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A novel imperialist competitivealgorithm for bi-criteria scheduling ofthe assembly flowshop problemE. Shokrollahpour a , M. Zandieh a & Behrouz Dorri aa Department of Industrial Management, Management andAccounting Faculty, Shahid Beheshti University, G.C., Tehran, Iran

Available online: 04 May 2010

To cite this article: E. Shokrollahpour, M. Zandieh & Behrouz Dorri (2011): A novel imperialistcompetitive algorithm for bi-criteria scheduling of the assembly flowshop problem, InternationalJournal of Production Research, 49:11, 3087-3103

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International Journal of Production ResearchVol. 49, No. 11, 1 June 2011, 3087–3103

A novel imperialist competitive algorithm for bi-criteria scheduling

of the assembly flowshop problem

E. Shokrollahpour, M. Zandieh* and Behrouz Dorri

Department of Industrial Management, Management and Accounting Faculty,Shahid Beheshti University, G.C., Tehran, Iran

(Received 10 August 2009; final version received 24 November 2009)

This paper deals with the two-stage assembly flowshop scheduling problem withminimisation of weighted sum of makespan and mean completion time as theobjective. The problem is NP-hard, hence we proposed a meta-heuristic namedimperialist competitive algorithm (ICA) to solve it. Since appropriate design ofthe parameters has a significant impact on the algorithm efficiency, wecalibrate the parameters of this algorithm using the Taguchi method. Incomparison with the best algorithm proposed previously, the ICA indicates animprovement. The results have been confirmed statistically.

Keywords: assembly flowshop; bi-criteria scheduling; makespan; mean comple-tion time; imperialist competitive algorithm

1. Introduction

A two-stage assembly flowshop scheduling problem is described as follows: in an assemblyflowshop there are two stages where the first stage consists of m machines and there is onlyone machine at the second stage. There are n jobs to be scheduled and each job needs mþ 1operation, the m of which are done at the first stage by m machine in parallel and then, afinal operation is done in the second stage.

Since the two-stage assembly flowshop reflects the real operation of several industries,it has received some attention recently. Lee et al. (1993) defined an application in a fireengine assembly plant. The body and chassis of fire engines are manufactured in parallel intwo different departments. After delivery of the engine and completion of body andchassis, they are assembled at the second stage. Potts et al. (1995) described anotherapplication in personal computer manufacturing where the main parts (such as the centralprocessing units, hard disks, monitors, keyboards, etc.) are produced at the first stage, andall of the components are assembled at the second stage.

Another use is in the area of queries scheduling on distributed database systems, asmentioned by Allahverdi and Al-Anzi (2006).

Different criteria can be considered for a two-stage assembly flowshop schedulingproblem. For instance, the performance measures of flowtime and completion time are thesame when jobs are ready at time zero. Tozkapan et al. (2003) reported the two-stageassembly scheduling problem with the total weighted flowtime objectives. They usedpermutation schedules for the problem with this objective.

*Corresponding author. Email: m_zandieh@sbu.ac.ir

ISSN 0020–7543 print/ISSN 1366–588X online

� 2011 Taylor & Francis

DOI: 10.1080/00207540903536155

http://www.informaworld.com

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The lower bound and dominance relation were developed and used in branch and

bound to solve this problem. Equality of flowtime and completion time should be noted

when jobs are ready at time zero. Al-Anzi and Allahverdi (2006) defined the same problem

with total completion time criterion. They proposed three algorithms, simulated annealing

(SA), tabu search (TS), and a hybrid tabu search and found optimal results for two

special cases. The comparison between these three meta-heuristics showed that hybrid tabu

is the best.Allahverdi and Al-Anzi (2007) defined this problem with bi-criteria of makespan and

mean completion time, they compared three meta-heuristics, SA, ant colony optimisation

(ACO) and self adaptive differential evolutionary algorithm (SDE), and SA outperforms

ACO and SDE. The assumption of no setup times or join with processing times can be

seen in Tozkapan et al. (2003) and Al-Anzi and Allahverdi (2007). Due to many

applications which require separate and non-zero setup times, this factor must be

considered for qualified solutions (Babu et al. 2003, Allahverdi et al. 2008). Allahverdi and

