Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime

Transcription

1 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Wagner Emanoel Costa Marco César Goldbarg Elizabeth G. Goldbarg Technical Report - UFRN-DIMAp RT - Relatório Técnico April Abril The contents of this document are the sole responsibility of the authors. O conteúdo do presente documento é de única responsabilidade dos autores. Departamento de Informática e Matemática Aplicada Universidade Federal do Rio Grande do Norte

2 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Wagner Emanoel Costa wemano@gmail.com Elizabeth G. Goldbarg beth@dimap.ufrn.br Marco César Goldbarg gold@dimap.ufrn.br Abstract. This paper presents a new hybridization of VNS and path-relinking on a particle swarm framework for the permutational fowshop scheduling problem with total flowtime criterion. The operators of the proposed particle swarm are based on path-relinking and variable neighborhood search methods. The performance of the new approach was tested on the bechmark suit of Taillard, and five novel solutions for the benchmark suit are reported. The results were compared against results obtained using methods from literature. Statistical analysis favors the new particle swarm approach over the other methods tested. Keywords: Flowshop, Scheduling, Total Flowtime, Heuristics, PSO. Resumo. Este documento apresenta uma nova hibridização entre VNS e pathrelinking dentro do framework de otimização por nuvem de partículas para o problema de minimização de total flowtime dentro do contexto de escalonamento flowshop permurtacional. Os operadores do método proposto são baseados em métodos de path-relinking e busca por variação de vizinhanças. O desempenho da nova abordagem foi testada usando o conjunto de instâncias do Taillard, e cinco novas soluções para o conjunto são reportadas. Os resultados são comparados contra resultados obtidos usando métodos da literatura. Análise estatística favorece a nova abordagem de nuvem de partículas contra os métodos testados. Palavras-Chave: Flowshop, Scheduling, Total Flowtime, Heuristicas, PSO. 1 Introduction The Permutational Flowshop Scheduling Problem (PFSP) with the total flowtime criterion (T F T ) is a NP-hard Combinatorial Optimization Problem [11, 15] that deals with the scheduling of a set of jobs, J, through a set of machines, M. Every job has to be processed Programa de Pós-graduação em Sistemas e Computação, UFRN Departamento de Informática e Matemática Aplicada - UFRN/CCET/DIMAp Departamento de Informática e Matemática Aplicada - UFRN/CCET/DIMAp 1

3 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 2 on all machines following the same machine sequence. Each job requires a definite machineprocessing time and can be processed on one machine at a time. Each machine processes only one job at a time. Once a machine starts processing a job, no preemption is allowed, and the machine becomes available as soon as the operation is finished. The goal is to produce a schedule of jobs such that the sum of completion times is minimized. Many methods have been proposed to address this problem. There are constructive methods [23, 20, 10, 21, 9, 18], local search method [5], iterated greedy approach [22], genetic algorithms [33, 31, 7], ant colonies [24, 34], particle swarm optimization [28, 19, 17], bee colony optimization [29], VNS [16, 3], hybrid EDA approaches [16, 32], hybrid discrete differential evolutionary algorithm [29] and parallel simulated annealing [4]. The particle swarm optimization (PSO) is a metaheuristic proposed by Eberhart & Kennedy [8] for continuous optimization. Due to its simplicity it has been adapted to many discrete optimization problems including the PFSP [28, 19, 17]. The PSO approach was based on models explaining the synchronous movements in a flock of birds. In those models the birds attempt to keep an optimal distance from each other so they can be close enough to profit over the discoveries and previous experience of a neighboring bird, and at the same time avoiding the competition for food [8]. The proposed discrete PSO is based on the work of Goldbarg et al. [14] for the Traveling Salesman Problem. Therefore, the cited metaphor is translated into an optimization method as follow. Agents, named particles, fly over the solution space. A particle occupies a position on the solution space. The position of a particle represents a valid solution currently under exam, it is encoded as a permutation of jobs (Π), and it has a objective value associated to it, named T F T (Π). Besides knowing their current position, the particle knows the best site (solution) it previously visited, knows the current position of a neighboring particle, and the best sites previously visited by a neighbor. On any given iteration of the method, each particle will compromise in doing one of the following actions: a) to explore the solution space on its own; b) to move towards the current site of a neighboring bird c) to move towards the best site previously visited by a neighboring bird. A probability is assigned to each possible action. During execution those probabilities are updated, the update process takes into account the quality of the last solution obtained through each action. The actions of the particles are implemented using search operator such as variable neighborhood search procedure (VNS) and path-relinking. Given a particle A, if A chooses to explore the search space on its own, A copies the configuration of the best previous site it visited Π A,Best to its current position Π A,Curr, a VNS procedure is executed over Π, the resulting solution becomes the current position of A, Π A,Curr. If the particle chooses to move towards a neighboring solution B, the second possible action, a combination of path-relinking procedure with VNS is executed. Initially, the path-relinking gradually transforms the current position Π A,Curr (solution) of particle A in the position Π B,Curr, occupied by the neighboring particle B. If during this process an intermediate solution Π A,Curr is found, such that the value of its objective function is smaller than the objective function values of Π A,Curr and Π B,Curr, then path-relinking is interrupted and the VNS procedure is executed over Π A,Curr, resulting in Π A,Curr. After VNS finishes, path-relinking is resumed transforming Π A,Curr into Π B,Curr. A new interruption may occur if a solution better than both Π A,Curr and Π B,Curr is found. The action concludes when no further improvement is found during path-relinking. The third action is implemented similarly to the second action, however in this case the target position is the best site visited by the neighboring particle B (Π B,Best ) instead of its current position (Π B,Curr ). The remainder of this article is organized into three sections. In Section 2 the description of the problem is given. Section 3 describes the hybrid algorithm proposed for the PFSP with

