Multi-objective DE and PSO Strategies for Production Scheduling

Jacomine Grobler, Student Member, IEEE, Andries P. Engelbrecht, Member, IEEE, V.S.S. Yadavalli

Abstract— This paper investigates the application of alterna-tive multi-objective optimization (MOO) strategies to a complexscheduling problem. Two vector evaluated algorithms, namelythe vector evaluated particle swarm optimization (VEPSO)algorithm as well as the vector evaluated differential evolu-tion (VEDE) algorithm is compared to a differential evolution-based modified goal programming approach. This paper isconsidered significant since no other reference to the applicationof vector evaluated algorithms in a scheduling environmentcould be found. Algorithm performance is evaluated on realcustomer data and meaningful conclusions are drawn withrespect to the application of MOO algorithms in a multiplemachine multi-objective scheduling environment.

I. INTRODUCTION

Production scheduling has fascinated researchers sincethe early 1950s. Customers increasingly expect to receivethe right product, at the right price, at the right time. Tomeet these requirements, manufacturing companies need toimprove their production scheduling performance.

The scheduling problem considered in this paper can bedescribed as a multi-objective flexible job shop schedul-ing problem with sequence-dependent set-up times and re-lease dates according to Graham et. al.’s three-field no-tation [1]. The assignment of auxiliary resources such astools and labour are also considered and makespan, totalearliness/tardiness and queue time need to be minimized [2].

Since the classical job shop scheduling problem, whichis significantly less complex than the proposed problem,can be classified as NP-hard [3], the problem addressed inthis paper can be considered to be too complex to obtainan optimal solution within reasonable computational time.Differential evolution [4] and particle swarm optimization [5]were thus considered to be suitable solution strategies. Thereexists significant research opportunities in expanding theseverely limited PSO scheduling literature [6]. Additionally,certain characteristics of the identified algorithms, includingfast convergence and inherent simplicity, can be desirableattributes of scheduling algorithms.

Loukil et. al. [7] provide a strong motivation for ad-dressing production scheduling as a multi-objective optimiza-tion (MOO) problem since more than one decision maker isoften involved in production decision making. Marketing is,for example, interested in maximizing customer satisfaction

Jacomine Grobler and V.S.S. Yadavalli is with the Department of Indus-trial and Systems Engineering at the University of Pretoria, South Africa(corresponding author to provide e-mail: jacomine@tuks.co.za).

Andries P. Engelbrecht is with the Department of Computer Science atthe University of Pretoria, South Africa.

The financial assistance of the National Research Foundation towardsthis research is hereby acknowledged. Opinions expressed in this paper andconclusions arrived at, are those of the authors and not necessarily to beattributed to the National Research Foundation.

by minimizing expected tardiness, while production manage-ment is concerned with minimizing makespan and subse-quently also work in progress and machine utilization. Workin progress, the number of jobs which are currently beingprocessed on the production floor, also impacts customersatisfaction. These objectives are often conflicting and it isthus in the best interests of all concerned to provide a set offeasible solutions representing trade-offs between the variousobjectives to be achieved.

The proposed scheduling problem has not yet been ad-dressed frequently in scheduling literature. A single objectivevariation of the problem has been solved by means of arandom keys GA algorithm [8] as well as various otherrule-based scheduling algorithms [9]. Grobler et. al. [10]developed a number of PSO-based heuristics based on al-ternative particle representations and mapping mechanisms.However, none of this work ([8], [9] or [10]) focusedexplicitly on the multi-objective aspects of the problem. Thepurpose of this paper is to investigate various PSO and DE-based MOO strategies for solving this complex combinatorialoptimization problem.

Two vector evaluated MOO algorithms, namely the vec-tor evaluated PSO (VEPSO) and the vector evaluated DE(VEDE) as well as a DE-based goal programming ap-proach (GP) were identified as suitable candidates for evalu-ation. When tested on real customer data the vector evaluatedalgorithms were not able to exceed the performance ofthe GP approach, yet highly satisfactory results were stillobtained in a fraction of the time required by the GPalgorithm.

Apart from the sheer complexity of the proposed problem,this paper is considered significant since no other reference tothe application of vector evaluated algorithms in a schedulingenvironment can be found in literature. Furthermore, boththe VEPSO and VEDE algorithms were extended to addressthree objective functions.

