Sustainability Formation of Machine Cells in Group Technology Systems Using Modiﬁed Artiﬁcial Bee Colony Algorithm

: The efﬁciency and sustainability of a cellular manufacturing system (CMS) in batch type manufacturing is highly valued. This is done using a systematic method of equipment into machine cells, and components into part families, based on the suitable similar criteria. The present work discusses the cell formation problem, with the objective of minimizing the cumulative cell load variation and cumulative intercellular moves. The quantity of parts, operation sequences, processing time, capacity of machines, and workload of machineries were considered as parameters. For the grouping of equipment, the modiﬁed artiﬁcial bee colony (MABC) algorithm is considered. The computational procedure of this approach is explained by using up to 40 machines and 100 part types. The result obtained from MABC is compared with the ﬁndings acquired from the genetic algorithm (GA) and ant colony system (ACS) in the literature.


Introduction
Group technology (GT) is a production strategy where the similar components are picked out and grouped together to gain the benefits of their similarities in design and/or production character [1]. Cellular manufacturing (CM) is based on the principles of group technology. Cellular manufacturing system plays a major role in the application to industries. It is an important technique to cope with the rapid changing industrial demands and new innovations. In CM, the equipment is located in close proximity, and sacrificed to make all necessary operations into the particular part family, and provide smooth flow of materials within the cell which lead to high productivity. The equipment in CM system permits the equipment to be changed or relocated whenever new part designs are incorporated, and product demand changes with minimum effort, in terms of cost and time [2]. GT concept has been implemented in various factories like machineries, automobiles, defense, and electrical industries [3]. There are many advantages in cellular manufacturing system (CMS) which include minimizing the processing cost, material handling cost, machine duplication cost, cycle times, material handling times, work-in-process inventory levels, factory space requirements, product defect rates, and machine idle times [4,5]. There have been many investigations done on part families and machine cell problems during the past three decades. Opitz [6] described a method of part classification and coding analysis for effective production planning and control. Burbidge [7] explained the material flow system with process organization and process to product organization in production flow analysis. The problem comprises of a group of equipment, group of components, and assigning of components and equipment into the part families. Rank order clustering method was proposed by King [8]. The main purpose is to devise a generalized procedure that would convert, in a finite number of iterations, any original machine-part matrix into a clustered diagonal form, if one exists. Srinivasan et al. [9] discussed the similarity coefficient method, which describes that this method is more flexible for incorporating production data, such as production volume, sequence of operation, and operational time, into the machine cell formation. Heuristic algorithms [10,11] have been implemented for many cell formation (CF) solutions. In the present days, most of the cell formation problems have been solved by using meta-heuristic algorithms. Meta-heuristic search approaches, such as simulated annealing [12][13][14] and a genetic algorithm [15], have also been used to solve various cell formation problems. Prabhakaran [16] has discussed the cell load variation and intercellular moves in cell formation problem using genetic algorithm (GA). Sarac and Ozcelik [17] formulated a genetic algorithm with proper parameters for manufacturing cell formation problems. In their paper, the cell formation in cellular manufacturing systems is considered with the objective of maximizing the grouping efficacy. Arkat et al. [18] employed the minimization of exceptional elements and voids in the cell formation problem using a multi-objective genetic algorithm, and presented a bi-objective mathematical model to simultaneously minimize the number of exceptional elements and the number of voids in the part machine incidence matrix. An e-constraint method is then applied to solve the model and to generate the efficient solutions. Other metaheuristics, like tabu search [19,20], ant colony optimization (ACO) [21,22], particle swarm optimization [23,24], and scatter search [25] have also been proposed for designing CMS. Chattopadhyayet et al. [26] discussed machine-part cell formation through visual decipherable clustering of self-organizing map. Their paper deals with the self-organizing map method, an unsupervised learning algorithm in artificial intelligence which has been used as a visually decipherable clustering tool of machine-part cell formation. Ghezavati and Saidi-Mehrabad [27] proposed an efficient hybrid self-learning method for stochastic cellular manufacturing problems. It addresses a new version of stochastic mixed-integer model to design cellular manufacturing systems under random parameters, described by continuous distributions. Ying-Chin and Ta-Wei [28] explained about the concurrent solution for intra-cell flow path layouts and I/O point locations of cells in a cellular manufacturing system. In this study, the authors propose a layout procedure that can solve these two problems at the same time, so that the sum of the inter-cell flow distance and the intra-cell flow distance can be minimized. Venkumar and Haq [29] discussed the complete and fractional cell formation using neural network methodology. The other recent approaches of CMS design include a new branch-and-bound algorithm [30], and the firefly-inspired algorithm [31]. Mohammad and Kamran [32] proposed an integrated bi-objective layout and cell formation problem. In this problem, the main aim is to minimize the total inter-and intra-cell material handling costs, and the second aim is to maximize the total similarity between machines. Behrang et al. [33] explained mathematical models in cellular manufacturing systems for clustering workers and machines in product mix variation case. Sakhaii et al. [34] developed a robust optimization approach for a new integrated mixed-integer linear programming model to solve a dynamic cellular manufacturing system with unreliable machines and a production planning problem simultaneously. Brown [35] proposed optimized manufacturing model to minimize costs associated with exceptional elements. Costs are minimized through intercellular transfer, machine duplication, and subcontracting. Kia et al. [36] explained a mixed-integer programming model for a multi-floor layout design of cellular manufacturing systems (CMS) in a dynamic environment. Yung Chin et al. [37] explained a design procedure for improving the effectiveness of fractal layouts formation aiming at minimization of routing distances. Zahra et al. [38] explained the task scheduling Non-polynomial (NP) hard problem in grid computing system using particle swarm optimization-gravitational emulation local search (PSO-GELS). In this problem, the PSO-GELS combined algorithm has provided better results than other approaches. Mohammad et al. [39] discussed the job scheduling problem in cloud computing, using the methodology of fuzzy theory and genetic algorithm. In this paper, the main aim is to perform optimal load balancing considering execution time and cost. Mohammad et al. [40] described a fuzzy reputation-based model for trust management in semantic P2P grids. Javanmardi et al. [41] present a hybrid job scheduling approach, which considers the load balancing of the system, and reduces total execution time and execution cost using genetic algorithm and fuzzy theory.
In the recent years, there are so many population based meta heuristic evolutionary algorithms used to solve cell formation problems due to their ease of implementation. However, these intelligence algorithms are sensitive to value and precision. It inspired us to use modified artificial bee colony (MABC) for cell formation problem. In this paper, the MABC algorithm, a meta heuristic population based algorithms is used in cell formation problem with the aim of reducing cumulative cell load variation and cumulative intercellular moves. The outcome of this approach is compared with that of both GA and ACO.