4 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 3 T F T criterion. Section 4 addresses the experiments carried out to tune parameters involved in proposed approach. Section 5 presents computational experiments comparing the proposed PSO with two state-of-the-art methods from literature. Statistical analysis over the results indicates significant differences favoring of the proposed PSO over two state-of-the-art approaches. Finally, some conclusions are presented in Section 6. 2 Problem As stated earlier, in the permutation flowshop context a set of jobs J = {1,..., n} will be processed by a set of machines M = {1,..., m} in sequence. Job j has a processing time of T jr on machine r, 1 j n, 1 r m. Let the permutation Π = {π 1, π 2,..., π n } denotes the job-processing order, and π i corresponds to the i-th job on the schedule Π. The completion time of job π i on machine r, denoted by C(π i, r), is given by the time elapsed since the first job begins to be operated on the first machine until job π i is completed on machine r. C(π i, r) is proper evaluated through Eq. (1) to (4), where Eq. (1) refers to the conclusion time of the first job scheduled on the first machine, and Eq. (2) refers to the conclusion time of the first job scheduled on the remainder machines. Analogously, Eq. (3) evaluates the completion time of job π i, 1 < i n, on the first machine (r = 1), whereas Eq. (4) evaluates the completion time of job π i, 1 < i n, on the remainder machines, 1 < r m. Based on Eq. (1) to (4), the total flowtime value of a given permutation Π (T F T (Π)) is defined as the sum of completion times on the last machine (Eq. 5). C (π 1, 1) = T π1,1 (1) C (π 1, r) = C (π 1, r 1) + T π1,r r {2,..., m} (2) C (π i, 1) = C (π i 1, 1) + T πi,1 i {2,..., n} (3) C (π i, r) =max {C (π i, r 1), C (π i 1, r)} + T πi,r i {2,..., n} r {2,..., m} (4) T F T (Π) = n C (π i, m) (5) i=1 3 Discrete Particle Swarm Optimization This section describes the proposed discrete PSO to optimize the total flowtime, and there are five subsections. The first subsection shows the pseudo-code of the proposed PSO and discusses four procedures to be defined. Each subsequent subsection explains in details each procedure and their tuning parameters.

