Download - Fuzzy resource-constrained project scheduling using taboo search algorithm

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: [email protected]: [email protected].

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.



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



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 activity durations on a given path.26 This approach has the advantage of beingrobust, intuitive, and efficient in computation with respect to its NP-hardness.

The main efforts of Wangs paper are as follows: Wang24 proposed the conceptof schedule risk based on possibility theory3 and developed an efficient fuzzy beamsearch algorithm to determine the schedule with the minimum schedule risk. Theobjective of this research is to develop a robust scheduling methodology basedon the fuzzy set theory for uncertain product development projects. The imprecisetemporal parameters related to the project are represented by fuzzy sets. A measure ofschedule robustness based on qualitative possibility theory27 is proposed to evaluatethe schedule performance. The proposed approach provides an alternative way thatevaluates the risk of poor schedule performances for uncertain product developmentprojects. Wang24 developed a fuzzy set approach to schedule product developmentprojects having imprecise temporal information. The objective is to determine astart time for each activity such that the fuzzy ready time, deadline, precedence, andresource constraints are satisfied. As the constraints are fuzzy, they are flexible inthe sense that their satisfaction depending on the choice of start times are degreeswithin the range [0, 1]. In addition, it may not be possible to find an acceptableschedule that partially satisfies all the constraints. Wang developed a new methodbased on possibility theory to determine the satisfaction degrees for the constraints.A beam search procedure, based on the generation of groups of activities and thedelay of which resolves the resource conflicts (i.e., delaying alternatives28), thatselects only the most promising nodes at each level of the search tree (the so-calledbeamwidth) for further expansion is developed to produce a set of fuzzy start timesfor each activity. Then, the crisp start time of each activity is determined based onpossibility theory, to maximize the satisfaction degrees of all fuzzy constraints. This



again illustrates our statement given above that a fuzzy schedule contains multiplecrisp schedules, the choice between which is at the discretion of management. Wangvalidated the procedure against a fuzzy version of the algorithm29 on 30, 60, and 90activity problems taken from the PSLIB test library.30

Atli and Kahraman25 used a resource allocation model to solve the PSP un-der fuzzy environment. They employed the uses of kangaroo algorithms and thefuzzy set theory to develop the F-RCPS model under uncertainty. They proposeda mathematical model to deal with PSP under vagueness and presented the frame-work of a heuristic approach to fuzzy RCPSP using a fuzzy parallel kangaroo andMinSlack scheduling method. We adopted the parallel kangaroo algorithm methodto F-RCPSP.

3. DEFINITIONS AND ASSUMPTIONS

In this section, some basic notions of the area of fuzzy theory that have beendefined by Dubois and Prade3 and Zimmermann31 are introduced. Then, projectnetwork is defined as a directed and acyclic graph under fuzziness. A fuzzy setcan be mathematically constructed by assigning to each possible individual in theuniverse of discourse a value representing its grade of membership in the fuzzyset. This grade corresponds to the individuals similarity to the concept representedby the fuzzy set. The fuzzy number A is a fuzzy set whose membership func-tion A(x) satisfies the following conditions: (i) A(x) is piecewise continuous; (ii) A(x) is a convex fuzzy subset; and (iii) A(x) is the normality of a fuzzy subset,implying that for at least one element x0 the membership grade must be 1, i.e., A(x0) = 1.

DEFINITION 1. Let R be the space of real numbers. A fuzzy set A is a set of orderedpairs{(x, A(x))|x R}, where A(x) : R [0, 1] and is upper semicontinuous.Function A(x) is called membership function of the fuzzy set.

DEFINITION 2. A convex fuzzy set, A, is a fuzzy set in which: x, y R, [0, 1], A (x + (1 ) y) min

[ A(x), A (y)

].

DEFINITION 3. A fuzzy set A is called positive if its membership function is such that A(x) = 0, x 0.

DEFINITION 4. TraFN is a convex fuzzy set, which is defined as: A = (x, (x)).Let a TraFN A be represented as a fuzzy number with membership function in



Figure 1. TraFN for project network activity.the form

A(x) =

x ab a , a x < b,1, b x < c,d xd c , c x < d,0, otherwise

(1)

is called a TraFN A = (a, b, c, d). Then for convenience, TraFN represented byfour real parameters a, b, c, d (a b c d) will be denoted by a trapezoidal(a,b, c, d) (see Figure 1).

DEFINITION 5. A TraFN A(a, b, c, d) is called positive TraFN if: (0 a b c d)

Centroid ( A) = (c2 + d2 + cd) (a2 + b2 + ab)

3[(c + d) (b + a)] . (2)

A TraFN may be represented by means of combinations of two triangular fuzzynumbers ( A1 and A3) and a single rectangular fuzzy number ( A2) (Figure 2).

We may write

Centroid ( A1) = a + 2b3 (3)

Centroid ( A3) = 2c + d3 (4)

Centroid ( A2) = b + c2 (5)

The value of centroid ( A) may be calculated by means of Equations 35.

3.1. Operation on TraFNs and Fuzzy Arithmetic

Dubois and Prade3 defined a number of operations on TraFNs. The followingare employed operations in the development of the proposed method. The most



1

0 a b c d x

2A

3A1A

A

( )A X

Figure 2. Membership function of TraFN A.

often used formulas of addition, subtraction, maximum, and minimum are in thefollowing, and suppose that two TraFNs are defined as A = (a1, a2, a3, a4) and B =(b1, b2, b3, b4). Applying the extension principle, the extended algebraic operationsof any two fuzzy activity times (FATs) are expressed as

Fuzzy addition: A B = (a1, b1, c1, d1) (a2, b2, c2, d2)

= (a1 + a2, b1 + b2, c1 + c2, d1 + d2) (6)

Fuzzy subtraction: A B = (a1, b1, c1, d1) (a2, b2, c2, d2)

= (a1 d2, b1 c2, c1 b2, d1 a2) (7)

Fuzzy maximum: max{ A, B} = [max{a1, b1}, max{a2, b2},

max{a3, b3}, max{a4, b4}] (8)

Fuzzy minimum: min{ A, B} = [min{a1, b1}, min{a2, b2},

min{a3, b3}, min{a4, b4}] (9)

A fuzzy algebra with i-level cut method, instead of fuzzy subtraction method,is proposed to avoid the inflation and unreasonable negative completing time. LetBi Ci = Ai to find Ci .

Bi Ci = [biL + ciL , biR + ciR ] = Ai = [aiL , aiR ], i [0, 1] (10)Ci = [aiL biL , aiR aiR ] (11)

The result of max{ A, B} and min{ A, B} from previous maximum operator for-mula is still a TraFN, but the real membership function of max{ A, B} and min{ A, B}may not be a TraFN anymore. In this paper, max i-level cut method is proposed to



Figure 3. Maximum Result of Ri = maxiP

[Ai1 ,

Ai2 ,

Ai3].

obtain more reasonable membership of earliest starting times and latest completingtimes. The operators are defined as

max{ Ai , Bi } = [max {aiL , biL }, max {aiR , biR }], i [0, 1] (12)min{ Ai , Bi } = [min {aiL , biL }, min {aiR , biR }], i [0, 1] (13)

Suppose m fuzzy numbers are Am, m = 1, 2, . . .M , the membership valuesof Am at i-level cut will be Aim = [ AimL, AimR], i [0, 1], m = 1, 2, 3, . . . ,M.Comparing all i-level cut values of Am fuzzy numbers at i-level and takingthe maximum value at each i-level cut. The set of maximum value is Ri =[ RiL , RiR ] = max

m{1,2,...,M}iP[ AimL, AimR]. An example of max i-level result Ri

with three fuzzy numbers is illustrated in Figure 3.