Problem Formulation
In this problem, the most cited performance measures, such as cumulative intercellular moves and cumulative cell load variation, are to be calculated. The constraint used is a minimum of two machines in a cell.

Cumulative Intercellular Moves
The movement of parts between the cells is known as intercellular movement. It deals with the impact of processing sequence and cell layout, which is explained [42].
where C k = cell number and processing k is carried out on component i, considering the operation sequence; C k+1 = cell number in which and processing k + 1 is carried out on component i, considering the operation sequence; k i = total number of processes carried out on component i to finish its necessary operations; C = cell counting; P = part counting.

Cumulative Cell Load Variation
The cell load variation is calculated by the difference between the equipment workload and the machine cell average load. A lower value of cell load variation is preferable, since it leads to efficient flow of parts and lowers the inventory level. The cumulative cell load variation is measured using the formula suggested [15].
where m = machine counting; c = cell counting; p = part counting.
where t ij = manufacturing time (hours/piece) of job j on equipment i; T i = available time on equipment i for the allotted time period; N j = manufacturing demand of part j for the allotted time period; is an m × c cell membership matrix, where x il = 1 if the i-th equipment is in cell 1, and 0 otherwise; M = [m lj ] is a c × p matrix of the average machine cell load. Where

The Combined Objective Function
where W 1 and W 2 are weights. By varying the values of W 1 and W 2 , the decision maker can decide which function should be given a high priority.

Existing ABC Algorithm
Karaboga [43][44][45][46] proposed that ABC algorithm has been generated by imitating the behavior of the honey bee swarm intelligence. This algorithm has been implemented for numerous engineering problems. Its development was instigated through the intelligent foraging character of bees in their colony, and their performance was measured by the benchmark optimization function. Kalayci et al. [47] clearly explained the behavior of bees to solve disassembled line balancing problems. Particularly, the number of control parameters in ABC is fewer than the other population-based algorithms. Moreover, the optimization characteristics of ABC are comparable, and in some cases, much better than the state-of-the-art meta-heuristics. Hence, ABC is applied to the many types of optimization problems.
In this problem, bees represent a solution. There are three types of bees available in the ABC colony. They are employed bees, onlookers, and scout bee. It is assumed that there is only one artificial employed bee for each food source. In other words, the number of employed bees in the colony is equal to the number of food sources around the hive. Employed bees go to their food source, and come back to the hive and dance in this area. The employed bee whose food source has been abandoned becomes a scout, and starts to search for a new food source. Onlookers watch the dances of employed bees, and choose food sources depending on dances. The main steps of the algorithm are given below: Initial food sources are produced for all employed bees.