5 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Pseudo-code Algorithm 1 exhibits the pseudo-code of the proposed method. The PSO procedure depends on other procedures, at first the pseudo-code of PSO is explained and the subsequent paragraphs detail the procedures on which PSO depends on. The first line of the algorithm initializes the set of particles (P articles), meaning that the position of each particle is set to an individual solution. The best visited site of a given particle is also initialized with its current position. The main body of PSO lies on the loop from lines 2 to 20. Usually the main loop on PSO is repeated until the stop criterion is satisfied, in this case the stop criterion was defined as a time limit of 0.4 n m seconds, as the fastest heuristics for minimizing T F T in PFSP context use the same stop criterion [16, 32, 31, 3]. In line 4, probabilities v a, v b and v c are defined in the procedure ComputeP robabilities standing for the probabilities of particle P choses the first, second or third action, respectively. In line 5 a random number is picked from the interval [0, 1] and stored in variable Action. If the value of Action is smaller or equal to v a (line 6), then particle P explores the search space in function ExploreAlone, otherwise P will move towards a neighbor or towards a best site of a neighbor. In both cases a destiny must be selected, this is done in lines 9 to 12. In line 9, the neighbor, of which either its current position or its best site will be used as a guide, is selected randomly. The loop from line 10 to line 12 ensures that a neighbor is selected. If the value of Action is greater than v a and smaller or equal to v b then particle P will move from its current position (Π P,Curr ) towards a site currently occupied by a neighbor (Π T arget,curr ). That move is made in procedure MoveT oneighbor, line 14. If the value of Action is greater than v b then P moves towards the best site a neighboring particle has occupied, line 16. The main loop finishes when the time limit is reached. In lines 21 to 26 the algorithm finds the best solution ever achieved by a particle and returns it in BestSol, line 27. The proposed discrete PSO depends on the implementation of the following procedures: a procedure to create an initial position for each particle; a procedure to define the initial probabilities associated with each possible action, and to update those probabilities (ComputeP robabilities); the explore alone procedure; a procedure that is capable of executing the movements towards a neighbor and the move towards the best site of a neighboring particle. The following subsections explain how each of these procedures were implemented and the parameters involved. 3.2 Initial positions The initialization is performed by a randomized version of the greedy algorithm H(1) presented by Liu & Reeves [20]. Several state-of-the-art approaches use either H(1) or a procedure based on this heuristics to provide initial solutions [5, 31, 3], for it was tested, in Dong et al. [5], that the methods based on the heuristic H(1), like the method H(x), are better providers for initial solutions than other tested heuristics. The method H(x) creates x different solutions by placing a distinct job on the first position and applying the greedy criterion of H(1). However, H(x) is a greedy procedure, after the first job is placed, it adds the remaining jobs to a solution deterministically according to the greedy function. Therefore, by using H(x)

6 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 5 Algorithm 1 Discrete PSO 1: Initialize the set of particles P articles 2: repeat 3: for each particle P with position Π P,Curr and best visited site Π P,Best do 4: (v a, v b, v c ) ComputeP robabilities() 5: Action random number [0, 1] 6: if Action v a then 7: ExploreAlone(P ) 8: else 9: T arget random number {1, 2,..., P articles } 10: while T arget = P do 11: T arget random number {1, 2,..., P articles } 12: end while 13: if v a < Action v b then 14: MoveT owards(π P, Π T arget,curr ) 15: else 16: MoveT owards(π P, Π T arget,best ) 17: end if 18: end if 19: end for 20: until time limit of 0.4 n m seconds is reached 21: BestSol Π 1,Best 22: for P = 2 to P articles do 23: if T F T (Π P,Best ) < T F T (BestSol) then 24: BestSol Π P,Best 25: end if 26: end for 27: return BestSol

