An electromagnetism-inspired method for a generalized flowshop problem

Transcription

1 Manufacturing Rev. 214, 1, 13 Ó M. Khalili, Publishe by EDP Sciences, 214 DOI: 1.151/mfreview/21413 Available online at: RESEARCH ARTICLE OPEN ACCESS An electromagnetism-inspire metho for a generalize flowshop problem Maji Khalili * Department of Inustrial Engineering, Islamic Aza University, Karaj Branch, Karaj, Iran Receive 3 November 213 / Accepte 17 August 214 Abstract There are two common simplifying assumptions mae in the scheuling of operations. One is that machines are available continuously, an the other is that once a job completes one process it is immeiately available for its next. Moreover, in flowshops, it is commonly assume that jobs must visit all machines. In practice, however, these assumptions may not be justifiable in some applications. This paper eals with a generalize flow shop (GFS) problem with machine availability constraints, transportation times between stages, an machine skipping. The electromagnetism-like metho (EM) has been successfully applie to some NP-har problems an this has motivate us to apply an assess the effectiveness of an EM algorithm in the GFS scenarios. Simulate annealing (SA) an a number of other well-recognize heuristics to the given GFS scheuling problem that minimizes two inepenent objective functions, namely the total tariness an the total completion time also has been applie. In orer to evaluate the performance of the propose EM an SA, a set of practical instances has been consiere. The relate results are analyze in two parts; in term of the objective functions, an the observe effects on variables in our instances. Extensive experiments an statistical analyses emonstrate that our propose EM is more efficient than SA an other heuristics with regars to the objective functions consiere in this paper. Key wors: Generalize flowshop scheuling, Transportation times, Machine availability, Electromagnetism-like metho 1. Introuction Prouction scheuling is efine as: etermining the process sequence by which a given set of parts are processe on a certain number of machines in orer to satisfy some performance measures uner the prouction constraints. Through scheuling, all parts in a manufacturing system can be processe by passing through each machine accoring to a preetermine sequence. Prouction scheuling evelops into a very active an relevant fiel of research after the first methoical stuy by Johnson [1]. Since, many papers with a variety of practical an impractical assumptions have been publishe; however, there always exists a gap between the theory an practice in the literature for this fiel. One of the most thoroughly stuie scheuling problems is the flow shop (FS) problem. In regular FS, we have a set of n jobs {J 1, J 2,..., J n }anasetofm machines {M 1, M 2,..., M m }. Each job has a set of m operations an each operation j can be one by one machine. All the jobs have the same processing routes, starting from machine 1 until finishing at machine m. Each job has a known an fixe processing time * Khalili.mj@gmail.com for operation j. P ij enotes the processing time of job i on machine j. By removing the restriction that all jobs nee to be processe on all machines, the regular FS converts to a more realistic problem [2]. In this problem, we still require jobs to move from the first to the last machine. Three types of ecision-making goals are prevalent in scheuling: (1) Efficient utilization of resources: minimizing completion time or makespan; (2) rapi response to emans, or minimizing the work-in-progress: mean completion time, mean flow time, or mean waiting time; an (3) close conformance to prescribe ealines: total tariness, maximum tariness, an the number of tary jobs. The tariness of each job is equivalent to the amount of time when the job is complete after its ue ate. The first an secon types of ecision-making goals are process oriente an the thir one, as we show in Figure 1, is consiere as a customer-oriente objective. In generalize flow shop (GFS), minimizing the total completion time (TCT) is synonymous to maximizing the throughput. Since the best sequence with respect to the TCT minimization may lea to a large number of jobs being complete after their ue ates, we also consier the total tariness (TT) minimization. The TT minimization is vital prominence in maketo-orer or just-in-time environments [3]. The main aim of This is an Open Access article istribute uner the terms of the Creative Commons Attribution License ( which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite.

