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Information Sciences 176 (2006) 237–262

www.elsevier.com/locate/ins

Integrating data envelopment analysisand analytic hierarchy for the facilitylayout design in manufacturing systems

Tijen Ertay a,*, Da Ruan b,*, Umut Rıfat Tuzkaya c

a Department of Managerial Engineering, Istanbul Technical University,

Macka 34367, Istanbul, Turkeyb Belgian Nuclear Research Centre (SCK-CEN), Reactor Physics & Myrrha Department,

Boeretang 200, B-2400 Mol, Belgiumc Department of Industrial Engineering, Yildiz Technical University, Barbaros Street,

Yildiz, _Istanbul 34349, Turkey

Received 11 June 2004; accepted 1 December 2004

Abstract

Facility layout design (FLD) has a very important effect on the performance of a

manufacturing system. The concept of FLD is usually considered as a multiobjective

problem. For this reason, a layout generation and its evaluation are often challenging

and time consuming due to their inherent multiple objectives in nature and their data

collection process. In addition, an effective facility layout evaluation procedure necessi-

tates the consideration of qualitative criteria, e.g., flexibility in volume and variety and

quality related to the product and production, as well as quantitative criteria such as

material handling cost, adjacency score, shape ratio, and material handling vehicle uti-

lization in the decision process. This paper presents a decision-making methodology

based on data envelopment analysis (DEA), which uses both quantitative and qualitative

criteria, for evaluating FLD. The criteria that are to be minimized are viewed as inputs

0020-0255/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2004.12.001

* Corresponding authors. Fax: +902122407260 (T. Ertay), Tel.: +32 14 332272; fax: +32 14

321529 (D. Ruan).

E-mail addresses: [email protected] (T. Ertay), [email protected] (D. Ruan).

mailto:[email protected]

mailto:[email protected]

238 T. Ertay et al. / Information Sciences 176 (2006) 237–262

whereas the criteria to be maximized are considered as outputs. A computer-aided

layout-planning tool, VisFactory, is adopted to facilitate the layout alternative design

process as well as to collect quantitative data by using exact and vague data by means

of fuzzy set theory. Analytic hierarchy process (AHP) is then applied to collect qualita-

tive data related to quality and flexibility. The DEA methodology is used to solve

the layout design problem by simultaneously considering both the quantitative and

qualitative data. The purposed integrated procedure is applied to a real data set

of a case study, which consists of 19 FLDs provided of the plastic profile production

system.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Decision-making; Facility layout; Robust design; AHP; DEA; VisFactory

1. Introduction

To operate production and service systems efficiently, systems should not

only have to be operated with optimal planning and operational policies, but

also be well designed. Optimal design of physical layout is an important issue

in the early stage of the system design [1]. As known, facilities layout problem

is concerned with the allocation of activities to space such that a set of criteria

is met and/or some objectives are optimized. To this end, the layout problemcan have different formulations, but it is usually abstracted as an optimization

problem. An assignment of the coordinates and orientations of components

that minimizes the cost and satisfies certain placement requirements is sought.

The problem can be viewed as a generalization of the quadratic assignment

problem and therefore belongs to the class of NP-hard problems. Conse-

quently, it is most unlikely that any exact solution to the general layout prob-

lem can be obtained in an amount of time that is bounded by a polynomial in

the size of the problem, resulting in prohibitive computation time for largeproblems. Therefore, algorithmic layout approaches for the larger problems

can be proposed to generate acceptable solutions. Some of these approaches

may be efficient for specific types of problems, but often place restrictions on

component geometry allowable degrees-of-freedom and the objective function

formulation. Others are applicable to wider variety of problems but may re-

quire prohibitively long computing time to solve even simplistic problems.

Hence, there is the potential for automated space layout products to play a

more significant role with the growing demand for computerized facilities plan-ning and management. Especially, commercial products have become available

based on some of these original algorithms for a long time. Most of them in-

volve over simplifying assumptions and request overwhelming computational

efforts such that they cannot be manipulated with ease in practice. These com-

mercial products have been incorporated automated algorithms within an

interactive framework. They are named as user-friendly approaches. How-

T. Ertay et al. / Information Sciences 176 (2006) 237–262 239

ever, user-friendly methods often do not capture all of the layout design

objectives.

Over the past two decades, most of facility layout approaches emphasized

on the design stage and very few results were accomplished in the evaluation

stage. The researches used to develop mathematical programming models or

simulation models to measure the performance of an operating system, whichmay or may not include the considerations on the layout design. The layout

evaluation is to investigate the characteristics of a layout alternative, under

the real constructions of time and information available, before the system

starts its operations; otherwise the relayout would incur higher expenses and

cause a loss of production time. The performance factors, which still provide

useful insight for the impacts resulting from a layout alternative, would be

valuable for evaluation of the layout alternatives [2]. While gathering the data

to evaluate the facility layout, natural vagueness associated with the input datamust be considered, because facility layout problem is an unstructured deci-

sion-making problem due to natural vagueness associated with the inputs to

the models. Hence, the used data in most of the layout models threat exactly,

but to increase the validity of the results, fuzzy linguistic variables and their

fuzzy relations must be considered. Therefore, the generation of model for

the layout design is a critical step because of its unstructured and vast nature.

The complexity increases further due to the multifactor influence on the devel-

opment of facility design for its placement on the floor. To cope with the dif-ficulties being the cause of this complexity, this paper purposes an integrated

framework based on the data envelopment analysis (DEA) methodology and

analytic hierarchy process (AHP). A software package, which operates under

the AutoCAD environment and is effective and user-friendly, is adopted to

constitute the layout alternative generation process as well as to collect the

quantitative performance data such as adjacency scores, shape ratios, material

handling cost and material handling vehicle utilizations. Besides, the proposed

framework includes the integration of various linguistic assessments to obtainthe required data for generating layout design alternatives and its impact on

the development of layout procedure. For example, some input data such

as activity relationships are considered as fuzzy numbers and a fuzzy deci-

sion-making system which consists of four main components as used in [3].