3.2. Ranking Trapezoidal Fuzzy Numbers

The ranking of fuzzy numbers is another important issue in the scheduling.Although many ranking methods have been proposed to date, there is no singleapproach that can produce a satisfactory result in every situation: some may generatecounter intuitive results and others are not discriminative enough.18 Ranking fuzzynumbers applied in fuzzy CPM/PERT are used to determine the earliest startingtime. Techniques for ranking fuzzy numbers are abundant in the literature.4 Let A =(a1, a2, a3, a4) and B = (b1, b2, b3, b4) are two TraFNs. If a1 b1, a2 b2, a3 b3, and a4 b4, the ranking of A and B is said that A is strongly greater than B. If oneof these four inequalities is not true, the comparison rule has to take the advantage ofweak comparison rule (WCR). The rule is so-called defuzzifying ranking method.But using a defuzzifying ranking method to obtain the maximum fuzzy number,the comparison result is the maximum value of fuzzy numbers participating incomparison. It cannot fully express the character of two or more fuzzy numbers. In



this study, we propose max i-level cut method. The result of this method seemsto be more reasonable than the defuzzying ranking method. Ranking methods areessential in fuzzy CPM (FCPM). Many methods of ranking fuzzy numbers havebeen proposed. However, certain shortcomings in some of those methods have beenreported in the literature. In those methods, the membership function should beknown. Our approach is very simple to apply, and it does not require knowing theexplicit form of the membership functions of the FATs. This section is about rankingtwo fuzzy numbers. For two real numbers, there is no problem in ordering themfrom smallest to largest. However, in the fuzzy case there is no universally acceptedway to do this. There are probably more than 40 methods proposed in the literatureof A B defining, for two fuzzy numbers A and B. A few key references on thistopic are given in Yager.32 He proposed a procedure for ordering fuzzy sets, inwhich a ranking index I (t) is calculated for the convex fuzzy number t from its cut t = [t l, tu ] according to the following formula:

I (t) = 1

0

12(t l + tu

)d (14)

For ranking fuzzy temporal parameters, it is required to compare fuzzy temporalparameters to generate a feasible schedule. For example, the start time assignmentfor an activity should satisfy the precedence constraints; i.e., the value that canbe assigned to the start time of an activity should be greater than or equal to thefinish times of all its predecessors. A number of approaches have been developed todetermine the ranking order between two fuzzy numbers in the literature.33,34 Thispaper applies the approach developed by Liou and Wang35 to derive the order amongfuzzy temporal parameters, due to its discriminating ability. It is also particularly easeof computation for the TraFN. Let Ai = (aij , bij , cij , dij ) be the FAT of activityAi .The DMs risk attitude index can be obtained by

=

i

j

(bij aij

)(bij aij

)+ (dij cij )AijACT

/t (15)

where ACT and t denote the set of all activities and the number of activities in anetwork. For a fuzzy number Ai with membership functions fAi (x), we define,

mi = min{x|fAi (x) = 1} + max{x|fAi (x) = 1} (16)

which is the center of the mean value of t . Considering two fuzzy numbers A andB, the case of I ( A) I ( B) shows that A B, and then max ( A, B) = A.

Where Ai and Aj are any fuzzy numbers in the set, the comparison of fuzzy nu-mbers has the following properties when obtaining ranking indices by Equation 14:

If R( Ai) > R( Aj ) Ai > Aj (17)International Journal of Intelligent Systems DOI 10.1002/int


If R( Ai) = R( Aj ) Ai = Aj (18)If R( Ai) < R( Aj ) Ai < Aj (19)

Then, the ranking value R (Ai) of the TraFN Ai can be obtained as follows:

R(Ai) = [ (di x1)

(x2 x1 ci + di)]+ (1 )

[1 (x2 ai)(x2 x1 + bi ai)

](20)

where is the DMs risk attitude index, x1 = min{a1, a2, . . . , an} and x2 = max{d1,d2, . . . , dn}. by using Equation 15 and taking the value calculated by Equation 20.One can easily calculate the ranking values of n TraFNs. On the basis of the rankingrules described above, the ranking of n TraFNs can then be effectively determined.

3.3. The Weak Comparison Rule

Assume that the expert expresses his/her optimistic and pessimistic informationabout parameter uncertainty on some prominent membership levels:

= 1; (x) = 1 Value x certainly belongs to the set of possible values; = ; > The expert estimates that value x with (x) has a good chance of belonging to the

set of possible values; = ; (x) < Value x with (x) < has a very small chance of belonging to the set of possible values;

i.e., the expert is willing to neglect the corresponding values of x with (x) < .

A TraFN defined as in Equation 1, i.e., A = {(x, A(x))|x X} where A isthe membership function of A is a special kind of a fuzzy set defined as a normalizedconvex fuzzy subset of real line R, i.e., X = R. Using the four-point convention, aTraFN A is then represented by the following list of symbols.

Let A = (a1, a2, a3, a4) and B = (b1, b2, b3, b4) be fuzzy numbers and letSL( A B), SR( A B), SL( A B).SR( A B) be the areas determined by theirmembership functions (Figure 4):

SL( A B) =U ( A, B)

( infxR

A infxR

B)d (21)

SR( A B) =V ( A, B)

(supxR

A supxR

B)d (22)

where

U ( A, B) = {| infxR

A infxR

B, 0 1} (23)

V ( A, B) = {| supxR

A supxR

B, 0 1} (24)

SL( A B) and SR( A B) have similar definitions. SL( A B) and SR( A B)are the areas that are related to the possibility of A B, where A and B arepotential realizations of A and B. The degree to which A B, denoted asC( A B),



( )x

( )LS A B ( )RS A B

( ) .LS B A R ( )S B A

A B

1a 2a 3a 4a1b 2b 3b 4b

1 =

0 =

Figure 4. The comparison of two TraFNs based on area compensation.

is computed as the sum of the areas in which A B minus a sum of areas in whichA B (Figure 4).

In the WCR, the degree C( A B) is defined as

C( A B) = 12 {SL( A B) + SR( A B) SL( B A) SR( B A)} (25)

It is now possible to define weak inequality, strict inequality, and equivalencebetween fuzzy numbers:

A B iff C( A B) 0 (26)A > B iff C( A B) > 0 (27)A B iff C( A B) = C( B A) = 0 (28)

Obviously, C( A B) = C( B A). Fortemps (1997) showed that the degreeC( A B) can be computed as

C(A B) = ( A) ( B) = E(A) + E( A)

2 E(

B) + E( B)2

(29)

where the mean value of a fuzzy number A is the interval E( A), E( A), E( A)and E( A) are the mean values related to the cumulative possibility function (leftspread), and to the cumulative necessity distribution (right spread), respectively. Thedefuzzification function ( A) for fuzzy number A in the four-point representation



x

1

0

( )x 1f 2

1t

Figure 5. Example of an infeasible fuzzy schedule.

can be computed as follows18:

( A) = 14(1 ) {( )(a1 + a4) + (1 )(a2 + a3)} (30)

3.4. The Strong Comparison Rule

Hapke and Slowinski19 give the following example to illustrate that the WCRmay lead to an unfeasible fuzzy schedule. Assume that the scheduling procedureconsiders an allocation of a resource to activity 2 in its ready time 2. Activity2 has a single predecessor activity 1, completed in f1. The WCR for checking ifactivity 1 has been completed by ready time 2 would yield the relation 2 f1 tobe true. For a very pessimistic realization of the duration of activity 1 (beyond t1),the resulting schedule would be infeasible because activity 2 would start before itspredecessor activity 1 was completed, a violation of the precedence constraints. Asa result activity 2 cannot be scheduled at 2 (see Figure 5). The strong comparisonrule (SCR) specifies that A = max ( A, B) = A.

