research article optimal computing budget allocation for...

Research ArticleOptimal Computing Budget Allocation for Ordinal Optimizationin Solving Stochastic Job Shop Scheduling Problems

Hong-an Yang, Yangyang Lv, Changkai Xia, Shudong Sun, and Honghao Wang

Department of Industrial Engineering, Northwestern Polytechnical University, Xi’an 710072, China

Correspondence should be addressed to Hong-an Yang; [email protected]

Received 5 January 2014; Revised 21 February 2014; Accepted 22 February 2014; Published 24 March 2014

Academic Editor: Massimo Scalia

Copyright © 2014 Hong-an Yang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We focus on solving Stochastic Job Shop Scheduling Problem (SJSSP) with random processing time to minimize the expected sumof earliness and tardiness costs of all jobs. To further enhance the efficiency of the simulation optimization technique of embeddingEvolutionary Strategy in Ordinal Optimization (ESOO) which is based onMonte Carlo simulation, we embed Optimal ComputingBudget Allocation (OCBA) technique into the exploration stage of ESOO to optimize the performance evaluation process bycontrolling the allocation of simulation times. However, while pursuing a good set of schedules, “super individuals,” which canabsorb most of the given computation while others hardly get any simulation budget, may emerge according to the allocatingequation of OCBA. Consequently, the schedules cannot be evaluated exactly, and thus the probability of correct selection (PCS)tends to be low.Therefore, we modify OCBA to balance the computation allocation: (1) set a threshold of simulation times to detect“super individuals” and (2) follow an exclusion mechanism to marginalize them. Finally, the proposed approach is applied to anSJSSP comprising 8 jobs on 8 machines with random processing time in truncated normal, uniform, and exponential distributions,respectively. The results demonstrate that our method outperforms the ESOO method by achieving better solutions.

1. Introduction

Most classical job shop scheduling problems assume that allthe problem data are fixed and known in advance. However,there are several inevitable stochastic factors in real man-ufacturing systems, for example, random processing time,machine breakdowns, and rush orders, amongwhich randomprocessing time is the most fundamental and representativeuncertain factor. Therefore, research in Stochastic Job ShopScheduling Problem (SJSSP) with random processing timehas a great importance in engineering applications. Eventhough, research works on SJSSP are fewer than those ondeterministic JSSP because of disadvantage caused by ran-dom processing time, for example, huge search space, lengthycomputation time, and challenging evaluation of schedules.

In general, three models are used to denote randomprocessing time: interval number method [1, 2], fuzzy theory[3–6], and stochastic method. However, the fluctuation anddistribution of random processing time are ignored by thefirst two mentioned methods, which leads to the inaccuratescheduling solutions. For this reason, an increasing number

of researchers use stochastic methods to denote randomprocessing time. Independent random distributions with aknown mean and variance are used to represent processingduration variability [7, 8]. Among all processing time dis-tributions, normal, exponential, and uniform distributionsare commonly found in SJSSP literatures [2, 9–12]. In orderto solve these NP-hard problems, many heuristic algorithmssuch as genetic algorithm [13], variable neighbourhoodsearch [14], and artificial bee colony algorithm [15] areintroduced to solve SJSSP.

However, random processing time of operations con-tributes to the randomness of completion time for each joband performance indicators of schedules, which leads to thedifficulty of evaluating feasible schedules in the evolutionof these heuristic algorithms. Stochastic simulation method,for example, Monte Carlo, which relies on repeated randomsampling to obtain statistic results, has been widely used tosolve performance evaluation [7, 9, 11].

Zhang and Wu [15] proposed that stochastic simulationdefinitely increased the computational burden because offrequent evaluations, especially when used in an optimization

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 619254, 10 pageshttp://dx.doi.org/10.1155/2014/619254

2 Mathematical Problems in Engineering

framework. Ho et al. [16] firstly developedOrdinal Optimiza-tion (OO) theory to obtain good enough solutions throughordinal comparisonwhile the value of a solutionwas still verypoor. Due to the fact that Evolutionary Strategy (ES) couldoptimize the sampling process in OO theory, Horng et al. [8]embedded ES in Ordinal Optimization (OO), abbreviated asESOO, to search for a good enough schedule of SJSSP usinglimited computation time.

Even though ESOO could significantly reduce the com-putation for SJSSP, the evaluation method in the explorationstage of ESOO is similar toMonte Carlo Simulation, in whichuniform computation is allocated to each schedule, regardlessof whether or not it should. Thus extra computation isallocated even after the Probability of Correct Selection(PCS) has already converged. Therefore, there is potential tofurther enhance ESOO’s efficiency by intelligently controllingevaluation process or by determining the optimal numberof simulation times among different schedules according totheir performances. Chen et al. [17] firstly proposed OptimalComputation Budget Allocation (OCBA) to enhance theefficiency of OO by allocating simulation times reasonably.In the recent years there are many articles about OCBA[18–20]. In order to optimize the computation allocation inthe evaluation process of solving SJSSP, we firstly proposean innovative hybrid algorithm of embedding OCBA intoESOO, abbreviated as ESOO-OCBA.

However, in the OCBA [17] deduction we find thehypothesis that the simulation times of the best individualoutnumber considerably the average ones is not the case inspecific SJSSP environment. As a result, “super individuals,”which could absorb most of the given computation in eachgeneration while others could hardly get any simulationbudget, may emerge according to the allocating equationof OCBA [17]. Thus we make improvements on OCBAto balance the computation allocation: (1) set a thresholdof simulation times to detect “super individuals,” (2) followan exclusion mechanism to isolate them, and (3) allocatethe existing computation budget to the remained individualsaccording to the dispatching rules of OCBA. The proposedmodification on OCBA is another contribution of this paper,which makes ESOO-OCBA suitable to solve SJSSP.