Soroush (2008) highlighted the significance of considering separate setup times. Allahverdi

and Al-Anzi (2009) published a paper to describe the two-stage assembly scheduling

problem for minimising the total completion time with setup times and in doing so, they

proposed three meta-heuristics. The algorithms are hybrid tabu search (Ntabu), SDE and

a new version of SDE (NSDE). Data for these algorithms were generated randomly. The

computational results showed that Ntabu outperforms others when there is no setup time

consideration, and SDE is good for such problems. Using dominance relation, a new

version of SDE, named NSDE, showed a better performance than the other two. Al-Anzi

and Allahverdi (2009) also reported a two-stage assembly flowshop with bi-criteria of

maximum lateness and makespan, and they proposed three meta-heuristics to solve the

problem. The proposed meta-heuristics are TS, particle swarm optimisation (PSO) and

SDE. The computational results show that both PSO and SDE perform much better than

TS and also statistical analysis demonstrates that PSO is superior to SDE.In this paper, we provide the two-stage assembly flowshop scheduling problem with a

weighted sum of makespan and mean completion time. We demonstrate the problem in

Section 2 and a new algorithm is described in Section 3. Section 4 presents an experimental

design which compared the results achieved by proposed ICA with those achieved by past

SA algorithm, the best algorithm proposed before. Finally, conclusion and future work are

given in Section 5.

2. Problem description

We assume that n jobs are simultaneously available at time zero and job processing is

non-pre-emptive. There are mþ 1 operations related to each job, m of which are completed

at the first stage on m parallel machines and the last operation performed at assembly

stage. The following assumptions are considered (Allahverdi and Al-Anzi 2007):

t½i, j � Operation time of the job in position i on machine j ði ¼ 1, . . . , n;

j ¼ 1, . . . ,mÞ.p½i � Operation time of the job in position i on the assembly machine

(i¼ 1, . . . , n).C½i � Completion time of the job in position iði ¼ 1, . . . , nÞ.

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Note that job k is completed after completing all of its operations t½k, j �ð j ¼ 1, . . . ,mÞ andp½k�, where the operation p½k� should start only after all operations t½k, j �ð j ¼ 1, . . . ,mÞ havebeen finalised. Potts et al. (1995) and Tozkapan et al. (2003) demonstrated the permutationschedules were used for problems with makespan and total flowtime criterion. Hence inthis paper we mention permutation scheduling, which means that the sequence of jobs onall machines, including the assembly machine, is the same. Al-Anzi and Allahverdi (2006)showed that the completion time of the job in position j is as follows:

C½ j � ¼ max maxk¼1,...,m

Xji¼1

t½i,k�

( ),C½ j�1�

( )þ p½ j �

C½0� ¼ 0

The mean completion time is

�C ¼1

n

Xni¼1

C½i �

and the makespan is Cmax ¼ C½n�.If the weight considered for the makespan is denoted by �, then the objective function

(OF ) is described as follows:

OF ¼ �Cmax þ ð1� �Þ �C

where 05�5 1. Note that, � ¼ 0, the objective function reduces to mean completiontime, and with � ¼ 1, the Cmax is the only objective function. The objective is to find asequence which minimises the value of OF.

No polynomial solution can be considered for this problem because it is known thatthe problem for � ¼ 0 (i.e. when the objective is to minimise the mean completion time) isNP-hard, even for the two-machine flowshop scheduling problem, i.e. when m¼ 1 (seeGonzalez and Sahni 1978). Thus, we present a meta-heuristic for solving this problem inthe next section.

3. The proposed imperialist competitive algorithm

3.1 Imperialist competitive algorithm in general

Imperialist competitive algorithm is proposed by Atashpaz and Lucas (2008). Theyshowed the algorithms capability in dealing with different types of optimisation problems(Atashpaz et al. 2008). We use this algorithm in a two-stage assembly flowshop schedulingproblem.

Similar to other evolutionary algorithms, this algorithm starts with an initialpopulation of solution which is named country. Some of the best countries in thepopulation are chosen to be the ‘imperialists’ and the rest are the ‘colonies’ of theseimperialists. All the colonies of initial population are distributed among the imperialistsbased on their power. A set of one imperialist and its colonies is called an ‘empire’.

The power of an empire which is equivalent to the fitness value in a genetic algorithm(GA) is inversely proportional to its cost. After distribution of all colonies amongimperialists, these colonies start moving towards their relevant imperialist country. Thetotal power of an empire relates to both the power of the imperialist country and the power

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of its colonies. This fact will be modelled by defining the total power of an empire byadding the percentage of the mean power of colonies to their imperialists. Then theimperialistic competition begins among all the empires. Any empire which is not strongenough to compete and cannot increase its power (or at least prevent decreasing it) will beeliminated. The imperialistic competition will lead slightly to an increase in the power ofpowerful empires and a decrease in the power of weaker ones. Weak empires will lose theirpower and finally they will collapse. The movement of colonies towards their relevantimperialists through the competition among empires and also the collapse mechanism willhopefully cause all the countries to converge to a state in which there is just one empire inthe world and all the other countries are colonies of that empire. In this ideal new world,colonies have the same position and power as the imperialist.