2 2 M. Khalili: Manufacturing Rev. 214, 1, 13 Efficient utilization of resources Process oriente Decision-making goals Minimization of work in process Costumer oriente Close conformance to prescribe ealines Figure 1. Classification of ecision-making goals in scheuling problems. consiering two inepenent objectives is to gauge the robustness of our propose EM by two essentially ifferent objectives an stuy as to whether the high performance of EM is transferable to other objective functions. In the other wors, we consier inepenently the total tariness an total completion time in orer to analyze the efficiency an robustness of our propose EM on each of these objectives against some existing methos. In real-worl situations machines are not continuously available ue to many reasons. For example it is possible that certain important jobs may have been promise uring the previous planning horizon. Another common reason for this unavailability coul be breakown (a stochastic event) or preventive maintenance (a eterministic event). A breakown makes the shop behavior har to preict, an thereby reuces the efficiency of the prouction system. Therefore, scheuling maintenance to reuce the breakown rate has commonly been recognize by the ecision makers. This paper investigates the eterministic case of machine unavailability before making ecisions for job sequence starts. In most stuies on scheuling it is assume that processing of a job i on machine j commences without elay after finishing its process on machine j 1. In practice, in most manufacturing an istribution systems, semi-finishe jobs are transferre from one processing facility to another by transporters such as robots an conveyors, an finishe jobs are elivere to warehouses or customers by vehicles such as trucks. The transportation time from machine j 1 to machine j is T f an transportation time from machine j to machine j 1 is T b. The time to loa an unloa the transporter is inclue in the transportation time. When transporter reaches machine j, it elivers the job to the machine if the setup time has been complete an machine is reay to receive the job. Otherwise, it places the job in a waiting line an immeiately starts its return to machine j 1. In other wors, once the transporter leaves the first machine, it always returns in time T f + T b to take the next job. This transportation time coul be either jobepenent or job in-epenent. We assume that all transportations are job-inepenent; an transportations between two machines are hanle by one transporter. This transporter can only carry one job at a time; therefore, a job may have to wait for the transporter before it is transporte. In regular flowshop, the earliest starting time of job j processe on machine i is compute by: s ji ¼ max C ji 1 ; C j i ; ð1þ where C ji 1 is the completion time of job j on machine i 1 (i.e., C ji 1 = s ji 1 + p ji 1 Þ an j is the preecessor of job j in the sequence. In this paper, flowshop with transportation times is consiere, on which the earliest starting time, s ji, is calculate by: n o s ji ¼ max C ji 1 þ T fj ; C j i 1 þ 2T fj þ T b j ; C j i ð2þ where T fj an T bj are transportation time from machine i 1 to machine i an return time of a transporter from machine i to machine i 1. Precisely, the term of C ji 1 þ T fj etermines the s ji when job j oes not wait for the transporter while the term of C j i 1 þ 2T fj þ Tb j is use when job j has to wait for the transporter. The term of C j i shows the availability time of machine i. Electromagnetism-inspire algorithms have recently been classifie as a meta-heuristic approach to tackle complex optimization problems. The motivation behin this meta-heuristic approach has risen from the attraction-repulsion mechanism of electromagnetic theories an this basic iea that in metaheuristics we esire to bring our search closer to a region with the superior objective function an at same time, go away from the region with the inferior objective function to move the solution graually towar the optimality. The EM shows very high performance than other meta-heuristics in NP-har problems [4, 5]. In our case, machine availability constraints an transportation times as well as skipping probability of some jobs from some stages are assume. To juge the performance of the propose EM, we go through the corresponing problem uner minimizing two separate objectives, total tariness from make-to-orer environments, an total completion times from make-to-stock environments. The GFS is categorize as a combinatorial optimization problem known to be strongly NP-har requiring an effective metaheuristic algorithm to be utilize to overcome the complexity of the problem consiere an solve it. Therefore, in this paper we have employe EM which is one of the recently introuce metaheuristic for the first time. This is the first time that the EM algorithm is being employe for the problem consiere GFS. Therefore, there is no similar EM in the literature to compare against. We have compare the most relevant an effective algorithms in the literature of the problem at han to prove the superiority of our algorithm. Meanwhile, base on your comments, we mentione our pioneering effort to employ EM in our problem in the paper. The purpose of this paper is to investigate a realistic flowshop case with three practical assumptions, transportation time, machine availability constraints an machine skipping. Doing so, we aim to establish a simple criterion to integrate GFS prouction scheuling with preventive maintenance (PM) activities to yiel a high level of machine availability noting that PM operations have priority over prouction operations. So PM

