Resource-constrained project scheduling problem with uncertain durations andrenewable resources

Weimin Ma Yangyang Che Hu Huang∗ Hua Ke

School of Economics and Management, Tongji University, Shanghai 200092, China

Abstract

Resource-constrained project scheduling problem is to make a schedule for minimization of the makespan subject toprecedence and resource constraints. In this paper, we consider an uncertain resource constrained project schedulingproblem (URCPSP) with renewable resource constraints, in which activity durations, with no historical data generally,are estimated by belief degrees. For purpose of managing these estimations in URCPSP, an uncertainty-theory-basedproject scheduling model is proposed. Furthermore, a genetic algorithm integrating a 99-method based uncertainsimulation is designed to search the quasi-optimal schedule. Numerical examples are also provided to illustrate theeffectiveness of the model and the algorithm.

Keywords: Resoure-constrained project scheduling, Uncertain duration, Genetic algorithm, Uncertainty theory,Renewable resource

1. Introduction

The resource-constrained project scheduling problem (RCPSP) has been a widely studied optimization problemsince it was initiated. The problem considers the minimization of the makespan subject to precedence and resourceconstraints [1]. As a job-shop generalization, RCPSP is NP-hard proved by Blazewicz et al.[2]. Most literature aboutRCPSP has been based on deterministic assumption that complete information of resource usage and deterministicactivity duration are given in advance. A feasible baseline schedule, i.e., a list of activity starting times and a definitemakespan, can be determined as a result. While in reality, project usually undergoes various disruptions such asfluctuant execution durations, unavailable resources, overdue materials, absent workers, etc, and the schedules gainedfrom deterministic models might perform very poor as a result [3]. In order to cope with more realistic circumstances,where the variability in durations should not be neglected, stochastic resource-constrained project scheduling problem(SRCPSP) has been developed [4, 5, 6, 7, 8, 9, 10, 11]. In these papers, the execution of a project in random contextis viewed as a dynamic decision process in which activity durations are represented as random variables. For theminimization of the expected makespan, different methods are proposed to generate scheduling policies, and thepolicies define which activities to start at each decision times (decision points, including the start of project and thecompletion points of activities) by applying the distributions of activities’ durations.

In real world, however, each project is somewhat unique and durations of its activities are usually lack of historicaldata. In this case, belief degrees given by experienced project managers or experts might be employed to estimatevalues or distributions. Surveys have shown that these human beings’ estimations generally possessed much widerrange of values than the real [12]. Therefore, these indeterminacies described by human belief degrees cannot betreated as random variables or fuzzy variables. When indeterminacy behaves neither randomness nor fuzzyness,uncertainty theory, initiated by Liu [13] and refined by Liu [14], can be applied. So far, the new theory has beensuccessfully applied in various problems: option pricing problem [15, 16], stock problem [17], production controlproblem [18, 19], inventory problem [20], assignment problem [21], supply chain pricing problem [22], etc. With

∗Corresponding author.Email:huanghu0213@163.com (Hu Huang)

Preprint submitted to Submitted to publication May 22, 2015

regard to project scheduling problem, Zhang and Chen [23] built an uncertain model to minimize expected projectmakespan under chance constraint of the total cost. Ji and Yao [24] considered an uncertain project schedulingproblem to make a schedule for allocating loans so that the total cost and the completion time of project are balancedunder some constraints. Ke [25] proposed a genetic algorithm-based hybrid approach to solve a project time-costtrade-off problem with uncertain measure. Ding and Zhu [26] considered two types of models for uncertain projectscheduling problems for different management requirements. Ke [27] and Ke et al. [28] further studied the projecttime-cost trade-off problem in environment involving simultaneous uncertainty and randomness.