The rest of this paper is organized as follows: Section 2defines the problem, presents a mathematical equation ofthe SJSSP, and improves a model for evaluating schedules.Section 3 outlines our ESOO-OCBA algorithm for findinga good enough schedule from the search space of SJSSP.Section 4 demonstrates the modifications on OCBA thatwe made. Section 5 shows and discusses the computationalexperiments and the results. Finally, in Section 6, we presentour concluding remarks and discuss several future researchdirections.

2. SJSSP Formulation andVarying Evaluation Model

2.1. SJSSP Formulation. The SJSSP studied in this paperconsists of a set of 𝑛 jobs, 𝐽 {𝐽

1, 𝐽2, . . . , 𝐽

𝑛}, and a set of 𝑚

machines,𝑀 = {𝑀1,𝑀1, . . . ,𝑀

𝑚}. Every job 𝐽

𝑖(1 ≤ 𝑖 ≤ 𝑛)

consists of ℎ𝑖operations, 𝑂

𝑖= {𝑂𝑖,1, 𝑂𝑖,2, . . . , 𝑂

𝑖,ℎ𝑖}, that need

to be processed in sequence. The operation 𝑂𝑖,ℎ

(1 ≤ ℎ ≤ ℎ𝑖)

denotes the ℎth operation on job 𝑖 and 𝑂𝑖,ℎ

∈ 𝑂𝑖must be

processed by a specified machine𝑀(𝑂𝑖,ℎ) ∈ 𝑀. Similarly, the

set of operations thatmust be processed on eachmachine,𝑀𝑢

(1 ≤ 𝑢 ≤ 𝑚), is denoted by 𝑂𝑀(𝑀𝑢).

The random processing time of 𝑂𝑖,ℎ

is denoted by 𝑝𝑖,ℎ,

which is a random variable following a given probabilitydistribution function with mean 𝑝

𝑖,ℎand variance 𝜎2

𝑖,ℎ. Let

𝑠𝑡𝑖,ℎ

be the start time of 𝑂𝑖,ℎ. The completion time and fixed

due date of job 𝐽𝑖∈ 𝐽 are denoted by𝐶

𝑖and 𝑑𝑖, respectively.𝑇

𝑖

and 𝐸𝑖stand for the tardiness and earliness cost of job 𝐽

𝑖. Set

𝑡𝑖and 𝑒𝑖as the tardiness and earliness penalty per unit time

for job 𝐽𝑖, respectively.

Let Ω denote the set of all the feasible and unfeasibleschedules; a feasible schedule 𝑆 should satisfy both prece-dence constraint and capacity constraint simultaneously.Without loss of generality, it is assumed that all data areintegers and no preemption is allowed. The goal of SJSSP isto find a feasible schedule 𝑆 ∈ Ω that minimizes the expectedsum of earliness and tardiness costs of all jobs.

Objective function:

Minimizecost = min𝑆∈Ω

𝐸{

𝑛

∑𝑖=1

(𝑇𝑖+ 𝐸𝑖)} . (1)

A feasible schedule 𝑆 ∈ Ω should be subject to

𝑇𝑖= 𝑡𝑖×max (0, 𝐶

𝑖− 𝑑𝑖) 𝑖 = 1, 2, . . . , 𝑛, (2)

𝐸𝑖= 𝑒𝑖×max (0, 𝑑

𝑖− 𝐶𝑖) 𝑖 = 1, 2, . . . , 𝑛, (3)

𝑠𝑡𝑖,𝑘+ 𝑝𝑖,𝑘≤ 𝑠𝑡𝑖,𝑘+1

𝑖 = 1, 2, . . . , 𝑛 ∧ 𝑘 = 1, 2, . . . , ℎ𝑖− 1

(4)

for all operations in the set 𝑂𝑀(𝑀𝑢):

𝑠𝑡𝑖,𝑘+ 𝑝𝑖,𝑘≤ 𝑠𝑡𝑗,ℎ∨ 𝑠𝑡𝑗,ℎ+ 𝑝𝑗,ℎ≤ 𝑠𝑡𝑖,𝑘

𝑖, 𝑗 = 1, 2, . . . , 𝑛 ∧ 𝑘 = 1, 2, . . . , ℎ𝑖∧ ℎ = 1, 2, . . . , ℎ

𝑗.

(5)

Constraints (2) and (3) represent the tardiness and earli-ness cost of each job, respectively. The precedence constraint(4) ensures that, for each pair of consecutive operations 𝑂

𝑖,𝑘

and 𝑂𝑖,𝑘+1

of the same job 𝑖, operation 𝑂𝑖,𝑘+1

cannot beprocessed before operation 𝑂

𝑖,𝑘is completed. The capacity

constraint (5) ensures that two operations of two differentjobs, 𝐽

𝑖and 𝐽𝑗, 𝑂𝑖,𝑘

and 𝑂𝑗,ℎ, in the set 𝑂𝑀(𝑀

𝑢) cannot be

processed simultaneously on machine𝑀𝑢.

2.2. Performance Evaluation Model Improved from the Objec-tive Function. To obtain a good statistical estimate for afeasible schedule, a large number of simulation replicationsare usually required for each schedule. However, the expectedobjective value of a feasible schedule is available only inthe form of a complex calculation via infinite simulatedreplications. Although infinite replications of simulation willmake the objective value of (1) stable, this calculatingmethodactually is intractable.

