Research Article A discrete group search optimizer for blocking flow shop multi-objective scheduling Advances in Mechanical Engineering 2016, Vol 8(8) 1–9 Ó The Author(s) 2016 DOI: 10.1177/1687814016664262 aime.sagepub.com Deng Guanlong, Zhang Shuning and Zhao Mei Abstract This article presents a multi-objective discrete group search optimizer for blocking flow shop multi-objective scheduling problem The algorithm is designed to search the Pareto-optimal solutions minimizing the makespan and total flow time for the flow shop scheduling with blocking constraint In the proposed algorithm, a diversified initial population is constructed based on the Nawaz–Enscore–Ham heuristic and its variants Unlike the original group search optimizer in which continuous solution representation is used, the proposed algorithm employs discrete job permutation representation to adapt to the considered scheduling problem Accordingly, operations of producer, scrounger, and ranger are newly designed An insertion-based Pareto local search is put forward in producer procedure, a crossover operation is introduced in scrounger procedure, and a local search based on the insert neighborhood is designed in ranger procedure A bunch of computational experiments and results show that the proposed algorithm is superior to two existing powerful meta-heuristics in terms of both inverted generational distance and set coverage Keywords Meta-heuristics, flow shop, multi-objective, blocking, scheduling Date received: 26 January 2016; accepted: 19 July 2016 Academic Editor: Xichun Luo Introduction Scheduling problems play an important role in industrial engineering and operation research and have attracted widespread scientific attention these years Two common problems which frequently appear in the scheduling literature are the permutation flow shop scheduling problem (PFSP)1,2 and job shop scheduling problem (JSP).3–5 The PFSP assumes that there are enough intermediate buffers for jobs between two consecutive machines, whereas the buffers are limited in real production If there is no buffer for a job and the next machine is busy, then the job has to stay on the incumbent machine and block itself Assume that the buffer size is zero between any two consecutive machines, a job is easy to block itself and hence the production is greatly delayed Such a problem is called the blocking flow shop scheduling problem (BFSP) The BFSP extensively exists in all sorts of industrial environments, such as petrochemical process, batch process, plastics molding, and steel manufacture.6,7 A great deal of research work has been done on the BFSP It was proved to be non-deterministic polynomial-time (NP)-hard for minimizing makespan for the case of more than two machines.8 In 2007, Companys and Mateo9 proposed a branch and bound method for makespan minimization in the BFSP, and they solved the problem instances with small sizes Hybridizing the School of Information and Electrical Engineering, Ludong University, Yantai, China Corresponding author: Deng Guanlong, School of Information and Electrical Engineering, Ludong University, Yantai 264025, China Email: dglag@163.com Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage) Downloaded from ade.sagepub.com by guest on November 22, 2016 Advances in Mechanical Engineering dynamic programming and branch and bound methods, Bautista et al.10 presented a bounded dynamic programming method, and the method was effective for small-size instances but powerless to solve instances with large sizes As regards the heuristic, there were profile fitting (PF),11 Nawaz–Enscore–Ham (NEH),12 minmax (MM), the hybridization of MM and NEH (MME), and the hybridization of PF and NEH (PFE).13 In recent years, meta-heuristics have been an extensively used method for the scheduling field To minimize the makespan in the BFSP, a genetic algorithm was proposed by Caraffa et al.,14 and a tabu search was presented by Grabowski and Pempera.15 Besides, some other algorithms, including the hybrid discrete differential evolution algorithm,16 the iterated greedy algorithm,17 and the hybrid harmony search algorithm,18 were put forward To minimize the total flow time in BFSP, Deng et al.19 introduced a hybrid discrete artificial bee colony algorithm which achieved good performance The above existing researches are oriented to a single criterion, while multi-objective scheduling problems have gained increasing focus since multiple criteria are usually encountered in real-life scheduling problems For the multi-objective JSP, Gao et al.20 proposed a Pareto-based grouping discrete harmony search algorithm and a discrete harmony search algorithm.21 For the multi-objective PFSP, Yenisey and Yagmahan22 provided an extensive review Although the multiobjective PFSP is widely studied by many researchers, there are less research reports for the multi-objective optimization in the BFSP This article considers the minimization of both makespan and total flow time and presents a multi-objective discrete group search optimizer (MDGSO) to search the Pareto-optimal solutions Figure Gantt chart of