a variable iterated greedy algorithm with differential evolution for the no-idle permutation...

Computers & Operations Research 40 (2013) 1729–1743

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Computers & Operations Research

0305-05

http://d

n Corr

E-m

panqua

ozge.bu

journal homepage: www.elsevier.com/locate/caor

A variable iterated greedy algorithm with differential evolutionfor the no-idle permutation flowshop scheduling problem

M. Fatih Tasgetiren a,n, Quan-Ke Pan b, P.N. Suganthan c, Ozge Buyukdagli a

a Industrial Engineering Department, Yasar University, Izmir, Turkeyb College of Computer Science, Liaocheng University, Liaocheng, Shandong, PR Chinac School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore

a r t i c l e i n f o

Available online 31 January 2013

Keywords:

Differential evolution algorithm

Iterated greedy algorithm

No-idle permutation flowshop

scheduling problem

Heuristic optimization.

48/$ - see front matter & 2013 Elsevier Ltd. A

x.doi.org/10.1016/j.cor.2013.01.005

esponding author.

ail addresses: [email protected] (M

[email protected] (Q.-K. Pan), [email protected]

[email protected] (O. Buyukdagli).

a b s t r a c t

This paper presents a variable iterated greedy algorithm (IG) with differential evolution (vIG_DE),

designed to solve the no-idle permutation flowshop scheduling problem. In an IG algorithm, size d of

jobs are removed from a sequence and re-inserted into all possible positions of the remaining

sequences of jobs, which affects the performance of the algorithm. The basic concept behind the

proposed vIG_DE algorithm is to employ differential evolution (DE) to determine two important

parameters for the IG algorithm, which are the destruction size and the probability of applying the IG

algorithm to an individual. While DE optimizes the destruction size and the probability on a continuous

domain by using DE mutation and crossover operators, these two parameters are used to generate a

trial individual by directly applying the IG algorithm to each target individual depending on the

probability. Next, the trial individual is replaced with the corresponding target individual if it is better

in terms of fitness. A unique multi-vector chromosome representation is presented in such a way that

the first vector represents the destruction size and the probability, which is a DE vector, whereas the

second vector simply consists of a job permutation assigned to each individual in the target population.

Furthermore, the traditional IG and a variable IG from the literature are re-implemented as well. The

proposed algorithms are applied to the no-idle permutation flowshop scheduling (NIPFS) problem with

the makespan and total flowtime criteria. The performances of the proposed algorithms are tested on

the Ruben Ruiz benchmark suite and compared to the best-known solutions available at http://soa.iti.

es/rruiz as well as to those from a recent discrete differential evolution algorithm (HDDE) from the

literature. The computational results show that all three IG variants represent state-of-art methods for

the NIPFS problem.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The processing order is the same for all jobs in a flowshopsituation. Furthermore, the job sequences are assumed to bepermutations; therefore, once a permutation is fixed for all jobson the first machine, this permutation is maintained for allmachines, which is the so-called permutation flowshop schedul-ing problem (PFSP). Although several performance measures existfor scheduling in a flowshop, the makespan criterion is the mostcommonly studied performance measure in the literature (see[1,2] for recent reviews of this problem).

This study focuses on solving a variant of the PFSP in whichidle times are not allowed on machines. The no-idle constraint

ll rights reserved.

. Fatih Tasgetiren),

u.sg (P.N. Suganthan),

refers to an important practical situation in the productionenvironment in which expensive machinery is employed [3],and idling of such expensive machinery is often undesirable. Forinstance, the steppers employed in the production of integratedcircuits via photolithography are clear examples. Certain otherexamples arise in industries where less expensive machinery isemployed. However, the machines cannot be stopped andrestarted. For example, ceramic roller kilns consume large quan-tities of natural gas when in operation. Idling is not an option inthis case because it takes several days to stop and restart the kilndue to notably large thermal inertia. In such cases, idling shouldbe avoided. Another practical example is the furnace used infiberglass processing in which glass batches are reduced tomolten glass. Because it takes three days to heat the furnace backto the required temperature of 2800 1F, the furnace should remainin operation during the entire production season. In addition tothe above examples, the three-machine flowshop production ofengine blocks in a foundry is presented in [4]. This example

http://soa.iti.es/rruiz


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Fig. 1. (a) Computation of FðpE1 ,k,kþ1Þ, (b) Computation of FðpE

2 ,k,kþ1Þ,

(c) Computation of FðpE3 ,k,kþ1Þ and (d) Computation of CmaxðpÞ.

M Fatih Tasgetiren et al. / Computers & Operations Research 40 (2013) 1729–17431730

includes the casting of sand molds and sand cores. The molds arefilled with metal in fusion, and the cores prevent the metal fromfilling certain species in the mold. The casting machines operatewithout idle times due to both economic and technologicalconstraints.

In a NIPFS problem, each machine must process jobs without anyinterruption from the start of processing the first job to thecompletion of the last job. Therefore, when needed, the start ofprocessing the first job on a given machine must be delayed to meetthe no-idle requirement. In this work, we denote the NIPFS problemwith the well-known three-fold notation of Fm=prmu,no�idle=Cmax.The computational complexity of the Fm=prmu,no�idle=Cmax pro-blem is briefly commented in [5]. The NP-Hardness of theF3=prmu,no�idle=Cmax problem was proven by [6,4]. Therefore, thisproblem has a significant importance both in theory and engineer-ing applications for development of effective and efficientapproaches to the problem discussed in this paper.

Despite its practical importance, the Fm=prmu,no�idle=Cmax

problem has not attracted much attention in the literature [7].A polynomial time algorithm for optimally solving the F2/prmu,no_idle/

PCj problem was presented in [8]. The makespan criter-

ion was studied for the first time in [9], whereas heuristicapproaches for the general m-machine NIPFS problem with themakespan criterion were examined in [10]. A branch-and-bound(B&B) method is also presented by [6] for the general m-machineNIPFS problem with the makespan criterion.

Certain mistakes in the paper by [8] were reported in [11]. TheF3=prmu,no�idle=Cmax problem was studied in [12], and the sameproblem was also studied in [4], in which a lower bound and anefficient heuristic are presented. This new heuristic favors the earliermethod of the authors [13], who published this work later on in [14].Kamburowski in [15] further enhanced the idea in [4] by proposing anetwork representation. A heuristic for the F3=prmu,no�idle=Cmax

problem based on the traveling salesman problem (TSP) was pro-posed in [14]. The F2=prmu,no�idle=Cmax and Fm=prmu,no�idle=Cmax problems were studied in two similar papers [12,16].In [17], Kalczynski and Kamburowski developed a constructiveheuristic, called the KK heuristic, for the Fm=prmu,no�idle=Cmax

problem with a time complexity of O(n2m). The authors alsopresented an adaptation of the Nawaz, Enscore and the Ham (NEH)heuristic [18] for the NIPFS problem. In addition, Kalczynski andKamburowski studied the interactions between the no-idle and no-wait flowshops in [17] as well. Recently, Baraz and Mosheiovintroduced an improved two-stage greedy algorithm named GH_BM,which consists of a simple greedy heuristic and an improvement stepbased on the adjacent pairwise interchange (API) method in [7].

In recent years, meta-heuristics have attracted increasingattention in the solution of scheduling problems because theyare able to provide high quality solutions with reasonablecomputational effort [19]. In addition to the above literature, intwo similar papers, Pan and Wang in [20,21] proposed discretedifferential evolution (DDE_LS) and a hybrid discrete particleswarm optimization (HDPSO) algorithm for the same problem.In both papers, a speed-up scheme for the insertion neighborhoodis proposed that reduces the computational complexity of a singleinsertion neighborhood scan from O(n3m) to O(n2m) when theinsertion is carried out in order. The speed-up that they proposedis based on the well-known accelerations presented by [22] forthe insertion neighborhood for the PFSP. In fact, both in DDE_LSand HDPSO, an advanced local search form, an IG algorithmproposed by [23], is employed as a local search. Both DDE_LSand HDPSO used the well-known benchmark suite of [24] bytreating these as NIPFS instances to test the results. In bothpapers, the authors tested the proposed methods against theheuristics in [7,25]. Recently, an IG_LS algorithm for the NIPFSproblem with the makespan criterion was presented in [3]. These

researchers employed their own benchmark suite and examinedthe performance of IG_LS in detail against the existing heuristicsand meta-heuristics from the literature, whereas a continuous DEalgorithm is presented in [26]. Most recently, a hybrid discretedifferential evolution algorithm (HDDE) was presented in [27] inwhich a speed-up method based on network representation isproposed to evaluate the entire insertion neighborhood of a jobpermutation. Through a detailed experimental campaign, it wasconcluded that the HDDE algorithm was superior to the existingstate-of-the-art algorithms from the literature.

In this paper, a vIG_DE algorithm and two other IG variants arepresented for comparison to the best-known solutions in [3] aswell as solutions presented by the HDDE algorithm in [27]. Thecomputational results show that all three IG variants representstate-of-art methods for the NIPFS problem.

The remaining part of the paper is organized as follows.Section 2 introduces the NIPFS problem. Section 3 presents the

M Fatih Tasgetiren et al. / Computers & Operations Research 40 (2013) 1729–1743 1731

vIG_DE algorithm and two other IG variants in detail, andSection 4 discusses the computational results in the context ofbenchmark problems. Finally, Section 5 presents the concludingremarks.

Fig. 2. (a) Computation of EðpF3 ,k,kþ1Þ, (b) Computation of EðpF

2 ,k,kþ1Þ,

(c) Computation of EðpF1 ,k,kþ1Þ and (d) Computation of CmaxðpÞ.

2. No-idle permutation flowshop scheduling problem

The NIPFS problem with n (j¼1,2,..,n) jobs and m (k¼1,2,..,m)machines can be defined as follows. Each job will be sequencedthrough m machines. p(j, k) denotes the processing time in whichthe setup time is included. At any time, each machine can processat most one job, and each job can be processed on at most onemachine. The sequence in which the jobs are to be processed isthe same for each machine. To follow the no-idle restriction, eachmachine must process jobs without any interruption from thestart of processing the first job to the completion of processing ofthe last job. The aim is then to find the same permutation on eachmachine to minimize the makespan. We follow the approach ofPan and Wang in [20,21] and elaborate the formulation of theNIPFS problem with the makespan and total flowtime criteria. Theformulation consists of forward and backward passes as well as acombined method to facilitate the speed-up, which will beexplained and illustrated with an example in Appendix A.