Regardless of the type of data, there is an element of vagueness or fuzziness

in it. Traditional layout methods treat these data as exact and so cannot satisfy

the desire of managers in a real layout problem. It is thus important to be

included the fuzzificated data in the developed framework. In the next step,AHP is applied to collect qualitative performance data such as volume flexibil-

ity, variety flexibility, production quality, and product quality. Later, the DEA

methodology is used to solve the layout design problem by simultaneously con-

sidering both the quantitative and qualitative performance data leading to the

determination of the more robust layout design alternatives. The proposed


integrated framework is successfully applied to a case study of the plastic pro-

file production system with its great efficiency and effectiveness.

2. Literature review

The facility layout problem is one of the best-studied fields to achieve its

goal of productivity and profitability. A number of formulations have been

developed for this problem. When the real shapes and sizes of the facilities

are disregarded, the facility layout problem is generally formulated as a qua-

dratic assignment problem (QAP) of allocating equal area facilities to discrete

locations on a grid with the objective of minimizing a given cost function. In

addition to QAP, the facility layout problem can be modeled as a quadratic

set-covering problem, linear integer programming, mixed integer program-ming, and graph theoretic problem [4]. However, because optimal methods

are limited in the number of departments, the substantial researchers have been

developing sub-optimal algorithms that can solve much larger problems. But

none of them is user-friendly and many of these algorithms based on software

implementation are difficult to use. Besides, Balakrishnan et al. [5] presented

FACOPT, the effective and user-friendly software using the simulated anneal-

ing and genetic algorithms to solve the facility layout problem. Computational

tests are also carried out to identify good parameter values and to compare theperformance of the two algorithms. From the viewpoint of facility layout, the

evaluation is time consuming due of its multiple objectives and data collection

procedures.

Yang et al. [6] used AHP to evaluate multiple-objective layout design alter-

natives generated from the Muther�s systematic layout planning (SLP) proce-

dure. Foulds and Partovi [7] used AHP to evaluate a closeness relationship

among planning departments for a layout problem. Cambron and Evans [8]

used the different computer-aided layout design methods to generate a set ofdesign alternatives and then these alternatives were evaluated by AHP accord-

ing to a set of design criteria. Shang and Sueyoshi [9] proposed a unified frame-

work to facilitate decision-making in the design and planning stage for the

problem of selecting the most appropriate flexible manufacturing system

(FMS). This framework contains three models: an AHP, a simulation module,

and an accounting procedure. These modules are unified through DEA. Both

AHP and simulation models are used to generate the necessary outputs for

DEA, whereas the accounting procedure determines the required inputs, suchas expenditures and resources for realizing the potential benefits. There is a lim-

ited literature to consider at the same time both AHP and DEA methodology

for the facility layout design. Yang and Kuo [10] proposed an AHP and DEA

approach to solve a facility layout design problem. A computer aided layout

planning tool, Spiral is used to generate a considerable numbers of layout

alternatives as well as to generate quantitative decision-making unit (DMU)


outputs. The qualitative output performance measures are weighted by AHP.

DEA is then used to solve the multiple-objective layout problems. However,

their approach only considers a constant input case that is different from the

standard DEA model because the cost associated with a change incurred at

the layout design stage is usually negligible. Therefore, a Banker–Charnes–

Cooper (BCC) model without inputs is adopted for solving the layout perfor-mance frontiers problem. In this paper, Charnes–Cooper–Rhodes (CCR) mod-

el by Charnes et al. [11] is applied to quantitative and qualitative data being

transformed the fractional program to an ordinary linear program.

As known, DEA allows each alternative to specify its own weights so as to

obtain its maximum efficiency score, which may result in a relatively high num-

ber of efficient alternatives and avoid DEA to appear as a robust approach in

determining the best alternative [12]. Hence, a minimax efficiency will be used

to discriminate between FLD alternatives that appear relatively efficient whenapplying the classical DEA model. Also many researchers applied fuzzy ap-

proaches to the facility layout planning and evaluation in literature. Wilhelm

et al. [13] used a fuzzy set approach in 1987 which is based on fuzzy linguistic

variables and fuzzy relations for layout analysis. Raoot and Rakshit [14] used

linguistic pattern for the multiple criteria facility layout problem and presented

a framework of an algorithm based on fuzzy linguistic variables for developing

and evaluating a layout. Another framework for fuzzy evaluation of the layout

designs is developed by Li et al. [15] to rate the various design alternatives gen-erated. They used a fuzzy weighted average to devise a fuzzy rating considering

three different major aspects for a candidate layout design. In this study, we

inspired from [3] which presents a distinct methodology to develop a crisp

activity relationship chart based on fuzzy set theory and the pairwise compar-

ison of AHP. Here, a similar approach is used for an integration of various lin-

guistic assessments to obtain the required data for generating a layout design

alternative procedure.

3. The robust layout framework

A robust layout is one that is good for a wide variety of demand scenarios

even though it may not be optimal under any specific demand scenarios [16]. A

robust layout procedure considers minimizing the total expected material han-

dling costs over a specific planning horizon. Most of the existing robust and

dynamic layout design procedures use a quadratic assignment problem(QAP) formulation. But, most researchers use heuristic approaches for a flex-

ible layout problem since QAP is an NP-hard problem. Besides, a flexible facil-

ity layout that allows the system to efficiently respond to the dynamic and

uncertain requirements is critical to achieving a cost effective system design.

A good layout is thus to determine a robust layout over a set of planning time,


which covers the total planning horizon. A planning time set is considered as a

consecutive time spans with a layout rearrangement occurring only at the

beginning of the time span. The number and length of these time spans are

determined based on the trade-off between material handling costs and facility

rearrangement costs. Thus, the aim is to modify the layout at the beginning of

each time span, but not to change the layout within these time spans. A purerobust layout strategy will have the time spans equal to a planning time set.

