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Applied Soft Computing 21 (2014) 139–148

Contents lists available at ScienceDirect

Applied Soft Computing

j ourna l h o mepage: www.elsev ier .com/ locate /asoc

ulti-objective fuzzy multiprocessor flowshop scheduling

. Behnamiana, S.M.T. Fatemi Ghomib,∗

Department of Industrial Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan, IranDepartment of Industrial Engineering, Amirkabir University of Technology, 424 Hafez Avenue, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 19 December 2011eceived in revised form 6 March 2014ccepted 22 March 2014vailable online 2 April 2014

eywords:

a b s t r a c t

This paper considers a bi-objective hybrid flowshop scheduling problems with fuzzy tasks’ operationtimes, due dates and sequence-dependent setup times. To solve this problem, we propose a bi-level algo-rithm to minimize two criteria, namely makespan, and sum of the earliness and tardiness, simultaneously.In the first level, the population will be decomposed into several sub-populations in parallel and eachsub-population is designed for a scalar bi-objective. In the second level, non-dominant solutions obtainedfrom sub-population bi-objective random key genetic algorithm (SBG) in the first level will be unified

uzzy hybrid flowshop schedulingi-objective optimizationareto optimal solutionarticle swarm optimizationarallel genetic algorithm

as one big population. In the second level, for improving the Pareto-front obtained by SBG, based on thesearch in Pareto space concept, a particle swarm optimization (PSO) is proposed. We use a defuzzificationfunction to rank the Bell-shaped fuzzy numbers. The non-dominated sets obtained from each of levelsand an algorithm presented previously in literature are compared. The computational results showedthat PSO performs better than others and obtained superior results.

ell-shaped fuzzy number

. Introduction

Scheduling problems are the allocation of machines to perform set of jobs in a period of time. In general, scheduling requiresoth sequencing and machine allocation decisions. When there isnly one machine, the allocation of that machine is completelyetermined by sequencing decisions. As a consequence, in theingle-machine model, no distinction exists between sequencingnd machine allocation. To appreciate that distinction, we mustxamine models with more than one machine. A hybrid flowshopcheduling (HFS) problem, as described by Linn and Zhang [1], con-ists of a series of production stages, each of which has severalachines operating in parallel. Some stages may have only oneachine, but at least one stage must have multiple machines. TheFS is an adequate model for several industrial settings such as

emiconductors, electronics manufacturing, airplane engine pro-uction, and petrochemical production [2]. The flow of products inhe plant is unidirectional. Each product is processed at only oneacility in each stage and at one or more stages before it exits thelant. The machines in stages can be identical, uniform, or unre-

ated and each job is processed by at most one machine at eachtage.

∗ Corresponding author. Tel.: +98 21 64545381; fax: +98 21 66954569.E-mail addresses: Fatemi@aut.ac.ir, fatemighomi@gmail.com

S.M.T. Fatemi Ghomi).

ttp://dx.doi.org/10.1016/j.asoc.2014.03.031568-4946/© 2014 Elsevier B.V. All rights reserved.

© 2014 Elsevier B.V. All rights reserved.

Many real-world problems involve simultaneous optimizationof several objective functions. In general, these functions oftencompete and conflict with themselves [3]. Many industries suchas aircraft, electronics, semiconductors manufacturing, etc., havetradeoffs in their scheduling problems where multiple objectivesneed to be considered in order to optimize the overall perfor-mance of system. For reflecting real-world situation adequately, weformulate the scheduling problems as bi-objective ones that simul-taneously minimize the maximum completion time (makespan),and the sum of the earliness and tardiness (ET) of jobs. The useof both objectives is well-justified in the practice, as makespanminimization implies the maximization of the throughput andthe second objective comes from the make-to-order philosophy inmanagement and production theory: an item should be deliveredexactly when it is required by the customer. Therefore, both earlyand tardy delivery of a task with respect to its due date is penal-ized. Interestingly, the objective of the ET problem fits perfectly tojust-in-time (JIT) production control policy where an early or a latedelivery of an order results in an increase in the production costs.

Fuzzy set theory has been studied extensively over the past 40years and many approaches have been proposed in this field. Mostof the early interest in fuzzy set theory pertained to representinguncertainty in the human cognitive processes. Fuzzy set theory isnow applied to problems in the various sciences and it is being

recognized as an important problem modeling and solution tech-nique [4]. Several successful applications and implementations offuzzy set theory in the production management have also beenreported. Fuzzy scheduling is a very important research topic not

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40 J. Behnamian, S.M.T. Fatemi Ghomi / A

nly because of the fuzzy nature of most real-world problems, butlso because there are still many open questions in this area.

Four important motivations that fuzzy set theory is relevant tocheduling are as follows [5].

In the scheduling environment, the information required to for-mulate an objective function, decision variables, constraints andparameters may be vague or not precisely measurable.Third, imprecision and vagueness as a result of personal bias andsubjective opinion may further dampen the quality and quantityof available information.A fuzzy scheduling algorithm constructs the real system flexibil-ity and models the uncertainty inherent in real environments.

This notice has led to the development of fuzzy scheduling. Themost obvious place” to introduce fuzzy concepts for modelingncertainty in scheduling algorithms is with a task’s operation andue date time. Typical scheduling algorithms assume this informa-ion to be crisp value; this may or may not be reliable. By modelinghese times of a task with a fuzzy number, a system designer canuild flexibility into the scheduling algorithm and reach to betterolution.

