Single machine scheduling problem with interval processing timesto minimize mean weighted completion time

Ali Allahverdi a,n, Harun Aydilek b, Asiye Aydilek c

a Department of Industrial and Management Systems Engineering, Kuwait University, P.O. Box 5969, Safat, Kuwaitb Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, P.O. Box 7207, Hawally 32093, Kuwaitc Department of Economics and Finance, Gulf University for Science and Technology, P.O. Box 7207, Hawally 32093, Kuwait

a r t i c l e i n f o

Available online 14 June 2014

Keywords:SchedulingSingle machineMean completion timeUncertaintyHeuristics

a b s t r a c t

The single resource scheduling problem is commonly applicable in practice not only when there is asingle resource but also in some multiple-resource production systems where only one of the resourcesis bottle neck. Thus, the single resource (machine) scheduling problem has been widely addressed in thescheduling literature. In this paper, the single machine scheduling problem with uncertain and intervalprocessing times is addressed. The objective is to minimize mean weighted completion time.The problem has been addressed in the literature and efficient heuristics have been presented. In thispaper, some new polynomial time heuristics, utilizing the bounds of processing times, are proposed. Theproposed and existing heuristics are compared by extensive computational experiments. The conductedexperiments include a generalized simulation environment and several additional representativedistributions in addition to the restricted experiments used in the literature. The results indicate thatthe proposed heuristics perform significantly better than the existing heuristics. Specifically, the bestperforming proposed heuristic reduces the error of the best existing heuristic in the literature by morethan 75% while the computational time of the best performing proposed heuristic is less than that of thebest existing heuristic. Moreover, the absolute error of the best performing heuristic is only about 1% ofthe optimal solution. Having a very small absolute error along with a negligible computational timeindicates the superiority of the proposed heuristics.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Scheduling decisions directly affect production costs and cus-tomer satisfaction. This is because the right scheduling decisionshelp reduce production costs as a result of better resourceutilization. This leads to shorter delivery time to customers, andhence, increased customer satisfaction.

Scheduling jobs (tasks) on a single machine (resource) is widelyapplicable in real life. Moreover, in many applications of multiple-machine production systems, one machine is bottle neck, andhence, the right scheduling decision on that particular machinegreatly affects the performance of the production system. There-fore, the problem of scheduling on a single machine is important,and hence, numerous researchers addressed this problem.

There are many applications of the single machine schedulingproblem where job processing times are known with certainty,e.g., Vilà and Pereira [14], Valente and Schaller [13], Kianfar and

Moslehi [4]. Therefore, the vast majority of research on the singlemachine scheduling problem has been devoted to the case ofdeterministic problem where job processing times are treated asknown and fixed values. Some researchers addressed the problemwhere job processing times are modeled as stochastic randomvariables with certain mean and variance, e.g., Iranpoor et al. [3].

For some scheduling environments, the exact probability distri-butions for processing times may not be known. A solution obtainedby assuming a certain probability distribution may not be even closeto the optimal solution for the realized processing times. It has beenobserved that although it is hard to obtain the exact probabilitydistributions of processing times before scheduling, it is relativelyeasier to obtain the upper and lower bounds of processing times inmany practical cases. Therefore, the bounds of processing times canbe utilized in finding a solution for the scheduling problem. Thisproblem is known as uncertain scheduling problemwith bounded orinterval processing times, Sotskov et al. [10].

Scheduling problems with uncertain and bounded processingtimes have also been addressed in the literature for other schedul-ing environments such as flowshops. For example, Allahverdiand Aydilek [1] addressed the two-machine flowshop schedulingproblem with interval processing times with the objective of

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Computers & Operations Research

http://dx.doi.org/10.1016/j.cor.2014.06.0030305-0548/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail addresses: ali.allahverdi@ku.edu.kw (A. Allahverdi),

aydilek.h@gust.edu.kw (H. Aydilek), aydilek.a@gust.edu.kw (A. Aydilek).