This strategy will have different demand levels in time spans and will have

choose the demand level to minimize the material handling cost, adjacency

scores and to maximize the material handling vehicle utilization.

In the literature, there are studies based on robust and dynamic layout de-

sign. For example, Montreuil and Laforge [17] considered a dynamic layout de-

sign problem given for a scenario tree of probable future scenarios. They used a

design skeleton approach to determine the layout. This approach allows thedepartments to have varying areas but not a fixed shape. Lacksonen [18] used

an approach to formulate a dynamic layout design problem. This approach

considers a two-step procedure similar to the idea of Montreuil�s study. How-ever, few effective and user-friendly approaches have been proposed for the ro-

bust facility layout design. There is a need for user-friendly software

incorporating effective and dynamic procedures. In other words, there is the

potential for automated space layout products to play a more significant role,

with the growing demand for computerized facility planning and management[19]. Some commercial facility management systems incorporate automated

algorithms to solve blocking activities. The blocking procedure can be classi-

fied into the two categories according to the types of the plane on which the

block layout is to be drawn. First category is based on block layouts drawn

on the plane called the grid-based plane, which is divided into squares or rect-

angles with a unit area by a grid. Second category is also based on block lay-

outs drawn on the continual plane, which is not divided by a grid. Most of

existing algorithms belong to the first category. Well-known algorithms suchas CRAFT, CORELAP, ALDEP, COFAD, CLASS, SHAPE, MULTIPLE,

and SABLE construct block layouts on the grid-based plane. These approaches

construct a block plan as the dual of a planar graph in which nodes represent

spaces and links represent required adjacencies [19,20]. Besides, if a block lay-

out is constructed on the second category, it is easier to make shapes of the

departments in the layout procedure, because shapes of the departments are

not constrained by a grid. Algorithms, which give a block layout in which

the facilities are of rectangular shapes, such as the one by Kim and Kim[21], construct block layout on the continual plane. But some of these algo-

rithms may give layouts with long or narrow departments in some conditions.

In this study, we present VisFactory, having inside a lot of heuristic ap-

proach that is effective and user-friendly. The software uses most of the above

determined approaches to effectively solve the facility layout design. Here, our

AHP methodology for qualitative data

evaluation

Qualitative performance data

input data to VisFactory

Layout alternative generation

Flow Distance Defuzzificated Adjacency Score

Material handling vehicle utilization

Shape Ratio

Handling cost

Quantitative performance data

Quality Flexibility

DEA methodology for the most robust layout design

Selection of the final layout design solution

Demand Forecasts Both exact and fuzzy

Fig. 1. The robust layout design framework.


layout problem requires a dynamic facility layout design to respond to the var-

iability of demand. Therefore, different time spans that have different demandstructures are considered in an alternative generation process. Thanks to the

software, by entering concerned data, alternative facility designs based on

the time span equal to a planning time set can be generated and some informa-

tion such as flow distances, handling costs, adjacency scores and material han-

dling vehicle utilizations can be obtained. Fig. 1 illustrates the robust layout

design framework, which is integrated with the VisFactory software based

on the DEA/AHP methodology.

3.1. Fuzzy set and decision-making in facility layout

Fuzzy set theory [22] deals with vague, imprecise and uncertain informa-

tion. A fuzzy set can be thought of a class of concepts/objects in which no

well-defined boundary exists between the concepts/objects that belong to the


class and those which do not belong [23]. If a collection of objects, X, is defined

as X = {xiji 2 N} then the fuzzy set A on X is defined by its membership func-

tion lA(x) that takes values in interval [0,1]. A is represented as A = {lA(x)/xjx 2 X}. In the framework of fuzzy set theory, linguistic variables in natural

language are usually used for complex systems, because of the difficulty to

describe them with quantitative terms [24,25].Here, using fuzzy set theory as decision-making system provides crisp out-

put that measures the facility planning rating in other words relationship level

(A: absolutely important, E: especially important, I: important, O: ordinarily

important, U: unimportant, X: undesirable). The obtained process of this rela-

tionship level within the existing knowledge and experience of facility layout

designers by a subjective approach depends on a fuzzy decision-making system

that consists of four main components: the fuzzification interface, the knowl-

edge base, the decision-making logic, and the defuzzification interface [3].These components can be applied to a facility layout problem. Firstly, their in-

put variables and values are determined. Then the collection of objects, the

membership functions and the linguistic values for these variables are decided

by experts that have knowledge about the system. After the fuzzification inter-

face is completed, decision-making logic is established by if-then rules. The fuz-

zy connective �and� is implemented as a fuzzy conjunction in a Cartesian

product space in which the input variables take in their respective universe

of discourses. The minimum operator is used. The membership value of thecontrol action of each rule is the minimum value of the input variables� mem-bership values.

At the last step fuzzy outputs are converted to crisp values by a

defuzzification method as the common used centre of area (COA) [26] as

follows.

R0 ¼P

iðlgR � RÞPilgR

ð1Þ

where R0 is the final crisp rating of the activity, g the fuzzy rating of the activity

for the rule in consideration, i the rules that are used in the activity, R the

numerical rating of the activity of the rule.

In our application two fuzzified variables, information flow (IF) and envi-

ronmental condition; (EC), are used for determining the activity relationships.

The number of communications between departments represent the IF vari-

able. Noise between departments can be considered as EC and measured indB. Also the weight factor (WF), numerical values between 0 and 1 are deter-

mined for the factors that are involved in decision-making process by using the

AHP methodology. WF is fuzzified using a set of membership functions. Then

the rules are defined to imitate the designers� decisions. At last, by using theserules, the min operator and defuzzification COA method, the designer can

obtain the exact relationship values.


3.2. VisFactory software tool

Engineering Animation Incorporated�s (EAI) VisFactory software [27] pro-cess has three main parts: FactoryFLOW, FactoryPLAN, and FactoryOPT.