The rest of the paper is organized as follows: Section IIdescribes the problem in more detail before Sections IIIand IV discuss the applicable literature. Section V describesthe priority-based algorithm on which this paper builds, whileSection VI describes the MOO strategies employed. Theresults are documented in Section VII before the paper isconcluded with Section VIII.

II. THE PROPOSED PROBLEM

The scheduling problem considered in this paper can bebest described by means of an example. To produce a specificproduct, two parts have to be manufactured and assembled.If each subassembly is defined as a different job, and each

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job consists of a number of operations representing the pro-duction processes through which each part has to be routed.Each operation can be performed on any machine from aset of primary resources. Tools and labour may be requiredand can be selected from a set of auxiliary resources. Theprocessing time of an operation includes sequence-dependentset-up times and is dependent on the resource on which itis produced. However, the actual production time for eachoperation may also be affected by scheduled maintenance,machine breakdowns and production calendars. Customerrequirements result in the subsequent schedules having tobe evaluated simultaneously with respect to makespan, totalearliness/lateness as well as total queue time [2].

The resulting problem can be classified as a multi-objective flexible job shop scheduling problem withsequence-dependent set-up times, release dates, auxiliary re-sources and resumable machine unavailability intervals [11].

III. MULTI-OBJECTIVE OPTIMIZATION

A. Introductory concepts

A multi-objective optimization problem can be formallydefined as follows:

minimize f(x)f(x)f(x)

subject to gm(xxx) ≤ 0, m = 1, . . . , ng

hm(xxx) = 0, m = ng + 1, . . . , ng + nh

xxx ∈ [xxxmin,xxxmax]nx (1)

where f(x)f(x)f(x) denotes the vector of objective functions to beminimized, gm and hm are respectively the inequality andequality constraints and xxx ∈ [xxxmin,xxxmax]nx represent theboundary constraints. A solution to a MOO problem can thusbe defined as a vector xxx that satisfies the constraints andoptimizes the vector function f(x)f(x)f(x) [12].

The purpose of MOO is to find a set of trade-off solutionsreferred to as the Pareto-optimal set, P , where

P ={xxx∗ ∈ �| � xxx ∈ � : x∗x∗x∗ ≺ xxx} (2)

and the dominance relation, ≺, indicates that solution xxx∗

dominates solution xxx, i.e. xxx∗ is not worse than xxx in allobjectives and xxx∗ is strictly better than xxx in at least oneobjective.

B. Multi-objective scheduling

The importance of multi-objective scheduling problemshave long since been recognized in literature and goodoverall reviews can be found in the works of Loukil andTeghem [13] and T’Kindt and Billaut [14]. However, due tothe extreme complexity of these combinatorial problems littleattention has been paid to multiple machine, multi-objectivescheduling problems. These problems allow the schedulingof an operation on any resource from a set of availableresources. Nonetheless a number of notable exceptions canbe listed in Table I from recent flexible job shop schedulingliterature.

From the algorithms listed in Table I two important conclu-sions can be drawn with respect to the use of multi-objective

TABLE I

MULTIPLE MACHINE, MULTI-OBJECTIVE SCHEDULING LITERATURE.

Algorithm Reference

Multi-objective simulated annealing Loukil et. al. [7]

(MOSA)

Tabu search (TS) Dauzere–Peres and Paulli [15]

Localization and EA approach Kacem et. al. [16]

Hybrid evolutionary algorithm Kacem et. al. [17]

and fuzzy logic

Simulated annealing-PSO algorithm Xia and Wu [18]

Multi-stage GA Zhang and Gen [19]

GA-bottleneck shifting hybrid Gao et. al. [20]

optimization techniques in multiple machine, multi-objectivescheduling problems. Firstly, the use of the weighted aggre-gation approach is quite common, as in [20], and secondlythe more sophisticated MOO techniques are difficult tounderstand and computationally complex [7]. There seemsto be a research opportunity in developing a multi-objectivemultiple machine scheduling algorithm which overcomes theproblems associated with weighted aggregation [21] whilestill being relatively simple to understand and implement.