2.1. Forward pass calculation

Let a job permutation p¼{p1,p2,y,pn} represent the scheduleof jobs to be processed and pE

j ¼ fp1,p2,. . .,pjg be a partialschedule of p such that j must be between 1 and n (1o jon).In addition, FðpE

j ,k,kþ1Þ refers to the minimum differencebetween the completion of processing the last job of pE

j onmachines kþ1 and k, which is restricted by the no-idle constraint.Then, FðpE

j ,k,kþ1Þ can be computed as follows:

FðpE1,k,kþ1Þ ¼ pðp1,kþ1Þ k¼ 1,2,:::,m�1 ð1Þ

FðpEj ,k,kþ1Þ ¼maxfFðpE

j�1,k,kþ1Þ�pðpj,kÞ,0gþpðpj,kþ1Þ

j¼ 2,3,::,n and k¼ 1,2,::,m�1 ð2Þ

Then, the makespan of job pn on machine m can be given by

Cðpn,mÞ ¼ CmaxðpEnÞ ¼

Xm�1

k ¼ 1

FðpEn,k,kþ1Þþ

Xn

j ¼ 1

pðpj,1Þ ð3Þ

Fig. 1a to d illustrates the calculation of the makespan for a3-job 3-machine problem.

Regarding the completion times (i.e., flowtimes because readytimes are zero), as the there is no idle time on machines, it istrivial to compute the completion time C(pj,m) of job pj on the lastmachine m, and it can be achieved by subtracting the processingtime of job jþ1 from the completion time of job jþ1 on the lastmachine m as follows:

Cðpj,mÞ ¼ Cðpjþ1,mÞ�pðpjþ1,mÞ j¼ n�1,n�2,::,1 ð4Þ

Thus, the total flowtime can be calculated as follows:

TFTðpÞ ¼Xn

j ¼ 1

Cðpj,mÞ ð5Þ

2.2. Backward pass calculation

Let pFj ¼ fpj,pjþ1,:::,png denote another partial schedule of p

such that j must be between 1 and n (1o jon). Let EðpFj ,k,kþ1Þ be

the lower bound for the minimum difference between the start ofprocessing the first job of pF on machines kþ1 and k, which isrestricted by the no-idle constraint. Then, EðpF

j ,k,kþ1Þ can be

calculated as follows:

EðpFn,k,kþ1Þ ¼ pðpn,kÞ k¼ 1,2,:::,m�1 ð6Þ

EðpFj ,k,kþ1Þ ¼maxfEðpF

jþ1,k,kþ1Þ�pðpj,kþ1Þ,0gþpðpj,kÞ

j¼ n�1,n�2,:::,1 and k¼ 1,2,::,m�1 ð7Þ

The completion time C(p1,m) of job p1 on the last machine m

can be given as follows:

Cðp1,mÞ ¼Xm�1

k ¼ 1

EðpF1,k,kþ1Þþpðp1,mÞ ð8Þ

Then, the makespan of the job p1 can be given in an alternativeapproach,

Cmaxðp1Þ ¼Xm�1

k ¼ 1

Eðp1,k,kþ1ÞþXn

j ¼ 1

pðpj,mÞ ð9Þ

Now, it is trivial to compute the completion time C(pjþ1,m) ofjob pjþ1on the last machine m, which can be obtained as follows:

Cðpjþ1,mÞ ¼ Cðpj,mÞþpðpjþ1,mÞ, j¼ 1,2,. . .,n�1 ð10Þ

Fig. 2a to d illustrates the calculation of the makespan for a3-job 3-machine problem.


Thus, the total flowtime can be calculated as follows:

TFTðpÞ ¼Xn

j ¼ 1

Cðpj,mÞ ð11Þ

Therefore, the objective of the NIPFS problem with the make-span/total flowtime criterion is to find a permutation p* in the setof all permutations P such that:

CmaxðpnÞrCmaxðpEnÞ or CmaxðpnÞrCmaxðpF

1Þ, 8pAP: ð12Þ

TFTðpnÞrTFT ðpEnÞ or TFTðpnÞrTFTðpF

1Þ, 8pAP: ð13Þ

Further, to be used in the speed-up method, let FðpFn,k,kþ1Þ

represent the minimum difference between the completions ofprocessing the last job of pF on machine kþ1 and k, which isrestricted by the no-idle constraint. Then, FðpF

j ,k,kþ1Þ can becalculated as follows:

FðpFn,k,kþ1Þ ¼ pðpn,kþ1Þ k¼ 1,2,:::,m�1 ð14Þ

FðpFn�1,k,kþ1Þ ¼maxfpðpn�1,kþ1Þ�EðpF

n,k,kþ1Þ,0g

þFðpFn,k,kþ1Þ: k¼ 1,2,:::,m�1 ð15Þ

FðpFj ,k,kþ1Þ ¼maxfpðpj,kþ1Þ�EðpF

jþ1,k,kþ1Þ,0gþFðpFjþ1,k,kþ1Þ

j¼ n�1,n�2,:::,1, k¼ 1,2,:::,m�1 ð16Þ

Fig. 3a to c illustrates the above calculation for a 3-job3-machine problem.

Fig. 3. (a) Computation of FðpE3 ,k,kþ1Þ, (b) Computation of FðpE

2 ,k,kþ1Þ and

(c) Computation of FðpE1 ,k,kþ1Þ.

2.3. Speed-up method for insertion neighborhood

The insertion neighborhood of a job permutation p is widelyused for flow shop scheduling problems in the literature [19].An insert move generates a new permutation by removing a jobfrom its original position u and inserting it into position v suchthat ve(u,u�1). A shortcut to evaluate the insertion neighbor-hood can be developed by following the reduction of computa-tional complexity presented by [22] for the PFSP with themakespan criterion. Subsequently, it is easy to extend to themakespan or total flowtime criterion as explained in the forwardand backward pass calculations for the NIPFS problem. Thefollowing speed-up method can be used to evaluate the insertneighborhood for the makespan or the total flowtime criterion:

1.
Let t¼1. 2. Let Dp¼{p1,p2,..,pn�1} be a partial permutation generated by
removing job pt from permutation p.a. Compute FðDpE

j ,k,kþ1Þ, EðDpFj ,k,kþ1Þ, and FðDpF

j ,k,kþ1Þfor k¼1,2,..,m�1, respectively.

b. Set FðDpFn,k,kþ1Þ ¼ EðDpF

n,k,kþ1Þ ¼ 0 for k¼1,2,..,m�1.
3. Repeat the following steps until all possible positions h of
Dp¼{p1,p2,..,pn�1} are considered such that hA{1,2,...,n}4-he{t, t�1}.a. Set DpE

h ¼DpEh�1 [ pt (Note that DpE

0 ¼f).b. Calculate FðDpE

h,k,kþ1Þ for k¼1,2,...,m�1.c. Set p¼DpE

h [ DpFh ¼ fp1,p2,. . .,png(Note that DpF

n ¼f).d. Then F(p,k,kþ1) can be given by

Fðp,k,kþ1Þ ¼maxfFðDpEh,k,kþ1Þ�EðDpF

h,k,kþ1Þ,0gþFðDpF

h,k,kþ1Þ for k¼1,2,...,m�1.e. Then the completion time of job pn on machine m is

Cðpn,mÞ ¼Xm�1

k ¼ 1

Fðp,k,kþ1ÞþXn

j ¼ 1

pðpj,1Þ

f. Or the completion time of job pj on the last machine m is

Cðpj,mÞ ¼ Cðpjþ1,mÞ�pðpjþ1,mÞ j¼ n�1,n�2, � � � ,1

Set t¼tþ1. If t4n then stop; otherwise go back to step 2.
4.
There are n iterations for Step 2 and Step 3, and both Step2 and Step 3 can be executed in time O(mn2). Therefore, thecomputational complexity of this speed-up method is O(mn2) forevaluating the entire insertion neighborhood of a permutation.The speed-up method is illustrated with an example instance inAppendix A.

3. Variable iterated greedy algorithm with differentialevolution

The IG algorithm is presented in Ruiz and Stutzle [23] and hassuccessful applications in discrete/combinatorial optimizationproblems. The IG algorithm is fascinating in terms of its ease ofuse by writing a few additional lines of codes. In this section, wefirst describe the traditional IG. Second, we summarize thevariable IG from the literature. Finally, the details of the proposedvIG_DE algorithm are given.

In general, an IG algorithm is either started with a randomsolution or a problem-specific heuristic, which is usually the NEHheuristic [18]. Next, the destruction–construction phase begins,which basically consists of removing d jobs from sequence and re-inserting them into remaining sequence of jobs. In this study, theIG algorithm starts with the initial solution generated by the NEHheuristic, and next a local search based on the insertion heuristicis applied. Next, a loop is started in such a way that the solution isdestructed and reconstructed by using the second phase of the


NEH heuristic, and then the local search based on insertionheuristic is applied again. This process is repeated until thetermination criterion is satisfied.

The NEH heuristic contains two phases. In the first phase, jobsare ordered in descending sums of their processing times. In thesecond phase, a job permutation is established by evaluating thepartial permutations based on the initial order of the first phase.Assume a current permutation is already determined for the firstk jobs, kþ1 and partial permutations are subsequently con-structed by inserting job kþ1 in kþ1 into possible positions ofthe current permutation. Among these kþ1 permutations, theone generating the minimum makespan or total flowtime is keptas the current permutation for the next iteration. Next, job kþ2from the first phase is considered, and the process continues untilall jobs have been sequenced.

Regarding the destruction and construction procedure denotedby DestructConstruct, in the destruction step, a given number d ofjobs are randomly chosen and removed from the solution withoutrepetition, thus resulting in two partial solutions. The first, withthe size d of jobs, is denoted as pR including the removed jobs inthe order in which they are removed. The second, with the sizen�d of jobs, is the original solution without the removed jobs,which is denoted as pD. Finally, in the construction phase, theNEH insertion heuristic is used to complete the solution. The firstjob pR

1 is inserted into all possible n�dþ1 positions in thedestructed solution pD thus generating n�dþ1 partial solutions.Among these n�dþ1 partial solutions, the best partial solutionwith the minimum makespan or total flow time is chosen andkept for the next iteration. Next, the second job pR

2 is consideredand the process continues until pR is empty or a final solution isobtained. Therefore, pD is again of size n.

