring the efficiency ofdecision making units. Euro. J. Oper. Res. 3, 339 (1979)

5. Cook, W.D., Seiford, L.M.: Data envelopment analysis (DEA)—Thirty years on. Euro.J. Oper. Res. 192, 1–17 (2009)

6. Dia, M.: A model of fuzzy data envelopment analysis. INFOR 42, 267–279 (2004)7. Färe, R., Grosskopf, S.: Network DEA. Socio-Econ. Plan. Sci. 34, 35–49 (2000)8. Guo, P.J.: Fuzzy data envelopment analysis and its application to location problems. Inf. Sci.

179, 820–829 (2009)9. Jahanshahloo, G.R., Soleimani-damaneh, M., Nasrabadi, E.: Measure of efficiency in DEA

with fuzzy input-output levels: A methodology for assessing, ranking and imposing ofweights restrictions. Appl. Math. Comput. 156, 175–187 (2004)

10. Kao, C.: Efficiency decomposition in network data envelopment analysis: A relational model.Euro. J. Oper. Res. 192, 949–962 (2009)

11. Kao, C.: Efficiency measurement for parallel production systems. Euro. J. Oper. Res. 196,1107–1112 (2009)

12. Kao, C., Hwang, S.N.: Efficiency decomposition in two-stage data envelopment analysis: Anapplication to non-life insurance companies in Taiwan. Euro. J. Oper. Res. 185, 418–429(2008)

13. Kao, C., Hwang, S.N.: Efficiency measurement for network systems: IT impact on firmperformance. Decis. Support Syst. 48, 437–446 (2010)

14. Kao, C., Lin, P.H.: Efficiency of parallel production systems with fuzzy data. Fuzzy Sets Syst.198, 83–98 (2012)

15. Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst.113, 427–437 (2000)

16. Kao, C., Liu, S.T.: Efficiencies of two-stage systems with fuzzy data. Fuzzy Sets Syst. 176,20–35 (2011)

17. Leon, T., Liern, V., Ruiz, J.L., Sirvent, I.: A fuzzy mathematical programming approach tothe assessment of efficiency with DEA models. Fuzzy Sets Syst. 139, 407–419 (2003)

18. Lertworasirkul, S., Fang, S.C., Joines, J.A., Nuttle, H.L.W.: Fuzzy data envelopment analysis(DEA): A possibility approach. Fuzzy Sets Syst. 139, 379–394 (2003)

19. Sexton, T.R., Lewis, H.F.: Two-stage DEA: An application to Major League Baseball.J. Prod. Anal. 19, 227–249 (2003)

20. Wen, M.L., Li, H.S.: Fuzzy data envelopment analysis (DEA): Model and ranking method.J. Comput. Appl. Math. 223, 872–878 (2009)

21. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)22. Zimmermann, H.Z.: Fuzzy Set Theory and Its Applications, 3rd edn. Kluwer-Nijhoff, Boston

(1996)

206 C. Kao

Chapter 10Network Fuzzy Data EnvelopmentAnalysis

Sebastián Lozano and Plácido Moreno

Abstract In this chapter a general approach to handle fuzzy data when the unitsunder analysis are formed by a network of processes is presented. ConventionalDEA assumes a single-process that consumes all the different inputs and producesall the different outputs. Network DEA, on the contrary, considers differentinterrelated processes, each one with its own inputs, its own outputs and, veryimportant, its own technology. This allows a more fine-grained analysis althoughat the expense of requiring more data. Conventional Network DEA approachesassume crisp data although recently two proposals have been made that can pro-cess fuzzy data in the special cases of a serial two-stage system and of parallelproduction processes. There is, however, a need to deal with general networks ofprocesses which can have fuzzy input or output data. In this chapter, several FuzzyDEA approaches are extended to Network DEA. The resulting models are illus-trated on a dataset from the literature.

Keywords Efficiency assessment � Network DEA � Fuzzy data

1 Introduction

Network DEA refers to a growing number of DEA approaches that instead ofassuming that all inputs and outputs are consumed and produced, respectively, by asingle process, the system can be modelled as formed by distinct sub-processes.Each sub-process is in itself a process that consumes a subset of the inputs and

S. Lozano (&) � P. MorenoDepartment of Industrial Management, University of Seville, Camino de losDescubrimientos, s/n 41012 Seville, Spaine-mail: slozano@us.es

P. Morenoe-mail: placidomb@us.es

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_10, � Springer-Verlag Berlin Heidelberg 2014

207

produces a subset of the outputs. A feature of Network DEA is the possibleexistence of intermediate products that are produced by a process and consumedby another. These endogenously generated-and-consumed products represent thelinks between the different stages or processes. Another key feature of NetworkDEA is that the technology, i.e. the production possibility set, is modelled at theprocess level. Thus, each process will have its own technology, with, for example,its own returns to scale assumption.

The number of Network DEA approaches has grown in the last decade, both atthe theory and at the application level. Seminal works were those of Färe andGrosskopf [3, 4], Sexton and Lewis [24] and Lewis and Sexton [17, 18]. The field,however, did not gain momentum until the relational network DEA approach ofKao and Hwang [12, 13] and Kao [10, 11] as well as the Network SBM approachof Tone and Tsutsui [25]. There are other Network DEA approaches like theweighted additive efficiency decomposition approach of Chen et al. [5] and Cooket al. [6] or the Network Slacks-Based Inefficiency (NSBI) approach of Fukuyamaand Weber [7] as well as many applications in different sectors, such as manu-facturing (e.g. [19]), supply chain management [1], transportation (e.g. [27]),tourism (e.g. [28]), finance (e.g. [2]), management (e.g. [8]), education (e.g. 22)and sports (e.g. [21]).

Although, in principle, different technologies, metrics and orientations can beused in Network DEA, for the sake of simplicity we will assume a VariableReturns to Scale (VRS) radial, input-oriented approach. First we will presentthe crisp data model formulation but, before, it is important to introduce appro-priate notation (see [20]). Thus, let us assume that there exist n DMUs all of whichare structurally homogeneous, i.e. all of them have the same number and type ofprocesses. Each process consumes a different subset of inputs and produces adifferent subset of outputs. Let I pð Þ be the set of exogenous inputs used in processp and, for each i 2 I pð Þ, let xp

ij denote the observed amount of exogenous input i

consumed by process p of DMU j. Similarly, let O pð Þ the set of final outputs ofprocess p and, for each k 2 O pð Þ, let yp

kj denote the observed amount of final outputk produced by process p of DMU j. Let PI ið Þ be the set of processes that consumethe exogenous input i and xij¼

P

p2PI ið Þxp

ij the total amount of exogenous input i

consumed by all processes of DMU j. Let PO kð Þ be the set of processes thatproduce the final output k and ykj¼

P

p2PO kð Þyp

kj the total amount of final output k

produced by all processes of DMU j.In addition to exogenous inputs and outputs, there exist R intermediate prod-

ucts. Let Pout rð Þ be the set of processes that generate the intermediate product r sothat for each p 2 Pout rð Þ let zp

rj the observed amount of intermediate product r

generated by process p of DMU j. Analogously, let Pin rð Þ be the set of processesthat consume the intermediate product r and for each p 2 Pin rð Þ let zp

rj the observed

amount of intermediate product r consumed by process p of DMU j. Let us assumethat an intermediate product r cannot be consumed and produced simultaneously

208 S. Lozano and P. Moreno

by a process, i.e. Pout rð Þ \ Pin rð Þ ¼ ; 8r. Also, without loss of generality, let usassume that

X

p2Pout rð Þzp

rj ¼X

p2Pin rð Þzp

rj 8r 8j

i.e. the intermediate products are completely generated and consumed within theown DMU. Finally, to facilitate the model formulation it is convenient to definethe sets Rout pð Þ and Rin pð Þ corresponding to the intermediate products producedand consumed, respectively, by a certain process p.

Note that the sets Pout rð Þ and Pin rð Þ (or, equivalently, Rout pð Þ and Rin pð Þ) jointlydetermine all the structure of intermediate flows within the system. Thus, forexample, a system consisting of just parallel process with no intermediate flows(R = 0) would have Rout pð Þ ¼ Rin pð Þ ¼ ; 8p. On the contrary, a typical multi-stage series system would have RoutðpÞ ¼ Rinðpþ 1Þ 1\p\P and Rin 1ð Þ ¼Rout Pð Þ ¼ ;.

To formulate the multiplier VRS radial input-oriented multiplier formulation ofrelational Network DEA model, letJ index of specific DMU being assessedui weight of exogenous input ivk weight of final output kwr weight of intermediate product rgp VRS free intercept variableEJ Efficiency of DMU J

1.1 Model I: Multiplier Form of Crisp Network DEA

EJ ¼ MaxX

k

X

p2PO kð Þvk yp

kJ þX

p

gp

s:t:X

i

X

p2PI ið Þui xp

iJ ¼ 1

X

k2O pð Þvkyp

kj þX

r2Rout pð Þwrz

prj þ gp �

X

i2I pð Þuix

pij �

X

r2Rin pð Þwrz

prj � 0 8j 8p

ui; vk;wr � 0 8i 8k 8r gp free 8p

This model looks for the values of the weights of the inputs, outputs andintermediate products that maximize the virtual output of the DMU J underassessment. A basic feature of the relational network DEA approach is that allprocesses that consume an input or intermediate product or produce an output or

10 Network Fuzzy Data Envelopment Analysis 209

intermediate product use the same weight for that input, output or intermediateproduct. As for the constraints, there are two types. One is that the virtual input ofthe DMU J is set to unity, as it is common in input-oriented DEA models. Thesecond set of constraints guarantees that the efficiency of all processes is boundedby unity. Adding these constraints for the different processes of each DMU leads toa set of constraints (not included in the model because they are redundant) whichguarantee that, with the weights chosen by DMU J, the efficiency of every DMU isnot greater than unity. Finally, note that, for each process p only its own subsets ofinputs, output and intermediate products are taken into account. Analogously, tocompute the virtual input and output only the processes that consume an input orproduce an output are taken into account.

To formulate the dual of model I, leth Uniform reduction factor of the inputs consumption of DMU Jkp

jIntensity variable of process p of DMU j

1.2 Model II: Envelope Form of Crisp Network DEA

EJ ¼ Min h

s:t:X

p2PI ið Þ

X

j

kpj xp

ij � hX

p2PI ið Þxp

iJ 8i

X

p2PO kð Þ

X

j

kpj yp

kj �X

p2PO kð Þyp

kJ 8k

X

p2Pout rð Þ

X

j

kpj zp

rj �X

p2Pin rð Þ

X

j

kpj zp

rj � 0 8r

X

j

kpj ¼ 1 8p

kpj � 0 8j8p h free

This envelopment form Network DEA model finds the maximum radial con-traction of the inputs consumed by DMU J by looking for an appropriate pointwithin the overall Production Possibility Set (PPS) [20]. Thus, the projectedoperation point must maintain the total amount of outputs produced by DMU J andbe such that for each intermediate product the total amount produced by thedifferent processes must be sufficient to satisfy the amounts needed by the differentprocesses that consume it. A basic feature of this type of Network DEA models isthat each process p has its own set of intensity variables kp

j . In other words, each

process has its own process PPS with its own Returns To Scale assumption. In theabove model it has been assumed that all processes operate under VRS.

210 S. Lozano and P. Moreno

2 Extension of Kao and Liu Approach to NetworkFuzzy DEA

Kao and Liu [16] and Kao and Lin [14] have extended Kao and Liu [15] FuzzyDEA approach to two-stage serial systems and to parallel production systems. Inthis section, the approach is extended to general networks of processes such asthose described in the previous section. The difference with the crisp NetworkDEA models I and II will lay on the consideration of fuzzy data for the inputs,outputs and intermediate products. Thus, let us assume that the inputs, outputs andintermediate products consumed or produced by each process are given as LR-typeFuzzy Numbers (LRFN)

eXpij ¼ xp

ij

� �L

; xpij

� �R

; bpij

� �L

; bpij

� �R� �

Lp

i;Rp

i

eYpkj ¼ yp

kj

� �L

; ypkj

� �R

; bbpkj

� �L

; bbpkj

� �R� �

Lp

k;Rp

k

eZprj ¼ zp

rj

� �L

; zprj

� �R

;^b

p

rj

� �L

;^b

p

rj

� �R� �

^Lp

r ;^R

p

r

where Lpi ;R

pi ;bLp

k;bRp

k;^L

p

r ;^R

p

r : 0; 1½ � ! 0; 1½ � are non-increasing, continuous shapefunctions and

Lpi 0ð Þ ¼ Rp

i 0ð Þ ¼ bLpk 0ð Þ ¼ bRp

kð0Þ ¼^L

p

r 0ð Þ ¼ ^Rp

r 0ð Þ ¼ 1 8i8k8r8p

Lpi 1ð Þ ¼ Rp

i 1ð Þ ¼ bLpk 1ð Þ ¼ bRp

k 1ð Þ ¼ ^Lp

r 1ð Þ ¼ ^Rp

r 1ð Þ ¼ 0 8i8k8r8p

The corresponding membership functions are of the type

leX xð Þ ¼

1 if xð ÞL� x� xð ÞR

L xð ÞL�x

bð ÞL� �

if xð ÞL� bð ÞL� x� xð ÞL

R x� xð ÞRbð ÞR

� �if xð ÞR� x� xð ÞRþ bð ÞR

0 otherwise

8>>>><

>>>>:

The a-cuts of eXpij, eY

pkj and eZp

rj are the intervals

eXpij

� �

a¼ eXp

ij

� �L

a; eXp

ij

� �U

a

� �

eXpij

� �L

a¼ xp

ij

� �L

�Lp�

i að Þ � bpij

� �L

eXpij

� �U

a¼ xp

ij

� �R

þRp�

i að Þ � bpij

� �R

10 Network Fuzzy Data Envelopment Analysis 211

eYpkj

� �

a¼ eYp

kj

� �L

a; eYp

kj

� �U

a

� �

eYpkj

� �L

a¼ yp

kj

� �L

�bLp�

k að Þ � bbpkj

� �L

eYpkj

� �U

a¼ yp

kj

� �R

þbRp�

k að Þ � bbpkj

� �R

eZprj

� �

a¼ eZp

rj

� �L

a; eZp

rj

� �U

a

� �

eZprj

� �L

a¼ zp

rj

� �L

�^Lp�

r að Þ � ^bp

rj

� �L

eZprj

� �U

a¼ zp

rj

� �R

þ^Rp�

r að Þ � ^bp

rj

� �R

where the inverse shape functions are defined as

L� að Þ ¼ sup h:L hð Þ� af g

R� að Þ ¼ sup h:R hð Þ� af g

For example, for trapezoidal and triangular fuzzy numbers

L hð Þ ¼ R hð Þ ¼ 1� h

L� að Þ ¼ R� að Þ ¼ 1� a

The Kao and Liu [15, 16] approach is a-level based, according to the classifi-cation of Fuzzy DEA approaches given by Hatami-Marbini et al. [9]. For each

a 2 0; 1½ � the corresponding a-cut of the efficiency of DMU J eEJ can be expressed as

eEJ

� �

a¼ eEJ

� �L

a; eEJ

� �U

a

� �

where the upper and lower limits can be computed using the following pair ofmodels.

2.1 Model III: Kao and Liu Upper Limit

eEJ

� �U

a¼ max

X

k

X

p2PO kð Þvk � eYp

kJ

� �U

aþX

p

gp

s:t:X

i

X

p2PI ið Þui � eXp

iJ

� �L

a

¼ 1

212 S. Lozano and P. Moreno

X

k2O pð Þvk � eYp

kJ

� �U

aþ

X

r2Rout pð Þwr � zp

rJ þgp

�X

i2I pð Þui � eXp

iJ

� �L

a�

X

r2Rin pð Þwr � zp

rJ � 0 8p

X

k2O pð Þvk � eYp

kj

� �L

aþ

X

r2Rout pð Þwr � zp

rj þ gp

�X

i2I pð Þui � eXp

ij

� �U

a�

X

r2Rin pð Þwr � zp

rj � 0 8j 6¼ J 8p

eZprj

� �L

a� zp

rj � eZprj

� �U

a8j8p8r 2 Rin pð Þ [ Rout pð Þ

ui; vk;wr � 0 8i 8k 8r

The above model corresponds to

eEJ

� �U

a¼ max

xp

ij

� L

a� x

p

ij� x

p

ij

� U

a

yp

kj

� �L

a� y

p

kj� y

p

kj

� �U

a

zp

rj

� �L

a� z

p

rj� z

p

rj

� �U

a

EJ

where the efficiency EJ is computed as per model I. The model takes into accountthat the maximum efficiency for DMU occurs when

xpiJ ¼ xp

iJ

� L

a 8p8i 2 I pð Þ

xpij ¼ xp

ij

� �U

a8j 6¼ J8p8i 2 I pð Þ

ypkJ ¼ yp

kJ

� U

a 8p8k 2 O pð Þ

ypkj ¼ yp

kj

� �L

a8j 6¼ J8p8k 2 O pð Þ

Since for the intermediate products it is not known in advance which value,between the corresponding lower and upper limits, leads to the maximum effi-ciency the corresponding amounts are left as variables in the model. Unfortunately,this makes model III non-linear although it can be easily linearised introducing thenew variables

bzprj ¼ wr � zp

rj 8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

and reformulating the corresponding constraints as

10 Network Fuzzy Data Envelopment Analysis 213

X

k2O pð Þvk � eYp

kJ

� �U

aþ

X

r2Rout pð Þbzp

rJ þ gp �X

i2I pð Þui � eXp

iJ

� �L

a�X

r2R pð Þbzp

rJ� 0 8p

X

k2O pð Þvk � eYp

kj

� �L

aþ

X

r2Rout pð Þbzp

rj þ gp �X

i2I pð Þui � eXp

ij

� �U

a�X

r2Rin pð Þbzp

rj � 0 8j 6¼ J 8p

wr � eZprj

� �L

a�bzp

rj�wr � eZprj

� �U

a8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

2.2 Model IV: Kao and Liu Lower Limit

fEJ

� �L

a¼ min h

s:t:X

p2PI ið Þkp

J � eXpiJ

� �U

aþX

p2PI ið Þ

X

j 6¼J

kpj � eX

pij

� �L

a� h

X

p2PI ið Þ

eXpiJ

� �U

a8i

X

p2PO kð Þkp

J � eYpkJ

� �L

aþX

p2PO kð Þ

X

j 6¼J

kpj � eY

pkj

� �U

a�

X

p2PO kð Þ

eYpkJ

� �L

a8k

X

p2Pout rð Þ

X

j

kpj � z

prj�

X

p2Pin rð Þ

X

j

kpj � z

prj� 0 8r

eZprj

� �L

a� zp

rj� eZprj

� �U

a8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

X

j

kpj ¼ 1 8p

kpj � 0 8j 8p h free

The above model corresponds to

eEJ

� �L

a¼ min

xp

ij

� L

a� x

p

ij� x

p

ij

� U

a

yp

kj

� �L

a� y

p

kj� y

p

kj

� �U

a

zp

rj

� �L

a� z

p

rj� z

p

rj

� �U

a

EJ

214 S. Lozano and P. Moreno

where the efficiency EJ is computed as per model II. In model IV it has been takeninto account that the minimum efficiency for DMU J occurs when

xpiJ ¼ xp

iJ

� U

a 8p8i 2 I pð Þ

xpij ¼ xp

ij

� �L

a8j 6¼ J8p8i 2 I pð Þ

ypkJ ¼ yp

kJ

� L

a 8p8k 2 O pð Þ

ypkj ¼ yp

kj

� �U

a8j 6¼ J8p8k 2 O pð Þ

Again, since for the intermediate products it is not known in advance whichvalue, between the corresponding lower and upper limits, leads to the minimumefficiency the corresponding amounts are left as variables in the model. Unfortu-nately, this makes model III non-linear although it can be easily linearisedintroducing the new variables

bzprj ¼ kp

j � zprj 8r 8j 8p

and reformulating the corresponding constraints asX

p2Pout rð Þ

X

j

bzprj �

X

p2Pin rð Þ

X

j

bzprj� 0 8r

kpj � eZ

prj

� �L

a�bzp

rj� kpj � eZ

prj

� �U

a8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

Kao and Liu [15, 16], in addition to estimating the overall efficiency of eachDMU, indicate how to estimate also the efficiency of the different processes.However, since one of the characteristics of multiplier-form DEA models (likeModel I or Model III) is that there can be alternative optimal solutions then it is notclear how to compute the efficiency of individual processes. Since this issue isopen and requires further research (even in the crisp data case) we will not try tocompute process efficiencies.

