Scenario-based Solution Approach for Uncertain

Scenario-based Solution Approach for Uncertain Resource Constrained Scheduling Problems Forhad Zaman, Saber Elsayed, Ruhul Sarker, and Daryl Essam Abstract—Many real-world decision problems involve uncertain parameters. The Resource Constrained Project Scheduling Problem (RCPSP) is one of those problems in which the activity durations are usually uncertain. Over the last decade, a good number of solution approaches have been developed to solve such problems, among them the population based algorithms received great attention. In the solution approaches, a large number of scenarios are usually evaluated which is computationally expensive. In this paper, as an attempt to reduce the computational time, we propose few alternative approaches and experiment them with an assumption that the uncertain parameters are random variables. For experimental study, these variables are generated using four different probability distributions. The proposed approaches are compared with the traditional scenario based approach by solving 10 well-known benchmark problems with 30 activities. The results revealed that it has advantages in terms of solution quality and computational time.

I. I NTRODUCTION Over the last few decades, the resource-constrained project scheduling problem (RCPSP) has been widely studied as an NP-hard optimization problem [1]. It aims to minimize the make-span time by scheduling different projects while satisfying a number of precedence and resource constraints. In most papers in the literature, RCPSPs have been solved using the deterministic approaches in which the activity durations are assumed to be known with certainty. However, in reality, the activity durations are often subject to a high level of uncertainty for various reasons such as process interruption, unavailability of resources, and unexpected delay of preceding activities. The RCPSPs with uncertain activities are stochastic problems which are much more challenging than their deterministic version [2]. During the last decade, many methods have been proposed to solve different uncertain real-world problems. Such problems have been broadly categorized into three types, predictive, proactive and reactive ones. In the predictive approach, the uncertainty of a problem is ignored and a deterministic value is predicted in advance. The proactive approach considers all the possible uncertainties of the stochastic variants of the problem. Lastly, in the reactive approach, a base-line solution is determined without considering any uncertainty, but the obtained solution is revised or re-optimized when a variant is changed from its base value [3]. From the above three methods, the proactive one is very popular for obtaining a robust solution for a RCPSP; for example, Ballestín [4] used a genetic algorithm (GA), BallestÃn et. al [5] a greedy randomized adaptive search procedure (GRASP), Bruni et al. [6] an adaptive robust optimization model, Ashtiani et al. [7] local-search procedure, and

Chakrabortty et al. [8] a heuristic-based approach (details of these algorithms are discussed in subsection II-B). Although the above procedures may obtain feasible or local (or, optimal in some cases) solutions for solving a RCPSP, they have some drawbacks to determine the level of uncertainty in a given solution. On the contrary, Tseng et al. [9] developed a scenario-based stochastic model for RCPSPs in which the problem was solved by considering all possible scenarios during project execution. In it, thousands of scenarios are generated for each uncertain parameter, and many fitness values (FVs) are evaluated by considering each scenario. Then, the expected FVs are determined by averaging all the obtained FVs of each scenario. The primary advantage of this approach is that the operator does not need to know the actual value of the uncertain parameter, but based on the scenario-model, all the uncertainties related to project completion could be determined [3]. However, evaluating such a huge number of scenarios is expensive in terms of computational time. In this paper, we present four alternative approaches that use limited number of scenarios to reduce the computational time by producing competitive solutions. These approaches use a single scenario at the initial stage but a large number of scenarios at the later stage of the search process. The single scenarios are generated based on the lower bound, upper bound, mean, and random values of the uncertain durations. It is noted here that the traditional approaches use a large number of scenarios for evaluating the fitness value in every generation. For detail evaluations, the scenarios are generated with random activity durations represented by four different probability distributions namely normal, beta, exponential and Poisson. That means, each of the four approaches are experimented with four cases of random variables. In this study, two multi-operator based evolutionary algorithms are used as the base search algorithm [10]. The performance of the proposed approaches is demonstrated by solving 10 well-known RCPSPs with 30 activities. The detailed results are compared with a traditional scenario based approach. The comparisons reveal that the proposed approaches significantly reduce the computational time without sacrificing the quality of solutions. Moreover, we found that the fitness values approximately follow a normal distribution, regardless the distribution followed to generate the samples of activity durations. The rest of this paper is organized as follows: a brief introduction of RCPSP and its solution methodologies are discussed in Section II; the proposed solution approach is shown in Section III; the experimental results are presented in section IV and finally, the findings are summarized in section