Regarding the local search algorithm, we use the referencedinsertion algorithm, called RIS, which was presented in our previousworks [28,29]. In the RIS procedure, pR is the reference permutation,which is the best solution obtained so far. For example, suppose thatthe reference permutation is given by pR

¼{3, 1, 5, 2, 4} and thecurrent solution by p¼{4, 2, 5, 1, 3}. The RIS procedure first finds job3 in the current permutation p. It removes job 3 from p and insertsit into all possible positions of p. Next, the RIS procedure finds job1 in the current permutation p. It removes job 1 and inserts it into

Fig. 4. Iterated greedy algorithm.

all possible positions of p. This procedure is repeated until pR isempty, and a new permutation of p with n jobs is obtained. Themain concept behind the RIS procedure is the fact that the jobremoval is guided by the reference permutation instead of a randomchoice. It should be noted that the speed-up method explained inSection 2.3 is employed in the NEH, DestructConstruct and RISprocedures. The IG algorithm we implemented in this paper isdenoted as IG_RIS. The pseudo-codes of the IG_RIS and the RISprocedures are given in Figs. 4 and 5, respectively.

In addition to the above, a variable IG algorithm (VIG_FL) isalso presented and implemented to solve the PFSP with the totaltardiness criterion in Framinan and Leisten [30]. The VIG_FLalgorithm is inspired from the variable neighborhood search(VNS) algorithm in [31]. By using the idea of neighborhoodchange of the VNS algorithm, a variable the IG algorithm isdeveloped. The maximum destruction size is fixed atdmax ¼ n�1. Next, the destruction size at the beginning is fixedat d¼1 and incremented by 1 (i.e., d¼dþ1), if the solution is notimproved until dmax ¼ n�1. If a solution improves in any destruc-tion size, it is again fixed at d¼1 and the search starts from thebeginning once again. In our implementation of the VIG_FLalgorithm, we employ our referenced insertion heuristic, RIS, asa local search as well. Note that the speed-up method explainedin Section 2.3 is employed in the NEH, DestructConstruct and RISprocedures again. The pseudo-code of the VIG_FL algorithmimplemented is given in Fig. 6.

Fig. 5. Referenced insertion scheme.

Fig. 6. Variable iterated greedy algorithm.

Fig. 7. Multi-vector chromosome representation.

Fig. 8. vIG_DE algorithm.


Differential evolution (DE) is an evolutionary optimizationmethod proposed by Storn and Price [32], and an excellent reviewof DE algorithms can be found in Das and Suganthan [33]. In thispaper, the standard differential evolution algorithm is modifiedsuch that the IG algorithm will be able to use a variabledestruction size and a probability of applying the IG algorithmto the associated individual in the target population, thus endingup with a variable iterated greedy algorithm guided by a DEalgorithm. NP denotes the population size in the target popula-tion. Therefore, the proposed algorithm is denoted as vIG_DE. Forthis purpose, we propose a unique multi-vector chromosomerepresentation given in Fig. 7.

In the chromosome/individual representation, the first vectorin the individual represents the destruction size (di) and theprobability (ri) of applying the IG algorithm, respectively. In otherwords, the first and second dimensions of xij vector will be xi1¼di

and xi1¼di, respectively. In addition, a permutation is assigned toeach individual in the target population, which represents thesolution of the individual. The basic concept behind the proposedalgorithm is that while DE optimizes the destruction size and theprobability, these optimized values guide the search for the IGalgorithm to generate trial individuals. In other words, a uniformrandom number r is generated. If r is less than the probabilityxi1¼di, the trial individual is directly produced by applying the IGalgorithm with the destruction size xi1¼di to the permutation pi

of the corresponding individual. In the initial population, thexi1¼di and xi1¼di parameters for each individual are establishedas follows: the destruction size is randomly and uniformlydetermined as xi1¼diA[1,n�1]. Next, the permutation for thefirst individual is constructed by the NEH heuristic. The remainingpermutations for the individuals in the target population arerandomly constructed, and the NEH heuristic is applied to each ofthem. Once the destruction size and permutation for eachindividual are constructed, the IG algorithm is applied to eachindividual at first glance. Next, the probability xi1¼di of applyingthe IG algorithm to each permutation is determined asxi1 ¼ di ¼ 1�f ðpiÞ=

PNPi ¼ 1 f ðpiÞ. By doing so, the higher the prob-

ability is, the higher the chance that the IG algorithm will beapplied to corresponding individual i. In the proposed vIG_DEalgorithm, mutant individuals are generated as follows:

vtih ¼ xt�1

ah þFr � ðxt�1bh �xt�1

ch Þ for h¼ 1,2, ð17Þ

where a, b and c are three individuals randomly chosen bytournament selection with a size of 2 from the target populationsuch that aabaca iA(1,...,NP) and (hA(1,2)). Fr40 is a mutationscale factor that affects the differential variation between twoindividuals. Next, an arithmetic crossover operator is applied toobtain the trial individual instead of the traditional uniformcrossover such that:

utih ¼ Cr � vt

ihþð1�CrÞ � xt�1ih for h¼ 1,2, ð18Þ

where Cr is a user-defined crossover constant in the range [0,1].During the reproduction of the trial population, parameter valuesviolating the search range are restricted to:

utih ¼ xmin

ih þðxmaxih �xmin

ih Þ � r for h¼ 1,2, ð19Þ

where xmini1 ¼ 1 and xmax

i1 ¼ n�1; xmini2 ¼ 0 and xmax

i2 ¼ 1; and r is auniform random number between 0 and 1.

It should be mentioned that at this stage, a trial vector utih is

generated, and the first dimension is truncated to an integer valueas di ¼ ut

i1

� �to determine the destruction size. Next, the second

dimension ri ¼ uti2 is used to determine whether the IG algorithm

will be applied to the permutation of the target individual togenerate the trial individual. The IG algorithm is then applied, andthe fitness value of the trial individual is computed if a uniformrandom number rori ¼ ut

i2. Afterwards, the selection is carriedout on the basis of the survival of the fittest among the trial andtarget individuals such that:

xti ¼

uti if f ðut

i Þo f ðxt�1i Þ

xt�1i otherwise

(ð20Þ

Note that the speed-up method explained in Section 2.3 isemployed in the NEH, DestructConstruct and RIS proceduresagain. Finally, the pseudo-code of the vIG_DE algorithm is givenin Fig. 8.

4. Computational results

The proposed algorithms were coded in Cþþ and run on anIntel Core 2 Quad 2.66 GHz PC with 3.5 GB memory. The crossoverprobability and mutation scale factor are taken as Cr¼0.9 andFr¼0.9, respectively, and the population size is fixed at NP¼30.For the IG_RIS algorithm, the destruction size was fixed at d¼4.We tested the performance of our algorithm on a benchmark suiteavailable in http://soa.iti.es/rruiz. The benchmark set is specifi-cally designed for the no-idle permutation flowshop schedulingproblem with the makespan criterion. It contains completecombinations of n¼{50,100,150,200,250,300,350,400,450,500}and m¼{10,20,30,40,50}. There are five instances per combina-tion; hence, there are 250 instances in total. Five runs werecarried out for each problem instance, the same as for othercompeting algorithms. Each run was compared to the best-knownsolution presented in http://soa.iti.es/rruiz. The average relativepercent deviation from the best-known solution is given asfollows:

Davg ¼XR

i ¼ 1

ðHi�BestÞ � 100

Best

� �=R ð21Þ

where Hi, Best, and R are the objective function values generatedby the proposed algorithms in each run, the best known solutionvalue, and the number of runs, respectively. In addition, Dmin,Dmax, and Dstd denote the minimum, maximum and standard



Ave

rage

Rel

ativ

e Pe

rcen

t Dev

iatio

n

With Speed-UpWithout Speed-Up

1.2

0.9

0.6

0.3

0.0

95% CI for the MeanInterval Plot of vIG_DE Algorithm with and without Speed-Up


deviation of five runs for each instance. As a termination criterion,the proposed algorithms were run for Tmax ¼ nðm=2Þ � t ms,where t¼30, which is the same as in [3] and [27]. In the resultspresented below, the algorithm XXXt denotes that the XXX algo-rithm was run for Tmax ¼ nðm=2Þ � t ms.

It should be noted that the main component of the algorithmspresented in this paper is the speed-up method employed. Tounderstand the impact of the speed-up method, we first run thevIG_DE algorithm with and without the speed-up method underthe termination criterion of Tmax ¼ nðm=2Þ � t ms, where t¼30.The computational results are given in Table 1. It is clear fromTable 1 that the speed-up method leads to significant improve-ments in all statistics. For example, Davg, Dmin, Dmax, and Dstd areimproved from 0.92%, 0.67%, 1.16% and 0.19% to �0.12%, �0.25%,0.02% and 0.11%, respectively. Furthermore, to determine if thedifferences in the average relative percent deviation (Davg) are

Table 1Computational results for vIG_DE with and without speed-up. t¼30.