REPEAT
Each employed bee goes to a food source in her memory and determines a neighbor source, then evaluates its nectar amount and dances in the hive.
Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbor around that, she evaluates its nectar amount.
Abandoned food sources are determined and are replaced with the new food sources discovered by scouts.
The best food source found so far is registered.
UNTIL (requirements are met)

Proposed Modified ABC Algorithm
The basic ABC algorithm was originally designed for continuous function optimization. But especially for the NP hard combinatorial optimization problems, some modifications are required in the basic ABC approach. The modified ABC approach (MABC) is described in the following subsections.

Representation
The length of a solution string represents the number of equipment considered in the problem. Each place in the solution string indicates the machine number, and the position number indicates the equipment present in that position. The position of any equipment represents its location. Example: 1 3 5 2 4 8 7 6 9 Here, the length of the solution string is 9, which indicates that the number of equipment considered in the problem is 9. The places in the solution string take the values 1 to 9, which indicate the equipment number. Position of equipment 1 is 1, 3 is 2, and so on.

Population Initialization
The size of the initial population is formulated randomly. An initial population of the desired size is generated randomly. In this proposed MABC, 40 solution strings are randomly generated and used as initial population. Each solution string is converted into a 3 × 3 matrix, and the objective value is calculated using the objective function equation.

Employed Bee Phase
The employed bees initialize food origin near their present locations. In this phase, insert and swap manipulators are used to produce nearby findings. The insert manipulator of a string, S i , is explained by taking away a bit from its initial location j, and inserting it into another location, k, such that (k ∈ {j, j − 1}), where as the swap manipulator generates a nearby of S i by exchanging two bits of S i in various positions. To improve the neighborhood structure and spread out the population, the following two neighboring approaches are utilized to generate neighboring food sources for the employed bees. Approach 1. Perform one insert operator to a solution S i .
Approach 2. Perform one swap operator to a solution S i .
One of the neighboring approaches is selected randomly to produce new food sources. If the new food source is better than the current food source, then the new food source is accepted.

Onlooker Bee Phase
In the basic ABC algorithm, an onlooker bee selects a food source, x i , depending on its winning probability value p i , which is similar to the wheel selection in genetic algorithms [16]. In this MABC, the tournament selection with the size of 2 is used, due to its simplicity and ability to escape from local optima. In the tournament selection, an onlooker bee selects a food source, x i , in such a way that two food sources are picked up randomly from the population, and compared to each other, and then the better one is chosen. In addition, the onlooker utilizes the same method as used by the employed bee to produce a new neighboring solution. If the new obtained food source is better than, or equal to the current one, the new food source will replace the current one, and become a new member in the population.

Scout Bee Phase
In the basic ABC algorithm, a scout produces a food source randomly in the predefined search scope. This will decrease the search efficacy, since the best food source in the population often carries better information than others during the evolution process, and the search space around it could be the most promising region. In the MABC algorithm, the scout generates a food source by utilizing the same method as used by the employed bee to produce a new neighboring solution on the best food source.

Steps for Computational Procedure of the Proposed MABC Algorithm
Step 1: Initialize the parameters SN, PS, Limit, and PL. (In MABC, SN is equal to PS and number of Onlookers.) Where SN denotes the size of the population, PS indicates the food source count, limit specifies control parameter and PL denotes the cycles count.
Step 3: Employed bees: For i = 1, 2, . . . , PS, repeat the following sub-steps: Produce a new solution S i * for the i-th employed bee who is associated with solution S i by using the strategy presented in the subsection employed bee phase, and evaluate the new solution.
If S i * is better than or equal to S i , let S i = S i *.
Step 4: Onlooker bee: For each i = 1, 2, . . . , PS, repeat the following sub-steps: Select a food source in the population for the onlooker bee, S i , by using the tournament selection presented in the subsection onlooker bee phase. Generate a new solution, S i *, for the onlooker by using the strategy presented in the sub section employed bee phase and evaluate it. If S i * is better than or equal to S i , let S i = S i *.
Step 5: Scout bee: For each i = 1, 2, . . . , PS, repeat the following sub-steps: If a solution S i in the population has not been improved during the last limit number of trials, abandon it. Generate a new solution onlooker by using the strategy presented in the subsection Employed bee phase on the best solution.
Step 6: Store the best result achieved so far.
Step 7: If the termination criterion is reached, return the best solution found so far; else go to step 3.

