Application of Genetic Algorithm in Generalized Machine Cell Formation Problem Manash Hazarika
Dipak Laha
Department of Mechanical Engineering Assam Engineering College Guwahati-781013, India
[email protected]
Department of Mechanical Engineering Jadavpur University Kolkata-700032, India
[email protected]
Abstract— In automated batch type production systems, machine-part cell formation problems (CFP) have long drawn attention of researchers. The objective of CFP in cellular manufacturing system (CMS) is to identity machine cells and part families in order to minimize the intercellular movements of parts as well as maximize the utilization of machines. Optimum cell formation results reduction in total production times, in-process inventories, material handling cost, labor cost/times, paper works, number of machine set-ups, set-up times. It also simplifies process plans, management and improves product quality, productivity, utilization of resources. Since the modern manufacturing machines are generally multifunctional, so the processing of parts can be performed by number of alternative routes. The objectives of this study is to determine the optimal processing route and balanced machine cells (to minimize cell load variation) incorporation with parts volume, process sequence for minimum intercellular movements of parts. A genetic algorithm metaheuristic approach is presented for a benchmark CFP. Computational results show that the proposed approach gives better result comparatively with the well-known existing methods in terms of total intercellular movements. Keywords— Cellular manufacturing system; Alternative process routings; Genetic algorithm; Route selection; Intercellular movement of parts
I. INTRODUCTION In CMS, machine-part CFP usually seeks to obtain a solution of completely independent machine cells where each machine cell is assigned with independent part families so that a cell can carry out all operations of that particular part family. But in actual practice, it is sometimes difficult to execute all the operations of a part family following a particular machine cell. Therefore, the principal objective of CFP in CMS is to minimize intercellular movements of parts and to maximize utilization of machines (Logendran, 1990). Since the CFP problem belongs to the class of NP- hard [1], heuristic and meta-heuristic approaches are mostly preferred to obtain
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optimal or near-optimal solution for this problem in reasonable computational time. In literature survey, we have seen that most of the cell formation methods have been applied for one process route, equal part volume without considering any sequence of processes [2-5]. But in today‟s CMSs, parts can be processed by multiple process routings, unequal part volume with sequence of processes [6]. In this paper, genetic algorithm approach is proposed for solving CFP with alternative routings, operation sequences of the parts and unequal part volumes. The remainder of this paper is organized as follows: Section 2 provides a brief literature review on CFPs and solution techniques. Sections 3 and 4 present brief description on CFPs and a genetic algorithm heuristic respectively. The computational results as well as a comparison to the existing methods and the proposed method are presented in Section 5 and finally, Section 6 concludes the paper. II. LITERATURE REVIEW In literature review, most of the CFPs have been studied considering single process routing, equal part volumes and no process sequences. But modern manufacturing machines are multifunctional and therefore, number of operations can be done in one machine. In such systems production processes can be perform by more than one process routes. Alternative routes give better cell configuration, increase flexibility in cell formation, and reduce intercellular movements of parts [7]. It also reduces number machines and provides better utilization of resources/or machines [2]. For cell formation in alternative routing conditions Kusiak and Cho [8], Chow and Hawaleshka [9], Gupta [10] recommended similarity coefficient methods. Chow and Hawaleshka [9] considered uneven part volume in their model,
whereas Gupta [10] considered process sequences and uneven part volumes also. Gupta [10] used Jaccard‟s similarity coefficient and recommend complete linkage clustering (CLINK) methods. In 1997, Wafik and Kim [7] used a generalize Jaccard‟s similarity coefficient for cell formation in alternative routes conditions. Yin and Yasuda [11] modified the Wafik and Kim [7] similarity coefficient with the integration of process sequence, part volumes, part processing time. Recently, Farouq [12] proposed another similarity coefficient with the integration of process sequence and part volumes. In 2016, Hazarika and Laha [13] find correlations of different machines in Euclidean distance matrix with parts volume and parts process sequences for alternative process route problems. They used single linkage (SLINK) clustering technique for machine cell formation. Since the CFP problem is seen to be a NP-hard [14], so researchers are showing more interest to implement metaheuristic algorithms due to its capability to search globally as well as locally to converge to the global optimum more efficiently, in reasonable computational time. During last two decades, researchers are significantly applying soft computing techniques in CFP. Different soft computing methods found in literature are fuzzy approach [15] and [16]; simulated annealing [17]; artificial neural networks [18] and [19]; genetic algorithm [20] and [5]; particle swarm optimization [21] and tabu search [20]. Recently, hybrid heuristics and meta-heuristics are being applied in CFPs such as hybrid grouping GA (HGGA) [22], hybrid GA (HGA) [23], GRASP [24], hybrid grouping based PSO (HGBPSO) [21], correlation analysis and relevance index (CARI) [25].
