# Fuzzy resource-constrained project scheduling using taboo search algorithm

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Fuzzy Resource-Constrained ProjectScheduling Using Taboo Search AlgorithmOmer Atli,1 Cengiz Kahraman21Aeronautics and Space Technologies Institute, Turkish Air Force AcademyYesilyurt, Bakirkoy, Istanbul 34149, Turkey2Department of Industrial Engineering, Istanbul Technical University, Macka,Istanbul 34367, Turkey

This paper proposes a mathematical model to deal with project scheduling problem under vague-ness and a framework of a heuristic approach to fuzzy resource-constrained project schedulingproblem (F-RCPSP) using heuristic and metaheuristic scheduling methods. Our approach is verysimple to apply, and it does not require knowing the explicit form of the membership functions ofthe fuzzy activity times. We first identify two typical activity priority rules, namely, resource overtime and minimum slack priority rules. They are used in the F-RCPS problem and in the initialsolution of Taboo search (TS) method. We improved the TS algorithm method for the solutionof F-RCPSP. Our objective is to check the performance of these rules and metaheuristic methodin minimizing the project completion time for the F-RCPS problems. In our study, we use trape-zoidal fuzzy numbers (TraFNs) for activity times and activity-on-nodes (AON) representation andcompute several project characteristics such as earliest, latest, and slack times in terms of TraFNs.The computational experiment shows that the performance of the proposed TS is better than theevaluation and light beam search algorithms in the literature. C 2012 Wiley Periodicals, Inc.

1. INTRODUCTION

Scheduling is deemed to be one of the most fundamental and essential basesof the project management science. With the globalization of the market econ-omy, competition between enterprises is becoming increasingly fierce; the sizesof projects are becoming increasingly large and the requirements of the projectmanagement are becoming increasingly high. The three variables plan and controlactivities, resources, and time effectiveness are the keys to ensure the project success.Project scheduling problem (PSP) as the core content of project management hasbeen extensively studied in recent decades, but its complexity, dynamic random-ness, fuzziness, and multiobjective attributes have not yet solved by a systematicmethod or theory. There is a big gap between the theoretical research and practical

Author to whom all correspondence should be addressed: e-mail: atliomer@gmail.com.e-mail: kahramanc@itu.edu.tr.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 27, 873907 (2012)C 2012 Wiley Periodicals, Inc.View this article online at wileyonlinelibrary.com. DOI 10.1002/int.21552

874 ATLI AND KAHRAMAN

application of PSP. This paper discusses the F-RCPS problems that exist widely inconstruction, software development, aircraft and ship manufacturing, designing acomputer system, planning a military invasion, introducing a new product, and othersingle- or small-batch production mode of the enterprise.

In the real world, many RCPS problems are often inherently uncertain dueto the vagueness of activity duration times. This uncertainty should be consideredin many realistic RCPS approaches. Traditionally, the uncertainty was handled bystochastic approaches using a probabilistic-based program evaluation and reviewtechnique (PERT) method. This kind of uncertainty is associated with randomness.However, in many situations, it is impossible to get the distribution of probabil-ities of activity duration times because such projects may not have been carriedout previously. The duration times have to be estimated by a Decision-Maker orProject Manager (DM/PM). Such human expertise often includes ambiguous infor-mation, which cannot be modeled using the probabilistic approaches. Some stochas-tic scheduling models are usually computationally too expensive and theoreticallytoo complex. Therefore, it is difficult to apply them for solving practical schedulingproblems. During project execution, however, duration time of project activities isalways vague in practice. Traditionally, the vagueness of activity duration time isassumed as randomness in PSP. However, sometimes activity duration time cannotbe described as a random variable. For instance, probability distributions for someactivity duration may be unknown due to the lack of statistical data. In this case,the fuzzy set theory may be a useful tool to study the PSP. However, little researchhas been performed on extending the RCPS problem with the fuzzy set theory. Afundamental approach to solve these problems is to apply fuzzy sets. Introductionof the fuzzy set theory by Zadeh in 1965 opened promising new horizons to dif-ferent scientific areas such as project scheduling. Fuzzy theory, with presumingimprecision in decision parameters and utilizing mental models of experts, is anapproach to adapt scheduling models into reality. The fuzzy set theory has provento be an effective way of handling such vague information. In addition, F-RCPSproblems are one of the nondeterministic polynomial time (NP)-hard classes due tothe complexity of the combinatorial nature. For even moderately sized problems,obtaining an optimal solution in reasonable computational time can be very difficult.However, heuristic-based approaches can produce a reasonable solution to F-RCPSproblems.