a BFSP example sequence is 1-2-3-4, which consequently results in the makespan 22 and total flow time 62 It is worth noting that a job sequence different from 1-2-3-4 probably causes the changes of makespan and total flow time, and there is contradiction between these two objectives Assuming that we have two different schedules pa and pb and the makespan of pa is less than that of pb , there is a possibility that the blocking time in schedule pa is more than that in schedule pb , and hence for some jobs, the completion time in schedule pa is more than that in schedule pb , making the total flow time of schedule pa more than that of schedule pb The minimization criteria in this article are makespan and total flow time For the considered multiobjective BFSP, the feasible solution is represented as job permutation p = fp(1), p(2), , p(n)g, where pj denotes a job number Let dp(j), i denote the departure time of job p(j) from machine Mi, then we can compute the departure time first for p(1), then for p(2), and so on until p(n) as follows dp(1), = ð1Þ dp(1), i = dp(1), iÀ1 + pp(1), i , i = 1, , m À ð2Þ dp(j), = dp(jÀ1), , j = 2, , n ð3Þ dp(j), i = maxfdp(j), iÀ1 + pp(j), i , dp(jÀ1), i + g, Multi-objective BFSP In BFSP, there are n jobs (job j, j = 1, 2, , n) and m machines (machine Mi, i = 1, 2, , m) Each job has to be processed first on machine M1, then on machine M2, , and finally on machine Mm The processing time of job j on machine Mi is known as p(j, i) The blocking constraint exists in the production process, which means there is no buffer between any two consecutive machines To be specific, the following assumptions are given: (1) at any time, a machine is able to process at most one job, and a job is able to be processed on at most one machine; (2) no job splitting is allowed; (3) all the jobs and machines are available at time zero; and (4) the set-up time, release time, and transfer time are omitted A Gantt chart of BFSP with four jobs and three machines is shown in Figure 1, where the blocking time is marked as shadow rectangles In Figure 1, the job j = 2, , n, i = 1, , m À dp(j), m = dp(j), mÀ1 + pp(j), m , j = 1, , n ð4Þ ð5Þ where dp(j), denotes the start time of p(j) on machine M1 Let f1 (p) and f2 (p) denote the objective value of makespan and total flow time of permutation p, respectively, then we have f1 (p) = dp(n), m f2 (p) = n X dp(j), m ð6Þ ð7Þ j=1 These two objective values can be computed in O(mn) time Let P denote the set of all the job permutations, the multi-objective BFSP is formulated as Downloaded from ade.sagepub.com by guest on November 22, 2016 Guanlong et al Minimizeff1 (p), f2 (p)g for all p P ð8Þ Pareto domination states that solution pa dominates solution pb if and only if 8i f1, 2g, fi (pa ) fi (pb ) and 9i f1, 2g, fi (pa )\fi (pb ) Solution pa is optimal in the Pareto sense if there is not any solution pb which dominates pa Pareto-optimal set is the set containing all Pareto-optimal solutions and Pareto front is the set of all objective values corresponding to the solutions in the Pareto-optimal set This article aims to solve the multi-objective problem by Pareto-optimal approach and propose a newly designed algorithm for the considered problem MDGSO The group search optimizer (GSO) was first presented for the continuous function optimization.23 The population of the GSO consists of three roles: producers, scroungers, and rangers, and each role has its specific updating mechanism Producers perform producing strategy by simulating the animal scanning mechanism, scroungers perform scrounging strategy by joining resources uncovered by others, and ranger search for the resources by random walks In each generation, the best member is selected as the producer, and certain individuals in the group are treated as scroungers, while the other individuals are handled as rangers Since the scheduling problem is different from the continuous function optimization problem, here individuals in GSO are designed as job permutations p Besides, the problem considered here is a multiobjective problem, so the mechanisms of producers, scroungers, and rangers need to be newly designed Population initialization In each generation, the proposed MDGSO maintains a population with size ps, which is denoted as PL = {L1, L2, , Lps} To get an initial PL with good performance, both the NEH and NEH_WPT24 heuristics are applied Specifically, we use the NEH heuristic to construct a permutation pNEH for the makespan criterion Then, the NEH_WPT heuristic is performed to generate a permutation pNEH WPT for the total flow time criterion Note that the NEH-WPT was validated as an effective heuristic for the total flow time criterion The two solutions are added into PL, and the remaining ps2 initial individuals are generated randomly Such an initial strategy is able to obtain an initial population with both diversity and quality The algorithm also maintains a set of non-dominated solutions, which is denoted as NS = {S1, S2, , Snb}, where nb is the incumbent size of NS The set NS is an independent set which stores the non-dominated solutions found by the algorithm so far After the Figure Solutions in population and non-dominated set initialization of PL, the NS is initialized by the nondominated solutions of PL Besides, each solution in PL is initially marked as ‘‘unsearched.’’ An example of PL and NS is shown in Figure 2, where ps = 20 and nb = Producer procedure The purpose of producers is to explore the neighborhood region of a relatively better solution For the multi-objective problem, the relatively better solutions are non-dominated solutions stored in NS So the producer is designed as follows: Step If there exists an ‘‘unsearched’’ solution Sk in NS, then let X = Sk, and go to Step Step Randomly select a solution Sh from NS, and let X = Sh Apply d insert moves to X and perform an insertion-based Pareto local search (IPLS) on X Step Perform IPLS on X If X is not updated, mark the corresponding Sk in NS with ‘‘searched.’’ In Step of the above procedure, the solution Sk is selected as follows: if there exists one and only one ‘‘unsearched’’ solution in NS, then the solution is assumed as Sk; if there exists more than one ‘‘unsearched’’ solutions in NS, then a randomly selected one among them is assumed as Sk In Step of the above procedure, the insert move means randomly selecting a job from the permutation and inserting it to another position of the permutation The IPLS searches the insert neighborhood of X in the way that X is updated if and only if a neighbor dominates X The searching process does not stop until a local optimum is found The steps of IPLS are as follows: Downloaded from ade.sagepub.com by guest on November 22, 2016 Advances in Mechanical Engineering Step Randomly generate a permutation pr = fpr1 , pr2 , , prn g, and let i = and j = Step Find the position of job prj in X, insert the job into the other n positions in X to obtain n solutions, and find the local non-dominated solution set (denoted as LNS here) of these n solutions Step If there exists an X LNS and X 0 X , then let LNS = LNS\X#, X = X , i = 1; else i = i + Step Let j = (j + 1) % n, update NS using LNS and mark all the newly added solutions in NS as ‘‘unsearched.’’ Step If i \ n, go to Step 2; else update NS using X and mark all the newly added solutions in NS as ‘‘unsearched.’’ Note that the producer procedure is applied for one time in each generation, and whenever a solution is newly added in NS, the algorithm does not apply the producing operation on it immediately, just marking it as ‘‘unsearched.’’ Scrounger procedure In each generation, all the individuals in the population are either selected as scrounger (with probability p) or ranger (with probability p) In MDGSO, the scrounger is designed by introducing the crossover operator in genetic algorithm Specifically, we randomly select a solution in NS and apply partially mapped crossover to it and the current scrounger individual Then, the current scrounger individual is updated according to the dominance relations of parent and offspring The steps of scrounger procedure are as follows: Step Randomly select a solution (denoted by Sk) in NS and perform partially mapped crossover on Sk and the current scrounger individual (denoted by Li) Denote the obtained offspring as X1 and X2 Step Update NS using X1 and X2 and mark all the newly added solutions in NS as ‘‘unsearched.’’ Step If Li X1 (A B means A dominates B) and Li X2 (see Figure 3(a)), then Li remains unchanged; if Li X1 (or X2) but Li does not dominate X2 (or X1) (see Figure 3(b)), then Li is replaced with X2 (or X1); if Li does not dominate X1 and Li does not dominate X2, then two cases exist: if X1 (or X2) dominates X2 (or X1) (see Figure 3(c)), then Li is replaced with X1 (or X2); else (see Figure 3(d)), Li is replaced with a randomly selected offspring Figure The dominance relation of offspring solutions (a) Li dominates both x1 and x2, (b) Li dominates x1 but not dominates x2, (c) Li does not dominates x1 or x2 while x1 dominates x2, (d) Li does not dominates x1 or x2 while no domination relation exists between x1 and x2 Downloaded from ade.sagepub.com by guest on November 22, 2016 Guanlong et al Ranger procedure In GSO, ranger is designed to perform random search; however, in the proposed MDGSO, the search procedure of ranger is not completely random but based on the solution set NS For a ranger Li, first a solution in NS is randomly selected Then, the insert neighborhood (denoted as Nb(X) for solution X) is searched and a descend direction is selected The search process iterates until the solution could not be improved in that descend direction At the end of ranger procedure, the incumbent ranger individual Li is replaced by the final solution found by the search process The whole steps are as follows: Step Randomly select a solution Sk in NS, and let X = Sk Step If there exists a solution X Nb(X ), and f1 (X )\f1 (X ), then go to Step 3; if there exists a solution X Nb(X ), and f2 (X )\f2 (X ), then go to Step 4; else stop Step Nb(X) = Nb(X)\X#, update NS using Nb(X) and mark all the newly added solutions in NS as ‘‘unsearched.’’ X = X , if there exists a solution X Nb(X ), and f1 (X )\f1 (X ), then go to Step 3; else go to Step Step Nb(X) = Nb(X)\X#, update NS using Nb(X) and mark all the newly added solutions in NS as ‘‘unsearched.’’ X = X , if there exists a solution X Nb(X ), and f2 (X )\f2 (X ), then go to Step 4; else go to Step Step Update NS using X If X is newly added in NS, denote it as ‘‘searched.’’ The incumbent ranger individual Li is replaced by X Procedure of MDGSO In the proposed MDGSO, the iteration is executed after the initial population PL and non-dominated solution set NS is generated In each iteration, first the producer procedure is performed, then for each individual in PL, either scrounger procedure (with probability p) or ranger procedure (with probability p) is performed There are only three parameters, ps, d, and p, to be determined in MDGSO, and the whole procedure is illustrated in Figure Computational experiments and results To validate the effectiveness of the proposed MDGSO, the well-known Taillard benchmark set which can be downloaded from the OR library (http://people.brunel.ac.uk/;mastjjb/jeb/info.html) was used This article tackled 90 instances, in which job number is from 20 to 100, and machine number is from to 20 It should be noted that all tested algorithms were programmed in C++ language, and a PC with Window operating Figure The flow chart of MDGSO system, Intel(R) Core(TM) i7-2600 CPU @3.06 GHz and GB RAM was used to execute the algorithms Performance measures There are various performance measures for multiobjective optimization to compare the performance of different algorithms We use the following two measures: Inverted generational distance (IGD).25 Let P* be a set of reference solutions and A be the nondominated solution set found by an algorithm The generalized distance of point x and point y is vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uX fi (x) À fi (y) 2 d(x, y) = t fimax À fimin i=1 ð9Þ where fi ( Á ) is the ith objective value, and fimax , fimin are the maximum value and minimum value of the ith objective for P* The IGD for A is computed as IGD(A, PÃ ) = Downloaded from ade.sagepub.com by guest on November 22, 2016 X d(x, y) jPÃ j y2PÃ x2A ð10Þ Advances in Mechanical Engineering The IGD index reflects the average value of the minimum distance of A and all solutions in P* Set coverage.26 Let A and B are two nondominated solutions, the set coverage C(A, B) represents the percentage of solutions in B that are dominated by at least one solution in A, computed as C(A, B) = jfx Bj9y A : y xgj j Bj ð11Þ Computational results In our computational experiments, the MDGSO was applied to solve each of the 90 instances with 10 replicates Note that for each instance, we could obtain a solution set for each run For convenience, all the solution sets of 10 replicates were gathered and a nondominated solution set A was obtained Two existing powerful meta-heuristics, non-dominated sorting genetic algorithm (NSGA-II)27 and bi-objective multistart simulated annealing algorithm (BMSA),28 were adapted for the considered problem here for comparison For each instance, we obtained the non-dominated solution set for each algorithm and then the performance measures were computed Note that since the Pareto front for each instance was not known, the reference solution set P* was formed by gathering all the non-dominated solution sets for all tested algorithms Regarding the parameter calibration for the MDGSO, the bigger the population size ps, the better the results are expected to be However, a bigger ps value will inevitably cause more computational expense Parameter d also affects the performance of the proposed algorithm If it is too big, the solution obtained by d insert moves in producer procedure will possibly lose good characteristics of the original solution If it is too small, the obtained solution will be possibly not able to escape the original local optimum As for parameter p which determines the probability that an individual is treated as scrounger, a relatively bigger value is a better choice since the ranger usually uses lower probability in GSO After some pilot experiments, we found that when the parameters were set as ps = 15, d = 6, and p = 0.8, the algorithm achieved relatively better performance Thus, in the computational experiments, the above parameter setting was used for the MDGSO while the parameters of the other two algorithms were set as the original papers in the literature The stopping criterion was set as 30mn ms The statistical results are given in Tables and 2, showing IGD values and set coverage values, respectively In the tables, the results are grouped by instance size for convenience, and bold numbers denote the better values Table The IGD values of the algorithms for instances grouped by different sizes Instance size IGD 20 20 10 20 20 50 50 10 50 20 50 50 10 50 20 Average MDGSO BMSA NSGA-II 0.00 0.00 0.00 0.01 0.04 0.02 0.00 0.01 0.02 0.01 0.00 0.00 0.00 0.11 0.13 0.04 1.89 0.43 0.32 0.33 0.01 0.00 0.00 0.25 0.28 0.14 3.31 1.12 0.87 0.67 IGD: inverted generational distance; MDGSO: multi-objective discrete group