fuzzy optimal model for resource-constrained construction scheduling

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FUZZY OPTIMAL MODEL FOR RESOURCE-CONSTRAINED

CONSTRUCTION SCHEDULING

By Sou-Sen Leu,1 An-Ting Chen,2 and Chung-Huei Yang3

ABSTRACT: Activity duration is uncertain due to the variation in the outside environment, such as weather,site congestion, productivity level, etc. Furthermore, resources for construction activities are limited in the realconstruction world so that scheduling must include resource allocation. A new optimal resource-constrainedconstruction scheduling model is proposed in this paper, in which the effects of both uncertain activity durationand resource constraints are taken into account. Fuzzy set theory is used to model the uncertainties of activityduration. A genetic algorithm-based searching technique is adopted to search for the fuzzy optimal projectduration under resource constraints. The model can effectively provide the optimal fuzzy profiles of projectduration and resource amounts under the constraint of limited resources.

INTRODUCTION

Since the late 1950s, the critical-path method (CPM) hasbeen intensively used by practitioners for planning and con-trolling large-scale projects in the construction industry. In atraditional CPM analysis, a major objective is to build up thefeasible duration required to perform a specific project underthe assumption of unlimited resources. In a real constructionproject, project activities must be scheduled under limited re-sources, such as limited crew sizes, limited equipmentamounts, and limited materials. Owing to the constraints ofresource availabilities, the scheduler must often take accountof the trade-off between available resources and activity du-rations. Resource-constrained scheduling problems have beenstudied intensively in the construction and manufacturing in-dustries because of practical applications. Several analytical orheuristic models have been generated to solve resource-con-strained scheduling problems. These models mainly focusedon deterministic situations. However, during project imple-mentation, many uncertain variables dynamically affect activ-ity durations. Examples of these variables are weather, spacecongestion, productivity level, etc. Currently, some systematicmethods, such as PERT, PNET, and Monte Carlo simulation,have been developed to deal with uncertain project durations.Nevertheless, the methods seldom take account of resourceconstraints, and optimal solutions are not generally exploredby these methods. Combining the aforementioned concepts todevelop a resource-constrained scheduling model under un-certainty would be beneficial to scheduling engineers in theforecast of a more realistic project duration. Engineers couldexplicitly analyze and quantify their combined impacts andincorporate these pieces of information in the project durationestimates. Such a model could generate additional manage-ment information, such as the impact of construction resourceson project schedules, level of sensitivity of an activity to spe-cific uncertainty variables, and the combinative effects. Thisis expected to give project management an insight into theproject schedule. Project management can take appropriatetimely action to reduce both unavoidable project delays andinefficient use of resources.

1Assoc. Prof., Dept. of Constr. Engrg., Nat. Taiwan Univ. of Sci. andTechnol., P.O. Box 90-130, Taipei, Taiwan, 10672.

2Master, Dept. of Constr. Engrg., Nat. Taiwan Univ. of Sci. and Tech-nol., P.O. Box 90-130, Taipei, Taiwan, 10672.

3Master, Dept. of Constr. Engrg., Nat. Taiwan Univ. of Sci. and Tech-nol., P.O. Box 90-130, Taipei, Taiwan, 10672.

Note. Discussion open until December 1, 1999. To extend the closingdate one month, a written request must be filed with the ASCE Managerof Journals. The manuscript for this paper was submitted for review andpossible publication on October 18, 1998. This paper is part of the Jour-nal of Computing in Civil Engineering, Vol. 13, No. 3, July, 1999.qASCE, ISSN 0887-3801/99/0003-0207–0216/$8.00 1 $.50 per page.Paper No. 19472.

J. Comput. Civ. Eng

This paper adopts a new approach, employing the uses ofgenetic algorithms (GAs) and fuzzy set theory, to develop theresource-constrained scheduling model under uncertainty.Fuzzy set theory has been used to model the uncertainty thatis associated with time elements in project networks. Previousstudies have successfully demonstrated the use of fuzzy settheory for estimating the project duration under uncertainty(Ayyub and Haldar 1984; Lorterapong 1994; Wu and Hadi-priono 1994). GAs were then used to allocate multiple avail-able construction resources to activities of a single project toachieve the objective of minimizing the project duration underuncertainty. In other words, the focus of the model is to op-timally allocate limited resources to project activities with theobjective of minimum project duration in an uncertain envi-ronment. In this article, case studies with standard test prob-lems will be presented to illustrate the performance and ac-curacy of the model.