vIG_DE without speed-up vIG_DE with speed-up

n m Davg Dmin Dmax Dstd Davg Dmin Dmax Dstd

50 10 0.54 0.40 0.67 0.12 0.03 �0.02 0.10 0.05

20 0.61 0.39 0.78 0.16 �0.04 �0.13 0.04 0.07

30 0.84 0.58 1.24 0.27 �0.17 �0.33 �0.04 0.11

40 0.79 0.51 1.06 0.21 �0.41 �0.64 �0.18 0.18

50 1.13 0.67 1.56 0.35 �0.16 �0.43 0.11 0.22

100 10 0.43 0.27 0.56 0.12 0.04 0.00 0.10 0.05

20 0.72 0.45 0.97 0.19 �0.09 �0.18 0.01 0.08

30 1.81 1.45 2.13 0.27 �0.19 �0.34 0.00 0.13

40 2.31 1.81 2.77 0.38 �0.65 �0.98 �0.40 0.24

50 2.58 2.25 2.93 0.27 �0.12 �0.40 0.17 0.22

150 10 0.02 0.01 0.03 0.01 0.00 0.00 0.00 0.00

20 1.02 0.80 1.23 0.17 0.05 �0.03 0.12 0.06

30 1.02 0.70 1.26 0.22 �0.18 �0.29 �0.11 0.08

40 2.62 2.13 3.13 0.38 �0.07 �0.31 0.17 0.20

50 2.01 1.58 2.39 0.33 �0.85 �1.14 �0.62 0.20

200 10 0.00 0.00 0.02 0.01 0.00 0.00 0.00 0.00

20 0.52 0.35 0.73 0.15 �0.07 �0.12 �0.01 0.05

30 1.05 0.79 1.34 0.24 �0.31 �0.44 �0.18 0.11

40 1.86 1.34 2.43 0.46 �0.26 �0.50 �0.04 0.18

50 2.01 1.43 2.43 0.41 �0.40 �0.55 �0.24 0.13

250 10 0.02 �0.01 0.05 0.02 �0.01 �0.01 �0.01 0.00

20 0.50 0.30 0.67 0.15 �0.03 �0.08 0.02 0.04

30 0.91 0.63 1.19 0.22 �0.14 �0.26 �0.02 0.10

40 1.65 1.28 2.05 0.30 0.02 �0.11 0.16 0.10

50 2.15 1.64 2.57 0.39 �0.60 �0.85 �0.31 0.22

300 10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

20 0.57 0.43 0.71 0.12 0.00 �0.08 0.09 0.06

30 0.89 0.66 1.11 0.18 0.02 �0.09 0.12 0.08

40 1.28 1.05 1.59 0.21 �0.34 �0.53 �0.16 0.15

50 2.05 1.54 2.58 0.38 �0.23 �0.59 0.24 0.34

350 10 0.05 0.02 0.08 0.03 0.00 0.00 0.00 0.00

20 0.46 0.27 0.64 0.16 �0.01 �0.06 0.05 0.04

30 0.83 0.63 1.00 0.15 �0.05 �0.15 0.05 0.08

40 1.16 0.89 1.40 0.20 0.08 �0.09 0.26 0.13

50 1.29 0.89 1.67 0.32 �0.44 �0.64 �0.21 0.17

400 10 0.04 0.00 0.08 0.03 0.00 0.00 0.00 0.00

20 0.44 0.29 0.58 0.12 0.04 �0.04 0.12 0.06

30 0.81 0.65 0.97 0.13 0.11 0.02 0.23 0.09

40 0.87 0.55 1.10 0.22 �0.04 �0.20 0.09 0.11

50 1.02 0.70 1.28 0.24 �0.25 �0.43 0.01 0.18

450 10 0.04 0.01 0.08 0.03 0.00 0.00 0.00 0.00

20 0.30 0.18 0.43 0.10 0.00 �0.07 0.05 0.05

30 0.58 0.43 0.74 0.12 �0.03 �0.15 0.10 0.10

40 0.76 0.51 1.02 0.20 �0.09 �0.24 0.05 0.12

50 1.04 0.67 1.42 0.31 �0.07 �0.35 0.32 0.26

500 10 0.02 0.01 0.04 0.01 0.00 0.00 0.00 0.00

20 0.11 0.05 0.22 0.07 �0.04 �0.06 �0.03 0.01

30 0.51 0.32 0.67 0.14 0.08 �0.01 0.19 0.09

40 0.80 0.51 1.05 0.21 0.10 �0.06 0.27 0.13

50 1.09 0.73 1.38 0.26 �0.03 �0.33 0.18 0.20

Overall Avg 0.92 0.67 1.16 0.19 �0.12 �0.25 0.02 0.11

Fig. 9. Internal plot of vIG_DE algorithm with and without speed-up.

statistically significant, we provide the interval plot of the vIG_DEalgorithms with and without speed-up, given in Fig. 9. Recall thatif the confidence interval (CI) of any two intervals do not coincide,then the means are statistically significant. From Fig. 9, it is clearthat the speed-up method has a significant impact on the solutionquality because the two confidence intervals (CIs) with andwithout speed-up do not coincide. In other words, the averagerelative percent deviations are statistically significant.

As mentioned previously, the HDDE algorithm is presented tosolve the same problem as in Deng and Gu [27]. In addition, theseresearchers re-implemented the IG_LS, DDE_LS and HDPSO algo-rithms in Deng and Gu [27] with the termination criterion ofTmax ¼ nðm=2Þ � t milliseconds, where t¼60. They claimed thatthe HDDE algorithm was superior to all of the IG_LS, DDE_LS andHDPSO algorithms.

For a detailed analysis, we also re-implemented the IG algo-rithm of Ruiz and Stutzle (denoted in this paper as IG_RIS) and thevariable IG algorithm of Framinan and Leisten (denoted in thispaper as VIG_FL) together with our vIG_DE algorithm. Note that inthe IG_RIS, VIG_FL and vIG_DE algorithms, we employed thespeed-up method given in Section 2.3 and the local searchpresented in Section 3. To avoid CPU time discussions, we fixedour termination criterion Tmax ¼ nðm=2Þ � t ms, where t¼30. Inother words, we compare our results to those of the HDDE,HDPSO and IG_LS algorithms with half of their CPU time require-ments. The computational results are given in Table 2.

Table 2 indicates that the proposed algorithms are superiorbecause they all generated better average relative percent devia-tions from the HDDE, HDPSO and IG_LS algorithms. In fact, thevIG_DE, IG_RIS and VIG_FL algorithms were able to furtherimprove the Davg values to �0.12, �0.04 and �0.06, respectively.It is also interesting to note that the IG_LS algorithm re-implemented in [27] was outperformed by the HDDE algorithm.However, in our re-implementation of the IG_RIS algorithm, thisapproach was superior to the HDDE, DDE_LS, HDPSO and IG_LSalgorithms because its average relative percent deviation was�0.04%. This result indicates the fact that our speed-up methodseemed to be much more effective than the one in [27]. Duringthese runs, we were able to further improve on most of the bestknown solutions reported both in [3] and [27].

To analyze the above results in Table 2, an interval plot of theexperimental results in Table 2 is given in Fig. 10. Note that if theconfidence intervals (CI) of any two intervals do not coincide,then the means are statistically significant. As can be observed inFig. 10, the means and CIs of the vIG_DE, IG_RIS and VIG_FL aresignificantly lower than those generated by the HDDE, DDE_LS,HDPSO and IG_LS algorithms.

Table 2Average relative percentage deviation Davg of algorithms.

HDDE60 DDE_LS60 HDPSO60 IG_LS60 vIG_DE30 IG_RIS30 VIG_FL30

50 10 0.20 0.42 0.43 0.49 0.03 0.04 0.08

20 0.29 0.28 0.60 0.55 �0.04 �0.06 0.04

30 0.25 0.33 1.00 0.78 �0.17 0.08 0.06

40 0.36 0.41 1.23 0.94 �0.41 0.13 �0.10

50 1.15 1.14 1.85 1.80 �0.16 1.06 0.56

100 10 0.10 0.13 0.24 0.14 0.04 0.06 0.06

20 0.09 0.31 0.62 0.52 �0.09 0.07 �0.04

30 0.50 0.51 1.18 1.05 �0.19 �0.17 0.05

40 0.07 0.33 1.48 1.21 �0.65 �0.41 �0.41

50 0.45 0.51 1.35 1.33 �0.12 0.35 0.49

150 10 0.01 0.01 0.02 0.01 0.00 0.00 0.01

20 0.43 0.41 0.54 0.54 0.05 0.04 0.12

30 0.14 0.23 0.57 0.61 �0.18 0.01 �0.07

40 0.25 0.51 1.31 1.09 �0.07 �0.05 0.29

50 0.17 0.18 1.33 0.90 �0.85 �0.46 �0.57

200 10 0.03 0.03 0.10 0.08 0.00 0.00 0.00

20 0.04 0.09 0.26 0.18 �0.07 0.00 �0.01

30 0.01 0.06 0.35 0.46 �0.31 �0.17 �0.20

40 0.10 0.15 0.58 0.60 �0.26 �0.29 �0.26

50 0.45 0.64 1.66 1.59 �0.40 �0.22 �0.17

250 10 0.00 0.00 0.04 0.02 �0.01 0.00 0.00

20 0.13 0.18 0.39 0.33 �0.03 0.05 0.02

30 0.00 0.19 0.61 0.41 �0.14 �0.06 �0.12

40 0.31 0.42 0.75 0.74 0.02 0.04 0.11

50 0.06 0.19 1.52 1.42 �0.60 �0.77 �0.49

300 10 0.00 0.00 0.00 0.01 0.00 0.00 0.00

20 0.12 0.15 0.29 0.30 0.00 �0.03 �0.03

30 0.30 0.38 0.60 0.69 0.02 0.00 �0.06

40 0.15 0.19 0.74 0.56 �0.34 �0.04 �0.17

50 0.10 0.24 1.19 1.12 �0.23 �0.27 �0.15

350 10 0.02 0.02 0.05 0.04 0.00 0.02 0.00

20 0.05 0.10 0.31 0.22 �0.01 0.02 �0.01

30 0.11 0.31 0.45 0.51 �0.05 �0.01 �0.08

40 0.31 0.37 0.79 0.89 0.08 0.02 0.01

50 0.18 0.33 1.26 0.95 �0.44 �0.40 �0.38

400 10 0.01 0.02 0.04 0.04 0.00 0.00 0.00

20 0.14 0.21 0.38 0.37 0.04 0.06 �0.01

30 0.23 0.26 0.51 0.47 0.11 0.05 �0.01

40 0.20 0.20 0.54 0.57 �0.04 �0.09 �0.15

50 0.08 0.11 0.74 0.71 �0.25 �0.28 �0.33

450 10 0.02 0.05 0.05 0.05 0.00 0.00 0.00

20 0.12 0.16 0.18 0.21 0.00 0.04 0.00

30 0.08 0.17 0.41 0.40 �0.03 �0.09 �0.12

40 0.01 0.14 0.34 0.37 �0.09 �0.02 �0.23

50 0.21 0.43 1.16 1.05 �0.07 �0.27 �0.43

500 10 0.01 0.01 0.03 0.02 0.00 0.01 0.00

20 0.04 0.08 0.14 0.16 �0.04 0.00 �0.04

30 0.13 0.16 0.24 0.27 0.08 0.04 0.00

40 0.13 0.32 0.63 0.55 0.10 �0.04 �0.04

50 0.17 0.35 0.71 0.66 �0.03 �0.08 �0.23

Overall Avg 0.16 0.25 0.64 0.58 �0.12 �0.04 �0.06

Ave

rage

Rel

ativ

e P

erce

nt D

evia

tion

VIG_FL30IG_RIS30vIG_DE30IG_LS60HDPSO60DDE_LS60HDDE60

0.75

0.50

0.25

0.00

95% CI for the MeanInterval Plot of Algorithms

Fig. 10. Internal plot of algorithms compared.