However, no research dealt with URCPSP by which we mean project scheduling problem with uncertain durationsand renewable resource constraints. This paper will fill the gap and deal with this type of resource-constrained projectscheduling problem. Professional managers’ or experts’ estimations are usually applied to estimate these uncertaindurations covering insufficient historical data and they can be characterized by uncertain variables. Furthermore, forpurpose of decoding schedule policies, an uncertain serial project generate scheme (US-SGS) is introduced. Sincethe makespan calculated based on expected value or medium number of each activity duration is usually less than thereal one [6], uncertain simulation based on 99-method is employed to make approximation of uncertain functions. AGA-based algorithm is designed to search for the quasi-optimal schedule afterwards.

The reminder of this paper is organized as follows: In Section 2, some basic concepts of uncertainty theory areintroduced. In Section 3, formulations of the URCPSP are detailed. Meanwhile, an uncertain expected value modelis established and transformed into a crisp form in some special case. Furthermore, a hybrid algorithm integrating99-method and genetic algorithm is proposed in Section 4. Numerical examples are presented in Section 5 to illustratethe effectiveness of the model and the algorithm. Relevant conclusions and prospective extensions are discussed inSection 6.

2. Preliminaries

In this section, some concepts and theorems of uncertainty theory are introduced to lay the foundation for theURCPSP modeling. Uncertainty theory is a branch of axiomatic mathematics for subjective uncertainty modeling,which has been well developed and applied in a wide variety of real problems.

Let Γ be a nonempty set, L a σ -algebra over Γ, and each element Λ in L is called an event. Uncertain measure isdefined as a function from L to [0,1]. Uncertain measure M is a set function defined over the following four axioms.

Axiom 1. (Normality Axiom) M{Γ}= 1.

Axiom 2. (Duality Axiom) M{Λ}+M{Λc}= 1 for any event Λ.

Axiom 3. (Subadditivity Axiom) For every countable sequence of events {Λi}, we have:

M{∞⋃

i=1

Λi} ≤∞

∑i=1

M{Λi}.

Axiom 4. (Product Measure Axiom) Let (Γk,Lk,Mk) be uncertainty spaces for k = 1,2, · · · . The product uncertainmeasure M is an uncertain measure satisfying

M

{∞

∏k=1

Ak

}=

∞∧k=1

Mk{Ak}

where Ak are arbitrarily chosen events from Lk for k = 1,2, · · · , respectively.

Definition 1. [13] An uncertain variable is a measurable function ξ from an uncertainty space (Γ,L,M) to the setof real numbers, i.e., for any Borel set B of real numbers, the set

{ξ ∈ B}= {γ ∈ Γ∣∣ ξ (γ) ∈ B}

is an event.

2

The uncertainty distribution is indispensable to establish the practical uncertain optimization problems model.

Definition 2. [13] The uncertainty distribution Φ of an uncertain variable ξ is defined by

Φ(x) =M{ξ ≤ x}

for any real number x.

An uncertainty distribution Φ is confirmed to be regular if its inverse function Φ−1(α) exists uniquely for eachα ∈ [0,1].

Definition 3. [13] Let ξ be an uncertain variable. The expected value of ξ is defined by

E[ξ ] =∫ +∞

0M{ξ ≥ r}dr−

∫ 0

−∞

M{ξ ≤ r}dr

provided that at least one of the above two integrals is finite.

Lemma 1. [14] Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then

E[ξ ] =∫ +∞

0(1−Φ(x))dx−

∫ 0

−∞

Φ(x)dx.

Lemma 2. [14] Let ξ be an uncertain variable with regular uncertainty distribution Φ. If the expected value exists,then

E[ξ ] =∫ 1

0Φ−1(α)dα.

Example 1. The expected value of the zigzag uncertain variable ξ = Z(a,b,c) is:

E[ξ ] =∫ 0.5

0((1−2α)a+2αb)dα +

∫ 1

0.5((2−2α)b+(1−2α)c)dα

=a+2b+ c

4.