LITERATURE REVIEW

Resource-constrained scheduling models can be categorizedinto two areas: (1) Deterministic scheduling; and (2) nonde-terministic scheduling. Currently, resource-constrained sched-uling models mostly focus on deterministic situations. Themost popular techniques of deterministic resource-constrainedscheduling models are analytical and heuristic methods.

Early attempts to solve deterministic resource-constrainedscheduling problems used mathematical models to obtain anoptimal solution. Integer linear programming, dynamic pro-gramming, as well as branch and bound were generally used(David 1973). Elmaghraby (1977) and Talbot (1982) used in-teger programming techniques to solve a resource-constrainedproblem under a certain environment. Johnson (1967) andStinson (1976) separately presented branch-and-bound solu-tions for deterministic resource-constrained scheduling prob-lems. However, resource-constrained scheduling problems be-long to one type of NP-hard problem. A great deal ofcomputational effort is required to solve problems of this kind.To avoid the problem of ‘‘combinatorial explosion,’’ heuristicrules were also used to solve the problems (Morse and White-house 1988; Tsai and Chiu 1996). To date, many heuristicscheduling rules have been proposed to solve deterministicresource-constrained scheduling problems, such as theMINSLK model (David and Patterson 1975), the ROT model(Elsayed 1982), the GENRES model (Whitehouse and Brown1979), and the three-heuristic model (Boctor 1990). Each heu-ristic model has its own philosophy, and they all try to increasethe possibility of obtaining the best solution.

Construction management has recently begun to pay atten-tion to nondeterministic scheduling due to many uncertain var-iables involved in construction operations. Nondeterministicscheduling models can be categorized according to the pres-

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ence or absence of resource constraints and the uncertaintytheories used (Fig. 1). Traditionally, uncertainties associatedwith project duration are modeled using probability theory.They can be classified into two subfields, depending onwhether resources are constrained or not. Classical resource-unconstrained scheduling models under uncertainty are PERTand Monte Carlo simulation, and they are the most widelyused in practice (Diaz and Hadipriono 1993). Ang (1975),Gong (1993), and several others also developed probabilisticresource-unconstrained scheduling models. When resourceconstraints are taken into consideration, only a few probabi-listic models have been developed. The model proposed byAhuja and Arunachalam (1984), and the DYNASTRAT by Pa-dilla and Carr (1991) are some examples of probabilistic re-source-constrained scheduling models.

On the other hand, it has been claimed that fuzzy set theoryis appropriately used to model uncertainty that is associatedwith time elements in project networks (Chanas and Kambu-rowski 1981; McCahon 1993; Lorterapong 1994). In practice,due to a sheer lack of information about activities, the valuesof project variables are often estimated by human experts.Many of the values are defined based upon fuzzy and/or in-complete information. This type of information might be bestmodeled by fuzzy set techniques instead of probabilistic ones.Similar to probabilistic scheduling models, fuzzy schedulingmodels can be categorized according to the presence or ab-sence or resource constraints (Fig. 1). Ayyub and Haldar(1984) developed fuzzy operations and relations to estimatethe possible duration of a single project without the consid-eration of resource constraints. The angular fuzzy set modelby Hadipriono and Sun (1990), the Fuzzy Modus Ponens de-duction technique by Wu and Hadipriono (1994), and thefuzzy network analysis model by Lorterapong and Moselhi(1996) are some examples of fuzzy scheduling models that didnot consider resource constraints. Furthermore, Wang et al.(1993) proposed a fuzzy resource-constrained project planningmodel in which the a-cut technique was used to analyze therisk level of completion time and cost. Lorterapong (1994)also proposed the FNET model in which a fuzzy heuristicmethod was developed to solve the resource-constrained proj-ect scheduling problem under uncertainty. Hapke and Slow-inski (1996) developed fuzzy priority heuristic rules to tacklethe resource-constrained scheduling problem.