3 M. Khalili: Manufacturing Rev. 214, 1, 13 3 Figure 2. 3D flowshop scheuling problem. operations must be scheule first, an then jobs sequence along with them. To solve such an NP-har problem [5], we propose to apply the EM. This propose algorithm tackles two inepenent objectives an we examine its robustness in ifferent situations. Finally, we compare these results with the results of a number of other relate algorithms. The following assumptions are usually characterize to GFS: each machine in each stage can process at most one job at a time, each job can be processe by at most one machine at a time in each stage, the process of a job on a machine cannot be interrupte, machines aren t available continuously an once a job completes one process it isn t immeiately available for its next (Figure 2). The problem uner consieration generalizes the problem stuie by the other paper. This generalization comes from assuming machine availability constraints. This paper consiers total completion minimization as well as total tariness, while the other paper minimizes makespan an total tariness. The rest of this paper is organize as follows: Section 2 gives literature review on the problem. We introuce the preventive maintenance an how to integrate it with prouction operations sequencing in Section 3. In Section 4, the propose algorithm is presente. In Section 5 we evaluate the algorithms by a benchmark. Finally, Section 6 conclues the paper. 2. Literature review The FS has been a very ynamic research omain in metaheuristic an heuristic techniques, principally because of the ifficulty encountere by exact methos to solve large or even meium instances. Salvaor [6] was the first who efine the FS problem. A etaile survey for the FS problem has been given by Linn an Zhang [7] an Wang [8]. Moursli an Pochet [9] presente a branch-an-boun algorithm to minimize the makespan in FS. Exact algorithm is applie to large FS problems; such an approach can take hours or ays to erive a solution so Sriskanarajah an Sethi [1] propose a heuristic algorithm for solving a special case of the FS problem. Guinet et al. [11] proposes heuristic techniques for a simplifie FS makespan problem with two stages an only one machine at stage two. Guinet et al. [11] also propose a heuristic for the makespan problem in a two-stage FS base on Johnson s rule [1]. To obtain a near optimal solution, meta-heuristic algorithms have also been propose. Nowicki an Smutnicki [12] propose a tabu search (TS) algorithm for the FS makespan problem. Gourgan et al. [13] an Zhang an Wu [14] presente several simulate annealing (SA)-base algorithms for the FS problem. A genetic algorithm has been wiely use in many previous works for the FS makespan problem, see e.g., Reeves [15]. Tavakkoli-Moghaam et al. [16] propose a memetic algorithm (MA) combine with a new local search engine, namely, neste variable neighborhoo search (NVNS), to solve a FS scheuling problem with processor blocking. The performance of the propose EM was verifie by a number of instances an then compare with genetic algorithms. Cheng et al. [17] aresse the earliness/tariness scheuling problem with ientical parallel machines, an they apply a genetic algorithm to solve this problem. Yang [18] was the first to use EM to train a neural network. The results show a great saving on the computation memories an time, an inicate that EM performe much better than genetic algorithm in fining the global optimum. For its merit of simple concept an economic computational cost, EM has been use in the areas of function optimization, fuzzy neural network training, project scheuling, an other combinatorial optimization fiels [5] but selom have use in scheuling problems, so we were motivate to solve our problem with this metho. Khalili an Tavakkoli- Moghaam [19] propose a new multi-objective electromagnetism algorithm for a bi-objective flowshop scheuling problem. Because of its significance both in inustrial an theory applications, the GFS has attracte the attention by many researchers. To obtain a near-optimal solution, meta-heuristic algorithms have also been propose. Different genetic algorithms (GA) are applie by Chen an Neppalli [2], Alowaisan an Allahveri [21]. Among the other metaheuristics, one coul refer the reaer to particle swarm optimization (PSO) by Pan et al. [22], simulate annealing (SA) by Fink an Voß [23], ant colony optimization (ACO) by Shyu et al. [24] an tabu search (TS) by Grabowski an Pempera [25]. Khalili [26] propose an iterate local search algorithm for flexible flow lines with sequence epenent setup times to minimize total weighte completion an also stuie multiobjective no-wait hybri flowshop scheuling problems to minimise both makespan an total tariness [27]. Next, briefly review the relate research on availability constraint an the traveling time between stages, which are other characteristics of the problem at han. Traveling times between stages in most investigations have been consiere as processing time. Hurink [28] assume that unlimite buffer spaces

4 4 M. Khalili: Manufacturing Rev. 214, 1, 13 exist between the machines, an all transportation is accomplishe through a single robot. Soukhal et al. [29] analyze two-machine FS scheuling problems with transportation an assume that finishe jobs are transporte from the processing facility an elivere to customers by trucks. In the other wors, both transportation capacity an transportation times were explicitly taken into account. Scheuling problems with availability constraints have been extensively examine by Ruiz et al. [3]. For more etails, a survey of existing methos for solving scheuling problems uner availability constraints as well as complexity results can be foun in Schmit [31]. Lee [32] hanle the preemptive FS problem with two machines an one unavailability perio first impose on machine 1 an then on machine 2 with the makespan objective. Both problems are prove to be NP-har in the orinary sense, an heuristics with error bouning analysis are propose. Blazewicz [33] investigate the two-machine problem with arbitrary number of unavailability perios on one machine, an prove that the makespan minimization problem is strongly NP-har. Breit [34] investigate the problem of scheuling n preemptable jobs in a two-machine FS where the first machine is unavailable for processing uring a given time interval. A more complex hybri FS problem is tackle in Allaoui an Artiba [35]. 