In Table 3, we provide the best makespan values found by thecompared algorithms. The makespan values for all instances werenot reported in [27]. As observed in Table 3, most of makespanvalues reported both in [3] and [27] are further improved by theproposed algorithms in this paper. Although the significantimprovements were achieved by half of their CPU time require-ment in Table 2 for the Davg values, we only provide the bestmakespan values of our peak runs for Tmax ¼ nðm=2Þ � t ms,where t¼60 in Table 3. Note that the best makespan values forthe HDDE, DDE_LS, HDPSO and IG_LS algorithms are obtainedthrough personal communication with the authors in [27]whereas the best values are taken directly from [3]. We furtheranalyze these results in Tables 4 to 8 in terms of the number ofimprovements (number of imp), number of equal (number ofequal) and number of worse (number of worse).

Table 4 shows the comparison of vIG_DE algorithms againstthe best in [3] and those in [27]. It can be observed that the

Table 3Makespan obtained by algorithms, t¼60.

Best HDDE60 DDE_LS60 HDPSO60 IG_LS60 vIG_DE60 IG_RIS60 VIG_FL60

I_7_50_10_1 4127 4127 4127 4127 4127 4127 4127 4127I_7_50_10_2 4283 4283 4283 4296 4291 4283 4283 4283I_7_50_10_3 3262 3263 3263 3281 3279 3262 3262 3262I_7_50_10_4 3219 3216 3231 3219 3224 3216 3219 3219

I_7_50_10_5 3470 3470 3470 3474 3473 3470 3471 3471

I_7_50_20_1 5647 5647 5647 5653 5647 5647 5647 5647I_7_50_20_2 5834 5820 5814 5830 5827 5818 5820 5820

I_7_50_20_3 5794 5793 5793 5794 5795 5793 5793 5793I_7_50_20_4 5803 5798 5803 5803 5804 5799 5798 5795I_7_50_20_5 4907 4881 4898 4918 4917 4884 4900 4897

I_7_50_30_1 7243 7256 7225 7260 7294 7223 7239 7239

I_7_50_30_2 7381 7351 7355 7400 7358 7351 7331 7330I_7_50_30_3 6902 6844 6887 6908 6880 6857 6900 6885

I_7_50_30_4 7624 7579 7590 7609 7611 7579 7580 7580

I_7_50_30_5 7340 7338 7365 7366 7378 7333 7366 7366

I_7_50_40_1 9264 9227 9169 9300 9266 9168 9167 9130I_7_50_40_2 10164 10116 10155 10181 10202 10137 10121 10117

I_7_50_40_3 9896 9791 9857 9980 9850 9782 9854 9836

I_7_50_40_4 9575 9607 9512 9613 9605 9523 9550 9533

I_7_50_40_5 9082 8967 8963 9037 9039 8968 8960 8957I_7_50_50_1 11652 11717 11744 11796 11703 11604 11753 11753

I_7_50_50_2 10946 10980 10942 10991 10985 10893 10942 10942

I_7_50_50_3 10960 10960 11020 10971 11085 10885 10955 10935

I_7_50_50_4 10026 10044 9970 10145 10211 9967 10030 10030

I_7_50_50_5 11380 11349 11386 11455 11494 11316 11365 11348

I_7_100_10_1 6575 6570 6570 6572 6575 6570 6575 6575

I_7_100_10_2 5798 5802 5803 5816 5802 5803 5808 5802

I_7_100_10_3 6533 6533 6533 6533 6533 6533 6533 6533I_7_100_10_4 6161 6171 6174 6174 6174 6158 6158 6158I_7_100_10_5 6654 6654 6654 6654 6654 6654 6654 6654I_7_100_20_1 8611 8606 8607 8671 8637 8606 8606 8606I_7_100_20_2 8223 8218 8218 8256 8272 8224 8241 8241

I_7_100_20_3 9057 9057 9059 9079 9057 9043 9055 9043I_7_100_20_4 9031 9029 9029 9029 9032 8972 8973 8970I_7_100_20_5 9126 9125 9117 9127 9124 9109 9109 9109I_7_100_30_1 11249 11228 11266 11305 11305 11210 11210 11202I_7_100_30_2 10989 10943 10972 10974 10988 10938 10938 10938I_7_100_30_3 10666 10674 10587 10735 10733 10571 10555 10555I_7_100_30_4 11175 11137 11169 11274 11218 11103 11097 11097I_7_100_30_5 11030 11065 11046 11088 11088 10983 10996 10985

I_7_100_40_1 12806 12721 12796 12813 12767 12606 12640 12640

I_7_100_40_2 13306 13295 13291 13404 13426 13117 13206 13202

I_7_100_40_3 12654 12574 12597 12726 12781 12488 12528 12504

I_7_100_40_4 12044 11934 11853 12016 12146 11781 11857 11829

I_7_100_40_5 12934 12911 12959 13000 12980 12920 12919 12919

I_7_100_50_1 16111 16035 16142 16173 16248 16019 16057 16050

I_7_100_50_2 15019 14800 14898 15100 15039 14787 14989 14916

I_7_100_50_3 17755 17798 17786 17844 17833 17585 17686 17650

I_7_100_50_4 16672 16703 16662 16752 16721 16684 16692 16667

I_7_100_50_5 14827 14948 14841 15044 14867 14802 14920 14854

I_7_150_10_1 10404 10404 10404 10404 10404 10404 10404 10404I_7_150_10_2 8824 8824 8826 8826 8826 8824 8826 8824I_7_150_10_3 9180 9180 9180 9180 9180 9180 9180 9181

I_7_150_10_4 10032 10032 10032 10032 10032 10032 10032 10032I_7_150_10_5 9870 9870 9870 9870 9870 9866 9870 9870

I_7_150_20_1 10768 10823 10826 10861 10873 10758 10800 10790

I_7_150_20_2 11718 11725 11725 11725 11720 11699 11704 11699I_7_150_20_3 12063 12058 12060 12083 12076 12046 12063 12060

I_7_150_20_4 10965 11001 10960 10975 10997 10936 10933 10933I_7_150_20_5 13210 13210 13210 13210 13223 13210 13210 13210I_7_150_30_1 15569 15548 15540 15592 15583 15505 15500 15500I_7_150_30_2 13747 13719 13738 13786 13766 13667 13699 13667I_7_150_30_3 14688 14688 14664 14778 14788 14651 14673 14651I_7_150_30_4 14627 14574 14555 14593 14609 14549 14549 14549I_7_150_30_5 15257 15259 15285 15290 15328 15265 15277 15277

I_7_150_40_1 16217 16114 16126 16267 16240 16025 16135 16120

I_7_150_40_2 18235 18238 18204 18264 18281 18122 18218 18173

I_7_150_40_3 16416 16434 16391 16551 16598 16356 16419 16381

I_7_150_40_4 14658 14647 14659 14874 14819 14648 14651 14640I_7_150_40_5 17298 17260 17318 17244 17337 17246 17288 17244I_7_150_50_1 20625 20388 20462 20703 20538 20364 20371 20367

I_7_150_50_2 19512 19389 19374 19514 19455 19121 19247 19227

I_7_150_50_3 19702 19655 19711 19857 19893 19447 19476 19358I_7_150_50_4 20355 20166 20329 20469 20388 20139 20237 20224

I_7_150_50_5 19611 19342 19424 19704 19644 19308 19429 19418


Table 3 (continued )


I_7_200_10_1 12155 12155 12155 12156 12155 12155 12155 12155I_7_200_10_2 12227 12227 12227 12227 12227 12227 12227 12227I_7_200_10_3 12595 12595 12595 12595 12595 12595 12595 12595I_7_200_10_4 12304 12301 12301 12301 12304 12301 12301 12301I_7_200_10_5 12076 12076 12076 12093 12086 12076 12076 12076I_7_200_20_1 14864 14877 14885 14927 14891 14864 14864 14864I_7_200_20_2 14134 14093 14095 14095 14095 14095 14126 14086I_7_200_20_3 16135 16115 16115 16133 16127 16115 16115 16115I_7_200_20_4 15972 15972 15972 15972 15972 15972 15972 15972I_7_200_20_5 14225 14214 14230 14181 14236 14175 14211 14211

I_7_200_30_1 17222 17116 17138 17200 17083 17053 17044 17044I_7_200_30_2 17126 16972 16991 16978 16988 16994 17019 17014

I_7_200_30_3 17529 17501 17516 17524 17621 17428 17428 17420I_7_200_30_4 20032 20020 20011 20005 20032 19991 19986 19986I_7_200_30_5 17995 18077 18049 18111 18067 17974 17988 17982

I_7_200_40_1 20128 20024 20019 20187 20103 19965 19953 19953I_7_200_40_2 21801 21743 21759 21830 21811 21724 21797 21763

I_7_200_40_3 20609 20642 20668 20724 20712 20564 20629 20624

I_7_200_40_4 17864 17624 17677 17734 17746 17507 17628 17548

I_7_200_40_5 21258 21234 21249 21269 21359 21237 21216 21209I_7_200_50_1 22912 22959 22865 23193 23217 22729 22632 22580I_7_200_50_2 23664 23631 23600 23742 23923 23488 23528 23509