4. PROJECT SCHEDULING AND THE EXTENDED FUZZY CPMAPPROACH

In this PSP, some major assumptions are made as follows: A project has itemsof N activities. The precedence or succeed relations for each activity are available.Resources quantity K for each activity is available. DM can get the informationabout the capacity of resources. The fuzzy operation time of each activity can berepresented as a TraFN, A = (a1, a2, a3, a4). The traditional CPM is an effectivetool frequently used to identify a critical path of a project while solving F-RCPS



problems. In the fuzzy CPM, fuzzy operations are employed to calculate the fuzzyearly start (FESTi) and the fuzzy late completion (FLCTi) times of an activity Aithrough the forward and backward pass in the project network. The CPM techniqueprovides for the computation of the network critical path, which consists of thesequence of project activities that determine the minimum required project time. Toidentify the critical path, first three characteristics of each event are determined: theearliest expected start time in the forward pass (from start toward end event); thelatest allowable start time TLi in the backward pass (from end toward start event);and the slack (float) of each event i. The slack of an event i is the possible delay inthe realization of event i that causes no delay in total project duration. The criticalpath is the path from start event to end event where the slacks of the events are allzeros. An increase in the duration of any activity along the critical path will certainlyincrease the total project duration. Besides event slack, the slack can be calculatedfor activity a = (i, j). Each of the events i and j of activity (i, j) possesses two possibletime values. Therefore, four relations are possible. Two of them are the total slackand free slack. The total slack is calculated as the amount of time the activity can bedelayed without impacting the latest allowable start time of subsequent activities.The free slack is calculated as the amount of time the activity can be delayed withoutimpacting the early start time of subsequent activities.

In this section, we use the extended fuzzy CPM/PERT approach to createthe computing procedure model project scheduling and risk index. The most oftenused operators for fuzzy PERT are addition, subtraction, maximum, minimum, andranking. Addition operator is applied to calculate the earliest completing times andoverall project completing time. Subtraction operator is used to compute the lateststarting and completing times. Maximum operator is applied for earliest starting andcompleting times. Minimum operator is applied for latest starting and completingtimes.

In this study, the FAT, denoted by FATi , of activity Ai in a project network isrepresented by the TraFN FATi = (aij , bij , cij , dij ), where aij and dij are minimumand maximum values of assessing activity time for Ai , whereas bij and cij are thefirst quartile and third quartile of activity time for Ai . If there is only one set of fourhistorical data, then aij , bij , cij , and dij can be sorted from minimum to maximum.Conversely, if one has no further information with respect to activity Ai , the FATFATi = (aij , bij , cij , dij ) can be evaluated subjectively by the DM based on his/herknowledge, experience, and subjective judgment. We will first give the notation:N The set of all nodes in a project network.Ai The activity node i.FATi The FAT ofAi .FESTj The earliest start fuzzy time of node j.FECTj The earliest completion fuzzy time of node j.FLSTj The latest start fuzzy time of node j.FLCTj The latest completion fuzzy time of node j.FTSi The total slack fuzzy time ofAi .S(j ) The set of all successor activities of node j.NS(j ) The set of all nodes connected to all successor activities of node j, i.e., NS(j ) = {k|Ajk

S(j ), k N}.



P (j ) The set of all predecessor activities of node j.NP(j ) The set of all nodes connected to all predecessor activities of node j, i.e., NP (j ) = {i|Aij

P (j ), i N}.PTi The ith path.PT The set of all paths in a project network.FCPM(Pk) The total slack fuzzy time of path Pk in a project network.

The important properties and theorem used in the fuzzy CPM are briefly in-troduced in the following. Set the initial node to zero for starting, i.e., FESTS =(0, 0, 0, 0) . In this study, we use AON representation for F-RCPS (Atli andKahraman 25). The computing procedure model is as follows:PROPERTIES 1a. Compute the earliest starting fuzzy time for each activity (FEST).The important properties of FCPM are as follows. First we set the initial arc tozero for starting, i.e., FESTS = (0, 0, 0, 0). Then, the following properties are true.Suppose there are N items in total project network, the first starting item is sand last completing item is e, the earliest starting time with a TraFN for item isFESTS = (0, 0, 0, 0). The earliest starting fuzzy time for each activity n is

FESTj = max{FESTi FATij |i NP (j ) , j = 1, j N

} (34)PROPERTIES 1b. Compute the earliest completing fuzzy time for each activity FECTi .We apply the equation FECTi = FESTi FATi , to compute the earliest completingfuzzy time for activity n.

FECTi = FESTi FATi (35)

PROPERTIES 2a. Compute the latest completing fuzzy time for each activity (FLCTj ).

To avoid the fuzzy number extending and unreasonable negative value afterfuzzy number subtraction operator, fuzzy algebra method is proposed to instead offuzzy subtraction method for computing the fuzzy latest dates of each activity.

FLCTj = min{

FLCTkFATjkk NS(j ),j = n, j N

}(36)

PROPERTIES 2b. Compute the latest starting fuzzy time for each activity (FLSTj ).Here, also use fuzzy algebra method to compute the latest starting fuzzy time foreach activity. The computing procedures are

FLSTj = FLCTjFATj (37)

We have computed the overall completing fuzzy time of total project. Theearliest and latest completing fuzzy time is the same for the last item activity in



project. The latest fuzzy completing time of the end item activity is based on theformula of forward method.

PROPERTIES 3. Compute the total slack fuzzy time for each activity (FTSij ).

FTSij = FLCTj(FESTi FATij ), 1 i < j n; i, j N (38)

FTSj = FLCTjFECTj = FLSTjFESTj (39)

PROPERTIES 4. Compute project scheduling risk index (PSRI). The definition of PSRIis the possibility which overall project completing fuzzy time is longer than projectcontract time. The relationship between the overall project completing fuzzy timeand project contract time is shown. We have derived the overall project completingfuzzy time. If project contract time is a crisp value, the PSRI can be computed asfollows:

FCPM(PK ) =

1 i < j ni, j PK

FTSij , PK PT (40)

In a project network, a path PC such that

FCPM(PC) = min{FCPM(PTi)|PTi PT} (41)

is a fuzzy critical path. In this section, a fuzzy critical path algorithm is utilized tofind a critical path of a project network in a fuzzy environment. The flowchart of thealgorithm is presented in Figure 6.