Mathematical Problems in Engineering 3

Therefore, depending on the amount of simulated repli-cations, (1) can be approximated as follows:

min𝑆∈Ω

𝐹𝐿(𝑆) =

1

𝐿

𝐿

∑𝑙=1

{

𝑛

∑𝑖=1

[𝑡𝑖𝑇𝑙

𝑖(𝑆) + 𝑒

𝑖𝐸𝑙

𝑖(𝑆)]} (6)

for a feasible schedule 𝑆; 𝐿 represents the number of itssimulation replications. 𝑡

𝑖𝑇𝑙

𝑖(𝑆) and 𝑒

𝑖𝐸𝑙

𝑖(𝑆) denote the tar-

diness and earliness costs of job 𝐽𝑖on the 𝑙th replication of

𝑆, respectively. 𝐹𝐿(𝑆) denotes the average sum of tardiness

and earliness costs of 𝑆 when the simulation length is 𝐿.Sufficiently large 𝐿will make the objective value of (6), 𝐹

𝐿(𝑆),

sufficiently stable. Let 𝐿𝑠= 105 represent the sufficiently large

𝐿 [8]. Let 𝐹𝐿𝑠(𝑆) represent the objective value of 𝑆 computed

by sufficiently exact evaluation model.

3. ESOO-OCBA Algorithm for SJSSP

3.1. Embedding the OCBA Technique into ESOO. To evaluatethe performance of a feasible schedule 𝑆 reliably needs acomplex calculation of 𝐹

𝐿𝑠(𝑆), not to even mention the

huge search space of SJSSP. ESOO could significantly reducethe computation for evaluation process [8]. However, inESOO, uniform computation is allocated to each individual,regardless of whether or not it should. This allocation cannotmeet the different demands of different individuals; thuscomputation allocated to each individual may be eitherinsufficient or redundant.

Ideally, overall simulation efficiency will be improved ifless computational effort is spent on simulating noncriticalschedules and more is spent on critical schedules. We wouldlike to improve the PCS in each generation by allocatingcomputation according to the performance of each schedule.Therefore, in this paper, OCBA technique is embedded intothe exploration stage of ESOO algorithm to intelligentlydetermine the optimal number of simulation times 𝐿 fordifferent individuals according to their performances.

SJSSP is a NP-hard problem, which reflects the real-world situations and always suffers from uncertain feature.Recently, many methods solving SJSSP suffer from lengthycomputation budget, because evaluating the objective of aschedule is already very time-consuming not to evenmentionthe extremely slow convergence of the heuristic techniques insearching through a huge search space. In order to overcomethe drawback of consuming much computation time, wepropose the ESOO-OCBA algorithm. In our ESOO-OCBAalgorithm, OO theory reduces the unbearable computationand OCBA technique allocates necessary computation toeach individual. The overall scheme of our algorithm isillustrated by Figure 1.

From the Figure 1, the NP-hard characteristic and therandomness of SJSSP contribute to many difficulties, forexample, huge search space, performance evaluation prob-lem, and slow convergence. OO theory contains two fun-damental Ideas: (1) ordinal comparison, that is, ordinal isused rather than cardinal optimization in order to reduce thesimulation times in evaluating schedules; (2) goal softening isused to decrease the degree of searching difficulty. It is provedthat ordinal comparison has an exponential convergence rate

[21, 22] and that goal softening can raise the Probability ofAlignment (PA) exponentially [23].

ESOO-OCBA algorithm consists of two stages. (1) Theexploration stage aims to find a subset of good samplesfrom the search space, where samples are evaluated by acrude evaluation model. Evolutionary strategy is employedin this stage to optimize the sampling process while OCBAis used to trade off the performance stability and save thecomputation of each individual by allocating the simulationtimes reasonably. (2) The exploitation stage that consists ofmultiple subphases is used to find out the good enoughindividual in the good sample subset. Individuals are selectedand eliminated by increasingly accurate evaluation models ineach subphase. The one with the smallest 𝐹

𝐿𝑠(𝑆) in the last

subphase, 𝑆∗, is the good enough schedule that we seek.

3.2. The Exploration Stage. In exploration stage, we use ES asa whole frame of sampling process. The number of neededsamples depends on the extent of goal softening, whichdecides the number of generations and the number of initialpopulation. In each generation of ES, each individual couldget a unique crude evaluation model developed from (6)with different simulation times 𝐿 allocated by OCBA. Theimplementation of exploration stage is as follows [8].

Precedence-Based Representation.We use a precedence-basedpresentation to define a chromosome with unpartitionedpermutation of 𝑙

𝑖repetitions of each job 𝐽

𝑖. The operations of

the same job should satisfy its precedence constraints to forman individual standing for a feasible schedule. This encodingmethod is employed due to its effectiveness on generatingfeasible individuals and on reducing the size of search space.

Initial Population. Each individual of the initial population isgenerated with a completely random method to enhance thevariety of the initial population.

Recombination. Generate offspring from the parents bydiscrete recombination [24]. Use repair operator to adjustinfeasible schedule when it appears [8].

Mutation.The insertionmutationmethod is adopted to breednew offspring. Delete all operations of one random selectedjob within a parent chromosome and then reinsert theminto the remained components randomly according to theprecedence constraints [8].

Selection. 𝜇 + 𝜆-selection mechanism is adopted in ourapproach. Select the best 𝜇 individuals from both the 𝜇

parents and 𝜆 offspring, according to the ranking of theapproximate fitness values obtained from the respective crudeevaluation model of each individual.

Termination.The ES is stopped when the number of genera-tions exceeds what we set; select the best𝑁 individuals as thegood sample set according to their performances.