3. Preventive maintenance Machine scheuling is concerne with the problem of optimally scheuling available machines. However, the majority of the scheuling literature carries a common assumption that machines are available at all times. This availability assumption may not be true in real inustrial settings, since a machine may become unavailable uring certain perios of time. For instance, a machine may not be available at the beginning of a planning horizon as it may continue to process the late jobs from the previous horizon. Similarly, when a machine breakown has occurre or a preventive maintenance activity has been scheule. A breakown causes the shop behavior ifficult to preict, an thereby reuces the efficiency of the prouction system. Therefore, scheuling maintenance to reuce the breakown rate has commonly been recognize by the ecision makers. It is known that maintenance plays an important role in many inustries because more reliable prouction systems with higher serviceable performance is what factories are mostly intening to reach; on the other han, maintenance systems are esigne to make sure that prouction facilities are serviceable an reliable to operate to achieve target prouction levels [3]. Maintenance policies influence the machine availability an the machine utilization ratio. Maintenance actions usually can be classifie into two major categories: corrective maintenance (CM) an preventive maintenance (PM). CM is the actions carrie out when a failure has alreay occurre. PM is the action taken on a system while it is still in service, but is carrie out in orer to keep the system at the esire level of performance. In PM, activities are conucte at fixe time intervals etermine for machines an facilities before a failure or breakown occurs. Here, the main question is whether PM ecisions an Table 1. Processing times (P i ) for single machine with n =4. Jobs (i) P i prouction sequencing shoul be jointly scheule. Ruiz et al. [3] investigate this issue an conclue that the optimal ecision is case epenent. Accoring to ifferent criteria, various PM policies are efine. The commonly use PM policy in inustry is preventive maintenance at fixe preefine time intervals [3]. PM operations are scheule in avance at pre-etermine time intervals without taking into account a probabilistic behavior for time to failure. The intervals between PM operations can be weekly, monthly or other perios epening on the system. In this policy, fixe time intervals (T PM ) are etermine an PM operations are carrie out exactly at those times. Our criterion works as follows: whenever a new job is to be processe on each machine, the completion time is compute. If this time excees the pre-efine T PM, then the process of the next job is postpone an the PM is carrie out first because PM operations have priority over processing. It shoul be pointe out that has been assume that the processing of a job cannot be interrupte (or preemption is not allowe or the jobs are nonresumable), an availability time of any machine is after finishing the process of the last job on that machine. The following example shows how PM operations scheuling an single machine scheuling are jointly consiere. Suppose a shop ecies to carry out PM at every 5 time units. The urations of these PM operations (D PM ) for the machine are 15 time units. The processing times are shown in Table 1. The urations of these PM operations (D PM ) for the machine are 15 time units. The processing times (P i ) are also shown in Table 1.Infact, will been scheule the jobs in the prouction horizon shown in Figure 3. Figure 4 epicts a sequence of all the jobs as {4, 1, 3, 2}. As it can be seen in this figure, the first maintenance is carrie out after processing jobs 4 an 1, the accumulate total processing time is = 45 an it is not possible to process the thir job of sequence (Job 3), because it has a processing time of 25 time units, which woul result in an accumulate total processing time of 7 units, which is longer than T PM =5. So, machine will be ile for 5 time units, an then PM operation begins at time 5 units, an lasts for 15 time units. After carrying out PM, Job 3 is processe. Then, once more it is not possible to process the next job (Job 2) since a processing time of = 12 will be accumulate. So, in the same manner, secon PM operation is carrie out from 1 to 115, then Job 2 is processe. 4. Propose algorithms 4.1. Heuristics In general, the heuristics applie to each ecision making goal, process or customer oriente, are ifferent. These two

5 M. Khalili: Manufacturing Rev. 214, 1, P i Machine PM PM Figure 3. Gantt chart of the prouction horizon after applying PM Time Machine J 4 J 1 PM J 3 PM J Time Figure 4. Gantt chart of the solution for the given example. groups have been introuce an explain the heuristics, which we have use in this stuy: 1. Heuristics to solve scheuling problem with emphasis on completion times such as SPT, LPT, NEH (Nawaz, Enscore an Ham), an (g/2, g/2) Johnson rule [1]. 2. Heuristics to solve scheuling problem with emphasis on tariness such as EDD, SLACK an NEH_EDD. The above seven heuristics are briefly escribe below. Shortest Processing time (or SPT): jobs are processe on machine 1 in increasing orer of the processing times on machine 1. That is, the process of jobs with shortest processing times is first starte. At subsequent machines, jobs are sorte in earliest reay time orer. LPT arranges jobs on machine 1 in escening orer of processing times of jobs on machine 1. (g/2, g/2) Johnson s rule, the sum of processing time of jobs on machine 1 to [m/2] an the sum over machine [m/2] + 1 to m are calculate to orer jobs on machine 1[1]. NEH, This heuristic can be ivie into three simple steps: 1. The total processing times for all the jobs on m machines are compute as follows: p i ¼ P m j¼1 P ij; i ¼ 1;...; n 2. Jobs are sorte in escening orer of P i. Then, the first two jobs (those with higher P i ) are taken an the two possible scheules containing them are evaluate. 3. Take job i (i =3,..., n) an fin the best scheule by placing it in all the possible i positions in the sequence of jobs that are alreay scheule. For example, if i = 4, the alreay built sequence woul contain the first three jobs of the sorte list calculate in step 2, then, the fourth job coul be place either in the first, secon, thir or the last position of the sequence. The best sequence of the four woul be selecte for the next iteration. EDD is a well known ispatching rule an orers the jobs accoring to imminent ue ates. SLACK, another name of this metho is the minimum slack because in this metho at time t, the job with the minimum value of j C j (s) is selecte, where C j (s) will be the completion time of job j 62 s if it is scheule at the en of sequence s. In NEH_EDD we consier the ue ates for efining an initial orer in which the jobs are consiere for insertion. The initial orer in NEH_EDD is base on the earliest ue ate ispatching rule that arranges jobs in ascening orer of their ue ates Simulate annealing Simulate annealing (SA) was first presente as a search algorithm for combinational optimizations (CO) problems in Cerny [36]. SA is moele after physical annealing of soli metal. In annealing, a metal is first heate to a high temperature an then coole own with a very slow rate to the room temperature. Sometimes, if cooling is not fast enough, quenching is one. In SA, solutions are ranomly generate from a set of feasible solutions. This process accepts not only those solutions that improve the objective function, but also those solutions, which o not improve objective function on the basis of transition probability (TP). Transition probability epens on the change in objective function an the annealing temperature. The main features of SA that make this algorithm more sophisticate are perturbation annealing scheule an the