(1)

Lemma 3. [14] Let ξ1,ξ2, · · · ,ξn be independent uncertain variables with regular uncertainty distributions Φ1,Φ2, · · · ,Φn,respectively. A function f (x1,x2, · · · ,xn) is strictly increasing with respect to x1,x2, · · · ,xm and strictly decreasing withrespect to xm+1,xm+2, · · · ,xn. Then ξ = f (ξ1,ξ2, · · · ,ξn) is an uncertain variable with inverse uncertainty distribution

Ψ−1(α) = f (Φ−1

1 (α), · · · ,Φ−1m (α),Φ−1

m+1(1−α), · · · ,Φ−1n (1−α)). (2)

Lemma 4. [29] Let ξi(i = 1,2, · · · ,n) be independent uncertain variables with regular uncertainty distribution-s Φ1,Φ2, · · · ,Φn respectively. If a function f (x1,x2, · · · ,xn) is strictly increasing with respect to x1,x2, · · · ,xm andstrictly decreasing with respect to xm+1,xm+2, · · · ,xn, respectively, then the expected value of uncertain variablesξ = f (ξ1,ξ2, · · · ,ξn) has an expected value as follows:

E[ξ ] =∫ 1

0f (Φ−1

1 (α), · · · ,Φ−1m (α),Φ−1

m+1(1−α),Φ−1n (1−α))dα (3)

provided that the expected value E[ξ ] exists.

3

3. Formulation and model

3.1. Problem description

The URCPSP considered in this paper is described as an activity-on-the-node network G(N,A), where N is the setof activities and A is the set of pairs of activities between which a zero-lag finish-start precedence relationship exists.Dummy activities i = 1 and i = N + 2 mark the beginning and the end of the project. The duration of each activitydi is assumed to be an uncertain variable whose distribution is given by experts’ or managers’ belief degrees, andthe resource requirement of resource k is indicated by rik. For renewable resource k, the availability Rk is a constant,which is constrained throughout the project. Each activity is executed in exclusive one mode and preemption is notallowed.

The execution of a project in uncertain context can be regarded as a dynamic decision process. The managermakes decisions on which activities to start at each decision times (the start of the project and the completion timesof activities) subject to the precedences and resource constraints. The solution of URCPSP is an activity order listwhich depicts the priority of each activity. Because of the uncertainty, the actual activity duration can only be attainedafter it is completed. According to the information incompleteness, the decision maker can merely apply partialinformation which is available before or at the time he makes decision. So the schedule generation scheme (SGS)used in deterministic condition may not always be feasible. In this paper, we develop a new procedure denoted asuncertain serial SGS (US-SGS) to decode the activity list λ . It works as the serial SGS, but adds the side constraint

si(λ )≤ maxλ j<λ i

(s j(λ )

)(4)

where λ i is the position of activity i in activity list λ and si is the starting time of activity i.The US-SGS works as follows:0: Given a network G(N,A). λ := activity list; λi:=activity in position i; d:= duration; E:= underway activity set;

U:= unexecuted activity set; S:= start time, F:= finish time, T:= Time point, num:= number of activities; K:= numberof renewable resources; rik:= demand of activity i for resource type k; Rk:= available amount of resource k.

1: E =∅, U = [1,2, · · · ,num], F = [in f , in f , · · · , in f ], S = [0,0, · · · ,0]2: im = 1, T = 03: While isempty (U) = 04: Sλim = max

i<imSλi ∨ max

( j,λim)∈AFj

5: if ∑i∈E

(rik)+ rimk ≤ Rk, k = 1,2 · · · K & Sλim ≤ T

6: F(λim) = T +d(λim)7: Put λim into E8: im = im+19: else10: T = min

i∈E(F(i))

11: Take out i from E and U if: F(i)i∈A ≤ T12: end13: end14: Return S and F

3.2. Expected value model

The URCPSP aims at minimizing the expected total project duration (makespan), in accordance with sn+2, denotedas start time of activity N +2, since it is a dummy end. Therefore the makespan Ft is:

Ft = sn+2

4

The mathematical model of the URCPSP can be established as follows:min E[Ft ]

subject to:si +di ≤ s j, ∀(i, j) ∈ A

∑i∈Pt

rik ≤ Rk, k = 1,2 · · · K, Pt = {i | si ≤ t < si +di}

In the above model, the objective function is to minimize expected makespan. The first constraint ensures that theearliest start time of activity j is forbidden to be smaller than the finish time of its predecessor activity i. The secondconstraint guarantees, for each time point t and resource type k, the resource demand should be below the resourcelimitation. Note that Pt represents the set of underway activities at time t, rik is the demand of activity i for resourcetype k, and Rk is the given available amount of resource k.

Because of the uncertain durations, the finish time of each activity is changeable and the makespan is an uncertainvariable. Given the activity list λ , the finish time of each activity can be attained as follows:

fi(λ ) = si + di

The critical problem is how si should be determined. If resource constraint is eliminated, the starting time ofactivity i, which is an uncertain variable as well, can be computed by

si(λ ) = max( j,i)∈A

(s j(λ )+ d j

)(5)

where λ is an activity list depicting the priority of each activity.When resource constraint is considered, however, this formula is no longer valid. An activity may be feasible in

term of precedence relation when all of its predecessors have been finished, while simultaneously infeasible for thelack of available resource. Namely, one activity has predecessors in both precedence relation logic and resource logic.Besides, a side constraint is added to decode the activity list. With the purpose of settling this problem, similar withArtigues and Roubellat [30], we define an extended precedence relation set as G(N,A∗) by adding extra resource flownetwork into the original activity-on-the-node network G(N,A). The namely resource flow network exists if there isa resource flow between two activities without precedence relations. Then the start time of activity i, a function ofactivity list λ and uncertain durations, can be attained by

si(λ ) = maxλ j<λ i

(s j(λ )

)∨ max

( j,i)∈A∗

(s j(λ )+ d j

)(6)

where λ i is the position of activity i in activity list λ .Given an activity list λ , one can obviously attain that si(λ )is monotone increasing of d j where λ j < λ i or ( j, i) ∈

A∗. Therefore, referring to Lemma.3, we have

Theorem 1. Suppose that the duration of activity i has regular distribution Φi and inverse uncertainty distributionΦ−1i , (α). Given the activity list λ , si(λ ) has an inverse uncertainty distribution Ψ

−1i (λ ,α) α ∈ (0,1], and we can

getΨ−1i (λ ,α) = max

λ j<λ i

(Ψ−1j (λ ,α)

)∨ max

( j,i)∈A∗

(Ψ−1j (λ ,α)+Φ

−1j (α)

)With Theorem.1, we can attain that the α-value of the makespan can be calculated by d=(Φ−1

d1(α),Φ−1

d2(α), · · · ,Φ−1

dn+2(α)).

Therefore, referring to Lemma.4, we have

Theorem 2. The expected makespan, depicted by the start time of dummy end activity, can be attained as follows:

E[Ft ] =∫ 1

0Ψ−1n+2(λ ,α)dα

5

The equivalent form of the uncertain model can be obtained as follows:min

∫ 10 Ψ

−1n+2(λ ,α)dα

subject to:si +di ≤ s j, ∀(i, j) ∈ A

∑i∈Pt

rik ≤ Rk, k = 1,2 · · · K, Pt = {i | si ≤ t < si +di}

With the crisp form of the expected model, 99-method can be applied to make approximation of the expectedmakespan.

min99∑

m=1Ψ−1n+2(λ ,m/100)/99

subject to:si +di ≤ s j, ∀(i, j) ∈ A

∑i∈Pt

rik ≤ Rk, k = 1,2 · · · K, Pt = {i | si ≤ t < si +di}

4. GA-based algorithms

To solve the RCPSP with uncertain durations, the expected duration or medium number are usually employed toreplace the random or uncertain numbers and then the deterministic model is applied. However, the optimal solutionattained from the deterministic model may perform very poor in real context. Meanwhile, the makespan, calculatedbased on the expected value of each activity duration, is often less than the real one. Stork [6] presented that theexpected makespan is more than 4% larger than the deterministic makespan with instances containing 20 activities.Moreover, coherently, Ballestin [7] revealed that this variation may extend to more than 10% when the number ofactivities increases to 120.