3.3. The Exploitation Stage. In the exploitation stage, weneed to find the best schedule from the 𝑁 schedules in the


Difficulties insolving SJSSP

Huge search spaceEvaluating methodSlow convergence

· · ·

Sufficient accurateevaluating modelThe goal S∗

Crudeevaluating

model

Simplify thedifficulties in

searching

Find evaluating tools

Sampling andevolving by ES

Good sample subsetKeep high PCS by OCBA technique

Exploration stage: select a good sample subset

Exploitation stage: narrow thesearch space step by step

Itera

tive u

se o

f OO

Optimizesampling process

Evaluatefitness value

OCBA technique

Allocate simulating timesL

Goal softening

The number ofneeded samples

Ordinalcomparison

Substage 1

High PA guaranteed byiterative OO theory

· · ·

Substage n

Sele

ct sc

hedu

les b

y in

crea

singl

yac

cura

te ev

alua

ting

mod

el

Exponentialconvergence

Satisfy softening level

Popsize and number of generations

Guarantee high confidencein containing good enough S

Objective functionwith varying

simulating times Lallocated by OCBA

With varying L

controlled by OCBA

With sufficientlarge L = Ls

With increasingly largeL in every subphase

Accelerate theprobability of

alignment

{S1, S2, . . . , SN}

Figure 1: Overall scheme of ESOO-OCBA algorithm for SJSSP.

good sample set obtained in the exploration stage. However,evaluating each of the 𝑁 schedules by sufficient accurateevaluation model (objective function with 𝐿 = 𝐿

𝑠) costs too

much computation.Thus the exploitation stage is divided intomultiple subphases according to the idea of iterative use ofordinal optimization [25]. The computational complexity ofexploitation stage can be dramatically decreased, as the sizeof the schedule set in each subphase has already been largelyreduced when the evaluation model is more refined.

The implementation of exploitation stage in [8] is adoptedin this paper. In each subphase, the objective functionequation (6) with various simulation times 𝐿 is used asan increasingly accurate evaluation model, where 𝐿 rangesfrom a given fixed 𝐿

𝑔(crude model in the first subphase)

to 𝐿𝑠(sufficient accurate model in the final subphase). The

remained schedules in the prior subphase are selected andsomeof themare eliminated according to their performances.In the last subphase, the one with the smallest 𝐹

𝐿𝑠(𝑆) is the

good enough schedule 𝑆∗ that we look for.

4. The OCBA Technique and Modifications

4.1. The OCBA Technique. OO theory usually allocatesuniform computation to each schedule, which can hardlyachieve the highest PCS within a given computation. OCBAtechnique [17] is employed to improve the PCS by allocating

simulation times to each schedule according to their per-formances. We define 𝐿

𝑖as the simulation times allocated

to schedule 𝑆𝑖and set 𝑇 as the total given simulation times

in each generation (i.e., total given computation). OCBAprovides a recipe of asymptotically optimal allocation ofsimulation times among schedules in each generation of theexploration stage in ESOO-OCBA algorithm:

𝐿1+ 𝐿2+ ⋅ ⋅ ⋅ + 𝐿

𝑘= 𝑇, (7)

𝐿𝑏= 𝜎𝑏√𝑘

∑𝑖=1,𝑖 = 𝑏

𝐿2𝑖

𝜎2𝑖

, (8)

𝐿𝑖

𝐿𝑗

= (𝜎𝑖/𝛿𝑏,𝑖

𝜎𝑗/𝛿𝑏,𝑗

)

2

, for 𝑖 = 1, 2, . . . , 𝑘, 𝑖 = 𝑗 = 𝑏, (9)

where 𝜎2𝑖is the observation variance of schedule 𝑆

𝑖which

can be approximated by sample variance and 𝑆𝑏is the best

schedule we observed. 𝐹𝐿 𝑖(𝑆𝑖) is the observed performance of

schedule 𝑆𝑖; 𝛿𝑏,𝑖= 𝐹𝐿𝑏(𝑆𝑏) − 𝐹𝐿 𝑖(𝑆𝑖).

4.2. Modifications on OCBA Technique. However, the afore-mentioned OCBA is observed to be unavailable to solve thespecific SJSSP in our experiments. While pursuing a goodset of schedules in the exploration stage, “super individuals”may emerge according to (7)–(9), which could absorb most


of the given simulation budget in each generation while otherindividuals could hardly get any simulation budget.

Without enough simulation times 𝐿, most of the averageindividuals in each generation cannot be compared with eachother exactly by (6). Therefore, this simulation distortionphenomenon will absolutely decrease the probability ofcorrect selection as some inferior schedules may be selectedwhile some good schedules may be eliminated.

We try to find the root cause of “super individuals” bydating back to the deduction of the original dispatching (7)–(9) [17]. After a series of deductions, Chen et al. [17] got (10)which could express the relationship between 𝐿

𝑏, 𝐿𝑖, and 𝐿

𝑗.

In order to simplify (10), Chen et al. [17] assumed 𝐿𝑏≫ 𝐿𝑖

according to (8). Therefore (10) could be simplified as (11),and then the ratio between 𝐿

𝑖and 𝐿

𝑗was deducted from (11)

which was expressed in (9). Consider the following:

exp(−𝛿2𝑏,𝑖

2 ((𝜎2𝑏/𝐿𝑏) + (𝜎2

𝑖/𝐿𝑖))) ⋅

𝛿𝑏,𝑖𝜎2𝑖/𝐿2𝑖

((𝜎2𝑏/𝐿𝑏) + (𝜎2

𝑖/𝐿𝑖))3/2

= exp(−𝛿2𝑏,𝑗

2 ((𝜎2𝑏/𝐿𝑏) + (𝜎2

𝑗/𝐿𝑗)))

⋅𝛿𝑏,𝑗𝜎2𝑗/𝐿2𝑗

((𝜎2𝑏/𝐿𝑏) + (𝜎2

𝑗/𝐿𝑗))3/2,

(10)

exp(−𝛿2

𝑏,𝑖

2 (𝜎2𝑖/𝐿𝑖)) ⋅

𝛿𝑏,𝑖𝜎2𝑖/𝐿2𝑖

(𝜎2𝑖/𝐿𝑖)3/2

= exp(−𝛿2𝑏,𝑗

2 (𝜎2𝑗/𝐿𝑗)) ⋅

𝛿𝑏,𝑗𝜎2𝑗/𝐿2𝑗

(𝜎2𝑗/𝐿𝑗)3/2.