IV. SUITABLE SOLUTION STRATEGIES

Recent investigations into the use of DE and PSO forscheduling ([18] and [22] to [25]) have resulted in thesetwo algorithms being identified as candidate algorithms forfurther investigation. This section describes the basics ofthese two popular solution strategies before the next sectionprovides more detail with respect to the actual implemen-tation of these algorithms in a multiple machine job shopenvironment.

A. Particle swarm optimization

PSO can be classified as a stochastic population-basedoptimization technique [5], which was developed as a modelof the flocking behaviour of birds. Since its development,the algorithm has established itself as a competitive solutionstrategy for a wide range of real-world problems. This sectionbriefly discusses the basics of the algorithm and providesdetails regarding the specific variation on the standard gbestalgorithm which was implemented.

1) Introductory concepts: In the gbest PSO algorithmeach potential problem solution is represented by the positionof a particle in multi-dimensional hyperspace. Throughoutthe optimization process velocity and displacement updatesare applied to each particle to move it to a different positionand therefore a different solution in the search space. Thevelocity of particle i in dimension j at time step t + 1 isgiven by:

vij(t + 1) =wvij(t) + c1r1j(t)[xij(t) − xij(t)]+

c2r2j(t)[x∗j (t) − xij(t)] (3)

where vij(t) represents the velocity of particle i in dimensionj at time t, c1 and c2 are the cognitive and social acceler-ation constants, xij(t) and xij(t) respectively denotes the

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1155

personal best position (pbest) and the position of particlei in dimension j during time t. x∗

j (t) denotes the globalbest position (gbest) in dimension j, w refers to the inertiaweight and r1j(t), r2j(t) is sampled from a uniform randomdistribution, U(0, 1).

The displacement of particle i at time t is defined as:

xxxi(t + 1) =xxxi(t) + vvvi(t + 1) (4)

This simultaneous movement of particles towards their ownprevious best solutions (pbest) and the best solution found bythe entire swarm (gbest) results in the particles convergingto one or more good solutions in the search space.

2) The guaranteed convergence PSO algorithm: Unfor-tunately, the gbest PSO developed in [5] has the potentialto stagnate on a solution which is not necessarily even alocal optimum [26]. This requires that special interventionsneed to be made. The guaranteed convergence particle swarmoptimization (GCPSO) algorithm of Van den Bergh andEngelbrecht [26] has been shown to address this problemeffectively and have thus been used in this paper. Thisalgorithm requires that different velocity and displacementupdates, respectively indicated by Equation (5) and Equa-tion (6), are applied to the global best particle:

vτj(t + 1) = − xτj(t) + x∗j (t) + wvτj(t)+

ρ(t)(1 − 2rj(t)) (5)

xτj(t + 1) =x∗j (t) + wvτj(t) + ρ(t)(1 − 2rj(t)) (6)

This in fact forces the gbest particle into a random searcharound the global best position. The size of the searchspace is adjusted on the basis of the number of consecutivesuccesses or failures of the particle, where success is definedas an improvement in the objective function value. It shouldbe noted that throughout the optimization run, only the gbestparticle is treated differently. All other particles utilize thevelocity update defined in Equation (8).

3) The Von Neumann PSO architecture: In recent years,the degree of social interaction between particles in the PSOalgorithm has received increasingly more attention. Fromthis research a number of alternative social network struc-tures have been developed to explore different informationexchange mechanisms between the particles within a swarm.A number of these social network structures, including thegbest, lbest, pyramid, star and Von Neumann structures, havebeen tested empirically by [27].

It is well known in PSO literature that the gbest PSOalgorithm converges fairly quickly [28]. This is due tothe fact that all particles are partially attracted to the bestposition found by the entire swarm since the beginning of theoptimization run. Depending on the problem, this relativelyfast loss of diversity can result in the algorithm finding asuboptimal solution within relatively few iterations.

The Von Neumann PSO organizes the particles into acubic structure according to the particle indices. Each par-ticle belongs to a neighbourhood consisting of its nearestneighbours in the cubic structure. Instead of being partiallyattracted to gbest, the velocity of a particle is influenced by

the best solution found thus far by the other particles in thesame neighbourhood. Since these neighbourhoods overlap,information about good solutions is eventually propagatedthroughout the swarm, but at a much slower rate. This resultsin more diversity being maintained and subsequent slowerconvergence. This, in turn, is thought to significantly improvethe algorithm’s chances of finding a good solution.

