The 2012 International Symposium on Semiconductor Manufacturing Intelligence Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem Wenqiang Zhang1, Mitsuo Gen2 and Jungbok Jo3* Graduate School of IPS, Waseda University, Fukuoka, Japan 2 Fuzzy Logic Systems Institute (FLSI), Fukuoka, Japan and Dept. of Industrial & Management Eng., Hanyang University, Korea 3 Division of Computer and Information Engineering, Dongseo University, Korea 1 * [email protected] Abstract Process planning and scheduling (PPS) is a most important, practical but very intractable problem in manufacturing systems. Many research works use multiobjective evolutionary algorithm (MOEA) to solve such problems; however, they cannot achieve satisfactory results in both quality and speed. This paper proposes a hybrid sampling strategy-based evolutionary algorithm (HSS-EA) to deal with PPS problem. HSS-EA tactfully unites the advantages of vector evaluated genetic algorithm (VEGA) and a sampling strategy according to Pareto dominating and dominated relationship-based fitness function (PDDR-FF). The sampling strategy of VEGA prefers the edge region of the Pareto front and PDDR-FF-based sampling strategy has the tendency converging toward the central area of the Pareto front. These two mechanisms preserve both the convergence rate and the distribution performance. Numerical comparisons show that the convergence performance of HSS-EA is better than NSGA-II and SPEA2 while the distribution performance is slightly better or equivalent, and the efficiency is obviously better than they are. Keywords: evolutionary algorithm, hybrid sampling, multiobjective optimization, process planning and scheduling 1. Introduction Process planning and scheduling (PPS) is to process a set of prismatic parts into completed products effectively and economically in a manufacturing system (MS). A prismatic part to be produced is generally described by features. For each feature, one or more corresponding operations are determined according to its feature geometry and available machining resources. Each operation includes the selection of machines, tools and tool access direction (TAD). There are precedence relationship constraints among operations according to the geometrical and technological consideration. Process planning is the determination of an optimal process plan, i.e. operations and their sequences, within the precedence relationship constraints and manufacturing resource (machine and tool) constraints. The scheduling is the allocation of the resources in the shop over time to manufacture the various parts [1]. PPS becomes more important for the effective allocation and utilization of resources in the MS. However, seeking an optimal solution rapidly and effectively from all of the permutations, combinations of all of the tasks, manufacturing resources according to specified criteria is very difficult for decision maker. Duos to its importance, practicality and difficulty, in the past decade, many research works have addressed the PPS problem. These approaches include simulated annealing (SA) algorithm [1], [2], tabu search algorithm [3], agent-based approach [4], [5], particle swarm optimization (PSO) algorithm [6], [7] and genetic algorithm (GA) [8-12]. Unfortunately, most of the above previous researchers concern single objective optimization rather than multiobjective optimization problems (MOOPs). In the real world, almost every problem involves optimization of several incomparable or incommensurable objectives simultaneously. Multiobjective evolutionary algorithms (MOEAs) have been recognized to be well-suited for solving MOOPs [13],[14]. As the first one of MOEAs, vector evaluated genetic algorithm (VEGA) divides the population into m sub-populations (m is number of objectives), each of which evolves toward a single objective [15]. The benefit of VEGA is the strong ability to converge to the edge region of the Pareto front by its simple sampling strategy and less time complexity [16]. However, the qualities (especially, diversity) of VEGA are not good because of the selection bias. Therefore, VEGA cannot be used to address multiobjective PPS problems in practical situation. As two classical MOEAs, NSGA-II [17] and SPEA2 [18] have been