6 6 M. Khalili: Manufacturing Rev. 214, 1, 13 Before move After move Figure 6. Single point operator. Before move After move Figure 7. Swap operator. Figure 5. General outline of a simulate annealing algorithm. transition probability. Perturbation generates a new solution, an the annealing scheule controls the initial temperature, final temperature an the rate of cooling, while transition probability help heuristic to escape local optima. SA is commonly sai to be the olest among the meta-heuristics an one of the first algorithms that ha an explicit strategy to avoi local optima. The funamental iea is to generate a new job sequence s by a ranom rule from the neighborhoo of present sequence x. This new sequence is accepte or rejecte by another ranom rule. A parameter t, calle the temperature, controls the acceptance rule. The variation between objective values of two caniate solutions is compute DC = obj(s) obj(x), where obj is the value of the objective function. If (DC, sequence s is accepte. Otherwise, sequence s is accepte with probability equal to exp(c=t i ). The algorithm procees by trying a fixe number of neighborhoo moves (max) attemperaturet i,while temperature is graually ecrease. The proceure is repeate until a stopping criterion is met. Moves resulting in solutions of worse quality (uphill move) than the current solution may be accepte to escape from local minimum. SA starts at a high temperature (T ), so most of the moves are accepte at first steps of the proceure. The probability of oing such a move is ecrease uring the search. Figure 5 shows the general outline of SA algorithm. The commonly use encoing scheme for FS problem is permutation of jobs which shows the job sequence on machine 1. It is known that initial solution can influence the quality of the solutions. A goo initial solution can result in a better an faster subsequent result. There are some alternative choices to consier ifferent parameters an ajust them by tuning an settings. The propose SA algorithm checks 1 neighbors at temperature t i (i.e., max =1). Moving operator generates a neighbor solution from current caniate solution by making a slight change in it. This operator must work in such way to avoi infeasible solutions. In this research, two ifferent move operators have been consiere: Swap operator (SO): the positions of two ranomly selecte jobs are swappe (Figure 6). Single point operator (SPO): the position of one ranomly selecte job is ranomly change (Figure 7). The algorithm evelope in this paper uses ifferent operators from the other paper. As mentione earlier, to avoi a local minimum, solutions with worse objective values are probabilistically accepte epening on the value of the temperature. When the proceure procee, the temperature is slightly lowere uner a mechanism which is calle cooling scheule. Here, exponential cooling scheule has been use, T i = a T i 1 (where a 2 (, 1) is temperature reucing rate), which is often believe to be an excellent cooling recipe Electromagnetism-like metho Birbil an Fang [37] propose the electromagnetism-like metho (EM), which is a flexible an effective populationbase algorithm to search for the optimal solution of global optimization problems. EM originates from the electromagnetism theory of physics by consiering potential solutions as electrically charge particles sprea aroun the solution space. This meta-heuristic metho utilizes an attraction-repulsion mechanism to move the particles towars optimality. Debels et al. [38] applie successfully a hybriization of the EM with scatter search for resource-constraine project scheuling problem. EM is useable for particular set of optimization problems with boune variables. Each caniate solution as a charge particle has been consiere. The charge of each caniate solution is relate to the objective function value. The size of attraction or repulsion over caniate solutions in the population is calculate by this charge. The irection of this charge for caniate solution i is etermine by vectorially aing the forces from all other solutions on the caniate solution i. In this mechanism, a caniate solution with goo objective function value attracts the other ones; caniate solutions with unfavourable objective function repel the other population members; an better the result of the objective function value the higher the size of attraction. EM has four phases incluing Initialization of algorithm, computation of total force exerte on each particle, movement along the irection of the force, an the local search Encoing scheme an initialization The most frequently use encoing for the flowshop problem is a simple permutation of the jobs. The relative orer of the jobs in the permutation inicates the processing orer of