During initial experimentation on real customer test data, itwas verified that a significant performance improvement wasindeed observed upon implementation of the Von Neumannsocial network structure. The resulting Von Neumann GCPSOalgorithm was subsequently used for all variations presentedin this paper.

B. Differential evolution

Similar to evolutionary algorithms, each candidate solutionin a DE algorithm is represented by a parameter vector. Thealgorithm attempts to move to different solutions within thesearch space by means of specially defined crossover andmutation operators. The algorithm’s success can be mainlyattributed to the use of the difference vector. Informationregarding the difference between two solutions in the searchspace is used to guide the algorithm towards better solu-tions [4].

More specifically, for each individual, i, in the populationthree different vectors are randomly selected from the currentpopulation, namely xxxr1

(t), xxxr2(t) and xxxr3

(t), where xrkj(t)denotes the jth dimension of the kth vector of individual i ofgeneration t and i �= r1 �= r2 �= r3. Then, for all dimensions,j, if r ∼ U(0, 1) ≤ pc or j = i ∼ U(1, ..., J)

cij(t) =xr1j(t) + F (xr2j(t) − xr3j(t)) (7)

otherwise cij(t) = xij(t), where pc is the probability ofreproduction, J is the number of dimensions and F is thescaling factor. If the fitness of ccci(t) is better than the fitnessof the ith individual of the original population, this individualis replaced by ccci(t) [21].

Similar to the PSO algorithm, a number of variationsof differential evolution have also been developed in re-cent years [4]. The rate of convergence and subsequentdecrease in population diversity can be controlled furtherby means of different trial vector selection mechanisms.For example, DE/rand/bin selects xxxr1

(t) randomly fromthe previous population and thereby maintains diversity forlonger, whereas DE/best/bin selects xxxr1

(t) as the populationmember with the lowest fitness function and subsequentlyobtains faster convergence. Experimenting with alternativeselection mechanisms resulted in DE/rand/bin being foundsuperior to DE/best/bin for the problems tested and thisvariant is again used throughout the rest of the paper.

V. THE PRIORITY-BASED SCHEDULING ALGORITHM

Although this paper focuses explicitly on investigatingalternative DE and PSO multi-objective strategies for theproposed problem, there are a number of algorithmic featureswhich remain constant regardless of the MOO strategy

1156 2008 IEEE Congress on Evolutionary Computation (CEC 2008)

employed. These include, amongst others, the problem rep-resentation as well as evaluation of the fitness function. Forthe MOO comparison in the next section, the priority-basedscheduling algorithm developed by Grobler et. al. [10] for theproposed problem was used. For the sake of convenience andcompleteness brief explanations as well as the pseudocoderequired for replication purposes are again provided below.

Multiple machine scheduling algorithms need to addresstwo important aspects, namely the allocation of operationsto resources as well as the sequencing of the allocatedoperations on the various resources. A two-part problemrepresentation should thus be used.

The representation of the priority-based algorithm consistsof a 2n-dimensional vector, where the sequencing variablesof dimensions 1 to n denote the priority values of each of theoperations to be scheduled. These priorities are used as inputto a schedule-building heuristic which attempts to scheduleeach operation at the earliest available time on its associatedresource.

Dimensions n+1 to 2n are used to represent the allocationof operations to resources. This is done by discretizing thesearch space. For each operation i, the ith dimension of thesearch space is divided into Mi intervals, where Mi denotesthe number of primary resources on which operation i canbe processed. Since each interval is associated with a uniqueinteger number or resource index, dimensions n + 1 to 2n

of the position vector can easily be interpreted as resourceallocation variables.

The pseudocode in Algorithm 1 describes the procedurefollowed to convert the problem representation into a feasibleschedule from which the fitness function can be evaluated.In addition, Figure 1 provides an example of the procedurefollowed to obtain QQQi, the set of possible starting timesof operation i on resource di, where the release date ofoperation i refers to the earliest time operation i is availablefor scheduling.