proven to be able to get better quality in solving MOOPs. NSGA-II can get the better quality owing to its Pareto ranking and crowding distance mechanism. SPEA2 depends on raw fitness assignment mechanism and density mechanism. Unfortunately, these two algorithms both need much CPU time. These drawbacks of NSGA-II and SPEA2 cause that they cannot be applied to solve multiobjective PPS problems online. Ho et al. propose a generalized Pareto-based scale-independent fitness function (GPSI-FF) to solve the large parameter optimization problem [19]. The GPSI-FF can obviously speed up the convergence rate [20], especially around the central area of the Pareto front. However, the lack of diversity preservation mechanism causes that GPSI-FF also cannot be chosen to deal with multiobjective PPS problems. Moreover, the difference between nondominated and dominated individuals can be decreased by GPSI-FF values. This disadvantage causes that more dominated individuals could be held in archive (external population) while nondominated ones would be removed from archive. It reduces the performance of archive, unless an enough large size of its archive is set to store a sufficient number of individuals. To increase the quality both of convergence and distribution, and to reduce the computation time, a new hybrid sampling strategy-based evolutionary algorithm (HSS-EA) is proposed to solve the multiobjective PPS problem. For covering the drawback of GPSI-FF, a new Pareto dominating and dominated relationship-based fitness function (PDDR-FF) is proposed to evaluate the individuals. It gives the sensible difference values between the nondominated and dominated individuals. Moreover, ones with different numbers of dominating are also set as different fitness values even though they are all nondominated individuals. The individuals locating around the central region of Pareto font will have smaller values than the edge points. The proposed method deliberately unites the two different mechanisms. One is the sampling strategy of VEGA with a preference for the edge region of the Pareto front, and the other is the sampling strategy of PDDR-FF with tendency converging toward the central area of the Pareto front. These two mechanisms not only preserve the convergence rate, but also guarantee the better distribution performance. Moreover, some problem-dependent crossover, mutation and local search methods are also used to improve the performance of the algorithm. The paper is organized as follows: Section 2 describes the PPS problem description and formulates the mathematical model; Section 3 presents the detailed HSS-EA approach; Section 4 gives Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem a discussion and analysis of numerical experiments results; finally, the conclusion and future work are given in Section 5. 2. PPS Problem 2.1 Problem Description The illustration of the PPS problem in a MS is shown in Fig. 1. In Fig. 1, there are a set of parts (four parts) to be processed by a number of machines (four machines) with a number of tools and different TADs. Each part has several operations (four parts have 4, 3, 3, and 4 operations respectively), and each operation can be performed on more than one suitable machines with different processing times. Table 1 shows the operation information of the 4 parts. Each column describes the part ID, operation ID, successors, operation name, TAD candidates, machine candidates, tool candidates and machining time, respectively. part1 part2 o12 o11 o14 o42 o33 o11 o41 o43 o42 o44 o32 o23 o44 part4 o31 o21 o13 o21 part3 o22 o12 o31 o33 o32 o41 o43 o22 o23 o13 o14 o21 Machine m1 o14 o12 Machine m2 o42 Machine m3 o44 o33 Machine m4 o41 o11 o32 o43 o23 o13 Makespan o22 Idle time and change time o31 0 Time Figure 1: Illustration of PPS problem Table 1: Operation information of part 1 to part 4 Part-ID Op-ID Successor Operations Part 1 Part 2 Part 3 Part 4 makespan and minimizing variation of workload for each machine are important and frequently-used. Reducing makespan means that the MS can get higher producing efficiency in a limited period. Decreasing variation of machine workload can balance the utilization of machines. The mathematical model of them is expressed in the