NOWADAYS, it is very important to develop effective and

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1 818 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 An Effective PSO-Based Hybrid Algorithm for Multiobjective Permutation Flow Shop Scheduling Bin-Bin Li, Ling Wang, and Bo Liu Abstract This paper proposes a hybrid algorithm based on particle swarm optimization (PSO) for a multiobjective permutation flow shop scheduling problem, which is a typical NP-hard combinatorial optimization problem with strong engineering backgrounds. Not only does the proposed multiobjective algorithm (named MOPSO) apply the parallel evolution mechanism of PSO characterized by individual improvement, population cooperation, and competition to effectively perform exploration but it also utilizes several adaptive local search methods to perform exploitation. First, to make PSO suitable for solving scheduling problems, a ranked-order value (ROV) rule based on a random key technique to convert the continuous position values of particles to job permutations is presented. Second, a multiobjective local search based on the Nawaz Enscore Ham heuristic is applied to good solutions with a specified probability to enhance the exploitation ability. Third, to enrich the searching behavior and to avoid premature convergence, a multiobjective local search based on simulated annealing with multiple different neighborhoods is designed, and an adaptive meta-lamarckian learning strategy is employed to decide which neighborhood will be used. Due to the fusion of multiple different searching operations, good solutions approximating the real Pareto front can be obtained. In addition, MOPSO adopts a random weighted linear sum function to aggregate multiple objectives to a single one for solution evaluation and for guiding the evolution process in the multiobjective sense. Due to the randomness of weights, searching direction can be enriched, and solutions with good diversity can be obtained. Simulation results and comparisons based on a variety of instances demonstrate the effectiveness, efficiency, and robustness of the proposed hybrid algorithm. Index Terms Adaptive meta-lamarckian learning, hybrid algorithm, local search, multiobjective optimization (MOO), Pareto front, particle swarm optimization (PSO), permutation flow shop scheduling. I. INTRODUCTION NOWADAYS, it is very important to develop effective and efficient scheduling technologies and approaches [1], [2], because production scheduling plays a key role in the manufacturing systems of an enterprise for maintaining a competitive position in the fast-changing market. The flow shop scheduling Manuscript received September 28, 2006; revised April 14, This work was supported in part by the National Science Foundation of China under Grants and , by the National 863 Hi-Tech R&D Plan under Grant 2007AA04Z155, and by the 973 Program 2002CB This paper was recommended by Associate Editor J. Lambert. The authors are with the Department of Automation, Tsinghua University, Beijing , China ( binbinlee@mails.tsinghua.edu.cn; wangling@ mail.tsinghua.edu.cn; liub01@mails.tsinghua.edu.cn). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TSMCA problem (FSSP) is a class of widely studied problems with strong engineering backgrounds, which currently represents nearly a quarter of manufacturing systems, assembly lines, and information service facilities [3]. The permutation FSSP (PFSSP), which is a simplification of FSSP, has earned a reputation for being a typical NP-complete combinatorial optimization problem [4]. Since the pioneering work of Johnson [5], PFSSP has received considerable theoretical, computational, and empirical research work [6]. In early research, generally, only one objective function is taken into account for PFSSP. Since the 1980s, as multiobjective optimization (MOO) problems are more and more confronted in practical manufacturing systems, much research work has been focused on multiobjective PFSSP to explore effective and efficient approaches. Due to its complexity, exact solution techniques, such as branch-and-bound and mathematical programming [7], are only applicable to small-scale problems. Thus, various heuristics have been proposed, including constructive heuristics, improvement heuristics, and their hybrids [8]. The constructive heuristics build a feasible schedule from scratch, mainly for solving the two- and three-machine scheduling problems, but the solutions obtained by constructive heuristics are not satisfactory although the processes are fast. Experimental results showed that the Nawaz Enscore Ham (NEH) heuristic [9] is currently one of the best constructive heuristics. The improvement heuristics are generally based on metaheuristics, such as simulated annealing (SA) [10], genetic algorithm (GA) [11], [12], tabu search algorithm (TSA) [13], evolutionary programming [14], and variable neighborhood search (VNS) [15], which start from one solution or a set of solutions and try to improve the solution(s) by applying some specific problem knowledge to approach the global or suboptimal optima. Improvement heuristics can obtain satisfactory solutions, but they are often time consuming and parameter dependent. Recently, hybrid algorithms have been a hot topic in many fields [2], [6], [12], [16] [19]. They assume that combining the features of different methods in a complementary way may result in more robust and effective optimization tools. Particularly, it is well known that the performances of evolutionary algorithms can be improved by combining with problem-dependent local searches. Hybrid evolutionary algorithms [sometimes called memetic algorithms (MAs)] [8], [20] may be considered as a union of a population-based global search and local improvements, which are inspired by Darwinian principles of natural evolution and Dawkins notion of a meme defined as a unit of cultural evolution that is capable of local refinements. So far, MAs have gained wide research on /$ IEEE