Traditionally, PERT, the critical path method (CPM), mathematical program-ming methods and heuristics have been applied to solve PM decision problems. Inparticular, when any of these models are used, the goals and model inputs are typi-cally assumed to be deterministic/crisp. In real-life PM decision problems, the inputdata or model parameters, such as resource demands and related cost coefficientsof the objective function and constraints, are frequently imprecise/fuzzy becausesome related information is incomplete or unavailable. Conventional heuristics andmathematical programming schemes clearly cannot solve all fuzzy programmingproblems. Hence, a new procedure based on Taboo search algorithm (TSA) andpriority heuristics is also developed for efficiently determining these decision vari-ables. In this method, activity duration times are described as fuzzy variables andresource-constrained software project or product scheduling problems are described

International Journal of Intelligent Systems DOI 10.1002/int

FUZZY RESOURCE-CONSTRAINED PROJECT SCHEDULING 875

as fuzzy programming problems. A new optimal F-RCPS model is proposed inthis paper. The fuzzy set theory is used to model the uncertainties of activity du-rations. A Taboo algorithm-based searching technique is adopted to search for thefuzzy optimal project duration under resource constraints. Most of RCPSP be-longs to NP-hard problem and it is hard to solve. The deterministic/crisp RCPSPand also F-RCPSP consists of scheduling activities on renewable resources avail-able in limited quantities. A classical objective function consists of optimizing theend date of the project, while respecting at the same time the precedence con-straints between activities and the resource constraints, i.e., at every time, the sumof the resource consumptions for the activities in process should not exceed theresource capacity. The environment of the project is dynamic. There is a vari-ety of uncertainties that make the traditional project scheduling model no longerreliable.

In this paper, a real case of a project schedule is considered as a fuzzy RCPSproblem. Most of these activity duration times have to be estimated by the DM byusing his/her knowledge and experience. This method is developed based on a num-ber of assumptions and definitions in fuzzy sets and project scheduling. In the fuzzyproject network considered in this paper, we assume that the durations of activi-ties are represented by trapezoidal fuzzy numbers (TraFNs). As activity durationsare estimated by human experts, sometimes under unique circumstances, projectmanagement is often confronted with vague and imprecise judgmental statementsthat are. For example, the duration of an activity may be expressed as more than2 days and less than 5 days. In those situations, which involve imprecision ratherthan uncertainty, the fuzzy scheduling literature recommends the use of fuzzy num-bers for modeling activity durations, rather than stochastic variables. The projectcharacteristics such as fuzzy earliest times and fuzzy project completion time arecalculated as TraFNs by forward pass and backward pass in fuzzy environment.In this paper, we proposed a modified Taboo search algorithm (TSA) to solve theRCPSP under fuzziness. The proposed approach aims to minimize the total projectcompletion time. An extensive experiment was conducted and the computationalresults show that the algorithm is effective for the proposed problem when comparedwith other algorithms. The advocates of the fuzzy activity duration approach arguethat probability distributions for the activity durations are unknown due to the lackof historical data.

The rest of the paper is organized as follows. In Section 2, we present a wideliterature summary and gap analysis for fuzzy project scheduling. Section 3 describesthe details of the problem and the assumptions. Then we describe fuzzy arithmeticand fuzzy ranking methods. In Section 4, we briefly describe project scheduling andgive the extended fuzzy CPM/PERT approaches. Section 5 develops the fuzzy linearprogramming models and procedures for solving the fuzzy RCPSP. Section 6 coversthe fuzzy heuristic approach and the Taboo algorithm approach. Section 7 presentsan example of an electronic product development process. And then we discuss theresults and findings for the practical application of the proposed approach. Finally,in the last section, we present concluding remarks together with suggestions aboutfurther research.