5. FUZZY RESOURCE-CONSTRAINED PROJECTSCHEDULING PROBLEM

While solving the F-RCPS problem, the following assumptions are taken intoconsideration: The processing time of each activity should be predefined. All theproject activities and specified project completion time are imprecise/fuzzy. A singleproject consists of a number of activities with fuzzy duration. The DM has alreadyadopted the pattern of TraFNs that determine the specified project completion timefor all activities. The precedence relationship among the activities should be fol-lowed, i.e., any activity cannot be completed without performing its precedence ac-tivity. The precedence relationship shows the relation between two activities, whichare predecessor and successor to each other. The maximum numbers of resourcesavailable should be predefined in limited quantities but renewable from period toperiod. Activities cannot be interrupted because there is only one execution modefor each activity. The main objective is to minimize the project duration. The mainobjective is to minimize the completion time or the makespan of the project.



START

Identify activities in project.

Establish precedence relationships of all activities.

Estimate the fuzzy activity time with respect to each activity by DM.

Construct project network.

Read Dataj = 1,2,3...Jr = 1,2,....,R

S( j ), S( j,s ), d( j ), K( r ), K( j,r )

Calculate project risk indeks for fuzzy CPM algorithm

Calculate Fuzzy CPM ValuesFEST, FECT, FLST, FLCT, FST

(SLACK)

Identify Critical and Non-Critical Activities and

Project Fuzzy Makespan

End

Figure 6. Fuzzy critical path algorithm.

5.1. Mathematical Formulation of F-RCPSP Model

In F-RCPSP, the activities are interrelated by two kinds of constraints. First,the precedence constraints which force an activity not to be started, before all itspredecessors have been finished. Second, processing of the activities is subject to theavailability of resources with limited capacities. While being processed, the activityj in project requires k units of resource k K during every period of its processingfuzzy time pj . Resource type k has a limited availability rjk at any point in time.Parameterspj , rjk,and k are assumed to be nonnegative, and all are integer. For the



start and end activities of a project, we have p0 = pn+1 = 0r0,k = rn+1,k = 0

}for all j I, where

i stands for node,jstands for activity index, and r stands for resource index. In thisformulation, the use of a dummy start as well as a dummy end activity is assumed.The set St denotes the set of activities that are in progress at time t . The study of anF-RCPS has been initiated in Hapke et al.36 and Hapke and Slowinski.16 They haveextended the priority rule-based serial and parallel scheduling schemes to deal withfuzzy parameters. Let us introduce some required notation:t Current time instant at which the allocation of resources to an activity is considered.A Set of activities to be scheduled.i Fuzzy ready time of activity i A.d Fuzzy duration of activity i A.s Fuzzy start time of activity i A.fi Fuzzy finish time of activity i A.E(t) Set of eligible activities at time t , i.e., the set of activities that have not yet been scheduled and

whose immediate predecessors have been completed by time t .E, (t) The subset of set E(t) of activities for which t , i.e., which are not ready at time t .S(t) The set of activities for which sl = t , i.e., which have been scheduled before (the notation

= refers to the strong comparison rule that was discussed in Section 3.4); setB = { fi : i S(t)} {i : i E, (t)}.

l The smallest value from the set B based on weak fuzzy number comparison.pj The processing time of activity j in project.rjk The scheduling activity j in project consuming resource units per period from resource k.k The maximum limited resource k only available with constant period availability.S The set of activities being in the progress in period p.

The definition of lis in terms of the WCR for comparing two fuzzy numbers.19We discuss the WCR in the previous section. The fuzzy time t moment is calculatedas the maximum over the current tand the least value l from the set B in the senseof the WCR. This is done to ensure feasibility of the schedule. Indeed, Hapke andSlowinski19 showed that the so-called strong comparison rule (SCR) must be appliedto ensure feasibility of schedules for every realization of fuzzy time parameters. TheSCR and an example showing that the application of WCR may lead to infeasibilitiesare given in Section 2. The mathematical models for the F-RCPSP can be stated asfollows:

Min

Ht=1

txn+1,t (42)

subject toHt=1

txj t + pi Ht=1

txit 0 (43)

nj=1

rjk

t+pi1u=t

xju

k(44)

Ht=1

xjt = 1 forj = 1, . . . , n (45)



H =v

j=1pj (46)

xjt ={1, If activity j finishes at time instant t

0, otherwise(47)

It should be clear to the reader that there is an immediate link between thebinary decision variables and the completion times of the activities. The finishingtime of activity j can be calculated by Equation 47. The objective function givenby Equation 42 seeks to minimize the total project time that is the sum of thecompletion time for all activities. It minimizes the completion time of the dummyend activity and thus the completion time of the project. Equation 45 specifies thatonly one completion time is allowed for every activity, which must lie in the intervalbetween the earliest and latest finishing time. The precedence constraints are givenby Equation 44. The completion time of any activity cannot exceed the start time(completion time minus duration) of its successor. The resource constraints for everyresource type k are specified in Equation 44 by considering for every time instant tand every resource type k, all possible completion times for all activitiesj such thatthe activity is in progress in period t . The weighted sum of the corresponding decisionvariables should not exceed the resource availability. The constraints in Equation 42are referred to limit the resource demand imposed by the activities being processed attimep to the available capacity. The constraints in Equation 43 are used to impose theprecedence relations between activities. In formulation 4247, xjt are 01 variables,which specify whether activity j finishes at time t or not. More specifically, for eachactivity j and for every feasible completion time t [EFTj , LFTj ], xjt is defined.The variable xjt can only be defined over the interval between the earliest and latestfinishing time of the activity under consideration. These limits are determined usingthe traditional forward and backward pass calculations. The backward pass recursionis started from a project length of T , which equals a feasible project length (e.g.,determined by any of the heuristic procedures).

5.2. Heuristics Based on Priority Rules

5.2.1. A Fuzzy Parallel Scheduling Procedure for Solving Problemm, 1||cpm, dj , j | Cmax Based on Priority Rules

For deterministic/crips RCPS problems heuristics based on priority rules havebeen tested by many researchers. However, there is no single priority heuristic thatalways gives a better result in every scheduling situation. It is useful to apply a setof priority rules simultaneously, to select the best result among them. In the scheme,two (resource over time, ROT and MinSlack) popular priority rules are employed,taking into account time, resources, and the successors of current scheduled activity.To employ these rules, the fuzzy early and late start times, and the fuzzy earlyand late finish times for each activity can be calculated by Equations 3440. Theheuristic methods for solving F-RCPS problems can be generally divided into serialand parallel. The serial approach derives its name from the fact that activities are



considered sequentially. The parallel approach considers in one moment all theactivities, which could be scheduled in that moment. In both approaches, the orderin which the activities are considered for scheduling is defined by a priority list.The main difference between them is the way of solving the resource conflicts. If anactivity cannot be scheduled in moment because of resource constraints, the serialprocedure looks for the earliest moment in which all the required resources areavailable. In the same situation, the parallel procedure takes another activity readyfor scheduling, with the next highest priority and tries to schedule it in momentt . Because of usually better performance, our interest is focused on the parallelprocedure. However, to extend scheduling procedures on fuzzy time parameters, oneshould solve the question how to determine the sequence of fuzzy time moments inwhich new activity (activities) is (are) to be considered for scheduling. It should bestressed that our goal is to construct a procedure that would give feasible schedulesfor every realization of fuzzy time parameters.