FactoryFLOW analyzes, compares, and improves laiyouts with respect to

material flow. This part integrates actual facilities drawings and material flowpaths with production and material handling data to generate flow, congestion,

and safety diagrams. FactoryPLAN is a planning tool for designing and ana-

lyzing layouts based on how desirable it is that activities be closed to each other

as determined by an assigned proximity value, the intensity of material flow be-

tween them or an aggregation of proximity and flow relationships. Factory-

OPT jump-starts the design process with a near-optimal block layout based

on the FactoryPLAN relationship and/or FactoryFLOW intensity data. Con-

sidering the data which are entered to the first two modules by the user, Fac-toryOPT uses the convenient algorithms which are embedded to the module to

find the near-optimal layout and generates node and block diagrams using ac-

tual space requirements information. Thanks to the VisFactory tool, alterna-

tive facility designs can be generated by entering concerned data and some

information such as flow distances, handling cost and adjacency scores can

be obtained.

3.3. Alternative layout design generation

The VisFactory tool [27] is used to efficiently investigate a large number of

layout design alternatives. The FactoryOPT module contains a number of

algorithms which are convenient for layout designing process as implied in

the beginning of Section 3. Software quickly generates alternative layouts using

these embedded algorithms and provides quantitative performance data that

will be the both input and output parameters for the DEA methodology. To

select the most robust layout design, it should be prevented that the final designsolution is trapped in a local optimum. Hence, the facility layout design alter-

natives considering as decision-making units (DMUs) in DEA methodology

should be at a minimum, 2–5 times the sum of the number of quantitative

and qualitative performance data.

3.4. AHP methodology for qualitative performance

The analytic hierarchy process is a multicriteria decision-making methodthat uses hierarchic or network structures to represent a decision problem

and then develops priorities for the alternatives based on the decision-maker�sjudgments throughout the system [28]. In general, the multicriteria methods

can be classified as weighting methods, outranking methods, goal and reference

point methods and value function methods. AHP is a kind of value function


method. The reason of adopting AHP especially for the qualitative perfor-

mance data is the fact that qualitative factors are often complicated and con-

flict. Also, the user acceptability and confidence in the analysis provided by the

AHP methodology is high when it is compared with other multiattribute deci-

sion approaches [29]. The other benefits of AHP include: providing a system-

atic way for subjective decision processes, serving sensitivity analysis, givinginformation about the evaluation criteria�s implicit weights, and providing bet-ter understanding and participation among the members of the decision-

making group and hence a commitment to the chosen alternative [30].

Although AHP is a popular and useful method, several shortcomings have

been reported in the literature and some modifications are suggested to deal

with these shortcomings. Firstly, instead of using an additive scale ranging

from 1 to 9, several alternative scales are proposed. One of the most known

scales is the geometric scale [31]. Other point is the method for estimatingthe priorities proposed by Saaty [32] that is called the eigenvector technique.

There are other alternative techniques and the most often discussed one is

the logarithmic least squares technique (LLST) [33]. May.be, the most contro-

versial issue in the use of AHP is the rank reversal phenomenon: the ranking of

alternatives determined by AHP may be altered by the addition of another

alternative for consideration. To overcome this problem, the concept of abso-

lute measurement [34] and multiplicative variant of AHP can be used and these

methods do not suffer from rank reversal [31]. One of the other criticisms isthat AHP is not an axiomatic framework. But Saaty [35] has provided the nec-

essary axioms, pertaining to reciprocal comparisons, homogeneity, indepen-

dence, and expectations. At last, AHP uses redundant judgments for

checking consistency, and this can exponentially increase the number of judg-

ments. And, also some methods have been developed to reduce it [36].

In this study, the aim of the AHP usage is to obtain the weights indicating

the relative importance of the facility layout alternatives for each criterion. At

each level, the user will be asked to determine a comparison matrix by compar-ing pairs of criteria whereas the alternatives at the lowest level are compared

against the standards that are established by the user for criteria. More num-

bers of alternatives, more convenient the rating method. Analytic focus of rat-

ing method enables decision-makers to evaluate a large number of alternatives

easily. In this method an element is compared against an ideal property and

generally, only the final alternatives of choice are absolutely measured.

The AHP approach consists of critical issues such as robustness and consis-

tency. The robustness can be verified by sensitivity analysis of the weights. Inother words, robustness is related with the sensitivity of eigenvector which

computes the relative ranking of the being evaluated criteria. The problem is

how sensitive the weights given by eigenvector components are slight changes in

the judgment values. Clearly, it is desirable that the weights do not fluctuate

widely with small changes in judgment [32]. In this study, Expert Choice package


program [37] is used for sensitivity analysis of pair-wise comparison model of

alternative layouts. Details of the analysis will be mentioned in Section 4.3.

Other critical issue is consistency which can be verified by the consistency

ratio. The consistency means that if aij = 2, ajk = 3, then aik must be equal to

6. AHP does not require that judgments be consistent or even transitive now

that the judgments are totally random in nature. Here, the important pointis the consistency of a matrix of such random judgments should be worse than

that of a matrix of informed judgments. The measure can be used to compare

and evaluate the goodness of the consistency of informed judgments.

The consistency index (CI) of a matrix of comparisons is given by

CI = (kmax � n)/(n � 1). Here, the kmax is the maximum eigenvalue and n is

the size of matrix. The consistency ratio (CR) is obtained by forming the ratio

of CI and random index (RI). DeSchutter has conjectured the following rela-

tionship between the index RI and n: RI = 1.98*[(n � 2)/n] where 1.98 is theaverage value of the ratio of each value computed so far from n = 3 to 15

divided by (n � 2)/n for the corresponding value of n [38].

Like any Expert Choice model, the ratings model has a goal, criteria, and

alternatives. However, the lowest levels of the model are nodes serving as scales

against which the alternatives will be measured under each criterion (or sub-cri-

terion). We refer to these as scales of intensity. In the model, the scale of inten-

sity for each criterion (sub-criterion) appears as a group of nodes under that

criterion. The intensity categories are prioritized through the usual pair-wisecomparison process. Alternatives do not appear within the main structure of

the three, but instead they are entered into the ratings spreadsheet. In a ratings

model, intensities are definite in place of the alternatives in the hierarchy. The

intensities are first constructed for each criterion (or the lowest level sub-crite-

rion) and then used to rate the alternatives [37].