V. II. RCPSP S AND M ETHODOLOGIES In this section, we provide an overview of RCPSPs and their solution methodologies. A. Problem Description A single RCPSP consists of D + 2 activities, and each activity must be completed in order to complete the entire project. There are two dummy activities, 0 and D + 1 those refer to the ‘start’ and ‘finish’ of the project, respectively. ˜ j , j = 1, 2, . . . , D, which are Each activity has duration, d subject to many uncertainties. The problem is formulated as an optimization problem with the aim to minimize project duration, subject to the precedence and resource completion constraints. The first constraints are used to ensure that activity j is not to be started, as long as any of its predecessors have not been finished. The second one is to use at each time for all the activities resource consumption to satisfy their maximum resource availability. There are two types of resources, renewable and non-renewable. The former one is limited to a particular time period and can be reused; e.g., machine and manpower. Whilst the non-renewable resources are used up throughout the project tenure, such as money. The optimization problem of RCPSP is defined as: Min. F TD+2 S.T: F Tj ≤ F Tj+1 − d˜j+1 , ∀j = {1, 2, . . . , D + 1}

rkj ≤ Rk , ∀k = {1, 2, . . . , K}

(1)

(2) (3)

j∈A(t)

F Tj ≥ 0, ∀j = {1, 2, . . . , D + 1}

(4)

where F TD+2 is the finish time (FT) of the last dummy activity, rkj is the required k th type of resource of the j th activity, which must be less than or equal to the maximum available k th type of resource, as Rk . K is the maximum number of resource types, and A(t) is a set of activities scheduled at time, t ≤ F TD+2 . In this research, we consider the durations, d˜j+1 , ∀j = {1, 2, . . . , D + 1} are uncertain, and it is assumed that an activity which is in-progress can not be interrupted, as long as it has not been completely finished. B. Literature Review RCPSPs have been widely studied in the literature with the assumption that their activity duration is known in advance. However, this fundamental assumption leads to poor performance, as durations are usually subject to a number of uncertainties that come from different directions, such as weather, unavailable resources, delayed delivery, the absence of workers. Consequently incorporating uncertainty of duration when solving uncertain RCPSPs has recently attracted many researchers and practitioners [2].

Over the last few years, many approaches have been proposed to solve such uncertain RCPSPs. Of them, metaheuristic-based approaches, such as GA [11], decompositionbased GA [12], hybrid GA (HGA) [13], differential evolution (DE) [14], fuzzy-clustering-based DE [15], improved particle swarm optimization (PSO) [16] and PSO-based hyper-heuristic [17] algorithms are very popular due to the complexity of the problem [2]. However, most the above-mentioned algorithms have their own pros and cons, some perform better for some problems but are inferior in others. Even for solving a single problem, an algorithm may perform well at an early stage of the process of evolution while it is inferior in other stages. To combine the strength of multiple algorithms and overcome their weakness, Saber et al. [10] proposed a consolidated optimization algorithm (COA) in which multiples algorithms were simultaneously performed while solving an RCPSP. The obtained results from his COA were compared with those of state-of-the-art algorithms, which revealed that it performed better for a wide-range of problems. Considering the uncertainties of activities duration in RCPSPs, Golenko and Gonik [18] developed a heuristic-based algorithm in which a zero-one integer programming approach is employed to solve them. In [19], RCPSP was solved in a stochastic programming context using a two-stage integer linear stochastic model. Recently, Bruni et al [6] developed an adjustable robust optimization model using such a two-stage stochastic programming approach. They also incorporated a decomposition technique to reduce the computational time. To address the uncertainty in RCPSPs, Zhao and Ke [20] solved the problems with the aim to maximize the net present value (NPV) of a project’s cash flows instead of the traditional minimization of make-span time. Ma et al. [1] solved uncertain RCPSPs using a GA in which activity duration was estimated based on an uncertainty-theory. However, estimating the actual values of duration is very challenging, and always has some errors. To overcome this estimation error, Tseng et al. [9] proposed a scenario-based-approach for uncertain RCPSPs, in which duration was represented as random variables. Each representation was called a scenario of an RCPSP. When FV of a sample solution was evaluated, NS number of possible scenarios were evaluated and their average value was considered as the expected FV of that sample solution. The primary advantage of this approach is that the operator does not need to know the actual information of the uncertain variants. However, a large number of scenarios (e.g., NS ) were generally required to obtain a robust solution for these scenario-based RCPSPs [3], [21]. Therefore, solving an RCPSP by considering many random scenarios of each activity is computationally expensive. The problem is even more complex when the NS -scenarios are considered at every generation of evolution, while solving it using any population-based method. This motivated us to propose an alternative approach for RCPSPs, in which the NS -scenarios of duration are considered only in the final generation, while their deterministic (or predicted) values are considered in other generations. In addition, it is not realis-