7 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 6 the number of different solutions is limited to the number of jobs (n) of the considered instance. Then, the number of particles of an instance with 20 jobs is limited to 20. In order to avoid this limitation, a randomized version of H(1) was implemented. The randomized version of H(1) builds at each step a subset with the three fittest jobs, according to the greedy criterion, and choose iteratively one of the three jobs to be added to to the solution. The job is chosen at random according to a uniform distribution. The greedy criterion of H(1) is explained next. The procedure H(1) weights down two criteria: weighted sum of machine idle time (IT ) and artificial flowtime (AT ). The idle time criterion for selecting job i when k jobs were already selected (IT ik ) is defined by Eq. (6), where w rk is calculated with Eq. (7). The term max {C(i, r 1) C(π k, r), 0} stands for the idle time of machine r. The weights, as defined on Eq. (7), stress that idle times on early machines are undesirable, for they delay the remaining jobs. Such stress is stronger if there are many unscheduled jobs (small value of k), and drops when the number k of scheduled jobs increases. IT ik = m w rk max {C(i, r 1) C(π k, r), 0} (6) r=2 w rk = m r + k(m r)/n 2 The artificial flowtime (AT ik ) of candidate job i after k jobs were scheduled refers to the T F T value obtained after including unscheduled job i, plus the time of an artificial job placed at the end of the sequence of jobs. The processing time of the artificial job on each machine is equal to the average processing time of all unscheduled jobs, excluding job i, on the correspondent machine. Both criteria, IT ik and AT ik are combined according to Eq. (8). 3.3 Defining and updating the probabilities f ik = (n k 2)IT ik + AT ik (8) The procedure ComputeP robabilities is responsible for defining the initial values for the probabilities and updating them for each particle. Initially, it assigns probability ρ to the action of exploring the search space on its own, and probability (1 ρ)/2 to each of the other action, that is, move towards a neighbor or towards a best site. Each time an action is performed the value of the solution obtained is stored in a specific variable, i.e. when a particle P explores the search space on its own, producing a solution Π, the value T F T (Π) is stored in variable T F T P,Alone, similarly if P moves towards a neighbor or towards a best site, the T F T values are stored in T F T P,Neighbor and T F T P,Site, respectively. Once particle P has performed each action at least once, ComputeP robabilities compares T F T P,Alone, T F T P,Neighbor and T F T P,Site, and assigns probability ρ to the action corresponding to the best T F T value, and probability (1 ρ)/2 for each of the other two possible actions. The value of ρ is a parameter to be tuned. 3.4 Exploring the search space on its own Whenever it is decided for a particle P to explore the search space on its own the procedure ExploreAlone(P ) is called. This procedure is based on the VNS4 procedure of [3]. The method VNS4 is a variable neighborhood search. It uses two neighborhood structures, job interchange and job insert. In the job interchange neighborhood, two jobs exchange (7)

8 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 7 positions, i.e. given two solutions Π A = {π A1, π A2,..., π An } and Π B = {π B1, π B2,..., π Bn }, they are neighbors if there are indices s and t, s t, such that π As = π Bt, π At = π Bs and c, c s, t, π Ac = π Bc. In the job insert neighborhood, a job is removed from its position in the solution and re-inserted in a different position, i.e. Π B = {π B1, π B2,..., π Bn } is a neighboring solution of Π A if there are indices s, t and c, s < t, such that one of the two possibilities is true: π Bs = π At and c, s c < t, π Bc = π Ac+1, or π Bt = π As and d, s < d t, π Bd = π Ad 1. VNS4 applies fourteen random insert moves on a solution as a Shake procedure, then it explores the job interchange neighborhood until no further improvement is possible. The first improvement policy is used in the job interchange, i.e. the current solution is replaced by the first neighbor whose T F T value is smaller than the former. After job interchange is fully explore, VNS4 starts using job insert neighborhood and as soon as a neighbor improving the current solution is found, VNS4 resumes the use of the interchange neighborhood. Different stop criteria were tested for the use of VNS4 within the proposed PSO. The experiment comparing all proposed stop criteria is detailed in Section 4. Algorithm 2 exhibits the pseudo-code of ExploreAlone(P ). In line 1, Π becomes a copy of the best solution found by particle P. The procedure VNS4 is applied over Π, line 2. After reaching a local optimum, Π is tested to verify if its T F T value is different from the T F T value of the other particles in their current position or in their best sites, line 3, if so the current position of P is updated, line 4, otherwise the current position of P remains unchanged. If there was an update in the current position of P then it is tested if this new site is better than the best site recorded for P (Π P,Best ), line 5, if so Π P,Best is updated, line 6. Algorithm 2 ExploreAlone(P ) 1: Π Π P,Best 2: V NS4(Π ) 3: if T F T (Π ) is different from any T F T of the other recorded positions then 4: Π P,Curr Π 5: if T F T (Π P,Curr ) < T F T (Π P,Best ) then 6: Π P,Best Π P,Curr 7: end if 8: end if 3.5 Moving towards a different site The last procedure discussed is M ovet osite, which is responsible for the movement of a particle towards the current site of a neighbor, or towards the best site a neighbor has ever achieved. Algorithm 3 shows the pseudo-code of MoveT osite. Initially, the solution Π is a copy of the current site of particle P (line 1). Within the loop from line 2 to 12, the procedure alternates between the use of a truncated path-relinking and VNS4, until no further improvement is possible. The VNS4 procedure is the same used in function ExploreAlone. In T runcatedp athrelinking, a path-relinking method is implemented. The path-relinking strategy was by Glover [13]. Given two solutions, an initiating solution, Π O, and a destiny solution, Π Destiny, the path-relinking strategy generates intermediate solutions by inserting into Π O properties of Π Destiny [12]. The sequence of solutions Π O, Π 1, Π 2,..., Π Destiny is called a path that links Π O to Π Destiny. The intermediate solutions are of interest for some of them may have better value of objective function (in this case T F T value) than both Π O and Π Destiny.