3.5. DEA methodology for the most robust FLDs

Data envelopment analysis is a linear programming based on technique

developed by Charnes et al. [11]. Until recently, DEA has been very sparingly

applied to justify a number of analyses and operations decisions related to the

advanced manufacturing system and technologies. For example, DEA has been

utilized for the evaluation of hybrid manufacturing layout in Shafer and

Brandford [39]. Khouja [40] proposed a decision model for technology selec-

tion problem using a two-phase procedure. Sheng and Sueyoshi [9] utilized a

combination of DEA and AHP for FMS evaluation and selection. In thisstudy, the AHP technique is used to identify qualitative benefits of FMS, which

is then incorporated into the DEA model. Ertay and Ruan [41] proposed a

decision model based on DEA for determining the most efficient number of

operators and the efficient measurement of operator allocation in cellular man-

ufacturing system (CMS).


Data envelopment analysis considers n decision-making units (DMUs) to be

evaluated (for this study, 19 FLDs), in which each DMU consumes varying

amounts of m different inputs to produce s different outputs. The relative effi-

ciency of a DMU is defined as the ratio of its total weighted output to its total

weighted input. In mathematical programming terms, this ratio, which is to be

maximized, forms the objective function for the particular DMU being evalu-ated. A set of normalizing constraints is required to reflect the condition that

the ratio of output over input for every DMU be less than or equal to unity.

The efficiency of a particular DMU0 is obtained by solving the following frac-

tional programming problem. This fractional formulation entails a nonlinear

objective function. In DEA the nonlinear objective function of maximizing

the ratio of output over input is linearized and solved via LP.

max h0

Pruryr0Pivixi0

subject to

PruryrjPivixij

6 1; j ¼ 1; . . . ; n

urvi P e > 0 r ¼ 1; . . . ; s; i ¼ 1; . . . ;m

ð2Þ

where h0 is the efficiency score of DMU0, yrj denotes amount of the output r

produced by the jth DMU, xij denotes amount of the input i used by the jth

DMU, and e is an infinitesimal positive number. The above fractional programis computed separately for each DMU, generating n sets of optimal weights.

The weights in the objective function are chosen to maximize the value of

the DMU�s efficiency ratio subject to the less-than-unity constraints. These

constraints ensure that the optimal weights for DMU in the objective function

(DMU0) do not denote an efficiency score greater than unity either for itself orfor the other DMUs. Besides, the fractional program is not used for actual

computation of the efficiency scores due to its nonlinear and nonconvex prop-

erties. Hence, the fractional program is transformed to an ordinary linear pro-

gram by letting lr = tur and xi = tvi, where t�1 ¼P

ivixi0. The linear programconstraints the weighted sum of inputs to be equal to unity and maximizes

the weighted sum of outputs of the DMU0 selecting appropriate values of lrand xi. In addition, defining a deviation variable for the DMU0 as d0, where

h0 = 1 � d0 relationship holds, the model can be expressed as follows:

min d0

subject toX

rxixi0 ¼ 1;

Xrlryrj �

X

i

xixij þ dj ¼ 0; j ¼ 1; . . . ; n

lr;xidj P e > 0; r ¼ 1; . . . ; s; i ¼ 1; . . . ;m; j ¼ 1; . . . ; n

ð3Þ


where dj, represents the deviation variable for the jth DMU. When applying

Eq. (3), DMU0 is efficient if and only if d0 = 0, thus d0 can be interpreted as

a measure of inefficiency. Besides, the flexibility for Eq. (3) to choose its input

and output weight (score) produces in general the efficient DMUs. This situa-

tion causes weighting a single input and/or output to appear efficient of weight-

ing a spread of inputs and outputs on an unrealistic weighting system toachieve efficiency. Therefore, the minimax efficiency is a practical method to

cope with the problem of multiple relatively efficient DMUs [42]. This formu-

lation is represented as:

min M

subject toX

rxixi0 ¼ 1;

Xrlryrj �

X

i

xixij þ dj ¼ 0; j ¼ 1; . . . ; n

M � dj P 0; j ¼ 1; . . . ; n

lr;xidj P e > 0; r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; j ¼ 1; . . . ; n

ð4Þ

where M is the maximum of all deviation variables. Here, DMU0 is minimax

efficient if and only if the value d0 corresponding to the solution that minimizes

M is zero. The efficiency score of DMU0 is defined as h0 = 1 � d0. Eq. (4) indi-

cates that if DMU0 is minimax efficient, it is also efficient in the classical DEA.

On the contrary, if DMU0 is found to be efficient in the classical DEA solutionit does not have to be minimax efficient since d0 = 0 does not necessarily result

in a minimum value of M.

4. A case study of the robust layout framework to a plastic profile

production system

The purposed robust layout framework is applied to the company, Sert Plas-tic Profile Industry Co., which has been active in plastic profile scope. The com-

pany tries to enter to the European market place. They intend in getting some

quality certificate to be able to sell their products to European Countries. To

this end, any production system must be revised. An important step of this

revision study is to solve problems caused by the facility design. By the robust

layout framework we aimed to get a facility design that produces quality prod-

ucts with in a flexible manner and low cost.

All produced extrusion products can be examined in the three sub-groups aswainscot, cable canal and pipe profiles. Besides, variety can be increased in

each group according to the color, the size, and the shape of profile. When con-

sidering all kinds of product demands, it is required to be frequently made a

Fig. 2. Current facility layout.


change in the production line. In the time spans that the demand is very high, it

cannot be answered to the customer�s needs. Because this situation is related tothe high setup times, the low production velocity causes the low production

flexibility being a result of the low production volume and the high production

variety in the production lines. Hence, the changes to be done in the design of

facility layout will be considered for surmounting with the above problems.