3.2 An ICA approach to two stage assembly flowshop

The implementation of this algorithm in assembly flowshop is as follows:

Begin ICA

(1) Initialise the empires.(2) Move the colonies toward their relevant imperialist (assimilating).(3) If there is a colony in an empire which has lower cost than that of imperialist,

exchange the positions of that colony and the imperialist.(4) Compute the total cost of all empires (related to the power of both imperialist and

its colonies).(5) Pick the weakest colony from the weakest empire and give it to the empire that has

the most likelihood to possess it (imperialistic competition).(6) Change some weakest colonies with new ones randomly (revolution).(7) Eliminate the powerless empires.(8) If stopping criteria met, stop, if not go to step 2.

End ICA

3.2.1 Generating initial empires

The main purpose of optimisation is to find an optimal solution; each solution in thisalgorithm is shown as a country. The term country in ICA stands for chromosome in GA.In an n-dimensional optimisation problem, a country is a 1� n array.

The array of country represents a sequence of jobs. Initial population is generated asfollows.

One initial sequence is obtained by ordering all of the jobs into increasing order of p½i �.A good solution will be obtained by ordering jobs based on the shortest processing time(SPT), when processing times on the assembly machine are larger than those of thefirst-stage machines. Generally for the mean completion time criterion the SPT ruleperforms well. By considering the first stage and ordering the jobs in increasing order ofmaxk¼1,...,m t½i,k�

� �the second initial solution is obtained.

Ordering the jobs in increasing order of

maxk¼1,...,m

t½i,k�� �

þ p½i �

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will yield a third sequence in which both stages are considered and the others were

generated randomly. OF function is considered as a cost function.Optimisation of an algorithm starts with generating initial population (countries) of

size pop. Nimp of the most powerful countries are selected to be imperialists. The remaining

Ncol of the population will be the colonies each belongs to imperialists. Thus we have two

types of countries; imperialist and colony.To form the initial empires, we distribute the colonies among imperialists based on

their power. To distribute the colonies among imperialists, we define the normalised cost

of an imperialist as follows:

NOFn ¼ maxifOFig �OFn

where OFn is the cost of nth imperialist and NOFn is its normalised cost. The normalised

power of each imperialist is shown below:

wn ¼NOFnPNimp

i¼1 NOFi

����������

On one side, the normalised power of an imperialist shows the number of colonies that

should be possessed by that imperialist. Thus, the initial number of colonies of an empire

will be

NColn ¼ roundfwn �Ncolg

where NColn is the initial number of colonies of nth empire. For distributing of colonies

among the imperialist we randomly choose NColn of the colonies and give them to it. The

imperialist and its colonies will form nth empire. Figure 1 shows the initial population of

each empire. As shown in Figure 1 bigger empires have a greater number of colonies while

.

.

.

.

.

Imperialist 1

Imperialist 2

Imperialist 3

Imperialist N

Colony 2Colony 3

Colony N

Colony 1

Figure 1. Generating the initial empires: the more colonies an imperialist possess, the bigger itsrelevant ? mark.

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weaker ones have less. In Figure 1 imperialist 1 has formed the most powerful empire and

has the greatest number of colonies.

3.2.2 Moving the colonies of an empire toward the imperialist (assimilating)

Imperialists countries started to improve their colonies. This fact has been modelled by

moving all the colonies toward the imperialist. Through this movement some parts of a

colony’s structure will be similar to the empire’s structure. The assimilating operator is

shown with an example in Figures 2 and 3.

. Select one cell randomly in imperialist array (cell 2, number 8).

. Find that number in the colony’s array and shift that until it comes to the same

position as in imperialist array (cell 6, number 8).. Put the right hand side number in the imperialist’s array (in this example it is 4),

at the right hand side of the shifted cell in the colony’s array (swap number 4 and 6).

3.2.3 Exchanging positions of the imperialist and a colony

Owing to movement towards the imperialist, a colony may reach a position with lower cost

than imperialist. In such a condition, the position of imperialist and colony are changed.