9 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 8 There are several types of path-relinking, e.g. forward path-relinking, backward and backand-forward path-relinking [25], among others. In forward path-relinking the initiating solution is the one with worse value of objective function assigned to it. In backward path-relinking the solution with the worse value is the destiny solution. In back-and-forward both trajectories, forward and backward, are examined. According to experiments reported in Ribeiro & Resende [25], the backward path-relinking often outperforms the forward one, whereas the back-andforward path-relinking produces solutions at least as good as either the forward or backward option in optimization problems and sometimes outperforms them, therefore the path-relinking implemented in this work. During the tuning experiments three strategies of inserting properties of Π Destiny into Π O were tested. The first one uses the job insert neighborhood, it takes π Destiny,n, the last job of the destiny solution, and finds its current position in the initiating solution(π O ). Then that job is re-inserted in the last position of Π O. This procedure is repeated until Π O is fully transformed into Π Destiny, or until an intermediate solution, Π Q,with T F T value better than T F T values of Π O and Π Destiny is found, in this case Π Q is returned. The second strategy uses job interchange neighborhood instead of job insert. In the third option, adjacent jobs were swaped. The three strategies stop when an intermediate solution better than both the initiating solution and the destiny one is found. Algorithm 3 MoveT osite(π P,Curr, Π Destiny ) 1: Π Π P,Curr 2: repeat 3: Continue F alse 4: Π T runcatedp athrelinking(π, Π Destiny ) 5: if T F T (Π ) < T F T (Π P,Curr ) and T F T (Π ) < T F T (Π Destiny ) then 6: V NS4(Π ) 7: if T F T (Π ) is different from any T F T of the other recorded positions then 8: Π P,Curr Π 9: Continue T rue 10: end if 11: end if 12: until Continue = F alse 13: if T F T (Π P,Curr ) < T F T (Π P,Best ) then 14: Π P,Best Π P,Curr 15: end if 4 Tuning experiments This section details the experiments to tune the parameter of the algorithm proposed in Section 3. This section is organized in six subsections. First, the methodology employed to tune parameters is described. The four subsections that follow the first one describe the experiments performed to tune each parameter. The last subsection presents an overall summary of the tuning experiments and recollects the final attribute of each parameter.

10 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Methodology The methodology consists of comparative experiments where each parameter is examined independently [1, 6]. A subset of Taillard s instances was used in the experimentation, as representative samples of the testbed. The complete dataset contains 120 randomly generated instances [27]. The number of jobs of Taillard s benchmark is in the set {20, 50, 100, 200, 500} and the number of machines in {5, 10, 20}. The subset utilized for each comparative experiment reported here comprises the five first test cases of each group of 50 and 100 jobs of Taillard s dataset, which made a total of 30 instances. Twenty independent executions of each algorithmic version were performed for each instance, each execution used a different seed value for the random number generator. The experiments were executed on a pentium Core 2 - Quad 2.4GHz (Q6600), 1GB RAM. Implementations were done in C++ using Gnu C++ compiler with -O2 flag. Results obtained during trials were transformed into relative percentage deviation (RP D), which was calculated with Eq. (9), wherein Best refers to the best solution found during experimentation for a given instance, and Heuristic refers to a solution obtained by one algorithm on the same instance. Because RP D is a dimensionless value resulting from a normalization procedure, the RP Ds from different instances were compared, treating RP D as a response variable similar to what is considered in [26]. RP D(%) = 100 Heuristic Best Best Solution (9) The Shapiro-Wilk test [2] indicates non-normal distributions were obtained in several tests with significance levels bellow Therefore, Kruskal-Wallis test [2], a non-parametric counterpart of ANOVA, was used to test if the results were statistically different. If the result from Kruskal-Wallis indicates differences between the compared algorithmic versions with significance level below or equal to 0.05 (5%), the median RP D value, among the 20 independent executions of the 30 tested instances, was used to ascertain which version is the best. Once the parameters were examined independently, they needed to have an initial value. These values are listed bellow: The number of particles used by PSO is initially equal to 15; The initial value of ρ is 0.60 (60%); The path-relinking operator uses the 2-swap as neighborhood structure; The remaining four subsections refer to experiments related to: 1. the stop criteria of VNS4; 2. the number of particles; 3. the value of ρ; 4. the neighborhood structure used by path-relinking.