7 M. Khalili: Manufacturing Rev. 214, 1, 13 7 x 1 x 3 F 23 F 13 Figure 9. Example of exertion of forces. x 2 F 3 Figure 8. Proceure of the local search. the jobs on the first machine in the shop. It is necessary to mention that a rawback of the algorithms propose for generalize flowshop problems is that they only orer jobs accoring to earliest reay time of jobs at the beginning of each stage. However in GFS problems, it is very likely to have some jobs with the same reay time at beginning of each stage. For example, all the jobs which skip the first stage woul have the reay time of zero at the beginning of the secon stage. So we nee to establish a clear criterion to orer these jobs. In this paper, if some jobs have the same reay time at the beginning of stage t (t =2,3,..., g), the same as their orer at stage t 1has been arrange. We subject all the compare algorithms to this criterion. Traitionally, in an EM, the initial population is generate ranomly. However, it is known that the initial solutions can affect on the quality of the results obtaine by the algorithms. The initialization proceure in such a complex combinatorial problem has to be mae with great care, to ensure convergence to esirable objective function values in a reasonable amount of time. Because of this, initial solutions for propose EM are generate by EDD an NEH_EDD for total tariness objective an SPT an NEH for the completion time objective. The population size (popsize-2) is ranomly generate. The objective function values of solutions are calculate an the best one is recore as x best Local search The propose EM is hybriize with a local search in orer to improve the performance of the algorithm. The proceure for this local search can be escribe as follows: the first job in the sequence of caniate solution i (x i1 ) is relocate to a new ranom position in the sequence. If this new sequence (v) results in a better makespan, the current solution (x i )isreplacebythe new sequence (v). This proceure iterates at most for all the subsequent jobs in sequence. If there is any improvement in the kth < n, the local search for the current solution terminates. Subsequently, the best solution is upate. Proceure for the local search is shown in Figure Computation of total forces The charge of each solution caniate i is calculate in relation to its objective function by:! fðx i Þ fðx best Þ q i ¼ exp n P popsize ; j¼1 f x j f ð xbest Þ 8ii¼ 1; 2;...; popsize: ð3þ This formula ensure that the better objective values are assigne higher charges. The total force F i exerte on caniate solution i is also calculate by the following formula: 8 >< F i ¼ >: X P j6¼i X p j6¼i q i q j x j x i jjx j x i jj ; 2 q i q j x i x j jjx j x i jj ; 2 f x j < f ð xi Þ; f x ð iþ < f x j : ð4þ A two-imensional example total force vector F i exerte on caniate solutions is shown in Figure 9. The force exerte by x 1 on x 3 is F 13 (repulsion: the objective function of x 1 is worse than that of x 3 ) an the force exerte by x 2 on x 3 is F 23 (attraction: the objective function of x 2 is better than that of x 3 ). F 3 is the total force exerte on x 3 by x 1 an x Moving by total forces Allthecaniatesolutionsaremovewiththeexceptionof the current best solution. The move for each caniate solution is in the irection of the total force exerteonitbyaranom step length. This length is generate from a uniform istribution between [, 1]. By selecting ranom length that caniate solutions have a nonzero probability to move to the unvisite solution space along this irection has been guarantee. Moreover, by normalizing the total force exerte on each caniate solution, we are able to avoi proucing infeasible solutions. Figure 1 shows a pseuo coe of our propose EM. 5. Experimental evaluation In this section, we apply a benchmark to evaluate the performance of our propose algorithms. They are implemente in MATLAB 7. an run on a Pentium IV PC with an Intel processor running at 3 GHz an 1 GB of RAM memory. Relative percentage eviation () for total completion time as a common performance measure to compare the methos has been use. The best solutions, Min sol, obtaine for each instance

8 8 M. Khalili: Manufacturing Rev. 214, 1, 13 Figure 1. Pseuo coe of the propose EM. are compute by any of the five algorithms. is obtaine by the following equation: ¼ Alg sol Min sol 1 ð5þ Min sol where Alg sol is the objective function value obtaine for a given algorithm an instance. Clearly, lower values of are preferre. In the case of the total tariness GFS problem, the best solution coul be zero (an therefore optimal), then in the above equation we will have ivision by zero. Moreover, if the best solution is a small value, the performance measure unerestimates an algorithm which obtains a solution slightly worse than the best. Therefore, a ifferent performance ratio is usually use in tariness cases to avoi these problems, which has been terme the relative eviation inex (RDI). The RDI is obtaine by: RDI ¼ Alg sol Min sol max sol Min sol 1: ð6þ With this measure, an inex between an 1 is obtaine for each metho such that a goo solution will have an inex very close to. Note that if the worst an the best solutions take the same value, all the methos provie the best (or same) solution an hence, the inex value will be (i.e., the best inex value) for all methos. Inee, the above formulas are two ifferent ways of normalization. They calculate the eviation from the best solution, which is always a positive number Data generation Data require for the given problem consist of the number of jobs (n), number of machines (m), range of processing times (p) an transportation times (T f an T b ), the reay times (r), skipping probability, time interval between two consecutive PM (T PM ) an uration of PM operations (D PM ). Our instances base on benchmark values [39] that are shown in Table 2 has been guarantee. We have n = {2, 5, 1, 2, 5} an m = {5, 1, 2}, which results in 15 combinations of n m. The processing time in Taillar s instances [39] are generate from a uniform istribution over the range [1, 99]. GFS is consiere by allowing

9 M. Khalili: Manufacturing Rev. 214, 1, 13 9 Table 2. Factor levels. Factor Levels Number of jobs Number of machines (m) Range of processing times (p) U [1, 99] Range of transportation times (T f + T b ) U [1, 3] Skipping probabilities.1.4 Range of time interval between two consecutive PM (T PM ) some jobs to skip some stages. The probability of skipping a stage is set at.1 or.4. The transportation times (T f an T b ) come from a uniform istribution in the range [1, 3], where the running average will work out about 3% of the processing time. T PM for each machine are obtaine from a uniform istribution in the range [2, 3]. D PM of each machine are istribute uniformly over three ranges [1, 5], [1, 99] an [1, 15], where the running average will be about 5%, 1% an 15% of the processing times, respectively. The ifferent levels of factors result in = 9 ifferent scenarios. Ten instances for each scenario, similar to a Taillar s benchmark have been prouce. Therefore, 9 1 = 9 instances have been. To generate ue ates for all n jobs an approach similar to [4] have been use. The following steps are applie to prouce