Based upon the aforementioned review, only a few fuzzyscheduling models under resource constraints have been pro-

208 / JOURNAL OF COMPUTING IN CIVIL ENGINEERING / JULY 1999

J. Comput. Civ. Eng.

posed until now. Nevertheless, most of the models are basedupon heuristic approaches. Heuristic rules may perform wellover a variety of problems and are widely used in practicalcases because of their simple format and efficiency in appli-cation; nevertheless, optimal solutions are not guaranteed. Fur-thermore, resource-constrained scheduling problems are a typeof multipeak search problem in which different combinationsof resource utilization may yield the same solutions. Such so-lutions may be of much use when they are required to factorin other planning considerations. Heuristic rules, nevertheless,generally only explore one or a few feasible solutions.

In this paper, the writers integrated ideas from the afore-mentioned research in a new approach using fuzzy set theoryand GAs to overcome the limitations of previous models. Thegoal of the paper is to establish a fuzzy optimal constructionscheduling model under resource constraints. In this model,the activity duration is first characterized by a fuzzy number.An acceptable risk level (i.e., a-cut level) is then defined asthe minimum condition that can be accepted. The GA tech-niques are further used to find optimal or near-optimal solu-tions within the fuzzy solution space.

The following sections briefly introduce the concepts offuzzy set theory and GAs that are used in this paper. Next, thecontents of the problem and its mathematical forms are de-scribed. A GA-based algorithm for solving the problem is thengiven. A standard test example involving different aspects ispresented to demonstrate the operation of the algorithm. Theconclusions are made in the last section.

FUZZY SET THEORY

Fuzzy set theory was developed specifically to deal withuncertainties that are not statistical in nature (Zadeh 1965; Klirand Yuan 1995). The definition of a fuzzy set can be stated asfollows: If X is a collection of objects denoted generically byx, then a fuzzy set A in X is defined as a set or ordered pairs[A = {(x, mA(x)) ux [ X}, where mA(x) is a membership func-tion for the fuzzy set A]. The membership function maps eachelement of X to a membership value between 0 and 1. Triangleand trapezoidal fuzzy numbers are commonly used (Fig. 2).Several fuzzy operations, such as union and intersection, havealso been proposed to manipulate relationships between fuzzynumbers. Among these operations, a-cut is an important con-cept in the fuzzy optimal scheduling model proposed in thispaper.

FIG. 1. Classes of Nondeterministic Scheduling Models

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FIG. 2. Examples of Membership Functions and a-Cut Level

Based upon the first decomposition theorem of fuzzy settheory, any fuzzy set can be associated to a collection of crispsets known as a-cut (Fig. 2). The a-level set or a-level cut ofA is the set aA = {(x, mA(x) $ a) ux [ X}, ; a [ [0, 1] (Klirand Yuan 1995). Every fuzzy set A can be expressed as

aA = a 3 A<a[[0,1]

This means that the membership function of A is the union ofall a-cuts, after each one is multiplied by a. The a-cut pro-vides a useful way for resolving a membership function interms of constituent crisp sets, as well as for synthesizing amembership function out of crisp sets. Based upon the a-cutconcept, uncertain durations represented by fuzzy numbers aretransformed into crisp sets. A deterministic resource-con-strained scheduling model is then used to find optimal solu-tions within different crisp sets. Finally, these crisp solutionsare unified into fuzzy optimal solutions.

GAs

GAs have been popularly used in many areas, such as con-strained or unconstrained optimization, scheduling and se-quencing, transportation, reliability optimization, artificial in-telligence, and many others (Goldberg 1989). Some researchhas been done in the optimization of construction schedulingusing GAs (Chua et al. 1995; Feng et al. 1997). GAs are sto-chastic search techniques based upon the mechanism of naturalselection and natural genetics. The operation of genetic algo-


rithms is shown in Fig. 3. In GAs, potential solutions to aproblem are represented as a population of chromosomes andeach chromosome stands for a possible solution at hand. Thechromosomes evolve through successive generations. Off-spring chromosomes are created by merging two parent chro-mosomes using a crossover operator, or modifying a chro-mosome using a mutation operator. During each generation,the chromosomes are evaluated on their performances withrespect to the fitness functions (i.e., objective functions). Chro-mosomes that are fitter have higher survival probabilities. Af-ter several generations, chromosomes in the new generationare identical, or certain termination conditions ar met. The fi-nal chromosomes hopefully represent the optimal or near-op-timal solutions to a problem.