Fig. 2 shows the current facility layout plan and Fig. 3 shows the operation

flow schema for three products.

4.1. Data collection for an alternative layout generation

To constitute the alternative layout designs with the VisFactory software

package, both FactoryFLOW and FactoryPLAN modules are considered for

entering the data related to the departments, products, and flows. The required

data about parameters for these modules are gathered from the historical data

in company records. And the opinions and experiences of the managers are

used to constitute the relationship scores between departments according to

fuzzy set theory.For the FactoryFLOW module, the data based on the number of facilities,

dimensions of departments, unit costs of moving materials between depart-

ments, raw material rates and quantities used in products, and quantities

and capacities of material handling vehicles are gathered. Also the production

quantities of wainscot, cable canal and pipe profiles are entered to software and

these quantities belong to the different time spans when the demands are rela-

tively high. After giving the data to the FactoryFLOW module, centres of

the departments are determined for the current layout in the AutoCAD

Extrusion Lines

Mixing Area

Raw Material Warehouse

Packaging Area

Cable Canal Warehouse

Receiving and Shipping Area

Wainscot Warehouse

Grinding Area

Pipe Profiles Warehouse

Fig. 3. Operation flow schema.

Fig. 4. The result screen of FactoryFLOW.


environment. At last the module calculates the total flow distance, the total

cost of material flows between departments, number of moves and total time

for the related layout design as shown in Fig. 4.


In the FactoryPLAN module, the closeness ratings among departments in

the facility are determined with the data to be entered to this module. This data

is related to the space requirements of departments and relationship levels be-

tween departments. The levels of relationship among the departments are: A,

E, I, O, U, X. Here, the relationship levels between departments are defined

by fuzzy sets. There are two variables that affect the relationships: informationflow (IF) and environmental conditions (EC) as mentioned in Section 3.1. For

all department pairs, weights of these factors (WF) are determined by the AHP

methodology. Membership functions of these factors are shown in Fig. 5.

Also, mathematical illustration of IF�s membership function is shown as an

example as follows. The determination of relationship level between depart-

ments 4 and 5 (extrusion lines and grinding area) is given as an example.

The weights of IF and EC related to departments 4 and 5 are determined as

0.67 and 0.33, respectively by the AHP methodology. Factor 1 (IF) has a valueof 11 communications per day, it belongs to fuzzy subset high (H) with a mem-

bership value of 0.67. Its weight factor (WF) has a value of 0.67 which belongs

to the fuzzy subset high (H) with a membership value of 1.00. Factor 2 (EC)

has a value of 87 dB, it belongs to fuzzy subset medium (M) with a membership

0

1

15 96 12 3

Low High Medium µg

IF(x)

0

1

11090 80 10070

High Low Medium µg

EC(x)

dB

0

1

0.45 0.30 0.60 Weight factor

0.15

Low High Moderate µg

WF(x)

# of communications per day

Fig. 5. Membership functions of information flow (IF), environmental conditions (EC) and inputs�weight factor (WF).

Table 1

If-then rules between factors and their weight factors

Rule 1 (IF) WF Rule 2 (EC) WF

L (Low) M (Medium) H (High) L M H

L U U O H U O O

M O I I M I I E

H E E A L E A A

Table 2

Relationship codes and space requirements of departments

Departments Required space (m2) 1 2 3 4 5 6 7 8 9

1 Receiving and shipping area 280 E U U U U E E E

2 Raw material warehouse 300 E U O O U U U

3 Mixing area 180 A I U U U U

4 Extrusion lines 1200 E A A A A

5 Grinding area 280 U O O O

6 Packaging area 240 I I I

7 Wainscot warehouse 270 U U

8 Cable canal warehouse 250 U

9 Pipe profile warehouse 200


value of 0.7. Its weight factor (WF) has a value of 0.33 which belongs to the

fuzzy subset moderate (M) with a membership value of 0.8. Then, if-then rules

and values of relationship levels (A = 10, E = 8, I = 6, O = 4 and U = 2) are

developed by the decision-maker. These rules are shown in Table 1. According

to Table 1, rating of Rule 1 is A and rating of Rule 2 is I.Using the minimum operator, Rule 1 has a membership value of the mini-

mum of 0.67 and 1.0. And Rule 2 has a membership value of the minimum

0.7 and 0.8. At the last step, the defuzzification is realized for the activity rela-

tionship value between the departments 4 and 5 as follows. Activity relationship

value = [(10*0.67) + (6*0.70)]/(0.66 + 0.70) = 10.8/1.36 = 7.9. This obtained

value is around the nearest integer value of 8, which corresponds to E relation-

ship code. By using this process, the relations among all departments are deter-

mined. Table 2 gives the relationship codes and space requirements of thedepartments.

By entering these relationship values to the FactoryPLAN module, the adja-

cency scores are calculated based on penalty scores indicating inappropriate

layouts.

4.2. Layout alternative generation: FactoryOPT

FactoryOPT determines the optimum location of activities within a hexag-onal adjacency graph based on the data contained in an aggregated proximity


and flow relationship matrix file. FactoryOPT uses the entered parameter val-

ues in these files and generates the first near-optimal layout. Then, the alterna-

tive layout generation is constituted by means of the changes in activity ratio of

shape and penalty cost of shape which are the parameters that can be adjusted

by the software user. A preliminary study selected 18 alternatives for a further

evaluation as shown in Fig. 6. Also, the data related with each alternative lay-out which will be used in DEA as input and output measures are obtained by

Fig. 6. Generated layout alternatives.


running the FactoryFLOW and FactoryPLAN modules for each alternative

layout.