2 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 819 a variety of problems, such as the traveling salesman problem (TSP) [21], graph bipartition problem [21], quadratic assignment problem [21], cellular mobile networks [22], and scheduling problems [23]. In [20] and [24], the authors discussed how to achieve a reasonable combination of global and local searches and how to make a good balance between exploration and exploitation. Some recent studies [20], [25] [27] have shown that the choice of local searches significantly affects the searching effectiveness and efficiency. In order to avoid the negative effects of incorrect local search, Ong and Keane [26] coined the term meta-lamarckian learning to introduce the idea of adaptively choosing multiple memes during an MA search in the spirit of Lamarckian learning and successfully solved continuous optimization problems by the proposed MA with multiple local searches. For a single-objective PFSSP, MAs have already been investigated in many studies. In [12], a multistep crossover fusion operator carrying a biased local search was introduced into GA. In [16], a hybrid GA was developed by replacing mutation with SA and by applying multiple crossover operations to subpopulations. In [17], two hybrid versions of GA were presented, i.e., genetic SA and genetic local search, where the improvement phases with SA as well as local search were performed before selection and crossover operations. In [18], a hybrid SA was integrated with features borrowed from the GA and local searches, which worked from a population of candidate schedules and generated new populations by applying suitable small perturbation schemes. In [19], a GA based on ordinal optimization was presented to ensure the quality of the solution found with a reduction in computation effort. For a multiobjective PFSSP, the hybridization with local search was first implemented as a multiobjective genetic local search (MOGLS) in [28], where a random weighted linear sum function was used to select parents and guide the local search. In [24], the MOGLS was modified, where only good individuals were chosen to perform local search in appropriate directions to balance genetic and local searches. In [29], an MOGLS was proposed by modifying the selection mechanism, where a pair of parents was randomly selected from a constantsize set of best solutions with respect to the current fitness function. In [30], Pareto dominance was adopted in MOGLS to classify the population and assign suitable fitness values to individuals, and then, a parallel multiobjective local search based on Pareto dominance relationship was implemented to intensify search in distinct regions. In [31], a multiobjective SA was developed, where a series of random weighted linear sum functions was used to guide the SA-based local search to obtain a list of potentially efficient solutions. In [32], a quantuminspired hybrid GA was proposed by fusing the Q-bit-based search and permutation-based GA search and by applying the nondominated sorting techniques to handle MOO. As an evolutionary technique, particle swarm optimization (PSO) was proposed mainly for continuous optimization problems [33]. Its development is based on the observations of the social behaviors of animals such as bird flocking. It is initialized with a population of random solutions, where each individual is assigned with a randomized velocity according to its own and its companions flying experiences. Then, the individuals, which are called particles, are flown through the solution space. PSO has memory; therefore, knowledge of good solutions is retained by all particles. Additionally, it has constructive cooperation between particles in the swarm. Due to the simple concept, easy implementation, and quick convergence, PSO has gained wide research in many fields [34], [35]. However, most of the published work on PSO is for continuous optimization problems [33], [34], whereas relatively little research work can be found for discrete optimization problems. In [36], a discrete binary PSO was proposed, which adopted 0-1 strings as the positions of particles. In [37], a PSO was presented to solve constraint satisfaction problems by redefining positions as permutations of some unique values. In [38], a general frame of discrete PSO was developed and illustrated by applying to TSP. As for scheduling problems, a hybrid PSO based on VNS [39] and a PSO-based MA [40] were proposed for PFSSP with single objective, and most recently, a PSO embedded with TS and specification of superior particle was proposed for bicriteria PFSSP [41]. In this paper, a hybrid optimization algorithm based on PSO (namely MOPSO) will be proposed for multiobjective PFSSP. In our MOPSO, a PSO-based parallel searching mechanism is used for exploration in continuous hyperspace, and several adaptive local searches are used for exploitation in permutation space. To make the standard PSO suitable for solving scheduling problems, a ranked-order value (ROV) rule based on random key technique [42] is presented to convert continuous position values of particles to job permutations. To stress exploitation, a local search based on the NEH heuristic is applied to good solutions with a specified probability. To enrich the searching behavior, SA with multiple neighborhoods is designed, and an adaptive meta-lamarckian learning strategy [26] is employed to decide which neighborhood will be used. To handle the optimization in the multiobjective sense, a weighted linear sum function with randomly generated weights is used to enrich searching directions and to obtain solutions with good diversity. Simulation results and comparisons demonstrate the effectiveness, efficiency, and robustness of the proposed MOPSO. The remaining contents are organized as follows. In Section II, MOO is stated mathematically, and some performance metrics for MOO are introduced. In Section III, the PFSSP and standard PSO are introduced. In Section IV, the MOPSO for multiobjective PFSSP is proposed after presenting solution representation, multiobjective handling technique, PSO-based search, and some local search elements. In Section V, we present the simulation results and comparisons. Finally, in Section VI, we end this paper with some conclusions and possible future work. II. MATHEMATICAL DESCRIPTION AND PERFORMANCE METRICS OF MOO A. Mathematical Description of MOO Problems Generally speaking, a MOO problem can be formulated as follows: Min f(x) =(f 1 (x),...,f n (x)) s. t. x X (1)

3 820 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 where f 1,...,f n are the objective functions and x is a feasible solution in the solution space X. For MOO optimizations, the following concepts are very important and useful. 1) Pareto dominance: A feasible solution x 1 is said to (Pareto) dominate another feasible solution x 2 (denoted as x 1 x 2 ) if and only if j {1, 2,...,n}, f j (x 1 ) f j (x 2 ) k {1, 2,...,n}, f k (x 1 ) <f k (x 2 ). (2) 2) Optimal Pareto solution: A feasible solution x 1 is said to be an optimal Pareto solution if and only if there is no feasible solution x 2 satisfying x 2 x 1. 3) Optimal Pareto set/nondominated solution set: The set containing all optimal Pareto solutions is defined as optimal Pareto set, whereas the set containing all nondominated solutions obtained by a certain algorithm is defined as nondominated solution set. 4) Optimal Pareto front: The optimal Pareto front (in objective space) is formed by those objective vectors corresponding to the solutions in the optimal Pareto set. B. Performance Metrics for MOO Algorithms Due to the multiobjective nature, the developed algorithm should efficiently and effectively find those solutions that satisfy multiple objectives. In other words, the obtained solutions should be of good proximity and diversity in the sense of MOO. Good proximity means that the algorithm is of excellent searching ability to obtain good solutions that are on or close to the optimal Pareto front. Good diversity means that the algorithm is capable of obtaining solutions that are diversely distributed for the decision maker to find a comparatively satisfactory one that is close to his preference. Suppose that E 1 and E 2 are nondominated solution sets obtained by two different MOO algorithms, next, we briefly describe some performance metrics used for comparing the performances of MOO algorithms in this paper. 