following notations. Indices i, k : indices of part, (i, k = 1, 2, , I ) . j, h : indices of operation for part i , ( j ,h=1,2, ,J i ) . m : index of machine, (m = 1, 2, , M ) . l : index of tool, (l = 1, 2, , L) . d : index of TAD, (d = 1, 2, , D) . Decision variables if oij is performed by m, M 1, xmij = 0, otherwise 1, if oij is performed by l , xlijT = 0, otherwise 1, if oij is performed by d , D xdij = 0, otherwise 1, if oij is performed directly before okh , yijkh = 0, otherwise Parameters I : number of parts. J i : number of operations for part M : number of machines. L : number of tools. D : number of TADs. Oi :set of operations for part oij : the j-th operation of part TAD candidates Machine candidates Tool Machining time candidates (time unit) mm : the m-th machine. tl : the l-th tool. o11 o12, o13 Milling +z m2, m3, m4 t6 , t7, t8 40, 40, 30 o12 o14 Milling -z m2, m3, m4 t6 , t7, t8 40, 40, 30 o13 o14 Milling +x m2, m3, m4 t6 , t7, t8 20, 20, 15 o14 -- Drilling +z, -z m1, m2, m3, m4 t2 12, 10, 10, 7.5 o21 o22, o23 Drilling +z, -z m1, m2, m3, m4 t1 12, 10, 10, 7.5 o22 -- Milling -x, +y, -y, -z m2, m3, m4 t12 20, 20, 15 o23 -- Milling +y m2, m3, m4 t5, t6, t11 18, 18, 13.5 o31 o32 Milling +z m2, m3, m4 t6 , t7, t8 20, 20, 15 o32 -- Milling -z m2, m3, m4 t6 , t7, t8 20, 20, 15 o33 o32 Milling +x, -x, +y, -z m2, m3, m4 t6 , t7, t8 15, 15, 11.25 o41 o43 Milling -y m2, m3 t6 , t9 12,15 o42 o44 Milling -y m2, m3 t9, t10 21,18 o43 -- Milling -z m2, m3 t3 18,25 o44 -- Milling +x, -x m2, m3 t1 , t3 27,25 i. i , Oi ={oij | j=1,2, ,J i } . i. ad : the d-th TAD. M ij : set of machines that can process oij . Am : set of operations that can be processed on machine m. rijh : precedence constraints. rijh = 1 , if oij is predecessor of oih ; 0, otherwise. M : machining time of oij by machine m. tmij MC t : machine change time. It is needed when two adjacent operations belong to 1) different parts, or 2) same part and different machines. The PPS problem herein is to determine a process plan and schedule, which tells, decision maker how, when and in which sequence to allocate these operations of parts to suitable manufacturing resources effectively. Moreover, this process plan and schedule should satisfy the multiply objectives concurrently while maintaining the feasibility of its. This problem can be defined as follows. Operation sequencing: determine the operation sequences for all the parts so that the precedence relationships among all the operations are not violated. Operation selection: determine the resources selection according to the feature geometry and available machining resources. Part scheduling: determine how and when to allocate the manufacturing resources to the parts. t TC : tool change time . It is needed when two adjacent operations belong to 1) different parts, or 2) same part and different machines, or 3) same part, same machine and different tools. 2.2 Mathematical Formulation processing time for an operation consists of the preparation time and the machining time for the operation. (2) t mijP = t mijPRE t mijM As the two objectives of PPS problems, minimizing t SC : set-up change time. It is needed when two adjacent operations belong to 1) different parts, or 2) same part and different machines, or 3) same part, same machine and different TADs. PRE t mij : preparation time of operation oij by machine m . The preparation time for an operation consists of machine change time, tool change time and set-up time for the operation. PRE (1) t mij = t MC t TC t SC P : processing time of operation oij by machine m . The t mij The 2012 International Symposium on Semiconductor Manufacturing Intelligence u m : workload of machine m . um = i =1 ji=1 t mijM xmijM I J (3) u : average workload of machine. u= 1 M M (4) u m =1 m C t mij : completion time of oij by machine m . Mathematical Model: The mathematical model for minimization of makespan and variation of workload can be stated as follows: C (5) min t = maxtmij M m ,i , j min wP = s.t. t C mkh 1 t M P mkh t M (6) (um u ) 2 m =1 C mij x M M mij mkh ijkh x y 0, i, j , k , h , m (7) rijh yihij = 0, (i, j ), h (8) yijij = 0, (i, j ) (9) (10) M M m =1 mij x = 1, (i, j ) M xmij = 0, i, j Am , m (11) yijkh 0,1 , i, j , k , h (12) x 0,1 , m, (i, j ) (13) C tmij 0, m, (i, j ) (14) M mij eval ( si )=q ( si ) 1 p ( si )1 , i=1,2, ..., popSize Equation (5) describes the one objective of minimization of makespan. Minimization of variation of workload is defined as Eq. (6). Equation (7) imposes that any machine cannot be selected for one operation until the predecessor is completed. The precedence constraint is defined as Eq. (8). Equation (9) ensures the feasible operation sequence. The feasible resource selection are defined as Eq. (10) and (11). Equations (12), (13) and (14) impose nonnegative condition. 