Computers & Operations Research 51 (2014) 200–207

minimizing makespan. They provided several polynomial timeheuristic algorithms and showed that one of the algorithms yieldsclose to the optimal solution. Some other researchers thataddressed the scheduling problems with interval job processingtimes include Lai et al. [6], Lai & Sotskov [5], Sotskov et al. [9,11],Sotskov and Lai [12], and Aydilek et al. [2]. It should be noted thatSotskov et al. [11] and Sotskov and Lai [12] used the stabilitymethod, by introducing a stability box which is a subset of stabilityregion, in identifying solutions. The objective of this paper,however, is to construct heuristic algorithms.

The performance measure considered in this paper is meancompletion time which is related to the work in process (WIP)inventory. The WIP inventory includes the set of partially com-pleted products either ongoing or awaiting processing. HoldingWIP inventory is costly due to inventory storage and handlingcosts, taxes and insurance costs, damage, loss and spoilage. There-fore, holding WIP inventory can be considerably costly and thiscost may even exceed the cost of holding finished goods inventory.Consequently, one of the main goals of the manager becomessustaining the production with the minimum level of WIP inven-tory. Therefore, many researchers worked on developing newmethods for handling WIP, e.g., Yang [15] and Massim et al. [7].The objective function of minimizing the mean completion timeminimizes the average WIP inventory during the entire productionprocess of jobs. Since the WIP inventory cost is an importantcomponent of production cost, the considered objective helpsreduce production cost, and hence, it increases the profit.

We consider the single machine scheduling problem withmean weighted completion time performance measure wherejob processing times are random and bounded. This problemwas recently addressed by Sotskov et al. [10] where they presentedsome dominance relations, and developed two efficient heuristics.By computational experiments, they indicated that the errors ofboth heuristics were close to the optimal solution. In this paper,we address the same problem and propose several new heuristics.We show that our newly proposed heuristics perform considerablybetter than those of Sotskov et al. [10] while computational timesof our proposed heuristics are less than those of their heuristics.

The rest of this paper is organized as follows. The next sectionbriefly describes the problem. The proposed heuristics are pre-sented in Section 3, and an illustrative example is provided in thesubsequent section. Computational experiments are explained inSection 5, comparison of the heuristics is performed in Section 6,and finally Section 7 concludes the paper.

2. Problem definition

It should be noted that minimizing total weighed completiontime and minimizing mean weighed completion time are equiva-lent performance measures. We use the second measure in thispaper. Let MWCT denote the mean weighted completion time.Besides, let MWCT (π) represent the mean weighted completiontime of a given sequence π.

The problem is to minimize the mean weighted completiontime in a single-machine scheduling environment. There is a set ofn available jobs waiting for processing and setup times areincluded in the processing times. There are no precedence rela-tionships between the jobs.

We assume that the job processing times are uncertain vari-ables with unknown probability distributions where only a lowerbound tLj Z0 and an upper bound tUj ZtLj of the processing time tjof job j (jA J ¼ f1; 2;…; ng) are known before scheduling. Let Cjstand for the completion time of job j. Let the bracket [j] denotethe job in position j in a given sequence. Then, the completion

time of the job in position j can be computed as

C½j� ¼ ∑j

i ¼ 1t½i�

The mean weighted completion time of a given sequence π canbe computed by taking the average of the weighted job comple-tion times in each position of the sequence π and given as follows:

MWCTðπÞ ¼ ∑n

j ¼ 1w½j�C½j�

!=n

where w½j� denotes the weight of the job in position j.Such a single machine problem can be denoted as

1jtLj rtjrtUj j∑wiCi where the first term denotes that the probleminvolves a single machine. The second term indicates that proces-sing times are uncertain variables with a value between some lowerand upper bounds. The last term specifies that the performancemeasure is to minimize weighted completion time which isequivalent to minimizing mean weighted completion time. Noticethat the problem 1jtLj rtjrtUj j∑wiCi can be considered as anuncertain single machine problem without any prior informationabout the probability distribution of the processing times. In thiscase, it is only known that the processing times of each job fallsbetween some given lower and upper bounds with probability one.