Fig. 1. When operations i − 3,i − 2 and i − 1 are already scheduled onresource di, the possible starting times of operation i (QQQi) are A, B andC.

VI. MULTI-OBJECTIVE PSO AND DE STRATEGIES

Over the years a number of papers have addressed theextension of both PSO and DE to multiple objectives. Acomprehensive review of multi-objective PSO algorithmscan be found in [29]. Three popular multi-objective DEalgorithms, from the past few years, include the pareto-basedmulti-objective DE algorithms of [30], [31] and [32].

For the purposes of this paper, two more general MOOstrategies were selected for comparison. Firstly, the vector

Algorithm 1: The Priority-based schedule buildingheuristic.Let ΦΦΦ be the partial schedule which contains scheduled1

operationsLet ΩΩΩ be the set of schedulable operations2

Let zi be the finishing time of operation i3

Let ti be the starting time of operation i4

Initialize ΦΦΦ = ∅5

Initialize ΩΩΩ to contain all operations without any6

predecessorswhile ΩΩΩ �= ∅ do7

Select i from ΩΩΩ as operation with the highest8

priorityDetermine QQQi (the set of possible starting times for9

operation i on resource di)while i ∈ ΩΩΩ do10

Set ti = min(QQQi)11

Calculate zi by incorporating production12

downtime and auxiliary resources into theschedule [10]if ti = min(QiQiQi) results in a feasible schedule13

thenDelete i from ΩΩΩ14

else15

Delete min(QQQi) from QQQi16

end17

end18

Insert i into ΦΦΦ19

for All successors j of i do20

if All other predecessors of operation j ∈ ΦΦΦ21

thenInsert j into ΩΩΩ22

end23

end24

end25

evaluated MOO approach showed potential due to its highlevel of simplicity and subsequent reduced computationalcomplexity compared to more sophisticated MOO tech-niques. Its frequent use in scheduling led to the selectionof GP as the second MOO strategy to be investigated.

A. The vector evaluated approach to multi-objective opti-mization

The vector evaluated approach can be classified asa criterion-based multi-objective strategy, where differentstages of the optimization process use different objec-tives [21]. This is achieved by assigning each objectivefunction to one of multiple populations for optimization.Information with respect to the different populations areexchanged in an algorithm-dependent fashion resulting in thesimultaneous optimization of the various objective functions.The advantage of this approach lies in reduced computationalcomplexity, which is a desirable property when solving acomplex combinatorial problem where the fitness function

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1157

evaluations are in themselves computationally cumbersome.The first vector evaluated algorithm was Schaffer’s vector

evaluated GA [33]. A number of years later the conceptwas transferred to the PSO and DE domains by Par-sopoulos et. al. [34], [35]. Their work focused on com-paring the vector evaluated PSO (VEPSO) algorithm tovarious aggregation-based approaches including bang-bangand dynamic weighted aggregation. The vector evaluatedDE (VEDE) algorithm proved to compare well with Scha-effer’s VEGA. These studies used two-objective problemsfrom the well-known MOO benchmark set of [12] forbenchmarking purposes.

In terms of the actual algorithms, VEDE and VEPSO differonly with respect to the information exchange mechanismused. VEPSO (Algorithm 2) makes use of an alternativevelocity update equation where (considering two swarms) theglobal best position of the first swarm is used in the velocityequation of the second swarm and vice versa. Thus,

vijs(t + 1) =wvijs(t) + c1r1j(t)[xijs(t) − xijs(t)]+

c2r2j(t)[x∗jms

(t) − xijs(t)] ∀s

(8)

where vijs(t), xijs(t) and xijs(t), respectively, denote thevelocity, position and personal best position of the jth

dimension of the ith particle of swarm s at time t andx∗

jms(t) is the jth dimension of the global best position of

swarm ms at time t.VEDE (Algorithm 3), on the other hand, migrates the best

individual between different populations and additionallymakes use of a dominance-based selection mechanism, wherean individual can only be replaced by a dominating offspring.

Both of the vector evaluated algorithms employed in thispaper make use of an external archive [36] of unlimited sizeto store all non-dominated solutions obtained throughout theoptimization process.