5.2.2. Starting Solution of the Resource-Constrained Scheduling Basedon the TS Method

In the only two kinds of methods presented here, research takes the heuristicmethod MinSlack as the starting solution of the resource constraint/allocation for theTS method. A simple heuristic method is generally used as the starting solution ofthe TS method, such as MinSlack LFT, ROT, and so on. It is observed that the rule ofMinSlack is better than other heuristic algorithms. Therefore, this research utilizesthe MinSlack and ROT rule to determine the priority of the resource-constrainedallocation. Priority given to the activity with the maximum ROT value. The ROTvalue is calculated by dividing the sum of each activitys resource requirementsby the duration of the activity and then finding the maximum ROT that an activitycontrols through the network on any one path. The flow of obtaining the startingsolution is shown in Figure 7.

5.3. Taboo Search Method and Proposed Taboo Search Algorithm

RCPSP is (nondeterministic polynomial-time) NP-Hard and F-RCPSP is alsoNP-Hard. In the early stage, during solving of the RCPSP, most researchers still usedto utilize the traditional heuristic rule to solve this problem. This rule basically tookthe mode of one by one analysis to solve problems according to the priority rulesdeveloped by other scholars. It was found that the solution quality was sometimesworse and the preferred solution was often achieved without this sort of priority.If one wanted to get the optimal solution, it may result in much longer executiontime. Therefore, the way how to obtain a near optimal solution efficiently withoutsacrificing the solution quality is important issue to investigate in solving NP-Hardproblem. Thus, the Taboo search (TS) method, a tool to solve the optimizationproblem, could be expected to be applied in this research to solve the projectmultiresource scheduling problem and obtain the optimal solution of this problem.TS have been proved that can get a good performance for solving combinatorialoptimization in many areas. Sometimes, it can obtain the better solution quality than



Figure 7. Flow chart of solving the starting solution by the MinSlack and ROT method.

some heuristic algorithms in some specified problems. The study will implementTSs concept as basis to develop a new algorithm to solve F-RCPSP in projects. Therelated study that the TS method was applied to in the scheduling optimization alsohad some achievements.



Taboo search algorithms (TSAs) have been broadly used in many areas such asplanning, scheduling and other optimization problems where conventionalmathematical techniques are inadequate. TSA was first developed by Glover in1975, based on Artificial intelligence systems. Each search iteration stands for apossible solution to a problem. After several iterations, the algorithm that hopefullyrepresents the optimal or near optimal solution to a problem can be found. TSAbelongs to the class of heuristic optimization techniques, and it is very useful whena large search space with knowledge of how to solve the problem is presented.A balance between exploitation and exploration in the search space is one of theimportant factors when using TSA. To provide this balance, determination of thedesign strategy for TSA parameters such as maximum iteration is one of the criticalissues. In this paper, we present a new approach based on the improved TSA forsolving F-RCPSP. In the following, we give the fundamentals of TSA.

5.3.1. The Neighborhood Solution Structure of the Resource ConstraintsScheduling Based on the TS Method

The neighborhood solution structure of the TS method has three commonmodes introduced in Section 3: (1) swap move, (2) insert move, (3) insert and swapmove. In this research, the objective value of the resource allocation is the priority ofeach activity of the project utilizing resources, which is a sequential array problem.The previous research presented that, for the array problem, the efficiency of theswap move was better than that of the insert move. But, during research, it wasfound that the move by using the swap move is less if the scale of the project islarge, however, an insert move is created by two swap moves and the move of theinsert move is greater than that of the swap move. Thus, this research utilizes suchtwo modes as the swap move and the insert move for the neighborhood solutionstructure of the resource scheduling problem, and the two moves are comparedwhen making a sensitivity analysis. So we are developed some swap rules for theproblem. A move (swap) is a transition from one feasible sequence to another byinterchanging the positions of two activities i and j . An objective function f isdefined as the project duration of a feasible sequence. The value of a move is thedifference between the objective function value of the feasible sequence yieldedby making the interchange and the objective function value of the current feasiblesequence. If the value of a move is negative, then the move is called an improvementmove.

The proposed method is based on a neighborhood search. This method usesmainly a local descent to avoid local locking. It enables solutions of better qualityregarding the criteria to be found. The algorithm is mainly composed of two blocks:the first one corresponds to the search of more efficient solutions inside a neighbor-hood. It consists in switching two activities stochastically chosen. If after a certainnumber of attempts, the solution is not improved the neighborhood is enlarged witha jump. The second block then corresponds to searching a new solution in a largerneighborhood to go out of local minima (as shown in the Figure 8). A jump consistsin switching two activities stochastically chosen three times. The number of searcheshas to be limited so as to limit the computing time.



Figure 8. Lateness possibility in case of conflict.

As regard to the complexity of the problem, multiobjective optimization prob-lems are very complex. The complexity plus the combinatorial aspect is a resultof there being no single optimal solution for these problems, but rather a set oftrade-offs called efficient solutions or optimal solutions.

5.3.2. Memory Structure of Taboo List for the F-RCPSP Based on the TS MethodThe memory structure of Taboo list is mainly used to memorize the several

precedent moves, i.e., to memorize the several precedent moving paths, which isdivided into long-term memory and short-term memory. At present, the simpleTS method with the short-term memory is usually applied, and there is additionalrelated research. Thus, this research utilizes the memory list with the short-termmemory. The Taboo list is mainly applied to memorize the precedent moves, andthus the memory structure of Taboo list depends on the structure of the neighborhoodsolution.

5.3.3. Termination Criterion of RCPSP Based on TS AlgorithmThe stopping rule of the TS method is flexible. If satisfying the condition that

searching objective is a criterion, the algorithm will set the stopping rule as thefollowing two conditions, either one of the following two conditions is satisfied andthen stop calculating:

1. Maximum number of computation for constraints;2. Maximum number of computation on the condition of not finding better objective value.

The Taboo list size is a parameter that is preset or determined experimentally.Our experiments have demonstrated that the better size of Taboo list should bearound activity size by

N 2. Taboo list is used to prevent a solution from

being revisited for a certain number of iterations. The activities of a project aredivided into two categories: critical and noncritical activities. An activity is classifiedas a critical activity if it is on the critical path of the project scheduled usingFCPM/FPERT. Otherwise it is a noncritical activity. A list named TaboolistC consistsof critical activities and another list named TaboolistNC consists of noncriticalactivities.

Before the scheduling process is started, the relative priority ranking of allactivities is done. The priority ranking is usually calculated with some heuristic



rules or formulas. Resources are allocated based on the priority ranking of activ-ities. A feasible activity is an activity for which the required preceding activitieshave been completed. At each scheduling instant, only the feasible activities areconsidered for resource allocation. At time t , the feasible activity with the high-est priority is scheduled if the available resources are sufficient. If the number ofresources available is inadequate for the next feasible activity at time t , then timet + 1 is considered. Note that activity preemption and partial resources assignmentsare not allowed. When resources are constrained, determination of the critical pathbecomes more difficult. These resource links are used to trace the use of resources,noting the significant resource dependencies that determine the project schedule andhence identify the critical path. We have added the use of additional links, which arereferred to as precedence links. Two unique heuristics are employed to provide start-ing solutions for TS. The MinSlack rule is very simple. The resource-unconstrainedFCPM/FPERT slack time for each activity is calculated. Then when resources areassigned, top priority is given to the activity with the smallest FCPM/FPERT calcu-lated slack time. In other heuristic rules, higher priority is given to the activity thathas a higher priority value. For each activity i, the priority rule is calculated as aweighted and scaled sum of two components, resource and time allocation factors.