3 Extension of Saati et al. Approach to NetworkFuzzy DEA

Saati et al. [23] propose a Fuzzy DEA approach that considers Triangular FuzzyNumbers (TFN). Therefore, let us assume that the inputs, outputs and intermediateproducts consumed or produced by each process are given as

eXpij ¼ xp

ij

� ��; xp

ij

� �0; xp

ij

� �þ� �

10 Network Fuzzy Data Envelopment Analysis 215

eYpkj ¼ yp

kj

� ��; yp

kj

� �0; yp

kj

� �þ� �

eZprj ¼ zp

rj

� ��; zp

rj

� �0; zp

rj

� �þ� �

whose corresponding a-cuts are

eXpij

� �

a¼ eXp

ij

� �L

a; eXp

ij

� �U

a

� �

eXpij

� �L

a¼ a � xp

ij

� �0þ 1� að Þ � xp

ij

� ��

eXUa ¼ a � xp

ij

� �0þ 1� að Þ � xp

ij

� �þ

eYpkj

� �

a¼ eYp

kj

� �L

a; eYp

kj

� �U

a

� �

eYpkj

� �L

a¼ a � yp

kj

� �0þ 1� að Þ � yp

kj

� ��

eYpkj

� �U

a¼ a � yp

kj

� �0þ 1� að Þ � yp

kj

� �þ

eZprj

� �

a¼ eZp

rj

� �L

a; eZp

rj

� �U

a

� �

eZprj

� �L

a¼ a � zp

rj

� �0þ 1� að Þ � zp

rj

� ��

eZprj

� �U

a¼ a � zp

rj

� �0þ 1� að Þ � zp

rj

� �þ

Of course, TFNs are just a special case of LRFNs where

xpij

� �0¼ xp

ij

� �L

¼ xpij

� �R

xpij

� ��¼ xp

ij

� �0� bp

ij

� �L

xpij

� �þ¼ xp

ij

� �0þ bp

ij

� �R

and

L hð Þ ¼ R hð Þ ¼ 1� h

L� að Þ ¼ R� að Þ ¼ 1� a

This approach is also a-level based. For each a 2 0; 1½ � an efficiency scoreEJ að Þ can be computed using the following model

216 S. Lozano and P. Moreno

3.1 Model V: Saati et al.

EJ að Þ ¼ maxX

k

X

p2PO kð Þvk yp

kJ þX

p

gp

s:t:X

i

X

p2PI ið Þui xp

iJ ¼ 1

X

k2O pð Þvk yp

kj þX

r2Rout pð Þwr � zp

rj þ gp �X

i2I pð Þui xp

ij �X

r2Rin pð Þwr � zp

rj� 0 8j8p

eXpij

� �L

a� xp

ij� eXpij

� �U

a8j 8p8i 2 I(p)

eYpkj

� �L

a� yp

kj� eYpkj

� �U

a8j 8p8k 2 O(p)

eZprj

� �L

a� zp

rj� eZprj

� �U

a8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

ui; vk;wr� 0 8i 8k 8r

Same as with the Kao and Liu upper limit model III, this corresponds to

eEJ

� �U

a¼ max

xp

ij

� L

a� x

p

ij� x

p

ij

� U

a

yp

kj

� �L

a� y

p

kj� y

p

kj

� �U

a

zp

rj

� �L

a� z

p

rj� z

p

rj

� �U

a

EJ

but maintaining xpij and yp

kj as variables also. Introducing new variables

bxpij ¼ ui xp

ij 8j8p 8i 2 I pð Þ

bypkj ¼ vk yp

kj 8j 8p 8k 2 O pð Þ

bzprj ¼ wr zp

rj 8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

leads to the following Linear Program (LP)

EJ að Þ ¼ maxX

k

X

p2PO kð Þbyp

kJ þX

p

gp

s.t.

10 Network Fuzzy Data Envelopment Analysis 217

X

i

X

p2PI ið Þbxp

iJ ¼ 1

X

k2O pð Þbyp

kj þX

r2Rout pð Þbzp

rj þ gp �X

i2I pð Þbxp

ij �X

r2Rin pð Þbzp

rj� 0 8j 8p

ui � eXpij

� �L

a� bxp

ij� ui � eXpij

� �U

a8j 8p 8i 2 I pð Þ

vk � eYpkj

� �L

a� byp

kj� vk � eYpkj

� �U

a8j8p 8k 2 O pð Þ

wr � eZprj

� �L

a�bzp

rj�wr � eZprj

� �U

a8j 8p 8r 2 Rin pð Þ [ Rout pð Þ

Summarising, the efficiency score computed by the Saati et al. approachcoincides with the Kao and Liu upper limit of model III. This is something thatwill be confirmed in the numerical results section.

4 Extension of Wang et al. Approach to NetworkFuzzy DEA

Wang et al. [26] proposed an a-level set approach which, contrary to the Kao andLiu [15] approach, uses the same crisp production frontier as reference for allDMUs and for all values of a. Such fixed production frontier is inferred from thebest performing values of the DMUs, which correspond to the a ¼ 0:0 upper limitfor the outputs and the a ¼ 0:0 lower limit for the inputs. For the intermediateoutputs the data are not fixed but they can vary between their a ¼ 0:0 lower andupper limits. For each a 2 0; 1½ � the corresponding a-cut of the efficiency of DMU

J eEJ can be expressed as

eEJ

� �

a¼ eEJ

� �L

a; eEJ

� �U

a

� �

where the upper and lower limits can be computed using the following pair ofmodels.

218 S. Lozano and P. Moreno

4.1 Model VI: Wang et al. Upper Limit

eEJ

� �U

a¼ max

X

k

X

p2PO kð Þvk � eYp

kJ

� �U

a

þX

p

gp

s:t:X

i

X

p2PI ið Þui � eXp

iJ

� �L

a¼ 1

X

k2O pð Þvk � eYp

kj

� �U

0:0þ

X

r2Rout pð Þwr � zp

rj þ gp�

�X

i2I pð Þui � eXp

ij

� �L

0:0�X

r2Rin pð Þwr � zp

rj� 0 8j8p

eZprj

� �L

0:0� zp

rj� eZprj

� �U

0:08j 8p 8r 2 Rin pð Þp [ Rout pð Þ

ui; vk;wr� 0 8i 8k 8r

4.2 Model VII: Wang et al. Lower Limit

eEJ

� �L

a¼ min h

s:t:X

p2PI ið Þ

X

j

kpj � eX

pij

� �L

0:0� h

X

p2PI ið Þ

eXpij

� �U

a8i

X

p2PO kð Þ

X

j

kpj � eY

pkj

� �U

0:0�

X

p2PO kð Þ

eYpkj

� �L

a8k

X

p2Pout rð Þ

X

j

kpj � z

prj �

X

p2Pin rð Þ

X

j

kpj � z

prj� 0 8r

eZprj

� �L

0:0� zp

rj� eZprj

� �U

0:08j 8p 8r 2 Rin pð Þ [ Rout pð Þ

X

j

kpj ¼ 1 8p

kpj � 0 8j 8p h free

10 Network Fuzzy Data Envelopment Analysis 219

The above two models can be linearised exactly the same as Kao and Liumodels III and IV, respectively.

5 Numerical Experiments

In this section the proposed models will be applied to a dataset from the literature.Although this chapter deals with general networks of processes, the only publishedNetwork Fuzzy DEA datasets are those in Kao and Liu [16] and Kao and Lin [14].However, although Kao and Lin [14] considers a parallel production systemproblem with fuzzy data, the specific dataset used to illustrate their approach is ofa shared-inputs type, i.e. the actual amounts of an input consumed by each processis not known but it is left to the DEA model the task of determining, within certainbounds, the share of the inputs that is supposedly consumed by each process. Sincethat type of Network DEA models is different to the one considered here thecorresponding dataset cannot be used for our purpose.

The dataset in Kao and Liu [16] corresponds to a simple two-stage system withtwo inputs, two intermediate products and two outputs, as shown in Fig. 1.Although the models formulated in this chapter can deal with VRS, for comparisonwith Kao and Liu [16] Constant Returns to Scale (CRS) will be assumed, whichmeans that variables gp should be dropped from multiplier formulations and

accordingly the convexity constraints on the intensity variables kpj should also be

dropped from the envelope formulations.Tables A.1, A.2, and A.3, in the appendix, show the TFN corresponding to the

inputs, intermediate products and outputs, respectively, of the 24 DMUs in Kao andLiu [16]. Tables 1 and 2 show, for different values of a 2 0; 1½ � the lower and upperlimits of the corresponding a-cuts computed by Kao and Liu models III and IV,respectively. There are some minor differences with respect to the results reportedby Kao and Liu [16]. The results obtained have been calculated with the datasetshown in Tables A.1, A.2 and A.3, which corresponds exactly to the dataset thatappears in Kao and Liu [16]. It seems that the results reported in that paper wereobtained using a dataset with more precision, as in Kao and Hwang [12].

Direct writen premiums

1 21j 1jZ Z=

Underwriting profits 2

1jY

Process 2

(Profit

generation)

Operating expenses 1

1jX

Insurance expenses 1

2 jX

Reinsurance premiums

1 22 j 2 jZ Z=

Investment profits 2

1jY

Process 1

(Premium

acquisition)

Fig. 1 Two-stage system from Kao and Liu [16]

220 S. Lozano and P. Moreno

Tab

le1

Upp

erli

mit

ofa-

cuts

ofef

fici

ency

aspe

rm

odel

III

Kao

and

Liu

DM

Ua¼

0:0

a¼

0:2

a¼

0:4

a¼

0:6

a¼

0:8

a¼

1:0

10.

904

0.86

20.

820

0.77

90.

741

0.70

02

0.79

50.

759

0.72

40.

691

0.65

90.

626

30.

861

0.82

40.

788

0.75

40.

721

0.69

04

0.42

60.

399

0.37

30.

348

0.32

60.

304

51.

000

1.00

00.

968

0.90

50.

847

0.79

26

0.51

00.

487

0.46

50.

444

0.41

60.

389

70.

375

0.35

30.

332

0.31

30.

295

0.27

78

0.37

10.

350

0.32

90.

310

0.29

20.

275

90.

294

0.27

80.

263

0.24

90.

236

0.22

410

0.63

60.

598

0.56

20.

529

0.49

70.

468

110.

217

0.20

40.

192

0.18

10.

170

0.15

912

0.94

20.

902

0.86

40.

828

0.79

30.

760

130.

276

0.26

00.

246

0.23

20.

219

0.20

714

0.39

20.

369

0.34

70.

327

0.30

70.

289

150.

794

0.75

30.

715

0.67

80.

644

0.61

216

0.43

30.

408

0.38

30.

361

0.33

90.

319

170.

484

0.45

60.

430

0.40

60.

383

0.36

118

0.35

00.

330

0.31

00.

292

0.27

50.

259

190.

497

0.48

00.

464

0.44

80.

433

0.41

320

0.72

50.

685

0.64

70.

609

0.57

30.

539

210.

247

0.23

50.

223

0.21

20.

201

0.19

122

0.78

30.

743

0.70

60.

671

0.63

80.

606

230.

556

0.52

20.

489

0.45

90.

431

0.40

424

0.17

60.

166

0.15

60.

147

0.13

90.

131

10 Network Fuzzy Data Envelopment Analysis 221

Tab

le2

Low

erli

mit

ofa-

cuts

ofef

fici

ency

aspe

rm

odel

IVK

aoan

dL

iu

DM

Ua¼

0:0

a¼

0:2

a¼

0:4

a¼

0:6

a¼

0:8

a¼

1:0

10.

500

0.53

50.

572

0.61

20.

654

0.70

02

0.44

70.

478

0.51

10.

547

0.58

50.

626

30.

549

0.57

50.

603

0.63

20.

661

0.69

04

0.21

30.

229

0.24

60.

265

0.28

40.

304

50.

563

0.60

30.

646

0.69

20.

741

0.79

26

0.27

90.

298

0.31

90.

341

0.36

40.

389

70.

203

0.21

60.

230

0.24

50.

261

0.27

78

0.20

30.

215

0.22

90.

243

0.25

90.

275

90.

163

0.17

40.

185

0.19

80.

211

0.22

410

0.33

70.

360

0.38

50.

411

0.43

90.

468

110.

116

0.12

40.

132

0.14

10.

150

0.15

912

0.55

90.

595

0.63

40.

676

0.72

00.

760

130.

153

0.16

30.

173

0.18

40.

195

0.20

714

0.21

20.

225

0.24

00.

256

0.27

20.

289

150.

460

0.48

90.

520

0.55

10.

580

0.61

216

0.23

30.

248

0.26

40.

281

0.30

00.

319

170.

266

0.28

30.

301

0.32

00.

340

0.36

118

0.18

90.

202

0.21

50.

229

0.24

30.

259

190.

309

0.32

80.

348

0.36

80.

391

0.41

320

0.39

50.

421

0.44

80.

476

0.50

70.

539

210.

145

0.15

40.

162

0.17

10.

181

0.19

122

0.51

00.

528

0.54

70.

566

0.58

60.

606

230.

291

0.31

10.

332

0.35

50.

379

0.40

424

0.09

50.

101

0.10

80.

115

0.12

30.

131

222 S. Lozano and P. Moreno

Tab

le3

Upp

erli

mit

ofa-

cuts

ofef

fici

ency

aspe

rm

odel

VI

Wan

get

al

DM

Ua

=0.

0a

=0.

2a

=0.

4a

=0.

6a

=0.

8a

=0.

10

10.

792

0.77

30.

754

0.73

60.

719

0.70

22

0.70

70.

691

0.67

40.

658

0.64

20.

627

30.

779

0.76

10.

743

0.72

50.

708

0.69

14

0.34

40.

336

0.32

80.

320

0.31

20.

305

50.

897

0.87

60.

855

0.83

50.

815

0.79

56

0.44

10.

431

0.42

10.

411

0.40

10.

391

70.

309

0.30

30.

297

0.29

10.

286

0.28

08

0.30

80.

302

0.29

60.

290

0.28

40.

278

90.

249

0.24

40.

239

0.23

50.

230

0.22

510

0.51

90.

509

0.49

90.

489

0.47

90.

469

110.

179

0.17

50.

172

0.16

80.

165

0.16

212

0.84

10.

825

0.80

90.

793

0.77

70.

761

130.

233

0.22

80.

224

0.22

00.

215

0.21

114

0.32

20.

316

0.30

90.

303

0.29

70.

292

150.

681

0.66

70.

654

0.64

10.

629

0.61

616

0.35

40.

347

0.34

00.

333

0.32

70.

320

170.

404

0.39

60.

388

0.38

00.

373

0.36

618

0.28

80.

282

0.27

60.

271

0.26

60.

261

190.

460

0.45

10.

443

0.43

40.

426

0.41

820

0.60

90.

597

0.58

50.

573

0.56

10.

550

210.

205

0.20

30.

200

0.19

80.

196

0.19

322

0.70

90.

690

0.67

20.

654

0.63

70.

621

230.

446

0.43

80.

429

0.42

10.

413

0.40

524

0.14

70.

144

0.14

10.

138

0.13

50.

132

10 Network Fuzzy Data Envelopment Analysis 223

Tab

le4

Low

erli

mit

ofa-

cuts

ofef

fici

ency

aspe

rm

odel

VII

Wan

get

al

DM

Ua¼

0:0

a¼

0:2

a¼

0:4

a¼

0:6

a¼

0:8

a¼

1:0

10.

494

0.50

60.

519

0.53

10.

544

0.55

72

0.44

00.

450

0.46

10.

473

0.48

40.

496

30.

487

0.49

90.

511

0.52

30.

536

0.54

94

0.21

30.

219

0.22

40.

230

0.23

50.

241

50.

563

0.57

70.

591

0.60

50.

620

0.63

56

0.27

90.

286

0.29

30.

300

0.30

70.

315

70.

203

0.20

70.

212

0.21

60.

220

0.22

58

0.20

30.

207

0.21

10.

215

0.21

90.

224

90.

163

0.16

60.

169

0.17

30.

176

0.18

010

0.33

70.

344

0.35

10.

358

0.36

50.

372

110.

116

0.11

90.

121

0.12

40.

126

0.12

912

0.55

40.

565

0.57

60.

588

0.60

00.

612

130.

153

0.15

60.

160

0.16

30.

166

0.16

914

0.21

20.

216

0.22

00.

224

0.22

90.

233

150.

449

0.45

80.

467

0.47

60.

486

0.49

616

0.23

30.

238

0.24

30.

248

0.25

30.

258

170.

266

0.27

10.

276

0.28

20.

287

0.29

318

0.18

90.

193

0.19

70.

201

0.20

50.

209

190.

304

0.31

00.

316

0.32

20.

329

0.33

520

0.39

50.

403

0.41

20.

420

0.42

90.

438

210.

145

0.14

70.

149

0.15

10.

153

0.15

522

0.47

80.

482

0.48

60.

490

0.49

40.

498

230.

291

0.29

70.

303

0.30

90.

315

0.32

124

0.09

50.

097

0.09

90.

101

0.10

30.

105

224 S. Lozano and P. Moreno

As advanced in Sect. 3, the efficiency scores computed by the Saati et al. modelV coincide with the upper limits of Kao and Liu shown in Table 1.

Tables 3 and 4 show the upper and lower limits, respectively, of the efficiencyscores, for the different possibility values a 2 0; 1½ �, computed by the Wang et al.models VI and VII. Note that for a = 1, the upper and lower limits in these tablesdo not coincide between themselves.

Let us make one final remark about a fact that may have been noticed by thereader and it is that, for almost none of the different approaches tried, no DMU isassessed as efficient, i.e. almost all efficiency scores are below unity, even fora ¼ 0:0 upper limits. This is not surprising and, actually, it is quite common inNetwork DEA because when each DMU consists of several processes then all ofthem must be efficient for the overall DMU to be efficient. That is a tall order thatdoes occur often. It occurs more often that one process of a DMU may be efficientbut not all the others.

6 Conclusions

This chapter has shown how to extend several Fuzzy DEA approaches to theNetwork DEA context. It has helped greatly the fact that the notation used for thecrisp Network DEA approach allows for a rather simple formulation of the modelfor general networks. Extending those formulations to handle fuzzy data is notstraightforward but not difficult, as the approaches shown in this chapter show.

Since Network DEA represents, in general, a more fine-grained level of analysiswhich can lead to more valid results (although at the expense of requiring moredetailed data), the possibility of applying a Network Fuzzy DEA approach forthose problems in which the data are uncertain contributes to enhance the use-fulness of the approach.

Acknowledgments This research was carried out with the financial support of the SpanishMinistry of Science grant DPI2010-16201, and FEDER.

Appendix

See Tables A.1, A.2, and A.3.