(11)

However, in our experiments, the performance of indi-vidual 𝑆

𝑖can be extremely similar to that of the best individual

𝑆𝑏; thus 𝛿

𝑏,𝑖= 𝐹𝐿𝑏(𝑆𝑏) − 𝐹𝐿 𝑖(𝑆𝑖) can be remarkably small. In

this condition, according to (9), the simulation times 𝐿𝑖of

individual 𝑆𝑖will outnumber considerably that of an average

individual 𝑆𝑗. Then we can get 𝐿

𝑏≈ 𝐿𝑖from (8); that is,

the assumption of Chen et al. [17], 𝐿𝑏≫ 𝐿𝑖, is not the case

in the specific SJSSP environment. As (10) cannot be useddirectly in dispatching simulation times for its complexity,modifications are made on the original dispatching ruleequations (7)–(9) to make the classical dispatching ruleavailable in SJSSP.

In order to diminish the negative influences from “superindividuals” in each generation, two steps must be taken: (1)set a threshold of simulation times to detect “super individu-als” and (2) follow an exclusion mechanism to isolate them.

Deducting from (7)–(9), (12) is used to allocate simulationtimes 𝐿

𝑖to each schedule 𝑆

𝑖:

𝐿𝑖= 𝑇 ×

𝜎2

𝑖𝛿2

𝑏,𝑗

𝜎2𝑗𝛿2𝑏,𝑖

× [

[

𝑘

∑𝑖=1,𝑖 = 𝑏,𝑖 = 𝑗

𝜎2𝑖𝛿2𝑏,𝑗


+ 𝜎𝑖(

𝑘

∑𝑖=1,𝑖 = 𝑏



)

1/2

+ 1]

]

−1

.

(12)

Modifications are made on (12) to realize the mentionedtwo Steps. Firstly, we set 𝐿

𝑠= 105 as the threshold of sim-

ulation times because it is the sufficient simulation times forevaluation.The individual will be seen as a “super individual”once it obtains more than 𝐿

𝑠simulation times. Secondly, we

define all the “super individuals” as a setΘ and the scale ofΘas num and then pick outΘ from the set of all the individuals.Then from the total given 𝑇, deduct the simulation timeswhich are occupied by “super individuals,” where we set thateach “super individual” occupies 𝐿

𝑠simulation times. Lastly,

allocate the existing simulation times to the individualsexcept for “super individuals” according to the dispatchingrules. Based on these steps, we improve (12) to (13) as follows:

𝐿𝑖= (𝑇 − num × 𝐿

𝑠)

×𝜎2𝑖𝛿2𝑏,𝑗


× [

[

𝑘

∑𝑖=1,𝑖 = 𝑏,𝑖 = 𝑗,𝑖∉Θ



+ 𝜎𝑖(

𝑘

∑𝑖=1,𝑖 = 𝑏



)

1/2

+ 1]

]

−1

.

(13)

Equation (13) is one of the main contributions of ourstudy. It can be used (i) to allocate computation in thesimulation optimization of SJSSP reasonably and effectivelyor (ii) to make simple estimates for designing experiments insolving SJSSP.

4.3. Implementation of the Modified OCBA. We adopt thecost-effective sequential approach based on OCBA which isdescribed as follows [17]: (1) 𝑛

0simulation replications for

each individual are conducted to get some initial informationabout the performance of each individual. As simulationproceeds, the sample means and sample variances of eachschedule are computed from all the data that are alreadycollected up. (2) According to this collected simulationoutput, an incremental computing budget, Δ, is allocated tothe set of all the individuals. Ideally, each new replicationshould bring us closer to the optimal schedules. (3) Thisprocedure is continued until the total given 𝑇 is exhausted


and then (13) can be improved as follows (𝑇𝑎denotes the

simulation time that has already been consumed):

𝐿𝑖= (𝑇𝑎+ Δ − num × 𝐿

𝑠)

×𝜎2

𝑖𝛿2

𝑏,𝑗


× [

[

𝑘

∑𝑖=1,𝑖 = 𝑏,𝑖 = 𝑗,𝑖∉Θ

𝜎2

𝑖𝛿2

𝑏,𝑗


+𝜎𝑖(

𝑘

∑𝑖=1,𝑖 = 𝑏

𝜎2

𝑖𝛿4

𝑏,𝑗


)

1/2

+ 1]

]

−1

.

(14)

The implementation of Optimal Computing Budget Allo-cation (OCBA) in each generation in the exploration stage ofESOO.

Step 1. Perform 𝑙th simulation replications for all individuals;𝑔 = 0; 𝐿𝑔

1= 𝐿𝑔

2= ⋅ ⋅ ⋅ = 𝐿

𝑔

𝑘= 𝑛0.

Step 2. If ∑𝑘𝑖=1𝐿𝑔

𝑖≥ 𝑇, stop.