GA-BASED RESOURCE-CONSTRAINEDSCHEDULING

Resource-constrained allocation belongs to one type of se-quencing problem. The basic work of resource-constrainedallocation is deciding the order in which to schedule the in-dividual activities. When using GAs to solve a resource-con-strained allocation problem, a string in the population repre-sents a possible sequence of activities, and each character inthe string stands for an activity ID (or name). An activity ina lower position has a higher priority of getting resources.Take a schedule network in Fig. 4 as an example. The chro-mosome in Fig. 5 represents a feasible solution to the schedulenetwork in Fig. 4. In the chromosome, Activity B is in a lower


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FIG. 3. Operation of GAs

FIG. 4. Example of Schedule Network

FIG. 5. Chromosome Structure of Resource-Constrained Al-location

position than Activity D, which means that Activity B hashigher priority than Activity D.

The genetic operators (i.e., crossover and mutation) used foractivity sequences in this paper are explained as follows. Theyare named as UX3 and UM3, respectively. The UX3 crossoveroperator performs the chromosome crossover by creating twoexclusive substrings from parent strings and then by randomlywriting the characters in the substrings directly to the offspringstrings. While the characters are written into the offspringstrings, activity precedence relationships are also taken intoaccount. After the crossover operation, all offspring chromo-somes are legal (i.e., fulfill the requirements of activity prec-edence relationships) and need not be fixed. The procedure ofthe UX3 crossover operator is shown in Fig. 6. First, a sub-



string, 3-4-5-6, is selected from Parent 1. Second, the char-acters that already exist in the substring (i.e., 3-4-5-6) are de-leted from Parent 2. The resulting sequence of characters is1-2-8-7-9. Third, an offspring chromosome is created by ran-domly writing the characters from two substrings directly tothe string. While the characters are written into the offspringstring, activity precedence relationships are also taken into ac-count. Using the same steps, the second offspring is producedfrom the same parents.

The basic concept of the UM3 mutation operator is similarto that of the crossover operator, but alteration of charactersis only within an individual chromosome. The mutation op-eration is shown in Fig. 7. First, a substring, 2-6-4-5, is se-lected from a parent chromosome. Second, the characters inthe substring are exchanged at random. When exchanging thecharacters, activity precedence relationships are also taken intoaccount. Third, the new substring (i.e., 6-4-5-2) is put backinto the parent string in the same position to obtain an off-spring chromosome for the next generation.

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FIG. 6. Crossover Operator Procedures

FIG. 7. Mutation Operator Procedures

Furthermore, to manipulate the generation of activity dura-tion within the acceptable risk level (i.e., a-cut level), one-cut-point (1-point) crossover and uniform mutation operatorswere used in the system. A one-cut-point method, which ran-domly selects one cut-point at parent strings and exchangesthe right parts of two parent strings to generate offspringstrings. Let two parent chromosomes be A = [a1, a2, a3, . . . ,an] and B = [b1, b2, b3, . . . , bn]. If they are crossed after thekth position, the resulting offspring are A9 = [a1, a2, a3, . . . ak,bk11, bk12, . . . , bn] and B9 = [b1, b2, b3, . . . bk, ak11, ak12, . . . ,an]. The number of offspring chromosomes produced in eachgeneration is based upon a predefined crossover rate. Uniformmutation replaces a gene (real number) with a randomly se-lected real number within a specified range. Let a chromosomethat is to be mutated be A = [a1, a2, a3, . . . , an]. A randomnumber k [ [1, n] is first selected based upon a predefinedmutation rate. An offspring A9 = [a1, a2, a3, . . . . . . , an] isa9,k

then produced. The value of is restricted to the lower anda9kupper bounds on the value ak.