Unfortunately, even just for precedence constrained jobs without any resource restrictions, the computation ofthe expected length of a critical path is NP-complete if each activity processing time distribution has two (discrete)values. Similar with the researches in stochastic cases, uncertain simulation is essential for approximation of theexpected makespan. Solving URCPSP is more difficult than the deterministic one. If all the uncertain durations areassumed to be real numbers, then the URCPSP is reduced to be a deterministic RCPSP, and consequently, NP-Hard.

In order to solve URCPSP, a heuristic algorithm integrated by 99-method and GA is introduced in this section.The chromosome is denoted by the activity list λ in which activities are arranged in different orders on the premisethat the precedence relationship is satisfied. Then, with a specific schedule generation scheme, a feasible solution canbe gained. Besides, the most essential part of the algorithm is the process of decoding in accordance with differentschedule generation schemes, which will be highlighted in this part.

The structure of the hybrid algorithm is as follows:Step 1: InitializationThe first step is to randomly generate initial population with size Popsize as parent population. Individual creations

rely on a list chromosome representation: an activity order list λ in which activities are arranged randomly on thepremise that the precedence relationship is satisfied.

Step 2: CrossoverCrossover is the main operation to produce the new chromosomes, and simultaneously expand the population. In

this paper, we implement one point crossover. First, the mother and father chromosomes are randomly chosen. Thecrossover points are randomly generated as well. Taking the position denoted by crossover point as boundary, theleft positions inherit exactly the same activity order from the mother. Subsequently, the remaining right positions arefilled according to the father by left-to-right scan on the premise that the precedence relationship is satisfied, skippingthe activities already taken from the mother.

Step 3: MutationMutation is significant for the purpose of population diversity. For the activity order list, mutation begins with

randomly selecting the activity which will be mutated. Mutation applied to the activity order list is built to keep list

6

precedence feasible. It determines the predecessor and successor of the selected activity first, takes their positions asbound, and then randomly chooses and inserts in a position in the range to generate a feasible child chromosome.

Step 4: DecodingIn this step, we employ the US-SGS to decode the activity list λ and approximate the fitness of each chromosome

by uncertain simulation based on 99-method, initiated by Liu [14] and successfully applied in project scheduling byZhang and Chen [23].

The uncertain simulation based on 99-method for E[Ft ] is presented in Algorithm 1.

Algorithm 1. (Uncertain simulation for E[Ft ])

1: Give an activity list λ ;2: Generate dm

1 ,dm2 , · · · ,dm

n according to the inverse uncertainty distributions of activities’ durations Φ−1d1

, Φ−1d2

,· · · , Φ

−1dn

, and denote dm=(Φ−1d1(m/100),Φ−1

d2(m/100), · · · ,Φ−1

dn(m/100)), m=1,2, · · · ,99, respectively;

3: Decode the activity list λ by using US-GSG for each dm, m=1,2, · · · ,99. Then the start time sm and finish timef m can be attained;

4: f mt = sm

n , m = 1,2, · · · ,99;

5: E[Ft ]=99∑

m=1f mt /99.

Step 5: SelectionWith the purpose of keeping the population size consistent, new parent population requires to be selected from

the combined population. This paper employs the roulette wheel selection, based on individuals’ fitness values.Individuals with better fitness are attached with larger probability to be selected in the roulette wheel. Additionally,the top-3 best-to-now chromosomes are specifically preserved in the population. As a consequence, selection processguarantees the high quality of population, meanwhile avoids the local optimum.

The framework of the algorithm is presented in Fig.1.