After that, the algorithm will continue by the imperialist in a new position and then

colonies start moving toward this position. Figure 4 depicts the position exchange between

a colony and the imperialist. In Figures 4 and 5 the best colony of the empire is shown in a

darker colour. This colony has a lower cost than that of the imperialist. Figure 5 shows the

whole empire after exchanging the position of the imperialist and that colony.

Figure 3. Swapping part of the assimilating operator.

Figure 2. Shifting part of the assimilating operator.

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3.2.4 Total power of an empire

Total power of an empire is mainly affected by the power of the imperialist country,though the power of the colonies of an empire has an effect, albeit negligible, on the totalpower of that empire. Therefore, the equation of total cost is:

TOFn ¼ OF ðimperialistnÞ þ �meanfOF ðcolonies of empirenÞg

where TOFn is the total cost of the nth empire and � is a positive number which isconsidered to be less than 1. A little value for � causes the total power of the empire to bedetermined by just the imperialist and increasing it will increase the role of the colonies indetermining the total power of an empire.

3.2.5 Imperialistic competition

As mentioned before, all empires try to possess the other empires’ colonies and controlthem. Through this imperialistic competition the power of weaker empires will decreaseand as a result the power of more powerful ones will increase. We model this competitionby just picking one of the weakest colonies of the weakest empires and making a

BestColony

Imperialist

Figure 4. Exchanging the positions of a colony and the imperialist.

Colony

Imperialist

Figure 5. The entire empire after position exchange.

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competition among all empires to possess this colony. Figure 6 illustrates the modelled

imperialistic competition. Based on their total power, in this competition, each of the

empires will have a likelihood of taking possession of the mentioned colonies. In other

words, these colonies will not be possessed by the most powerful empires; however, these

empires will be more likely to possess them.To start the competition, first, the possession probability of each empire should be

found based on its total power.The normalised total cost is simply obtained by

NTOFn ¼ maxifTOF gi � TOFn

where TOFn andNTOFn are respectively total cost and normalised total cost of nth empire.

Having the normalised total cost, the possession probability of each empire is given by

Wwn¼

NTOFnPNimp

i¼1 NTOFi

����������

To divide the mentioned colonies among empires based on the possession probability of

them, we form the vector W as

W ¼ ½Ww1,Ww2

,Ww3, . . . ,WwNimp

�

Then we create a vector R with the same size as W whose elements are uniformly

distributed random numbers.

R ¼ ½r1, r2, r3, . . . , rNimp�

r1, r2, r3, . . . , rNimp� Uð0, 1Þ

Weakest Colony in Weakest Empire

Empire 1

Empire 2

Empire 3

Empire N

The Weakest Empire

Imperialist 1

Imperialist N

Imperialist 3

Imperialist 2 . . . . . . .

. . . .

Figure 6. Imperialistic competition. The more powerful an empire is, the more likely it will possessthe weakest colony of the weakest empire.

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Vector D is obtained by subtracting R from W.

D ¼W� R ¼ ½D1,D2,D3, . . . ,DNimp� ¼ ½Ww1

� r1,Ww2� r2,Ww3

� r3, . . . ,WwNimp� rNimp

�

Referring to vector D, we will hand the mentioned colonies into an empire whose relevantindex in D is maximum.

3.2.6 Revolution

In each iteration we select some of the weakest colonies and replace them with new ones,randomly. The replacement rate is named as the revolution rate.

3.2.7 Eliminating the powerless empires

Powerless empires will collapse in the imperialistic competition and their colonies will bedistributed among other empires. In modelling collapse mechanism different factors can bedefined for considering an empire powerless. In this paper, we assume an empire collapseswhen it loses all of its colonies.

3.2.8 Stopping criteria

The algorithm continues until no iteration is remaining or just one empire exists in theworld.

4. Experimental design

4.1 Data generation and settings

An experiment was conducted to test the performance of the imperialist competitivealgorithm. Following Allahverdi and Al-Anzi (2007), the data required for a problemconsists of the number of jobs, number of machines in the first stage and �, as presented inTable 1.

The processing times were randomly generated from a uniform distribution [1, 100] onall m machines at the first stage, as well as the assembly machine at the second stage.

4.2 ICA parameters tuning

Appropriate design of the parameters and operators has a significant impact on theefficiency of the imperialistic competitive algorithm. In this section, we study thebehaviour of the different parameters of the proposed ICA.

Table 1. Factor levels.