search optimizer; BMSA: bi-objective multi-start simulated annealing algorithm; NSGA: non-dominated sorting genetic algorithm Table The set coverage values of the algorithms for instances grouped by different sizes Instance size 20 20 10 20 20 50 50 10 50 20 50 50 10 50 20 Average MDGSO (A) vs BMSA (B) MDGSO (A) vs NSGA-II (C) C(A, B) C(B, A) C(A, C) C(C, A) 0.01 0.01 0.01 0.82 0.50 0.53 0.96 0.88 0.84 0.51 0.00 0.00 0.00 0.13 0.40 0.41 0.01 0.08 0.08 0.12 0.03 0.01 0.02 0.98 0.81 0.93 1.00 1.00 1.00 0.64 0.00 0.00 0.00 0.00 0.18 0.03 0.00 0.00 0.00 0.02 MDGSO: multi-objective discrete group search optimizer; BMSA: bi-objective multi-start simulated annealing algorithm; NSGA: non-dominated sorting genetic algorithm Figure The non-dominated solutions obtained by the three algorithms for Ta36 Downloaded from ade.sagepub.com by guest on November 22, 2016 Guanlong et al Figure The non-dominated solutions obtained by the three algorithms for Ta86 From Table 1, we can see that for the instances with 20 jobs, the differences between the MDGSO and the other two algorithms are not significant, which is probably because of the simplicity of the small-size instances However, for the instances with 50 and 100 jobs, the IGD values of the MDGSO are clearly greater than the other two algorithms, which indicates the superiority of the MDGSO We note that for some groups, the IGD values of the MDGSO are nearly zero, which means that the solution sets obtained by the MDGSO are very close to the reference solution sets Table also indicates the superiority of the BMSA over the NSGA-II Table shows the differences of the algorithms’ performances in another facet For the comparison between the MDGSO and the BMSA, the C(A, B) value is greater while the C(B, A) value is smaller, which indicates that the MDGSO is superior to the BMSA Similarly, the values in the other two columns validate the superiority of the MDGSO over the NSGA-II To show the differences among the algorithms more clearly, we draw the non-dominated solution set of each algorithm for instance Ta36 and Ta86 in Figures and 6, respectively Note that the non-dominated solution set of each algorithm is formed by the results of 10 replicates It is seen from these two figures that the nondominated solutions found by the MDGSO have better diversity as well as better quality than those found by any of the other algorithms As an example, the Gantt chart of a non-dominated solution for instance Ta01 is shown in Figure 7, where the makespan is 1380 and the total flow time is 15,042 Conclusion This article has presented an MDGSO for the multiobjective optimization in blocking flow shop Figure Gantt chart of a non-dominated solution for instance Ta01 Downloaded from ade.sagepub.com by guest on November 22, 2016 Advances in Mechanical Engineering scheduling A population initialization method based on NEH heuristic and its variants has been designed to generate diversified solutions Besides, we have designed the strategies of producers, scroungers, and rangers by hybridizing some local search methods, and we have conducted a multitude of experiments minimizing both the makespan and total flow time objectives in blocking flow shop scheduling The computational results have shown that the proposed algorithm is superior to both the NSGA and BMSA in terms of IGD and set coverage In future, we will focus on adapting the discrete GSO to other complex scheduling problems, such as the no-wait job shop problem, the flexible flow shop problem, and stochastic scheduling problem 10 11 12 13 Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article 14 15 Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors appreciate the support of National Natural Science Foundation of China (Grant No 61403180), the Project for Introducing Talents of Ludong University (LY2013005), National Natural Science Foundation of China (Grant No 51407088), and Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (Grant No BS2015DX018) 16 17 18 19 References Garey MR, Johnson DS and Sethi R The complexity of flowshop and jobshop scheduling Math Oper Res 1976; 1: 117–129 Ruiz R and Stutzle T A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem Eur J Oper Res 2007; 177: 2033–2049 Gao KZ Effective ensembles of heuristics for scheduling flexible job shop problem with new job insertion Comput Ind Eng 2015; 90: 107–117 Gao KZ, Suganthan PN, Chua TJ, et al A two-stage artificial bee colony algorithm scheduling flexible jobshop scheduling problem with new job insertion Expert Syst Appl 2015; 42: 7652–7663 Gao KZ, Suganthan PN, Pan QK, et al An effective discrete harmony search algorithm for flexible job shop scheduling problem with fuzzy processing time Int J Prod Res 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multi- objective JSP, Gao et al.20 proposed a Pareto-based grouping discrete harmony search algorithm and a discrete harmony... algorithm for the flowshop scheduling problem with blocking Omega 2011; 39: 293–301 Wang L, Pan QK and Tasgetiren MF A hybrid harmony search algorithm for the blocking permutation flow shop scheduling