1) ONVG: For an obtained nondominated solution set E 1, the metric overall nondominated vector generation (ONVG) [43] is defined as E 1, which is the number of distinct nondominated solutions in the set. 2) CM: In [44], a C metric (CM) is used to compare E 1 and E 2, which maps the ordered pair (E 1,E 2 ) to the interval [0, 1] to reflect dominance relationship between solutions in the two sets. C(E 1,E 2 )= {x 2 E 2 : x 1 E 1, x 2 x 1 } / E 2. (3) If all the solutions in E 2 are dominated by those in E 1, then C(E 1,E 2 )=1. Conversely, if all the solutions in E 1 are dominated by those in E 2, then C(E 1,E 2 )=0. It should be noticed that the sum of C(E 1,E 2 ) and C(E 2,E 1 ) is not always equal to 1, because there may exist some solutions in E 1 and E 2 that are not dominated by each other. 3) Distance Metrics (D av,d max ): In [45] and [46], the following two distance metrics were used to measure the performance of nondominated solution set E 1 relative to a reference set R of the optimal Pareto front: D av = min d(x, x R )/ R (4) x E 1 x R R { } D max = max min d(x, x R ) x R R x E 1 where d(x, x R ) = max j=1,...,n {(f j (x) f j (x R ))/ j }, x E 1, x R R, j is the range of f j values among all the solutions in E 1 and R, D av is the average distance from a solution x R R to its closest solution in E 1, and D max represents the maximum of the minimum distance from a solution x R R to any solution in E 1. Obviously, smaller D av and D max values correspond to a better approximation to the optimal Pareto front. When comparing the metrics of two nondominated solution sets E 1 and E 2, if the optimal Pareto front is not known, we will combine the two sets and select all the nondominated solutions to form set R. 4) TS: In [47], the following spacing metric was used to measure how evenly the solutions are distributed: (5) E 1 TS = 1 (D i D) E 1 2 /D (6) i=1 where D = E 1 i=1 D i/ E 1 and D i is the Euclid distance in objective space between the solution i and its nearest solution. The smaller the metric is, the more uniformly the solutions distribute. 5) MS: This metric was defined in [47] to measure how well the optimal Pareto front is covered by the obtained nondominated solutions in set E 1 through the hyperboxes formed by the extreme function values observed in the optimal Pareto front and E 1 MS = 1 n n j=1 ( max E 1 i=1 f j(x i ) min E 1 F max j F min j ) 2 i=1 f j(x i ) (7) where f j (x i ) is the jth objective of solution x i and Fj max and Fj min are the maximum and minimum values of the jth objective in the optimal Pareto front, respectively. The larger the metric is, the more the optimal Pareto front is covered by solutions in set E 1. When comparing the metrics of two sets E 1 and E 2, if the optimal Pareto front is not known, we will combine the two sets and select all the nondominated solutions to form the optimal Pareto front. 6) AQ: This metric was proposed to measure the quality of the solution set, which was originally expressed in the form of weighted Tchebysheff function [48]. However, the function may hide certain aspects about the quality of the solution set because poor performance with respect to proximity could be compensated by good performance in the distribution of solutions. Therefore, diversity indicators of spread and space are added to the formulation to overcome the limitation, and a

4 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 821 metric is given by the average value of a scalarized function over a representative sample of weight vectors. where AQ = λ Λ s a (f, z 0, λ,ρ)/ Λ (8) s a (f, z 0, λ,ρ) = min i max [ ( λj fj (x i ) zj 0 )] n ( + ρ λ j fj (x i ) zj 0 ) j j=1 and Λ= λ=(λ 1,...,λ n ) λ j {0, 1/r, 2/r,...,1}, n λ j =1 j=1 where z 0 is a reference point in the objective space that is set to (0, 0) and (0, 0, 0) for two- and three-objective problems, respectively, and ρ is a sufficiently small number which is set to 0.01 here. Moreover, r is a parameter changed as the number of objectives, which is often set to 50 and 10 for two- and threeobjective problems, respectively. A smaller metric corresponds to a better solution set. 7) RT: To reflect the efficiency of a MOO algorithm, the running time (RT) of the algorithm is also used as a key metric. The aforementioned metrics are used to measure the performances of MOO algorithms from different aspects: The ONVG metric reflects the number of nondominated solutions, CM shows the dominance relationship between two compared solution sets, the distance metrics reflect the closeness of the obtained solution set to the optimal Pareto front, the Tan s spacing (TS) metric evaluates the distribution of solutions in the set, the maximum spread (MS) metric measures the range that the solution set covers the optimal Pareto front, the average quality (AQ) metric evaluates the comprehensive quality of all the solutions, and the RT metric reflects the efficiency. III. STATEMENT OF PFSSP AND BRIEF INTRODUCTION TO PSO A. Statement of PFSSP The PFSSP with J jobs and M machines is commonly defined as follows [3]. Each of the J jobs is to be sequentially processed on machine 1,...,M. The processing time p i,j of job i on machine j is given. At any time, each machine can process at most one job, and each job can be processed on at most one machine. The sequence in which the jobs are to be processed is the same for each machine. The goal is to find a sequence or a set of sequences for processing J jobs on M machines so as to optimize a set of predefined objectives. Denote S i (σ) and C i (σ) as the start and completion times of job i in a solution σ (a permutation-based schedule), respectively, then C i (σ) =S i (σ)+ M j=1 p i,j when preemption is not allowed. If there exists a due date d i for job i, then L i (σ) =C i (σ) d i is defined as the lateness of job i. Thus, the tardiness and earliness of job i can be defined as T i (σ) = max{l i (σ), 0} and E i (σ) = max{ L i (σ), 0}, respectively. Moreover, an indicator function U i (σ) can be used to denote whether job i is tardy (U i (σ) =1) or not (U i (σ) =0). The following objectives appear most frequently in practice and in the literature [3], where ω i is the weight associated to job i: 1) maximum completion time or makespan C max (σ) = max i C i (σ); 2) mean (weighted) completion time C (ω) (σ) =(1/J) J i=1 ω ic i (σ); 3) maximum tardiness T max (σ) = max