PROBLEM DEFINITION AND GA-BASED SOLVER

The operational architecture of the GA-based fuzzy sched-uling model is shown in Fig. 8. The assumptions of the modelare made as follows: (1) Activities in the network diagramcannot be split; (2) once a resource is occupied by an activity,the resource will not be released until the activity is done; (3)


the supplies and demands of resources are kept constant duringthe project implementation and their values are crisp; (4) ac-tivity duration is characterized by a fuzzy number due to en-vironmental variation. The basic logic of the model is ex-pressed mathematically as follows.

amin T

a a aT = max{ t 1 d u i = 1, 2, . . . , n} (1)i i

subject toa a at 2 t 2 d $ 0, ;j [ S (2)j i i i

r # b , k = 1, 2, . . . , m (3)kO ad kiat [Ai ati

a at , d $ 0, i = 1, 2, . . . , ni i

where aT = project duration at a specific a-cut level; ati =starting date of activity i at a specific a-cut level; adi = durationof activity i at a specific a-cut level; Si = set of successorsof activity i; = resource demand of kth resource of activ-rad kiity i at the duration of di under a specific a-cut level; bk =resource limit of kth resource; = set of on-going activitiesAatiat date ti under a specific a-cut level; m = total number ofresource types; and n = total number of activities.

Eq. (1) indicates the computation for fuzzy project duration.Eq. (2) states that the difference in occurrence times of two


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FIG. 8. Operation of Fuzzy Optimal Resource-Constrained Scheduling Model

connected nodes must be at least as great as the duration ofthe connecting activity. Eq. (3) ensures that resources utilizedwill not exceed available resources.

Combining the aforementioned resource-constrained sched-uling algorithms with a-cut concepts and GAs, the operationof the fuzzy optimal resource-constrained construction sched-uling model is described as follows (Fig. 8). The model con-sists of four subsystems: (1) Activity duration generation sub-system; (2) activity order generation subsystem; (3) evaluationsubsystem; and (4) selection subsystem. Each has its own pur-pose. The activity duration generation subsystem was createdto manipulate the generation of activity duration. It uses 1-point crossover and uniform mutation operators to generatefeasible child chromosomes, restricted to the fuzzy durationmembership functions and the accepted risk level (i.e., a-cutlevel). A pool of chromosomes is created to represent possibleactivity durations. Each character in a chromosome represents



the duration of a corresponding activity, whose value is re-stricted within the pessimistic and optimistic values, basedupon the a-cut level. Next, at the activity order generationsubsystem, another pool of chromosomes is created to repre-sent the possible priority order of getting resources for projectactivities. The activity order generation subsystem uses UX3crossover and UM3 mutation to generate feasible chromo-somes. The details of UX3 and UM3 as well as the structureof ordering chromosomes were discussed in the previous sec-tion. The ordering and activity duration chromosomes are thenaggregated in pairs into longer chromosomes. Each possibleproject duration is then calculated at the evaluation subsystem.The calculation is based upon the values in the aggregatedchromosomes, while also considering the activity precedencerelationships [defined in (2)], and the resource constraints [de-fined in (3)]. The calculated project duration (called fitnessvalues in terms of GAs) will be used for the selection of sur-

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FIG. 9. Test Problem [Adapted from Lorterapong (1994)]

FIG. 10. Fuzzy Activity Durations for Test Problem

viving chromosomes for the next generation, according to theobjective function in (1). In the final step, the surviving chro-mosomes for the next generation are selected at the selectionsubsystem according to the roulette wheel principle. Thismeans that the selection probability for a chromosome k isproportional to the ratio of fj, where fk is the fitnesspop size–f /(k j=1

value of the chromosome k.The fuzzy optimal scheduling system was implemented us-

ing Microsoft Visual Basic 5.0. All data associated with thesystem were stored in Microsoft Access 97. These data arerelated to network information and GA operational parameters.Network information includes fuzzy activity duration mem-bership functions, precedence relationships, as well as resourcedemands and supplies. The network data were input througha graphical user interface, which was designed using VisualBasic 5.0 GA operational parameters related to the number ofgenerations, population sizes, and crossover and mutation rateswere input through dialogs. Output data were exported to Mi-crosoft Excel 97 for data analysis and graphical plotting.