Let n be the number of activities in a project, I = {0, 1, 2, 3 . . . n} be the set ofall activities, P be the set of all permutations defined on ith, any permutation p Pis defined as an n vector (p (1), p (2), p (3). . . .p (n)). When the order of activitiesin a permutation p is consistent with the precedence relationships in the project, thepermutation p is called a feasible sequence. That means activities can be completedin the same order as in the sequence. Let F be the set of all feasible sequences, thenF is a subset of P . A project can be scheduled under the resource constraints andhence the project duration can be determined. The goal of TS is to find a feasiblesequence in F , which gives the optimal or near-optimal schedule.

A starting solution is a feasible sequence obtained from a heuristic approach andthe starting solution will be improved by the TSA. As quality of a solution obtainedby TS is considerably affected by the starting solution, we use the MinSlack ruleand other priority rules to try different starting solutions. When resource constraintsare included, in general, when a critical activity is delayed, the successors of thedelayed activity will also be delayed. Consequently, the completion time of theresource-constrained project will be delayed with a high probability.

In Figure 9, a flowchart summarizing the basic steps of the proposed approachis presented. Application of TS to fuzzy activity duration PSP is implemented. TheTSA for solving the research problem is initialized with the initial population ofgenerated from ROT and MinSlack priority rules randomly.

The procedure of TSA is summarized as follows (for details, see Atli26):Step 1. Identify activities in project:

Establish precedence relationships of all activities. Estimate the FAT with respect to each activity by DM. Construct project network.



Figure 9. Flowchart summarizing the basic steps of the proposed Taboo approach.

Step 2. Read data file.

Step 3. Use fuzzy critical path algorithm (in Figure 6). Calculate project parametersand risk index for fuzzy CPM algorithm related to comparison two and more TraFN.

Step 4. Choose and use initial solution method (MinSlack and ROT priority rules inFigure 7).

Define a starting feasible sequence, denoted as S with priority rules.



Step 5. Use TSA (the basic steps of the proposed approach Figure 9).

Step 6. Initialize variables used in TS:

Create initial schedule for TS previous Step 4. At the beginning the Taboo list be empty. Select values for Taboo list size (N 2) Select values for stopping criteria maximum iterations.

Step 7. Create a candidate list that consists of number of swap moves:

Two activities are randomly selected and the positions of these two activities are inter-changed.

Proceed swap rules.

Step 8. Choose the best admissible candidate move.

Step 9. Make the best admissible candidate move.

Step 10. Check stopping criteria.

If iter < Max Iter then schedule is reported as the solution, else go to Step 11.

Step 11. Update Taboo restrictions and aspiration criteria and go to Step 6.

The solution determined using the TS also depends on the starting solution. Toexplore diversified paths so that a better solution may be found, one can repeat theprocedure described above several times with different starting solutions.

6. AN APPLICATION PROBLEM DESCRIPTION ANDMATHEMATICAL FORMULATION MODEL

6.1. Example of an Electronic Product Development

In this section, an electronic product development project is used to illustratethe approach developed in Wangs paper.25 The entire project is divided into sevenphases: system requirements analysis, system definition, requirements allocation,preliminary design, detailed design, engineering model testing and evaluation, andqualification tests. For the demonstration purpose, we simplify the entire projectwith hundreds of activities into 51 activities. In addition, a project usually involvesseveral disciplines. We aggregate these disciplines into four types: systems engineers(r1), software engineers (r2), hardware engineers (r3), and supporting engineers(r4). Hardware engineers include mechanical engineers and electrical engineers.Supporting engineers consist of the disciplines that act as a supporting role during



Figure 10. Project network scheme.

the product development; such as manufacturing, cost, logistics, test, reliability,maintainability, marketing, and so on.

Wangs paper are used a flat fuzzy number which represented by six tuples(a1, a2, a3, a4, a5, a6) while a1 and a6 are the lower and upper modal values. Andwe decided to use that illustrative example. However, a flat fuzzy number can beconsidered a special case of the TraFN where the lower and upper modal values aresame (a1,a6). But in case of the difference of that problem, we use TraFNs for activ-ities. In the scheduling, a TraFN can be represented by four tuples (a1, a3, a4, a6),where a1and a4 are the lower and upper bounds of the support of fuzzy numberA. Here we exempted a2 and a5 so a flat fuzzy numbers are transformed to theTraFNs. The project presented here is composed of 51 + 2 activities. The prece-dence relations among activities are represented by a precedence network, as shownin Figure 10. A project consists of (j + 2) activities where each activity has to beprocessed to complete the project. The (j + 2) activities compose of a set of activitiesj = {0, 1, 2 . . . j + 1}, where activities 0 (S) and (j + 1) (E) are dummy and haveno duration and no resources. These activities represent the start and end activitiesof the project. There exists a set of renewable resources that are represented byr = {1, 2 . . . k}. The illustrative project steps for installation are determined to beas shown in Table I. Any project consists of number of activities. These activitiesare represented in a network by nodes. Therefore, the project defined in Table I andall of the durations are in man-hours. The fuzzy project ready time is set to (0, 0,0, and 0). In the project, four different kinds of renewable resources are requiredthroughout the project. The amount of these resources is limited and its resourceavailability is 6, 6, 5, 4, respectively, in each period. Each activity resource usageand maximum capacitated is showed in Table I.

The results of application of FCPM to the sample project are shown inTable II.

In Wangs paper, the parameters for running the genetic algorithm were set asfollows: population size is 10, number of generations is 100, mutation rate is 0.01,



Tabl

eI.

The

corr

espo

ndin

gac

tivity

info

rmat

ion

N 524

4

66

54

Tra

FN N

o

Act

.Su

cc.

Act

.Su

cces

sor

No.

TraF

Nr jk

No

Succ

esso

ract

ivity

No

TraF

Nr jk

No.

Succ

esso

rac

tivity

S0

00

00

00

01

227

67

78

04

02

130

15

66

73

00

01

328

1012

1214

22

44

130

210

1212

135

33

22

45

291

22

44

22

26

3132

3339

4042

34

55

64

24

21

630

911

1113

42

30

134

47

88

10

42

22

16

3110

1313

145

03

21

375

24

46

50

00

17

328

1010

110

04

21

386

34

45

42

22

18

336

88

94

32

01

357

23

34

43

32

49

1011

1334

34

45

42

33

136

82

33

50

42

01

1435

46

67

22

04

141

93

44

62

15

31

1536

13

34

00

20

141

104

55

64

00

01

1237

34

45

00

22

141

117

99

105

00

01

1638

12

23

10

31

141

123

44

53

00

11

1639

23

34

00

22

141

131

22

33

20

11

1640

11

12

40

43

345

4649

141

22

33

03

21

1641

34

45

04

00

143

151

22

24

22

21

1742

12

23

33

00

144

163

44

55

22

02

1819

431

22

33

03

02

4546

174

55

65

24

21

2044

34

45

23

41

152

184

55

64

24

31

2045

45

56

22

42

147

195

66

75

23

43

2122

2346

45

56

52

23

148

205

77

94

00

01

2447

34

45

20

42

152

213

55

75

14

41

2448

57

78

20

23

150

224

55

60

30

01

2449

68

810

42

32

151

231

22

35

11

11

2550

23

35

42

33

152

248

1010

122

15

33

2627

2851

24

47

42

33

153

254

55

62

15

31

30E

00

00

00

00

264

66

80

04

31

29



Tabl

eII

.Su

mm

ary

ofr

esu

ltsu

tiliz

ing

FCPM

.