International Journal of Intelligent Systems DOI 10.1002/int

876 ATLI AND KAHRAMAN

2. LITERATURE

Scheduling concerned with fuzziness is still a new and challenging field withonly a limited number of published papers. In the following, we briefly summarizethe related literature. A method that utilizes fuzzy sets in a project network analysisis presented by several researchers, for example, Dubois and Prade,13 Chanasand Kamburowski,4 Chanas,5 Chanas and Zielinski,6 Buckley,7 Mares,8 Gazdik,9McCahon and Lee,10 Tsujimura et al.,11 Slyeptsov and Tyshchuk,12 Liang and Han,13Chen and Hsueh,14 and Nasution.15,16

Dubois and Prade13 calculate the latest allowable start time in the backwardpass. They argue that the imprecision must accumulate and there is a risk of countingit twice in the course of calculating the latest allowable start time. The methodthat Chanas and Kamburowski4 developed is analogous to CPM or PERT, but,because activity durations are fuzzy sets on time space, they called the method fuzzyPERT. Its goal is to determine the fuzzy project completion time; thus there wasno mention of backward pass and identification of a critical path. Chanas5 statesthat the fuzzified backward pass cannot be performed. To execute the backwardpass, he proposes to initiate the calculation with independently fixed deterministicor fuzzy target time. He defines the degree of criticality of an event. Chanas andZielinski6 propose a method to undertake critical path analysis of the network withfuzzy. Mares8 introduce the fuzzification of not only durations of the activities, butalso the structure of the network. He also considers the general case, i.e., whenall components of the network are vague simultaneously. Gazdik9 hints at variousfuzzifications of the network; however, he does not elaborate, except for the fuzzyduration of the activity. No backward pass is performed. He suggests that the criticalpath be found by enumerating all possible paths. McCahon and Lee10 performforward and backward calculation and propose the application of the comparisonmethod, instead of using the extended max and extended min. Tsujimura et al.11consider the case where the triangular fuzzy numbers to be used in the networkwere supposed to be given by several experts; thus these numbers were greatlydifferent. For computation they determine, according to some criteria, the so-calledmajor and minor triangular fuzzy numbers for each activity time. Slyeptsov andTyshchuk12 present an efficient computation method of fuzzy times for the lateststart and finish times of operations in the fuzzy network problems. The fuzzy settheory is used to tackle problems where a source of vagueness is involved. Linguisticterms can be properly represented by the approximate reasoning of the fuzzy settheory.

A method that utilizes fuzzy sets in an F-RCPSP is presented by some re-searchers, for example, Hapke et al.,17 Hapke and Slowinski,18 Hapke andSlowinski,19 Lorterapong,20 Ozdamar and Alanya,21 Wang and Fu,23 Wang,24 andAtli and Kahraman.25 We summarize these works in the following.

The study of a fuzzy model of resource-constrained project scheduling (F-RCPS) was initiated in Hapke et al.17 and Hapke and Slowinski.18 They haveextended the priority rule-based serial and parallel scheduling schemes to deal withfuzzy parameters. Hapke and Slowinski18 applied 12 dispatching rules with thefuzzy set theory to generate a set of schedule and selected the schedule with the

International Journal of Intelligent Systems DOI 10.1002/int

FUZZY RESOURCE-CONSTRAINED PROJECT SCHEDULING 877

minimum fuzzy makespan. Hapke and Slowinski19 discussed the application of sim-ulated annealing for solving the multiobjective fuzzy resource-constrained projectscheduling problem (F-RCPSP). Lorterapong20 presented heuristic-basedapproaches applying fuzzy sets to deal with fuzzy activity duration times of aproject. The procedure is an adaptation of the Pareto simulated annealing proceduredeveloped by Czyzak and Jaskiewicz22 for solving crisp multiobjective combina-torial problems. The procedure has been incorporated in an integrated softwarepackage.19 For details, we refer to Hapke and Slowinski.19 A typical problem wherethe use of fuzzy modeling is recommended by Ozdamar and Alanya21 is softwaredevelopment. A project involving the customization of a software application basedupon the needs of a client is unique and involves more risk compared to repeti-tive standard tangible projects. Ozdamar and Alanya21 illustrated the use of fourpriority-based heuristics: the standard minimum slack (MinSlack) rule, the latestfinish time rule, the maximum number of immediate successor rule and a minimumrisk rule on a case study. The authors mention that a fuzzy logic approach to activitymodeling will allow the project manager to be provided with a range of scenariosrather than a single one in the preplanning phase. As a second motivation for usingfuzzy numbers rather than stochastic variables, the authors note that they are dealingwith subjective evaluations of human behavior-related quantities. The authors usea six-point membership function, which allows for easy computation of the sum ofthe...

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