CASE STUDY AND ANALYSIS OF FINDINGS

In this section, a test case adapted from Lorterapong (1994)will be discussed. The result from the fuzzy optimal resource-


constrained scheduling model raised in this paper will betested and verified, and it will be compared with that fromLorterapong’s FNET model. Later, the influence factors of thefuzzy optimal scheduling model and their impacts will be ex-amined.

The schedule example is a seven-activity CPM network.The precedence relationships of the network and the resourcedemands of activities are depicted in Fig. 9. The duration ofeach activity is assumed to be uncertain, and is expressed indifferent types of fuzzy membership functions (Fig. 10). Ac-tivities ‘‘start’’ and ‘‘end’’ are dummy activities. Each activityhas a crisp resource demand. It is assumed that 13 resourceunits are available. Table 1 shows the result from the Lorter-apong’s (1994) FNET model. The project duration after re-source allocation, based upon the FNET, was calculated to be(20, 23, 29, 32).

To obtain good performance using GA search, tuning of theparameters, crossover rates, and mutation rates are indispen-sable. In this case, we tested the GA-based system with variousvalues of the crossover and mutation rates to find better valuesfor them. These tests were performed with crossover rates of0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9 and mutation ratesof 0.2, 0.4, and 0.6. Experimental results from the fuzzy op-


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TABLE 1. Result from FNET [Adapted from Lorterapong(1994)]

Activity(1)

Fuzzyduration

(2)Fuzzy start

(3)Fuzzy finish

(4)

Start (0, 0, 0, 0) (0, 0, 0, 0) (0, 0, 0, 0)A (3, 4, 4, 5) (0, 0, 0, 0) (3, 4, 5, 5)B (5, 5, 5, 5) (0, 0, 0, 0) (5, 5, 5, 5)C (6, 8, 10, 12) (3, 4, 4, 5) (9, 12, 14 17)D (5, 5, 7, 7) (9, 12, 14, 17) (14, 17, 21, 24)E (6, 6, 8, 8) (14, 17, 21, 24) (20, 23, 29, 32)

End (0, 0, 0, 0) (20, 23, 29, 32) (20, 23, 29, 32)

timal scheduling model show that, whenever the crossover rateis between 0 and 1, optimal solutions can be reached. Fig. 11shows the search histories of the fuzzy scheduling model withmutation rates of 0.2, 0.4, and 0.6 when the a-cut level is 1.In the proposed fuzzy scheduling model, the population size,crossover rate, and mutation rate in the fuzzy schedulingmodel were defined as 50, 0.4, and 0.6, respectively. Optimalsolutions with different a-cut values were determined in a sim-ilar way. To compare with the result from the FNET model,maximum project duration with different a-cut values was alsocalculated. Similar to the procedure discussed previously, onlymaximum project duration becomes the search objective. Theminimum and maximum project durations are both shown inTable 2. The results from both our model and the FNET modelare shown in Fig. 12.

Comparing both results, the maximum project durations ofboth models are slightly different in some details, but bothvalues are approximately equal (Fig. 12). However, from theviewpoint of minimum project durations, the fuzzy optimalscheduling model proposed in this paper found better solutionsthan the FNET did. The FNET model adopted fuzzy set op-erations (such as !, etc.) to search for a feasible˜˜max, min,fuzzy project duration under crisp resource conditions. Thedefinitions of !, and can be found in Lorterapong’s˜˜max, min(1994) paper. Moreover, Lorterapong developed the heuristicrules, negative impact and expected delay, to decide appropri-ate fuzzy activity schedules under resource constraints. How-ever, these heuristic rules apparently do not lead to optimalsolutions. For example, based upon the result from the FNET(Table 1), the starting dates of activities A, B, and E are (0,0, 0, 0), (0, 0, 0, 0), and (14, 17, 21, 24), respectively. Thismeans that activities A and B have first priority to get re-sources as the project starts. Their total resource demand is11, which is under the available resource amount (i.e., 13). Inthe case of the combinations of activities A and E, as well asB and E, their resource requirements are 13 and 12, respec-tively. These activity combinations are also satisfying solu-tions. Based upon the heuristic rules raised in the FNET, thesetwo activity combinations were denied first priority in gettingresources. However, the optimal model raised in this papershows that activities A and E need to start first to obtain theoptimal solution.