tic that each activity duration has been implemented before and its probability distribution function (PDF) is assumed known based on historical data, as some of the activities of a project might be totally new and their distribution are unknown. Therefore, considering a certain distribution while generating random numbers of an uncertain activity duration may lead to poor performance. Based on our knowledge, no research has dealt with an unknown probability distribution for uncertain activity durations. In this research, we represent the durations with NS -scenarios those follow a number of probability distributions. III. S OLUTION M ETHODOLOGY In this research, RCPSP is considered to be stochastic, specifically its parameters of the duration of each activity. To solve this problem, we use a scenario-based approach, in which Ns scenarios for each uncertain variant are generated using different probability distribution, as discussed in subsection III-B. The proposed approach starts with an initial population with NP random individuals. In the subsequent generations, new offspring are generated based on COA, as discussed in section III-A. Two strategies are used to evaluate solutions. The first strategy is the traditional one, in which every individual of a population, at each generation, are evaluated considering Ns scenarios of the durations. In the second strategy, the individuals of a population up to the last generation are evaluated considering a single value of the durations e.g., its minimum, mean or maximum value, while in the final generation, those are evaluated considering Ns scenarios. The details of evaluating the FVs are presented in subsection III-C. Algorithm 1 shows the steps of the proposed approach, with more detail given in the following subsections. Algorithm 1 Pseudo-code of the solution approach Require: Maximum number of generations, NG ; population size, NP ; and number of scenarios, NS . 1: Randomly generate initial population of size NP . 2: for p = 1 : NP do Loop for the population 3: Evaluate the expected FV and constraint violation (CV) of xp individuals, as discussed in Algorithm 2. 4: end for 5: for g = 1 : NG do Loop for the iteration/generation 6: Generate new NP offspring using the COA algorithm, as discussed in subsection III-A. 7: Evaluate the expected FV and CV of each individual as in steps 2 to 4. 8: Determine best individuals from both parents and offspring based on a greedy selection approach [22]. 9: end for A. COA Although a large number of attempts to solve different sets of RCPSPs have been carried out, there are still some drawbacks. The main drawback of the existing algorithms

is required an excessive parameter tuning to determine the best algorithm for each individual problem. To overcome this drawback, Elsayed et. al. [10] recently proposed a COA, in which two algorithms were considered simultaneously in a general framework, and each used multiple search operators. Due to the success of COA in solving deterministic RCPSPs, in this research, we employ COA to solve the scenario-based uncertain RCPSPs. The COA starts with a common population scheme and their individuals are evolved by two multi-operator evolutionary algorithms (MOEAs): multi-operator GA (MOGA) and multioperator DE (MODE). The new individuals are generated simultaneously using both algorithms under two different subpopulations. A local heuristic search is applied to every new individual. Then, a greedy selection scheme is used to obtain the best solutions, and those survive to the next generation. The process is continued until a cycle is completed. In the next cycle, a random algorithm is applied to generate the new offspring while the other one is turned off. The main advantage of this process is to increase the diversity of the search space during the evolution process, as one algorithm may work better in the early stage of the evolution but be poorer in the later stages. The details of the algorithm could be found in [10]. B. Scenarios Generation ˜ j ∀j) of each activity are In this research, durations (d considered uncertain, which is represented by a number of scenarios (Ns ) . It is assumed that the uncertain parameters, ˜ j ∀j follow one of the following four probability distributions: d (i) normal, (ii) beta, (iii) exponential, and (iv) Poisson. In the following subsections, we show the process of generating ˜ j ∀j using such distributions. scenarios of d 1) Normal Distribution: The normal distribution (also called Gaussian distribution) is the most common form of distribution, because many real-world problems can be modeled ˜ j ∀j using the normal with it. The random scenarios of the d distribution are generated as: ˜ r,j = N (μj , σj ) , ∀r = 1, 2, . . . , Ns d