11 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Number of VNS iterations within PSO The first experiment performed concerns the stop criteria of VNS4. In the work of Costa et al. [3], it is conjectured that a hybrid genetic algorithm with VNS4 had poor performance on instances with 100 jobs or greater due to the time spent by VNS4 to find a local optimum. The second stop criterion in consideration for VNS4 is to constrain the number of improvements over a single solution, therefore stopping the procedure before it reaches a local optimum common to both neighborhoods. This criterion is inspired by the work Xu et al. [31], where a hybrid genetic algorithm limited the number of improvements made by a VNS approach (named E-VNS). For instance if the limit is set at 60 improvements, the VNS is interrupted as soon it improves a solution by the 60 th time. In total six versions VNS4 were tested, five constrained in the number of improvements over a solution that they can produce, and one, the original unconstrained version. The limits of the constrained versions ranged from 60 to 100 improvements over a given solution. The results of the experiments are listed on Table 1. According to the Kruskal-Wallis test there are differences between the different implementations with p-value lower than 10 3, these differences favors the unconstrained version as it is the one with the lowest RP D value for median. Therefore, the unconstrained version of VNS4 was adopted. Tabela 1: Median RPD values for different stop criteria for VNS4 within PSO. stop criterion for VNS4 Number of Improvements Median RP D(%) Number of Improvements Unconstrained Median RP D(%) Number of particles The second parameter tuned was the size of the set of particles, that is the number of particles present in PSO. As stated in Section 3, the initial position of a particle is defined by a randomized version of H(1), and the initial best site of each particle is the same as its initial position. Nine different sizes for the set of particles were tested starting from ten particles going up to a total of fifty particles. The median values obtained in the experiments are recorded in Table 2. According to the Kruskal-Wallis test, there are significant differences between the performances (p-value lower than 10 3 ). The PSO version with ten particles exhibited the best performance among the tested options; therefore the number of particles was then reduced to ten. 4.4 The value of ρ The parameter ρ is used in procedure ComputeP robabilities to define the probability assigned to each possible action of a given particle. During the first two tuning experiments that value was fixed at 0.60 (60%). The third aimed at tuning the value of ρ. Six values for ρ were

12 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 11 Tabela 2: Median RP D values for different sizes of the set of particle. Number of particles present in PSO Number of Particles Median RP D(%) Number of Particles Median RP D(%) Number of Particles Median RP D(%) tested, ranging from 40% to 90%. The experiments were conducted according to the methodology described in Subsection 4.1; the median RP D values obtained for the tested values of ρ are summarized in Table 3. The results of the Kruskal-Wallis test did not point out significant differences between the results obtained in the experiments (p-value equals to ). This fact indicates that the value of ρ did not influence on the performance of the implemented PSO. Therefore the value of ρ remained set to 60%. Tabela 3: Median RP D values for different values of ρ. The value of ρ Value of ρ 40% 50% 60% Median RP D(%) Value of ρ 70% 80% 90% Median RP D(%) Neighborhood structure of path-relinking The last tuning experiment regards the neighborhood structure used for path-relinking. Three neighborhood structures were under consideration: 2-swap, job insert and job interchange. The Kruskal-Wallis test returned p-value equals to indicating that there is no significant difference of performance was detected between the tested neighborhoods. Therefore, the neighborhood structure used by the path-relinking operator (2-swap) was maintained. Table 4 reports the median RP D concerning this experiment. Tabela 4: Median RP D values associated with the use of different neighborhood structures within path-relinking Neighborhood structure used for path-relinking Neighborhood 2-Swap Job Insert Job Interchange Median RP D(%)