Machine setup time is a significant factor for production sched-ling in many factories. In sequence-dependent scheduling, the

ength of time required to do the setup depends on both the priornd the current job to be processed [6]. Due to great saving whenetup times are explicitly included in scheduling decisions, weake into account the existence of sequence-dependent setup timesn our problem. Sequence-dependent setup times (SDST) hybridowshop scheduling can be found in a vast number of industries.umerous examples are given in the literature, including the plas-

ics manufacturing, rolling slitting in the paper industry [7] andafer testing in semiconductor manufacturing [8]. Since recently,

onsidering SDST becomes popular researches’ area [9], accordingo taxonomy of Quadt and Kuhn [10] only few studies consideredetup times.

Our goal in this paper is to develop a two level algorithm toi-objective fuzzy hybrid flowshop that gives a set of compromisenon-dominated) solutions, so that, these solutions should repre-ent a good approximation to the Pareto optimal front.

The paper has the following structure. Section 2 gives litera-ure review. Section 3 presents mathematical modeling. Section

is the bell-shaped fuzzy number description and its arithmetic.ection 5 introduces the two-level algorithm and characteristicsf the proposed algorithms. Section 6 presents the experimentalesign which compares the algorithms’ results. Finally, Section 7 isevoted to conclusions and future works.

. Literature review

Ishibuchi and Murata [11] fuzzified the due dates of jobs.n their work, the satisfaction degrees for job completion timesre described using fuzzy sets, and the objective is to obtain aob sequence whose completion time has a maximum degreef satisfaction. In that study, a multi-objective genetic algorithms developed to handle these fuzzy scheduling objectives. Thencertainty of processing time has been described as triangleuzzy numbers by Liu and Gu [12] in the flowshop schedulingnvironment under uncertainty on fuzzy programming theory.alasubramanian and Grossmann [13] presented a mixed integer

inear programming models for the flowshop scheduling problems

ith uncertainty in the processing time. The uncertainty of taskurations used from the fuzzy set theory concepts. They predicthe optimistic and pessimistic values of makespan. Peng and Song14] proposed a hybrid intelligent algorithm to solve three types of

Soft Computing 21 (2014) 139–148

fuzzy flowshop scheduling models. In that study the authors aimedto demonstrate how the hybrid intelligent algorithm can be usedfor managing fuzzy scheduling on the flowshop problems. Celanoet al. [15] used a concept of agreement index to solve the sched-uling problems with fuzzy due dates considering the fuzzy-sumand fuzzy maximum among triangular fuzzy numbers.

Zheng and Gu [16] considered no-wait flowshop productionscheduling within finite intermediate storage under uncertaintyin which fuzzy processing time was conducted. The maximummembership function of mean value was applied to convert thefuzzy scheduling model to an accurate optimization problem. Animproved simulated annealing algorithm was proposed to verifyand optimize the scheduling model. Wu and Gu [17] used the fuzzynumbers to denote the uncertainty of processing time in flow-shop scheduling problem, and developed the multiple objectivesscheduling model for flowshop scheduling problems with fuzzyprocessing time. In that research, the multiple objectives are con-verted into single one by weighted method, and the fuzzy optimalproblem then transformed to determinate one by Zimmermannalgorithm. Finally, Wu and Gu [17] proposed a genetic algorithmfor solving the problems. Temiz and Erol [18] applied fuzzy con-cept to the flowshop scheduling problems. A branch-and-boundalgorithm was used for three-machine flowshop problems withfuzzy processing time. In that study, fuzzy arithmetic on fuzzynumbers is used to determine the minimum completion time.Proposed algorithm gets a scheduling result with a membershipfunction for the final completion time. Wang et al. [19] consideredthe fuzzy flowshop scheduling problem with distinct due window.For the fuzzy flowshop scheduling, first the authors applied tri-angular fuzzy number as processing time and established a fuzzyprogramming model. Then they transformed fuzzy objection func-tion into deterministic function. In that paper, a genetic algorithmbased on the fuzzy logic controller is developed. Xu and Gu [20]considered a fuzzy scheduling model for flowshop problems withuncertain processing time based on fuzzy programming theory,in which the uncertain processing can be dealt with by the fuzzyoperators and fuzzy model can be transformed to the optimal pro-gramming. For solving the scheduling problem, they combined animmune algorithm and a branch and bound method in a hybridalgorithm. Petrovic and Song [21] proposed a new approach basedon Johnson’s algorithm for the two-machine flowshop problem inthe presence of uncertainty. It is assumed that the processing timesof jobs on the machines are described by triangular fuzzy sets andused ˛-cuts of the fuzzy processing times. Kilic [22] proposed anant colony optimization metaheuristic in the flowshop schedulingproblem with fuzzy processing times and flexible due dates. Wangand Yang [23] proposed a particle swarm optimization algorithmfor the flowshop scheduling problems with uncertain processingtimes. In that study, the uncertain processing times are representedby triangular fuzzy numbers. They designed an integrated objec-tive function to embody the bi-criteria schedule, which involvesthe fuzzy makespan and the robustness of makespan. The sched-uling problem with fuzzy processing times and fuzzy due datesare concerned in Lai and Shu [24] study. To obtain an optimalschedule which minimizes tardiness fuzzy-valued objective func-tions, they invoked the ant colony algorithm to solve the optimalproblem. Bozejko et al. [25] proposed a genetic algorithm for theproblems with fuzzy data and block properties, which enable theinter-island genetic operator to distribute calculations and use localsearch process apart of classic operators. Javadi et al. [26] developeda fuzzy multi-objective linear programming model for solving themulti-objective no-wait flowshop scheduling problem in a fuzzy

environment. The proposed model in this paper attempts to simul-taneously minimize the weighted mean completion time and theweighted mean earliness. Zhou and Gu [27] considered a type ofno-wait flowshop scheduling problem with fuzzy due date. For

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J. Behnamian, S.M.T. Fatemi Ghomi / A

he mentioned problem, first they established a sort of schedulingodel based on evolution games on the premise of finite rational-

ty in which scheduling task model is mapped to games model, andhen proposed a hybrid solving algorithm based on genetic algo-ithm and competitive strategy. Wu [28] concerned the flowshopcheduling problems with fuzzy processing times and fuzzy dueates. In this paper, the objective function is taken as the weightedum of fuzzy earliness and fuzzy tardiness which the fuzzy ear-iness and fuzzy tardiness are proposed based on the concepts ofubtraction and maximum of any two fuzzy numbers.