i T i (σ); 4) mean (weighted) tardiness T (ω) (σ) =(1/J) J i=1 ω it i (σ); 5) maximum earliness E max (σ) = max i E i (σ); 6) mean (weighted) earliness E (ω) (σ) =(1/J) J i=1 ω ie i (σ); 7) maximum flow-time F max (σ) = max i (C i (σ) S i (σ)); 8) mean (weighted) flow-time F (ω) (σ) =(1/J) J i=1 ω i[c i (σ) S i (σ)]; and 9) number of tardy jobs N T (σ) = J i=1 U i(σ). Obviously, some of the aforementioned objectives are conflicting, that is to say, an improvement in one objective may induce a detriment to another objective. Considering that there is usually no such a solution that is best under all objectives, it requires the scheduling algorithms to find a set of nondominated solutions with good proximity to the optimal Pareto front [24], [28] [32], [49]. Moreover, these nondominated solutions should distribute as uniformly as possible in objective space. In addition, how to efficiently evaluate the quality of the obtained nondominated solutions should be addressed. In this paper, the metrics described in Section II will be used to test and compare the performances of scheduling algorithms for multiobjective PFSSP. B. Brief Introduction to PSO In a PSO system [33], it starts with a random population (swarm) of individuals (particles) in the search space and works on the social behavior in the swarm. The position and the velocity of the ith particle in the d-dimensional search space can be represented as X i =[x i,1,x i,2,...,x i,d ] and V i = [v i,1,v i,2,...,v i,d ], respectively. Each particle has its own best position (pbest) P i =[p i,1,p i,2,...,p i,d ] corresponding to the personal best objective value obtained so far at time t. The global best particle (gbest) is denoted by P g, which represents the best particle found so far at time t in the entire swarm. The new velocity of each particle is calculated as follows: v i,j (t +1)=wv i,j (t)+c 1 r 1 (p i,j x i,j (t)) +c 2 r 2 (p g,j x i,j (t)) (9) where j =1, 2,...,d; c 1 and c 2 are acceleration coefficients; w is the inertia factor; and r 1 and r 2 are two independent random numbers uniformly distributed in the range of [0, 1]. Thus, the position of each particle is updated in each generation according to the following: where j =1, 2,...,d. x i,j (t +1)=x i,j (t)+v i,j (t +1) (10)

5 822 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 Fig. 1. Representation of position information and the corresponding ROV (job permutation). Generally, the value of each component in V i can be clamped to the range [ v max,v max ] to control the excessive roaming of particles outside the search space. Then, the particle flies toward a new position according to (10). This process is repeated until a user-defined stopping criterion is reached. We refer to [33] for more details about PSO. IV. MOPSO FOR MULTIOBJECTIVE PFSSP In this section, the MOPSO for multiobjective PFSSP will be presented and explained in detail. A. Solution Representation and Conversion The encoding scheme based on job permutation [3] has been widely used in permutation-based GA for PFSSP. For example, for a three-job flow shop problem, permutation [2 3 1] corresponds to the processing sequence from job-2, job-3, to job-1. However, the position values of every particle in PSO are continuous values, i.e., the position values do not directly represent a schedule. Therefore, we present a ROV rule based on random key technique [42] to convert the position X i =[x i,1,x i,2,...,x i,j ] of a particle to a job permutation σ i = {j 1,j 2,...,j J }. The ROV rule is described as follows. The smallest position value of a particle is first picked and assigned a smallest rank value 1. Then, the second smallest position value is picked and assigned rank value 2. With the same way, all the position values will be handled to convert the position information of a particle to a job permutation. A simple example with six jobs is provided to show the ROV rule in Fig. 1. In the instance, the position is X i =[0.06, 2.99, 1.86, 3.73, 2.13, 0.67]. Because x i,1 =0.06 is the smallest position value, x i,1 is picked first and assigned rank value 1, then x i,6 =0.67 is picked and assigned rank value 2. Similarly, the ROV rule assigns rank values3to6tox i,3, x i,5, x i,2, and x i,4, respectively. Thus, based on the ROV rule, the job permutation is obtained, i.e., σ i = {1, 5, 3, 6, 4, 2}. Notice that, in MOPSO, we will also apply some permutation-based local searches to stress the exploitation, but these local searches are not directly applied to position information. Therefore, after a local search, the new position of the particle should be adjusted to guarantee that the permutation converted from the new position is the same as the new permutation obtained by a local search. Due to the mechanism of the ROV rule, the adjustment is very easy to implement. In Fig. 2, suppose that the SWAP neighborhood [3] is used as the local search for job permutation, obviously, swapping jobs 5 and 6 corresponds to swapping position values 2.99 and As for Fig. 2. Swap-based local search for job permutation and the corresponding adjustment for position information. other neighborhood-based local searches, such as INVERSE and INSERT [3], the adjustment is similar. Remark: In MOPSO, PSO-based search in continuous position space and local search in job permutation space are both applied. Both of them may improve the solutions. By using the ROV rule, information can be passed and shared between two different search spaces. That is, better job permutation can be obtained by converting the improved particle based on the ROV rule, and better particle can be obtained by adjusting its position value based on the ROV rule according to the improved job permutation. Therefore, the PSO-based search in continuous space and the local search in discrete space are fused. B. Multiobjective Handling Technique A classical method to handle MOO is to convert the multiobjective problem to a single-objective problem by combining the various criteria into a single scalar value. The most common way is to set weight to each criterion and to add them all together using an aggregating function. Such method has been successfully applied on a variety of multiobjective PFSSPs [24], [28], [29], [31]. Because this method is very effective and easy to implement, we will adopt the following random weighted linear sum evaluation function in MOPSO to handle multiple objectives: f(x) = n α i f i (x). (11) i=1 α 1,...,α n are nonnegative weights associated to the n objectives and subject to n i=1 α i =1. Obviously, each weight vector (α 1,...,α n ) corresponds to a different certain search direction. Considering that we try to obtain multiple nondominated solutions of the multiobjective PFSSP, weights are randomly generated as follows to enrich the search directions [24], [28], [41]: / n α i = RND i RND j (12) j=1 where RND i,i=1,...,n, are mutually independent and uniformly distributed on [0, 1].