13 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Overall summary Four distinct parameters of PSO were examined independently. In each experiment diverse attribute values for each parameter were compared. According to a non-parametric statistical test, from the four conducted experiments, two exhibited significant evidences of differences of performance between the tested attributes (the stop criterion of VNS4 and the number of particle), whereas for the other two parameters, ρ and the neighborhood structure used during path-relinking, no significant difference was indicated by the test. The final configuration of PSO parameters is as follows: VNS4 stops only when it reaches a local optimum common to both job interchange and insert neighborhoods; the number of particles is set to ten; the value of ρ remained at 0.60 (60%); the 2-swap neighborhood structure is the one used to perform path-relinking. 5 Comparison with methods from literature Experiments were performed comparing the performance of the proposed discrete PSO with the performance of two state-of-the-art methods, namely VNS4 [3] and the asynchronous genetic algorithm, AGA [31]. The authors of [31] kindly provided the source code of AGA. The comparison is done by using the Kruskal-Wallis statistical test over the results. There are also comparisons with the best solutions found by the proposed approach to the best solutions from state-of-the-art. Each best know result was tagged to identify the method that found it. The tags are: EDA-VNS [16], VNS [16], HGLS [31], SAwGE [4], hdde [30], DABC [30], PHEDA [32], AGA [31] and VNS4 [3]. The remainder of this section is organized as follows: first the method AGA is described. The method VNS4 was detailed in Section 3.4, so after Section 5.1, there is a subsection detailing the experiment, its methodology, summaries of the results and the statistical analysis of the data. 5.1 Asynchronous genetic algorithm (AGA) The method AGA is a hybrid genetic algorithm with a population of 40 solutions, one of them is the best solution from the H (n/m) heuristic, where n/m are created using the H criterion, followed by interchange local search, the other 39 are randomly generated. At each iteration all solutions undergoes an E-VNS procedure (a specific VNS approach), crossover followed by another execution of E-VNS. The E-VNS uses both job insert local and job interchange neighborhoods. When using job insert, the job re-inserted into another position, π i, is randomly selected. The position where π i will be re-inserted is also randomly selected. If such attempt improves the current solution, the new solution is accepted and a new iteration of job insert occurs. E-VNS executes 50 iterations of job insert. After that, 50 iterations using job interchange occurs, again the jobs to be interchanged are randomly selected. If any improvement is achieved while using interchange neighborhood the solution is accepted and E-VNS resumes job insert local search. The number of times job insert can be resumed is limited by a randomly selected value. Each time E-VNS is called, a new limit is randomly picked between 10 and 60; this allows E-VNS to terminate with a solution that is not a local optimum.

14 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime Results and statistical analysis The experiments were executed on a pentium Core 2 - Quad 2.4GHz (Q6600), 1GB RAM. The three methods, discrete PSO, VNS4 and AGA were encoded in C++. The experiments were performed over all 120 instances from Taillard s benchmark [27]. Thirty independent executions of each method were performed for each instance. The three methods share the same stop criterion, i.e. a time limit of 0.4 n m seconds. The results on instances with n = 20 jobs are not summarized in tables because most of the trials returned the best known results for them. From the 30 instances with n = 20 jobs the proposed PSO method found the best known solution for all of them on all of its independent executions. The AGA heuristics found the best known solution for 29 instances on all of its trials, and failed to do so once for one instance 1. The procedure VNS4 converged to the best known solutions on all of its trials on 20 cases, and failed to do so on 10 cases 2. For the remaining 90 instances, a summary of the results, consisting of the minimum, average, maximum and standard deviation values obtained by each algorithm, are summarized in Tables 5 to 7. Each line of these tables reports the instance name (Instance), the best T F T value reported in the literature for the instance (Best). The next column reports the tag of the stateof-the-art method which achieved the best known value (Algorithm), this column is followed by the minimum (Min), average (Ave), maximum (Max) and standard deviation (S.D.) values obtained by each method after 30 independent trials on the corresponding instance. Solutions presenting T F T value equal or better than the best known solution reported in the literature are indicated by a star symbol ( ). The best minimum and average values among the three tested methods are indicated in bold face. For the 30 instances with n = 50 jobs (Table 5), among the tested method, the proposed discrete PSO found the best minimum value on 24 instances, and it achieved the best average value on 27 instances. The method VNS4 found the minimum value on 5 cases and achieved the best average value on 2 cases. In the set of instances with n = 100 jobs, Table 6, discrete PSO found the minimum value on 23 cases and exhibited the best average on 26 cases. The AGA heuristics achieved the minimum value for 1 case (ta082). VNS4 achieved the best minimum on 6 cases and the best average on 4 cases. For the 30 instances with n 200 jobs (Table 7), discrete PSO achieved the minimum solution on 7 instances and achieved the best average on 8 instances. VNS4 exhibited the best minimum value on 23 cases, and the best average on 22 cases. Comparing the minimum value obtained by the three tested methods with the best known results, the discrete PSO found 5 novel solutions with T F T value better than those reported in the problem literature, all of them for instances with n = 500 jobs (supposedly the hardest ones). The discrete PSO also found 10 solutions with T F T values equal to the best reported in literature. The method AGA achieved 2 solutions with T F T value equal to the best known marks from literature for n = 50 jobs. VNS4 found 7 solutions, 1 solution for n = 200 jobs and 6 on cases with n = 500 jobs. In order to conduct the statistical analysis, the results from 90 instances were converted into RP D, using the same methodology of Section 4.1 and the statistical tests were applied to the transformed data. A statistical summary of the data is presented in Table 8. It contains the minimum RP D value, first quartile, median, third quartile, maximum, average and standard 1 The instance named ta The instances named ta001, ta003, ta014, ta016, ta018, ta021, ta022, ta029 and ta030.