10 Network Fuzzy Data Envelopment Analysis 225

Tab

leA

.1In

puts

data

ofK

aoan

dL

iu[1

6]pr

oble

m

DM

Ux1 1j

���

x1 1j

��

0x1 1j

��þ

x1 2j

���

x1 2j

��

0x1 2j

��þ

111

1311

7812

5663

667

371

72

1305

1381

1472

1278

1352

1441

311

1211

7712

5555

959

263

14

568

601

641

561

594

633

563

3166

9971

4131

6733

5135

726

2483

2627

2800

631

668

712

718

5319

4220

4713

7714

4315

218

3615

3789

3994

1787

1873

1974

914

9515

6716

5290

695

010

0110

1243

1303

1373

1238

1298

1368

1118

7219

6220

6864

167

270

812

2473

2592

2732

620

650

685

1324

8126

0927

3913

0113

6814

3614

1328

1396

1466

940

988

1037

1520

7721

8422

9361

965

168

416

1152

1211

1272

395

415

436

1713

8214

5315

2610

3210

8511

3918

720

757

795

520

547

574

1915

115

916

717

318

219

120

138

145

152

5053

5621

8084

8825

2627

2214

1516

910

1023

5154

5727

2829

2415

516

317

122

323

524

6

226 S. Lozano and P. Moreno

Tab

leA

.2In

term

edia

tepr

oduc

tsda

taof

Kao

and

Liu

[16]

prob

lem

DM

Uz1 1j

���

z1 1j

��

0z1 1j

��þ

z1 2j

���

z1 2j

��

0z1 2j

��þ

170

4174

5179

4380

985

691

22

9469

1002

010

681

1712

1812

1932

345

1347

7650

9152

956

059

74

2999

3174

3383

351

371

395

535

335

3736

239

680

1657

1753

1869

692

1197

4710

390

900

952

1015

710

193

1068

511

262

613

643

678

816

473

1726

718

199

1082

1134

1195

910

945

1147

312

093

521

546

575

1078

3282

1086

5348

150

453

111

6890

7222

7612

613

643

678

1290

0094

3499

4310

6711

1811

7813

1323

913

921

1461

777

181

185

214

7034

7396

7766

442

465

488

1599

1110

422

1094

371

274

978

616

5331

5606

5886

382

402

422

1773

1876

9580

8032

534

235

918

3453

3631

3813

946

995

1045

1910

8311

4111

9645

848

350

620

300

316

331

124

131

137

2121

422

523

638

4042

2249

5254

1314

1523

233

245

257

4749

5124

452

476

499

611

644

675

10 Network Fuzzy Data Envelopment Analysis 227

Tab

leA

.3O

utpu

tsda

taof

Kao

and

Liu

[16]

prob

lem

DM

Uy1 1j

���

y1 1j

��

0y1 1j

��þ

y1 2j

���

y1 2j

��

0y1 2j

��þ

193

098

410

4964

468

172

62

1160

1228

1309

788

834

889

327

729

331

262

265

870

14

234

248

264

167

177

189

574

1978

5183

6937

0939

2541

846

1619

1713

1826

392

415

442

721

3622

3923

6041

943

946

38

3720

3899

4110

593

622

656

999

510

4310

9925

226

427

810

1619

1697

1789

529

554

584

1114

1814

8615

6617

1819

1215

0215

7416

5986

790

995

813

3432

3609

3789

212

223

234

1413

3214

0114

7131

633

234

915

3191

3355

3523

528

555

583

1681

285

489

718

719

720

717

2990

3144

3301

353

371

390

1865

869

272

715

516

317

119

493

519

544

4446

4820

337

355

372

2526

2721

4851

536

66

2278

8286

44

423

11

117

1819

2413

514

214

915

1617

228 S. Lozano and P. Moreno

References

1. Alfonso, E., Kalenatic, D., López, C.: Modeling the synergy level in a vertical collaborativesupply chain through the IMP interaction model and DEA framework. Ann. Oper. Res. 181,813–827 (2010)

2. Avkiran, N.K.: Opening the black box of efficiency analysis: an illustration with UAE banks.Omega 37, 930–941 (2009)

3. Färe, R., Grosskopf, S.: Productivity and intermediate products: a frontier approach. Econ.Lett. 50, 65–70 (1996)

4. Färe, R., Grosskopf, S.: Network DEA. Socio-Econ. Plan. Sci. 34, 35–49 (2000)5. Chen, Y., Cook, W.D., Li, N., Zhu, J.: Additive efficiency decomposition in two-stage DEA.

Eur. J. Oper. Res. 196, 1170–1176 (2009)6. Cook, W.D., Zhu, J, Bi, G., Yang, F.: Network DEA: Additive efficiency decomposition. Eur.

J. Oper. Res. 207, 1122–1129 (2010)7. Fukuyama, H., Weber, W.L.: A slacks-based inefficiency measure for a two-stage system

with bad outputs. Omega 38(5), 398–409 (2010)8. Guan, J., Chen, K.: Measuring the innovation production process: a cross-region empirical

study of China’s high-tech innovations. Technovation 30, 348–358 (2010)9. Hatami-Marbini, A., Emrouznejad, A., Tavana, M.: A taxonomy and review of the fuzzy data

envelopment analysis literature: two decades in the making. Eur. J. Oper. Res. 214, 457–472(2011)

10. Kao, C.: Efficiency decomposition in network data envelopment analysis: a relational model.Eur. J. Oper. Res. 192, 949–962 (2009)

11. Kao, C.: Efficiency measurement for parallel production systems. Eur. J. Oper. Res. 196,1107–1112 (2009)

12. Kao, C., Hwang, S.N.: Efficiency decomposition in two-stage data envelopment analysis: anapplication to non-life insurance companies in Taiwan. Eur. J. Oper. Res. 185, 418–429(2008)

13. Kao, C., Hwang, S.N.: Efficiency measurement for network systems: IT impact on firmperformance. Decis. Support Syst. 48(3), 437–446 (2010)

14. Kao, C., Lin, P.H.: Efficiency of parallel production systems with fuzzy data. Fuzzy Sets Syst.198, 83–98 (2012)

15. Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst.113, 427–437 (2000)

16. Kao, C., Liu, S.T.: Efficiencies of two-stage systems with fuzzy data. Fuzzy Sets Syst. 176,20–35 (2011)

17. Lewis, H.F., Sexton, T.R.: Network DEA: efficiency analysis of organizations with complexinternal structure. Comput. Oper. Res. 31, 1365–1410 (2004)

18. Lewis, H.F., Sexton, T.R.: Data envelopment analysis with reverse inputs and outputs.J. Prod. Anal. 21, 113–132 (2004)

19. Liu, S.T., Wang, R.T.: Efficiency measures of PCB manufacturing firms using relational two-stage data envelopment analysis. Expert Syst. Appl. 36, 4935–4939 (2009)

20. Lozano, S.: Scale and cost efficiency analysis of networks of processes. Expert Syst. Appl.38(6), 6612–6617 (2011)

21. Moreno, P., Lozano, S. (in press) A network DEA assessment of team efficiency in the NBA.Ann. Oper. Res. doi:10.1007/s10479-012-1074-9

22. Rayeni, M.M., Saljooghi, F.H.: Network data envelopment analysis model for estimatingefficiency and productivity in universities. J. Comput. Sci. 6(11), 1252–1257 (2010)

23. Saati, S., Memariani, A., Jahanshahloo, G.R.: Efficiency analysis and ranking of DMUs withfuzzy data. Fuzzy Optim. Decis. Making 1, 255–267 (2002)

24. Sexton, T.R., Lewis, H.F.: Two-stage DEA: an application to major league baseball. J. Prod.Anal. 19, 227–249 (2003)

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25. Tone, K., Tsutsui, M.: Network DEA: a slacks-based measure approach. Eur. J. Oper. Res.197, 243–252 (2009)

26. Wang, Y.M., Greatbanks, R., Yang, J.B.: Interval efficiency assessment using dataenvelopment analysis. Fuzzy Sets Syst. 153, 347–370 (2005)

27. Yu, M.M.: Assessment of airport performance using the SBM-NDEA model. Omega 38,440–452 (2010)

28. Yu, M.M., Lee, B.C.Y.: Efficiency and effectiveness of service business: evidence frominternational tourist hotels in Taiwan. Tour. Manage. 30, 571–580 (2009)

230 S. Lozano and P. Moreno

Chapter 11An Application of Fuzzy DataEnvelopment Analysis Approachfor Occupational Safety

Esra Bas

Abstract In this chapter, an integrated fuzzy data envelopment analysis (DEA) isproposed for evaluating different departments of a company with respect to theoccupational safety investment. The inputs in the DEA methodology are occu-pational safety investments classified as A, B, C, and D-type investments accordingto their level in hierarchy of preventive and protective measures. The outputs in theDEA methodology are occupational safety related performance measures andbusiness environment related performance measures. The t-norm and t-conormfuzzy relations are used for the defuzzification of the fuzzy CCR-based DEAmodel.

Keywords Data envelopment analysis � Occupational safety

1 Introduction

Data envelopment analysis is a nonparametric analysis technique for the relativeefficiency evaluation of different units in a system. Although DEA has diverseapplications in different areas, its application in safety performance measurementhas been relatively limited [1–7].

In this chapter, a Decision Making Unit (DMU) in the DEA methodology isassumed to be a department in a company, and each department under consider-ation is evaluated with respect to their performance for occupational safetyinvestment. With this concern, inputs are defined as occupational safety invest-ment types based on hierarchy of preventive and protective measures, and outputsare defined as occupational safety related performance measures and business

E. Bas (&)Istanbul Technical University, Department of Industrial Engineering,Macka, Istanbul, Turkeye-mail: atace@itu.edu.tr

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_11, � Springer-Verlag Berlin Heidelberg 2014

231

environment related performance measures. Since inputs and outputs are subject touncertainty and vagueness, fuzzy approach is proposed to address these issues, andtwo CCR-based models are developed based on t-norm and t-conorm fuzzyrelation defuzzification to find the relative efficiency of each department.

The structure of the chapter is as follows: In Sect. 2, the methodology forintegrating occupational safety with DEA is introduced, and the details are given.In Sect. 3, the CCR-based model in crisp case is introduced and its fuzzy coun-terparts are developed based on t-norm and t-conorm fuzzy relation. In Sect. 4, ahypothetical example is introduced including three departments of a company withtheir hypothetical data, the models are solved, and the results are discussed.Finally, in Sect. 5, conclusions and potentials for the next research are discussed.

2 Integration of Occupational Safety and DEA

In this section, integration of the hierarchy of preventive and protective measures(PPM) for occupational safety with DEA methodology is proposed for perfor-mance measurement of different departments with respect to occupational safetyinvestment. Figure 1 provides the general methodology.

2.1 Hierarchy of Preventive and Protective Measuresfor Occupational Safety

Hierarchy of preventive and protective measures describes the measures in orderof implementation and effectiveness. In this chapter, each hierarchy is defined asA, B, C and D-type measures.

A-type measures:Eliminate risk at source

B-type measures:Control risk at source

C-type measures:Minimize risk at source

D-type measures:Personal protective equipment

A-type investments:Investments to “eliminate risk at source”

B-type investments:Investments to “control risk”at source

C-type investments:Investments to “minimize risk at source”

D-type investments:Investments for “personal protective equipment”

Business environment related performance measures

Occupational safety related performance measures

Hierarchy of PPM Inputs for DEA Outputs for DEAFig. 1 The methodology forintegration of occupationalsafety and DEA (measuresand inputs adapted fromAlli [8])

232 E. Bas

2.1.1 A-Type Measures: Eliminate Risk at Source

A-type measures involve engineering and organizational measures to thoroughlyeliminate risk at source. One typical example for this type of measure would be touse machine guards to totally prevent the interaction of people with the machine.Another example could be to change the processes of a manufacturing line tototally prevent the emission of some caustic gases (adapted from Alli [8]).

2.1.2 B-Type Measures: Control Risk at Source

B-type measures involve engineering and organizational measures for thoseoccupational safety risks that cannot be totally eliminated such that the risk can becontrolled by other means to decrease its occurrence or effect. An example for thistype of measure could be to use machine guards when required. Another examplecould be to change the processes of a manufacturing line to decrease the emissionof some caustic gases (adapted from Alli [8]).

2.1.3 C-Type Measures: Minimize Risk at Source

C-type measures involve mostly organizational measures for those occupationalsafety risks with a high probability of occurrence such that their occurrence ortheir effect in case of their occurrence is minimized (adapted from Alli [8]). Safetymanagement practices and first-aid measures are typical examples for this type ofmeasure.

2.1.4 D-Type Measures: Personal Protective Equipment

D-type measures involve those types of equipment to be worn by the employeessuch as hard hats, welding glasses to protect the body from the effect of any hazard(adapted from Alli [8]).

2.2 Inputs for DEA

The inputs for DEA are life-cycle costs of preventive and protective measuresoutlined in Sect. 2.1 and costs of possible changes to the business processes.

2.2.1 Life-cycle Costs of Preventive and Protective Measures

Each type of measure outlined in Sect. 2.1 requires some non-recurring costs suchas initial costs, while some of these measures require additionally recurring costs

11 An Application of Fuzzy Data Envelopment 233

such as operating and maintenance costs, periodic review for conformance andperiodic training. Since the time to evaluate the performance measure of adepartment can be at any time with regards to the efficiency of a measure, theninitial costs and some operating and maintenance costs will be sunk costs at thetime of evaluation. However, it is recommended that all possible life-cycle costs ofan investment are considered including sunk costs and future costs and to find thetotal present value of these costs, where the present time represents the currenttime of evaluating the performance of a department. It should also be noted thatthis approach may generally not be practical, and those costs incurred at the timeof evaluation can be considered for the relative efficiency.

It is important to detail the investments related to preventive and protectivemeasures for occupational safety. Tong and Ding [6] proposed to determine thesafety costs per man, since the number of staff is likely to differ betweendepartments. Some examples of investments for preventive and protective mea-sures are as follows (adapted from Tong and Ding [6]):

A-type investment:Purchasing costs of machine guards.Initial system costs for business process reengineering.B-type investment:Organizational safety management costs to control risks at sourceInitial system costs for business process change to control risk at sourceCosts for periodic maintenance of machines and equipmentCosts for periodic inspection of machines and equipmentCosts of control of workers for conformance to safety measuresCosts for safety communication to workersInitial and periodic training costs for management and workers.C-type investment:Organizational safety management costs to minimize risks at sourceCosts of passive guards such as warning linesCosts of first-aid preparation and implementationCosts of emergency plans such as evacuation plans.D-type investment:Costs of personal protective equipment per manCosts of training per man for personal protective equipment.

2.2.2 Costs for Business Processes Change

Additional investments that are much harder to predict in monetary values are theadaptation for the changes to business processes as a result of introducing a pre-ventive and protective measure. Some measures have no effect to the businessprocesses, while others may require radical investment as a result of reengineeringin the business processes.

234 E. Bas

2.3 Outputs for DEA

The outputs to be considered in DEA methodology are occupational safety relatedperformance measures and general business environment related performancemeasures.

2.3.1 Occupational Safety Related Performance Measures

The preventive and protective measures outlined in Sect. 2.1 are expected toimprove occupational safety related environment by decreasing hazard, decreasingthe number of non-fatal and fatal accidents and near-accidents [2, 6]. Othermeasures such as safety culture are also proposed to be evaluated. Although someof the performance measures are to be represented in numerical values, someothers such as safety culture are hard to be evaluated in numerical values.

2.3.2 Business Environment Related Performance Measures

The preventive and protective measures outlined in Sect. 2.1 are expected not onlyto improve occupational safety related environment, but also general businessenvironment by decreasing damage, increasing productivity and employee satis-faction and decreasing rate of errors. Fernández-Muñiz et al. [9] studied therelation between occupational safety and firm performance, and found thatinvestment in occupational safety has a positive effect on quality of products andservices, customer satisfaction and firm reputation. However, the findings fromFernández-Muñiz et al. [9] cannot be included in our model, since these outputsare not at departmental level, rather at firm level.

2.4 Uncertainty Nature in the Inputs and Outputs

Some of the DEA inputs and outputs outlined in Sect. 2 are hard to be representedin monetary values, rather to be represented with vague information, while othersrepresented in monetary values have some uncertainty, since the analysis periodcovers future costs that are hard to be predicted at the time of evaluation. Thus, anuncertainty approach is indispensable to consider uncertainty nature of future costsand vagueness of some evaluation. Since fuzzy approach has been effectively usedto include vague and uncertain information in different applications, it is alsoproposed to be used in the proposed methodology. Tables 1 and 2 give someexamples of inputs and outputs with fuzzy nature.

11 An Application of Fuzzy Data Envelopment 235

3 Model Formulation

3.1 Model Formulation in Crisp Case

In this section, a CCR-based DEA model will be proposed to evaluate eachdepartment under consideration for their relative efficiency with respect to occu-pational safety investment (adapted from Cooper et al. [10]):

Model I

max h ¼X

r

lOS;ryOS;r;0 þX

r

lB E;ryB E;r;0

s:t:X

i

mA;ivA;i;0 þX

i

mB;ivB;i;0 þX

i

mC;ivC;i;0 þX

i

mD;ivD;i;0

þX

i

mBPC;ivBPC;i;0 ¼ 1

X

r

lOS;ryOS;r;j þX

r

lB E;ryB E;r;j

�X

i

mA;ivA;i;j þX

i

mB;ivB;i;j þX

i

mC;ivC;i;j þX

i

mD;ivD;i;j

þX

i

mBPC;ivBPC;i;j; j ¼ 1; 2; . . .; n

mA;i; mB;i; mC;i; mD;i; mBPC; lOS;r; lBE;r � 0; 8i; r

vA;i : Weight assigned to the ith A-type investmentvB;i : Weight assigned to the ith B-type investmentvC;i : Weight assigned to the ith C-type investmentvD;i : Weight assigned to the ith D-type investment

Table 1 Some examples of inputs with fuzzy nature

DEA inputs Fuzzy nature

Costs businessprocesseschange

Investments for adaptation of the businessprocesses as a result of reengineering

Fuzzy nature with hard toget crisp information

Investments for reorganization Fuzzy nature with hard toget crisp information

Table 2 Some examples of outputs with fuzzy nature

DEA outputs Fuzzy nature

Occupational safety relatedperformance measures

Increase in level of perceptionof safety culture

Fuzzy nature with vagueinformation

Business environment relatedperformance measures

Increase in employeesatisfaction

Fuzzy nature with vagueinformation

236 E. Bas

vBPC;i : Weight assigned to the ith business process change costxA;i;j : Costs assigned to DMUj for the ith A-type investmentxB;i;j : Costs assigned to DMUj for the ith B-type investmentxC;i;j : Costs assigned to DMUj for the ith C-type investmentxD;i;j : Costs assigned to DMUj for the ith D-type investmentxBPC;i;j : Costs assigned to DMUj for the ith business process changeyOS;r;j : Occupational safety related performance improvement of DMUj related

to the rth occupational safety related performance measureyBE;r;j : Business environment related performance improvement of DMUj

related to the rth business environment related performance measurelOS;r : Weight assigned to the rth occupational safety related performance

measure (decision variable)lBE;r : Weight assigned to the rth business environment related performance

measure (decision variable)

In Model I, j = 0 implies the DMU under consideration. Thus, the model providesthe results for DMU under consideration, and it should be repeated for each DMU.

3.2 Model Formulation in Fuzzy Case

Hatami-Marbini et al. [8] provided a taxonomy and review of the fuzzy DEAliterature and discussed different fuzzy DEA methods. They classified methodsinto tolerance approach, a-level based approach, fuzzy ranking approach andpossibility approach.

There are two types of uncertainty of inputs and outputs of the model proposedin this chapter: The quantitative factors with some uncertainty as outlined inTable 1 and qualitative factors as outlined in Table 2. Cook and Seiford [12]discussed qualitative factors as ordinal variables. In the following model, it isassumed that qualitative factors are represented as linguistic variables, and t-normand t-conorm fuzzy relations are used for the defuzzification of the model based onInuiguchi et al. [13] and Bas and Kahraman [14].