(5)

where NS is the number of scenarios, μj and σj are the mean and standard deviation of the duration of the j th activity. In this research, we consider μj = dj and σj = 0.5dj ∀j = 1, 2, . . . , 30, where dj is the given duration for a RCPSP. 2) Beta Distribution: This is another type of continuous distribution, and is very popular in RCPSPs [23]. It is generally defined in the interval between 0 to 1, and is parametrized by two positive shape parameters, α and β. The random scenarios ˜ j ∀j are generated using the beta distribution as: of the d rj = B(α, β)

(6)

˜ r,j = LBj + (U Bj − LBj ) × rj d

(7)

where rj ∈ [0, 1] is a beta random number, and LBj and U Bj are the lower and upper limits of the duration of the j th activity. We assume, α = 2, β = 5, LBj = max (0, (μj − σj )) and U Bj = (μj + σj ). 3) Exponential Distribution: The exponential distribution can be used when there is a constant rate of failure. It is a special case of the gamma and Weibull distributions. ˜ r,j ∀j are generated using the exponential Random durations, d distribution as: ˜ r,j = E(μj ) d

(8)

where μj is the same as the above mean value of the duration of the j th activity. 4) Poisson Distribution: This is a discrete distribution which models the number of events based on a constant rate of occurrence. It is also used as an approximation of the binomial distribution, in which the number of trails is large and the probability of success becomes low. Using this discrete distribution, the scenarios are generated, as: ˜ r,j = P (μj ) d

(9)

Note that the random samples obtained from the other three distributions (except Poisson) are continuous values. For simplicity and to use the COA algorithm, the random samples ˜ r,j = of durations are rounded to their nearest integer, i.e.,d ˜ dr,j . C. Evaluating FVs and CVs The FVs and CVs of an RCPSP are evaluated considering ˜ r,j ∀j, r. In this research, we evalNs −scenarios of duration, d uate them using two different strategies: first is the traditional one which considers NS −scenarios at every generation to evaluate the FVs and CVs of every individual in a population. On the contrary, the second one evaluates them by considering ˜ j ∀j up to (NG − 1) generations, a deterministic value of d while those are evaluated considering NS −scenarios in the final (= NG ) generation. The FV of an individual is evaluated using Eqn. (1), while the CV is calculated as follow [3]:

CVi =

K k=1

E 0, He (xi ) max 0, Gk (xi ) +

(10)

e=1

where Gk and He the inequality and equality constraints of ith individual (i.e., xi ), respectively, and K and E the numbers of inequality and equality constraints for a RCPSP, respectively. The pseudo-code of evaluating the FVs and CVs of an individual is shown in Algorithm 2. IV. E XPERIMENTAL R ESULTS In this section, we solve an instance of RCPSPs using our proposed approach in the uncertain environment. An instance was taken from PSPLIB and it contains 10 problems, with each has 30 activities (j30). Each problem considers 1000 scenarios (i.e., NS = 1000) of each activity-duration

Algorithm 2 Pseudo-code of FVs and CVs evaluations Require: xp , ∀p = 1, 2, ..., NP is an individual of a population, and NS is number of scenarios. 1: switch strategy do 2: case strategy-1 3: Generate the NS -scenarios for each d˜r,j ∀j, r = 1, 2, ..., NS , considering a certain PDF, as discussed in subsection III-B. 4: for r = 1 : NS do Loop for the scenario 5: Evaluate the F Vr and CVr , considering rth scenario for each d˜r,j ∀j. 6: end for 7: Based on the values of F Vr and CVr , r = 1, 2, .., NS , determine the expected FV and CV for of the xp individual. 8: 9: 10: 11: 12: 13: 14:

case strategy-2 if current_generation < NG then Consider a single scenario of duration, i.e., either d˜j = LBj or d˜j = μj or d˜j = U Bj . Evaluate the deterministic FVs and CVs for the xp individual. else Performs steps 3 to 7. end if

˜ r,j ∀j = 1, 2, . . . , D , these are generated using one of the d four probability distributions. The problems are solved using the COA algorithm with considering the following cases: •

•

•

•

•

˜ r,j ∀j are considered at the final case-A: NS -scenarios of d ˜ r,j ∀j are used generation, while the minimum values of d in other generations; ˜ r,j ∀j are considered at the final case-B: NS -scenarios of d ˜ r,j ∀j are used generation, while the average values of d in other generations; ˜ r,j ∀j are considered at the final case-C: NS -scenarios of d ˜ r,j ∀j are used generation, while the maximum values of d in other generations; ˜ r,j ∀j are considered at the final case-D: NS -scenarios of d ˜ r,j ∀j are used in generation, while random values of d other generations; and ˜ r,j ∀j are considered at every case-E: NS -scenarios of d generation during the evolutionary process.