3. Hybrid Sampling Strategy-based Evolutionary Algorithm (HSS-EA) 3.1 Pareto dominating and dominated relationship based fitness function (PDDR-FF) The fitness assignment strategy plays an important role in MOEAs. The GPSI-FF makes the best use of Pareto dominance relationship to evaluate individuals. The GPSI-FF of an individual si is calculated by the following function: eval ( si )= p ( si )q ( si )c , i=1,2, ..., popSize (15) where p(si) is the number of individuals, which can be dominated by the individual si, q(si)is the number of individuals which can dominate the individual si, and c is population size. The bigger value is better. (a) GPSI-FF mainly lie in the number of individuals which are dominated by this individual, p(si), because c is a constant and q(si) is 0. Moreover, the individuals locating around the central region of the Pareto front with bigger domination area can dominate individuals more than edge region. It causes that the higher GPSI-FF values can tend the solutions toward the central of the Pareto front. However, the difference between nondominated and dominated individuals can be decreased by GPSI-FF values (as shown in Fig. 2a). In Fig. 2a, individual A has bigger fitness value than B and C, while B>C. In fact, nondominated individuals C should be better than B because B is a dominated individual. This unclear difference between nondominated and dominated individuals causes that more dominated individuals could be kept while nondominated ones would be removed in the evolving process. It can reduce the convergence performance of algorithm, unless an enough large size of its archive is set to store a sufficient number of individuals. For covering the disadvantage of GPSI-FF, a new PDDR-FF -based fitness function is proposed to evaluate the individuals. The PDDR-FF of an individual si is calculated by the following function: The smaller value is better. According to the Eq. (16), PDDR-FF can set the obvious difference values between the nondominated and dominated individuals. If the individual belongs to nondominated one, its fitness value will not exceed one. The fitness value of individual which is dominated by other will exceed one. Furthermore, ones with different numbers of dominating are also given the different fitness values even though they are all nondominated individuals. It is obvious that the nondominated individuals locating around the central region of Pareto font with bigger domination area will have smaller values (near to 0) than the edge points (near to 1). From Fig. 2b, individual C is better than B while C is worse than B in Fig. 2a. Moreover, nondominated individual A has smaller value than C because the p(A) > p(C). Moreover, the total time complexity of the PDDR-FF is O(mN2). 3.2 The Proposed Algorithm In selection phase of EA, the PDDR-FF based sampling strategy has the advantage with the tendency converging toward the center area of the Pareto front, but drawback to the edge region. It causes bad distribution performance. The sampling strategy of VEGA prefers the edge rather than certer regions of Pareto front that it causes VEGA cannot achieve better distribution performance. So it is natural, reasonable and possible to combine these two methods to improve the overall performance and reduce the computation time of the algorithm. Figure 3 shows the description of HSS-EA. (b)PDDR-FF Figure 2: The comparisons of GPSI-FF and PDDR-FF In GPSI-FF, the fitness values of nondominated individuals (16) Figure 3: The description of HSS-EA Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem The strong convergence capability of VEGA and PDDR-FF ensures that the HSS-EA has the ability