No

FEST

FECT

FLST

FLCT

SLAC

KN

oFE

STFE

CTFL

STFL

CTSL

ACK

S0

00

00

00

07

70

077

77

00

777

70

077

2767

8787

105

7394

9411

342

9898

152

5010

510

515

86

311

1185

10

00

05

66

77

70

077

70

66

827

70

077

2871

9393

113

8110

510

512

736

9393

148

5010

510

515

87

70

077

25

66

715

1818

207

06

682

57

1818

927

70

077

2981

105

105

127

8210

710

713

150

105

105

158

5410

710

715

97

70

077

315

1818

2019

2323

265

321

2195

47

2626

997

33

380

3082

107

107

131

9111

811

814

454

107

107

159

6711

811

816

87

70

077

415

1818

2022

2626

305

718

1892

47

2626

997

70

077

3182

107

107

131

9212

012

014

570

120

120

170

8413

313

318

06

113

1388

522

2626

3024

3030

364

726

2699

41

3030

101

77

00

7732

8210

710

713

190

117

117

142

7212

212

217

083

132

132

178

59

1515

886

2430

3036

2734

3441

41

3030

101

36

3434

104

77

00

7733

9111

811

814

497

126

126

153

6711

811

816

876

126

126

174

77

00

777

2734

3441

2937

3745

36

3434

104

32

3737

106

77

00

7734

9712

612

615

310

013

013

015

876

126

126

174

8113

013

017

77

70

077

829

3737

4531

4040

502

446

4611

4 1

949

4911

66

99

985

3510

013

013

015

810

413

613

616

581

130

130

177

8813

613

618

17

70

077

929

3737

4532

4141

512

545

4511

31

949

4911

67

08

884

3692

120

120

145

9312

312

314

984

133

133

180

8813

613

618

16

113

1388

1029

3737

4533

4242

513

237

3710

62

642

4211

07

70

077

3790

117

117

142

9312

112

114

783

132

132

178

8813

613

618

15

915

1588

1133

4242

5140

5151

612

642

4211

01

651

5111

77

70

077

3882

107

107

131

8310

910

913

485

134

134

180

8813

613

618

14

627

2798

1229

3737

4532

4141

502

147

4711

41

651

5111

76

610

1085

3982

107

107

131

8411

011

013

584

133

133

179

8813

613

618

14

726

2697

1331

4040

5032

4242

531

949

4911

61

651

5111

76

99

985

4010

413

613

616

510

513

713

716

788

136

136

181

9013

713

718

27

70

077

1432

4141

5133

4343

541

949

4911

61

651

5111

77

08

884

4182

107

107

131

8511

111

113

685

133

133

179

9013

713

718

24

626

2697

1540

5151

6141

5353

631

651

5111

71

453

5311

87

70

077

4285

111

111

136

8611

311

313

990

137

137

182

9313

913

918

34

626

2697

1641

5353

6344

5757

681

453

5311

89

5757

121

77

00

7743

8611

311

313

987

115

115

142

9313

913

918

396

141

141

184

46

2626

9717

4457

5768

4862

6274

957

5712

13

6262

125

77

00

7744

105

137

137

167

108

141

141

172

108

151

151

192

113

155

155

195

59

1414

8718

4457

5768

4862

6274

957

5712

13

6262

125

77

00

7745

105

137

137

167

109

142

142

173

9614

114

118

410

214

614

618

87

14

479

1948

6262

7453

6868

813

6262

125

468

6813

07

70

077

4610

914

214

217

311

314

714

717

910

214

614

618

810

815

115

119

27

14

479

2053

6868

8158

7575

904

6868

130

1375

7513

57

70

077

4711

314

714

717

911

615

115

118

410

815

115

119

211

315

515

519

57

14

479

2153

6868

8156

7373

886

7070

132

1375

7513

57

52

279

4810

513

713

716

711

014

414

417

590

137

137

182

9814

414

418

77

70

077

2253

6868

8157

7373

877

7070

131

1375

7513

57

42

278

4911

014

414

417

511

615

215

218

598

144

144

187

108

152

152

193

77

00

7723

5875

7590

5977

7793

1375

7513

516

7777

136

77

00

7750

116

152

152

185

118

155

155

190

108

152

152

193

113

155

155

195

77

00

7724

5977

7793

6787

8710

516

7777

136

2887

8714

47

70

077

5111

815

515

519

012

015

915

919

711

315

515

519

512

015

915

919

77

70

077

2567

8787

105

7192

9211

144

100

100

154

5010

510

515

86

113

1387

E12

015

915

919

712

015

915

919

712

015

915

919

712

015

915

919

77

70

077

2667

8787

105

7193

9311

328

8787

144

3693

9314

87

70

077



and crossover rate is 0.9. After 10 random runs, the obtained best fuzzy projectduration is (175; 203; 231; 231; 257; 283) with the robustness measure 0.4933 andthe plausible project durations 256.8 (257). Wangs obtained schedule is listed inTable III (for more detail, see Wangs study): the sensitivity analysis of resourceavailability to the schedule robustness and the plausible project duration. Projectmanagers may consider increasing an additional hardware engineer (i.e., R = 6; 6;6; 4), if he/she feels that the risk of late project is high under the current resourceavailability (R = 6; 6; 5; 4). In our work, we found is the DMs risk attitude index,0,509477. The MinSlack and ROT rules are used to find a starting feasible sequencefor TS. The project duration for the sequence, is determined using the procedurepresented in the beginning of this section. The middle resulting data required by thetwo priority rules (MinSlack and ROT) are listed in Tables III.

The results of application of TS (Max iteration: 100) to the sample project areshown in Table IV.

The proposed method using TS makes it possible to improve fuzzy projectscheduling under resource constraints. The model can effectively provide the op-timal fuzzy profiles of project duration and resource amounts under the constraintof limited resources. The computational experiment shows that the performanceof the proposed TS is better than evaluation and light beam search algorithms ap-pearing in the literature, there is no significant difference between the proposedcomparison method and weakstrong comparison rules on the performance of thealgorithm.

7. COMPUTATIONAL EXPERIMENT

The proposed Taboo and overhaul algorithm for the F-RCPSP was codedusing object-oriented programming language Visual C# 2008 for a personnel laptoprunning under the Windows 7 operating system. Because of Wangs study, primarycomputational experiments were conducted to test the proposed GA approach on20 sets of benchmark problems randomly selected from the project schedulingproblem library (PSPLIB)30 to evaluate its efficiency. The set of problems containsfour problem types including 32 activities and 62 activities. Each problem typecontains 10 problems. Due to all the benchmarking problems created from PSPLIBare deterministic, he randomly fuzzified the problem data. For each crisp durationti , six numbers used to define the fuzzified activity duration ti were randomlygenerated from the interval ( xti, ti + (1 )xti) and ranked in ascending order. Inhis experiment, was set to 0.8. The project ready-time and deadline were fuzzifiedin the similar way.25 We could not benchmark as Wang randomly fuzzified theproblem data. So we decided to use similar example (J30X 1.RCP, J60X 1.RCP,J90X 1.RCP and X = 1,2,3) from the PSPLIB. But we fuzzified the repeatedproblem data, which were deterministic/crisp durations. We use MinSlack rules forthe randomly selected 9 problems initial solutions and solve the TSA.