In fact, a resource-constrained scheduling problem can beconsidered as a multipeak optimization problem without re-gard to deterministic or fuzzy scheduling. Its objective func-tion is characterized by a number of sharp peaks, which leadto global or local optima. For the same project, different com-binations of resource utilization may yield the same optimalvalues. Furthermore, a set of activity schedules could producethe same or different resource utilization profiles but with thesame objective value. In solving resource-constrained sched-uling problems, there is a significant difference between theGA-based approach and the heuristic approach. In general, theheuristic approach generates only a single feasible schedule.In contrast, the GA-based approach may recommend several



FIG. 11. Search Histories of Fuzzy Scheduling (a-cut 5 1)

TABLE 2. Pessimistic and Optimistic Project Duration Valueswith Different a-Cut

a(1)

Minimum projectduration

(2)

Maximum projectduration

(3)

0.0 17 320.2 18 310.4 18 310.6 19 300.8 19 301.0 19 30

FIG. 12. Fuzzy Membership Functions of Both Results

feasible or near-optimal schedules to decision makers. Suchsolutions may be of much use when they are required to factorin other planning considerations.

In this example case, several activity combinations have thesame objective values (Table 3). From Table 3, the combina-tions, start-E-A-C-B-D-end, start-A-E-C-B-D-end, and start-A-E-B-C-D-end, have the same results. Activities A and Eneed to start first to obtain the optimal solution. Otherwise, ifactivity E does not start simultaneously with activity A (e.g.,the project starts with activities A and B), resources requiredby the combinations of activities C and E, or D and E, exceedthe available resources. The project must be extended due tolack of resources. Actually, a resource-constrained schedulingmodel needs to simultaneously take account of precedence re-lationships and resource constraints. Possible activity combi-nations can be huge. The GA-based search method providesbetter exploitation and exploration of the solution space thanother methods, such as random search and hill-climbing search(Michalewicz 1994). Moreover, a fuzzy resource-constrainedscheduling problem needs to tackle an even larger number ofpossible activity combinations. The GA-based search provides

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TABLE 3. Results with Different Activity Combinations afterResource Allocation

a(1)

Start(2)

A(3)

B(4)

C(5)

D(6)

E(7)

End(8)

(a) Resource usage priorities: start → E → A → C → B → D → end(early start)

1.0 0 0 6 6 14 0 190.8 0 0 6 6 14 0 190.6 0 0 6 6 14 0 190.4 0 0 6 6 13 0 180.2 0 0 6 6 13 0 180.0 0 0 6 6 12 0 17

(b) Resource usage priorities: start → A → E → C → B → D → end(early start)

1.0 0 0 6 6 14 0 190.8 0 0 6 6 14 0 190.6 0 0 6 6 14 0 190.4 0 0 6 6 13 0 180.2 0 0 6 6 13 0 180.0 0 0 6 6 12 0 17

(c) Resource usage priorities: start → A → E → B → C → D → end(early start)

1.0 0 0 4 6 14 0 190.8 0 0 4 6 14 0 190.6 0 0 4 6 14 0 190.4 0 0 4 6 13 0 180.2 0 0 4 6 13 0 180.0 0 0 3 6 12 0 17

more computational efficiency for this kind of complex sched-uling problem.

Basic influence factors of the fuzzy resource-constrainedscheduling model are schedule networks, fuzzy membershipfunctions concerning the duration of project activities, resourceavailabilities, and resource requirements. The following sec-tions depict the impacts of these factors.

Schedule Networks

As mentioned above, the major task of resource-constrainedscheduling is deciding the priority of getting resources for in-dividual activities. The priorities depend upon the structure ofa schedule network. Precedence relationships among projectactivities are an especially significant factor. Different activityprecedence relationships for the same project will yield dif-ferent fuzzy optimal schedule profiles.

Fuzzy Membership Functions for Activity Durations

In the test case, different membership functions for activitydurations were used to generate the fuzzy project duration pro-files. It was found that different function shapes have an im-pact on the profiles. At a simple extreme, every activity ownsa crisp membership function; the fuzzy project duration profilebecomes constant without regard to different a-cut values. Ata complex extreme, every activity owns a different member-ship function; the fuzzy project duration profile becomes ir-regular. Like the above-mentioned case, each activity owns adifferent membership function (including triangle, trapezoid,rectangle, etc.), and the output membership function becomesirregular (Fig. 12).