3. Heuristics

When tLj ¼tUj for all j¼1,2, …, n, the problem reduces to thedeterministic single machine scheduling problem for which anoptimal solution can be obtained by the weighted shortest proces-sing time (WSPT) rule, Pinedo [8]. However, for at least some jobs, thelower bound is different from the upper bound. Moreover, it is notpossible to know the exact value of the processing time tj before theprocessing of job j has been completed. Yet, a decision on when toprocess job j has to be made before the observation of tj. Hence, adecision on the timing of the jobs can be made only using the lowerand upper bounds, tLj and tUj , which are the only available data on theprocessing time of job j. Therefore, several heuristics are generatedusing the lower and upper bounds, tLj and tUj , and these heuristics aredescribed below.

For heuristics AA1–AA5, WSPT rule is applied to the problem byusing atj in place of job processing times where atj ¼ ½tLj þβðtUj�tLj Þ�=wj for each job jAf1;…;ng and for a given value of β whichindicates the weight assigned to the lower and upper bounds. Thesequence obtained becomes one of our proposed heuristics. Theheuristic sequence AA1 is obtained when β¼0. Similarly, the heuristicsequences AA2–AA5 are obtained when β¼0.25, β¼0.50, β¼0.75 andβ¼1. It should be noted that while AA1 uses only the information oflower bounds, AA5 uses only the information about upper bounds.AA3 gives equal weights to the lower and upper bounds. On the otherhand, AA2 gives higher weight to lower bounds and AA4 gives higherweight to upper bounds.

Additional heuristics are obtained by using the bounds. Heur-istic GA is obtained by assigning the geometric average of tUj =wj

and tLj =wj to atj. More specifically, the heuristic sequence of GA is

obtained when atj ¼ ðtLj tUj Þ1=2=wj. Another heuristic, HA, is obtainedby using the harmonic average of the bounds such that the atj ¼½2ð tLj tUj Þ=ðtLj þtUj Þ�=wj. In addition to GA and HA, GC and HC areobtained by using the complement such that the atj for GC isatj ¼ ½þ tLj þ tUj �ðtLj tUj Þ1=2�=wj and the atj for HC is atj ¼ ½tLj þtUj �2ðtLj tUj Þ=ðtLj þtUj Þ�=wj.

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207 201

Two more heuristics, U1 and U2, are obtained by using only asingle observation of the processing times. Initially both U1 and U2are set to the AA3 sequence. After the observation of processingtimes, tj, the observed processing times are mapped to the interval[0,10] such that the mapped processing time is tpj ¼ 10ðtj�tLj Þ=ðtUj �tLj Þ. The tpjdata is sorted and the largest 20% and the smallest20% are trimmed. The statistic st is obtained by finding the meanof the trimmed data and it is used to update the sequences U1 andU2. If st is less than 4, U1 is updated with HA sequence and U2 isupdated with GA sequence. If st is greater than 6, U1 is updatedwith HC sequence and U2 is updated with GC sequence. Otherwise,no updating is performed. The values of 4 and 6 are obtainedbased on simulations and numerical results. Please note that if asingle observation of processing times does not exist, then, theheuristics U1 and U2 are reduced to AA3.

The 20 percentage for trimming the data has been obtainedbased on extensive computational experiments. Specifically, valuesof 5, 10, 15, 20, 25, 30 have been tested for the trimmingpercentage, and the value of 20% has been selected. Small valueshave been observed not to be very productive, and large valueshave been observed to be misleading as large portion of datawould be lost. In other words, if the percentage of trimming is lessthan 20%, the statistic st has been observed to be more volatile,and less informative.

The detailed steps of heuristics U1 are given as follows:

Step 1: For all j, set atj¼[tjLþ0.5(tjU�tjL)]/wj

Step 2: Sort atj in ascending order to obtain the sequence π1Step 3: Set U1¼π1Step 4: For all j, compute tpj ¼ 10ðtj�tLj Þ=ðtUj �tLj ÞStep 5: Trim the smallest and largest 20% of the values in givenin Step 4, and find st, the mean of the remaining valuesStep 6: If sto4, let atj¼[2(tjLtjU)/(tjLþtj

U)]/wj. Go to step 8Step 7: if st46, let atj¼[tjLþtj

U�2(tjLtjU)/(tjLþtjU)]/wj.

Step 8: Sort atj in ascending order to obtain the sequence π2Step 9: Set U1¼π2

It should be noted that the tj value in Step 4 is any value withinthe interval [tjL, tjU]. The steps of U2 are the same as those of U1except Steps 6 and 7, which are replaced by the following steps:

Step 6: If sto4, let atj¼(tjLtjU)1/2/wj. Go to step 8Step 7: if st46, let atj¼[tjLþtj

U�(tjLtjU)1/2]/wj.