Step 3. Increase the computing budget (i.e., number ofadditional simulation times by𝐿 and compute the newbudgetallocation, 𝐿𝑔+1

1, 𝐿𝑔+1

2, . . . , 𝐿

𝑔+1

𝑘, using (14).

Step 4. Perform additional max(0, 𝐿𝑔+1𝑖

− 𝐿𝑔

𝑖) simulations for

schedule, 𝑖, 𝑖 = 1, . . . , 𝑘, 𝑔 = 𝑔 + 1. Go to Step 2.In the OCBA steps above, 𝑔 is the iteration number and

𝑘 is the number of initial population. What needs to beremarked is the best individual 𝑏 which may change fromiteration to iteration.

5. Computational Results and Discussion

5.1. SJSSP Test Instance with Three Processing Time Distri-butions. In order to demonstrate the computational qualityand efficiency of our ESOO-OCBA algorithm, numericalexperiments on SJSSP comprising 8 jobs on 8 machines [8]have been carried out. (𝑎

𝑖,ℎ, 𝑝𝑖,ℎ, 𝜎2𝑖,ℎ) is given in Table 1, which

is used to denote the operating environment of operation𝑂𝑖,ℎ. 𝑎𝑖,ℎ

denotes the processing sequence of 𝑂𝑖,ℎ, 𝑝𝑖,ℎ

and 𝜎2𝑖,ℎ

denote the mean and variance of stochastic processing time𝑝𝑖,ℎ, respectively. The due dates 𝑑

𝑖of each job 𝐽

𝑖are given

in Table 2. The tardiness penalty per unit time 𝑡𝑖and the

earliness penalty per unit time 𝑒𝑖for each job 𝑖 are set to 1.

Three distributions of random processing time on themachines are used to test the computational efficiency andthe obtained schedules quality of our algorithm. The firstdistribution is truncated normal distribution with mean𝑝𝑖,ℎ

and variance 𝜎2𝑖,ℎ. The second distribution is uniform

distribution in the interval [𝑝𝑖,ℎ− 3 × 𝜎

𝑖,ℎ, 𝑝𝑖,ℎ+ 3 × 𝜎

𝑖,ℎ]. The

third distribution is exponential distribution with mean 𝑝𝑖,ℎ.

In the exploration stage of ESOO-OCBA, we set thenumber of initial population as 𝜇 = 1000 and the numberof offspring as 𝜆 = 2000. In order to compare the efficiencybetween ESOO-OCBA and ESOO [8], the same total givensimulation times in each generation of the exploration stageare set as 𝑇 = 368 × 𝜆, and the number of generations is

set as 𝑘max = 100. It is well understood that a small initialsimulation time of each individual, 𝑛

0, can contribute to

more flexibility for better allocation of the computing budget.Nevertheless, if 𝑛

0is too large, we may waste too much

computation budget in simulating nonpromising designs.Intuitively, if the total computing budget, 𝑇, is very large,the effect of 𝑛

0should be less important. In the dispatching

process of OCBA technique in each generation, we know that𝑇 is very large and so we set 𝑛

0= 33.

In addition, the selection of incremental computingbudget, Δ, is typically problem-specific. A large Δ can lead towaste of computation time to obtain an unnecessarily highconfidence level. Nevertheless, if Δ is too small, we needto carry out the budget allocation problem many times. Soaccording to the total computing budget,𝑇, we setΔ = 66000.𝐿𝑠= 105 is set as the threshold for detecting “super individ-

ual.” We start from randomly generating 𝜇 individuals as theparent population. After an evolution of 𝑘max generations, werank all the remained 𝜇 + 𝜆 = 3000 individuals (parents andoffspring) based on their performances and select the best𝑁 = 1000 individuals.

In the exploitation stage of ESOO-OCBA, we adopt allthe parameters used in related works [8]. Table 3 shows thenumber of subphases, the simulation length, and the numberof candidate schedules in each subphase. In the last subphase,we compute the exact object value 𝐹

𝐿𝑠(𝑆) of the 𝑁

6= 7

candidate schedules. The one with the smallest 𝐹𝐿𝑠(𝑆) is the

good enough schedule 𝑆∗ that we look for.

5.2. Test Results of Modified OCBA. In order to show theadvantages of our modifications on OCBA, we choose arandom generation from the exploration stage of ESOO-OCBA in solving SJSSP with truncated normal distributedprocessing time. We compare our modified OCBA withclassical OCBA [17] by allocating simulation times to allindividuals (𝜆 = 2000), respectively. The test results areshown in Figures 2, 3, and 4.

Figure 2 shows the general results of the realized allo-cation by the two analyzed OCBA techniques consideringthe different individuals, allocated simulation times, andperformances as comparison criteria. Figure 3 describes therelation between the simulation times and the individualperformances in detail and also helps to understand thedifferences between classical OCBA and modified OCBA.Figure 4 illustrates the distribution of simulation times ofeach individual, demonstrating the improvements of modi-fied OCBA compared with traditional OCBA in dispatchingsimulation times.

In Figure 3, we can see that, with the individual perfor-mance (objective value) rising, the simulation times allocatedto the better individuals increase under the influence of theallocationmechanisms in both classical andmodifiedOCBA.This increment meets the demand of more simulation timeswhen better individuals need to be evaluated exactly in theevaluation process.

Also, for most of the individuals, more simulation timesare allocated by our modified OCBA than by classical OCBA.The reason of this phenomenon lies in the mentioned “super


Table 1: Operation environment vector of SJSSP with 8 jobs and 8 machines.