In the simple flowshop problems, each machine operationenter includes just one machine. If at least one machine cen-er includes more than one machine, the scheduling problemecomes a flexible flowshop problem. Hong and Chen [29] demon-trated how discrete fuzzy concepts could easily be used in theriskandarajah and Sethi’s algorithm [30] for managing uncertainexible-flowshop scheduling. Hong et al. [31] extended applicationf Palmer algorithm for managing uncertain scheduling in fuzzyexible flowshops with more than two machine centers. In theirork, also, a heuristic algorithm is then designed since optimal

olutions seem unnecessary for uncertain environments. For theexible flowshops with more than two machine centers, Hong andang [32] generalized discrete fuzzy concepts to continuous fuzzy

omains. They used triangular membership functions to examinerocessing time uncertainties and use the triangular fuzzy LPT algo-ithm to allocate jobs, and then use the triangular fuzzy Palmerlgorithm to deal with sequencing the tasks. Recently Niu et al.33] proposed a novel cultural algorithm based on differential evo-ution to solve the hybrid flowshop scheduling problems with fuzzyrocessing time. In the proposed solving algorithm, the authorsade use of the mutation and crossover operations of differential

volution in order to enhance the performance of algorithm.In the multiprocessor scheduling environment, Yang et al. [34]

onsidered task scheduling in the heterogeneous computing envi-onment. This paper presented a Bayesian optimization algorithmn which scheduling is divided into two phases. In the first phase,ccording to the task graph of multiprocessor scheduling problems,ayesian networks have been used to assign tasks to the differentrocessors. Then in this study, in the second phase, the sequencef assigned tasks on the same processor is set by the heuristicpproach.

This review clearly reveals that there was no multi-objectivepproach implemented with fuzzy arithmetic to solve the hybridowshop scheduling.

. Mixed integer linear programming

In the following, we introduce the characters of HFS problemonsidered in this paper.

1) The production line consists of g consecutive production stages.Stage t has a set of mt identical machines, t = 1, . . ., g.

2) In each stage t, there are mt ≥ 1 parallel identical machines, withmt ≥ 2 for at least one stage, t = 1, . . ., g.

3) There are n independent jobs that are available at time 0, to bescheduled without preemption. The jobs can wait in betweenstages and the intermediate storage is unlimited. Each job hasto be processed serially through the t stages.

4) A machine can process only one job at a time. Furthermore, a job

can be processed by any of the machines and will be processedby a single machine in each stage.

5) Processing and setup times are a bell-shaped fuzzy number.6) The setups are assumed to be sequence-dependent.

Soft Computing 21 (2014) 139–148 141

Before modeling the problem as mixed integer linear program-ming (MILP), input parameters and decision variables are definedbelow.

n number of true jobs to be scheduledmt number of machines at stage tpt

iprocessing time for job i at stage t,

Cti

completion time of job i at stage t,Cmax maximum completion time of all jobs,Ei earliness of job i,Ti tardiness of job i,di due date of job i,st

ijsequence dependent setup time from job i to job j at stage t,

L denotes the large positive number

xtij

{1 if job i is scheduled immediately before job j at stage t0 otherwise

,

The earliness and tardiness of job j are defined asEj = max(0,dj − Cj) and Tj = max(0,Cj − dj), respectively, whereCj is the completion time of job j. In particular, the objectivefunction for ET measurement can be written as

f =n∑

j=1

(Ej + Tj). (1)

Note that, we introduced the dummy jobs 0 and n + 1. Theprocessing time of these jobs is set at 0. Now we can model theHFS by above introduced nomenclature.

Z = Min

⎛⎝Cmax,

n∑j=1

(Ej + Tj)

⎞⎠ (2)

s.t. Cgi

+ Ei − Ti = di, i = 1, 2, . . ., n, (3)

Ei≥di − Cgi

, i = 1, 2, . . ., n, (4)

Ti≥Cgi

− di, i = 1, 2, . . ., n, (5)

n+1∑j=1

xtij ≤ 1, i = 1, 2, . . ., n, t = 1, 2, . . ., g, (6)

n∑i=0

xtij ≤ 1, j = 1, 2, . . ., n, t = 1, 2, . . ., g, (7)

n∑j=1

xt0j ≤ mt, t = 1, 2, . . ., g, (8)

Ctj − Ct

i + L(1 − xtij)≥pt

i + stij, i, j

= 1, 2, . . ., n, t = 1, 2, . . ., g, (9)

Ct+1i

− Cti + L(1 − xt

ij)≥pti + st

ij, i, j = 1, 2, . . ., n,

t = 1, 2, . . ., g − 1, (10)

{C1

i− C1

0≥M1p1i,

Cti

− Ct−1i

≥Mtpti,

i = 1, 2, . . ., n, t = 1, 2, . . ., g, (11)