The complexity of the proposed heuristic algorithms AA1–AA5,GA, HA, GC, HC, U1, and U2 is O(n log(n)).

Sotskov et al. [10] addressed our problem and proposed twoheuristics called S(T)&SUM and S(T)&PROD. Their heuristics of S(T)&SUM and S(T)&PROD are referred to as heuristics S1 and S2,respectively, in this paper. We compare the performances of ourproposed heuristics with those of S1 and S2 in Section 6.

4. An illustrative example

An example is presented in order to illustrate how theproposed heuristics are obtained, how the optimal solution isdetermined, and how the heuristics are compared with theoptimal solution.

Consider a problem with eight jobs for which the lower andupper bounds on processing times along with the weights aregiven in Table 1. The values in Table 1 are randomly generatedfrom a uniform distribution, U(1, 50), and rounded.

We may have many scenarios of processing times between thelower and upper bounds. For the problem given in Table 1, threedifferent scenarios are presented in Table 2.

The optimal sequence changes with the scenario. Thus, ingeneral, it is not possible to find a sequence which gives theminimum mean weighted completion time for all the scenarios.For each of the three scenarios, the optimal sequence can becalculated using the WSPT rule. Given that all the processing timesare known now, the optimal solution can be obtained by applyingthe WSPT rule to the realized processing times. Table 3 presentsthe optimal solutions and optimal weighted mean completiontime for the three scenarios listed in Table 2. Table 3 reveals thatthe optimal solution and the optimal MWCT may change from onescenario to another. For example, while the WSPT rule gives anMWCT of 614.4 in scenario 1, it produces an MWCT of 509.1 inscenario 2 and 587.6 for scenario 3.

Mean weighted completion times for the sequences given inTable 3 for the three different scenarios are presented in Table 4. InTable 4, columns 2–4 represent scenarios (1–3), rows representsequences and the optimal values are given in bold. For example,the optimal sequence is (3,8,5,6,7,1,2,4) for the scenario 1 with anMWCT of 614.4. However, this sequence does not remain optimal forthe scenarios 2 and 3. Similarly, the sequence (8,3,7,5,6,1,2,4) is optimalonly for the third scenario withMWCT of 587.6. On the other hand, thesequence (3,8,7,5,6,1,2,4) is only optimal for the second scenario.However, as can be seen from Table 4, MWCT value of this sequenceis very close to the optimal solution for the third scenario.

Table 5 displays the sequences proposed by the 11 heuristics forthe problem given in Table 1. It can be seen that some of theheuristics produce the same sequence for this specific problemsince the problem size is relatively small.

Table 6 displays theMWCT values of the heuristics in Table 5 forthe scenarios given in Table 2. The performances of the proposedheuristics change with the scenarios. However, since we know theoptimal solution for each scenario, which is given in Table 3, wecan calculate the error of our heuristics. For example, the heuristicHA gives the optimal solutions for scenario 3, and hence, the HAerror for this retaliation is zero. On the other hand, the heuristicsAA4, AA5, GC, HC, U1, and U2 give the optimal solution for scenario1. Notice that for scenario 2, none of the heuristics give theoptimal solution. However, the error of the heuristic AA3 is thesmallest among the proposed heuristics for this particular

Table 1Upper and lower bounds of job processing times and weights.

Job j 1 2 3 4 5 6 7 8

tLj 44 34 8 8 24 29 3 1

tUj 50 42 16 19 25 35 12 4

wj 9 7 26 1 14 12 3 5

Table 2Three different scenarios, for the example given in Table 1.

Scenario 1 2 3 4 5 6 7 8

1 46 41 15 8 25 34 10 42 45 37 9 13 25 31 5 23 48 39 16 17 24 33 4 3

Table 3Optimal sequences for the three different scenarios.