Before proceeding with the model, some basic preliminaries with respect tot-norm and t-conorm fuzzy relations are given. An a�cut of a fuzzy set ~A isdefined as ~A

� �a¼ x 2 X l~A xð Þ� a

��

� �and a strict a�cut of a fuzzy set ~A is defined

as ~A� �

a¼ x 2 X l~A xð Þ[ a��

� �[13]. If � is a crisp binary relation, ~� is a fuzzy

extension of the crisp binary relation � , ~A and ~B are normal and compact fuzzysets, and T = min is a t-norm, and S = max is a t-conorm, then the followingidentities are equivalent [13, 14]:

l ~� T ~A; ~B� �

� a if and only if inf ~A� �

a� sup ~B� �

a ð1Þ

l ~� S~A; ~B� �

� a if and only if sup ~A� �

1�a� inf ~B� �

1�a ð2Þ

11 An Application of Fuzzy Data Envelopment 237

When fuzzy set ~A is a strictly convex and normal fuzzy number, then (2) willalso be true for a-cut as well as for strictly a�cut [13]. We assume that allparameters are fuzzy variables, since the model can easily be relaxed for the inputsand outputs with crisp nature:

Model II

max h ¼X

r

lOS;r~yOS;r;0 þX

r

lB E;r~yB E;r;0

s:t:X

i

mA;i~vA;i;0 þX

i

mB;i~vB;i;0 þX

i

mC;i~vC;i;0 þX

i

mD;i~vD;i;0

þX

i

mBPC;i~vBPC;i;0 ¼ 1

X

r

lOS;r~yOS;r;j þX

r

lB E;r~yB E;r;j

�X

i

mA;i~vA;i;j þX

i

mB;i~vB;i;j þX

i

mC;i~vC;i;j þX

i

mD;i~vD;i;j

þX

i

mBPC;i~vBPC;i;j; j ¼ 1; 2; . . .; n

mA;i; mB;i; mC;i; mD;i; mBPC;lOS;r; lBE;r � 0; 8i; r

In model II, the quantitative factors with uncertainty can be represented astriangular or trapezoidal fuzzy numbers, while the qualitative factors can be rep-resented as linguistic variables, which are then converted into respective triangularor trapezoidal fuzzy numbers. Model III and Model IV provide the model based ont-norm and t-conorm fuzzy relations as in Bas and Kahraman [14]:

Model III

max h1 ¼X

r

lOS;rinf ~yOS;r;0� �

aþX

r

lBE;rinf ~yBE;r;0� �

a

s:t:X

i

mA;isup ~xA;i;0

� �aþX

i

mB;isup ~xB;i;0

� �aþX

i

mC;isup ~xC;i;0

� �a

þX

i

mD;isup ~xD;i;0� �

aþX

i

mBPC;isup ~xBPC;i;0� �

a¼ 1

X

r

lOS;rinf ~yOS;r;j

� �aþX

r

lBE;rinf ~yBE;r;j

� �a

�X

i

mA;isup ~xA;i;j

� �aþX

i

mB;isup ~xB;i;j

� �a

þX

i

mC;isup ~xC;i;j

� �aþX

i

mD;isup ~xD;i;j

� �a

þX

i

mBPC;isup ~xBPC;i;j

� �a0 j ¼ 1; 2; . . .; n

mA;i; mB;i; mC;i; mD;i; mBPC;i; lOS;r; lBE;r � 0; 8i; r

238 E. Bas

Model IV

max h1 ¼X

r

lOS;rsup ~yOS;r;0

� �1�aþ

X

r

lBE;rsup ~yBE;r;0

� �1�a

s:t:X

i

mA;iinf ~xA;i;0� �

1�aþX

i

mB;iinf ~xB;i;0� �

1�aþX

i

mC;iinf ~xC;i;0� �

1�a

þX

i

mD;iinf ~xD;i;0� �

1�aþX

i

mBPC;iinf ~xBPC;i;0� �

1�a¼ 1

X

r

lOS;rsup ~yOS;r;j

� �1�aþ

X

r

lBE;rsup ~yBE;r;j

� �1�a

�X

i

mA;iinf ~xA;i;j

� �1�aþ

X

i

mB;iinf ~xB;i;j

� �1�a

þX

i

mC;iinf ~xC;i;j

� �1�aþ

X

i

mD;iinf ~xD;i;j

� �1�a

þX

i

mBPC;iinf ~xBPC;i;j

� �1�a0 j ¼ 1; 2; . . .; n

mA;i; mB;i; mC;i; mD;i; mBPC;i; lOS;r; lBE;r � 0; 8i; r

The solutions of Model III and Model IV give the result for each DMU suchthat h ¼ h1; h2½ �. If h ¼ 1; 1½ �, then the DMU is said to be efficient, otherwiseinefficient.

4 A Hypothetical Example

We assume that a company has initiated an occupational safety investment plan,and three departments of this company are to be evaluated with respect to theirrelative efficiency. Tables 3 and 4 provide the DEA inputs and outputs with theirrespective crisp and trapezoidal fuzzy numbers. Table 5 is used for the conversionof linguistic variables into trapezoidal fuzzy numbers.

Model III and Model IV have been solved with MS Excel solver by using thedata in Tables 3 and 4. The results provide that Department 1 and Department 3are efficient for all a-cut values. Although Department 2 is found to be inefficientfor all a-cut levels, its relative efficiency varies according to the a-cut levels.Figure 2 provides the triangular fuzzy number representing the relative efficiencyvalues of Department 2.

According to the results in Fig. 2, the most possible relative efficiency value ofDepartment 2 is 0.89286 with the membership value 1, while the highest possiblevalue is 0.91667. The evaluation for the relative efficiency of Department 2 can bebased on different interpretations. The decision maker can view the results fromFig. 2 as different uncertainty levels, thus consider any of the a-cut levels as the

11 An Application of Fuzzy Data Envelopment 239

risk position, and choose one of them for a decision. As another interpretation, thedefuzzification of the triangular fuzzy number is possible to get a final singlerelative efficiency value for Department 2. However, in any case, Department 2 isfound to be inefficient, while Department 1 and Department 3 are efficient.

Table 3 A hypothetical case study with DEA inputs

Department 1 Department 2 Department 3

DEA inputs/life-cycle costs of preventive and protective measuresA-type measure: Initial system

costs for business processreengineering

$50.000 $75.000 $75.000

B-type measure:Organizational safety

management costs

5.000/month5 months

7.000/month5 months

7.500/month5 months

B-type measure : Costs for safetycommunication to workers(CSCW)

$2.000/month5 months

$2.500/month5 months

$1.500/month5 months

D-type measure:Costs of personal

protectiveequipment

$1.000/man3 workers

$1.000/man5 worker

$1.000/man4 workers

DEA inputs/costs for business processes changeCosts for reorganization ($50.000,

$75.000,$75.000,$100.000)

($80.000,$100.000,$100.000,$120.000)

($80.000,$100.000,$100.000,$120.000)

Table 4 A hypothetical case study with DEA outputs

Department 1 Department 2 Department 3

DEA outputs/occupational safety related performance measuresSavings from decrease in the

number of non-fatal andfatal accidents

$50.000 $10.000 $22.500

Savings from decrease in thenumber of near-accidents

$30.000 $10.000 $20.000

Increase in perception level ofsafety culture

Medium(0.4, 0.5, 0.6, 0.7)

Medium(0.4, 0.5, 0.6, 0.7)

Medium(0.4, 0.5, 0.6, 0.7)

DEA outputs/Business environment related performance measuresSavings from decrease in rate

of errors$10.000 $15.000 $20.000

Increase in employeesatisfaction

Medium(0.4, 0.5, 0.6, 0.7)

Medium(0.4, 0.5, 0.6, 0.7)

Medium High(0.5, 0.6, 0.7, 0.8)

240 E. Bas

5 Conclusions

In this chapter, an integrated DEA methodology has been proposed for the eval-uation of departments of a company with respect to the occupational safetyinvestment. The proposed model considers the uncertainty in the inputs and out-puts of the CCR-based DEA model, and proposes two models based on t-norm andt-conorm fuzzy relation defuzzification. The results from two models with the datafrom a hypothetical example give a triangular fuzzy number for the relative effi-ciency value for a department. Although the proposed model considers uncertaintyand vagueness in the evaluation and give fuzzy numbers for the relative efficiencyof DMUs, the defuzzification of the fuzzy numbers converted from linguisticvariables to evaluate qualitative factors should be discussed. In this chapter, eachDMU has been evaluated independently for qualitative factors by using linguisticvariables instead of ordinal variables. This approach has been found to be prac-tical, since each department is expected to be evaluated by the department

Table 5 An example for aset of linguistic variables(Li [15])

Lingustic variable Trapezoidal fuzzy number

Extremely low (EL) (0, 0, 0.1, 0.2)Very low (VL) (0.1, 0.2, 0.3, 0.4)Low (L) (0.2, 0.3, 0.4, 0.5)Medium low (ML) (0.3, 0.4, 0.5, 0.6)Medium (M) (0.4, 0.5, 0.6, 0.7)Medium high (MH) (0.5, 0.6, 0.7, 0.8)High (H) (0.6, 0.7, 0.8, 0.9)Very high (VG) (0.7, 0.8, 0.9, 1.0)Extremely high (EG) (0.8, 0.9, 1.0, 1.0)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.89 0.895 0.9 0.905 0.91 0.915 0.92

t-norm

t-conorm

Relative efficiency of Department 2

µ

Fig. 2 The triangular fuzzynumber representing therelative efficiency ofDepartment 2

11 An Application of Fuzzy Data Envelopment 241

managers who know the performance measures more in detail, but have noinformation related to the other departments. Although independent auditing isgenerally possible, some qualitative factors such as employee satisfaction are stillto be evaluated by the department managers.

As the next research, the practicality of the approach should be tested for large-scale problems with many DMUs. New decision rules should also be defined incase of triangular fuzzy numbers with the relative efficiency value 1 for at leastsome a-cut values.

References

1. Egilmez, G., McAvoy, D.: Benchmarking road safety of U.S. States: a DEA-based malmquistproductivity index approach. Accid. Anal. Prev. (2013). doi:10.1016/j.aap.2012.12.038

2. El-Mashaleh, M.S., Rababeh, S.M., Hyari, K.H.: Utilizing data envelopment analysis tobenchmark safety performance of construction contractors. Int. J. Project Manage. 28, 61–67(2010)

3. Hermans, E., Brijs, T., Wets, G., Vanhoof, K.: Benchmarking road safety: lessons to learnfrom a data envelopment analysis. Accid. Anal. Prev. 41, 174–182 (2009)

4. Nissi, E., Rapposelli, A.: Analysing industrial accidents in European countries using dataenvelopment analysis. In: Parodi, G., Sciulli, D. (eds.) Social exclusion, AIEL Series inLabour Economics, Springer-Verlag Berlin Heidelberg (2012)

5. Peng, M., Song, L., Guohui, L., Sen, L., Heping, Z.: Evaluation of fire protection performanceof eight countries based on fire statistics: an application of data envelopment analysis, FireTechnology (2012), Springer Science ? Business Media, LLC. Manufactured in The UnitedStates, doi: 10.1007/s10694-012-0301-x

6. Tong, L., Ding, R.: Efficiency assessment of coal mine safety input by data envelopmentanalysis. J. China Univ. Min. Technol. 18, 88–92 (2008)

7. Yan, L., Tong, W., Hui, D., Zongzhi, W.: Research and application on risk assessment DEAmodel of crowd crushing and trampling accidents in subway stations. Procedia Eng. 43,494–498 (2012)

8. Alli, B.O.: Fundamental principles of occupational health and safety, 2nd edn. InternationalLabor Office, Geneva (2008)

9. Fernández-Muñiz, B., Montes-Peón, J.M., Vázquez-Ordás, C.J.: Relation betweenoccupational safety management and firm performance. Saf. Sci. 47, 980–991 (2009)

10. Cooper, W., Seiford, L., Tone, K.: Data envelopment analysis: a comprehensive text withmodels, applications, references and DEA solver software. Kluwer Academic Publishers,Boston/Dordrecht/London (2000)

11. Hatami-Marbini, A., Emrouznejad, A., Tavana, M.: A taxonomy and review of the fuzzy dataenvelopment analysis literature: two decades in the making. Eur. J. Oper. Res. 214, 457–472(2011)

12. Cook, W.D., Seiford, L.M.: Data envelopment analysis (DEA) -Thirty years on. Eur. J. Oper.Res. 192, 1–17 (2009)

13. Inuiguchi, M., Ramik, J., Tanino, T., Vlach, M.: Satisficing solutions and duality in intervaland fuzzy linear programming. Fuzzy Sets Syst. 135, 151–177 (2003)

14. Bas, E., Kahraman, C.: Fuzzy capital rationing model. J. Comput. Appl. Math. 224, 628–645(2009)

15. Li, D.-F.: Compromise ratio method for fuzzy multi-attribute group decision making. Appl.Soft Comput. 7, 807–817 (2007)

242 E. Bas

Chapter 12Integrating Fuzzy Intermediate Factorsin Supply Chain Efficiency Evaluation

Qiong Xia, Liang Liang and Feng Yang

Abstract Effective supply chain management (SCM) depends on the reasonableperformance evaluation to the entire supply chain. Data envelopment analysis(DEA) has been widely developed to measure the performance of supply chains.Although there are sufficient researches on performance evaluation to supplychains, all of these precursory surveys assume that the inputs and outputs involvedin performance evaluation are crisp data. However, in most situations only fuzzyinformation on intermediate factor is available. The current chapter integratesfuzzy intermediate factors in supply chain efficiency evaluation and proposes afuzzy supply chain data envelopment analysis (FSCDEA) model based on previoussupply chain DEA models. A triangular fuzzy membership function is attached onthe fuzzy intermediate factors. According to the operational rules on triangularfuzzy numbers, we turn the FSCDEA model into a linear programming. Finally weuse the proposed FSCDEA model to assess the operational efficiency of a group ofbank branches. Three theorems about the FSCDEA efficiency are proposed andvalidated. The advantages of the FSCDEA model are also discussed.

Keywords Efficiency evaluation � Data envelopment analysis (DEA) � Fuzzy �Supply chain management

L. Liang � F. Yang (&)School of Management, University of Science and Technology of China, No. 96 JinzhaiRoad, Hefei 230026 Anhui Province, People’s Republic of Chinae-mail: fengyang@ustc.edu.cn

Q. XiaSchool of Liberal Arts and Economics, Hefei University of Technology, No. 193 TunxiRoad, Hefei 230009 Anhui Province, People’s Republic of China

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_12, � Springer-Verlag Berlin Heidelberg 2014

243

1 Introduction

Effective supply chain management (SCM) has proven to be a valid mechanism foran organization to provide high quality or quantity of products and services withthe least consumption of capital, labors and other costs. As a part of supply chainmanagement, performance evaluation to the entire supply chain plays a vital rolein examining how effectively a supply chain system operates. An acceptableperformance evaluation tool designed for supply chains can identify whether lessinitial inputs can be used to produce more final outputs in a specific supply chain,and subsequently help decide how to realize it.

There are many optimization criteria to describe the performance of a supplychain, such as total cost, total profit, inventory levels, fill rate, stockout probability,product demand variance, system capacity and others which are introduced in awide range of literatures on supply chain management. Most researches on thetopic deal with problems concerned with only isolated parts of those performancecriteria or strategic issues on how to improve them.

Recently performance evaluation to supply chains has been frequently dis-cussed literarily, although it has not been regarded as an important strategy formanagerial improvement by some corporations such as General Motors (Ross1998). Data envelopment analysis (DEA), a mathematical programming tooldesigned for performance evaluation for a group of homogeneous decision makingunits (DMUs) with multiple inputs and multiple outputs [1], has now used in theSCM to examine if a supply chain system transforms its initial inputs to the finaloutputs with high efficiency.

Liang et al. [2] link DEA and SCM initially to propose an additive efficiencymodel to measure the overall performance of two-stage supply chains. Further-more, Chen et al. [3] and Liang et al. [4] use game-based models to present theefficiency conflict among supply chain members. Kao [5, 6] first consider theefficiency evaluation to parallel systems, and proposes an efficiency decompositiontechnique to measure the integrated efficiency of series systems and parallel sys-tems. Additive efficiency technique used in supply chain performance diagnosisusually results in nonlinear mathematical programming which is difficult to solve,and Chen et al. [7] develop a proportioning resource approach to turn the nonlinearprogramming into a linear one. Although Chen and Zhu [8] and Kao [5] usedifferent models to measure the overall efficiency of two-stage supply chains, theirequivalence can be proven by Chen et al. [9]. Beyond what has been stated, Zhaand Liang [10] use a multiplicative efficiency model and Du et al. [11] use abargaining game to discuss the cooperation and conflict of supply chain members.Yang et al. [12] propose a formal definition to supply chain production possibilityset and supply chain efficiency can be measured based on production possibilityset. The existing studies make it possible to probe in the performance analysis ofmore complex systems than supply chains. Tone and Tsutsi [13] and Cook et al.[14] propose two types of network DEA models based on slack-based measureapproach and additive efficiency decomposition technique.

244 Q. Xia et al.

As discussed above, although there are sufficient researches on performanceevaluation to supply chains, all of these precursory surveys assume that the inputsand outputs involved in performance evaluation are crisp data. Intermediatefactors, which is not only the outputs of the upstream stage but also the inputs ofthe downstream stage in supply chains, play a vital role in performance evaluationto supply chains. In practice, the exact values of intermediate factors are usuallyunknown because of internal transactions, commercial secrets or immeasurablefactor such as service quality. As a result, only fuzzy information on intermediatefactor is available. Therefore considering the fuzzy intermediate factors inperformance evaluation to supply chains is meaningful.

The current chapter integrates fuzzy intermediate factors in supply chain effi-ciency evaluation and proposes a fuzzy supply chain data envelopment analysis(FSCDEA) model based on previous supply chain DEA models. A triangular fuzzymembership function is attached on the fuzzy intermediate factors. According tothe operational rules on triangular fuzzy numbers and calculation rules on fuzzylinear programming, we turn the FSCDEA model into a linear programming.

Finally we use the proposed FSCDEA model to assess the operational efficiencyof a group of bank branches. The example of banking supply chains validates thethree theorems ratiocinated by our FSCDEA model, and reveals some advantagesof our model such as inclusion of traditional SCDEA model, robustness ofefficiency results, and detecting the source of inefficiency.

2 Supply Chain DEA Model

We start our study with the efficiency evaluation for supply chains where allfactors (inputs, outputs, and intermediate factors) are crisp data. Suppose there areN homogeneous two-stage supply chains under evaluation as shown in Fig. 1. Allof supply chains consume the same types of inputs to generate the same types ofoutputs. For the jth supply chain, Stage S is denoted by Sj and Stage B is Bj. Theinitial inputs of Sj are Xj and the outputs of Bj are Yj. In particular, the outputs of Sj

are also the inputs of Bj. We call them intermediate factors in this chapter anddenote them as Zj. All of the inputs, outputs and intermediate factors are multi-dimensional, known and crisp. Denote Xj = (x1j, x2j,…, xsj), Yj = (y1j, y2j,…, ymj),Zj = (z1j, z2j,…, zpj).

The DEA efficiency score for the subsystem Sk can be calculated using thetraditional CCR model [1] as follows:

Fig. 1 A two-stage supplychain

12 Integrating Fuzzy Intermediate Factors in Supply Chain Efficiency Evaluation 245

ESðkÞ ¼ maxuSZk

vSXk

subject to

uSZj

vSXj� 1; j ¼ 1; . . .;N

uS; vS� 0

ð1Þ

Similarly, the DEA efficiency scores for the subsystem Bk can be calculatedusing the following programming problem:

EBðkÞ ¼ maxuBYk

vBZk

subject to

uBYj

vBZj� 1; j ¼ 1; . . .;N

uB; vB� 0

ð2Þ

As Liang et al. [2] have pointed out, the traditional CCR model regard thesupply chain as a black-box as described as Fig. 2, and will overrate the overallefficiency of the supply chains.

The following programming provides the CCR efficiency of the kth two-stagesupply chain.

ECCRðkÞ ¼ maxuCCRYk

vCCRXk

subject to

uCCRYj

vCCRXj� 1; j ¼ 1; . . .;N

uCCR; vCCR� 0

ð3Þ

We use the following Supply chain DEA (SCDEA) model to evaluate theoverall efficiency of the kth two-stage supply chain as follows.

ESCðkÞ ¼ maxuSCYk

vSCXk

subject to

cSCZj

vSCXj� 1; j ¼ 1; . . .;N

uSCYj

cSCZj� 1; j ¼ 1; . . .;N

uSC; vSC; cSC � 0

ð4Þ

Fig. 2 The black-boxstructure of the supply chain

246 Q. Xia et al.

The equivalent linear programming of (4) is written as the following SCDEAmodel.

ESCðkÞ ¼ max lSCYk

subject to

mSCXk ¼ 1

gSCZj � mSCXj� 0; j ¼ 1; . . .;N

lSCYj � gSCZj� 0; j ¼ 1; . . .;N

lSC; mSC; gSC � 0

ð5Þ

3 Integrating Fuzzy Intermediate Factors in Scdea Model

In the current section we consider a fuzzy supply chain DEA (FSCDEA) problemwhere the intermediate factors are fuzzy values while the initial inputs and finaloutputs remain crisp values.

In an actual supply chain, the existence of fuzzy intermediate factors may bedue to incomplete information disclosure, especially for the connected transactionsamong supply chain members. In general, the information disclosure of a supplychain member does not include the detail of connected transactions. As a result,the accurate values of intermediate factor for other supply chains are not available.For example, open information reveals that the sale of one intermediate factor is1,000 dollars, but does not report the accurate values of the amount and unit priceof the intermediate factor. The unit price of the factor in open market is betweenone dollar and two dollars. Thus we can conclude the fuzzy amount of the factor is[500, 1,000]. Accordingly, the existence of fuzzy intermediate factors is prevalentin the efficiency analysis of supply chains.