6 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 823 In MOPSO, the aforementioned weight vectors are different for different solutions in the swarm. When the swarm is updated, P s (the swarm size) different weight vectors are generated, which means that P s search directions are utilized in every iteration of the PSO-based evolution. Also, the updating process of each particle is governed by its own evaluation function. Moreover, such a random weighted linear sum evaluation function is also used in all the multiobjective local searches embedded in MOPSO. In addition, an elite set is used to store the obtained nondominated solutions. Once a new solution is generated, it will be compared with all the solutions in the current elite set. If it is dominated by any solution in the set, it will be discarded; otherwise, it will be added to the set, and all the solutions in the elite set that are dominated by the new solution will be removed. C. PSO-Based Search As for the PSO-based search, it applies (9) and (10) to adjust the positions of all particles of the swarm to perform exploration in continuous space. After adjustment, the ROV rule is used to convert the positions to permutations so as to calculate the scheduling objective values. Then, different weight vectors generated randomly are used to evaluate P s new particles and to update their corresponding pbest. When updating the pbest of a particle, the current solution and current pbest are compared. If they have domination relationship, then the nondominated one is used as pbest. If they are not dominated by each other, we will update pbest with the one that has a smaller weighted linear sum under the weight vector in use for the current particle. In this way, pbest is always the optimal Pareto solution found by the particle during the searching procedure, and it will be employed to guide the evolution of the particle in the next iteration. After that, a new weight vector generated randomly is used to evaluate all the pbest and the current gbest to update the gbest. Due to the randomness of the weight vector, more nondominated solutions can be obtained. in the sequence of jobs that are already scheduled. The best partial sequence is selected for the next iteration. Step 4) If k<j, go to step 3); otherwise, terminate the local search. In MOPSO, all pbests of particles in the swarm will perform the aforementioned local search with a predefined probability p ls. By adjusting p ls, it is easy to control the exploitation process. 2) SA-Based Multiobjective Local Search Combining Adaptive Meta-Lamarckian Learning Strategy: In MOPSO, we also design an SA-based multiobjective local search with multiple different neighborhoods for all pbests with a certain probability p SA to enrich the local searching behavior and to avoid premature convergence [51]. The considered neighborhoods include SWAP, INSERT, and INVERSE [3]. In addition, the adaptive meta-lamarckian learning strategy [26] is employed to decide which neighborhood will be used. To provide a compromise between effectiveness and efficiency, J(J 1) steps of metropolis sampling are performed at each temperature. According to the adaptive meta-lamarckian learning strategy [26], the SA-based search is divided into training and nontraining phases. During the training phase, for each pbest of the swarm, SA uses all different neighborhoods for a Metropolis sampling process at the initial temperature. Then, the reward η of each neighborhood for each pbest is determined as follows: η = pf cf / (J(J 1)) (13) where pf and cf are the weighted linear sum values of the old and the best permutations found by SA based on a certain neighborhood during the sampling process. After the reward of each neighborhood for each pbest is determined, the utilizing probability p ut of each neighborhood for each pbest is adjusted as follows: D. Multiobjective Local Search In MOPSO, the following multiobjective local search elements are embedded in the parallel evolutionary framework of PSO. 1) NEH-Based Multiobjective Local Search: The NEH heuristic has been regarded as one of the best heuristics for PFSSP. According to the improved NEH-heuristic in [40] and [50], we propose the following NEH-based multiobjective local search for multiobjective PFSSP. Step 1) Given a sequence of all jobs σ. Step 2) Take the first two jobs from σ and evaluate the two partial possible schedules. Select the better sequence with a smaller weighted linear sum value as a current sequence. Note that the weighted linear sum function is fixed during the local search. Step 3) Let k = k +1. Take job k from σ, and find the best schedule with the smallest weighted linear sum value by placing it in all the possible k positions / K p ut,i = η i η j (14) j=1 where η i is the reward value of the ith neighborhood and K is the number of neighborhoods. During the nontraining phase, according to the utilizing probability p ut of each neighborhood for each pbest, the roulette wheel rule [52] is used to decide which neighborhood will be used for the SA-based local search. If the ith neighborhood is used, its reward will be updated by η i = η i + η i, where η i is the reward value of the ith neighborhood calculated during the nontraining phase. Then, the utilizing probability p ut of each neighborhood for each pbest should be adjusted once again for the next generation. In addition, MOPSO applies exponential cooling schedule, i.e., t k = λt k 1. After the aforementioned SA-based local search, the pbest of each particle will be updated, and the gbest should also be updated.