processing time is 0 (zero); a5,2(3) =3(4), i.e., third operation is performed in machine M5 for the part P2 on route 3 and here processing time is 4 and so on. The separate machine-part incidence matrix and processing time matrix is shown in TABLE II and TABLE III respectively. Empty elements means no operation is there. TABLE I. Part (annual demand) 1(6)
A CFP with alternative part routes and machines/or processes sequence taken from Chang et al. [31] is shown in TABLE I. The problem has uneven part volume (annual demand), single batch and machine process capacities.
3
4
5
6
7
8
9
10
1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 3 1 2
0(0) 0(0) 0(0) 0(0) 0(0) 1(4) 0(0) 2(4) 1(4) 2(3) 0(0) 0(0) 0(0) 0(0) 0(0) 3(3) 1(1) 0(0) 0(0) 0(0) 3(4) 4(4) 0(0) 0(0) 0(0)
5(3) 0(0) 2(4) 0(0) 2(4) 0(0) 2(3) 0(0) 0(0) 0(0) 4(4) 1(4) 4(4) 4(4) 3(3) 0(0) 0(0) 0(0) 2(3) 1(3) 0(0) 0(0) 0(0) 5(4) 0(0)
3(4) 0(0) 0(0) 3(4) 1(4) 4(4) 3(3) 0(0) 0(0) 3(3) 2(4) 4(3) 2(4) 2(3) 1(3) 0(0) 4(4) 0(0) 5(3) 4(4) 0(0) 1(4) 1(4) 3(3) 3(3)
1(3) 1(4) 0(0) 0(0) 0(0) 2(3) 4(4) 3(4) 2(3) 0(0) 0(0) 0(0) 0(0) 0(0) 4(3) 0(0) 2(4) 0(0) 0(0) 0(0) 4(3) 0(0) 0(0) 0(0) 0(0)
0(0) 0(0) 3(3) 1(4) 3(4) 0(0) 0(0) 0(0) 0(0) 0(0) 5(3) 2(3) 0(0) 5(4) 0(0) 0(0) 0(0) 2(4) 3(4) 2(3) 0(0) 0(0) 0(0) 1(3) 0(0)
2(3) 2(3) 0(0) 0(0) 0(0) 3(3) 5(3) 4(4) 3(4) 4(4) 0(0) 0(0) 0(0) 0(0) 0(0) 5(4) 3(4) 5(4) 0(0) 3(3) 5(3) 0(0) 2(4) 4(3) 0(0)
0(0) 3(3) 0(0) 0(0) 5(4) 0(0) 1(4) 0(0) 4(4) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 1(4) 0(0) 0(0) 0(0) 0(0) 1(3) 2(3) 3(4) 0(0) 0(0)
4(4) 4(4) 0(0) 0(0) 0(0) 5(4) 0(0) 1(4) 5(4) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 2(4) 0(0) 4(3) 0(0) 0(0) 2(4) 2(3) 4(3) 0(0) 4(4)
0(0) 0(0) 1(4) 4(3) 0(0) 0(0) 0(0) 0(0) 0(0) 1(3) 3(3) 5(4) 3(3) 3(4) 2(4) 0(0) 0(0) 3(3) 1(3) 0(0) 0(0) 0(0) 0(0) 0(0) 1(4)
0(0) 0(0) 4(3) 2(3) 4(3) 0(0) 0(0) 0(0) 0(0) 0(0) 1(4) 3(3) 1(4) 1(3) 5(4) 4(3) 0(0) 1(3) 4(4) 0(0) 0(0) 0(0) 0(0) 2(3) 2(3)
300
300
300
300
300
300
300
300
300
300
4(14)
5(20) 6(6)
7(18) 8(14)
9(12)
10(6) Capacity limit
TABLE II Part (annual demand) 1(6) 2(18)
3(20)
5(20) 6(6)
8(14)
9(12)
10(6)
The CFP is generally formulated as „m×p‟ incidence matrix (where, m is the number of machines and p is the number of parts). Here, columns stand for machines and rows stands for parts. Elements of the matrix represent operation indices of parts. For example, in TABLE I, a3,1(2) =0(0), i.e., no operation is performed in machine M3 for the part P1 on route 2 and so
2
3(20)
7(18)
III. PROBLEM DESCRIPTION
1
2(18)
4(14)
Heuristic technique for alternative process routings, uneven parts volume and process sequential CFPs found in literature are simulated annealing [26-27], fuzzy approach [16], tabu search [28], branch and bond [29] and genetic algorithm [30].