4.3. AHP for qualitative performance data

Qualitative performance data are obtained by the AHP methodology andsolved by Expert Choice Package Program. Qualitative data include both as-

pects of ‘‘flexibility’’ and ‘‘quality.’’ Flexibility involves two sub-criteria. The

first is ‘‘volume flexibility’’ based on ones of future expansion. The second is

‘‘variety flexibility’’ related to the capability to perform a variety of products

under the different operating conditions. Quality involves also two sub-criteria

based on ‘‘product’’ and ‘‘production’’. Production quality is influenced by the

locations according to each other of the departments. The mixer ratios,

the heating levels, and the mixture recipe also influence product quality. Theimportance rates of variety and volume flexibilities are determined as 0.697

and 0.303 and the importance rates of production and product qualities are

determined as 0.455 and 0.545 by using the Expert Choice program. Different

rating scales are prepared for each of these four criteria by the pair-wise com-

parison matrices. For example, to realize the variety flexibility, three conditions

must be provided: (a) the greatness of shape ratio of the department 3, (b) the

greatness of shape ratio of the department 4, and (c) the adjacency of the

departments 3 and 4. According to the above-provided conditions, eight differ-ent intensities are determined by the pair wise comparison method as shown in

Fig. 7.

For all stages of the hierarchy, the consistency ratios are calculated. Incon-

sistencies of volume flexibility, variety flexibility, production quality, and prod-

uct quality sub-criteria matrices are 0.04, 0.03, 0.03, and 0.03, respectively.

According to these results, the inconsistency ratios are below 10% and the

matrices that are prepared for criteria and sub-criteria are consistent. To mea-

sure the performance of alternative layouts, the rating method is used. Thismethod is preferred because there are many layout alternatives. While the alter-

native number increases, the inconsistency ratio will increase for pair wise

Fig. 7. Intensities of sub-criteria for variety flexibility.


comparison matrices. Table 3 gives the scales for criteria and resulting weights

for alternatives.

Also the sensitivity analysis was performed to determine how sensitive the

results are to changes in priority of the criteria. Below, sensitivity analysis of

layout alternatives for modifications in the priorities of production quality

and product quality criteria is shown. The Expert Choice program is used toconduct gradient sensitivity analysis for first nine alternatives according to

the sub-criteria of quality.

The overall effect of product quality is around 54.5%. At this value, the

alternative 7 and alternative 12 have the biggest priority 8.5%, the alternative

9 has 7.2% priority and alternatives 3, 10 and 16 have 6.5% priority. As

the product quality value is increasing, beginning from the 56.8% point, the

Table 3

Scale for criteria and resulting weights for alternatives

Alternative

layouts

Flexibility Alternative

layouts

Quality

Total

weight

Variety

flexibility

(0.697)

Volume

flexibility

(0.303)

Total

weight

Product

quality

(0.545)

Production

quality (0.455)

Current 0.086 8 4 7 0.085 7 8

16 0.086 8 4 12 0.085 7 8

14 0.086 8 4 9 0.072 6 8

12 0.086 8 4 3 0.065 8 4

5 0.086 8 4 16 0.065 8 4

10 0.086 8 4 10 0.065 8 4

11 0.086 8 4 14 0.064 5 8

6 0.072 8 3 11 0.064 5 8

9 0.067 7 4 4 0.064 5 8

2 0.034 6 2 17 0.064 5 8

18 0.034 6 2 2 0.048 7 4

15 0.034 6 2 5 0.048 7 4

13 0.034 6 2 15 0.045 7 3

3 0.031 6 1 1 0.041 6 5

4 0.024 5 2 6 0.036 6 4

7 0.024 5 2 13 0.036 6 4

17 0.024 5 2 Current 0.022 4 4

8 0.011 2 2 18 0.018 3 4

1 0.011 2 2 8 0.013 3 2

Scale for volume flexibility

4 3 2 1

0.565 0.262 0.118 0.055

Scale for variety flexibility, product quality and production quality

8 7 6 5 4 3 2 1

0.331 0.230 0.157 0.106 0.071 0.048 0.033 0.024

Fig. 8. Layout alternatives performance sensitivity analysis.


alternative 9 has a smaller priority than the alternatives 3, 10 and 16.

When product quality effect arrives 69.4%, the alternatives 3, 10 and 16

have the biggest priority. Also, when we decrease the product quality value,

begging from the 50.3% point, the alternatives 4, 11 and 14 have a biggerpriority than the alternatives 3, 10, and 16. The representations given in

Fig. 8 (performance sensitivity analysis) show the behavior of the alternatives

with respect to each other which is given in a compact and very explanatory

manner.

According to this explanation, the gradient sensitivity analysis illustrates

that the alternatives 3, 10 and 16 are affected by the variation (increasing) in

the product priority rather than other alternatives. Also sensitivity analysis

for flexibility represents changes in priorities. This means that alternativesare sensitive to little changes in criteria priorities. But, while the weight of prod-

uct quality is between 54.5% and 69.4% and the weight of variety flexibility is

between 64.4% and 69.7%, the chosen robust layout alternative does not

change.

4.4. DEA model for both quantitative data and qualitative data

In a number of previous DEA evaluation models, the criteria that are to beminimized are viewed as inputs and the criteria to be maximized are considered

as outputs [43]. This situation is also used in evaluation of the facility layout

designs. DEA is performed by employing material handling cost, adjacency

scores as the input variables, and shape ratio, utilization of material-handling


vehicle, flexibility, and quality as the output variables. Table 4 lists the data set

for the 18 facility layout alternatives and the current layout.

The correlation of inputs and the correlation of outputs are checked and the

results are shown in Table 5. The strongest correlation, 0.264 is between quality

and flexibility outputs. Even this one is not meaningful and all inputs and out-

puts are convenient for DEA. There is not a necessity to remove any criterionout or to add new one in.

The DEA Eqs. (3) and (4) are applied to the data set of 18 facility layout

alternatives and the current layout. Table 6 shows the efficiency scores obtained

using DEA.