𝑀1

𝑀2

𝑀3

𝑀4

𝑀5

𝑀6

𝑀7

𝑀8

𝐽1

3,70,140 2,80,160 1,90,180 6,50,100 4,40,80 8,60,120 5,70,140 7,50,100𝐽2

1,80,160 2,40,80 3,50,100 5,90,180 4,40,80 7,50,100 6,60,120 8,40,80𝐽3

1,50,100 2,40,80 3,80,160 5,60,120 4,70,140 6,40,80 8,40,80 7,70,140𝐽4

2,60,120 1,50,100 3,60,120 4,70,140 7,80,160 5,40,80 6,50,100 8,80,160𝐽5

4,50,100 3,50,100 2,70,140 1,40,80 7,50,100 5,60,120 6,90,180 8,60,120𝐽6

2,60,120 3,80,160 1,90,180 5,70,140 6,50,100 4,40,80 8,80,160 7,90,180𝐽7

1,40,80 3,60,120 4,40,80 2,80,160 5,60,120 7,70,140 8,50,100 6,60,120𝐽8

2,90,180 1,70,140 3,50,100 4,60,120 5,90,180 7,80,160 6,40,80 8,40,80

Table 2: Due dates for each job.

𝐽𝑖

𝐽1

𝐽2

𝐽3

𝐽4

𝐽5

𝐽6

𝐽7

𝐽8

𝑑𝑖

490 510 540 500 540 470 530 560

20004000

60008000

10000

0500

100015002000

Individual

performance

Simulation times

Indi

vidu

al n

umbe

r

Classical OCBAModified OCBA

106

105

104

103

102

101

Figure 2: 3D view of the comparison between modified OCBA andclassical OCBA.

3000 4000 5000 6000 7000 8000 9000 10000Individual performance

Sim

ulat

ion

times

Modified OCBAClassical OCBA

10001000

Super individuals10

6

105

104

103

102

101

Ls = 105 n0 = 33

n0 = 33

Figure 3: Relationship between individual performance and allo-cated simulation times.

individuals” produced by the classical OCBA dispatchingrules (see (7)–(9)). Besides, for the total of individuals(𝜆 = 2000) in the generation, the only two labeled “superindividuals” absorb 85.75% of the total simulation times,

0 200 400 600 800 1000 1200 1400 1600 1800 2000Individual number

Sim

ulat

ion

times

Modified OCBAClassical OCBA

10001000

106

105

104

103

102

101

Ls = 105

n0 = 33

Figure 4: Distribution of allocated simulation times for individuals.

while all the other individuals are allocated with low share(14.25%) of total simulation times.

In Figure 4, a more detailed representation of the distri-bution of simulation times allocated to individuals can beobserved: in the test results of classical OCBA, the simulationtimes allocated to different individuals have no apparentalterations, even if they have a huge performance gap. It is alsoclear that 92.85% of all the individuals (1857 of the total 2000)are allocated between 33 and 100 simulation times by classicalOCBA. This lack of simulation times leads to a simulationdistortion in the evaluation process, not being able to detectthe variations in different individual performances, whichabsolutely decreases the PCS in this generation.

For modified OCBA, 97.85% of all the individuals (1957of the total 2000) are allocated from 33 to 1000 simulationtimes. After we use the threshold 𝐿

𝑠= 105 to limit the “super

individuals,” a larger portion of the total simulation times areallocated to average individuals (74.04%).This situation leadsto high performed individuals getting muchmore simulationtimes even if they are slightly better (this effect can beobserved in Figure 3). This improved allocation contributes


Table 3: Number of candidate schedules and simulation times in each subphase.

Subphase 𝑖 1 2 3 4 5 6𝑁𝑖

1000 368 135 50 18 7𝐿𝑖

1000 2718 7389 20085 54598 100000

Table 4: The good enough schedule 𝑆∗, corresponding 𝐹𝐿𝑠(𝑆∗) obtained by ESOO-OCBA, and the percentage improvements relative to

ESOO.

Distribution The good enough schedule 𝑆∗ ESOO-OCBA ESOO [8] % improvement CPU-T (s)