Scenario Optimal solution WMCT

1 3 8 5 6 7 1 2 4 614.42 3 8 7 5 6 1 2 4 509.13 8 3 7 5 6 1 2 4 587.6

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207202

example. Specifically, the value of MWCT of the heuristic AA3 is509.8 while the optimal value of MWCT is 509.1. It should benoticed that the heuristic U1 (similarly U2) initially is set to thesequence AA3 and it is updated by using the only observationscenario 1 as explained in the steps of the U1 given in the earliersection.

5. Computational experiments

The computations have been conducted on a PC with Intel Core2 Duo CPU T8300 processor of 2.4 GHz with 2 GB RAM.

In order to evaluate the proposed heuristics, first the upper andlower bounds of processing times are generated by using themethods described in Section 6. Next, the weights of jobs aregenerated from a uniform distribution between 1 and 50. Theseare the weights used by Sotskov et al. [10].

In order to evaluate the performance of each heuristic, wecompare its performance with that of the optimal solution of therealized processing times. Once all the jobs are processed on themachine, all the processing times are known and hence theoptimal solution can be obtained by applying the WSPT rule tothe realized processing times.

The heuristics are evaluated for different n values. The parameter nis set to 100, 200, 300, 400, 600, 800, and 1000. Heuristic perfor-mances are evaluated based on the criteria of absolute error (Error).

Error ¼ 100� MWCT of the heuristic�MWCT of the optimal sequenceMWCT of the optimal sequence

� �

The performance of the heuristics is compared using a total of16,000 replicates. First, for given values of n, D (which defines thegap between the lower and upper bounds of processing times),and distribution, a problem with lower and upper bounds ofprocessing times and weights of the jobs are generated. Then,for the problem, 80 scenarios of processing times between thebounds are generated using the given distribution. Next, thisproblem generation is repeated for 200 times, and the averagesof these results are obtained in order to get the final results. Thereason for selecting a large value for the generation of the problemis to make sure that a heuristic performs well under different

Table 4MWCT values of the given sequences for scenarios in Table 2 2.

Sequence Scenario

1 2 3

3 8 5 6 7 1 2 4 614.4 513.9 596.33 8 7 5 6 1 2 4 624.8 509.1 587.98 3 7 5 6 1 2 4 628.4 510.0 587.6

Table 5The sequences obtained by the proposed heuristics.

Heuristic Sequence

AA1 8 3 7 5 6 2 1 4AA2 8 3 5 7 6 1 2 4AA3 3 8 5 7 6 1 2 4AA4 3 8 5 6 7 1 2 4AA5 3 8 5 6 7 1 2 4GA 8 3 5 7 6 1 2 4GC 3 8 5 6 7 1 2 4HA 8 3 7 5 6 1 2 4HC 3 8 5 6 7 1 2 4U1 3 8 5 6 7 1 2 4U2 3 8 5 6 7 1 2 4

Table 6MWCT values of the given heuristics for different scenarios.

Scenario Heuristic

AA1 AA2 AA3 AA4 AA5 GA GC HA HC U1 U2

1 634.3 620.3 616.6 614.4 614.4 620.3 614.4 628.4 614.4 614.4 614.42 512.3 510.6 509.8 513.9 513.9 510.6 513.9 510.0 513.9 513.9 513.93 589.5 589.6 589.9 596.3 596.3 589.6 596.3 587.6 596.3 596.3 596.3

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

δ

Ave

rage

Erro

r

AA2AA3AA4U1U2S1S2

Fig. 2. Average error versus δ under Sotskov's environment.

100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5

4

Number of Jobs

Ave

rage

Err

or

AA2AA3AA4U1U2S1S2

Fig. 1. Average error versus number of jobs under Sotskov's environment.

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207 203

values of bounds and weights. Hence, we generate a total of 16,000problems for each combination of n, D, and distribution.

6. Heuristics comparison

Heuristics are evaluated under two different experimentalenvironments. The first environment is the one used by Sotskovet al. [10]. In other words, the same settings and parameter valuesare used. The second environment is a generalization of the oneused by Sotskov et al. [10]. Under both environments, among theproposed heuristics of AA1–AA5, HA, HC, GA, GC, U1, and U2, theheuristics AA2, AA3, AA4, U1, and U2 performed better than theothers, and hence, only these better performing heuristics aregoing to be compared with the existing heuristics S1 and S2.