B. Modified goal programming

Goal programming (GP) [37] is based on the assump-tion that a user has sufficient knowledge of the problemdomain to specify suitable target values for each of theobjective functions to be optimized. An aggregate objectivefunction consisting of the sum of the deviations betweenthe actual values obtained and the specified targets is thenminimized [37].

The user, in fact, attempts to direct a single objectiveoptimization algorithm towards a pre-specified point on thePareto front. Due to the complexity of MOO problems it canhappen that the user defines targets which denote a positionin the objective space which is actually dominated by thePareto front. In this case, it is desirable for the algorithmto continue optimizing until convergence is obtained on anearby solution located on the Pareto front. To achieve this,the aggregated fitness function, f4, can be given by

f4 =

I∑i=1

|(fi − gi)| + βfi (9)

Algorithm 2: The vector evaluated particle swarm opti-mization (VEPSO) algorithm.

Initialize three swarms, S1S1S1,S2S2S2 and S3S3S31

t = 12

while t < Imax do3

for All swarms s do4

f(x∗sx∗sx∗s(t)) = mini(fs(xsxsxs(t)))5

for All individuals i do6

if fs(xisxisxis(t)) < fs(xisxisxis(t)) then7

xisxisxis(t) = xisxisxis(t)8

end9

end10

if mini(fs(xsxsxs(t))) < fs(x∗sx∗sx∗s(t)) then11

f(x∗sx∗sx∗s(t)) = mini(fs(xsxsxs(t)))12

end13

Update the external archive to include all14

non-dominated solutions in SSSs

end15

for All swarms s do16

for The gbest particle τ do17

for All dimensions j do18

vτjs(t) = −xτjs(t) + x∗jms

(t) +19

wvτjs(t) + ρ(1 − 2r2j)xτjs(t) = x∗

js(t)+wvτjs(t)+ρ(1−2r2j)20

end21

end22

for All particles i such that i �= τ do23

for All dimensions j do24

vijs(t) = wvijs(t) + c1r1j(xijs(t) −25

xijs(t)) + c2r2j(x∗jms

(t) − xijs(t))xijs(t) = xijs(t) + vijs(t)26

end27

end28

end29

t = t + 130

end31

where fi denotes the ith fitness function, gi is the target valueof the ith fitness function and β is selected as sufficientlysmall.

Goal programming has the advantage that the algorithmfocuses explicitly on a single objective throughout the entireoptimization process. However, this results in only onesolution being obtained as output to the scheduling algorithm.Obtaining the same number of solutions as can be obtainedwith a single iteration of a vector evaluated algorithm, canindeed be a time-consuming process. Furthermore, the per-formance of a GP-based MOO algorithm is highly dependenton the type of single objective optimization algorithm used.

To ensure a fair comparison between GP and the othervector evaluated algorithms tested, a number of single objec-tive optimization algorithms including Norman and Bean’srandom keys GA [8], a priority-based PSO [10] and apriority-based DE was tested on the 56-operation problem.

1158 2008 IEEE Congress on Evolutionary Computation (CEC 2008)

Algorithm 3: The vector evaluated differential evolution(VEDE) algorithm.

Initialize three populations, P1P1P1,P2P2P2 and P3P3P31

t = 12

while t < Imax do3

for All populations s do4

Update the external archive to include all5

non-dominated solutions in PsPsPs

f(x∗sx∗sx∗s(t)) = mini(fs(xsxsxs(t)))6

end7

for All populations s do8

x∗sx∗sx∗s(t) = x∗

msx∗

msx∗

ms(t)9

end10

for All populations s do11

for All individuals i do12

Randomly select 3 individuals from13

population s, xr1sxr1sxr1s(t), xr2sxr2sxr2s(t) and xr3sxr3sxr3s(t),such that r1 �= r2 �= r3 �= i

Randomly select one of the dimensions R14

for All dimensions j do15

if lijs ≤ pc or j = R then16

cijs =17

xr3js(t) + F (xr1js(t) − xr3js(t))else18

cijs = xijs(t)19

end20

end21

if cisciscis ≺ xisxisxis(t) then22

xisxisxis(t) = cisciscis23

end24

end25

end26

t = t + 127

end28

The resulting conclusion was that the priority-based DEoutperforms the other two algorithms and it was subsequentlyemployed as part of the modified GP approach described inAlgorithm 4.