I_7_200_50_3 22615 22561 22670 22817 22927 22431 22471 22438

I_7_200_50_4 24140 24221 24146 24460 24342 23969 24092 24022

I_7_200_50_5 24424 24334 24408 24578 24496 24275 24374 24355

I_7_250_10_1 16640 16640 16640 16640 16640 16639 16640 16640

I_7_250_10_2 15483 15476 15476 15476 15476 15476 15476 15476I_7_250_10_3 14872 14872 14872 14872 14872 14872 14872 14872I_7_250_10_4 15247 15248 15247 15250 15248 15247 15250 15250

I_7_250_10_5 15026 15026 15026 15026 15026 15026 15026 15026I_7_250_20_1 17613 17633 17622 17622 17631 17577 17583 17578

I_7_250_20_2 17692 17684 17684 17713 17698 17683 17683 17683I_7_250_20_3 17534 17543 17562 17559 17567 17500 17522 17487I_7_250_20_4 17651 17646 17646 17673 17666 17645 17647 17647

I_7_250_20_5 17274 17297 17296 17298 17298 17277 17277 17277

I_7_250_30_1 21920 21920 21920 21970 21928 21920 21920 21920I_7_250_30_2 21946 21876 21994 22155 22148 21853 21890 21884

I_7_250_30_3 20196 20096 20122 20124 20142 20111 20107 20077I_7_250_30_4 19886 19807 19851 19887 19885 19794 19821 19809

I_7_250_30_5 20991 20910 20911 20987 20997 20906 20909 20907

I_7_250_40_1 22871 22870 22874 23045 23075 22820 22957 22899

I_7_250_40_2 24193 24247 24241 24271 24287 24119 24152 24149

I_7_250_40_3 24412 24482 24502 24556 24567 24353 24394 24394

I_7_250_40_4 24913 24876 24858 24904 24909 24748 24824 24824

I_7_250_40_5 23536 23562 23571 23602 23631 23506 23478 23471I_7_250_50_1 28662 28898 28807 29126 29143 28563 28610 28599

I_7_250_50_2 24932 24930 24864 25272 25239 24577 24770 24480I_7_250_50_3 26973 26678 26811 27145 27264 26512 26563 26528

I_7_250_50_4 25971 25565 25577 25927 25861 25611 25612 25608

I_7_250_50_5 27670 27542 27511 27635 27655 27389 27393 27393

I_7_300_10_1 17498 17498 17498 17498 17498 17498 17498 17498I_7_300_10_2 17350 17350 17350 17350 17350 17350 17351 17351

I_7_300_10_3 18627 18627 18627 18627 18628 18627 18628 18627I_7_300_10_4 16941 16941 16941 16941 16941 16941 16941 16941I_7_300_10_5 17524 17524 17524 17524 17524 17521 17524 17524

I_7_300_20_1 18873 18884 18876 18885 18929 18837 18870 18859

I_7_300_20_2 22032 22035 22035 22056 22050 22032 22032 22032I_7_300_20_3 20268 20278 20284 20302 20309 20230 20235 20235

I_7_300_20_4 19516 19497 19484 19514 19515 19490 19484 19484I_7_300_20_5 20705 20707 20716 20732 20726 20705 20705 20705I_7_300_30_1 26565 26529 26562 26644 26568 26501 26530 26530

I_7_300_30_2 24290 24350 24331 24398 24408 24267 24261 24261I_7_300_30_3 24386 24381 24423 24456 24466 24370 24369 24369I_7_300_30_4 23728 23761 23742 23785 23823 23710 23727 23726

I_7_300_30_5 22638 22630 22649 22632 22680 22568 22558 22558I_7_300_40_1 26817 26762 26731 26956 26891 26599 26640 26639

I_7_300_40_2 29332 29316 29359 29478 29474 29158 29175 29173

I_7_300_40_3 25534 25391 25382 25591 25518 25362 25598 25565

I_7_300_40_4 27648 27565 27546 27648 27660 27479 27498 27498

I_7_300_40_5 28872 28812 28873 29047 28976 28760 28810 28810

I_7_300_50_1 31860 31755 31873 32129 32445 31667 31775 31683

I_7_300_50_2 29834 29515 29665 29930 30017 29490 29676 29657

I_7_300_50_3 30867 30892 30851 31034 30937 30731 30747 30747

I_7_300_50_4 32402 32405 32497 32695 32681 32265 32350 32346

I_7_300_50_5 29512 29359 29205 29532 29265 29051 29239 29239

I_7_350_10_1 19302 19300 19302 19302 19302 19302 19302 19302




I_7_350_10_2 21319 21319 21319 21320 21320 21316 21319 21320

I_7_350_10_3 21330 21330 21331 21331 21331 21330 21330 21330I_7_350_10_4 21759 21759 21759 21759 21759 21759 21759 21759I_7_350_10_5 20591 20591 20591 20591 20591 20591 20591 20591I_7_350_20_1 25417 25417 25417 25417 25417 25413 25415 25413I_7_350_20_2 27185 27185 27185 27207 27185 27185 27185 27185I_7_350_20_3 22906 22906 22899 22951 22965 22880 22907 22907

I_7_350_20_4 22994 22985 22975 23098 23077 22968 22971 22970

I_7_350_20_5 22778 22750 22750 22762 22751 22746 22750 22750

I_7_350_30_1 25382 25393 25449 25518 25609 25226 25355 25275

I_7_350_30_2 27773 27812 27765 27834 27864 27744 27740 27740I_7_350_30_3 27775 27673 27674 27704 27690 27653 27657 27657

I_7_350_30_4 29358 29306 29305 29379 29347 29295 29295 29295

I_7_350_30_5 25240 25209 25268 25346 25325 25230 25221 25211

I_7_350_40_1 29381 29282 29324 29507 29569 29182 29241 29241

I_7_350_40_2 29163 29239 29154 29397 29366 29043 29024 29024I_7_350_40_3 36287 36368 36368 36392 36405 36247 36253 36249

I_7_350_40_4 34788 34744 34759 34810 34836 34644 34677 34669

I_7_350_40_5 29847 29905 29966 30078 30035 29840 29879 29814I_7_350_50_1 32559 32375 32768 33071 33023 32144 32435 32178

I_7_350_50_2 33454 33167 33276 33464 33516 32911 33042 33032

I_7_350_50_3 34982 34908 34903 35294 35138 34718 34735 34697I_7_350_50_4 37210 37081 37161 37424 37369 37009 37031 36996I_7_350_50_5 35710 35588 35686 35758 35548 35390 35349 35348I_7_400_10_1 25244 25238 25245 25255 25245 25238 25238 25238I_7_400_10_2 23001 23001 23001 23001 23001 23001 23001 23001I_7_400_10_3 23665 23665 23665 23665 23665 23665 23665 23665I_7_400_10_4 23275 23277 23277 23277 23277 23275 23275 23275I_7_400_10_5 21956 21956 21956 21959 21956 21956 21956 21956I_7_400_20_1 27704 27696 27704 27704 27704 27686 27686 27686I_7_400_20_2 28092 28092 28128 28123 28149 28088 28092 28092

I_7_400_20_3 26254 26274 26351 26389 26382 26224 26224 26224I_7_400_20_4 25164 25177 25188 25282 25244 25155 25168 25168

I_7_400_20_5 24753 24711 24716 24742 24760 24688 24707 24702

I_7_400_30_1 29487 29473 29468 29486 29502 29405 29446 29444

I_7_400_30_2 29295 29296 29341 29349 29395 29217 29274 29241

I_7_400_30_3 28725 28793 28703 28873 28857 28733 28691 28633I_7_400_30_4 31324 31381 31378 31371 31405 31279 31281 31278I_7_400_30_5 34543 34579 34574 34600 34593 34535 34534 34533I_7_400_40_1 37552 37587 37540 37714 37546 37440 37498 37464

I_7_400_40_2 33936 33912 33889 33938 33924 33805 33839 33839

I_7_400_40_3 34548 34498 34545 34618 34692 34482 34499 34491

I_7_400_40_4 35406 35403 35374 35531 35450 35306 35262 35245I_7_400_40_5 32853 32936 32857 32901 32862 32739 32777 32776

I_7_400_50_1 38000 38006 37938 38251 38152 37825 37871 37827

I_7_400_50_2 38411 38342 38336 38460 38446 38237 38275 38275

I_7_400_50_3 38337 38122 38154 38372 38259 37880 37930 37912

I_7_400_50_4 40578 40521 40572 40863 40683 40465 40523 40480

I_7_400_50_5 35968 35921 35761 36158 36167 35516 35628 35527

I_7_450_10_1 23989 23989 23994 23989 23989 23987 23987 23987I_7_450_10_2 26277 26277 26277 26277 26277 26277 26277 26277I_7_450_10_3 25849 25849 25849 25849 25849 25849 25849 25849I_7_450_10_4 26910 26911 26910 26911 26910 26910 26911 26910I_7_450_10_5 25191 25191 25191 25191 25191 25191 25191 25191I_7_450_20_1 27533 27522 27521 27524 27524 27514 27527 27521

I_7_450_20_2 27924 27930 27924 27933 27933 27924 27930 27924I_7_450_20_3 28811 28808 28821 28816 28874 28770 28772 28769I_7_450_20_4 28485 28494 28461 28524 28508 28446 28459 28446I_7_450_20_5 28543 28585 28589 28623 28625 28541 28541 28539I_7_450_30_1 35197 35222 35224 35290 35262 35127 35123 35123I_7_450_30_2 32639 32544 32584 32687 32675 32517 32565 32561

I_7_450_30_3 32131 32057 32013 32148 32174 32021 32036 32001I_7_450_30_4 33769 33754 33734 33831 33736 33700 33731 33731

I_7_450_30_5 33732 33755 33789 33853 33953 33667 33664 33658I_7_450_40_1 39797 39677 39641 39659 39668 39562 39596 39557I_7_450_40_2 36174 36137 36228 36311 36333 36020 36050 36050