7 824 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 Fig. 3. Framework of MOPSO for multiobjective PFSSP. E. MOPSO for Multiobjective PFSSP Based on the ROV rule, the multiobjective handling technique, the PSO-based search, and multiobjective local searches, the framework of MOPSO is shown in Fig. 3. It can be seen that not only does MOPSO apply the PSObased parallel evolution mechanism to perform exploration for all particles within continuous space but also it applies problem-dependent multiobjective local searches to perform exploitation for pbest in permutation space. Due to the balance of exploration and exploitation, it is helpful for MOPSO to obtain nondominated solutions with good proximity to the optimal Pareto front of multiobjective PFSSP. In addition, weight vectors are randomly generated for evaluation in the sense of multiobjectiveness, which is helpful in enriching the searching direction to obtain nondominated solutions with good diversity. V. S IMULATION AND COMPARISON In this section, we will test the performances of MOPSO based on some testing problems, including some multiobjective scheduling problems with small sizes found in the literature and some randomly generated instances with large sizes. All implemented algorithms are programmed under Mathworks

8 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 825 TABLE I RESULTS ON TESTING PROBLEMS FOUND IN THE LITERATURE Company s Matlab and simulated on a personal computer with Pentium IV 2.40-GHz CPU, 512-MB RAM, and Windows XP operating system. A. Multiobjective PFSSPs With Small Size in the Literature In this part, based on some well-known multiobjective PFSSPs with different small scales and different types of objectives from [53] [62], we test the performance of MOPSO to demonstrate its generality and effectiveness. Because these problems are comparatively easy to solve, we set the parameters of MOPSO as follows: P s =20, w =1.0, c 1 = c 2 = 2.0, x min =0, x max =4.0, v min = 4.0, v max =4.0, p ls = 0.1 (the probability to apply NEH-based search), p SA =0.05 (the probability to apply SA-based search), initial temperature T 0 =3.0, annealing rate λ =0.9, and the maximum generation is 50. Table I shows the results of ten experiments corresponding to the multiobjective PFSSPs in the literature [53] [62]. From Table I, it can be obviously found that the nondominated solutions obtained by MOPSO are identical or even better than those from the literature. Therefore, it is concluded that our proposed algorithm is effective for the problems with different kinds of objectives. B. Randomly Generated Multiobjective PFSSPs With Large Size 1) Testing Problems: In this part, we will test the performances of MOPSO based on some randomly generated multiobjective PFSSPs in large size, which represent a class of typical and widely studied PFSSPs [24], [29] [32]. The scheme for generating instances is as follows. The processing time of each job in every machine is uniformly distributed in interval

9 826 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 TABLE II STATISTICAL PERFORMANCE METRICS COMPARISONS WHEN CONSIDERING f =(C max,t max) [1, 99], whereas the due date of each job is uniformly distributed in interval [Q(1 a (b/2)),q(1 a +(b/2))], where a and b represent the tardiness factor of jobs and the dispersion range of due dates, respectively, and Q is following lower bound of makespan, which is estimated according to [62] as follows: Q = max max 1 j M max i J i=1 M p i,j, j=1 j 1 p i,j + min p i,l + min i i l=1 M l=j+1 p i,l. (15) Similar to [30], four scenarios about due dates are considered in this paper, where each scenario is determined by a different combination of the values of a and b. In general, the due dates are more restrictive when a increases, whereas the due dates are more diversified when b increases. Scenario 1) low tardiness factor (a =0.2) and small due date range (b =0.6); Scenario 2) low tardiness factor (a =0.2) and wide due date range (b =1.2); Scenario 3) high tardiness factor (a =0.4) and small due date range (b =0.6); Scenario 4) high tardiness factor (a =0.4) and wide due date range (b =1.2). 2) Implementation of Algorithms: As introduced before, the MOGLS proposed by Ishibuchi and Murata (denoted as IM- MOGLS) also adopted random weighted linear sum function to handle multiple objectives and guide the local search [24], [28], and IM-MOGLS was tested to be very effective and viewed as a typical and benchmark approach for multiobjective PFSSPs. Therefore, here, we will compare the performances of MOPSO with IM-MOGLS. The parameters of MOPSO and IM-MOGLS are fixed for all the simulation experiments that follow. For MOPSO, the parameters are set as the same values as those in [40]: swarm size P s =20, w =1.0, c 1 = c 2 =2.0, x min =0, x max =4.0, v min = 4.0, v max =4.0, p ls =0.1, p SA =0.05, T 0 =3.0, and λ =0.9. For IM-MOGLS, we specify the parameters according to [24]: population size N pop = 200, crossover probability p crossover =0.9, mutation probability per string p mutation =0.6, the number of elite solutions N elite =10,the number of neighbors to be examined k neighbor =2, the tournament size in the selection of initial solutions N tournament =5, and local search probability p LS =0.8. We refer to [24] and [28] for the details about IM-MOGLS. In order to perform a fair comparison, we use a predetermined maximum evaluation of solutions (50 000) as the stopping criterion. Fig. 4. Pareto fronts obtained by MOPSO and IM-MOGLS. Next, based on the metrics introduced in Section II, we will evaluate the performances of MOPSO and compare it with IM-MOGLS by simulation with some two- and three-objective PFSSPs. 3) Simulation Results: Two-objective PFSSP with f =(C max, C): Here, we test the performances of two algorithms based on some twoobjective PFSSPs, i.e., makespan and average completion time. Moreover, a total of 20 instances of 20 jobs and 10 machines are randomly generated for testing by both MOPSO and IM- MOGLS. The statistical performance metrics of 20 independent experiments are shown in Table II, and one set of typical running results by the two algorithms is shown in Fig. 4. From Table II, it can be seen that both of the two algorithms can achieve good results for the given multiobjective PFSSPs. However, there are still some differences between them, which are revealed by the statistical performance metrics. From the ONVG metric, it can be seen that MOPSO obtains few optimal Pareto solutions than IM-MOGLS. However, the solutions obtained by MOPSO are of better proximity than those obtained by IM-MOGLS. From CM, it can be seen that more than half of the solutions obtained by IM-MOGLS are dominated by those obtained by MOPSO, whereas only few solutions obtained by MOPSO are not nondominated solutions compared with those obtained by IM-MOGLS. Also, from Table II, it can be seen that the D av and D max metrics of MOPSO are less than those of IM-MOGLS. Because D av and D max metrics measure the distance between the obtained solution set and the optimal Pareto solution set in average and max min senses, respectively, it can be concluded that MOPSO is able to obtain solutions that are closer to the optimal Pareto solutions than IM-MOGLS. As for the diversity, it can be seen that the TS metrics of