the ue ates [41]: 1. Compute the total processing time of each job on all g stages. p i ¼ Xg p it ; 8 i 2 N: ð7þ t¼1 2. Determine a ue ate for each job: i ¼ ðp i Þð1 þ ranom 3Þ; 8 i 2 N; ð8þ where ranom is a ranom number from a uniform istribution over the range [, 1] Parameter tuning U [2, 3] Range of uration of PM operations (T PM ) U [1, 5] U [1, 99] U [1, 15] The quality of algorithms is significantly influence by the values of parameters. In this section, the behavior of ifferent operators an parameters of SA an the propose EM have been stuie. In orer to tune the algorithms, we apply a full factorial esign in the esign of experiment (DOE) approach [42]. Five instances for each combination of n an m, which results in a total of 75 instances have been ranomly generate. The stopping criterion is n m.2 s of the computational time. This criterion allows for more time as the number of jobs or machines increases Simulate annealing The propose SA has three parameters, initial solution, {initial temperature, cooling rate} an the move operator. The consiere levels of the parameters are: Initial solution (IS): two levels (SPT, NEH). {Initial temperature (T ), cooling rate (a)}: three levels ({5,.985}, {1,.98}, {2,.97}). Move operator (MO): two levels (SO, SPO). So, = 12 ifferent SAs are obtaine by these levels an all the 75 instances are solve by them. The results are analyze by means of the analysis of variance (ANOVA) technique. Three main hypotheses, incluing normality, homogeneity of variance an inepenence of resiuals have been examine an no bias have been foun to question the valiity of the experiments. The means plot an least significant ifferences (LSD) intervals (at the 95% confience level) for the levels of IS, {T, a} anmo parameter are shown in Figures 11 13, respectively. Figure 1 illustrates that the initial solution of NEH provies statistically better results than SPT. In Figure 11 can be seen that there is statistically significant ifference between two move operators an that SPO is superior an also in Figure 12 can be seen T =5ana =.985 results in statistically better output than either of the other two sets. As the results of the analysis, the parameters IS =NEH,MO =SPO an (T, a) = (5,.985) have been set Electromagnetism-like metho One of the avantages of the propose EM is that it has only one parameter, popsize (number of population). The consiere levels for popsize are 2, 4, 6 an 8. All 75 instances are solve by the EM algorithm obtaine by the above values of popsize. The results are assesse by means of Analysis of Variance (ANOVA) technique. The means plot an least significant ifferences (LSD) intervals for various levels of popsize parameter factor are shown in Figure 14. This figure reveals that the number of population of four provies statistically better results than other values of popsize = 2,6, Experimental results In this subsection, the authors inten to compare our propose EM with other existing methos. As escribe before, this phase inclues two subsections each of which consiers one objective function. In each subsection, the propose EM

10 1 M. Khalili: Manufacturing Rev. 214, 1, NEH SPT Figure 11. Means plot an LSD intervals for the initial solution of SA SO SPO Figure 12. Means plot an LSD intervals for the move operator of SA. 1.8 Table 3. Average relative percentage inex ðrpiþ for the algorithms groupe by n an m. Instances EDD SLACK NEH_EDD SA EM Average with some well-known heuristics as well as SA for associate objective functions has been compare. In the first subsection, the EM is compare with SA, EDD, SLACK an NEH_EDD, an in the next stage it is compare with SA, SPT, (g/2, g/2) Johnson s rule, an NEH [1]. In each subsection, the effects of variables such as problem size an numbers of machines on the performance of the algorithms are investigate. The stopping criterion is n m.2-s computation time. As mentione earlier, the total completion time (TCT) belonging to the process oriente goals, an the total tariness (TT) belonging to the costumer oriente goals inepenently to analyze the efficiency an robustness of our propose EM on each of these objectives against some existing methos has been consiere Analysis of the total tariness.4.2 (1,.98) (2,.97) (5,.985) Figure 13. Means plot an LSD intervals for the initial parameters (temperature, cooling rate) in SA Popsize Figure 14. Means plot an LSD intervals for number of population in the propose EM. In this section, the authors compare our propose EM algorithm with SA, NEH_EDD, SLACK an EDD with the aim of minimizing the total tariness. The results of the experiments, average for each combination of n an m are shown in Table 3. As expecte, the meta-heuristics perform better than the heuristics, an the worst performing algorithms in almost all the instances are EDD an SLACK with RDI of 89.32% an 92.79%, respectively. The propose EM provies best results among these algorithms with the RDI of 5.21%. For further analysis, the ANOVA has been carrie out. The means plot for the ifferent algorithms with the least significant ifference (LSD) intervals are shown in Figure 15.Asitis seen, the propose EM provies statistically better results than the other methos. NEH_EDD supersees the traitional EDD an SLACK. EDD an SLACK are statistically the same. The results also show that EM statistically outperforms the other algorithms. It is interesting to see the performance of NEH in comparison with two other heuristics. NEH_EDD significantly supersees EDD an SLACK. In aition, we carry out a two-way ANOVA for each variable to fin the interaction between them an performance of the algorithms. Figure 16 shows the tren of the quality of the methos evaluate as