to converge to the true Pareto front both in central and edge regions. The preferences for the edge area of the Pareto front in VEGA and the central area of the Pareto front in PDDR-FF guarantee that the HSS-EA distributes along the Pareto front evenly. Moreover, less computing time makes that HSS-EA has higher efficiency than other approaches. Since the multiobjective PPS problem belongs to the classical combinatorial optimization and NP-hard problems, higher efficacy and efficiency make that HSS-EA seems well suited to solve such kind of problems. In addition, some problem-dependent crossover, mutation and local search methods are used to improve the performance of the algorithm. 3.3 Main Framework of HSS-EA The evolving process of one generation of HSS-EA is shown in Fig. 4. A(t) represents the archive at generation t and P(t) represents the population at generation t. The solution procedure of one generation includes 4 phases. The smallest |A(t)| individuals in A’(t) are copied to form A(t+1). This archive updating mechanism likes a elitist sampling strategy to keep the better individuals with better PDDR-FF values. 3.4 Genetic Coding The chromosome consists of multiple vectors, i.e. operation sequence vector (v1), machine vector (v2), tool vector (v3) and TAD vector (v4). The detailed initialization process of a chromosome includes three steps. Step 1: Deciding operation sequence Operations are randomly selected from eligible operations to generate feasible operation sequences. After one operation being selected, it will be removed from the precedence graph and the eligible operations will be updated. In the same manner, the feasible operation sequence can be easily generated. Step 2: Assigning resources Machine and tool are selected randomly for each operation from available machine and tool candidates. TAD is also selected randomly from all TAD candidates for each operation. The generated chromosome is shown in Fig. 5. 1 node ID j : operation o21 sequence v1(j): machine v2(j): m4 2 3 4 5 6 7 8 9 10 11 12 13 14 o42 o41 o43 o23 o44 o31 o22 o11 o13 o33 o32 o12 o14 m3 m3 m3 m4 m2 m2 m2 m4 m3 m2 m4 m3 m3 tool v3(j): t1 t10 t6 t3 t11 t1 t6 t12 t8 t8 t6 t6 t7 t2 TAD v4(j): -z -y -y -z +y -x +z -z +z +x -x -z -z -z Figure 5: A chromosome of the example Step 3: Generating schedule After generating the chromosome, the schedule can be generated. When generating it, an operation can be started whenever its predecessor has been finished and the machine to process it is available. 3.5 Genetic Operators Figure 4: The evolving process in one generation Phase 1: Selection (sampling strategy of VEGA) In this phase, the sampling strategy of VEGA is used to select the better individuals into part of mating pool. For two objectives PPS problem, individuals are selected with replacement according to objective 1 into sub population 1 while ignoring objective 2 until the size of the sub population 1 (half of population size) is reached. In the same manner, individuals are selected for objective 2 into sub population 2 without considering objective 1 until sub population 2 is full. Phase 2: Generation of mating pool (hybrid sampling) In this study, the sub populations and A(t) are combined to form a mating pool. In the mating pool, sub-pop-1 stores the good individuals for one objective, and sub-population 2 holds the good individuals for the other objective. The archive saves the individuals with good PDDR-FF values. Since there are two objectives in this PPS problem, the sizes of those two sub populations and archive are set as the same as half the population size. Therefore, in the mating pool, one-third of the individuals serve one objective, one-third the other objective, and the left one-third both the two objectives. The archive mechanism tries to cover the selection bias of VEGA. These three parts of the mating pool make the solutions converge to the Pareto front evenly. Phase 3: Reproduction and local search Some problem-dependent crossover and mutation operators are used to reproduce new chromosomes. Moreover, a local search mechanism is proposed to improve the quality of individuals after the reproduction process. Phase 4: Archive maintenance (PDDR-FF based Elitist sampling strategy) The individuals of A(t) and P(t) are combined to form a temporary archive A’(t). Thereafter, the PDDR-FF values of all individuals in A’(t) are calculated and sorted in a ascending order. 