Factor Levels

Number of jobs Small: 20–40. Large: 60–80Number of machines constant 4–6–8Alpha (�) 0.2–0.5–0.8

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There are several ways to statistically calibrate the parameters of the proposedalgorithm. We used the Taguchi method. In this method orthogonal arrays are used tostudy a large number of decision variables with a small number of experiments. Taguchicreated a transformation of the repetition data to another value which is the measure ofvariation. The transformation is the signal-to-noise (S/N) ratio, which explains why thistype of parameter design is called a robust design (Phadke 1989, Al-Aomar 2006). Here,the term ‘signal’ denotes the desirable value (response variable) and ‘noise’ denotes theundesirable value (standard deviation). So the S/N ratio indicates the amount ofvariation present in the response variable. The goal is to maximise the signal-to-noiseratio.

Taguchi classifies objective functions into three categories:

(1) The smaller-the-better type.(2) The larger the-better type.(3) The nominal-is-best type.

Since almost all objective functions in scheduling are classified in the smaller-the-bettertype, the corresponding S/N ratio (Phadke 1989) is

S=Nratio ¼ �10Log1

k

Xki¼1

ðOFiÞ2

!

As mentioned earlier, in this study, the control factors are: zeta ð�Þ, the revolution rate(RR), the number of countries (pop), number of iterations (IT ) and the number ofimperialists (Nimp). Different levels of these factors are shown in Tables 2 and 3 and theresults for each level are shown in Figures 7 and 9.

The associated degree of freedom for these four factors is 7. Therefore, the selectedorthogonal array should have a minimum of eight rows and four columns to accommodatethe four factors. From the standard table of orthogonal arrays, L9ð3

4Þ is selected as thefittest orthogonal array design that fulfils all our minimum requirements.

The orthogonal array L9ð34Þ is presented in Table 4.

Table 2. Factors and factor levels for small problems.

Factors Level

Zeta (�) 0.04–0.05–0.06Revolution rate 0.2–0.3–0.4Number of imperialists 5–8–10Number of (countries, iterations) (100, 200)–(200, 500)–(300, 700)

Table 3. Factors and factor levels for large problems.

Factors Level

Zeta (�) 0.04–0.05–0.06Revolution rate 0.1–0.2–0.3Number of imperialists 5–8–10Number of (countries, iterations) (200, 400)–(300, 700)–(400, 700)

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A set of 12 instances was generated as follows. The set of instances comprises eight

combinations of n and m, i.e. n¼ {20, 40, 60, 80} and m¼ {4, 6, 8}. The processing times

were generated from uniform distributions between the interval (1, 100).There are three replicates for each combination, summing up to 12 instances.In order to conduct the experiments, we implemented ICA in MATLAB 7.6 run on a

personal computer with a 2.0GHz Intel Core 2 Duo processor and 2GB RAM memory.

RR Nimp (pop,IT)Zeta3.53.73.94.14.34.54.74.95.15.3

RP

D

Figure 8. The mean RPD plot for each level of the factors for small and medium problems.

–62.4

–62.35

–62.3

–62.25

–62.2RR Nimp (pop,IT)Zeta

S/N

Rat

io

Figure 7. The mean S/N ratio plot for each level of the factors for small and medium problems.

Table 4. The orthogonal array L9(34).

Trial Zeta RR Nimp (pop, IT)

1 1 1 1 12 1 2 3 23 1 3 2 34 2 1 3 35 2 2 2 16 2 3 1 27 3 1 2 28 3 2 1 39 3 3 3 1

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We use the relative percentage deviation (RPD) for the makespan as a common

performance measure to compare the methods. The best solutions obtained for each

instance (denoted Minsol) are calculated. RPD is obtained from the formula

RPD ¼A lgsol�Minsol

Minsol� 100

where A lgsol is the OF obtained for a given algorithm and instance. Clearly, lower values

of RPD are preferable.In order to do so, we analyse the results of the experiment using a different measure,

i.e. the response variable (RPD). The results for each level are shown in Figures 8

and 10. This analysis strongly supports our decision with respect to the optimal level for

factors �, revolution rate, number of imperialists and (number of countries, number of

iterations). It finally turns out that levels 2, 2, 2 and 3 (for small and medium size

problems) and Levels 2, 1, 3 and 3 (for large problems) are the preferable levels for

factors, respectively.To explore the relative significance of individual factors in terms of their main effects

on the objective function, analysis of variance (ANOVA) was conducted. The results of the

analysis are presented in Tables 5 and 6. The number of (countries, iterations) factor has

the greatest effect on the quality of the algorithm with a relative importance of 89.05% for

small problems and 93.03% for large problems.