Except the actual supply chains, the operational process as described in Fig. 3can be also detected inside an enterprise. While evaluating the performance ofsome homogeneous enterprises, we often find out that the operational process ofthose enterprises can be decomposed into a serial of sub-manufactures or sub-services. The intermediate factors among the sub-manufactures or sub-services areoften ignored because of the inferior importance and statistical difficulty. As aresult, we have to use fuzzy numbers to characterize the intermediate factors.

Fig. 3 A two-stage supplychain with fuzzy intermediatefactors

12 Integrating Fuzzy Intermediate Factors in Supply Chain Efficiency Evaluation 247

The above discuss discovers that the fuzzy intermediate factors occur not onlyin actual supply chains but also within independent enterprises. We regard such atwo-stage serial operational process as a supply chain, and develop correspondingDEA-based approach to evaluate its relative efficiency in the following paragraphs.

We rewrite Programming (5) by integrating the fuzzy intermediate factors asfollows:

EFðkÞ ¼ max lFYk

subject to

mFXk ¼ 1

gF~Zj � mFXj ~� 0; j ¼ 1; . . .;N

lFYj � gF~Zj ~� 0; j ¼ 1; . . .;N

lF; mF; gF � 0

ð6Þ

Here ~Zj are symmetrical triangular fuzzy numbers denoted as ~Zj ¼ ðcj; wjÞ. Thesymbol ‘‘ ~� ’’ means ‘‘almost smaller than or equal to’’.

The membership function of the symmetrical triangular fuzzy number isdefined as

PZjðxÞ ¼

0; x� cj � wj

Lðcj � x

wjÞ; cj � wj\x\cj

1; x ¼ cj

Lðx� cj

wjÞ; cj\x\cj þ wj

0; x� cj þ wj

8>>>>>>>>>><

>>>>>>>>>>:

ð7Þ

Let LðyÞ ¼ 1� y, we have

PZjðxÞ ¼

0; x� cj � wj

x� cj þ wj

wj; cj � wj\x\cj

1; x ¼ cj

cj þ wj � x

wj; cj\x\cj þ wj

0; x� cj þ wj

8>>>>>>>>>><

>>>>>>>>>>:

ð8Þ

Figure 4 describes the membership function of Eq. (8):Guo [15] has provided a definition as follows.

Definition 1 Given two symmetrical triangular fuzzy number Z1 ¼ ðc1;w1ÞLandZ2 ¼ ðc2;w2ÞL, the relation Z1 ~� hZ2 (0\h� 1) hold if and only if the followinginequalities are true for any possibility level k 2 ½h; 1�

248 Q. Xia et al.

Z1 � L�1ðkÞw1� Z2 � L�1ðkÞw2

Z1 þ L�1ðkÞw1� Z2 þ L�1ðkÞw2

ð9Þ

According to Definition 1, we rewrite (6) as the following programming.

EFðk; hÞ ¼ max lFYk

subject to

mFXk ¼ 1

gFðcj � L�1ðhÞwjÞ � mFXj� 0;¼ 1; . . .;N

gFðcj þ L�1ðhÞwjÞ � mFXj� 0; j ¼ 1; . . .;N

ð10Þ

lFYj � gFðcj � L�1ðhÞwjÞ� 0; j ¼ 1; . . .;N

lFYj � gFðcj þ L�1ðhÞwjÞ� 0; j ¼ 1; . . .;N

lF; mF ; gF � 0

Here h is the given possibility level by a specific decision-maker. L-1(.) is theinverse function of L(.). According to L(y) = 1 - y, we have L-1(x) = 1 - x. Thus(10) is equivalently transformed into the following FSCDEA model (11).

EFðk; hÞ ¼ max lFYk

subject to

mFXk ¼ 1

gFðcj � wj þ hwjÞ � mFXj� 0; j ¼ 1; . . .;N

gFðcj þ wj � hwjÞ � mFXj� 0; j ¼ 1; . . .;N

lFYj � gFðcj � wj þ hwjÞ� 0; j ¼ 1; . . .;N

lFYj � gFðcj þ wj � hwjÞ� 0; j ¼ 1; . . .;N

lF ; mF ; gF � 0

ð11Þ

Fig. 4 The membershipfunction

12 Integrating Fuzzy Intermediate Factors in Supply Chain Efficiency Evaluation 249

The optimal objective value of Programming (11) reports the efficiency score ofthe kth supply chain with a given possibility level h. For example, EF (k, 1) is theefficiency score under the situation where the values of the intermediate factors areselected as the midpoint of the fuzzy interval with full possibility.

Next we give three theorems about the FSCDEA efficiency scores. Thesetheorems are easy to be proven, so that we omit the proofs.

Theorem 1 EF (k, h1) C EF (k, h2) if h1 C h2.Theorem 1 discovers that EF (k, h) is an increasing function with parameter

h. As a result, max EF (k, h) = EF (k, 1) and min EF (k, h) = EF (k, 0). In fact, EF

(k, 1) and EF (k, 0) can be regarded as the upper and lower limit of the efficiencyinterval of the kth supply chain.

Theorem 2 EFðk; 1Þ ¼ EUF ðkÞand EFðk; 0Þ ¼ EL

FðkÞ. Here EUF ðkÞ and EL

FðkÞ arethe optimal values of the following two programming models which are used todetermine the efficiency interval of the kth supply chain.

EUF ðkÞ ¼ max lFYk

subject to

mFXk ¼ 1

gFcj � mFXj� 0; j ¼ 1; . . .;N

lFYj � gFcj� 0; j ¼ 1; . . .;N

lF; mF; gF � 0

ð12Þ

ELFðkÞ ¼ max lFYk

subject to

mFXk ¼ 1

gFðcj þ wjÞ � mFXj� 0; j ¼ 1; . . .;N

lFYj � gFðcj � wjÞ� 0; j ¼ 1; . . .;N

lF; mF; gF � 0

ð13Þ

Theorem 3 EFðk; 1Þ ¼ ESCðkÞ. Here ESCðkÞ is determined by (5).If h = 1, the fuzzy value of the intermediate factor turns into a crisp value,

which can be observed from Fig. 4. Thus programming (11) is equivalent to (5).Thus we can conclude that the SCDEA model as expressed by (5) is a special-ization of our FSCDEA model (11).

4 Numerical Illustration

In this section we use the proposed models to assess the operational efficiency of17 bank branches of China Construction Bank in Anhui province, P. R. China. Theoperational process of a commercial bank can be regarded as two-stage serial

250 Q. Xia et al.

process. In the first stage of banking operation, fixed assets (FA), employee (EM),expenditure (EX) are used to create desirable outputs such as credit (CR) andinterbank loan (IL). In the second stage, credit (CR) and interbank loan (IL) aretransformed into final outputs, namely, loan (LO) and profit (PR). Thus the initialinputs are FA, EM and EX, the final outputs are LO and PR, and the intermediatefactors are CR and IL.

China construction bank (CCB) is one of the largest state-owned commercialbanks of China. There are 31 branches of CCB in Anhui province, i.e., Hefei,Bengbu, Huainan, Huaibei, Maanshan, Tongling, Wuhu, Anqing, Huangshan,Fuyang, Suzhou, Chuzhou, Luan, Xuancheng, Chizhou, Chaohu and Bozhou. Inthis application, 17 branches of CCB are included in the evaluation, and the dataare shown in Tables 1 and 2. The data of the initial inputs and the final outputs arefrom Annual Report (2004) of China Construction Bank in Anhui Province. Thedata of the intermediate factors are adapted from that file.

Table 3 reports the efficiency scores of the 17 bank branches with differenth. Some worthful facts can be observed from Table 3.

(1) The efficiency scores are increasing with h, namely, EF (k, h1) C EF (k, h2) ifh1 C h2. For any bank branch k, the maximal FSCDEA efficiency is EF (k, 1)which is equivalent to the crisp SCDEA efficiency. Accordingly, defuzzing thevalues of the intermediate factors is a feasible strategy to enhance the oper-ational efficiency of the banking supply chains.

Table 1 Data of the initial inputs and the final outputs

No. Bank branch Initial inputs Final outputs

FA(¥108)

EM(103)

EX(¥108)

LO (¥108) PR(¥108)

1 Hefei 1.0168 1.221 1.2215 122.1954 3.75692 Bengbu 0.5915 0.611 0.4758 19.4829 0.66003 Huainan 0.7237 0.645 0.6061 34.4120 0.77134 Huaibei 0.5150 0.486 0.3763 15.2804 0.32035 Maanshan 0.4775 0.526 0.3848 34.9897 0.84306 Tongling 0.6125 0.407 0.3407 32.5778 0.46167 Wuhu 0.7911 0.708 0.4407 30.2331 0.67328 Anqing 1.2363 0.713 0.5547 20.6013 0.48649 Huangshan 0.4460 0.443 0.3419 8.6332 0.1288

10 Fuyang 1.2481 0.638 0.4574 9.2354 0.301911 Suzhou 0.7050 0.575 0.4036 12.0171 0.313812 Chuzhou 0.6446 0.432 0.4012 13.8130 0.377213 Luan 0.7239 0.510 0.3709 5.0961 0.145314 Xuancheng 0.5538 0.442 0.3555 13.6085 0.361415 Chizhou 0.3363 0.322 0.2334 5.9803 0.092816 Chaohu 0.6678 0.423 0.3471 9.2348 0.200217 Bozhou 0.3418 0.256 0.1594 2.5326 0.0057

12 Integrating Fuzzy Intermediate Factors in Supply Chain Efficiency Evaluation 251

Table 2 Data of the fuzzy intermediate factors

No. Bank branch CR (¥108) IL (¥108)

cj - wj cj wj cj ? wj cj - wj cj wj cj ? wj

1 Hefei 134 167 33 200 6.6 8.3 1.7 102 Bengbu 40 50 10 60 1.4 1.8 0.4 2.23 Huainan 38 48 10 58 2.7 3.4 0.7 4.14 Huaibei 28 35 7 42 1.8 2.3 0.5 2.85 Maanshan 40 50 10 60 4.4 5.5 1.1 6.66 Tongling 18 23 5 28 1 1.2 0.2 1.47 Wuhu 31 39 8 47 0.8 1.1 0.3 1.48 Anqing 30 37 7 44 3.3 4.1 0.8 4.99 Huangshan 17 21 4 25 0.6 0.7 0.1 0.8

10 Fuyang 36 45 9 54 1.4 1.7 0.3 2.011 Suzhou 31 38 7 45 1.8 2.2 0.4 2.612 Chuzhou 24 30 6 36 1.9 2.3 0.4 2.713 Luan 21 27 6 33 1.0 1.3 0.3 1.614 Xuancheng 18 22 4 26 0.8 1 0.2 1.215 Chizhou 13 16 3 19 0.4 0.5 0.1 0.616 Chaohu 17 22 5 27 0.9 1.2 0.3 1.517 Bozhou 10 13 3 16 0.3 0.4 0.1 0.5

Table 3 Efficiency scores of bank branches with different h

No Bank branch h = 0 h = 0.1 h = 0.3 h = 0.5 h = 0.7 h = 0.9 h = 1

1 Hefei 0.6700 0.6980 0.7570 0.8202 0.8881 0.9612 1.00002 Bengbu 0.3022 0.3148 0.3414 0.3699 0.4005 0.4335 0.45103 Huainan 0.2858 0.2980 0.3238 0.3514 0.3810 0.4130 0.43004 Huaibei 0.1926 0.2009 0.2184 0.2371 0.2572 0.2789 0.29045 Maanshan 0.4881 0.5089 0.5526 0.5995 0.6499 0.7042 0.73306 Tongling 0.3242 0.3387 0.3691 0.4018 0.4369 0.4748 0.49487 Wuhu 0.3433 0.3580 0.3889 0.4221 0.4578 0.4962 0.51668 Anqing 0.1958 0.2041 0.2217 0.2406 0.2608 0.2826 0.29429 Huangshan 0.0894 0.0934 0.1017 0.1107 0.1203 0.1307 0.1362

10 Fuyang 0.1438 0.1498 0.1624 0.1760 0.1906 0.2063 0.214611 Suzhou 0.1718 0.1791 0.1944 0.2108 0.2284 0.2475 0.257512 Chuzhou 0.2069 0.2156 0.2340 0.2537 0.2748 0.2977 0.309713 Luan 0.0859 0.0895 0.0971 0.1052 0.1140 0.1234 0.128414 Xuancheng 0.2243 0.2338 0.2537 0.2751 0.2981 0.3229 0.336015 Chizhou 0.0938 0.0979 0.1066 0.1160 0.1261 0.1369 0.142616 Chaohu 0.1300 0.1356 0.1473 0.1599 0.1735 0.1881 0.195817 Bozhou 0.0539 0.0563 0.0613 0.0668 0.0726 0.0789 0.0822

252 Q. Xia et al.

(2) There is at least one banking supply chain to be determined as efficient whenh = 1. However, if h \ 1, probably all of the banking supply chains areinefficient according to our FSCDEA model. The fact discovers that the sourceof inefficiency includes not only the inferior operational performance but alsothe inaccurate estimation to the intermediate factors.

(3) Although the FSCDEA efficiency scores are changing with varying h, the rankorders of all bank branches remain steady. For example, Hefei branch (#1) isalways the best one, and Bozhou branch (#17) is always the worst. This factshows that our FSCDEA model can generate robust results, and does notdisturb the efficiency evaluation by SCDEA model.

5 Conclusions

The current chapter discovers that the fuzzy intermediate factors appear in manytwo-stage serial operational processes such as supply chains or independententerprises. We regard such a two-stage serial operational process as a supplychain, and develop corresponding DEA-based approach, namely, FSCDEA model,to evaluate its relative efficiency. We integrate fuzzy intermediate factors into thetraditional SCDEA model, and use symmetrical triangular fuzzy numbers tocharacterize the intermediate factors. According to fuzzy arithmetic rules, we turnthe FSCDEA model to a linear programming with parameter h. A numericalexample of banking supply chains illustrates our model and reveals someadvantages of our model. Firstly, the traditional SCDEA model is a specializationof our FSCDEA model. Secondly, our method discovers that the source of inef-ficiency includes not only the inferior operational performance but also the inac-curate estimation to the intermediate factors. Thirdly, our FSCDEA model cangenerate robust results.

Future research along with this topic may be more interesting, and somedirections are worth investigating as follows. A possible extend of our FSCDEAmodel is to analyze the relationship between subsystem efficiency and supplychain efficiency with the existence of fuzzy intermediate factors. The conflictbetween two subsystems’ efficiencies is also worth learning, and integrating gametheory into the FSCDEA model is helpful to understand such a conflict. In addi-tion, more forms of fuzzy number and fuzzy arithmetic rules may be introduced tothe analysis process to generate valuable decision information.

Acknowledgments The authors would like to thank Ministry of Education of China (Programfor New Century Excellent Talents in University) and National Natural Science Foundation ofChina (Grant nos. 71271195, 71121061 and 71090401) for their financial support.

12 Integrating Fuzzy Intermediate Factors in Supply Chain Efficiency Evaluation 253

References

1. Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units.Eur. J. Oper. Res. 2, 429–444 (1978)

2. Liang, L., Yang, F., Cook, W., Zhu, J.: DEA models for supply chain efficiency evaluation.Ann. Oper. Res. 145(1), 35–49 (2006)

3. Chen, Y., Liang, L., Yang, F.: A DEA game model approach to supply chain efficiency. Ann.Oper. Res. 145(1), 5–13 (2006)

4. Liang, L., Cook, W., Zhu, J.: DEA models for two-stage processes: game approach andefficiency decomposition. Naval Res. Logist. 55(7), 643–653 (2008)

5. Kao, C.: Efficiency decomposition in network data envelopment analysis: A relational model.Eur. J. Oper. Res. 192(3), 949–962 (2009)

6. Kao, C.: Efficiency measurement for parallel production systems. Eur. J. Oper. Res. 196(3),1107–1112 (2009)

7. Chen, Y., Liang, L., Zhu, J.: Equivalence in two-stage DEA approaches. Eur. J. Oper. Res.193, 600–604 (2009)

8. Chen, Y., Zhu, J.: Measuring information technology’s indirect impact on firm performance.Inf. Technol. Manag. J. 5(1–2), 9–22 (2004)

9. Chen, Y., Cook, W., Li, N., Zhu, J.: Additive efficiency decomposition in two-stage DEA.Eur. J. Oper. Res. 196(3), 1170–1176 (2009)

10. Zha, Y., Liang, L.: Two-stage cooperation model with input freely distributed among thestages. Eur. J. Oper. Res. 205(2), 332–338 (2010)

11. Du, J., Liang, L., Chen, Y., Cook, W., Zhu, J.: A bargaining game model for measuringperformance of two-stage network structures. Eur. J. Oper. Res. 210(2), 390–397 (2011)

12. Yang, F., Wu, D., Liang, L., Bi, G., Wu, D.: Supply chain DEA: production possibility setand performance evaluation model. Ann. Oper. Res. 185(1), 195–211 (2011)

13. Tone, K., Tsutsui, M.: Network DEA: A slack-based measure approach. Eur. J. Oper. Res.197, 243–252 (2009)

14. Cook, W., Zhu, J., Bi, G., Yang, F.: Network DEA: additive efficiency decomposition. Eur.J. Oper. Res. 207(2), 1122–1129 (2010)

15. Guo, P.: Fuzzy data envelopment analysis and its application to location problems. Inf. Sci.179, 820–829 (2009)

16. Ross, D.F.: Competing Through Supply Chain Management. Kluwer Academic Publishers,Boston (1998)

254 Q. Xia et al.

Chapter 13Supplier Evaluation and Selection Usinga FDEA Model

Atefeh Amindoust and Ali Saghafinia

Abstract Due to the growth of global outsourcing, supplier evaluation andselection is one of the strategic decisions for purchasing management in the supplychain. In this chapter, we address the important attributes through the literaturethat constituent suppliers should possess in order to achieve the successful supplychain. These attributes (criteria) are obtained using an Affinity Diagram. Then, acommittee of decision makers is formed to provide linguistic ratings to the can-didate suppliers for the selected criteria. The linguistic ratings are then transformedinto fuzzy numbers and fed into a fuzzy DEA (FDEA) model based on the a- cutapproach assessment of candidate suppliers. A hypothetical application is providedto demonstrate the applicability and feasibility of the method.

Keywords Supplier selection � Affinity diagram � FDEA

1 Introduction

A supply chain is a network of organizations that are involved in different pro-cesses and activities that produce value in the form of products and services in thehands of the ultimate consumer [1]. So, the role of suppliers is a key one for thesuccess of the supply chain and therefore supplier selection is a vital decision insupply chain management. In the supplier selection process based on decisionmakers’ opinion, they normally prefer to verify the suppliers in linguistic terms

A. Amindoust (&)Centre of Product Design and Manufacture (CPDM), University of Malaya, 50603 KualaLumpur, Malaysiae-mail: Atefeh_Amindoust@yahoo.com

A. SaghafiniaIslamic Azad University Majlesi Branch, Electrical Engineering Department, Esfahan, Irane-mail: saghafi_ali@yahoo.com

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_13, � Springer-Verlag Berlin Heidelberg 2014

255

instead of numerical form. Using lingusitic terms is a simple and tangible way forthem to express their perceptions and a feasible way of securing the company’sinformation. So, the supplier selection decision is involved with a high degree ofvagueness and ambiguity in practice. To deal with this uncertainty, fuzzy settheory is an efficient tool and some researchers have used fuzzy concepts in thisarea [2–10]. In addition, the nature of supplier selection is a multi-criteria decision.So, a DEA approach based on multiple inputs and outputs is one of the appropriatetools in such decisions [11–17]. But, the DEA has a strong shortcoming that it doesnot cope with fuzzy data (input and output). Some researchers have proposedvarious fuzzy DEA (FDEA) models for dealing with the impreciseness andambiguity in the DEA (readers are referred to [18]). One of the most popularFDEA models is the a-cut based approach which has been applied in this work.This FDEA model is implemented to rank the suppliers in a hypothetical appli-cation to show the applicability.

2 Identification of Supplier Selection Criteria

In order to analyze the supplier selection problem, the key criteria for supplierselection must be identified. In this research, the Affinity diagram technique hasbeen used to identify these criteria. The Affinity Diagram is one of the sevenquality management and planning tools and is used to generate ideas for decisionmaking through brainstorming, surveys, and interviews [19]. First, the existingsupplier selection papers in the literature were gathered. Since the fuzzy envi-ronment has been considered for supplier selection in this work, the fuzzy supplierselection papers have been focused. So, 40 supplier selection papers were collectedand the supplier selection criteria in these papers were derived from the literatureas shown in Table 1.