Truncated normal

𝑂4,1𝑂7,1𝑂2,1𝑂1,1𝑂1,2𝑂2,2𝑂5,1𝑂4,2𝑂7,2𝑂1,3

2089 2280 8.38 392.12

𝑂3,1𝑂4,3𝑂2,3𝑂7,3𝑂7,4𝑂7,5𝑂7,6𝑂8,1𝑂2,4𝑂4,4

𝑂5,2𝑂1,4𝑂1,5𝑂1,6𝑂7,7𝑂4,5𝑂3,2𝑂2,5𝑂3,3𝑂4,6

𝑂7,8𝑂5,3𝑂8,2𝑂8,3𝑂1,7𝑂4,7𝑂5,4𝑂3,4𝑂8,4𝑂1,8

𝑂2,6𝑂5,5𝑂6,1𝑂5,6𝑂2,7𝑂4,8𝑂6,2𝑂2,8𝑂8,5𝑂3,5

𝑂8,6𝑂3,6𝑂3,7𝑂8,7𝑂5,7𝑂6,3𝑂5,8𝑂8,8𝑂3,8𝑂6,4

𝑂6,5𝑂6,6𝑂6,7𝑂6,8

Uniform

𝑂7,1𝑂6,1𝑂4,1𝑂7,2𝑂2,1𝑂7,3𝑂4,2𝑂7,4𝑂5,1𝑂2,2

2452 2778 11.74 405.22

𝑂6,2𝑂4,3𝑂1,1𝑂8,1𝑂4,4𝑂7,5𝑂7,6𝑂6,3𝑂3,1𝑂7,7

𝑂6,4𝑂2,3𝑂8,2𝑂4,5𝑂6,5𝑂7,8𝑂2,4𝑂1,2𝑂1,3𝑂3,2

𝑂6,6𝑂4,6𝑂5,2𝑂2,5𝑂8,3𝑂8,4𝑂3,3𝑂5,3𝑂6,7𝑂1,4

𝑂2,6𝑂6,8𝑂5,4𝑂2,7𝑂1,5𝑂4,7𝑂8,5𝑂8,6𝑂5,5𝑂4,8

𝑂3,4𝑂5,6𝑂3,5𝑂2,8𝑂1,6𝑂8,7𝑂8,8𝑂3,6𝑂1,7𝑂1,8

𝑂3,7𝑂5,7𝑂5,8𝑂3,8

Exponential

𝑂7,1𝑂4,1𝑂5,1𝑂7,2𝑂3,1𝑂4,2𝑂7,3𝑂3,2𝑂5,2𝑂7,4

2590 2683 3.47 381.17

𝑂7,5𝑂4,3𝑂7,6𝑂3,3𝑂5,3𝑂4,4𝑂2,1𝑂4,5𝑂3,4𝑂4,6

𝑂3,5𝑂2,2𝑂5,4𝑂5,5𝑂8,1𝑂1,1𝑂1,2𝑂3,6𝑂8,2𝑂2,3

𝑂4,7𝑂5,6𝑂7,7𝑂2,4𝑂3,7𝑂4,8𝑂2,5𝑂6,1𝑂5,7𝑂3,8

𝑂7,8𝑂8,3𝑂8,4𝑂5,8𝑂1,3𝑂8,5𝑂6,2𝑂2,6𝑂6,3𝑂1,4

𝑂8,6𝑂6,4𝑂2,7𝑂8,7𝑂1,5𝑂2,8𝑂1,6𝑂8,8𝑂1,7𝑂6,5

𝑂1,8𝑂6,6𝑂6,7𝑂6,8

to a reliable evaluation which guarantees a high PCS in eachgeneration, at the same time reflecting the whole idea ofOCBA which essentially is better individuals are allocatedwith more simulation times.

5.3. Test Comparisons and Performance Evaluation. In thissection, we show the test results of the proposed ESOO-OCBA algorithm and demonstrate the schedule qualitycomparing with the ESOO algorithm [8]. The followingcomputational results are conducted in Visual C++ 2010 ona Dual-Core E6600/2GB RAM/Windows XP.

Because of the random nature of the considered problemwe also have repeated the simulation process for 10 simulationruns. We have found that after the 10 simulation runs theresult changes a little. Table 4 shows the best objective values,the best schedule performance obtained by ESOO-OCBAand ESOO, respectively. The processing sequence of thebest schedules is showed in the table; 𝑂

𝑖,𝑗denotes the 𝑗th

operation on job 𝑖. Data from truncated normal distribution,uniform distribution, and exponential distribution are allshowed in Table 4. As can be observed, our ESOO-OCBAalgorithm outperforms the ESOO algorithm for these threedistributions in the quality of the results.

In order to compare our algorithm ESOO-OCBA, weadopt the same total simulation times as ESOO. However,because of the computational burden caused by the embed-ded OCBA technique, the overall consumed CPU times inour experiments are slightly longer than the consumed byESOO (within 6 minutes), but still short enough to apply ouralgorithm in real time.

Also, as we can see from Figure 5, the convergence rate inthe exploration stage of ESOO-OCBA is significantly fasterthan that in ESOO.Here in ESOO-OCBA and ESOO randomprocessing time is obtained from truncated normal distribu-tion. This result demonstrates that introducing OCBA intoESOO algorithm really improves the efficiency of ESOO.

6. Conclusion

To cope with the computationally intractable SJSSP, wefirstly embed OCBA technique into the exploration stageof ESOO algorithm to further enhance ESOO’s efficiencyby intelligently allocating simulation times according toindividual performance. However, “super individuals,” whichlead to a simulation distortion in the evaluation process,may emerge according to the classical OCBA. Then we seta threshold to constrain the simulation times allocated to


0 10 20 30 40 50 60 70 80 90 1002000220024002600280030003200340036003800

Generations

Best

indi

vidu

al p

erfo

rman

ce

Convergence rate comparison(truncated normal distribution)

ESOOESOO-OCBA

Figure 5: Convergence rate comparison between ESOO-OCBA andESOO.

“super individuals,” by which more simulation times can beallocated to other average individuals. The improvementson classical OCBA optimize the simulation times allocationmechanism, which guarantee a high probability of correctselection in each generation of the evolution in explorationstage.

The proposed algorithm ESOO-OCBA is applied to aSJSSP comprising 8 jobs and 8 machines with random pro-cessing time in truncated normal, uniform, and exponentialdistributions. The simulation test results obtained by ESOO-OCBA are compared with ESOO algorithm, demonstratingthat our algorithm has superior performances in the aspectof schedule quality, and ourmodifications onOCBA aremorereasonable in allocating computation in the evaluation.

The future research on SJSSP can be conducted from thefollowing aspects.

(1) It is worthwhile to consider other types of random-ness in job shops, for example, rush orders andmachine breakdowns.

(2) It is worthwhile to consider a new global computationallocation mechanism (i.e., the breadth versus depthapproach [26]) as OCBA technique only allocatesthe computation within each generation. Ideally, thetotal computation can be largely reduced by allocatingcomputation globally.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

Theauthors gratefully appreciate the suggestions and detailedexperiment data from Shih-Cheng Horng. This researchhas been supported by the Graduate Starting Seed Fundof Northwestern Polytechnical University (no. Z2014105),

the National Natural Science Foundation, China (no.50705076), and the Programme of Introducing Talents ofDiscipline to Universities (B13044).

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