The results of application to the sample project are shown in Table V.



Tabl

eII

I.Se

nsiti

vity

anal

ysis

ofr

eso

urc

eav

aila

bilit

yto

sche

dule

perfo

rman

ce.

Wan

gs

Stud

yR

esou

rce

avai

labi

lity

(6,6,5

,4)(7,

6,5,

4)(6,

7,5,4)

(6,6,

6,4)

(6,6,5

,5)Fu

zzy

proje

ctdu

ratio

n(17

5,23

1,23

1,28

3)(17

4,229

,229,2

80)

(175,2

31,23

1,283

)(16

5,21

9,21

9,27

0)(16

9,22

4,22

4,27

5)R

obu

stne

ssm

easu

re0.

4933

0.52

700.

4933

0.66

220.

5946

Plau

sible

pro

jectd

urat

ion

257

253

257

236

245

Our

Stud

y

Min

slack

(181,

240,

240,

294)

(181,2

40,24

0,294

)(18

1,240

,240,2

94)

(167,

221,

221,

272)

(175,2

33,23

3,286

)Ta

bu

Min

Sla

ck(17

1,22

6,22

6,27

9)(17

8,236

,236,2

89)

(172,2

28,22

8,282

)(16

1,21

7,21

7,26

8)(16

6,220

,220,2

71)

ROT-

R1

(179,

237,

237,

293)

(178,2

35,23

5,290

)(17

9,237

,237,2

93)

(165,

219,

219,

272)

(173,2

30,23

0,285

)Ta

buRO

T-R

1(17

2,22

7,22

7,27

9)(17

0,223

,223,2

73)

(171,2

26,22

6,279

)(16

2,21

4,21

4,26

5)(16

0,212

,212,2

63)

ROT-

R2

(171,

226,

226,

279)

(176,2

33,23

3,287

)(17

8,236

,236,2

89)

(167,

221,

221,

272)

(172,2

29,22

9,281

)Ta

buRO

T-R

2(17

4,23

0,23

0,28

3)(17

0,223

,223,2

73)

(172,2

27,22

7,279

)(16

5,21

9,21

9,27

0)(16

8,221

,221,2

71)

ROT-

R3

(179,

237,

237,

293)

(179,2

37,23

7,293

)(17

9,237

,237,2

93)

(169,

223,

223,

276)

(170,

226,

226,

280)

Tabu

ROT-

R3

(172,

228,

228,

282)

(171,2

25,22

5,276

)(17

1,226

,226,2

79)

(162,

214,

214,

265)

(162,2

13,21

3,263

)RO

T-R

4(17

8,23

6,23

6,28

9)(17

6,234

,234,2

89)

(178,2

36,23

6,289

)(16

7,22

1,22

1,27

2)(17

2,229

,229,2

81)

Tabu

ROT-

R4

(171,,

226,

226,

279)

(170,2

23,22

3,273

)(17

4,229

,229,2

81)

(161,

215,

215,

265)

(162,2

14,21

4,265

)



Tabl

eIV

.Th

ere

sults

oft

aboo

sear

chto

the

sam

ple

projec

t.FC

PM

Criti

calP

ath

S-1-

2-4-

5-6-

7-10

-11

-15-

16-1

7-19

-20-

23-2

4-26

-28-

29-3

0-33

-34-

35-4

0-48

-49-

50-5

1-E

Fuzz

yPr

oject

Com

plet

ion

Tim

e(12

0,15

9,15

9,19

7)In

itial

Solu

tions

(MinS

lacka

nd

ROT)

Min

Slac

kFe

asib

leSc

hedu

leS-

1-2-

4-3-

5-6-

7-10

-9-1

1-8-

14-1

3-12

-15-

16-1

7-18

-19-

20-2

2-21

-23-

24-2

6-28

-27-

25-2

9-30

-39-

41-3

3-38

-34-

35-4

2-43

-31-

36-

32-3

7-40

-48-

49-5

0-45

-46-

47-4

4-51

-EM

inSl

cak

Feas

ible

Solu

tion

(181,

240,

240,

294)

Rot

Feas

ible

Sche

dule

for

Rl

S-1-

2-3-

4-5-

6-7-

9-10

-8-1

4-12

-13-

11-1

5-16

-17-

18-1

9-21

-22-

20-2

3-24

-25-

26-2

8-27

-29-

41-3

0-39

-42-

38-3

3-34

-31-

35-3

6-43

-32

-37-

40-4

4-48

-49-

50-4

5-46

-47-

51-E

Rot

Feas

ible

Solu

tion

for

Rl

(179,

237,

237,

293)

TABU

Sear

chSo

lutio

nTa

buM

inSl

ack

Feas

ible

Sche

dule

S-1-

2-4-

3-5-

6-7-

10-9

-8-1

1-14

-12-

13-1

5-16

-17-

18-1

9-22

-20-

21-2

3-24

-26-

27-2

8-25

-29-

34-3

2-31

-36-

38-3

3-42

-43-

30-3

5-41

-39

-37-

40-4

8-49

-45-

50-4

6-47

-44-

51-E

Tabu

Min

Slca

kFe

asib

leSo

lutio

n(17

1,22

6,22

6,27

9)TA

BUR

otFe

asib

leSc

hedu

lefo

rR

lS-

1-2-

4-3-

5-6-

7-10

-9-8

-11-

12-1

3-14

-15-

16-1

8-17

-19-

20-2

2-21

-23-

24-2

7-28

-25-

26-2

9-34

-35-

30-4

3-42

-31-

41-3

2-33

-39-

38-

36-3

7-40

-48-

44-4

9-50

-45-

46-4

7-51

-ETA

BUR

otFe

asib

leSo

lutio

nfo

rR

l(17

1,22

6,22

6,27

9)



Tabl

eV.

Ben

chm

ark

val

uefo

rth

esa

mpl

epr

ojec

ts.Pl

ausib

leB

estV

alue

Mak

espa

nCP

U-T

ime

[sec]

FCPM

Min

Slac

kTa

buM

inSl

ack

Def

uzzi

edVa

lue

Diff

eren

cefro

mTh

eB

estV

alue

CPU

-Tim

e

[sec]

J301

143

0.30

(38,3

8,38

,38)

(43,43

,43,

43)

(43,43

,43,

43)

430

0,06

9J3

021

380.

02(34

,34,

34,3

4)(38

,38,

38,3

8)(38

,38,

38,3

8)38

00,

018

J303

172

0.01

(72,7

2,72

,72)

(72,7

2,72

,72)

(72,7

2,72

,72)

720

0,01

5J6

011

77(77

,77,

77,7

7)(86

,86,

86,8

6)(86

,86,

86,8

6)86

90,

047

J602

165

(65,6

5,65

,65)

(65,6

5,65

,65)

(65,6

5,65

,65)

650

0,03

1J6

031

60(60

,60,

60,6

0)(60

,60,

60,6

0)(60

,60,

60,6

0)60

00,

031

J901

173

(67,6

7,67

,67)

(86,86

,86,86

)(85

,85,

85,8

5)85

120,

046

J902

196

(96,9

6,96

,96)

(96,9

6,96

,96)

(96,9

6,96

,96)

960

0,09

3J9

031

81(81

,81,

81,8

1)(82

,82,82

,82)

(82,8

2,82

,82)

821

0,04

6

Internat

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