I_7_450_40_3 37917 37794 37909 37895 37877 37811 37829 37829

I_7_450_40_4 37774 37770 37785 37821 37878 37606 37610 37596I_7_450_40_5 35870 35773 35877 35965 35966 35712 35731 35731

I_7_450_50_1 37913 37847 37976 38396 38382 37563 37442 37442I_7_450_50_2 43449 43503 43559 43702 43735 43392 43447 43431

I_7_450_50_3 44275 44157 44250 44421 44387 44087 44010 44010I_7_450_50_4 41295 41421 41598 41945 41813 41036 41044 41044

I_7_450_50_5 41348 41180 41207 41292 41386 40923 41047 41044

I_7_500_10_1 28839 28840 28840 28839 28840 28839 28840 28840

I_7_500_10_2 27924 27923 27924 27923 27923 27923 27923 27923




I_7_500_10_3 27349 27349 27349 27349 27349 27349 27349 27349I_7_500_10_4 27575 27575 27575 27575 27575 27575 27575 27575I_7_500_10_5 27457 27457 27457 27457 27457 27457 27457 27457I_7_500_20_1 35973 35948 35948 35948 35954 35948 35948 35948I_7_500_20_2 34134 34129 34134 34174 34196 34129 34129 34129I_7_500_20_3 31114 31102 31123 31136 31134 31066 31069 31065I_7_500_20_4 30916 30905 30912 30929 30944 30900 30900 30900I_7_500_20_5 33776 33782 33776 33791 33795 33768 33768 33768I_7_500_30_1 36381 36417 36413 36438 36459 36337 36345 36345

I_7_500_30_2 39381 39367 39374 39365 39384 39357 39357 39356I_7_500_30_3 39261 39290 39311 39297 39300 39226 39256 39250

I_7_500_30_4 34003 33972 34000 34069 34049 33918 33942 33927

I_7_500_30_5 38368 38390 38372 38370 38382 38340 38353 38348

I_7_500_40_1 40793 40768 40834 40868 40836 40708 40732 40685I_7_500_40_2 44170 44181 44172 44202 44195 44099 44144 44144

I_7_500_40_3 40523 40485 40537 40592 40659 40366 40357 40357I_7_500_40_4 41992 42094 42198 42253 42296 41917 41886 41886I_7_500_40_5 36448 36343 36386 36704 36548 36312 36449 36351

I_7_500_50_1 46461 46331 46459 46555 46524 46238 46268 46263

I_7_500_50_2 43461 43552 43538 43658 43662 43281 43379 43379

I_7_500_50_3 45484 45409 45394 45590 45756 45206 45164 45164I_7_500_50_4 42620 42687 42781 42744 42775 42417 42454 42454

I_7_500_50_5 43346 43224 43277 43563 43399 43145 43061 43048

Table 4Comparison of vIG_DE with competing algorithms.

No of imp No of equal No of worse

Best 198 46 6

HDDE 183 49 18

DDE_LS 194 46 10

HDPSO 215 33 2

IG_LS 215 33 2

Table 5Comparison of our algorithms with HDDE.


vIG_DE 183 49 18

IG_RIS 146 52 52

VIG_FL 160 50 40

Table 6Comparison of our algorithms with DDE_LS.


vIG_DE 194 46 10

IG_RIS 164 47 39

VIG_FL 178 48 24

Table 7Comparison of our algorithms with HDPSO.


vIG_DE 215 33 2

IG_MFT 203 37 10

VIG_FL 207 37 6

Table 8Comparison of our algorithms with IG_LS.


vIG_DE 215 33 2

IG_MFT 201 37 12

VIG_FL 206 37 7


vIG_DE algorithm was able to further improve 198 out of 250solutions reported in [3], whereas it was able to further improve183, 194, 215 and 215 out of the 250 best-known solutionsprovided in [27]. The best algorithm in [27] was the HDDE. It isclear that the vIG_DE algorithm was able to further improve 183out of 250 solutions provided in [27].

Table 5 shows the comparison of our algorithms against theHDDE algorithm in [27]. It can be observed that the vIG_DE,IG_RIS and VIG_FL algorithms were able to further improve 183,146 and 160 out of the 250 best-known solutions provided in[27], respectively.

Table 6 shows the comparison of our algorithms against theDDE_LS algorithm in [27]. It can be observed that the vIG_DE,IG_RIS and VIG_FL algorithms were able to further improve 194,164 and 178 out of the 250 best-known solutions provided in[27], respectively.

Table 7 shows the comparison of our algorithms against theHDPSO algorithm in [27]. It can be observed that the vIG_DE,IG_RIS and VIG_FL algorithms were able to further improve 215,203 and 207, respectively, out of the 250 best-known solutionsprovided in [27].

Table 8 shows the comparison of our algorithms against theIG_LS algorithm in [27]. It can be observed that the vIG_DE, IG_RISand VIG_FL algorithms were able to further improve 215, 201 and206, respectively, out of the 250 best-known solutions providedin [27].

All of these analyses clearly show that the proposed algo-rithms are superior to those presented both in [3] and [27]. Inaddition to the makespan criterion, we provide the total flowtime

values as the best known solutions in Table 9 for furtherresearchers to test their algorithms with Tmax ¼ nðm=2Þ � t ms,where t¼60.

Table 9Best-known solutions of vIG_DE algorithm for total flowtime criterion, t¼60.