Start

Initialization

Crossover

Mutation

Selection

End

Yes

Terminating condition

Decod

ing

(99

-meth

od

)No

Figure 1: Outline of the hybrid GA algorithm

7

5. Numerical example

In this section, we consider a project schedule problem containing 30 activities (J30) and 4 renewable resources.For assumption of uncertainty, the durations are estimated by experts and characterized as uncertain variables. In-terested readers can consult [12] for more details about how to use experts’ belief degrees to estimate uncertaindistributions. Information of the numerical instance including durations, resource utilizations and successor activitiesis presented in Table 1, and correspondingly, the structure of the project is provided in Fig.2.

Table 1: The information of the test instances

Activity Duration Expected Duration R1 R2 R3 R4 Successors1 0 0 0 0 0 0 2, 3, 42 Z(5,8,10) 7.75 4 0 0 0 6, 11, 153 L(3,5) 4 10 0 0 0 7, 8, 134 Z(4,6,7) 5.75 0 0 0 3 5, 9, 105 L(2,4) 3 3 0 0 0 206 Z(6,8,9) 7.75 0 0 0 8 307 Z(3,5,6) 4.75 4 0 0 0 278 Z(7,9,13) 9.5 0 1 0 0 12, 19, 279 L(1,3) 2 6 0 0 0 14

10 Z(6,7,9) 7.25 0 0 0 1 16, 2511 Z(5,9,11) 8.5 0 5 0 0 20, 2612 L(7,9) 8 0 7 0 0 1413 Z(4,6,7) 5.75 4 0 0 0 17, 1814 Z(2,3,6) 3.5 0 8 0 0 1715 Z(6,10,12) 9.50 3 0 0 0 2516 Z(7,11,12) 10.25 0 0 0 5 21, 2217 Z(3,6,7) 5.5 0 0 0 8 2218 Z(3,5,6) 4.75 0 0 0 7 20, 2219 Z(1,3,4) 2.75 0 1 0 0 24, 2920 Z(5,7,8) 6.75 0 10 0 0 23, 2521 Z(1,2,4) 2.25 0 0 0 6 2822 Z(5,7,8) 6.75 2 0 0 0 2323 Z(2,3,5) 3.25 3 0 0 0 2424 Z(11,13,14) 12.75 0 9 0 0 3025 Z(1,3,4) 2.75 4 0 0 0 3026 Z(5,7,8) 6.75 0 0 4 0 3127 Z(6,8,11) 8.25 0 0 0 7 2828 Z(1,3,4) 2.75 0 8 0 0 3129 Z(5,7,8) 6.75 0 7 0 0 3230 Z(10,12,13) 11.75 0 7 0 0 3231 Z(10,12,13) 11.75 0 0 2 0 3232 0 0 0 0 0 0

Note: 1, 32 are dummy activity, and the limits of the four resource are (12, 13, 4, 12)

To minimize the expected makespan, the following model can be applied.min E[F32]

subject to:si +di ≤ s j, ∀(i, j) ∈ A

∑i∈Pt

rik ≤ Rk, k = 1,2 · · · 4, Pt = {i | si ≤ t < si +di}

8

4

5

7

8

9

11

13

14

15

16

18

19

20

21

23

25

26

27

12

2

10

29

17

28

3230

24

31

1 3

6

22

Figure 2: The activity-on-the-node network of the project

The algorithm is coded by MATLAB and runs on a PC with core i3 with 2.3GHz clock speed and 2G RAM. Thequasi-optimal schedules and expected makespan (EM) attained from experiments are presented in Table 2, resultingfrom 10 times’ run of the hybrid intelligent algorithm with 1000 cycles for the expected models under differentparameter settings.