10 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 827 TABLE III METRIC COMPARISON FOR SCENARIO 1WHEN CONSIDERING f =(C max,t max) MOPSO are smaller than that of IM-MOGLS, which indicates that the solutions obtained by MOPSO have a more uniform distribution than those of IM-MOGLS. On the contrary, the MS metrics of MOPSO are smaller than those of IM-MOGLS, showing that the solution set obtained by MOPSO covers less of the real optimal Pareto front than that by IM-MOGLS. However, it should be noted that proximity and diversity are two important features for MOO, and they may influence each other to some extent. Therefore, a metric considering only proximity or diversity cannot comprehensively reflect the performance of an MOO algorithm. Thus, we use the AQ metric that considers both the proximity and diversity for comparison. From Table II, it can be seen that the AQ values of MOPSO are less than those of IM-MOGLS. Therefore, it is concluded that MOPSO is a more effective MOO algorithm than IM-MOGLS. Comparing their efficiencies, the RT of MOPSO is much less than that of IM-MOGLS, which means that MOPSO can explore the same number of solutions in a shorter time or search more solutions in a given time than IM-MOGLS. Therefore, it is also concluded that MOPSO is a more efficient MOO algorithm than IM-MOGLS. Two-objective PFSSP with f =(C max,t max ): Here, we consider some PFSSPs with objectives of makespan and maximum tardiness. As we know, under different scenarios (with different a s and b s), the magnitude and diversity of the due date may be different, which will result in problems with different degrees of difficulty. Therefore, we compare our proposed MOPSO and IM-MOGLS on 20-job and 10-machine PFSSPs under four different scenarios, and for each scenario, five instances are randomly generated. The performance metrics of the two algorithms are shown in Tables III VI. From Tables III VI, we can draw similar conclusions as those for PFSSP with f =(C max, C). Although the number of the nondominated solutions obtained by MOPSO is slightly less than that of IM-MOGLS, the nondominated solutions obtained by MOPSO can dominate nearly all the solutions obtained by IM-MOGLS in almost every case. Moreover, those solutions obtained by MOPSO are closer to the optimal Pareto solutions than those obtained by IM-MOGLS, which is revealed by both average distances and max-min distances between the obtained solution sets and the optimal Pareto solution set. As for the diversity performance, MOPSO can obtain solutions that distribute more uniformly than IM-MOGLS in 6 out of 20 instances and can cover more area of the optimal Pareto front than IM-MOGLS in 2 out of 20 instances. When considering both diversity and proximity performances, the AQ values of MOPSO are always smaller than those of IM-MOGLS, showing that MOPSO has a better comprehensive performance than IM-MOLGS. Moreover, the RT of MOPSO is only about 14% of that of IM-MOGLS. Therefore, it is concluded once again that MOPSO is an effective and efficient algorithm for multiobjective PFSSPs. A set of typical running results by the two algorithms in one experiment is shown in Fig. 5, which also demonstrates the aforementioned conclusion. Moreover, it can also be found that all performance metrics of MOPSO are similar for different scenarios, and the standard deviations of performance metrics are very small. Therefore, it is concluded that MOPSO is of good robustness, i.e., it can obtain solution sets with good proximity and diversity in the sense of MOO for scheduling problems with different degrees of difficulty. Three-objective PFSSP with f =(C max, C,T max ): In this section, we apply MOPSO and IM-MOGLS to the 20 instances under four scenarios generated in the last section, considering three objectives, i.e., makespan, average completion time, and maximum tardiness. In Table VII, the statistical performance metrics of five instances in each of the four scenarios are illustrated.

11 828 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART A: SYSTEMS AND HUMANS, VOL. 38, NO. 4, JULY 2008 TABLE IV METRIC COMPARISON FOR SCENARIO 2WHEN CONSIDERING f =(C max,t max) TABLE V METRIC COMPARISON FOR SCENARIO 3WHEN CONSIDERING f =(C max,t max) From Table VII, it can be seen that both MOPSO and IM- MOGLS can obtain a large number of nondominated solutions in a more complex space. The solutions obtained by MOPSO are less but closer to optimal Pareto solutions than that of IM-MOGLS, i.e., the solutions obtained by MOPSO dominate most of the solutions obtained by IM-MOGLS. Although the diversity performances of solutions obtained by MOPSO are not as good as that of the solutions obtained by IM-MOGLS, the solutions obtained by MOPSO are better than that of IM- MOGLS on the comprehensive performance, considering both proximity and diversity. Furthermore, MOPSO is more efficient than IM-MOGLS. To sum up, our proposed MOPSO can also obtain satisfactory nondominated solutions with good proximity and diversity for three-objective PFSSPs. In a word, the proposed hybrid algorithm based on PSO for multiobjective PFSSPs is of good performance in terms of effectiveness, efficiency, and robustness. VI. CONCLUSION AND FUTURE WORK This paper presented a hybrid algorithm based on PSO for multiobjective PFSSP, whose characteristic features mainly lie

12 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 829 TABLE VI METRIC COMPARISON FOR SCENARIO 4WHEN CONSIDERING f =(C max,t max) TABLE VII STATISTICAL PERFORMANCE METRIC COMPARISON FOR SCENARIOS 1 4 WHEN CONSIDERING f =(C max, C,T max) Fig. 5. Pareto fronts obtained by MOPSO and IM-MOGLS. in two aspects: 1) The hybridization of PSO-based evolutionary search with problem-dependent local searches, which is helpful in balancing the exploration and exploitation to obtain solutions with good proximity. In particular, a ROV rule was presented to convert the continuous positions to job permutations to make PSO suitable for solving PFSSP; a NEH-based multiobjective local search and an adaptive SA-based local search were proposed and probabilistically applied to all good particles to enhance exploitation. 2) The evaluation based on random weighted linear sum function to handle multiple objectives, which is helpful in enriching the search directions and obtaining solutions with good diversity. Simulation results and comparisons with several performance metrics demonstrated the effectiveness, efficiency, and robustness of the proposed algorithm. In our future work, we will study some Pareto-based PSO algorithms and adaptive algorithms based on PSO with sequentially decreasing inertia weight and PSO with constriction factor [35] for multiobjective scheduling problems, and we will also apply the proposed algorithm to other multiobjective combinatorial optimization problems. ACKNOWLEDGMENT The authors would like to thank the Editor-in-Chief, Associate Editor, and the anonymous referees for their constructive comments on the earlier version of this paper.