–67.33

–67.32

–67.31

–67.3

–67.29

–67.28

–67.27

–67.26

–67.25Zeta RR Nimp (pop,IT)

S/N

Rat

io

Figure 9. The mean S/N ratio plot for each level of the factors for large problems.

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

zeta

RP

D

RR Nimp (pop,IT)

Figure 10. The mean RPD plot for each level of the factors for large problems.

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4.3. Experimental results

Using average percentage error (Error) and standard deviation (Std) we compared the

performance of the ICA and SA, the best algorithm proposed by Allahverdi and Al-Anzi

(2007).The percentage error is defined as:

100�ðOF of the heuristic�OF of the best heuristicÞ

ðOF of the best heuristicÞ

There are 36 combinations for the different values of n, m and �. Twenty-five replicates

were generated for each combination and therefore, a total of 900 instances were generated

and evaluated.The results of the computational experiments are shown in Figures 11–13. Figure 11

demonstrates that in problems with m¼ 4, 6 ICA performs better than SA, Figure 12

shows that in problems with n¼ 20, 40, 60 ICA outperforms SA and Figure 13 indicates

that in problems in which �¼ 0.5, 0.8, ICA performs better. It means that ICA shows

better performance when the weight of makespan is higher than mean completion time.Figure 14 indicates the standard deviation (Std), in which ICA has less Std in most of

the problems (when �¼ 0.5, 0.8) and SA has less Std when �¼ 0.2.The t-test is selected as a method for statistical testing of these data. Hypothesis testing

for all test problems is as follows:

Null hypothesis: The average error of SA¼ the average error of ICA.Alternative hypothesis: The average error of ICA5 the average error of SA.

The computed p-value for this problem is 0.02. In other words the null hypothesis was

rejected for all combinations at a 98% significance level. Thus, the average error of ICA is

statistically smaller than that of SA. Generally, in most of the problems ICA performs

better than SA.

Table 5. ANOVA table for the S/N ratio of small problems.

Factors DF SS MS F Percent of X Cumulative

Nimp 2 0.016194 0.008097 1.6907 39.5 39.5(pop, IT ) 2 0.020021 0.01001 2.0902 49.5 89.0Error 4 0.002098 0.000524 0.1095 11.0 100.0

Total 8 0.038313 0.004789

Table 6. ANOVA table for the S/N ratio of large problems.

Factors DF SS MS F Percent of X Cumulative

Nimp 2 0.002693 0.001346473 0.9693 23.1 23.1(pop, IT ) 2 0.007904 0.003952223 2.8450 69.9 93.0Error 4 0.000258 6.44971E-05 0.0464 7.0 100.0

Total 8 0.011113 0.001389171

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The computational time comparison can be seen in Figures 15 and 16 where it is shownthat, approximately in all the test problems, ICA takes twice as much time as SA. SinceICA is a population-based meta-heuristic and SA is point-based, the time differentiation isaxiomatic. Inasmuch as the longest time is less than two minutes, this gap is not important.

0.2 0.5 0.80

0.02

0.04

0.06

0.08

0.1

0.12

Err

or

ICA

SA

α

Figure 13. Overall error for different values of �.

0

0.02

0.04

0.06

0.08

0.1

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2

m

Err

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SA

4 6 8

Figure 11. Overall error for different values of m.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 20 40 60 80n

Err

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Figure 12. Overall error for different values of n.

3100 E. Shokrollahpour et al.

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30

40

50

60

70

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m

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Figure 16. Average completion time for different values of m.

0

0.01

0.02

0.03

0.04

0.05

0.2 0.5 0.8

Std ICAStd

SAStd

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Figure 14. Overall standard deviation for different values of �.

20

40

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100

120

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Figure 15. Average completion time for different values of n.

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5. Conclusions and future work

The purpose of the current study was to compare imperialist competitive algorithm withthe best algorithm proposed before (SA), in a two-stage assembly flowshop schedulingproblem with a weighted sum of makespan and mean completion time. For betterperformance of the algorithm, its parameters were calibrated by the Taguchi method, andthe conducted t-test showed that the average error of ICA is smaller than that of SA.

In future researches it might be possible to use different assimilation operators in ICA,or different ways for generating initial population. Antagonist objectives for this problemcan be considered in future works.

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