VII. EMPIRICAL RESULTS

Optimatix is a South African-based company which spe-cializes in providing customized software solutions. Forperformance evaluation purposes, three test problems cor-responding to the requirements of Optimatix in terms ofproblem size and complexity, were derived and extendedfrom actual customer data. All three data sets are availablefrom the corresponding author.

An important issue relating to performance evaluation, isthe selection of suitable parameter values for the algorithmswhich are compared. A parameter derivation study wasconducted on the 56-operation problem and the resultingparameter values which were used for comparison purposesare listed in Table II. The number of individuals in thepopulation is denoted by ns and a and b denote the interval

Algorithm 4: The modified GP approach.

Let Oji be the value taken by objective j if objective i1

is minimizedfor All criteria i do2

OOOi = min fi3

end4

for All criteria j do5

Tjmin = min∀i OOO

j6

T jmax = max∀i OOO

j7

�j =T j

max−Tj

min

p8

end9

T1 = T 1min −�110

for All points i do11

T1 = T1 + �112

T2 = T 2min −�213

for All points j do14

T2 = T2 + �215

T3 = T 3min −�316

for All points k do17

T3 = T3 + �318

PFPFPF = PFPFPF ∪ {min∑I

i=1 |(fi − gi)| + βfi}19

end20

end21

end22

sizes within which the decision variables are initialized.The size of the discretization intervals, γi, of operation i

is dependent on D, where

γi =2a

XiD(10)

Xi is the number of resources on which operation i maybe scheduled and m −→ n indicates that the associatedparameter is decreased linearly from m to n over 95% ofthe total number of iterations, Imax.

TABLE II

PRIORITY-BASED PSO AND DE PARAMETERS.

Parameter Value used

General parameters

ns 27

a 500

b 1500

D 3

Imax 200

PSO-specific parameters

c1 2.0 −→ 1.0

c2 2.8 − c1

w 0.8 −→ 0.4

DE-specific parameters

pc 0.75 −→ 0.25

F 0.75 −→ 0.125

Measuring the performance of multi-objective optimiza-

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1159

tion algorithms is significantly more complex than simplyconsidering accuracy or speed of convergence, which arecommonly considered to be suitable measures of perfor-mance associated with single objective algorithms. Insteadthree aspects need to be considered, namely the minimizationof the distance between the solutions obtained and theactual Pareto front, the maximization of the diversity of thePareto front and the ability to maintain already found non-dominated solutions [21]. Comprehensive reviews of MOOperformance measures can be found in [38] and [21].

Three performance measures were used in this pa-per to compare the different MOO strategies. The S-metric [38], [39] measures the size of the region dominatedby the Pareto front based on a reference vector consistingof the maximum value in each objective (Table III). The S-metric of the set, PFPFPF , with respect to fffref can be describedas the Lebesgue integral of the set R(PFPFPF ,fffref ), where

R(PFPFPF ,fffref ) = ∪∀fff∈PFPFPF R(fff,fffref ) (11)

and

R(fff,fffref ) ={f ′f ′f ′|f ′f ′f ′ ≺ fffref and fff ≺ f ′f ′f ′, f ′f ′f ′ ∈ Rnk} (12)

TABLE III

THE REFERENCE VECTORS USED FOR EACH OF THE PROBLEMS.

Problem Vector56-operation problem {1000, 4000, 1000}100-operation problem {2000, 5000, 2000}256-operation problem {2500, 12000, 5000}

The other two measures include the size of the approxi-mated Pareto fronts (N(PFPFPF )) as well as the extent of thePareto fronts (χ(PFPFPF )) [21], where

χ(PFPFPF ) =

√√√√N(PFPFPF )∑k=1

max{|fk(xxx) − f ′k(xxx)| : fff,f ′f ′f ′ ∈ PFPFPF}.

(13)

The actual results of the multi-objective comparison arerecorded in Tables IV through VI, where the VEPSO, VEDEand DE-based GP algorithm results were each recorded over30 independent simulation runs. Throughout the rest of thissection, μ and σ respectively denote the mean and standarddeviation associated with the corresponding performancemeasure.