Machine (processing time)
Part route
Part route 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 3 1 2
Machine 1
2
3
4
5
3
1 1
2 2 1 2 2 1 2 4 1 4 4 3 3 1 2 1 3 4
3 1 4 3
5
2 2 3 1 3
2 4 3 2
3 2 4 2 2 1
4
4
2
Indices and parameters: i Index for machines j Index for parts r Index for routes
8
3
4 4
1
10
1 4
4 2 4
5 1 4
1 5 1 3 5 3 3 2
5
2 3 2
9
5 3 5 4 3 4
4
5
7
5 2
5 4 1 1 3 3
6
5 3 5 3 5 2 4
1
2 4
1 2 3
1 3 1 1 5 4
3 1
1 4
1
2 2
2 3 4 4
Rj k C M P Dj Lk Uk ti,j(r) Ti Tk
Total number of routes of part j Index for cells Total number of cells Total number of machines Total number of parts Demand rate of part j Minimum number of machines in cell k Maximum number of machines in cell k Processing time on machine i of part j in route r Allowable processing time of machine i Cell load variation of cell k
Constraints: 1. Assignment of one machine to only one cell (4)
TABLE III.
2(18)
3(20) 4(14)
5(20) 6(6)
7(18) 8(14)
9(12)
10(6)
Part route 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 1 2 3 1 2 3 1 2
Machine processing time 1
2
3
4
18
24
18 24
72 72 80 60 56 56 42 80 80 24 24 18 54 18 42 42 48 48
54 72 80 60
5
54 72 72 60 80 56 42
42 80 60 24 18 18
18
72
72
8
18
24 24
18
72 72 56 42 36 48 18
3.
(5) Total processing time of machines for selected routes
10
72 54
54 54 54
IV. GENETIC ALGORITHM Genetic algorithm (GA) is a stochastic based global search optimization algorithm technique guided by natural evolution principle. It is initialized with a set of random solutions, called as initial population and then executes selection, reproduction, crossover and mutation sequentially for a fixed number of operations. Few numbers of members are selected from population with the help of fitness value or best neighborhood solution. A new set of population is generated from selected members by crossover and mutation, for the next iteration. In this manner the process is repeated for a certain number of iterations. The basic steps of the GA are as follows: Step I.
80 80 56
Step II.
56 56 42 60 80 18 24 24
24
56 56 42
9
72 60 60 56 56 56
36
24
7
60 60
42 56 48 48 18 18
6 18 18
Lower bound and upper bound of a cell size
(6)
Decision variables: xi,k =1 if machine i is assigned to cell k; otherwise 0 yr,j =1 if route r of part j is assigned; otherwise 0 σj(k,k΄) =1 if successive operations of part j is done on cells k and k΄; otherwise 0
Part (annual demand) 1(6)
2.
72
72 42
36 36 48
42 42
42 56
24
18 18
48 36 36 24
Step III.