Cti ≥Ct

0, i = 1, 2, . . ., n, t = 1, 2, . . ., g, (12)

Cmax≥Cti , i = 1, 2, . . ., n, t = 1, 2, . . ., g, (13)

Ei≥0, i = 1, 2, . . ., n, (14)

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42 J. Behnamian, S.M.T. Fatemi Ghomi / A

Ti≥0, i = 1, 2, . . ., n, (15)

Cti ≥0, i = 1, 2, . . ., n, t = 1, 2, . . ., g. (16)

xtij ∈ {0, 1}, i, j = 1, 2, . . ., n, t = 1, 2, . . ., g, (17)

xtij = 0, i = j, t = 1, 2, . . ., g, (18)

The pairs of constraints (4), (15) and (5), (16) assure the properinimization of the earliness and the tardiness in objective func-

ion (2) and constraint (3) reflects the earliness or tardiness for eachart with respect to the defined due date. Relations (6) and (7) guar-ntee that each job is scheduled on one and only one machine inach stage. Constraint set (8) ensures that mt machines are sched-led in each stage. Relation (9) assures that completion time ofach job that immediately precedes another job are greater than orqual to the sum of processing and setup times of that job on allachines. Relation (10) calculated the completion time of each job

t each stage according to its processing time and setup time pervi-us stage. Constraint sets (11) and (12) ensure that the completionime of a job at stage t that does not visit stage t, is set to the job’sompletion time at stage t − 1. In Relation (11), L denotes the largeositive number and value Mt is calculated by (19) similar to thepper bound introduced in Rios-Mercado and Bard [35].

M1 =n∑

i=1

(p1

i + maxj ∈ {0,1,...,n}

s1ji

)

Mt = Mt−1 +n∑

i=1

(pt

i + maxj ∈ {0,1,...,n}

stki

) (19)

Relation (13) calculates the makespan value. Relations (14)–(18)epresent the state of the variables. Another important issue is theue dates of the jobs. To generate tight due dates of all n jobs, weropose the following steps:

Compute the total processing time of each job on all g stages.

i =g∑

t=1

pti , ∀i ∈ n (20)

Compute the average setup time for all possible subsequent jobsnd sum it for all g stages.

i =g∑

t=1

∑nk=1,k /= is

tik

n − 1, ∀i ∈ n (21)

Determine a due date for each job.

i = (1 + random × 3) × (g/mint ∈ g(mt)) × (si + pi), ∀i ∈ n (22)

here random is a random number from a uniform distributionver range (0,1).

. Bell-shaped fuzzy numbers

The bell function depends on three parameters a, b, and c asiven by

(x; a, b, c) = 1

1 +∣∣(x − c)/a

∣∣2b(23)

here the parameter b is usually positive. The parameter c locateshe center of the curve. In this fuzzy number, we can state theollowing definitions.

Soft Computing 21 (2014) 139–148

• Support of a fuzzy set is all the elements in x (universe of discourse)that have nonzero membership values form the support of thefuzzy set.

• Height of a fuzzy set is the largest membership value of a fuzzy set.• Core of a fuzzy set is defined for four different situations. If the

membership function of a fuzzy set reaches its maximum at onlyone element of the universe of discourse, the element is called thecenter of the fuzzy set. If the membership function of a fuzzy setachieves its maximum at more than one element of the universeof discourse and all these elements are bounded, the middle pointof the element is the center. If the membership function of a fuzzyset attains its maximum at more than one element of the universeof discourse and not all of the elements are bounded, the largestelement is the center if it is bounded; otherwise, the smallestelement is the center. For practical applications, bell-shaped orsymmetric-triangular fuzzy numbers are the most popular rep-resentations whenever input or output is a fuzzy quantity orlinguistically represented.

4.1. Fuzzy arithmetic on bell-shape fuzzy numbers

For fuzzy arithmetic, we proposed the following formulas forthe elementary operations between three field fuzzy numbers. Forgiven two fuzzy numbers n1 = (a1,b1,c1) and n2 = (a2,b2,c2), the sumof the two bell-shaped numbers cab be written as follows.

n1 + n2 = (a1, b1, c1) + (a2, b2, c2) = (a1 + a2, b1 + b2, c1 + c2).

(24)

If we making use of the opposite −n of the bell-shaped fuzzynumber n, which can be defined as

−n2 = (−a2, b2, c2).

Now, we can deduce the following formula from sum equationfor the subtraction:

n1 − n2 = (a1, b1, c1) + (−a2, b2, c2) = (a1 − a2, b1 + b2, c1 + c2).

(25)

Note that in the case of subtraction of three field fuzzy numbers,the fuzzy numbers n1 and n2 have to be of opposite three field typesto guarantee the closure of the operation.

Based on extension principle, the multiplication of numbers n1and n2 can be formulated as following.

n1 · n2 = (a1, b1, c1).(a2, b2, c2) = (a1a2, a1b2 + a2b1, a1c2 + a2c1).

(26)

In the multiplication of numbers n1 and n2, if we making use ofthe 1/n as

1/n2 = (1/a2, 1/c2, 1/b2).

We can deduce the following formula from multiplication equa-tion for the division.

n1/n2 = (a1, b1, c1)/(a2, b2, c2) =(

a1/a2, (a1c2 + a2b1)/a22, (a1b2

+a2c1)/a22

). (27)

J. Behnamian, S.M.T. Fatemi Ghomi / Applied Soft Computing 21 (2014) 139–148 143

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5.1. Encoding scheme

In this problem, encoding represents the scheduling in a string.The proposed representation to solve the problem is based on

Fig. 1. Fuzzy arithmetic

To generalize the mention equations for the sum operator, weonsider the sum of the first two numbers as new number N1 thenhe third number is added to this number.