Resource Availabilities

The same test problem from Lorterapong (1994) was usedto analyze the impact of resource availabilities. The resourceavailabilities changed to 11 and 12. The results are depictedin Fig. 13. Resource availabilities have a significant impact onproject scheduling. Project duration with fewer available re-sources is generally longer than that with more resources with-

J. Comput. Civ. Eng

FIG. 13. Fuzzy Membership Functions with Different AvailableResources

TABLE 4. Multiple Resource Requirements for Test Case

Resource(1)

Start(2)

A(3)

B(4)

C(5)

D(6)

E(7)

End(8)

1 0 6 5 7 9 7 02 0 7 6 9 12 11 0

out regard to deterministic or fuzzy scheduling. However, forfuzzy scheduling, the optimistic and pessimistic margins of thefuzzy project duration membership function may not movesimultaneously when available resources change. In the testcase, the optimistic margin (i.e., left margin) shifts to the right;while the pessimistic margin (i.e., right margin) stays at thesame location when the available resources change from 13 to12. However, when available resources change from 13 to 11,both optimistic and pessimistic margins simultaneously moveto the right. Analyzing the chromosomes that were output fromthe GA-based fuzzy resource-constrained scheduling model indetail, it was found that for fuzzy scheduling both activitycombinations and activity durations have interactive impact onboth maximum and minimum project durations under resourceconstraints. With the same available resources, different activ-ity combinations with varied activity durations may yield thesame optimistic and pessimistic margins. Moreover, differentactivity combinations with dissimilar activity durations mayyield the same optimistic or pessimistic margins under the con-straints of different available resources.

Multiple Resource Requirements

If multiple resources are required, project duration willchange, depending on the resource availabilities and require-ments. Take the fuzzy schedule network from Lorterapong(1994) as an example, where one more resource was added tothe case (Table 4). The resource availabilities are constant overthe project duration and the upper limits of two resource sup-plies are assumed to be 13 and 15. The GA-based fuzzy sched-uling system was modified to take account of multiple re-sources. After many trials, the population, crossover rate, andmutation rate were defined to be 50, 0.4, and 0.6, respectively.With different a-cut values, the fuzzy membership functionprofile was obtained and is shown in Fig. 14. In general, theproject duration profile under multiple resource requirementsmoves to the right due to resource conflict (Fig. 14). In thegiven example, if only a single resource (i.e., resource 1) isconsidered, A and B, A and E, as well as B and E are thesatisfying activity combinations as the project starts. Whenmultiple resources are considered, only the combination of ac-tivities A and B is satisfied. Other activities need to wait untilresources are released by previous activities. In conclusion, the


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FIG. 14. Fuzzy Membership Functions with Multiple ResourceRequirements

project duration becomes longer when multiple resources aretaken into account.

CONCLUSIONS

This paper addressed a GA-based fuzzy resource-con-strained scheduling model that incorporates an efficient com-putational technique for resource allocation and a more suit-able way of modeling uncertainty in network analysis.Compared with traditional heuristic models (such as Lortera-pong’s FNET), the GA-based fuzzy scheduling model has sev-eral advantages. First, unlike heuristic models, it is not nec-essary for the GA-based model to commit to any particularheuristic rules. Because of this, the GA-based model has moreflexibility when solving complex resource-constrained sched-uling problems. Second, the GA-based fuzzy schedulingmodel can explore and exploit several near-optimal solutions,which are generally not provided by heuristic models. Thesenear-optimal solutions may provide more information for de-cision making, when other planning considerations are re-quired. Moreover, this paper adopts fuzzy set theory to de-velop a framework to perform resource-constrained schedulingin an uncertain environment. Compared with a deterministicmodel, the fuzzy model can reflect the degree of the uncer-tainty of the input data. In the course of future research, afunctional user interface will be required for practical use ofthe GA scheduler. Moreover, clear guidelines on GA param-eters, such as crossover rates, mutation rates, and so on, willbe of great use to practitioners.

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