Table 2: The top 10 quasi-optimal schedules

Seq Popsize pc pm Quasi-optimal schedules EM1 20 0.8 0.4 1,3,4,2,8,10,13,7,18,11,12,5,16,9,27,14,15,19,20,26,17,21,22,29,6,25,23,28,31,24,30,32 65.93002 30 0.7 0.4 1,3,4,8,7,2,13,10,9,18,11,16,19,12,27,26,5,14,21,15,20,17,22,6,28,31,29,23,24,25,30,32 66.11303 20 0.8 0.3 1,4,3,8,2,13,5,10,18,7,11,16,12,9,27,26,14,21,19,15,20,17,22,28,29,31,25,23,24,6,30,32 66.11304 20 0.8 0.4 1,3,4,13,8,2,10,18,7,11,16,12,27,9,26,14,5,21,15,19,20,17,22,28,31,6,29,23,24,25,30,32 66.11305 30 0.7 0.4 1,4,3,2,8,7,10,13,18,11,19,12,5,16,9,27,14,15,20,17,22,29,21,26,23,28,31,6,24,25,30,32 66.14406 30 0.7 0.4 1,3,4,2,8,10,13,7,18,9,5,16,12,11,27,14,15,20,19,17,22,29,25,21,26,23,28,31,24,6,30,32 66.14407 30 0.6 0.3 1,4,3,7,2,13,8,10,5,11,18,16,19,12,9,27,14,15,20,17,22,21,29,26,23,25,28,31,24,6,30,32 66.15008 20 0.8 0.3 1,3,4,8,2,13,10,11,7,18,12,16,27,9,14,5,15,19,20,17,22,25,26,29,21,23,28,31,6,24,30,32 66.15009 30 0.6 0.2 1,4,3,13,2,8,10,18,7,11,16,9,12,5,27,14,15,21,20,26,17,19,22,28,6,31,29,23,25,24,30,32 66.2120

10 30 0.6 0.2 1,3,4,2,13,8,5,10,7,18,11,16,12,27,9,14,15,21,20,17,26,19,22,28,31,29,23,6,24,25,30,32 66.2120

Similar with [31], the quality of solution can be obtained via the deviation of the worst solution from the optimalsolution attained in the 10 times’ run and the results are presented in Table 3. The results reveal that the deviationdenoted by percent error does not exceed 5%, which implies the effectiveness of the algorithm integrating uncertainsimulations.

6. Conclusion

In this paper, an uncertain model was proposed to deal with uncertain resource-constrained project schedulingproblem, in which the durations, with no historical data generally, were described as uncertain variables. Furthermore,

9

Table 3: The effectiveness of the GA

Gen Popsize pc pm The best The worst Average Error(%)1000 30 0.6 0.2 66.2120 68.0190 66.7959 2.73%1000 30 0.6 0.3 66.1500 69.4150 67.5808 4.94%1000 30 0.7 0.4 66.1130 69.1600 66.5831 4.61%1000 20 0.7 0.2 66.4450 68.4580 67.2897 3.03%1000 20 0.8 0.3 66.1130 67.8900 66.9114 2.69%1000 20 0.8 0.4 65.9250 67.7930 66.7275 2.83%

Note: error(%)= T he worst−T he bestT he best

an uncertain serial schedule generate scheme (US-SGS) for activities with uncertain durations was introduced. Forpurpose of capturing quasi-optimal strategies, we transformed the proposed uncertain project scheduling model into acrisp equivalence and designed a GA-based hybrid algorithm integrating uncertain simulation. A numerical examplewas provided to illustrate the effectiveness of the model and algorithm as well.

This paper involved merely one type of indeterminacy, while the real world project might be complicated thatrandomness and uncertainty might co-exist. Future research can extend the RCPSP to twofold indeterminacy andcorrespondingly uncertain random variable can be applied. In addition, this paper assumed that each activity wasexecuted with one mode, multi-mode RCPSP in uncertain environment can be taken into consideration in furtherdevelopment.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Nos.71371141,71071113,71001080),a Ph. D. Programs Foundation of Ministry of Education of China (No.2010007211011), Shanghai Philosophical andSocial Science Program (No.2010BZH003), and the Fundamental Research Funds for the Central Universities.

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