The Affinity diagram exercise was conducted with a survey using the afore-mentioned criteria. The results of the Affinity diagram show that there are fourcategories of criteria which must be looked for in supplier selection and evalua-tion. The first category is comprised of the economic characteristics which arecritical for a supplier selection decision in any organization. The economic cate-gory includes related criteria such as quality, cost/price, delivery, financial capa-bility, flexibility, and service. The second category is comprised of technologyfocused characteristics which are an essential for suppliers in order to improveorganizational performance. Different technology related criteria fall into thiscategory including innovation, technical problem-solving, technical employee andequipment, allocated capacity, involvement in new product development, pro-ductivity, produce knowledge, assistance knowledge, and current manufacturingfacilities. The third category is comprised of environmental characteristics whichare vital for green supply chains. There are a lot of related criteria for the envi-ronmental category such as environmental management systems, green design,green production plans, cleaner production, safety and environmental concerns,

256 A. Amindoust and A. Saghafinia

Tab

le1

Sup

plie

rse

lect

ion

crit

eria

inli

tera

ture

Ref

eren

ces

Sup

plie

rS

elec

tion

Cri

teri

a

[20]

Cos

t(u

nit

pric

e,co

stre

duct

ion

plan

);Q

uali

ty(i

nter

val

reje

ctio

nra

te,c

usto

mer

reje

ctio

nra

te);

Del

iver

y(l

ead

tim

e,fl

exib

ilit

y);

Org

aniz

atio

nal

cult

ure

and

stra

tegy

(man

agem

ent

capa

bili

ty,

stra

tegi

cfi

t);

Tec

hnic

alca

paci

ty(i

nnov

atio

n,te

chni

cal

prob

lem

-sol

ving

)[2

1]T

echn

olog

ical

leve

l;E

cono

mic

alsi

tuat

ion;

Pro

duct

ion

capa

city

;M

arke

tsh

are;

Qua

lity

leve

l;D

eliv

ery

rate

;C

ost

redu

ctio

n;P

art

quot

atio

n;In

vest

men

tco

st;

Pro

ject

tim

e[2

2]Q

uali

ty(q

uali

type

rfor

man

ce,q

uali

tyco

ntai

nmen

tan

dV

DC

Sfe

edba

ck);

pric

e&

term

s(p

rice

,ter

ms,

resp

onsi

vene

ss,l

ead

tim

e,V

MI/

VO

Ihu

bS

etU

pco

st);

supp

lych

ain

supp

ort

(pur

chas

eor

der

reac

tive

ness

,ca

paci

tysu

ppor

t&

flex

ibil

ity,

deli

very

/VM

Iop

erat

ion)

;te

chno

logy

(tec

hnic

alsu

ppor

t,de

sign

invo

lvem

ent,

EC

N/P

CN

proc

ess)

[23]

Qua

lity

;pr

ice;

deli

very

[24]

Pro

duct

qual

ity;

Ser

vice

;D

eliv

ery;

Pri

ce[2

5]P

rodu

ctqu

alit

y;R

elat

ions

hip

clos

enes

s;D

eliv

ery

perf

orm

ance

;P

rice

[9]

Rel

atio

nshi

pcl

osen

ess;

repu

tati

onan

dpo

siti

onin

indu

stry

;pe

rfor

man

cehi

stor

y;co

nflic

tre

solu

tion

;de

live

ryca

pabi

lity

[26]

Qua

lity

;fl

exib

ilit

y;de

live

ry;

supp

lier

’ste

chno

logy

;jo

int

grow

th;

rela

tion

ship

buil

ding

;co

stof

prod

uct;

cost

ofre

lati

onsh

ip;

supp

lyco

nstr

aint

;bu

yer–

supp

lier

cons

trai

nt;

supp

lier

’spr

ofile

[27]

Qua

lity

;P

rice

;D

eliv

ery;

Tec

hniq

ue[2

8]P

rice

;qu

alit

y;de

live

ry[8

]D

eliv

ery

(com

plia

nce

wit

hdu

eda

te,l

ead

tim

e,fi

llra

te,fl

exib

ilit

y);

Ser

vice

(rel

iabi

lity

,em

path

y,re

spon

sive

ness

,ass

uran

ce);

Pro

duct

(pro

duct

rang

,new

prod

ucta

vail

abil

ity,

recy

cled

mat

eria

ls,e

rgon

omic

feat

ures

);Q

uali

ty(q

uali

tyco

ntro

lrej

ecti

onra

te,c

usto

mer

reje

ctio

nra

te);

Cos

t(p

urch

ase

pric

e,lo

gist

ics)

[29]

Cos

t;R

efer

ence

s;Q

uali

tyof

prod

uct;

Del

iver

yti

me

(day

s);

Inst

itut

iona

lity

;E

xecu

tion

tim

e(y

ears

)[3

0]C

ost;

key

qual

ity

char

acte

rist

ics;

proc

essi

ngfl

exib

ilit

y;on

-tim

ede

live

ry;

resp

onse

toch

ange

[31]

Cos

t;de

live

ry;

qual

ity

[32]

Pro

duct

qual

ity;

Ser

vice

qual

ity;

Del

iver

yti

me,

Pri

ce[3

3]C

ost;

Del

iver

y;Q

uali

ty[6

]U

nit

cost

;qu

alit

y;pe

rcen

tof

onti

me

deli

very

;m

anag

emen

tst

abil

ity,

mut

ual

trus

t;st

reng

thof

geog

raph

ical

loca

tion

;in

tern

atio

nal

com

mun

icat

ion

[34]

Use

ofen

viro

nmen

tfr

iend

lyte

chno

logy

;E

nvir

onm

ent

frie

ndly

mat

eria

ls;

Gre

enm

arke

tsh

are;

Par

tner

ship

wit

hgr

een

orga

niza

tion

s;M

anag

emen

tco

mm

itm

ent;

Adh

eren

ceto

envi

ronm

enta

lpo

lici

es;

Invo

lvem

ent

ingr

een

proj

ects

;S

taff

trai

ning

;L

ean

proc

ess

plan

ning

;D

esig

nfo

ren

viro

nmen

t;E

nvir

onm

enta

lce

rtifi

cati

onan

dpo

llut

ion

cont

rol

init

iati

ves

(con

tinu

ed)

13 Supplier Evaluation and Selection Using a FDEA Model 257

Tab

le1

(con

tinu

ed)

Ref

eren

ces

Sup

plie

rS

elec

tion

Cri

teri

a

[35]

Pro

duci

ngcr

itic

al/s

afet

y;P

rodu

cing

sim

ilar

part

;T

echn

ical

empl

oyee

and

equi

pmen

t;P

rodu

ctio

nca

paci

ty;

Tes

tca

pabi

lity

;M

anag

ing

dive

rsifi

cati

on;

Des

ign

and

impr

ovem

ent;

Fin

anci

al;

Pri

cepo

licy

;U

sing

cert

ifica

tes;

Dis

patc

hpr

oble

ms;

Pac

king

;tr

ansp

orta

tion

;G

eogr

aphi

cal

loca

tion

;W

ork

safe

ty;

Env

iron

men

tal

effe

cts

[36]

Net

cost

,ne

tre

ject

ions

,ne

tla

tede

live

ries

[37]

Org

aniz

atio

n,F

inan

cial

perf

orm

ance

,S

ervi

cequ

alit

y,T

echn

olog

y,S

ocia

lre

spon

sibi

lity

&en

viro

nmen

tal

com

pete

ncie

s[3

8]D

eliv

ery

(com

plia

nce

wit

hdu

eda

te,

com

plia

nce

wit

hqu

anti

ty);

cost

;qu

alit

y(r

emed

yfo

rqu

alit

ypr

oble

ms,

reje

ctio

nra

tefr

omQ

C)

[28]

Qua

lity

;C

ost;

Del

iver

y;S

ervi

ce;

Tec

hnic

alan

dpr

oduc

tion

capa

bili

ty,

Rel

atio

nco

mbi

nati

on;

Org

aniz

atio

nal

man

agem

ent

[39]

On

tim

ede

live

ry;

clos

enes

sof

rela

tion

ship

wit

hth

esu

ppli

er;

supp

lier

’spr

oduc

tqu

alit

y;su

ppli

er’s

tech

nolo

gica

lca

pabi

lity

;co

st[4

0]C

ost

(pro

duct

,fr

eigh

tco

st,

cust

omdu

ties

);qu

alit

y(r

ejec

tion

,pr

oces

sca

pabi

lity

,qu

alit

yas

sess

men

t);

serv

ice

(on

tim

ede

live

ry,

tech

nica

lsu

ppor

t,re

spon

seto

chan

ges,

ease

ofco

mm

unic

atio

n);

risk

(geo

grap

hica

llo

cati

on,

poli

tica

lst

abil

ity,

econ

omy)

[41]

Val

ue-a

ddin

gpr

acti

ces

toa

firm

toen

sure

the

profi

tabi

lity

ofsu

pple

r,R

elat

ions

hip,

Del

iver

yre

liab

ilit

y,Q

uali

ty,

Sat

isfy

cust

omer

need

s,F

lexi

bili

ty,

Ser

vice

,C

omm

unic

atio

n,M

anag

emen

t,G

reen

desi

gn,

Env

iron

men

tal

cert

ifica

tes,

Gre

enpr

oduc

tion

plan

,C

lean

erpr

oduc

tion

,G

reen

purc

hasi

ng,

Lif

ecy

cle

asse

ssm

ent,

Env

iron

men

tal

man

agem

ent

syst

em,

R&

Dca

pabi

lity

,In

nova

tion

[42]

Qua

lity

;P

rice

;L

ocat

ion;

Fin

ance

;F

acil

ity;

Pro

duct

ivit

y;L

ong-

term

rela

tion

ship

capa

bili

ty;

Tec

hnic

alca

pabi

lity

;M

anag

eria

lor

gani

zati

on;

Qui

ckre

spon

sefo

rre

quir

emen

ts[4

3]P

rodu

ctqu

alit

y;O

nti

me

deli

very

;P

rice

/cos

t;S

uppl

ier’

ste

chno

logi

cal

leve

l;F

lexi

bili

ty[4

4]P

rodu

ctqu

alit

y(pr

oduc

tper

form

ance

-rel

iabi

lity

and

accu

racy

-,le

velo

fte

chno

logy

);D

eliv

ery(

cond

itio

nof

prod

ucts

onar

riva

l,on

-tim

ede

live

rype

rfor

man

ce,a

ccur

acy

infi

llin

gor

ders

,ord

ercy

cle

tim

e,ab

ilit

yto

fill

emer

genc

yor

ders

,acc

urac

yin

bill

ing

and

cred

it);

Pri

ce/C

ost(

pric

eof

prod

ucts

and

serv

ices

,fina

ncia

lstr

engt

h,co

stha

rnes

sca

pabi

lity

);S

ervi

ce(p

osts

ales

assi

stan

cean

dsu

ppor

t,ab

ilit

yan

dw

illi

ngne

ssto

assi

stw

ith

the

desi

gnpr

oces

s,ea

seof

com

mun

icat

ion)

[45]

Man

agem

ent

and

orga

niza

tion

;qu

alit

y;te

chni

cal

capa

bili

ty;

prod

ucti

onfa

cili

ties

and

capa

bili

ties

;fi

nanc

ial

posi

tion

;de

live

ry;

serv

ices

;re

lati

onsh

ips;

safe

tyan

den

viro

nmen

tal

conc

erns

and

cost

[46]

New

proj

ect

man

agem

ent;

Sup

plie

rm

anag

emen

t;Q

uali

tyan

den

viro

nmen

tal

man

agem

ent;

Pro

duct

ion

proc

ess

man

agem

ent;

Tes

tan

din

spec

tion

man

agem

ent;

Cor

rect

ive

&pr

even

tive

acti

ons

man

agem

ent;

War

rant

yco

stra

tio;

Def

ect

rati

o;Q

uali

tym

anag

emen

tN

ewpr

ojec

tm

anag

emen

t;S

uppl

ier

man

agem

ent;

Qua

lity

and

envi

ronm

enta

lm

anag

emen

t;P

rodu

ctio

npr

oces

sm

anag

emen

t;T

est

and

insp

ecti

onm

anag

emen

t;C

orre

ctiv

e&

prev

enti

veac

tion

sm

anag

emen

t;W

arra

nty

cost

rati

o;D

efec

tra

tio;

Qua

lity

man

agem

ent

[47]

Sup

plie

rse

lect

ion

proc

ess

inC

lose

d-L

oop

Sup

ply

Cha

in(C

LS

C)

netw

ork,

Pro

pose

afr

amew

ork

for

supp

lier

sele

ctio

ncr

iter

iain

Rev

erse

Log

isti

c(L

R)

base

don

supp

lier

rela

ted,

part

rela

ted,

and

proc

ess

rela

ted

cate

gori

es

(con

tinu

ed)

258 A. Amindoust and A. Saghafinia

Tab

le1

(con

tinu

ed)

Ref

eren

ces

Sup

plie

rS

elec

tion

Cri

teri

a

[48]

Pro

fit,

Qua

lity

,Del

iver

y,S

ervi

ce,E

nvir

onm

enta

lcom

pete

ncie

s,E

nvir

onm

enta

lman

agem

ents

yste

m,T

heri

ghts

ofst

ockh

olde

rs,W

ork

safe

ty&

Lab

orhe

alth

[49]

Dis

crim

inat

ion;

Abu

seof

hum

anri

ght;

Chi

ldla

bor;

Lon

gw

orki

ngho

urs;

Unf

air

com

peti

tion

(soc

iety

);po

llut

ion(

envi

ronm

enta

l)[5

0]O

rgan

izat

ion;

Fin

anci

alpe

rfor

man

ce;

Ser

vice

qual

ity;

Tec

hnol

ogy;

Gre

enco

mpe

tenc

ies

[51]

Sup

plie

rsco

ndit

ions

(bus

ines

sre

lati

onsh

ips,

fina

ncia

lsi

tuat

ions

,com

pany

’sty

pes,

com

pany

orga

niza

tion

);P

rice

&D

eliv

ery

(pri

ce,p

roce

ssin

gti

me

ofor

deri

ng,

flex

ibil

ity

ofor

der

alte

ring

,de

live

ryon

tim

e,m

anuf

actu

ring

flex

ibil

ity)

;Q

uali

ty(p

rodu

ctqu

alit

y,pr

oduc

tre

liab

ilit

y,co

ntin

uing

impr

ovem

ent

abil

ity)

;P

rofe

ssio

nal

tech

niqu

es(m

anuf

actu

ring

faci

lity

and

prod

ucti

vity

,te

chni

que

capa

bili

ty,

desi

gnan

dde

velo

pmen

tab

ilit

y)[5

2]E

cono

mic

(pre

ssur

eov

erth

efo

odm

arke

t,av

aila

bili

tyof

raw

-mat

eria

l,tr

ansp

orta

tion

cost

s,st

orag

eco

sts,

stru

ctur

eof

coop

erat

ives

,ge

nera

lde

man

d,su

pply

ing

cost

s);S

ocia

l(fi

nanc

ing

avai

labi

lity

,pla

nted

area

,pro

fita

bili

ty);

Tec

hnol

ogic

al(p

rodu

ctiv

ity,

ener

gyef

fici

ency

,cru

shin

gco

sts,

prod

ucer

know

ledg

e,as

sist

ance

know

ledg

e)[5

3]P

rice

;Q

uali

ty;

Del

iver

y;T

rans

port

atio

nco

st;

Tec

hnol

ogy;

Pro

duct

ion

syst

emfl

exib

ilit

y[5

4]C

ost;

Qua

lity

reje

ctio

n;P

erce

ntag

eof

late

deli

vere

dit

em;

Gre

enho

use

gas

emis

sion

[55]

Cos

t;D

eliv

ery;

Qua

lity

13 Supplier Evaluation and Selection Using a FDEA Model 259

life cycle assessment, green supply chain management, product recycling, pollu-tion, greenhouse gas emission, and resource consumption. The fourth category iscomprised of social characteristics which are important with growing of sustain-ability issues. Different social related criteria fall into this category includingsocial- responsibilities, work safety & labor health, and the interests and rights ofemployees and the stakeholders.

3 Addressing Uncertainty in Supplier Selection

Zadeh (1965) introduced fuzzy set theory to cope with the imprecision anduncertainty which is inherent to the human judgments in decision making pro-cesses through the use of linguistic terms and degrees of membership [56]. Sincein this work the supplier’s performance has been considered in linguistic terms onthe basis of decision makers’ opinion, fuzzy set theory is applied to handle thesubjectivity. To express fuzzy sets from a mathematical point of view, consider aset of objects X. A set is represented as follows:

X ¼ x1; x2; . . .. . .. . .; xn ð1Þ

where, xi is an element in the set X. A membership value (l) expresses the grade ofmembership related to each element xi in a fuzzy setA, which is shown as follows:

A ¼ l1ðx1Þ; l2ðx2Þ; . . .. . .. . .; lnðxnÞ ð2Þ

3.1 Fuzzy Membership Functions

In this chapter, triangular type membership functions are used for the estimation ofthe supplier’s performance with respect to the criteria. A triangular fuzzy numberis represented as ~w ¼ ðal; am; auÞ in Fig. 1 and the triangular membership functionis defined as Eq.(3).

1

)(xTµ

xla ma ua

Fig. 1 The triangular fuzzymembership function

260 A. Amindoust and A. Saghafinia

l~wðxÞ ¼

0 if x\al

1am � al

ðx� alÞ if al� x� am

1am � au

ðx� auÞ if am� x� au

0 if x [ au

8>>>>>>><

>>>>>>>:

ð3Þ

Linguistic variables are used to measure the magnitude of a piece of data. Inthis chapter, a group of linguistic variables is utilized for the supplier’s perfor-mance with respect to criteria including weakly preferred (WP), moderately pre-ferred (MP), strongly preferred (SP), and extremely preferred (EP) as seen inFig. 2. The fuzzy numbers which are related to these linguistic terms are shown inTable 2.

4 The Fuzzy DEA model

DEA is a non-parametric programming method which was proposed by Charneset al. [57] (CCR) and developed by Banker et al. (1984) [58] to evaluate therelative efficiency of decision making units (DMUs) by crisp inputs and outputs.

0 2 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

performance scale

Deg

ree

of m

embe

rshi

p fu

nctio

n WP MP SP EPFig. 2 The membershipfunctions for the supplier’srating

Table 2 The linguistic terms and fuzzy numbers for the supplier’s rating

Linguistic variables Corresponding triangular fuzzy number

Weakly Preferred (WP) (0, 2, 4)Moderately Preferred (MP) (2, 4, 6)Strongly Preferred (SP) (4, 6, 8)Extremely Preferred (EP) (6, 8, 10)

13 Supplier Evaluation and Selection Using a FDEA Model 261

But today, in real applications the inputs and outputs are not usually precise andcan be represented by fuzzy sets.

Some researchers have proposed various fuzzy methods for dealing with thisimpreciseness and ambiguity in DEA. Since the original study by Sengupta (1992),there has been a continuous interest and increased development in the fuzzy DEAliterature [59]. Hatami et al. (2011) reviewed the fuzzy DEA methods and pre-sented a taxonomy by classifying the fuzzy DEA papers published over the pasttwo decades into four primary categories, namely, the tolerance approach, the a-level based approach, the fuzzy ranking approach, and the possibility approach;and a secondary category to group the pioneering papers that did not fall into thefour primary classifications [18].