N1 = n1 + n2 = (a1, b1, c1) + (a2, b2, c2)= (a1 + a2, b1 + b2, c1 + c2),

N2 = n1 + n2 + n3 = N1 + n3= (a1 + a2, b1 + b2, c1 + c2) + (a3, b3, c3)= ((a1 + a2) + a3, (b1 + b2) + b3, (c1 + c2) + c3) ,

N3 = n1 + n2 + n3 + n4 = N2 + n4

(28)

nd so on. In the same way for three number multiplications, wean write:

N1 = n1 · n2 = (a1, b1, c1) · (a2, b2, c2)= (a1a2, a1b2 + a2b1, a1c2 + a2c1),

N2 = n1 · n2 · n3= N1 + n3 = (a1a2, a1b2 + a2b1, a1c2 + a2c1) · (a3, b3, c3)

N3 = n1 · n2 · n3 · n4 = N3 · n4

(29)

For all cases and for n numbers, the generalized results can beone as calculate in above. In (Fig. 1), the fuzzy arithmetic on bell-haped fuzzy number are shown.

.2. Fuzzy ranking function

The best sequence is the job sequence with minimum objec-ive functions. In our problem, the obtained objective functionsre still a fuzzy number. Ranking is an important issue in fuzzyheory because solving a decision problem with fuzzy quantitiessually requires choosing among different fuzzy numbers. To solvehe problem, we have to compare the quality of feasible schedules.his emerges the need to identify an appropriate ranking method.lthough many methods have been suggested in the literature, it

s hard to find a method which can provide satisfactory resultsn most cases [36]. In this study, we use the ranking function toefuzzification the fuzzy numbers.

. Bi-level algorithm

The idea of this algorithm is decomposing the population intoeveral sub-populations to avoid loss the diversity of evolution-ry algorithm in the premature convergence of the search. So inhe first level, after generating the population, a parallel genetic

ll-shape fuzzy numbers.

algorithm is used to solve the sub-populations. Since in this level,sub-populations are considered separately and they do not sharewith each other, it might lose the chance to obtain a better solution.Consequently, in the second level, the Elitism strategy for globalarchive is used to collect the best non-dominated solution fromall sub-populations as a global Pareto archive. The global Paretoarchive is expected to improve the solution quality and main-tain the diversity. Here, the concept of Pareto dominance is usedto handle the achievement of solutions in sense of bi-objectiveoptimization. For this reason, we proposed a particle swarm opti-mization (PSO) in the second level.

Fig. 2 illustrates the procedure of solution improvement in theobjective space.

Fig. 3 shows the logical relation between level 1 and level 2 asthe steps of solving algorithms.

Let us now introduce the encoding scheme and discuss the prop-erties of proposed algorithms in each level.

Fig. 2. The procedure of solution improvement in the objective space.

144 J. Behnamian, S.M.T. Fatemi Ghomi / Applied Soft Computing 21 (2014) 139–148

Fig. 3. The logical relation betw

Job1 Job3Job2 Job43

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Fig. 4. Representation of solutions.

oding all jobs as genes in a 1-by-n string representation. Hence, solution for the problem of assigning jobs to machines is repre-ented by an array whose length is equal to the number of jobs.hus, the digit in the ith chromosome will be equal to the machineumber to which the ith job is assigned. An example for repre-entation is shown in Fig. 4. In this example, there are four jobsnd number of identical parallel machines at the first stage is 2. Ashown in this figure, job 2 and job 3 are assigned to machine 2; alsoob 1 and job 4 are assigned to machine 1. The sequence of the firsttage is represented by the number of genes from the left side tohe right side.

.2. Level 1: Parallel genetic algorithm

The basic idea of parallel genetic algorithm in the first level iso decompose the population into several sub-populations, ran-omly, which are assigned different weights by scalarizing multiplebjectives into the single objective [37]. Each sub-population (SP)s just like a squad team with a pre-assigned goal and hopefullyvery sub-population concentrates on specific exploring space andhus the diversity of the population can be kept among these sub-opulations. Genetic algorithms (GA) have shown great advantages

n solving the combinatorial optimization problems in the view ofts characteristic that has high efficiency and that is fit for practicalpplication [38].

The framework of sub-population bi-objective random keyenetic algorithm (SBG) is shown in Algorithm 1:

lgorithm 1. Sub-population bi-objective random key geneticlgorithm.1: Representation: Encode the problem into a search solution.2: Initialization3: Parameters setting: Set the number of sub-population in

parallel, number of individuals in each sub-population, runtime (Rtime), crossover rate, mutation rate, andreproduction rate.

4: divide population into ns sub-populations5: Assign weight to each objective6: Initial population generation: based on RKGA concept7: Evaluation8: Selection9: Crossover operation10: Mutation operation11: Evaluate solutions12: Update global archive: Transfer to archive the solution

with maximum elitism or the sets of solutions on the firstfront.

13: Generate next generation14: Stopping criteria: Terminate the algorithm if the stopping

criterion is met; else return to step 7.