Instance Best Instance Best Instance Best Instance Best Instance Best

I_7_50_10_1 125,380 I_7_150_10_1 895,229 I_7_250_10_1 2385,078 I_7_350_10_1 3512,173 I_7_450_10_1 5533,817

I_7_50_10_2 131,385 I_7_150_10_2 748,122 I_7_250_10_2 2106,167 I_7_350_10_2 4264,790 I_7_450_10_2 6131,390

I_7_50_10_3 106,832 I_7_150_10_3 766,768 I_7_250_10_3 2036,196 I_7_350_10_3 3838,239 I_7_450_10_3 5974,770

I_7_50_10_4 101,154 I_7_150_10_4 870,653 I_7_250_10_4 2016,989 I_7_350_10_4 4285,421 I_7_450_10_4 6647,632

I_7_50_10_5 107,635 I_7_150_10_5 855,290 I_7_250_10_5 1966,408 I_7_350_10_5 3807,928 I_7_450_10_5 5673,018

I_7_50_20_1 223,411 I_7_150_20_1 1066,328 I_7_250_20_1 2799,970 I_7_350_20_1 5523,837 I_7_450_20_1 7248,737

I_7_50_20_2 224,312 I_7_150_20_2 1103,487 I_7_250_20_2 2699,738 I_7_350_20_2 5989,808 I_7_450_20_2 7356,184

I_7_50_20_3 226,144 I_7_150_20_3 1175,110 I_7_250_20_3 2704,526 I_7_350_20_3 4776,536 I_7_450_20_3 7860,688

I_7_50_20_4 227,380 I_7_150_20_4 1068,552 I_7_250_20_4 2765,178 I_7_350_20_4 4959,059 I_7_450_20_4 7518,296

I_7_50_20_5 178,918 I_7_150_20_5 1390,521 I_7_250_20_5 2655,471 I_7_350_20_5 4842,744 I_7_450_20_5 7474,315

I_7_50_30_1 298,825 I_7_150_30_1 1617,787 I_7_250_30_1 3864,142 I_7_350_30_1 5979,224 I_7_450_30_1 10,270,198

I_7_50_30_2 298,813 I_7_150_30_2 1474,905 I_7_250_30_2 3542,779 I_7_350_30_2 6507,562 I_7_450_30_2 9209,464

I_7_50_30_3 292,663 I_7_150_30_3 1647,655 I_7_250_30_3 3428,481 I_7_350_30_3 6469,955 I_7_450_30_3 9397,035

I_7_50_30_4 319,430 I_7_150_30_4 1575,960 I_7_250_30_4 3370,868 I_7_350_30_4 7015,586 I_7_450_30_4 9790,031

I_7_50_30_5 299,626 I_7_150_30_5 1640,144 I_7_250_30_5 3702,931 I_7_350_30_5 5800,519 I_7_450_30_5 9450,992

I_7_50_40_1 395,014 I_7_150_40_1 1856,681 I_7_250_40_1 4158,216 I_7_350_40_1 6954,417 I_7_450_40_1 12,610,909

I_7_50_40_2 444,201 I_7_150_40_2 2050,045 I_7_250_40_2 4271,224 I_7_350_40_2 7141,816 I_7_450_40_2 11,153,309

I_7_50_40_3 416,128 I_7_150_40_3 1896,527 I_7_250_40_3 4457,779 I_7_350_40_3 9289,344 I_7_450_40_3 11,798,619

I_7_50_40_4 405,692 I_7_150_40_4 1682,298 I_7_250_40_4 4580,109 I_7_350_40_4 9052,676 I_7_450_40_4 11,798,390

I_7_50_40_5 389,638 I_7_150_40_5 2026,015 I_7_250_40_5 4243,709 I_7_350_40_5 7352,826 I_7_450_40_5 10,912,158

I_7_50_50_1 512,272 I_7_150_50_1 2516,185 I_7_250_50_1 5570,902 I_7_350_50_1 8367,007 I_7_450_50_1 12,011,731

I_7_50_50_2 475,389 I_7_150_50_2 2300,230 I_7_250_50_2 4724,396 I_7_350_50_2 8674,513 I_7_450_50_2 14,690,680

I_7_50_50_3 493,319 I_7_150_50_3 2329,927 I_7_250_50_3 5150,808 I_7_350_50_3 9133,211 I_7_450_50_3 14,564,560

I_7_50_50_4 429,130 I_7_150_50_4 2449,533 I_7_250_50_4 4808,325 I_7_350_50_4 9898,786 I_7_450_50_4 13,652,052

I_7_50_50_5 507,519 I_7_150_50_5 2354,078 I_7_250_50_5 5166,847 I_7_350_50_5 9275,077 I_7_450_50_5 13,590,028

I_7_100_10_1 371,774 I_7_200_10_1 1358,622 I_7_300_10_1 2733,950 I_7_400_10_1 5391,735 I_7_500_10_1 7598,974

I_7_100_10_2 343,116 I_7_200_10_2 1377,316 I_7_300_10_2 2707,337 I_7_400_10_2 4599,658 I_7_500_10_2 7102,394

I_7_100_10_3 384,173 I_7_200_10_3 1404,810 I_7_300_10_3 3103,430 I_7_400_10_3 5166,180 I_7_500_10_3 6957,682

I_7_100_10_4 338,206 I_7_200_10_4 1366,985 I_7_300_10_4 2723,606 I_7_400_10_4 4928,345 I_7_500_10_4 7062,995

I_7_100_10_5 376,015 I_7_200_10_5 1339,260 I_7_300_10_5 2884,897 I_7_400_10_5 4511,915 I_7_500_10_5 7290,714

I_7_100_20_1 587,306 I_7_200_20_1 1865,167 I_7_300_20_1 3392,341 I_7_400_20_1 6702,405 I_7_500_20_1 10,777,501

I_7_100_20_2 572,831 I_7_200_20_2 1720,128 I_7_300_20_2 4025,018 I_7_400_20_2 7101,551 I_7_500_20_2 9682,988

I_7_100_20_3 624,842 I_7_200_20_3 2094,286 I_7_300_20_3 3667,443 I_7_400_20_3 6275,051 I_7_500_20_3 9058,512

I_7_100_20_4 613,650 I_7_200_20_4 2152,333 I_7_300_20_4 3563,901 I_7_400_20_4 5845,455 I_7_500_20_4 8726,791

I_7_100_20_5 652,339 I_7_200_20_5 1840,898 I_7_300_20_5 3794,552 I_7_400_20_5 5836,249 I_7_500_20_5 10,223,564

I_7_100_30_1 871,063 I_7_200_30_1 2392,036 I_7_300_30_1 5417,342 I_7_400_30_1 7765,965 I_7_500_30_1 11,648,780

I_7_100_30_2 804,380 I_7_200_30_2 2321,316 I_7_300_30_2 4857,488 I_7_400_30_2 7412,412 I_7_500_30_2 12,888,556

I_7_100_30_3 789,281 I_7_200_30_3 2422,092 I_7_300_30_3 5076,261 I_7_400_30_3 7214,769 I_7_500_30_3 13,051,895

I_7_100_30_4 836,288 I_7_200_30_4 2863,910 I_7_300_30_4 4822,848 I_7_400_30_4 8505,084 I_7_500_30_4 10,403,033

I_7_100_30_5 822,142 I_7_200_30_5 2511,521 I_7_300_30_5 4429,028 I_7_400_30_5 9358,018 I_7_500_30_5 12,323,304

I_7_100_40_1 1015,413 I_7_200_40_1 3020,448 I_7_300_40_1 5670,453 I_7_400_40_1 10,803,267 I_7_500_40_1 14,019,665

I_7_100_40_2 1046,051 I_7_200_40_2 3182,705 I_7_300_40_2 6349,566 I_7_400_40_2 9479,914 I_7_500_40_2 15,153,155

I_7_100_40_3 989,818 I_7_200_40_3 3144,740 I_7_300_40_3 5389,312 I_7_400_40_3 9788,011 I_7_500_40_3 13,706,925

I_7_100_40_4 945,786 I_7_200_40_4 2550,159 I_7_300_40_4 6088,014 I_7_400_40_4 9861,591 I_7_500_40_4 14,577,131

I_7_100_40_5 1053,204 I_7_200_40_5 3179,470 I_7_300_40_5 6431,843 I_7_400_40_5 9088,311 I_7_500_40_5 11,902,207

I_7_100_50_1 1328,844 I_7_200_50_1 3535,086 I_7_300_50_1 7187,544 I_7_400_50_1 11,166,074 I_7_500_50_1 16,633,031

I_7_100_50_2 1231,619 I_7_200_50_2 3753,945 I_7_300_50_2 6705,587 I_7_400_50_2 11,237,488 I_7_500_50_2 15,315,588

I_7_100_50_3 1486,911 I_7_200_50_3 3469,069 I_7_300_50_3 6966,348 I_7_400_50_3 10,887,948 I_7_500_50_3 16,252,743

I_7_100_50_4 1411,469 I_7_200_50_4 3758,949 I_7_300_50_4 7339,019 I_7_400_50_4 12,045,944 I_7_500_50_4 15,497,661

I_7_100_50_5 1241,533 I_7_200_50_5 3793,772 I_7_300_50_5 6557,870 I_7_400_50_5 10,494,339 I_7_500_50_5 15,283,675


5. Conclusions

In this paper, we present a variable iterated greedy algorithmin which the parameters (the destruction size to be used in thedestruction–construction phase of iterated greedy (IG) algorithmand the probability of applying the iterated greedy algorithm toan individual) are optimized by the differential evolution algo-rithm. Furthermore, the traditional IG and a variable IG from theliterature are re-implemented as well. A unique multi-vectorchromosome representation is presented in such a way that thefirst vector represents the destruction size and the probability,which is a DE vector, whereas the second vector simply consists ofa job permutation assigned to each individual in the targetpopulation. The proposed algorithms are applied to the no-idlepermutation flowshop scheduling (NIPFS) problem with themakespan and total flowtime criteria. The performances of theproposed algorithms are tested on the Ruben Ruiz benchmarksuite and compared to the best-known solutions available athttp://soa.iti.es/rruiz as well as solutions from a recent discrete

differential evolution algorithm (HDDE) in the literature. Thecomputational results show that these algorithms are state-of-the-art methods for the NIPFS problem. Ultimately, 198 out of 250instances were further improved. In comparison to the recentHDDE algorithm, 183 out of 250 solutions provided were alsofurther improved by significant margins.

In future work, multi-objective algorithms will be developedto handle the no-idle permutation flowshop scheduling problemwith some conflicting performance measures.

Acknowledgments

M. Fatih Tasgetiren acknowledges the support provided by theTUBITAK (The Scientific and Technological Research Council ofTurkey) under grant #110M622. In addition, this research ispartially supported by National Science Foundation of Chinaunder Grant 61174187.



Appendix A. An example of the speed-up procedure

Job j
Machine k
1
2 3
p1k
3 3 2 p2k 4 1 3 p3k 2 3 3 p4k 2 2 3
1.
Suppose that the permutation is given by p¼{1,2,3,4},Let t¼1.
2.
Let Dp¼{2,3,4} be a partial permutation generated by removingjob 1 from p. Then
a. Calculate FðDpEj ,k,kþ1Þ, EðDpF

j ,k,kþ1Þ, and FðDpFj ,k,kþ1Þ,

respectively.FðDpE

1,1,2Þ ¼ 1; FðDpE1,2,3Þ ¼ 3; FðDpE

2,1,2Þ ¼ 3; FðDpE2,

2,3Þ ¼ 3; FðDpE3,1,2Þ ¼ 3; FðDpE

3,2,3Þ ¼ 4;EðDpF

3,1,2Þ ¼ 2; EðDpF3,2,3Þ ¼ 2; EðDpF

2,1,2Þ ¼ 2; EðDpF2,

2,3Þ ¼ 3; EðDpF1,1,2Þ ¼ 5; EðDpF

1,2,3Þ ¼ 1;FðDpF

3,1,2Þ ¼ 2; FðDpF3,2,3Þ ¼ 3; FðDpF

2,1,2Þ ¼ 3;FðDpF

2,2,3Þ ¼ 4; EðDpF1,1,2Þ ¼ 3; EðDpF

1,2,3Þ ¼ 4b. Let FðDpF

4,k,kþ1Þ ¼ EðDpF4,k,kþ1Þ ¼ 0, k¼1,2

3.
Repeat the following steps until all possible positions h ofDp¼{p1,p2,..,pn�1} are considered such that hA{1,2,...,n}4he{t,t�1}.
Case 1. Insert job 1 into position 2 of Dp:

a. Let DpE2 ¼DpE

1 [ pt ¼ f2,1g, then FðDpE2,1,2Þ ¼ 3

FðDpE2,2,3Þ ¼ 2

b. Let p¼DpE2 [ DpF

2 ¼ f2,1,3,4g then

Fðp,1,2Þ ¼max FðDpE2,1,2Þ�EðDpF

2,1,2Þ,0� �

þFðDpF2,1,2Þ ¼maxf3�2,0gþ3¼ 4:

Fðp,2,3Þ ¼max FðDpE2,2,3Þ�EðDpF

2,2,3Þ,0� �

þFðDpF2,2,3Þ ¼maxf2�3,0gþ4¼ 4

c. The completion time (makespan) of job p4 on machine3 is

Cðp4,3Þ ¼ Fðp,1,2ÞþFðp,2,3Þþ11¼ 19

d. The completion time of job p3 on machine 3 is

Cðp3,3Þ ¼ Cðp4,3Þ�pðp4,3Þ ¼ 19�3¼ 16,

e. The completion time of job p2 on machine 3 is

Cðp2,3Þ ¼ Cðp3,3Þ�pðp3,3Þ ¼ 16�3¼ 13

f. The completion time of job p1 on machine 3 is

Cðp1,3Þ ¼ Cðp2,3Þ�pðp2,3Þ ¼ 13�2¼ 11


a. let DpE3 ¼DpE

2 [ pt ¼ f2,3,1g, then FðDpE3,1,2Þ ¼ 3

FðDpE3,2,3Þ ¼ 2

b. Let p¼DpE3 [ DpF

3 ¼ f2,3,1,4g then

Fðp,1,2Þ ¼maxfFðDpE3,1,2Þ�EðDpF

3,1,2Þ,0g



3,2,3Þ,0gþFðDpF3,2,3Þ

¼maxf2�2,0gþ3¼ 3

c. The completion time of job p4 on machine 3 is



Cðp3,3Þ ¼ Cðp4,3Þ�pðp4,3Þ ¼ 17�3¼ 14,


Cðp2,3Þ ¼ Cðp3,3Þ�pðp3,3Þ ¼ 14�2¼ 12


Cðp1,3Þ ¼ Cðp2,3Þ�pðp2,3Þ ¼ 12�3¼ 9


a. Let DpE4 ¼DpE

3 [ pt ¼ f2,3,4,1g, then FðDpE4,1,2Þ ¼ 3

FðDpE4,2,3Þ ¼ 3

b. Let p¼DpE4 [DpF

4 ¼ f2,3,4,1gthen


4,1,2Þ,0g



4,2,3Þ,0g


c. The completion time of job p4 on machine 3 is



Cðp3,3Þ ¼ Cðp4,3Þ�pðp4,3Þ ¼ 17�2¼ 15,


Cðp2,3Þ ¼ Cðp3,3Þ�pðp3,3Þ ¼ 15�3¼ 12


Cðp1,3Þ ¼ Cðp2,3Þ�pðp2,3Þ ¼ 12�3¼ 9

4.
Set t¼tþ1. If t4n then stop; otherwise go back to step 2.
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