6.1. Sotskov et al. [10] environment

First, a random integer C is generated between 1 and 200 usinga uniform distribution. Next, the lower bound, tLj , is obtained byusing the equation tLj ¼C(1�δ⧸100) and the upper bound, tUj , isobtained by using the equation tUj ¼C(1þδ⧸100). The parameter δis set to 0.1, 0.5, 1, 5, 10, 15, 25, 50, 75, and 100. This is theenvironment that Sotskov et al. [10] have conducted their experi-ments in order to evaluate the performances of their heuristics S1and S2. For a fair comparison, we also used the same environmentto compare the performance of the proposed heuristics in thispaper with those of S1 and S2 of Sotskov et al. [10].

The heuristics are compared by using the average errors sincethe standard deviations were very small compared with theaverage error. The average error results of the heuristics, for thisenvironment, are summarized in Figs. 1 and 2 with respect to thenumber of jobs and δ, respectively. It should be noted that theresults for δ values of 75 and 100 are not included in Fig. 2 sinceotherwise it becomes difficult to see the differences between theheuristics for δ¼0.1, 0.5, 1, 5, 10, 15, 25.

The results for S1 and S2 in Figs. 1 and 2 are almost on the top ofeach other. The heuristic S1 performs slightly better than S2, as alsowas shown by Sotskov et al. [10], for all δ values except for δ¼100,and error of S1 is about half of that of S2 for δ¼100. In the figures, theresults for the compared proposed heuristics (AA2, AA3, AA4, U1, U2)are exactly the same for a given n or δ. This is due to the fact thatunder this environment the lower and upper bounds are perfectlycorrelated, and hence, they give the same results.

As can be seen from Figs. 1 and 2, the proposed heuristicsoutperform the best heuristic of Sotskov et al. [10], namely S1.More specifically, the overall average error of S1 and the proposedheuristics are 3.9 and 0.4 respectively. In general, it can be seenfrom Fig. 1 that the performance of the heuristics is not sensitive tothe number of jobs. On the other hand, as expected, the averageerrors of all the heuristics increase as δ increases as can be seen inFig. 2. However, the increase in the error of proposed heuristics ismuch less than that of S1 and S2.

Finally, for a fair comparison of the heuristics, the computa-tional time (CPU) should be considered in addition to the averageerror. It should be noted that the CPU times of the heuristics arenot significant as even for the largest problem size considered, e.g.,1000 jobs, the CPU time of S1 and S2 is less than 2 min. On theother hand, the CPU time of the proposed heuristics for the largestproblem size considered is much less than this value, i.e., it is lessthan 2 s. Therefore, the proposed heuristics are superior to thebest existing heuristic in the literature, i.e., S1.

6.2. Generalized environment

The upper bound of processing time for job j, tjU, is generatedfrom a uniform distribution between 1 and 100. The lower bound,tLj , is also generated from a uniform distribution between (tjU–D)and tj

U where D is set at five different values of 20, 30, 40, 50, and60. Next, using different distributions (uniform, normal, positivelinear, negative linear, positive exponential and negative exponen-tial), instances of processing times between the correspondinglower and upper bounds are generated. These distributionsinclude both symmetric (normal, uniform) and skewed cases(positive and negative linear and exponential distributions), seeAppendix A. It is true that only lower and upper bounds are knownfor processing times without the knowledge of the distribution.However, it is not reasonable just to consider uniform distribution,between the lower and upper bounds, in the evaluation of a givenheuristic. Therefore, a representative set of distributions has to beconsidered while evaluating the performances of proposed heur-istics. For example, a heuristic might perform well for a givenspecific distribution, e.g., uniform. However, it might not performwell for another distribution.

This environment is a generalization of the environment ofSotskov et al. [10] since the lower and upper bounds in theenvironment of Sotskov et al. [10] are perfectly correlated. Inother words, the correlation coefficient of the bounds is one.However, in the generalized environment presented in this sub-section, the bounds are not perfectly correlated. Moreover, fiveadditional distributions are used to generate processing times inthe generalized environment while only the uniform distributionis used in the environment of Sotskov et al. [10]. Among theseadditional five distributions, four of them are skewed.