The a-level based approach of fuzzy DEA has been applied in this work. TheCCR model with fuzzy inputs and outputs can be written as shown in (4) tomeasure the efficiency of the supplier with respect to a set of peer suppliers in themodel below:

MaxZs ¼XC

J¼1

usJ~ysJ

s:t:XL

l¼1

vsl~xsl ¼ 1

XC

J¼1

usJ~ykJ �XL

l¼1

vsl~xsk� 0 k ¼ 1; 2; . . .:; S

usJ ; vsl� e l ¼ 1; 2; . . .:; L; J ¼ 1; 2; . . .:;C

ð4Þ

Where, zs is the efficiency score of supplier s; the subscript l represents theinputs; subscript J represents the outputs; subscript s ¼ 1; 2; . . .; S represents theDMUs; the variables uJs and vls are the weights of the inputs and the outputs,respectively; yJs is the amount of output s provided by unit J; xls is the amount ofinput l used by unit s; the symbol e is the non-Archimedean constant; and ‘*’shows the fuzziness. For the sth supplier if the efficiency score is 1, it is said to beefficient; otherwise, it is inefficient. Fuzzy numbers in this formula are representedby triangular fuzzy numbers. Then, Eq. (4) can be written as follows:

262 A. Amindoust and A. Saghafinia

~xsl ¼ ðxlsl; x

msl ; x

uslÞ

~ysJ ¼ ðylsJ ; y

msJ ; y

usJÞ

Max Zs ¼XC

J¼1

usJðylsJ ; y

msJ ; y

usJÞ

s:t:XL

l¼1

vslðxlsl; x

msl ; x

uslÞ ¼ 1

XC

J¼1

usJðylkJ ; y

mkJ ; y

ukJÞ �

XL

l¼1

vslðxlsk; x

msk; x

uskÞ� 0 k ¼ 1; 2; . . .:; S

usJ ; vsl� e l ¼ 1; 2; . . .:; L; J ¼ 1; 2; . . .:;C

ð5Þ

Now the fuzzy CCR model can be transformed into an interval programmingmodel by using the a-cut method as follows [33]:

Max Zs ¼XC

J¼1

usJðð1� aÞylsJ þ aym

sJ ; ð1� aÞyusJ þ aym

sJÞ

s:t:XL

l¼1

vslðð1� aÞxlsl þ axm

sl ; ð1� aÞxusl þ axm

slÞ ¼ 1

XC

J¼1

usJðð1� aÞylkJ þ aym

kJ ; ð1� aÞyukJ þ aym

kJ�

XL

l¼1

vslðð1� aÞxlsk þ axm

sk; ð1� aÞxusk þ axm

sk� 0

XC

J¼1

�

k ¼ 1; 2; . . .:; S; usJ ; vsl� e; l ¼ 1; 2; . . .:; L; J ¼ 1; 2; . . .:;C;

ð6Þ

The above model can be divided into two models which give an upper boundand a lower bound of efficiency respectively [33] as follows:

Max Zs ¼XC

J¼1

usJðð1� aÞyusJ þ aym

sJÞ

s:t:XL

l¼1

vslðð1� aÞxusl þ axm

slÞ ¼ 1

XC

J¼1

usJðð1� aÞyukJ þ aym

kJ�

XL

l¼1

vslðð1� aÞxusk þ axm

skÞ� 0 k ¼ 1; 2; . . .:; S

usJ ; vsl� e l ¼ 1; 2; . . .:; L; J ¼ 1; 2; . . .:;C

ð7Þ

13 Supplier Evaluation and Selection Using a FDEA Model 263

Max Zs ¼XC

J¼1

usJðð1� aÞylsJ þ aym

sJÞ

s:t:XL

l¼1

vslðð1� aÞxlsl þ axm

slÞ ¼ 1

XC

J¼1

usJðð1� aÞylkJ þ aym

kJ�

XL

l¼1

vslðð1� aÞxlsk þ axm

skÞ� 0 k ¼ 1; 2; . . .:; S

usJ ; vsl� e l ¼ 1; 2; . . .:; L; J ¼ 1; 2; . . .:;C

ð8Þ

5 An Illustrative Example

In this section, an illustration is utilized to show the application of the FDEAmodel. Suppose that X Company is a producer of electrical products. A committeeof decision makers tries to find the appropriate suppliers among five candidatesuppliers. The suppliers’ ratings are derived from the decision makers’ preferencesas shown in Table 3.

The obtained fuzzy numbers from Table 3 are substituted into Eqs. (7) and (8),respectively to get the upper bound and lower bound efficiency values as presentedin Tables 4 and 5.

The upper bound and lower bound efficiency values have been measured fordifferent a levels (a ¼ 0; 0:5; 1) as seen in Tables 6 and 7.

Table 3 The suppliers’ ratings with respect to the criteria

Criteria

Economic Technological Environmental Social

Suppliers S1 EP(6, 8, 10) SP(4, 6, 8) EP(6, 8, 10) SP(4, 6, 8)S2 SP(4, 6, 8) SP(4, 6, 8) MP(2, 4, 6) MP(2, 4, 6)S3 EP(6, 8, 10) MP(2, 4, 6) SP(4, 6, 8) MP(2, 4, 6)S4 SP(4, 6, 8) WP(0, 2, 4) MP(2, 4, 6) WP(0, 2, 4)S5 MP(2, 4, 6) MP(2, 4, 6) WP(0, 2, 4) WP(0, 2, 4)

Table 4 Upper bound efficiency values of suppliers based on the a value

Input Outputs

Suppliers 1 ðð1� aÞ1:0þ 0:8aÞ ðð1� aÞ0:8þ 0:6aÞ ðð1� aÞ1:0þ 0:8aÞ ðð1� aÞ0:8þ 0:6aÞ1 ðð1� aÞ0:8þ 0:6aÞ ðð1� aÞ0:8þ 0:6aÞ ðð1� aÞ0:6þ 0:4aÞ ðð1� aÞ0:6þ 0:4aÞ1 ðð1� aÞ1:0þ 0:8aÞ ðð1� aÞ0:6þ 0:4aÞ ðð1� aÞ0:8þ 0:6aÞ ðð1� aÞ0:6þ 0:4aÞ1 ðð1� aÞ0:8þ 0:6aÞ ðð1� aÞ0:4þ 0:2aÞ ðð1� aÞ0:6þ 0:4aÞ ðð1� aÞ0:4þ 0:2aÞ1 ðð1� aÞ0:6þ 0:4aÞ ðð1� aÞ0:6þ 0:4aÞ ðð1� aÞ0:4þ 0:2aÞ ðð1� aÞ0:4þ 0:2aÞ

264 A. Amindoust and A. Saghafinia

Table 5 Lower bound efficiency values of suppliers based on the a value

Input Outputs

Suppliers 1 ðð1� aÞ0:6þ 0:8aÞ ðð1� aÞ0:4þ 0:6aÞ ðð1� aÞ0:6þ 0:8aÞ ðð1� aÞ0:4þ 0:6aÞ1 ðð1� aÞ0:4þ 0:6aÞ ðð1� aÞ0:4þ 0:6aÞ ðð1� aÞ0:2þ 0:4aÞ ðð1� aÞ0:2þ 0:4aÞ1 ðð1� aÞ0:6þ 0:8aÞ ðð1� aÞ0:2þ 0:4aÞ ðð1� aÞ0:4þ 0:6aÞ ðð1� aÞ0:2þ 0:4aÞ1 ðð1� aÞ0:4þ 0:6aÞ 0:2a ðð1� aÞ0:2þ 0:4aÞ 0:2a1 ðð1� aÞ0:2þ 0:4aÞ ðð1� aÞ0:2þ 0:4aÞ 0:2a 0:2a

Table 6 Upper bound efficiency values and ranking of suppliers

Efficiency Ranking

a ¼ 0 a ¼ 0:5 a ¼ 1

Suppliers S1 1.00000 1.00000 1.00000 1S2 1.00000 1.00000 1.00000 1S3 1.00000 1.00000 1.00000 1S4 0.80000 0.77778 0.75000 2S5 0.75000 0.71429 0.66667 3

Table 7 Lower bound efficiency values and ranking of suppliers

Efficiency Ranking

a ¼ 0 a ¼ 0:5 a ¼ 1

Suppliers S1 1.00000 1.00000 1.00000 1S2 1.00000 1.00000 1.00000 1S3 1.00000 1.00000 1.00000 1S4 0.66667 0.71429 0.75000 2S5 0.50000 0.60000 0.66667 3

Table 8 Upper bound super-efficiency values and ranking of suppliers

Efficiency Ranking

a ¼ 0 a ¼ 0:5 a ¼ 1

Suppliers S1 2. 0000 2.33333 3.00000 1S2 1.50000 1.66667 2.00000 3S3 1.54440 1.66667 2.00000 2S4 1.06667 1.08885 1.12500 4S5 1.00000 1.00000 1.00000 5

13 Supplier Evaluation and Selection Using a FDEA Model 265

As seen in the above Tables, the efficiencies of the first three suppliers (S1, S2,S3) are equal to 1 and for the two remaining suppliers, the efficiencies are lessthan 1. So, the suppliers S1, S2, and S3 are efficient and S4 and S5 are inefficient.Since this work is based on the DEA-CCR model which is weak in discriminatingbetween efficient suppliers [60], the super-efficiency model is used to rank thesuppliers as shown in Tables 8 and 9.

From Tables 8 and 9, it is found that the first supplier is the best one. Looking atTable 3, it can be seen that the ratings of this supplier with respect to economic,technological, environmental, and social criteria are better than those of the othersuppliers which is a validation of the method. Similar to the efficiency results, thesuper-efficiency results for different a levels are the same which is another vali-dation for the method.

6 Conclusion

In this chapter, we address the supplier selection problem under fuzzy environ-ments and present various fundamental criteria that can be used by decisionmakers for evaluation purposes in this area. These criteria are obtained using theAffinity Diagram and through the supplier selection literature. Moreover, the mainstrength of our work is the investigation of the FDEA model based on the a-cutapproach for supplier selection under uncertainty. Suppliers can be evaluated forthe relevant criteria in terms of linguistic preferences and the use of the modelbecause of its ability to evaluate alternatives based on multi-criteria. The next stepof our work involves designing a FDEA based model which considers the relativeimportance of criteria in the selection process.

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Table 9 Lower bound super-efficiency values and ranking of suppliers

Efficiency Ranking

a ¼ 0 a ¼ 0:5 a ¼ 1

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266 A. Amindoust and A. Saghafinia

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13 Supplier Evaluation and Selection Using a FDEA Model 269

Chapter 14Supplier Performance Evaluation Usinga Hybrid Fuzzy Data EnvelopmentAnalysis Approach

Anjali Awasthi, Khoshrow Noshad and Satyaveer Singh Chauhan

Abstract Supplier performance evaluation plays a prime role in supplier qualitydevelopment and controlling the cost of quality in supply chains. In this bookchapter, we present a multistep approach based on fuzzy data envelopment anal-ysis (DEA) approach for supplier quality evaluation. The criteria for supplierperformance evaluation are obtained using Delphi technique. The hierarchy for thecriteria and preferential relations between them are developed using the principlesof analytic hierarchy process (AHP). Supplier performance evaluation is per-formed using fuzzy data envelopment analysis. Cross-efficiencies for performanceevaluation minimize distance from the Ideal DMUs and maximize distance fromthe anti-ideal DMUs. A numerical illustration is provided.

Keywords DEA � Cross-efficiency � AHP � Delphi technique � Supplier per-formance evaluation � Fuzzy number � Linguistic data

1 Introduction

Supplier performance evaluation is vital for identifying supplier strengths andweaknesses, supplier development, and controlling cost of quality in supply chains[1, 2]. Poor quality suppliers not only lead to low productivity but also high return

A. Awasthi (&)CIISE—EV 7.640, Concordia University, Montreal, Canadae-mail: awasthi@ciise.concordia.ca

K. NoshadCIISE—EV 10.154, Concordia University, Montreal, Canadae-mail: noshadravan1@yahoo.com

S. S. ChauhanDecision Sciences, JMSB, Concordia University, Montreal, Canadae-mail: sschauha@alcor.concordia.ca

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8_14, � Springer-Verlag Berlin Heidelberg 2014

271

and repair costs due to product failure in the hands of the customer which isdetrimental to the business performance and economic goals of organizations.Therefore, it is important to timely and accurately assess the performance ofsuppliers to develop a good supplier base and ensure good product quality andservice quality to customers.

Supplier performance evaluation is a multicriteria decision process that gen-erally involves measuring supplier performances against a set of criteria. Initially,only economic criteria were used for evaluation purposes but in the recent yearsenvironmental and social criteria are also being assessed due to rising environ-mental costs and GHG emissions. Supplier quality evaluation criteria were firstproposed by Dickson in 1966 [3], who identified quality, delivery, performancehistory, warranties and claim policies, production facilities and capacity, price,technical capability and financial position as the main criteria. Weber et al. [4]propose net price, delivery, quality, production facilities and capacity, geograph-ical location, technical capability, and management and organization. Ramanathan[5] suggests the use of costs of manufacturing, quality costs, technology and after-sales service as supplier quality evaluation criteria. Huang and Keskar [6] pro-posed comprehensive metrics for supplier quality evaluation of original equipmentmanufacturers (OEMs) which can be categorized into ‘‘reliability, responsivenessand flexibility’’ in the product-related division; ‘‘cost and financial’’ and ‘‘assetsand infrastructure’’ in the supplier-related division; and safety and environment inthe society division.

The commonly used methods for supplier performance evaluation reported inliterature are AHP [7, 8], ANP [9], Goal Programming [10], Case Based Rea-soning, DEA [11–17], Total cost of ownership [5, 18], Mathematical programming[8], Artificial Intelligence [19, 20] etc. For a detailed review of methods supportingsupplier selection, please refer to de Boer et al. [21].

In this chapter, we are using Data Envelopment Analysis [22] for evaluating theperformance of suppliers. DEA and its hybrids have been widely used for supplierperformance evaluation [23]. The traditional efficiency calculation in DEA relieson self-evaluation and assigns the most favorable set of input and output weights toDMUs to maximize their efficiency. This often leads to several DMUs beingefficient making it difficult to be discriminated. Under this situation, peer evalu-ation is recommended. In other words, the performance of a DMU should be notonly determined in terms of its optimistic efficiency but also through cross-effi-ciencies obtained from the weights determined by other DMUs [24]. Therefore, wewill use cross-efficiencies to compare the performance of suppliers in ourapproach.

The cross-efficiency evaluation was initially investigated by Sexton et al. [25]and then by Doyle and Green [26] who proposed aggressive and benevolent for-mulations for cross-efficiency. Wang and Chin [27] propose some alternativemodels for DEA cross-efficiency evaluation. Hatami-Marbini et al. [28] present anideal-seeking fuzzy data envelopment analysis framework. Mirhedayatian et al.[29] present an integrated fuzzy data envelopment analysis and TOPSIS approachfor welding process selection for repairing nodular cast iron engine block. Wang

272 A. Awasthi et al.

et al. [24] present a cross-efficiency evaluation approach based on ideal and anti-ideal decision making units. Our approach for cross-efficiency evaluation is basedon the concept of ideal and anti-ideal DMUs proposed in Wang et al. [24] whichwill be coupled with Delphi technique and weight restrictions obtained from AHPfor supplier performance evaluation.

The rest of the book chapter is organized as follows. In Sect. 2, we present theproblem definition. The solution approach is presented in Sect. 3. Section 4 con-tains the numerical application. The conclusions and future works are provided inSect. 5.

2 Problem Definition

The research questions we are addressing in this chapter are:

• Which criteria to use for supplier performance evaluation?• How to identify relationships and structure the criteria?• How to determine the importance of criteria?• How to compare the performance of suppliers using linguistic data?

Our final goal is to develop a comprehensive solution approach that addressesthese questions for supplier performance evaluation.

3 Solution Approach

The proposed approach for supplier performance evaluation consists of three steps:

1. Identification of performance evaluation criteria using Delphi Technique.2. Structuring the criteria and identifying relationships using Analytical Hierarchy

Process.3. Generating criteria weights and assessing performance of suppliers using Data

Envelopment Analysis.

3.1 Criteria Identification Using Delphi Technique

The Delphi method is a group decision making process. It involves a committee ofexperts who review a given list of information in several rounds. The informationpresented at each round is the analysis of the results obtained in the previousround. The process is reiterated as long as desired or converges when commonagreement is reached on the presented results [30].

14 Supplier Performance Evaluation 273

The Delphi technique was applied with three supply chain experts from aca-demia in this research. The criteria obtained for supplier performance evaluationare:

• Number of defects (or Reject rate)• Delivery Time• Response Time (to change, production disruption, new requirements from

customers)• Late/Missed Deliveries• Sales• Costs (Material, Production including Labor, Technology, Space and facility

maintenance, Warehousing, Testing and inspection, Transportation, Return andRepair, Training)

• Emissions• Energy Usage• Health and Safety standards• Community engagement• Business ethics and integrity

It can be seen that the criteria mentioned above are heterogeneous in nature. Forexample, Sales is an economic criteria whereas Product Quality, Service Qualityare technical criteria. Emissions and energy usage can be grouped into environ-mental criteria and Equity into social criteria. Therefore, in this research we areevaluating the performance of suppliers from social, economic, technical andenvironmental perspective.

3.2 Criteria Structuring and Ranking Using AnalyticalHierarchy Process

In order to structure the supplier quality evaluation criteria and identify their inter-relationships, we relied on the principles of a multicriteria decision makingtechnique called Analytical Hierarchy Process (AHP). The AHP technique wasdeveloped by Saaty [31] for evaluating and selecting alternatives against a set ofselected criteria. AHP has been applied in several decision making situations. Thestrength of AHP is that it organizes tangible and intangible factors in a systematicway, and provides a structured yet relatively simple solution to the decision-making problems In addition, by breaking a problem down in a logical fashionfrom the large, descending in gradual steps, to the smaller and smaller, one is ableto connect, through simple paired comparison judgments, the criteria at differentlevels of the hierarchy. AHP requires pairwise comparison matrices betweenelements of same level scored using a scale of 1–9 (1 being equally preferred, 9being extremely preferred) and applies eigenvector method to generate criteria(or sub-criteria) weights. These weights are then adjusted depending upon the

274 A. Awasthi et al.

relationship between various criteria and sub-criteria in the hierarchical diagramobtained from AHP.

The results of this step are shown in Table 1. It can be seen that the criteriaobtained from Delphi have been grouped into four main categories namely tech-nical, economical, environmental and social. Each criterion has also beendecomposed into several sub-criteria (Column 3). The classification of criteria asinput-type (I) or output-type (O) is done in column 4. Finally, the weights ofvarious criteria for any supplier k are presented in column 5.

In our chapter, the decision makers provide their preferences for criteria (orsub-criteria) values using rules which are much easier to generate than assigningnumerical weights. This preferential ranking is done using a three level scalenamely more important, equally important, and less important. For example, ifinput criteria u1k and u2k are rated equally important for supplier k, then thedifference between their weights is set equal to 0 i.e. u1k-u2k = 0. If u1k is moreimportant than u2k, then u1k -u2k [ 0 and u1k -u2k \ 0 in the contrary case.Applying these rules, the decision making committee preferences for the differentcriteria presented in Table 1 are presented below.

Table 1 Supplier performance metrics

Category Criteria Sub-criteria Type Weight(Supplierk)

Technical(C1)

Productquality (C1.1)

Number of defects (or Rejectrate) (C1.1)

O u1k

Service Quality(C1.2)

Delivery Time (C1.2.1) O u2k

Response Time (C1.2.2) O u3k

Late/Missed Deliveries(C1.2.3)

O u4k

Economic(C2)

Sales (C2.1) Sales (C2.1) O u5k

Costs (C2.2) Material (C2.2.1) I v1k

Production (C2.2.2) I v2k

Warehousing (C2.2.3) I v3k

Testing and inspection(C2.2.4)

I v4k

Transportation (C2.2.5) I v5k

Return and Repair (C2.5.6) I v6k

Training (C2.5.7) I v7k

Environmental(C3)

EnvironmentalPerformance (C3)

Emissions (C3.1) O u6k

Energy Usage (C3.2) O u7k

Social(C4)

Social Performance (C4) Health and Safety standards(C4.1)

O u8k

Community engagement (C4.2) O u9k

Business ethics and integrity(C4.3)

O u10k

O: Output, I: Input

14 Supplier Performance Evaluation 275

• Product quality is more important than service quality (i.e. u1k-u2k-u3k-

u4k [ 0).• Sales is more important than cost (i.e. u5k-v1k-v2k-v3k-v4k-v5k-v6k-v7k [ 0.• Material, Production, Transportation, Warehousing Cost, Return and Repair,

Testing and Inspection Cost are more important than Training Cost (i.e.v1k ? v2k ? v3k ? v4k ? v5k ? v6k-v7k [ 0).