The parameters of both COA and RCPSPs are kept the same values as in [10]. However, the maximum fitness function evaluations (FFEs) are set to 1000, as the algorithm requires evaluating a large number FFE to solve NS - scenarios for each individual. Fifteen independent runs were performed for each test case of each problem, and solutions are recorded for further analysis. The algorithm was run until it reached the maximum number of generation (NG ), and it was implemented on a desktop computer with a 3.4 GHZ Intel Core i7 processor and 16 GB of RAM using the Matlab (R2017a) environment.

Table I: FVs of RCPSPs when durations considered their minimum, mean and maximum values

Min. Mean Max.

2 32 47 64

Mean FVs 3 4 27 41 47 62 67 84

of different 5 6 23 32 39 48 57 66

problems 7 8 38 39 60 53 82 74

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1 43.78 5.12 0.77 43.78 5.05 0.51 43.97 5.17 0.77 43.64 5.04 0.49 43.95 5.16 4.78

2 46.37 5.21 0.74 46.37 5.22 0.49 46.36 5.19 0.74 46.42 5.28 0.47 46.56 5.21 2.57

3 44.86 5.65 0.72 44.50 5.51 0.47 44.65 5.50 0.72 44.73 5.56 0.47 44.67 5.49 2.55

4 60.90 6.60 0.82 61.27 6.50 0.53 61.40 6.42 0.83 61.01 6.56 0.53 61.47 6.45 2.78

5 37.55 4.89 0.73 37.84 5.00 0.46 37.91 5.01 0.76 37.51 4.88 0.47 37.74 4.91 2.13

6 46.30 5.82 0.74 47.10 5.62 0.45 46.91 5.62 0.72 46.24 5.77 0.46 46.25 5.91 2.74

7 56.68 7.13 0.72 56.62 7.20 0.45 56.66 7.16 0.73 56.50 7.26 0.46 56.57 7.13 2.94

8 53.16 6.26 0.75 52.84 6.15 0.48 52.96 6.19 0.76 52.66 6.19 0.48 52.59 6.15 2.70

9 48.89 5.46 0.74 48.73 5.38 0.46 48.66 5.38 0.74 48.50 5.36 0.47 48.83 5.39 2.64

10 44.20 5.17 0.74 44.82 5.11 0.45 44.95 5.09 0.73 44.10 4.90 0.47 44.38 5.16 2.54

A. Analyses of Different Cases The effect on determining expected FVs by considering different scenarios of duration with different kind of evolution is analyzed. We assume the samples of duration are normally distributed and their FVs also follow the same distribution. Before considering any of these scenarios, we solved all RCPSPs using the minimum (LBj ∀j), mean (μj ∀j) and ˜ r,j ∀j. After solving each maximum (U Bj ∀j) values of d problem 15 times, Table I shows the mean FVs of different RCPSPs for different activity durations. It is seen that the FVs correspondingly varied with duration, i.e., the greater the ˜ r,j ∀j, the greater the make-span time (i.e., FVs), values of d and due to uncertainty, the FVs could vary within the range that reported in Table I. ˜ r,j ∀j are considered with different Now, the scenarios of d kinds of evaluation (i.e., cases-A to E) for solving different RCPSPs. Table II shows the results of the expected FVs (i.e., Expe.), standard deviation (Std.) of the expected FVs, and the computational time (in minutes) of all cases. Although the FVs and standard deviations of all cases are almost the same, the computational times significantly vary from case-E to the others. This means that evaluating the expected FVs with NS scenarios at each generation during the process of evolution is unnecessary. On the contrary, when the FVs are evaluated using the same NS scenarios at only the final generation, and with only a single value (either its minimum, mean or maximum value) at other generations, the same quality solutions were obtained with significantly less computational time. Fig. 1 shows the PDFs of the FVs at the final generation for all the cases. It is seen that their PDFs approximately fit (details are discussed in subsection IV-C) to a normal distribution, even when the scenarios are considered at final generation only (i.e., case-A to case-D).