The average error results of the proposed heuristics aresummarized in Figs. 3–8 with respect to the six distributionsconsidered. In general, as expected, the performances of AA2 andAA4 are sensitive to the distribution of processing times since theratio of the weights of lower and upper bounds is 1–3. However,the weights of the bounds are the same for AA3, and hence, itsperformance is not that sensitive. Thus, in general, AA3 performsbetter than AA2 and AA4. Moreover, the heuristics U1 and U2perform better than AA3. The remaining analysis is conducted forthe overall average errors across all the distributions.

The average errors of the proposed and existing heuristics overall the distributions are summarized in Fig. 9 with respect to thenumber of jobs. It is clear from the figure that the errors of theproposed heuristics are much smaller than those of S1 and S2 in

100 200 300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

Number of Jobs

Ave

rage

Erro

r

AA2AA3AA4U1U2

Fig. 3. Average error versus number of jobs, positive exponential.

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207204

the general environment case as well. The comparison among theproposed heuristics can be better seen if S1 and S2 are excludedfrom the graph, and this is shown in Fig. 10. It is obvious from thefigure that AA3, U1, and U2 perform much better than AA2 and

AA4. Moreover, the performance of U2 is better than those of AA3and U1. As can be seen from the figure, while the performance ofthe proposed heuristics is not sensitive to the number of jobs, theperformance of S1 and S2 get slightly better as n increases.

100 200 300 400 500 600 700 800 900 10001

1.5

2

2.5

3

3.5

4

4.5

Number of Jobs

Ave

rage

Err

or

AA2AA3AA4U1U2

Fig. 4. Average error versus number of jobs, negative exponential.

100 200 300 400 500 600 700 800 900 10001.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

Number of Jobs

Ave

rage

Erro

r

AA2AA3AA4U1U2

Fig. 5. Average error versus number of jobs, uniform.

100 200 300 400 500 600 700 800 900 10000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Number of Jobs

Ave

rage

Err

or

AA2AA3AA4U1U2

Fig. 6. Average error versus number of jobs, normal.

100 200 300 400 500 600 700 800 900 10000.8

1

1.2

1.4

1.6

1.8

2

Number of Jobs

Ave

rage

Err

or AA2AA3AA4U1U2

Fig. 7. Average error versus number of jobs, positive linear.

100 200 300 400 500 600 700 800 900 1000

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Number of Jobs

Ave

rage

Erro

r

AA2AA3AA4U1U2

Fig. 8. Average error versus number of jobs, negative linear.

100 200 300 400 500 600 700 800 900 10001

2

3

4

5

6

7

Number of Jobs

Ave

rage

Err

or

AA2AA3AA4U1U2S1S2

Fig. 9. Average error versus number of jobs under the general environment.

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207 205

Fig. 11 summarizes the results with respect to the values of Dwhich indicates the gap between the lower and upper bounds. It isexpected that as the gap increases, the uncertainty in processing

times increases, and hence, the performance of a heuristic dete-riorates. This can also be seen in the figure. However, theperformances of the proposed heuristics are not as adverselyaffected as those of the existing heuristics S1 and S2. In order tosee the effect of the gap on the performances of the proposedheuristics better, the results for S1 and S2 are removed and this ispresented in Fig. 12. The comparison of the proposed heuristicswith respect to D values is similar to the one with respect to thenumber of jobs.

In summary, the overall average errors of AA1, AA2, AA3, AA4,AA5, HA, HC, GA, GC, U1, U2, S1 and S2 are 3.96, 1.68, 1.32, 1.75, 2.64,1.90, 1.92, 2.87, 2.41, 1.24, 1.12, 4.94, and 5.15, respectively. It shouldbe noted that the CPU time of the proposed heuristics is almost thesame and it is less than 2 s for the largest problem size considered.Even though the CPU time of S1 and S2 is not an issue since it isless than 2 min for the largest problem size considered, it is atleast 50 times that of the proposed heuristics. Moreover, theoverall error of the best existing heuristic in the literature, S1, isat least four times that of the best proposed heuristic of U2, thus,the heuristic U2 is recommended.