After weight value assignment, the corresponding scalarizedbjective value of the two objectives in the sub-population shoulde determined. For this reason, we combined min–max bi-objective

een level 1 and level 2.

method with the weighting method to generate various Paretosolutions. Thus, the objective function value is defined by:[

w

(f1(x) − f ∗

1f ∗1

)p

+ (1 − w)

(f2(x) − f ∗

2f ∗2

)p]1/p

, (30)

where f ∗1 and f ∗

2 will be used to denote the individual minima ofeach respective objective function, and 0 ≤ w ≤ 1. W denotes theweight (or relative importance) given to Cmax and ET penalties. Theexponent p gives different ways of calculating the distance. Themost frequently used values for p are 1 for the simplest formula-tion, 2 for the Euclidean distance, and ∞ for Tchebycheff norm. Inthis paper, we assume p value is 1. Because the scales of the twoobjectives are different, the objective values with this method arenormalized. Note that, all instances are solved using 10 differentseeds for each algorithm and the minimum solution in all runs isused as f* for each objective. Then they are replaced in Eq. (30).

5.3. Level 2: Discrete particle swarm optimization

In this section, we proposed a particle swarm optimization inorder to further expand the searching process until the final near-optimal solution is found. The method is based on the concept ofPareto dominance and it explores several regions of the solutionspace. This method starts with an initial set S of non-dominatedsolutions and each generation tries to improve first level solutions.Then algorithm creates new Pareto archive in this level and updatesthis archive with the new Pareto solutions until stopping criteria ismet.

After studying the social behavior of birds, Kennedy and Eber-hart [39] originated PSO in 1995. According to what scientists havefound, in order to search for food, each member in a swarm of birdsdetermines its velocity based on their personal experiences as wellas information gained through interaction with other members ofthe swarm. This idea was the main principle for PSO. Each bird,called particle, flies through the solution space of the optimizationproblem searching for the optimum solution and thus its positionrepresents a potential solution for the problem. The following is themathematical concepts of PSO.

Suppose a D-dimensional searching space and a swarm of Nparticles searching for the globally optimal solution within thesearching space. Three D-dimensional vectors are assigned to eachparticle, position, denoted by Xi = (xi1, xi2, . . ., xid), velocity, denotedby Vi = (vi1, vi2, . . ., vid) and best personal position, denoted byPi = (pi1, pi2, . . ., pid).

In a continuous searching space, each dimension of the posi-tion vector corresponds to the value of a decision variable for theproblem. In other words, the position of each particle is a potentialsolution for the problem at hand, and the fitness of this particle(solution) can be calculated by putting these values into a pre-determined objective function. When the fitness is more desirable

in terms of the objective function, the particle’s position is better.Velocity vector represents the distance a particle will traverse ineach dimension in each iteration of the algorithm. Best personalposition vector is the best visited position for a particle. Particles

pplied

aso

v

wcn

a

x

apbcrtii

A

ii

v

x

5

f

A

5

s

of these factors can have at least two levels. These levels are listedin Table 1.

In general, 20 combinations of above levels are tested.

Table 1Factor levels.

Factor Levels

Number of jobs 6 30 100Number of machines Constant: 1

Variable: Uniform(1, 4)2Uniform(1, 10)

10

J. Behnamian, S.M.T. Fatemi Ghomi / A

lso need to be aware of the best global position visited by the wholewarm which is denoted by Pg = (pg1, pg2, . . ., pgd). The new velocityf each particle is calculated as follows:

id(k + 1) = ω vid(k) + �1r1 [pid(k) − xid(k)] + �2r2[pgd(k) − xid(k)

](31)

here �1 and �2 are constants called acceleration coefficients, ω isalled the inertia factor, and r1 and r2 are two independent randomumbers uniformly distributed in the range [0,1].

Thus, the position of each particle is updated in each generationccording to Eq. (32).

id(k + 1) = xid(k) + vid(k + 1) (32)

In the standard PSO, Eq. (31) is used to calculate the new velocityccording to its previous velocity and to the distance of its currentosition from both its own best historical position and its neigh-ors’ best position. Generally, the value of each component in vian be clamped to the range [−vmax,vmax] to control the excessiveoaming of particles outside the search space. Then, the particle fliesoward a new position according to Eq. (32). This process, as shownn Algorithm 2, is repeated until a user-defined stopping criterions reached.

lgorithm 2. Basic PSO structure.1: Initialize a population of particles with random positions

and velocities, where each particle contains d variables.2: Evaluate the objective values of all particles. Let pbest of

each particle and its objective value be equal to its currentposition and objective value, and let gbest, and its objectivevalue be equal to the position and objective value of thebest initial particle.

3: Update the velocity and position of each particle accordingto Eqs. (28) and (29).

4: Evaluate the objective values of all particles.5: For each particle, compare its current objective value with

the objective value of its pbest. If the current value is better,then update pbest and its objective value with the currentposition and objective value.

6: Determine the best particle of the current swarm with thebest objective value. If the objective value is better thanthe objective value of gbest, update gbest and its objectivevalue with the position and objective value of the currentbest particle.

7: If a stopping criterion is met, output gbest and its objectivevalue; otherwise, go to Step 3.

According to the Kashan and Karimi [40], the process of generat-ng a new position for a selected individual in the swarm is depictedn the following equations.

i(k + 1) = ω ⊗ vi(k) ⊕ r1 ⊗ [pi(k) xi(k)] ⊕ r2 ⊗[pg(k) xi(k)

](33)

i(k + 1) = xi(k) ⊕ vi(k + 1) (34)

.3.1. The subtract operatorThe subtraction of the current position of the ith particle, xi(k);

rom a desired position pi(k) (or pg(k)) can be calculated as follows.

lgorithm 3. The subtract operator.1: Create an array of n elements.2: If content of the corresponding element in xi(k) is different

from the desired one, that element gets its value from pi(k)(or pg(k)).