The first DEA model denotes that the 9 alternatives (excluding the current

layout) are relatively efficient. To improve the discriminating power of DEA,

Table 4

Inputs and outputs of DEA

DEA inputs DEA outputs

Cost ($) Adjacency score Shape ratio Flexibility Quality Hand-carry utility

1 20309.56 6405.00 0.4697 0.0113 0.0410 30.89

2 20411.22 5393.00 0.4380 0.0337 0.0484 31.34

3 20280.28 5294.00 0.4392 0.0308 0.0653 30.26

4 20053.20 4450.00 0.3776 0.0245 0.0638 28.03

5 19998.75 4370.00 0.3526 0.0856 0.0484 25.43

6 20193.68 4393.00 0.3674 0.0717 0.0361 29.11

7 19779.73 2862.00 0.2854 0.0245 0.0846 25.29

8 19831.00 5473.00 0.4398 0,0113 0.0125 24.80

9 19608.43 5161.00 0.2868 0.0674 0.0724 24.45

10 20038.10 6078.00 0,6624 0.0856 0.0653 26.45

11 20330.68 4516.00 0.3437 0.0856 0.0638 29.46

12 20155,09 3702.00 0.3526 0.0856 0.0846 28.07

13 19641.86 5726.00 0.2690 0.0337 0.0361 24.58

14 20575.67 4639.00 0.3441 0.0856 0.0638 32.20

15 20687.50 5646.00 0.4326 0.0337 0.0452 33.21

16 20779.75 5507.00 0.3312 0.0856 0.0653 33.60

17 19853.38 3912.00 0.2847 0.0245 0.0638 31.29

18 19853.38 5974.00 0.4398 0.0337 0.0179 25.12

Current 20355.00 17402.00 0.4421 0.0856 0.0217 30.02

Table 5

Correlations between DEA inputs and outputs

Cost Shape ratio Flexibility Quality

Adjacency score 0.198

Flexibility 0.035

Quality �0.17 0.264

Hand-carry utility 0.065 0.114 0.156

Table 6

Efficiency scores that are obtained by DEA

Alternatives Efficiency score Minimax efficiency score Efficiency score of min M�0.3*d01 0.985 0.952 0.932

2 0.988 0.959 0.952

3 0.997 0.933 0.926

4 0.949 0.872 0.872

5 1.000 0.794 0.794

6 0.973 0.897 0.897

7 1.000 0.794 0.793

8 0.857 0.787 0.776

9 0.889 0.775 0.775

10 1.000 0.847 0.811

11 0.998 0.900 0.900

12 1.000 0.868 0.868

13 0.776 0.776 0.776

14 1.000 0.970 0.970

15 1.000 1.000 0.994

16 1.000 1.000 1.000

17 1.000 0.973 0.924

18 0.852 0.796 0.785

Current 1.000 0.806 0.806


the minimax efficiency scores are computed. The third column of Table 6 indi-

cates the minimax efficiency scores of the facility layout alternatives. According

to the minimax efficiency scores, the second DEA model denotes that two alter-

natives are relatively efficient. Although the minimax efficiency formulation re-

duces the number of efficient facility layout alternatives from 9 to 2, but, it still

has deficiency in determining the best facility layout alternative. To overcome

this situation, the objective function of Eq. (4) is modified as ‘‘minM � kd0’’

where k 2 (0,1) is a constant that is determined by trial-and-error in a way toobtain a single relatively efficient alternative. This change in objective function

results in an unfavorable consideration for each facility layout alternative under

evaluation by minimizing negative d0. The fourth column of Table 6 provides

the efficiency scores for k = 0.3. The efficiency scores obtained using ‘‘min -

M � 0.3*d0’’ as the objective function are less than or equal to the efficiency

scores computed employing ‘‘minM’’. The results showed that the alternative

16 is the best facility design. The company�s existing layout seemed inefficient.

5. Conclusion remarks

This study addresses the evaluation of the facility layout design by develop-

ing a robust layout framework based on the DEA/AHP methodology with the


VisFactory tool. In the proposed framework, fuzzy sets are considered while

data gathering for the VisFactory tool related to activity relationships. How-

ever, because of the requirements of the tool, activity relationship data are en-

tered after defuzzified. Then, flow distances, handling cost, adjacency scores

and material handling vehicle utilization data are obtained as crisp values

for the 18 alternatives which are proposed by VisFactory. Because of the ob-tained crisp values for alternative layouts, parameters to be used for DEA

are considered as crisp values. But in the future studies, fuzzy set theory can

be embedded to the DEA models. In this way, uncertainties will be also con-

sidered in DEA [44].

Moreover, the quantitative data used in VisFactory are accepted as valid

parameters because of gathering from historical data in the company records.

Also, sensitivity analysis is applied for qualitative data in the AHP methodol-

ogy. Sensitivity analysis for quality factor is explained, as an example for thevalidity of a qualitative parameter. According to changes in weight of sub-cri-

teria, the priority of alternatives can be changed. The intervals of weights for

sub-criteria that do not change the robust layout are determined.

The proposed framework is applied to a real data set consisting of the 18

facility layout alternatives. As a result of application of DEA including both

quantitative and qualitative data, the 9 alternatives are determined as relatively

efficient. To increase discriminating power among these 9 alternatives, the con-

cept of minimax efficiency is employed. The minimax efficiency formulationboth improves the discriminating power of DEA and prevents the frequently

encountered problem of unrealistic weight allocation. Thus, a real case-study

illustrated the effectiveness of the proposed framework. By means of the pro-

posed robust facility layout framework, the firms can provide efficient solutions

for their dynamic layout problem. Moreover, the framework presented in this

paper can easily be implemented on a personal computer and offers a system-

atic guidance to the decision makers in planning the dynamic layout design

under fuzzy environment. In the future research, this approach could be devel-oped towards finding a scientific method. This study�s nature includes an

appropriate peculiarity for this development procedure. Especially, a real

case-study indicated the effectiveness of the existing framework raises the value

of this research from the point of view of practicability and supports aspect

being scientific of the research in the future.

Acknowledgments

The authors would like to acknowledge Managers of Sert Plastic Profile

Industry Co for supporting this research and providing all facilities necessary

to get required data.


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