3.5.1 Crossover operator In this study, one-cut point order-based crossover is adopted. This crossover is mainly applied according to operation sequence vector. The locus of machine vector, tool vector and TAD vector are changed with same alleles of operation sequence vector Notation pc is defined as crossover probability. The algorithm is described as follows. Step 1: According to the length of a parent, a cut point is generated randomly. Then, each parent is divided into two parts, a left side and a right side. Step 2: The left side of offspring 1 comes from the left side of parent 1. The operator constructs the right side of offspring 1 according to the order of operations in parent 2. According to this order, the operator constructs the right side of offspring 1 with operations of parent 2, whose operation IDs are the same as operations of the right side in parent 1. Step 3: Offspring 2 is created in the same manner. 3.5.2 Mutation operator Three resource mutation operators proposed by Zhang et al. [8] are used. Notation pm are defined as mutation probability. Machine mutation changes the machine to perform an operation so as to reduce the machine change times. The algorithm is described as follows. Step 1: A node ID is chosen randomly, and the corresponding machine Mold of this node ID is marked as the old machine. Step 2: A new machine Mnew is chosen randomly from the alternative machine set of this node ID to replace the old machine. Step 3: Mark all the other machines in the machine vector whose old machine is Mold. If there exists an The 2012 International Symposium on Semiconductor Manufacturing Intelligence alternative machine Mnew in any one of v2, Mold is replaced with Mnew. Tool mutation uses the similar manner to operate on tool vector after the machine mutation. TAD mutation also uses the similar mechanism to operate on TAD vector after both the machine mutation and the tool mutation. 3.5.3 Selection operator In this study, a binary tournament selection (BTS) with replacement is used. Two individuals are chosen randomly from the population and the better (smaller) one is copied into the mating pool. 3.6 Local Search The local search is proposed to reduce the changeovers (the machine change times, tool change times and TAD change times). It is applied for every chromosome after the reproduction. The pseudocode of local search is listed in Fig. 6. HSS-EA, GPSI-FFGA, NSGA-II and SPEA2 are run 30 times to compared the results with each other. It should be noted that the same selection, crossover, mutation, local search, and different fitness functions and archive mechanisms for four methods are used. Moreover, the parameters of all four methods are the same, except for the size of archive. The archive sizes of HSS-EA and GPSI-FFGA are set to be half the population size, 50, while of NSGA-II and SPEA2 are set to be the same as the population size, 100. GPSI-FFGA uses the same framework as HSS-EA, except the different fitness function. 4.1 Performance Measures Let Sj be a solution set for each method (j=1, 2, 3, 4). PF* is a known reference Pareto solutions. Because no researches have published their reference Pareto solutions for this problem, therefore, PF*in this study comes from combining all of the obtained Pareto set with 30 runs by 4 methods. In this study, the following three performance measures are considered. Coverage C(S1,S2) is the percent of the individuals in S2 which are weakly dominated by S1 [21]. The larger C(S1,S2) means that S1 outperforms S2 in convergence. Generational distance GD(Sj) finds an average minimum distance of the solutions of Sj from PF* [22]. The smaller GD of Sj means better Sj in approaching PF*. Spacing SP(Sj) is the standard deviation of the closest distances of individuals by Sj [23]. Smaller SP means better distribution performance. The C, GD are used to verify convergence performance while SP is used to check the distribution performance. 