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Table II: Statistics of the FVs for RCPSPs with different cases

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Fig. 1: PDF of the FVs of RSPSP (problem no.-1) at final generation when solving using different cases Table III: p(h0 )−values of the Friedman test for different problems Criterion FVs Std. Problem> FVs Std.

1 0.00 (1) 0.22 (0) 6 0.00 (1) 0.00 (1)

p(h0 ) values of the 2 3 0.44 (0) 0.08 (0) 0.60 (0) 0.55 (0) 7 8 0.17 (0) 0.01 (1) 0.16 (0) 0.71 (0)

problem 4 0.30 (0) 0.15 (0) 9 0.04 (1) 0.20 (0)

5 0.04 (1) 0.15 (0) 10 0.00 (1) 0.00 (1)

B. Statistical Test In this section, the obtained results from different cases are statistically compared using two well-known non-parametric tests: Friedman and Wilcoxon sign tests. Firstly, we mutually compare the results of all cases for each run for every problem using the Friedman test. We assume the null hypothesis (h0 ) that there is no significant difference between the specified populations at the 5% significance level. If h0 = 0, it indicates that the null hypothesis cannot be rejected, while h0 = 1 means the null hypothesis can be rejected. Table III lists the p−value of different problems in different cases based on the criteria of FVs, standard deviation and computational time. The binary values within the parenthesis represent the status of the null hypothesis. It is seen that there is a significant difference in a few problems. However, based on the results presented in Table III, it is difficult to distinguish which results are significantly different than others. Therefore, we use the Wilcoxon nonparametric test to analyze the results of each problem at each run (i.e., 10 problems × 15 runs = 150 values) in each case against the traditional scenario-based method i.e., case-E. We define three notations, such that ‘+’ means that case-E is significantly better than other cases in terms of expected FVs in the final generation, ‘−’ is significantly worse and ‘≈’ is no significant difference. We use 5% significance level for all comparisons. Table IV shows the statistics of the obtained

results for different comparisons. It is seen that the results obtained from cases B and D have no significant differences than those obtained from the traditional approach of case-E. However, when RCPSPs are solved considering the minimum ˜ r,j ∀j, i.e., case-A and case-C, there or maximum values of d is a significant difference among their solutions from those obtained in case-E.

˜ r,j ∀j are uncertain, and As mentioned, we consider that d assume they follow a certain PDF. For testing purposes, we considered the four probability distributions: normal, beta, exponential and Poisson. To demonstrate the effect on final ˜ r,j ∀j using differexpected FVs for generating scenarios of d ˜ r,j ∀j ent PDFs, we use case-B, in which NS scenarios of d are independently generated at the final generation, while their mean values are used in the previous generations. The expected FVs of each individual in the final population are determined using the process described in subsection III-C. To compare the distributions of the obtained FVs with the ˜ r,j ∀j, a quantile-quantile (QQ) plot [24] input distributions of d is used. The QQ-plot is a well-known graphical technique to determine whether two data sets follow a common distribution or not. When the condition is true, the points approximately fall along a reference linear line. When the points deviate from the reference line, the two data sets may not fall in the same distribution. The greater the departure from the linear line, it is correspondingly likely that the two sets of data have come from different distributions [24]. Figs. 2 to 5 show the sample QQ-plots of problem-1, ˜ r,j ∀j are generated using normal, where the scenarios of d beta, exponential and Poisson distributions respectively. For each, the obtained FVs are assumed to follow the above four distributions. It is seen from Fig. 2, that the FVs approximately ˜ r,j ∀j which are fall with the reference samples of the d normally distributed. It is also seen from Figs. 3 to 5, that the FVs approximately lie with the reference line of normal ˜ r,j ∀j are distribution, even when the input samples of the d generated using different distributions. Therefore, it can be said that the obtained points of the FVs always follow a normal distribution (approximately), regardless of the distribution used ˜ r,j ∀j. to generate the samples of d To demonstrate the numerical errors of the obtained FVs with a reference line of an ideal distribution, mean-squareerror(MES) is calculated as: NS 2 1 F Viref − F Vitrue Ns i=1

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Fig. 2: QQ Plot of the FVs when the samples of durations are generated using a normal distribution

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Worse 80 87 89 56


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Cases E vs. A E vs. B E vs. C E vs. D