7. Conclusions

We have addressed the single machine scheduling problemwith uncertain and bounded processing times to minimize meanweighted completion time. The problem was earlier addressed inthe literature and the best heuristic was found to be S1. In thispaper, we have proposed new heuristics by utilizing the lower andupper bounds of processing times. Computational experimentshave been conducted under a generalized simulation environmentand five additional distributions, including some skewed distribu-tions, have been considered. The results showed that the bestproposed heuristic U2 performs significantly better than the bestexisting heuristic S1. Specifically, the heuristic U2, on average,reduces the error of S1 by 77% with a negligible computationaltime. More specifically, the computational time of U2 is much lessthan that of S1. Besides, the overall absolute error of U2 is only 1.1%of the optimal solution. It should be noted that the reason for theheuristic S1 taking more computational time is that the heuristicS1 also considers robust schedule criterion.

As pointed out in Section 3, if a single observation of processingtimes does not exist, then the heuristic U2 reduces to the heuristicAA3. In this case, the heuristic U2 is also recommended as theheuristic U2 decreases the error of S1 by 73%.

Acknowledgments

This research was supported by Kuwait University ResearchAdministration Grant no. EI02/12.

Appendix A

The distributions which are used in the paper in order tosimulate the processing times between the lower and upperbounds are summarized below. In total, six different distributionswere considered which are uniform, normal, linear (positive andnegative), and exponential (positive and negative). Note thatamong these distributions, linear and exponential distributionsare skewed, and the skewness of exponential distributions aremore than that of the linear distributions. The remaining twodistributions, uniform and normal, are symmetric distributions.The probability density functions (pdf) for these distributionsexcept uniform are given below.

100 200 300 400 500 600 700 800 900 10001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Number of Jobs

Ave

rage

Err

or

AA2AA3AA4U1U2

Fig. 10. Average error versus number of jobs under the general environmentexcluding S1 and S2.

20 25 30 35 40 45 50 55 600

2

4

6

8

10

12

D

Ave

rage

Erro

r

AA2AA3AA4U1U2S1S2

Fig. 11. Average error versus D under the general environment.

20 25 30 35 40 45 50 55 600

0.5

1

1.5

2

2.5

3

3.5

4

D

Ave

rage

Err

or

AA2AA3AA4U1U2

Fig. 12. Average error versus D under the general environment excluding S1 and S2.

A. Allahverdi et al. / Computers & Operations Research 51 (2014) 200–207206

Normal distribution

For the normal distribution, job processing times are distrib-uted with a mean of μ¼ tUj þtLj =2 and a standard deviation ofσ ¼ tUj �tLj =6. We truncated the normal distribution. In otherwords, the lower and upper bounds of the normal distributionare set at the lower and upper bounds of the processing times, andnot at negative and positive infinities as in ordinary normaldistribution. Hence, if the generated processing time, tj, is smallerthan the lower bound, tLj , or larger than the upper bound, tUj , thenthe processing time, tj, is regenerated. Note that, the probability ofthe processing time, tj, being outside the boundaries is less than0.3 percent.

Exponential distribution (positive and negative)

The pdf for the truncated exponential distribution is f ðxÞ ¼ λeλx=

ðeλtUj �eλtLj Þ for xAðtLj ; tUj Þ and zero elsewhere. The parameter λ is

taken as 0.20 for positive exponential distribution, and �0.20 fornegative exponential distribution.

Linear distribution (positive and negative)

The pdf for the positive linear distribution is f ðxÞ ¼ 2ðx�tLj Þ=ðtUj �tLj Þ2 for xAðtLj ; tUj Þ and zero elsewhere. On the other hand,

f ðxÞ ¼ 2ðtUj �xÞ=ðtUj �tLj Þ2 forxAðtLj ; tUj Þ and zero elsewhere is thepdf for the negative linear distribution.

In order to generate a random value which has an F distribu-tion, we used the fact that if U has a uniform distribution, then

F �1ðUÞ has an F distribution. In other words, if U has a uniformdistribution, then the inverse of the cumulative distributionevaluated at U has an F distribution. More specifically, let x be auniform random value generated over the interval [0, 1]. Then,

F �1ðxÞ gives a data generated from the F distribution where thecumulative distribution function is FpðxÞ ¼

R xtL f pðyÞdy and the

corresponding probability density function is f pðyÞ. This approach

is used while generating the linear and exponential randomvalues.

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