4.2 Discussion of the Results Figure 7 shows the 50% attainment surface by using HSS-EA, GPSI-FFGA, NSGA-II and SPEA2 with 30 runs. From Fig. 7, it is clear that HSS-EA is better than GPSI-FFGA, NSGA-II and SPEA2 around both central and edge region of the Pareto front. Figure 7: 50% attainment surface by HSS-EA, GPSI-FFGA, NSGA-II and SPEA2 Figure 6: Pseudocode of local search 4 Experiments and Discussion Because it is very difficult to get real world problem data on PPS, 4 parts (each part has 20, 16, 14 and 7 operations, respectively) problem suggested by Li and Mcmahon [2] and Li et al. [10] are used. All the simulation are performed on Core 2 Quad processor (3.0 GHz clock), and the program is written in C# language. The adopted parameters are listed as follows: population size, 100; maximum generation, 2000; archive size, 50; crossover probability, 0.70; mutation probability, 0.60. Figure 8 shows the numerical comparison of the box-and-whisker plots for C, GD and SP and the CPU times by HSS-EA, GPSI-FFGA, NSGA-II and SPEA2. From Fig. 8a,b,c, it is easy to see that the HSS-EA is slightly better than GPSI-FFGA, and is obviously better than NSGA-II and SPEA2 on C measure. The GD measure as shown in Fig. 8d also indicates that HSS-EA can get smaller value than NSGA-II and SPEA2 while almost the same as GPSI-FFGA. The distribution performances, SP, as shown in Fig. 8e indicates that HSS-EA is obviously better than GPSI-FFGA, while better than NSGA-II and SPEA2. Without special mechanism to preserve the diversity evenly, HSS-EA can also achieve satisfactory distribution performance. From the comparisons and analysis as shown in Fig. 8f, it is clear that HSS-EA is much faster than NSGA-II and SPEA2. Hybrid Sampling Strategy-based Multiobjective Evolutionary Algorithm for Process Planning and Scheduling Problem In general, the convergence and distribution performance of HSS-EA is better than GPSI-FFGA while the efficiency is equivalent, and the convergence and distribution performance is also better than NSGA-II and SPEA2 and the efficiency is obviously better than them. [2] [3] [4] [5] [6] [7] [8] [9] Figure 8: C, GD, SP and CPU times by HSS-EA, GPSI-FFGA, NSGA-II, and SPEA2 Such better convergence and distribution performances should mainly attribute to the hybrid sampling strategy of VEGA's preference for the edge region of the Pareto front and PDDR-FF's tendency converging toward the center area of the Pareto front. They preserve better performances both in efficacy and efficiency. Especially, these two mechanisms can also keep diversity evenly without special distribution mechanisms like NSGA-II and SPEA2. [10] [11] [12] 5 Conclusions [13] In this study, a hybrid sampling strategy-based evolutionary algorithm (HSS-EA) approach was proposed to solve the multiobjective PPS problem while considering minimization of makespan and minimization of variation of machine workload. For covering the disadvantage of GPSI-FF, a new Pareto dominating and dominated relationship-based fitness function (PDDR-FF )was proposed to evaluate the individuals. The proposed method tactfully united the advantages of VEGA and PDDR-FF to solve multiobjective PPS problem. The sampling strategy of VEGA has a preference for the edge region of the Pareto front and the PDDR-FF-based sampling strategy has the tendency converging toward the center area of the Pareto front. The proposed method could preserve both the convergence and distribution performances while reducing the computation time. The binary tournament selection, problem-dependent order-based crossover, resource-based mutation, local search could improve the abilities of exploration and exploitation. Complete numerical comparisons indicted that HSS-EA was better than GPSI-FFGA in efficacy while the efficiency was equivalent, and both convergence and distribution performance was also better than NSGA-II and SPEA2 and the efficiency was obviously better than NSGA-II and SPEA2. Acknowledgement This work is partly supported by the Dongseo Frontier Project Research Fund of Dongseo University in 2011 and the KOFST project in 2010-2011 at Hanyang University. [14] References [1] Y. Zhang, A. Saravanan, and J. Fuh, “Integration of process planning and scheduling by exploring the flexibility of process planning,” [15] [16] [17] [18] [19] [20] [21] [22] [23] International journal of