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14 LI et al.: EFFECTIVE PSO-BASED HYBRID ALGORITHM 831 [53] R. T. Nelson, R. K. Sarin, and R. L. Daniels, Scheduling with multiple performances measures: The one machine case, Manage. Sci., vol. 2, no. 4, pp , Apr [54] J. G. Shanthikumar, Scheduling n jobs on one machine to minimize the maximum tardiness with minimum number tardy jobs, Comput. Oper. Res., vol. 10, no. 3, pp , [55] L. N. Van Wassenhove and L. F. Gelders, Solving a bicriterion scheduling problem, Eur. J. Oper. Res., vol. 4, no. 1, pp , Jan [56] M. M. Koksalan, M. Azizoglu, and S. K. Kondakci, Heuristics to Minimize Flowtime and Maximum Tardiness on a Single Machine, ser. CMME Working Paper Series 9. West lafayette, IN: Krannert School Manag., Purdue Univ., [57] W. J. Selen and D. Hott, A mixed integer goal programming formulation of the standard flow shop scheduling problem, J. Oper. Res. Soc., vol.37, no. 12, pp , Dec [58] J. C. Ho and Y. L. Chang, A new heuristic for n-job, m-shop problem, Eur. J. Oper. Res., vol. 52, no. 2, pp , May [59] C. Rajendran, Two-stage flowshop scheduling problem with bicriteria, J. Oper. Res. Soc., vol. 43, no. 9, pp , Sep [60] C. Rajendran, Heuristics for scheduling in a flowshop with multiple objectives, Eur. J. Oper. Res., vol. 82, no. 3, pp , May [61] C. J. Liao, W. C. Yu, and C. B. Joe, Bicriteria scheduling in the two machines flow shop, J. Oper. Res. Soc., vol. 48, no. 9, pp , Sep [62] H. Hoogeven, Single machine bicriteria scheduling, Ph.D. dissertation, Univ. Eindhoven, Eindhoven, The Netherlands, Ling Wang received the B.Sc. and Ph.D. degrees from Tsinghua University, Beijing, China, in 1995 and 1999, respectively. Since 1999, he has been with the Department of Automation, Tsinghua University, where he became an Associate Professor in He has published two books, namely, Intelligent Optimization Algorithms With Applications (Springer Press, 2001) and Shop Scheduling With Genetic Algorithm (Springer Press, 2003). He has also authored more than 150 refereed international and domestic academic papers. He has been the Reviewer for the European Journal of Operational Research, Computers and Operations Research, Computers and Industrial Engineering, Information Science, Journal of Global Optimization, Engineering Applications of Artificial Intelligence, andinternational Journal of Advanced Manufacturing. He is also the Associate Editor of the International Journal of Metaheuristics and an Editor Board Member of the International Journal of Automation and Control, European Journal Industrial Engineering, International Journal of Soft Computing, International Journal of Memetic Computing, the Open Operations Research Journal, etc. His current research interests are mainly in optimization theory and algorithms and production scheduling. Dr. Wang is a member of the Emergent Technology Technique Committee of the IEEE Computational Intelligence Society. He has also been the Reviewer for many international journals, such as the IEEE TRANSACTIONS ON SYSTEMS,MAN, AND CYBERNETICS, IEEE TRANSACTIONS ON EVOLUTION COMPUTATION, IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND ENGINEERING, and IEEE TRANSACTIONS ON NEURAL NETWORKS. Hewas the recipient of the Outstanding Paper Award in the International Conference on Machine Learning and Cybernetics organized by the IEEE Systems, Man, and Cybernetics Society in 2002 and of the National Natural Science Award (First Place) nominated by the Ministry of Education of China in He was the Young Talent of Science and Technology of Beijing City in Bin-Bin Li received the B.Sc. degree in automation and the M.Sc. degree in control science and engineering from Tsinghua University, Beijing, China, in 2004 and 2006, respectively. He is currently a Ph.D. candidate at the Division of Systems Engineering, Boston University, Boston, USA. His research interests include optimization and decision theory, with main applications in communication and sensor networks. Bo Liu received the B.S. degree in electrical engineering from Xi an Jiaotong University, Xi an, China, in 2001 and the Ph.D. degree from Tsinghua University, Beijing, China, in He is currently a joint Postdoctoral Research Staff with the Centre for World Food Studies, Faculty of Economics and Business Administration, Vrije Universiteit, Amsterdam, and with the Center for Chinese Agricultural Policy, Chinese Academy of Sciences, Beijing. His research interests include production scheduling, decision analysis, and support systems for agricultural economics.