8 60 24 18 24 54
Step IV. Step V. Step VI.
Generation of initial population (random solution) Calculation of fitness values of each chromosome in the population Selection of best populations based on fitness values Generation of new set of population from best individuals by crossover Mutation Obtaining the best solution after executing a number of iterations (put a number to stop the process)
A.
Objectives of problems The goals of the proposed approach is to minimized the intercellular movement of each part j (Ij) as a result of minimum intercellular movements (Z) and minimum cell load variation (T) of the system; these can be formulated as (1) (2) (3)
Encoding and generation of initial population GA is initializes with the generation of a set of random solutions. In this study, each random solution of the initial population is encoded with integers and it is varies from 1 to the number of maximum possible groups (depends upon the maximum numbers of machines in a cell). In each chromosome or individual, integers indicate which machine is assigned to which group of machine cell. For example, for a problem of 10 machines the maximum numbers of machines in a cell is four. Therefore total cell numbers will be 10/4 3 (rounded to next integer). Fig. 1 shows a randomly generated solution or chromosome for three machine cells. Here machine M2, M3, M5 and M10 are assigned to machine cell 1, machine M1, M7 and M8 are assigned to machine cell 2 and machine M4, M6
and M9 are assigned to machine cell 3. Fig.1 shows a randomly generated solution for 10 machines cell formation problem. M1 2
M2 1
M3 1
M4 3
M5 1
M6 3
M7 2
M8 2
M9 3
M10 1
Offspring 1
2
1
1
3
1
3
3
1
3
2
Offspring 2
1
2
2
1
3
2
2
2
3
1
Fig. 3. Two offspring generated after the crossover of parent 1and parent 2
Fig. 1. Randomly generated solution for 10 machines CFP
B. Evaluation of fitness of each individual population Computation of fitness value of each chromosome or random solution in the population is a criterion of the selection process to assess the high probability of selecting the candidate solution to the next iteration. The larger fitness value is having the higher probability of survival for the next generation. The objective functions of each individual in the population are measured by the intercellular movements and cell load variation which have to be minimized. C. Selection or sorting population Few numbers of chromosomes are selected with greater fitness values (called parent chromosome) from the populations through which generate a new set of solutions (called offspring). There are numbers of selection procedures; in this study the roulette-wheel-selection procedure has been used. D.
Generation of new set of population from best individual by crossover operator Crossover is the process of generation of offspring from more than one parent chromosome. The most useful crossover operators are single point, two-point, uniform crossover operator. In this work, single point crossover was used. For the 10 machine CFP, cross-over point of two selected chromosomes, say, parent 1 and parent 2 is taken at a randomly selected location and as a result, another two new chromosomes say, offspring1 and offspring 2 are generated. Fig. 2 shows two parent chromosomes and their crossover points and Fig. 3 presents two offspring (offspring 1 and offspring 2) generated after the crossover of parent 1 and parent 2. Parent 1
2
1
1
3
1
3
2
2
3
1
1
3
2
Crossover point
Parent 2
1
2
2
1
3
2
3
Fig. 2. Two parent chromosomes and their crossover points
E. Mutation Mutation is an operator which is required to introduce diversities in populations from one generation to another generation in order to overcome the local optimum. It swaps randomly selected gene values (or bit value) in the chromosomes with a probability ratio equal to the mutation probability ratio. There is no guarantee that mutation gives always positive directions towards the optimal solution. For example, consider a chromosome as shown in Fig. 4; the mutation point is 6th gene. The new chromosome after mutation is shown Fig. 5. 1
2
2
1
3
2
2
2
3
1
3
1
Fig. 4. A chromosome before mutation
1
2
2
1
3
1
2
2
Fig. 5. Previous chromosome after mutation
F. Obtaining the best solution To obtain the feasible optimum solution, repeat the above process for a certain number of iterations. The number of iterations are generally depends upon the size of problem. G. GA Parameters The proposed GA starts by generation of initial population. After the generation of initial population, GA executes loops for the selection of feasible solutions. It also executes crossover and mutations within the loop sequentially. In this study, we used single point operators for both crossover and mutation. The proposed GA searches better solutions having minimum intercellular movements and cell load variation and ignores inferior solutions within the vicinity and creates a new updated better solution in each iteration process. To run successfully the algorithm, following parameters have a crucial role for optimum solutions. The selected parameters of the proposed GA are as follows:
Crossover operator: single-point crossover Chromosome length: equal to total number of machines in the matrix Maximum cell numbers: depends upon cell size Selection: Rank-based roulette wheel selection Population size: Probability of mutation: Number of generations: Number of selected chromosomes: Number of trials: 10
To compare the performance of the proposed method with intercellular movements and cell load variation with computational time, the algorithms generated by Zhao & Wu [32] and Chang et al. [31] are considered. The comparative computational results are shown in T ABLE V. From TABLE V, it is seen that our approach gives better result in total intercellular movements in reasonable computational time. But in terms of cell load variation, the proposed approach is inferior. So, we think that our algorithm is more efficient in cases of minimum intercellular movements and computational efforts.