• Technical and Economic Performance are more important than Social andEnvironmental Performance (i.e.u1k ? u2k ? u3k ? u4k ? u5k-u6k-u7k-u8k-

u9k-u10k [ 0).• Environmental performance is more important than social performance (i.e.

u6k ? u7k-u8k-u9k-u10k [ 0).

These relations will be used for restricting weights of inputs and outputs ofsuppliers from seeking hypothetical or non-realistic values in DEA.

3.3 Transforming Fuzzy Supplier Inputs-Outputs into CrispNumbers

Most of the supplier performance evaluation criteria are qualitative in nature. Forexample, it is much easier to measure service quality as good, very good than innumerical values. However, to use them in quantitative approaches such as DEA,we would have to transform them into crisp numbers. To transform a fuzzy tri-angular number ~a ¼ a1; a2; a3ð Þ into a crisp number a, we will employ the fol-lowing equation:

a ¼ a1 þ 4a2 þ a3

6ð1Þ

For example, for a fuzzy triangular number ~a ¼ 1; 1; 3ð Þ, the crisp output a isgiven by

a ¼ 1þ 4�1þ 3ð Þ=6 ¼ 1:333

Table 2 presents the various linguistic ratings used for supplier performanceassessment criteria and their associated crisp values obtained using Eq. (1).

Once, the crisp values for different supplier performance evaluation criteria areobtained, they can be treated easily through Data Envelopment Analysis in thenext step.

276 A. Awasthi et al.

3.4 Cross-efficiency Evaluation Using Ideal and Anti-idealDecision Making Units

Cross-efficiency evaluation is a DEA technique in which each DMU has multipledifferent efficiency scores (obtained with respect to self-evaluation and peer-evaluation) which are averaged to reflect the overall performance of the DMU.Based on average cross-efficiencies, the different DMUs are compared and ranked.

Consider a set of n suppliers, with each supplier Sj(j = 1,2,..,n) using m inputsxij(i = 1,..,m) and s outputs yrj(r = 1,2…,s). For a supplier k under consideration,the relative efficiency score using the original CCR model is defined by:

max hkk ¼Ps

r¼1urkyrkPm

i¼1vikxik

s:t: hjk ¼Ps

r¼1 urkyrjPmi¼1 vikxij

� 1; j ¼ 1; 2; ::; n

urk � 0; r ¼ 1; 2::; s

vik� 0; i ¼ 1; 2::;m

ð2Þ

where urk is the weight of the rth output value and vik is the weight of the ith inputof the kth supplier. The objective of above model is to find a set of input and outputweights that are most favorable to supplier k. Using the Charnes and Coopertransformation [22], model (2) can be transformed into the following linearprogram.

max hkk ¼Ps

r¼1urkyrk

s:t:Pm

i¼1vikxik ¼ 1

Xs

r¼1

urkyrj �Xm

i¼1

vikxij� 0; j ¼ 1; 2; ::; n

urk � 0; r ¼ 1; 2::; s

vik� 0; i ¼ 1; 2::;m

ð3Þ

Table 2 Linguistic ratingsand their crisp values

Linguistic rating Fuzzy triangularnumber (a1, a2, a3)

Crisp number~a

Very low (VL) (1, 1, 3) 1.333Low (L) (1, 3, 5) 3Medium (M) (3, 5, 7) 5High (H) (5, 7, 9) 7Very high (VH) (7, 9, 9) 8.67

14 Supplier Performance Evaluation 277

Let u�rk and v�ik represent the optimal solution to (3). The optimistic efficiency orCCR efficiency obtained from solving (3) using u�rk and v�ik is denoted by h�kk. Thisefficiency reflects the self-evaluation of supplier k. The cross-efficiency of supplier

k with respect to peer j is denoted by hjk where hjk ¼Ps

r¼1u�rkyrjPm

i¼1v�ikxij

. Model (3) will be

solved for each supplier k and generate n input weights and n output weightsrespectively. Each supplier k will have (n-1) cross-efficiency plus an optimisticefficiency. These efficiencies form a cross-efficiency matrix as shown in Table 3below where hkk ¼ h�kkðk ¼ 1; 2; ::; nÞ are the optimistic efficiencies of then suppliers.

Our DEA model for cross-efficiency evaluation is inspired by the concept ofideal and anti-ideal decision making units commonly used in multicriteria decisionmaking [24, 28, 29]. An ideal DMU consumes minimum inputs and generatesmaximum outputs whereas the anti-ideal DMU consumes maximum inputs andgenerates minimum outputs. A decision making unit that is close to the perfor-mance of ideal DMU and farthest from the performance of anti-ideal DMU isdeemed efficient. Based on these distances, we generate a closeness coefficientratio (CCk) for each supplier k. Our goal is to maximize this closeness coefficientratio (Eq. 6) for each supplier k. Using these concepts, we present the followingformulations:

Inputs and outputs of IDMU (Here, Ideal Supplier or IS)

xmini ¼min

jfxijg; i ¼ 1; 2; ::m

ymaxi ¼max

rfyrjg; r ¼ 1; 2; ::s

ð4Þ

Inputs and outputs of AIDMU (Here, Anti-ideal Supplier or AIS)

xmaxi ¼ max

jfxijg; i ¼ 1; 2; ::m

ymini ¼ min

rfyrjg; r ¼ 1; 2; ::s

ð5Þ

Let us denote the distance between supplier k and ideal supplier (IS) by d(k, IS)and the distance between supplier k and anti-ideal supplier (AIS) by d(k, AIS).These distances are calculated as follows:

Table 3 Cross-efficiency matrix for n DMUs (suppliers)

Targetsupplier

1 2 – n Average cross-efficiency

1 h11 h12 – h1n1n

Pnk¼1 h1k

2 h21 h22 – h2n1n

Pnk¼1 h2k

– – – – – –– – – – – –n hn1 hn2 – hnn

1n

Pnk¼1 hnk

278 A. Awasthi et al.

dðk; ISÞ ¼Xs

r¼1

urkðymaxr � yrkÞþ

Xm

i¼1

vikðxik � xmini Þ; k ¼ 1; 2; ::; n

dðk;AISÞ ¼Xs

r¼1

urkðyrk � yminr Þþ

Xm

i¼1

vikðxmaxi � xikÞ; k ¼ 1; 2; ::; n

ð6Þ

The closeness coefficient ratio (CCk) of supplier k with respect to IS and AIS isgiven by:

CCk ¼dðk;AISÞ

dðk; ISÞ þ dðk;AISÞ ; k ¼ 1; 2; ::; n ð7Þ

Ps

r¼1urkðyrk�ymin

r ÞþPm

i¼1vikðxmax

i �xikÞPs

r¼1urkðymax

r �yrkÞþPm

i¼1vikðxik�xmin

i ÞþPs

r¼1urkðyrk�ymin

r ÞþPm

i¼1vikðxmax

i �xikÞ;

k ¼ 1; 2; ::; n

Our goal is to maximize CCk for each supplier k in order to make it as close tothe performance of IS and as far from the performance of AIS. The higher thevalue of RC, the more efficient the supplier is. The weights urk and vik that helpachieve this objective for supplier k are obtained from model (8) as follows.

max

RCk ¼Ps

r¼1urkðyrk�ymin

r ÞþPm

i¼1vikðxmax

i �xikÞPs

r¼1urkðyrk�ymin

r ÞþPm

i¼1vikðxmax

i �xikÞþPs

r¼1urkðymax

r �yrkÞþPm

i¼1vikðxik�xmin

i Þ

s:t:Ps

r¼1urkyrk � h�kk

Pm

i¼1vikxik ¼ 0

Ps

r¼1urkyrj �

Pm

i¼1vikxij� 0; j ¼ 1; 2; ::; n

urk � 0; r ¼ 1; 2::; svik� 0; i ¼ 1; 2::;m

ð8Þ

Model (8) can be written in linear form as follows:

maxXs

r¼1

urkðyrk � yminr Þ þ

Xm

i¼1

vikðxmaxi � xikÞ

s:t:Xs

r¼1

urkðyrk � yminr Þ þ

Xm

i¼1

vikðxmaxi � xikÞ þ

Xs

r¼1

urkðymaxr � yrkÞ þ

Xm

i¼1

vikðxik � xmini Þ ¼ 1

Xs

r¼1

urkyrk � h�kk

Xm

i¼1

vikxik ¼ 0

Xs

r¼1

urkyrj �Xm

i¼1

vikxij� 0; j ¼ 1; 2; ::; n

urk � 0; r ¼ 1; 2::; s

vik� 0; i ¼ 1; 2::;m

ð9Þ

14 Supplier Performance Evaluation 279

Model (9) can be simplified as:

maxXs

r¼1

urkðyrk � yminr Þ þ

Xm

i¼1

vikðxmaxi � xikÞ

s:t:Xs

r¼1

urkðymaxr � ymin

r Þ þXm

i¼1

vikðxmaxi � xmin

i Þ ¼ 1

Xs

r¼1

urkyrk � h�kk

Xm

i¼1

vikxik ¼ 0

Xs

r¼1

urkyrj �Xm

i¼1

vikxij� 0; j ¼ 1; 2; ::; n

urk � 0; r ¼ 1; 2::; s

vik� 0; i ¼ 1; 2::;m

ð10Þ

4 Numerical Application

In this section, we present the application of the proposed fuzzy DEA approach forperformance evaluation of three suppliers. Table 4 presents the linguistic evalu-ations for the three suppliers on the proposed criteria by a committee of decisionmaking experts.

The crisp numbers for the fuzzy evaluations presented in Table 4 are given inTable 5. The last two columns of Table 5 present the ideal and the anti-idealsolution (suppliers) for the various criteria obtained from Delphi technique. It canbe seen that the ideal solution (IS) consumes minimum inputs to generate maxi-mum outputs whereas for the anti-ideal solution (AIS), the opposite holds true.

To obtain the optimistic efficiencies of the three suppliers, we apply modelEq. (3). The results yield that h�11 ¼ 1; h�22 ¼ 1; h�33 ¼ 1. Based on these results, wecan say that the three suppliers are efficient but are not able to distinguish betweentheir performances. Therefore, we will apply the proposed fuzzy DEA approachfor cross-efficiency evaluation which is based on maximizing the distance from theanti-ideal solution (supplier) and minimizing the distance from the ideal solutionusing model Eq. (10). The resulting input–output weights for the three suppliersare presented in column 2 of Table 6. It can be seen that very few criteria arecontributing to the overall decision making (C2.1, C2.2.1, C2.2.3, C2.2.4, othersbeing 0). Column 3 contains the weights obtained when the weight restriction isstrictly positive ([0), therefore, we set all the input and output criteria weights [ewhere e ¼ 0:0001. It can be seen that criteria weight results (column 3) are nowdifferent from that of the proposed model (column 2). Therefore, our proposedmodel will have different results depending on the weights (strictly 0 or not) andtherefore, the decision makers should interpret the weighting results cautiously andapply whichever weight value restriction applies to their problem.

280 A. Awasthi et al.

Table 4 Supplier performance ratings

Criteria Criteriatype

Supplier linguistic ratings

S1 S2 S3

C1.1 O M VH MC1.2.1 O L H VLC1.2.2 O VH VL HC1.2.3 O VL VH VHC2.1 O VL L LC2.2.1 I L M HC2.2.2 I VL VL VLC2.2.3 I VL VL VLC2.2.4 I M M LC2.2.5 I H L VLC2.2.6 I M VL HC2.2.7 I L VL MC3.1 O VL H MC3.2 O H H HC4.1 O L M MC4.2 O L VH VHC4.3 O L M VL

Table 5 Crisp performance values, ideal and anti-ideal suppliers

Criteria Crisp ratings Ideal and anti-ideal suppliers

S1 S2 S3 IS AIS

C1.1 5 8.67 5 8.67 5C1.2.1 3 7 1.33 7 1.33C1.2.2 8.67 1.33 7 8.67 1.33C1.2.3 1.33 8.67 8.67 8.67 1.33C2.1 1.33 3 3 3 1.33C2.2.1 3 5 7 3 7C2.2.2 1.33 1.33 1.33 1.33 1.33C2.2.3 3 3 5 3 5C2.2.4 5 5 3 3 5C2.2.5 7 3 1.33 1.33 7C2.2.6 5 1.33 7 1.33 7C2.2.7 3 1.33 5 1.33 5C3.1 1.33 7 5 7 1.33C3.2 7 7 7 7 7C4.1 3 5 5 5 3C4.2 3 8. 67 8. 67 8.67 3C4.3 3 5 1.33 5 1.33

14 Supplier Performance Evaluation 281

Table 7 presents the average cross-efficiency results obtained from model (7). Itcan be seen that supplier 1 performs best followed by suppliers 2 and 3.

In order to see the influence of preferential relations obtained from AHP oninput–output weights on cross-efficiency results and resulting ranking, we con-ducted another set of experiment. Table 8 presents the input–output weightsobtained from DEA by eliminating input–output weight restriction. The resultinginput–output weights for the three suppliers are presented in column 2 of Table 8.It can be seen that very few criteria are contributing to the overall decision making(C1.2.3, C2.2.2, C2.2.3, C2.2.5, C3.2 others being 0). Column 3 contains theweights obtained when the weight restriction is strictly positive ([0), therefore, weset all the input and output criteria weights [e where e ¼ 0:0001. It can be seenthat criteria weight results (column 3) are now different from that of the proposed

Table 6 Input-output weights

Criteria Weights

urk � 0; r ¼ 1; 2::; s

vik � 0; i ¼ 1; 2::;m

urk � e; r ¼ 1; 2::; s

vik � e; i ¼ 1; 2::;m

e ¼ 0:0001

S1 S2 S3 S1 S2 S3

C1.1 0 0 0 0.0001 0.0001 0.0001C1.2.1 0 0 0 0.0001 0.0001 0.0001C1.2.2 0 0 0 0.0758 0.0001 0.0001C1.2.3 0 0 0 0.0001 0.0001 0.1347C2.1 0.1763 0.2724 0.2724 0.0001 0.0988 0.0001C2.2.1 0.1763 0 0 0.0001 0.0001 0.0001C2.2.2 0 0 0 0.0001 0.0001 0.8693C2.2.3 0 0.2724 0 0.2192 0.0001 0.0001C2.2.4 0 0 0.2724 0.0001 0.0001 0.0032C2.2.5 0 0 0 0.0001 0.0001 0.0001C2.2.6 0 0 0 0.0001 0.0001 0.0001C2.2.7 0 0 0 0.0001 0.22587 0.0001C3.1 0 0 0 0.0001 0.0001 0.0001C3.2 0.042 0 0 0.0001 0.0001 0.0001C4.1 0 0 0 0.0001 0.0001 0.0001C4.2 0 0 0 0.0001 0.0001 0.0001C4.3 0 0 0 0.0001 0.0001 0.0001

Table 7 Cross efficiency results

S1 S2 S3 Average cross-efficiency

Rank

S1 1 0.93 0.667 0.867 1S2 0.444 1 0.6 0.681 2S3 0.267 0.6 1 0.622 3

282 A. Awasthi et al.

model (column 2). Also, the results are different from Table 6 (with input outputweight restrictions from AHP).

Table 9 presents the cross efficiency results without AHP weight restriction. Oncomparing the results of Table 7 and Table 9, we can see that the ranking forsuppliers 2 and 3 is different although the best supplier remains the same. Thisimplies that the final results of our model are sensitive to weight restrictions.However, the results with AHP weight restrictions are more realistic as the input–output weights are context dependent in practical situations and therefore shouldconform to the decision maker preferences.

Table 8 Input-output weights (without inputs-output weight restrictions)

Criteria Weights

urk � 0; r ¼ 1; 2::; s

vik � 0; i ¼ 1; 2::;m

urk � e; r ¼ 1; 2::; s

vik � e; i ¼ 1; 2::;m

e ¼ 0:0001

S1 S2 S3 S1 S2 S3

C 1.1 0 0 0 0.0003 0.0003 0.0003C1.2.1 0 0 0 0.0001 0.0001 0.0001C1.2.2 0 0 0 0.0001 0.0001 0.0001C1.2.3 0 0.07627 0.1344 0.0001 0.0001 0.0001C2.1 0 0 0 0.1756 0.2684 0.27C2.2.1 0 0 0 0.175 0.0042 0.0014C2.2.2 0 0 0.8703 0.0001 0.0001 0.0001C2.2.3 0.5 0.2204 0 0.0001 0.2636 0.0001C2.2.4 0 0 0 0.0001 0.0001 0.268C2.2.5 0 0 0.0026 0.0001 0.0001 0.0001C2.2.6 0 0 0 0.0001 0.0001 0.0001C2.2.7 0 0 0 0.0001 0.0001 0.0001C3.1 0 0 0 0.0001 0.0002 0.0002C3.2 0.2142 0 0 0.0414 0.0001 0.0001C4.1 0 0 0 0.0001 0.0001 0.0001C4.2 0 0 0 0.0001 0.0001 0.0001C4.3 0 0 0 0.0001 0.0001 0.0001

Table 9 Cross efficiency results (without inputs-output weight restrictions)

S1 S2 S3 Average cross-efficiency

Rank

S1 1.00 1 0.599 0.866 1S2 0.154 1 0.599 0.584 3S3 0.152 1 1.00 0.716 2

14 Supplier Performance Evaluation 283

5 Conclusions and Future Works

Supplier performance evaluation plays a critical role in supplier quality develop-ment and controlling cost of quality in supply chains. In this chapter, we presenteda hybrid approach based on Delphi technique, AHP and fuzzy DEA for supplierperformance evaluation. The Delphi technique and AHP are used to identify andstructure the criteria. Fuzzy DEA is used to generate cross-efficiencies computedusing the concept of ideal and anti-ideal decision making units on linguistic input–output data for supplier performance evaluation. The strength of the proposedapproach is its ability to structure the criteria, rationalize supplier input–outputweights through preferential relations obtained from principles of AHP, and abilityto treat qualitative (or linguistic) data on supplier inputs-outputs in DEA.

The limitation of our approach is that the generated criteria weights will havedifferent results if the weight values are strictly positive ([e over 0, e being a smallpositive number), therefore, the decision makers should interpret the weightingresults cautiously and apply whichever weight value restriction applies to theirproblem.

The next step of our work involves:

• Comparing the proposed approach with other standard cross-efficiencyapproaches.

• Investigating the impact of defuzzification techniques on cross-efficiencyresults.

• Calculating cross-efficiency under criteria correlation.

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Biography of Editors

Ali Emrouznejad is a reader in OperationsResearch and Management at Aston BusinessSchool, UK. His areas of research interestinclude performance measurement andmanagement, efficiency and productivity analysisas well as data mining. Dr Emrouznejad serves onthe editorial board of several scientific journals; heis editor of Annals of Operations Research,associate editor of Socio-Economic PlanningScience, senior editor and one of the foundingmembers of the Data Envelopment AnalysisJournal, associate editor of the IMA Journal ofManagement Mathematics and guest editor to

several special issues of journals including Journal of Operational ResearchSociety, Journal of Medical Systems and International Journal of EnergyManagement Sector. He is co-founder of Performance ImprovementManagement Software (www.DEAsoftwarte.co.uk). Further to this Ali’s website(www.DEAzone.com) has internationally recognised as the main source for DataEnvelopment Analysis.

Madjid Tavana is a professor of BusinessSystems and Analytics and the LindbackDistinguished Chair of Information Systems andDecision Sciences at La Salle University, wherehe served as Chairman of the ManagementDepartment and Director of the Center forTechnology and Management. He is aDistinguished Research Fellow at KennedySpace Center, Johnson Space Center, NavalResearch Laboratory at Stennis Space Center,and Air Force Research Laboratory. He wasrecently honored with the prestigious Space Act

A. Emrouznejad and M. Tavana (eds.), Performance Measurement with FuzzyData Envelopment Analysis, Studies in Fuzziness and Soft Computing 309,DOI: 10.1007/978-3-642-41372-8, � Springer-Verlag Berlin Heidelberg 2014

287

Award by NASA. He holds a MBA, PMIS, and PhD in Management InformationSystems and received his Post-Doctoral Diploma in Strategic Information Systemsfrom the Wharton School at the University of Pennsylvania. He is the Editor-in-Chief of Decision Analytics, International Journal of Applied Decision Sciences,International Journal of Management and Decision Making, International Journalof Strategic Decision Sciences, and International Journal of EnterpriseInformation Systems. He has published several books and over one hundred andtwenty research papers in academic journals.

288 Biography of Editors