15 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 14 Tabela 5: Summaries of results for instances with n=50 jobs. Minimum value (Min), average (Avg), maximum (Max) and standard deviation (S.D.) included. Discrete PSO AGA VNS4 Instance Best Algorithm Min Average Max S.D. Min Average Max S.D. Min Average Max S.D. tai SAwGE tai hdde tai hdde tai SAwGE tai SAwGE tai hdde tai SAwGE tai PHEDA tai HGLS tai PHEDA tai SAwGE tai HGLS tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai HGLS tai DABC tai VNS tai EDA-VNS tai HGLS tai PHEDA tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE

16 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 15 Tabela 6: Summaries of results for instances with n=100 jobs. Minimum value (Min), average (Avg), maximum (Max) and standard deviation (S.D.) included. Discrete PSO AGA VNS4 Instance Best Algorithm Min Average Max S.D. Min Average Max S.D. Min Average Max S.D. tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE

17 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 16 Tabela 7: Summaries of results for instances with n {200, 500}. Minimum value (Min), average (Avg), maximum (Max) and standard deviation (S.D.) included. Discrete PSO AGA VNS4 Instance Best Algorithm Min Average Max S.D. Min Average Max S.D. Min Average Max S.D. tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai AGA tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai SAwGE tai VNS tai SAwGE tai SAwGE tai VNS tai VNS tai VNS tai VNS tai SAwGE tai AGA tai VNS tai SAwGE tai VNS

18 Hybridizing VNS and path-relinking on a particle swarm framework to minimize total flowtime 17 deviation values measured for each method. The analysis disregarded only the thirty instances used to tune the parameters of PSO, as they would constitute a bias favoring PSO. The Shapiro- Wilk normality test indicated, with associated p-value lower than 10 3, that the results of each algorithm were not normally distributed. The comparison analysis used the Kruskal-Wallis statistical test. The Kruskal-Wallis test returned a p-value lower than 10 3 indicating significant differences between the methods and favoring the discrete PSO approach as it is the one with lowest RP D median, indicated with bold face in Table 8. The method VNS4 is ranked second as it presents RP D median lower than the one produced by AGA. Tabela 8: Statistical summary of the RP D values obtained by each method. Summary Discrete PSO AGA VNS4 Minimum First quartile Median Average Third quartile Maximum Std. deviation Conclusions In this paper a discrete Particle Swarm Optimization method (PSO) was proposed to minimize the total flowtime criterion in a permutational flowshop scheduling environment. The proposed method uses operators that combine VNS and truncated path-relinking procedures to explore the search space. The proposed discrete PSO was compared to two state-of-the-art approaches VNS4 [3] and AGA [31]. The experiment was conducted over the 120 instances from Taillard s benchmark set [27]. During the trials, five novel solutions for the Taillard s dataset were found by the proposed discrete PSO. The Kruskal-Wallis test was used to verify if there were significant differences between the examined methods. The results of that test indicated that there are differences between the methods and favored the proposed PSO over VNS4 and AGA methods, for when one compares the RP D measurements of the examined methods, discrete PSO has the lowest median value. Acknowledgements The authors want to thank Xu, Xu & Gu who kindly provided the source code of AGA used in this study. This work was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq - Brasil, under Grants /2009-0, / and /