V. COMPUTATIONAL RESULTS The source of benchmark problem is Chang et al. [31] and the machine-part incidence matrix with processing time is given in TABLE I. The proposed method was coded in MATLAB R2010a and executed on a processor with Intel Core i5 CPU with 8 GB RAM at 3.30 GHz speed. The solution for cell formation with best routes and cell load variations for processing times of the propose approach is given in TABLE IV. Here we considering same weight factor to minimum intercellular movement of parts and minimum cell load variation. Total intercellular movement =40+18+14+6=78 Maximum cell load variation (i.e., processing time) =298-122=176min. And the computational time is 0.755526 seconds.
VI. CONCLUSIONS Study of metaheuristic algorithms is very important for solving cellular manufacturing system problems since it is difficult to generate optimal solutions, especially for largesized cell formation problems in reasonable computational times using heuristic methods. This paper considers the machine cell formation problem with the objective of minimizing intercellular movements and cell load variation in alternative routes of parts and process sequential situation. In this study, a metaheuristic genetic algorithm is presented to solve CFP. Computational results and comparisons with wellknown metaheuristic algorithms are reported in Section V. The computational results reveal that the proposed heuristic produces best solution in terms of intercellular movement.
TABLE IV. Parts
Routes
1(6) 3(20) 4(14) 7(18) 9(12) 2(18) 5(20) 6(6) 8(14) 10(6) Machine load
2 2 2 2 1 2 2 2 1 2
1 0(0) 0(0) 1(4) 1(1) 3(4) 0(0) 0(0) 0(0) 0(0) 0(0) 122
Machine cell 1 4 6 7 1(4) 2(3) 3(3) 4(4) 5(3) 1(4) 2(3) 3(4) 4(4) 2(4) 3(4) 0(0) 4(3) 5(3) 1(3) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 0(0) 5(4) 0(0) 0(0) 0(0) 0(0) 254 190 298
8 4(4) 0(0) 5(4) 0(0) 2(4) 0(0) 0(0) 0(0) 4(3) 4(4) 194
2 0(0) 2(3) 0(0) 0(0) 0(0) 0(0) 1(4) 4(4) 0(0) 0(0) 164
Machine cell 1 3 5 9 0(0) 0(0) 0(0) 3(3) 0(0) 0(0) 0(0) 0(0) 0(0) 4(4) 0(0) 0(0) 0(0) 0(0) 0(0) 3(4) 1(4) 4(3) 4(3) 2(3) 5(4) 2(3) 5(4) 3(4) 0(0) 2(4) 3(3) 3(3) 0(0) 1(4) 282 212 224
10 0(0) 0(0) 0(0) 0(0) 0(0) 2(3) 3(3) 1(3) 1(3) 2(3) 192
Intercellular movement for annual demand of each parts 40 18
14 6
References Algorithms generated by Chang et al. (2004) Zhao & Wu (2000) Our
TABLE V. Total intercellular movements 88 82 78
Maximum cell load variation 118 112 176
Computational time in seconds 0.03 519.4 0.755
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