Table IV: Wilcoxon sign test for case-E vs. different cases


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Fig. 3: QQ Plot of the FVs when the samples of durations are generated using a beta distribution where F V true are the actual FVs after solving RCPSPs when ˜ r,j ∀j generated using a certain considering NS -scenarios of d ref are the ideal points of the FVs generated distribution. F V using the same distribution. Table V shows the MSEs for different probability distributions for the first problem. The rows represent the input ˜ r,j ∀j that are generated using a distribution, e.g., samples of d ˜ r,j ∀j are generated the first row indicates the scenarios of d using normal distribution, those of the second are generated using the beta, third using exponential and fourth using Poisson distribution. The columns represent the assumptions of the FVs’ following a distribution after solving RCPSPs. For example, in the first column, it is assumed that the obtained FVs are normally distributed, while those in the second, third and fourth are beta, exponential and Poisson distribution, respectively. Each cell of Table V indicates the MSEs of such

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Normal 97.97% 100% 0.00% 99.34%

Fitness Value Beta Exponential 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Poisson 2.03% 0.00% 100% 0.66%

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distribution. Since the durations of activity for an RCPSP are highly unlikely to follow the exponential distribution, we can say that the FVs of an uncertain RCPSP always follow a normal distribution. V. C ONCLUSION AND F UTURE W ORK

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1



80

Poisson 69.82 37.93 298.41 76.68

150

80

Quantiles of normal Distribution 100

Fitness Value Beta Exponential 95.17 1863.27 29.61 904.28 693.09 3296.72 164.61 1945.80


Fig. 4: QQ Plot of the FVs when the samples of durations are generated using an exponential distribution

80

Normal Beta Exponential Poisson

Normal 44.28 7.91 499.52 58.16

Table VI: Statistics of FVs which follow a distribution

durations

100



150

80 60 40 20 20

40

60

80


Fig. 5: QQ Plot of the FVs when the samples of durations are generated using a Poisson distribution

a combination; for example, the first cell (44.28) represents ˜ r,j ∀j are generated using a the MSEs when the samples of d normal distribution and the obtained FVs for problem-1 are fitted with the normal distribution. From the table, it is seen that the MSE is minimum when the random durations are generated using a beta distribution, and maximum when those are generated using an exponential distribution. In addition, it is evident that regardless the distribution of scenarios for ˜ r,j ∀j, the FVs approximately fit to the normal distribution d with minimum errors. However, when the random durations is generated using an exponential distribution, the FVs are most likely to follow a Poisson distribution. Table VI shows that the statistics of the MSEs for all ten RCPSPs, in which it is also seen that the FVs most likely follow a normal distribution for all cases, with the exception of ˜ r,j ∀j that are generated using the exponential the scenarios of d

In this research, we solved a number of uncertain RCPSPs using a scenario-based approach. Activity durations were considered uncertain, and they were represented as a number of scenarios which generated using different probability distributions. Although in a traditional scenario-based-approach, the scenarios are considered in every generation of evolution, we only considered them in the final generation. Based on the comparisons, it was evident that the same quality solutions, with significantly less computation could be obtained using our proposed approach. From statistical analyzes, it was found that there was no significant difference among the solutions obtained from either traditional and proposed solution approaches. In addition, when the scenarios of durations are generated using any of four different probability distributions, the obtained FVs after solving RCPSPs, approximately fitted to a normal distribution. The performance of our proposed approach was tested by solving j30 activities of 10 problems. In future, more complex problems can be considered, along with continuous values of durations. ACKNOWLEDGMENT This research has been funded by Australian Research Council (ARC) Discovery Project DP170102416. R EFERENCES [1] W. Ma, Y. Che, H. Huang, and H. Ke, “Resource-constrained project scheduling problem with uncertain durations and renewable resources,” International Journal of Machine Learning and Cybernetics, vol. 7, no. 4, pp. 613–621, Aug 2016. [2] S. Creemers, “Minimizing the expected makespan of a project with stochastic activity durations under resource constraints,” Journal of Scheduling, vol. 18, no. 3, pp. 263–273, Jun 2015. [3] M. Zaman, S. M. Elsayed, T. Ray, and R. A. Sarker, “Evolutionary algorithms for power generation planning with uncertain renewable energy,” Energy, vol. 112, pp. pp. 408–419, 2016. [4] F. Ballestín, “When it is worthwhile to work with the stochastic rcpsp?” Journal of Scheduling, vol. 10, no. 3, pp. 153–166, Jun 2007.

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