3: If xi(k) is exactly equal to pi(k) or pg(k), we omit the termsthat contain the subtract operator.

4: For those elements that have the same content m xi(k) andpi(k) (or pg(k)), their corresponding jobs are listed based onLPT order and are assigned to machines successively,whenever a machine becomes free.

.3.2. The multiply operatorBy multiplication process we can add exploration ability to PSO

ystem via generating the different binary vectors for two different

Soft Computing 21 (2014) 139–148 145

vectors. In this paper, the ⊗ operator is equivalent to Hadamardproduct and acts as follows.

Algorithm 4. The multiply operator.1: Create an array of n elements.2: The Hadamard product of two 1-by-n arrays A and B is

denoted by AB and is an 1-by-n array given by (A.B)t = atbt.3: For those elements are zero, their corresponding jobs are

listed based on LPT order and are assigned to machinessuccessively, whenever a machine becomes free.

5.3.3. The add operatorAs shown in Algorithm 5, to add xi(k) with pi(k) (or pg(k)), we

use two-point crossover operator that typically is used in geneticalgorithms. Crossover is viewed as the operator that has the mis-sion of interchanging structural information developed during thesearch.

Algorithm 5. The add operator.1: Create an array of n elements.2: Select two positions (cut points) from particle chain

randomly, and exchange dimensions between these twopositions

3: Select one of two new offsprings randomly obtained fromcrossover operator as the result of add operator.

6. Computational results

In this section, we are going to compare the output of each levelof our proposed algorithm with cultural differential evolution (CDE)of Niu et al. [33]. We also interested to justify the proposing ofstep 2. Therefore the results of each level are also embedded in thecomputational results. All algorithms are coded in MATLAB 7 andrun with an Intel Pentium IV dual core 2.5 GHz PC at 896 MB RAMunder a Microsoft Windows 7.

6.1. Data generation and settings

An experiment was conducted to test the performance of thehybrid algorithm. Data required for a problem consists of the num-ber of jobs, number of stages, number of machines in each stage,range of processing times, and the range of sequence-dependentsetup times [6]. The ready times for stage 1 are set to 0 for all jobs.The ready times at stage t+1 are the completion times at stage t, sothere is no need this data to be generated. Processing times are dis-tributed uniformly over two ranges with a mean of 60: (50,70) and(20,100). The cores of setup times are uniformly distributed from12 to 24 which are 20% to 40% of the mean of the processing time[40]. The setup time matrices are asymmetric and satisfy the trian-gle inequality. The setup time characteristics follow Rios-Mercadoand Bard [35].

The problem data can be characterized by five factors, and each

Number of stages 2 4 8Core of processingtimes

Uniform(50, 70) Uniform(20, 100)

Skipping probability 0.00 0.05 0.40

146 J. Behnamian, S.M.T. Fatemi Ghomi / Applied Soft Computing 21 (2014) 139–148

Table 2Evaluation of non-dominated solution for algorithms grouped by problem size and index type.

Problem size Algorithm & index

MID SNS RAS

CDE SBG PSO CDE SBG PSO CDE SBG PSO

6 × 2 × 6 1293.88 1210.22 944.11 185.67 159.39 148.25 0.28 0.52 0.416 × 4 × 4 565.22 1019.14 994.87 88.66 51.59 58.99 0.38 0.12 0.336 × 8 × 2 2039.73 1942.75 1725.67 13.13 11.62 10.38 0.95 1.09 0.436 × 8 × 4 1098.07 1057.97 923.45 11.18 33.59 15.12 0.20 0.16 0.3830 × 2 × 6 5502.70 6013.47 5952.29 570.14 460.49 496.05 0.68 1.51 0.7930 × 2 × 10 3912.80 2769.79 2242.98 314.38 79.04 268.16 0.53 0.40 0.1930 × 4 × 1 5425.95 5826.06 5363.22 358.96 236.86 285.14 1.58 1.64 1.2630 × 4 × 4 6533.87 7358.82 6010.16 300.90 212.78 290.85 1.42 1.90 1.5230 × 4 × 10 3364.77 3002.89 2928.90 148.26 72.15 126.55 1.78 1.18 1.2430 × 8 × 2 1694.37 2490.52 2139.27 122.04 143.26 183.20 0.72 1.08 0.8130 × 8 × 4 9453.39 7309.97 6622.68 359.58 431.50 429.95 1.49 1.19 1.5230 × 8 × 10 1851.67 1825.05 1788.10 251.15 49.43 144.42 1.23 0.38 1.25100 × 2 × 2 5212.46 4932.68 4917.22 123.84 606.79 118.12 0.03 0.22 0.17100 × 2 × 10 4182.20 3968.42 3665.58 118.53 1284.45 97.84 1.44 1.83 1.21100 × 4 × 1 13,729.36 14,043.99 12,764.28 572.75 629.44 553.08 1.13 0.85 0.97100 × 4 × 4 7067.40 7300.66 6300.14 46.98 53.66 62.10 0.13 0.11 0.09100 × 4 × 10 3696.97 3679.27 2515.42 205.57 612.21 318.78 1.42 1.27 1.04