11 M. Khalili: Manufacturing Rev. 214, 1, RPI EDD SLACK NEH_EDD SA EM Figure 15. Means plot an LSD intervals (at 95% confience level) for the type of the algorithm factor. RPI Number of jobs EDD SLACK NEH_EDD Figure 16. Means plot an LSD intervals for the interaction between the algorithm type an the number of jobs. SA EM Table 4. Average relative percentage eviation ðrpiþ for the algorithms groupe by n an m. Instances SPT Johnson s NEH SA EM Average RPI Number of stages EDD SLACK NEH_EDD Figure 17. Means plot an LSD intervals for the interaction between the algorithm type an the number of stages. the number of jobs increases. EM an SA are the same in the case of n = 2 an the greatest ifference between SA an EM is shown when n = 1 an again the same in the case of n = 5. As illustrate in Figure 17, the higher the number of stages, the better the heuristics performs, while there is no change of the metrics for EM an SA. Figure 16 epicts the means plot an LSD intervals for the interaction between the various algorithm as a functions of the number of stages. It is interesting to see in this figure that the propose EM shows a robust performance in almost all cases against the number of stages. SA EM SPT Joh. NEH SA EM Figure 18. Means plot an LSD intervals (at 95% confience level) for the type of the algorithm factor Analysis of the total completion times In the total completion time objective, the authors compare the results of SA, NEH, an EM. First, the algorithms in terms of the selecte objective function (i.e., total completion time) have been analyze, an then effects of variable, such as problem size an numbers of machines, on the performance of the algorithms are examine. measure to compare the algorithms has been use. The stopping criterion is again n m.2 s of the computational time. EM outperforms the other algorithms again, which support its robustness (Table 4). Here an analysis similar to the previous section has been applie. The means plot for the ifferent algorithms with the LSD intervals are shown in Figure 18. The Johnson s rule an SPT have a poor performance while the propose EM gives excellent results than the heuristics an the SA. Figure 19 shows the quality tren of the evaluate methos as the number of jobs increases. As seen, there is almost no effect on the performance of all five algorithms with regars to the problem size. Only SPT in comparison with the (g/2, g/2) Johnson s rule improves when the number of jobs increases. As shown in Figure 2, increasing the number of

12 12 M. Khalili: Manufacturing Rev. 214, 1, stages results in better performance for the heuristics. It is noteworthy to see that EM again shows a robust performance in almost all cases with the number of jobs as the variable. It is conclue that the algorithms perform ifferently in various situations. In particular, the number of jobs, as a factor, has noticeable impact on the performance quality of the algorithms. 6. Conclusion Number of jobs In this paper, an EM for a generalize flow shop (GFS) problem with machine availability constraints an transportation times between stages to minimize two inepenent objectives has been propose. EM an also SA as well as a number of other well-known heuristics to minimize total tariness an total completion times, inepenently. The results were analyze in terms of objective functions an the effects of variable factors on the instances consiere in this paper. The results show that the propose EM performs very well. This is the first time that an EM has been applie to GFS problems, an the computational results show it hols a goo promise. In fact, the overall performance of our propose metho can be regare as a very goo meta-heuristic algorithm since it oes not make the use of problem specific knowlege such as the critical paths concept or extensive spee-ups as use in other high-performing algorithms. SPT John. Figure 19. Means plot an LSD intervals for the interaction between the algorithm type an the number of jobs Number of stages NEH SA EM SPT John. Figure 2. Means plot an LSD intervals for the interaction between the algorithm type an the number of stages. NEH SA EM In moern manufacturing systems such as agile an flexible manufacturing systems, there is an increasing eman for customize proucts, which are prouce in smaller lot sizes than before. Therefore, there appear to be an increase focus on fining new methos that have the most of the strategic avantages of a GFS but also can provie some of the operational avantages of an assembly line without flexibility limitations of Cellular Manufacturing Systems. Virtual Cellular Manufacturing Systems (VCMSs) are a new manufacturing technology generate from the changing an ynamic marketing environment, which has gaine momentum uring the last ecae. Therefore, extensions of the propose EM to VCMS consiering other objectives or features, such as sequence-epenent setup times, are possible. For further insight, it will be interesting to work on other iterative algorithms or on hybriization of the propose EM with other algorithms to achieve even better results. References 1. S.M. Johnson, Optimal two an three-stage prouction scheules with set up times inclue, Naval Research Logistics Quarterly 1 (1954) W.H. Yang, A stuy on the intelligent neural network training using the electromagnetism algorithm, Unpublishe Master Thesis, Dept. of Inustrial Engineering an Management, I-Shou University, Kaohsiung County, Taiwan, R. Ruiz, T. Stützle, An Iterate Greey heuristic for the sequence epenent setup times flowshop problem with makespan an weighte tariness objectives, European Journal of Operational Research 187, 3 (28) M. Khalili, M.J. Tarokh, B. Naeri, Using electromagnetism algorithm for etermining the number of kanbans in a multistage supply chain system, Journal of Inustrial Engineering 6 (21) P. Wu, W.-H. Yang, N.-C. Wei, An electromagnetism algorithm of neural network analysis an application to textile retail operation, Journal of the Chinese Institute of Inustrial Engineers 21 (24) M.S. Salvaor, A solution to a special case of flow shop scheuling problems, in: S.E. Elmaghraby (E.), Symposium of the Theory of Scheuling an its Applications, Springer, New York, 1973, pp R. Linn, W. Zhang, Hybri flow shop scheuling: a survey, Computers & Inustrial Engineering 37, 1 2 (1999) H. Wang, Flexible flowshop scheuling: optimum, heuristics, an artificial intelligence solutions, Expert Systems 22, 2 (25) O. Moursli, Y. Pochet, A branch-an-boun algorithm for the hybri flowshop, International Journal of Prouction Economics 64, 1 3 (2) C. Sriskanarajah, S.P. Sethi, Scheuling algorithms for flexible flowshops: worst an average case performance, European Journal of Operational Research 43, 2 (1989) A. Guinet, M.M. Solomon, P.K. Keia, A. Dussauchoy, A computational stuy of heuristics for two-stage flexible flowshops, International Journal of Prouction Research 34, 5 (1996) E. Nowicki, C. Smutnicki, The flow shop with parallel machines: a tabu search approach, European Journal of Operational Research 16, 2 3 (1998)