TABLE IV

INVESTIGATING ALTERNATIVE MULTI-OBJECTIVE OPTIMIZATION

STRATEGIES FOR THE 56-OPERATION PROBLEM.

Strategy VEPSO VEDE GPS(PFPFPF ) μ 1.00 × 109 9.07 × 108 1.26 × 109

σ 1.19 × 107 1.67 × 107 3.96 × 107

N(PFPFPF ) μ 69 10.20 10.03σ 17.74 2.70 1.56

χ(PFPFPF ) μ 29.12 18.33 30.07σ 2.36 1.91 1.57

For all the problems tested, the modified GP approach issuperior with respect to the S-metric. This measure provides

TABLE V

INVESTIGATING ALTERNATIVE MULTI-OBJECTIVE OPTIMIZATION

STRATEGIES FOR THE 100-OPERATION PROBLEM.

Strategy VEPSO VEDE GPS(PFPFPF ) μ 8.54 × 109 7.90 × 109 9.36 × 109

σ 4.75 × 107 1.27 × 108 1.52 × 108

N(PFPFPF ) μ 209.5 29.10 17.57σ 27.25 4.94 2.25

χ(PFPFPF ) μ 46.16 35.90 36.39σ 1.01 1.91 1.39

TABLE VI

INVESTIGATING ALTERNATIVE MULTI-OBJECTIVE OPTIMIZATION

STRATEGIES FOR THE 256-OPERATION PROBLEM.

Strategy VEPSO VEDE GPS(PFPFPF ) μ 2.83 × 1010 2.29 × 1010 4.59 × 1010

σ 1.51 × 109 1.12 × 109 2.12 × 109

N(PFPFPF ) μ 155.97 18.30 16.13σ 34.32 3.58 2.01

χ(PFPFPF ) μ 94.84 62.11 85.62σ 4.38 8.72 2.72

an indication of both the quality of the obtained Paretofront with respect to the actual Pareto front as well asthe diversity of the solutions obtained. By comparing thediversity measures of GP to the other two approaches itbecomes evident that this superior performance with respectto the S-metric is mostly due to the quality of the Paretofront, since the χ values indicate that VEPSO outperformsGP with respect to Pareto front diversity for two out of thethree problems tested.

The results thus seem to indicate that GP favours exploita-tion of the search space while VEPSO facilitates greaterexploration. This outcome makes sense when the structureof these two algorithmic approaches are considered. GPrepeatedly solves a significantly easier single objective opti-mization problem, whereas VEPSO and VEDE attempt toconverge simultaneously on a number of different Paretosolutions. The effect on exploitation of either the migration ofgood individuals to a different population, as in VEDE, or theabsence of a gbest velocity component (with respect to thecurrent swarm in VEPSO) should also not be underestimated.

Apart from the extreme computational complexity of themodified GP approach, another concerning aspect becomesevident. The GP algorithm is executed 27 times, where eachiteration attempts to zoom in on a different uniformly distrib-uted point on the Pareto front. However, when all dominatedsolutions were removed from the resulting Pareto front,only 10.03, 17.57 and 16.13 solutions were obtained overthe 30 simulation runs for the three problems tested. Thisindicates that on average, 16.97, 9.43 and 10.87 solutionswere dominated by one or more other solutions obtained bythe algorithm. This has dramatic implications for algorithmefficiency indicating that up to 63% of computational time iswasted by the inability of GP to use information regardingprevious non-dominated solutions during the search process.

Finally, it should be noted that in terms of the actualnumber of solutions obtained, VEPSO dramatically outper-

1160 2008 IEEE Congress on Evolutionary Computation (CEC 2008)

forms both of the other algorithms and that VEDE performssignificantly poorer on all three test problems with respectto all of the utilized performance measures.

VIII. CONCLUSION

This paper investigated the application of two alternativeMOO strategies in a complex scheduling environment. Twovector evaluated algorithms, namely the VEPSO and VEDEwere compared to a DE-based modified goal programmingapproach. When tested on real customer data a number ofinteresting conclusions could be drawn.

Significant future research opportunities exist in improvingthe exploitation capabilities of the vector evaluated algo-rithms. A more in depth investigation into the large differencein performance between the VEPSO and VEDE algorithmscould also prove to be insightful.

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