177653154

6

ttampbpaotsSaifstiepcu

100 × 8 × 2 8089.13 8396.82 7664.01

100 × 8 × 4 17,082.11 20,458.71 12,825.42

100 × 8 × 10 2437.35 2485.36 2730.46

.2. Algorithm parameters

The efficiency of the algorithms depends on the choice ofhe best parameters in order to prevent premature convergence,o ensure diversity in the search space, to intensify the searchround interesting regions, etc. There are several parameters thatay influence on the performance of the algorithm. For exam-

le, the larger population size may find better solution qualityut cost larger computational expense. When the number of sub-opulations is larger, it may have better diversity. However, it maylso be a trade-off that to reduce the number of generations. Inrder to reduce the number of combinations, some parameters ofhe algorithms such as numbers of sub-population and populationize are fixed a priori based on the claims suggested in the literature.o, the numbers of sub-population and population size are set to 50nd 2500, respectively (50 individuals in each sub-population). Thenitial population is generated randomly. An elitist strategy is usedor reproduction. Each chromosome is decoded and the resultingchedule is evaluated for the objective value. Crossover and muta-ion are the most important parts of a GA. The crossover is usedn a GA to exchange information among chromosomes when gen-

rating offspring. Mutation serves to prevent all solutions in theopulation from falling into a local optimum. There are numerousrossover and mutation methods. Chromosomes with lower val-es are more desirable, so x% of the chromosomes with the lowest

Fig. 5. Plots of evaluation metric value for the interaction

.95 146.66 162.93 0.59 0.77 0.64

.24 911.12 1162.92 0.46 0.47 0.99

.65 625.41 141.28 0.44 0.47 0.46

objective values are automatically copied to the next generation.Parametrized uniform crossover is used to select y% of the chro-mosomes in the next generation. For each chromosome in the nextgeneration, the following is performed. Two chromosomes in thecurrent generation are selected at random. For each job, a randomnumber is generated. If the value is less than 0.7 (following Kurz andAskin [41]), the value from the “first” chromosome is copied to thenew chromosome, otherwise the value from the “second” chromo-some is selected. The remaining (1 − (x + y))% of the next generationis filled through “immigration”, in which new chromosomes arerandomly generated. The above procedure is repeated until we arefairly sure that the population has settled into a good location inthe search space. In SBG, procedure continues until 50 generationshave been examined without finding an improved schedule.

The x and y values were selected using empirical testingapproach to find the best tuning parameters. In other words, wehave applied parameters tuning only for the crossover rate (Cr) andmutation rate (Mr) in SBG considering the following ranges:

• Cr: three levels (0.70, 0.75 and 0.80),• Mr: two levels (0.01, 0.02 and 0.05),

Nine different crosses are obtained by these levels. We generate6 instances, 2 small, 2 medium and 2 large, for each combination ofdata factor levels. The preferable levels for crossover rate for small,

between the type of algorithm and number of jobs.

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J. Behnamian, S.M.T. Fatemi Ghomi / A

edium and large instances are 0.80, 0.80 and 0.75, respectively.egarding to the mutation rate these levels are 0.02, 0.05 and 0.05.

.3. Stopping rules

The stopping criterion used when testing all instances with thelgorithms is set to a CPU time limit fixed to

(n2 ×

∑gt=1mt/g

)×

0 ms. This stopping criterion is not only responsive to the numberf stages, but also is sensitive to the rise in the number of jobs andumber of machines at each stage.

.4. Evaluation metric

For non-dominated solutions evaluation, at first, we needeters that provide quantitative measures to compare different

ets of Pareto solution. For this reason, there are several approachesn the multi-objective literature. For this reason, in this paper, wesed the following indices that introduced in [42].

1) MID (mean ideal distance): The closeness between the Paretosolution and ideal point (0, 0) which is defined as follows.

MID =∑n

i=1ci

n(35)

where n is the number of non-dominated solutions and

ci =√

f 21i

+ f 22i

.

2) SNS: The spread of non-dominated solution, as a diversity mea-sure, can be expressed by the following relation:

SNS =√∑n

i=1(MID − ci)2

n − 1(36)

3) RAS: The rate of achievement to two objectives simultaneouslywhich is represented in Relation (38).

RAS =∑n

i=1

[((f1i − Fi)/Fi

)+

((f2i − Fi)/Fi

)]n

(37)

where Fi = min{f1i, f2i}.

Note that the lower value of MID and RAS, the better of solutionuality we have. The higher value of SNS, the better of solutionuality we have (more diversity in obtained solution).

.5. Numerical results

In order to evaluate the efficacy and performance of the algo-ithms proposed in this paper, 20 problems are used. Table 2epresents the values of the four metrics.

As it can be seen in Fig. 5, the PSO is generally superior to theBG and CDE when index MID is concerned. With index SNS, SBGn average provides the better result than other algorithms. Also,

hen index RAS is concerned, the results of algorithms CDE andBG have nearly similar performance and PSO is better than others.

. Conclusions and future works

This paper considered the hybrid flowshops with fuzzy bell-haped processing times, sequence-dependent setup times and dueates. To minimize makespan and sum of the earliness and tardi-ess of jobs, we proposed a bi-level scheduling algorithm. In thislgorithm, in level 1, the population is decomposed into severalub-populations and genetic algorithm is designed using a scalar bi-

bjective concept. Multiple objectives are combined with min–maxethod then each sub-population evolves separately until to obtain

good approximation of the Pareto-front. In the second level, non-ominant solutions are unified as one big population. In this stage,

[

Soft Computing 21 (2014) 139–148 147

for improving the Pareto-front which obtains from first level, basedon the search in Pareto space concept, we developed a particleswarm optimization in the global population. The non-dominatedsets obtained from each level are compared with the adaption ofdifferential evolution proposed in a lately published. It is foundthat most of the solutions in the net non-dominated front areyielded by our proposed algorithm. As future research direction, thescheduling with other system characteristics, which has not beenincluded in this paper, such as reentrant processing and limitedintermediate buffers can be a practical extension.

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