production research, vol.41, no.3, pp.611–628, 2003. W. Li and C. McMahon, “A simulated annealing-based optimization approach for integrated process planning and scheduling,” International Journal of Computer Integrated Manufacturing, vol.20, no.1, pp.80–95, 2007. H. Yan, Q. Xia, M. Zhu, X. Liu, and Z. Guo, “Integrated production planning and scheduling on automobile assembly lines,” IIE Transactions, vol.35, no.8, pp.711–725, 2003. T. Wong, C. Leung, K. Mak, and R. Fung, “An agent based negotiation approach to integrate process planning and scheduling,” International journal of production research, vol.44, no.7, pp.1331–1351, 2006. W. Shen, L. Wang, and Q. Hao, “Agent-based distributed manufacturing process planning and scheduling: a state-of-the-art survey,” IEEE Transactions on Systems, Man, and Cybernetics, Part C: Applications and Reviews, vol.36, no.4, pp.563–577, 2006. Y. Guo, W. Li, A. Mileham, and G. Owen, “Optimisation of integrated process planning and scheduling using a particle swarm optimisation approach,” International Journal of Production Research, vol.47, no.14, pp.3775–3796, 2009. Y. Guo, W. Li, A. Mileham, and G. Owen, “Applications of particle swarm optimisation in integrated process planning and scheduling,” Robotics and Computer Integrated Manufacturing, vol.25, no.2, pp.280–288, 2009. F. Zhang, Y. Zhang, and A. Nee, “Using genetic algorithms in process planning for job shop machining,” IEEE Transactions on Evolutionary Computation, vol.1, no.4, pp.278– 289, 1997. X. Zhang and H. Yan, “Integrated optimization of production planning and scheduling for a kind of job-shop,” The International Journal of Advanced Manufacturing Technology, vol.26, no.7, pp.876–886, 2005. L. Li, J. Fuh, Y. Zhang, and A. Nee, “Application of genetic algorithm to computer-aided process planning in distributed manufacturing environments,” Robotics and Computer Integrated Manufacturing, vol.21, no.6, pp.568–578, 2005. M. Salehi and A. Bahreininejad, “Optimization process planning using hybrid genetic algorithm and intelligent search for job shop machining,” Journal of Intelligent Manufacturing, vol. 22 no. 4, pp. 643–652 , 2011. W. Zhang and M. Gen, ”Process planning and scheduling in distributed manufacturing system using multiobjective genetic algorithm”, TEEE: IEEJ Transactions on Electrical and Electronic Engineering, vol. 5, no. 1, pp. 62-72, 2010. M. Gen and R. Cheng: Genetic Algorithms and Engineering Design, 432pp, John Wiley and Sons, New York, 1997. M. Gen, R. Cheng and L. Lin: Network Models and Optimization: Multiobjective Genetic Algorithm Approach, 710pp, Springer, London, 2008. J. Schaffer, “Multiple objective optimization with vector evaluated genetic algorithms,” Proceedings of the 1st International Conference on Genetic Algorithms table of contents, pp.93–100, L. Erlbaum Associates Inc. Hillsdale, NJ, USA, 1985. W. Zhang: Research on Scheduling Problems in Machining and Assembly Processes using Multiobjective Evolutionary Algorithms, PhD Thesis, Waseda University, 160pp, July 2011. K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, vol.6, no.2, pp.182–197, 2002. E. Zitzler, M. Laumanns, and L. Thiele, “SPEA2: Improving the Strength Pareto Evolutionary Algorithm,” Evolutionary Methods for Design, Optimisation, and Control, 2001. S. Ho, L. Shu, and J. Chen, “Intelligent evolutionary algorithms for large parameter optimization problems,” IEEE Transactions on Evolutionary Computation, vol.8, no.6, pp.522–541, 2004. W. Zhang and M. Gen, ”An Efficient Multiobjective Genetic Algorithm for Mixed-model Assembly Line Balancing Problem Considering Demand Ratio-based Cycle Time”, Journal of Intelligent Manufacturing, vol. 22, no. 3, pp. 367-378, 2011. E. Zitzler and L. Thiele, “Multiobjective evolutionary algorithms: A comparative case study and the strength pareto approach,” IEEE transactions on Evolutionary Computation, vol.3, no.4, pp.257–271, 1999. D. Van Veldhuizen, Multiobjective Evolutionary Algorithms: Classifications, Analyses, and New Innovations, Ph.D. thesis, Air Force Inst Of Tech Wright-Pattersonafb Oh School Of Engineering, 1999. J. Schott, “Fault tolerant design using single and multicriteria genetic algorithm optimization,” Master’s thesis, MIT, 1995.

© Copyright 2019