Results and Discussion
The proposed algorithm is coded in C++ language on a personal computer with a 1.3 GHz Pentium IV CPU processor (Intel, Bengaluru, India) and 16 GB RAM memory (Intel, Bengaluru, India). Different parameters, such as machine workload, operation sequences, unit processing times on every equipment, and manufacturing quantity, among others, are considered in this study. From the literature [16], 20 data sets have been taken, and the experimental works have been carried out using the proposed MABC algorithm, and the outcomes were compared with those of the other two approaches, GA [16] and ACO [21]. The parameter settings selected for the problem, its implementation, and the results obtained are discussed below: MABC Parameter Setting: For the given problem structure, the parameter settings are shown in Table 1. The modified ABC algorithm is implemented, and the results are simulated for different problem sizes. The result obtained for the MABC problem, with respect to the problem sizes considered, are given below. The problem sizes considered for the present investigation is 8 × 20, 10 × 12, and 10 × 15. Figure 1 shows the MABC convergence curve for intercellular moves of problem size 8 × 20. As can be seen in the figure, the increase of cycles minimizes the intercellular movement. Further, the result indicates that number of cells consisting of 2 indicates the minimum intercellular moves. Normally, the increase of cycles tries to minimize the intercellular move. Whereas for the 3 cell problem, the trend is almost linear up to 300 cycles, after which it tries to minimize, up to 400 cycles. The 400 to 500 cycles result does not indicate the possible variations. That is, variations obtained for cycles 400-500 are very minimal. For the cell number 4, drastic changes are observed in intercellular moves from the cycles 100 to 200, and minimum variation from the cycle of 200 to 500. the literature [16], 20 data sets have been taken, and the experimental works have been carried out using the proposed MABC algorithm, and the outcomes were compared with those of the other two approaches, GA [16] and ACO [21]. The parameter settings selected for the problem, its implementation, and the results obtained are discussed below: MABC Parameter Setting: For the given problem structure, the parameter settings are shown in Table 1. The modified ABC algorithm is implemented, and the results are simulated for different problem sizes. The result obtained for the MABC problem, with respect to the problem sizes considered, are given below. The problem sizes considered for the present investigation is 8 × 20, 10 × 12, and 10 × 15. Figure 1 shows the MABC convergence curve for intercellular moves of problem size 8 × 20. As can be seen in the figure, the increase of cycles minimizes the intercellular movement. Further, the result indicates that number of cells consisting of 2 indicates the minimum intercellular moves. Normally, the increase of cycles tries to minimize the intercellular move. Whereas for the 3 cell problem, the trend is almost linear up to 300 cycles, after which it tries to minimize, up to 400 cycles. The 400 to 500 cycles result does not indicate the possible variations. That is, variations obtained for cycles 400-500 are very minimal. For the cell number 4, drastic changes are observed in intercellular moves from the cycles 100 to 200, and minimum variation from the cycle of 200 to 500.     Figure 3 shows the MABC convergence curve for intercellular move for the problem size 10 × 12. In the above figure for the cell number 2, 3, and 4, the curve is almost linear from 100 to 300 cycles. But for the cell number 5, there is drastic change of curve from 400 to 500 cycles.       Figure 3 shows the MABC convergence curve for intercellular move for the problem size 10 × 12. In the above figure for the cell number 2, 3, and 4, the curve is almost linear from 100 to 300 cycles. But for the cell number 5, there is drastic change of curve from 400 to 500 cycles.         Figure 6 shows the MABC convergence curve for cell load variation of problem size 10 × 15. In the above figure, for the cell number 2, there are some improvements in minimizing cell load variation from cycle 100 to 300, and considerable improvement from cycle 300 to 500. For the cell numbers 3, 4, and 5, the curves are located very close to each other.
Comparison of results for minimum-sized problems, medium-sized problems, and bigger-sized problems from the literature [16] are mentioned in the following tables. Tables 2-4 show the results that were obtained by the other methods in the literature using genetic algorithm (GA) [16], ant colony optimization (ACO) [21], results acquired by MABC, and MABC CPU time for intercellular move and cell load variations.     Figure 6 shows the MABC convergence curve for cell load variation of problem size 10 × 15. In the above figure, for the cell number 2, there are some improvements in minimizing cell load variation from cycle 100 to 300, and considerable improvement from cycle 300 to 500. For the cell numbers 3, 4, and 5, the curves are located very close to each other.
Comparison of results for minimum-sized problems, medium-sized problems, and bigger-sized problems from the literature [16] are mentioned in the following tables. Tables 2-4 show the results that were obtained by the other methods in the literature using genetic algorithm (GA) [16], ant colony optimization (ACO) [21], results acquired by MABC, and MABC CPU time for intercellular move and cell load variations.  Figure 6 shows the MABC convergence curve for cell load variation of problem size 10 × 15. In the above figure, for the cell number 2, there are some improvements in minimizing cell load variation from cycle 100 to 300, and considerable improvement from cycle 300 to 500. For the cell numbers 3, 4, and 5, the curves are located very close to each other.
Comparison of results for minimum-sized problems, medium-sized problems, and bigger-sized problems from the literature [16] are mentioned in the following tables. Tables 2-4 show the results that were obtained by the other methods in the literature using genetic algorithm (GA) [16], ant colony optimization (ACO) [21], results acquired by MABC, and MABC CPU time for intercellular move and cell load variations. From Table 2, it has been inferred that there are 5 different matrix-sized problems and 21 cells. The cumulative intercellular moves provide better results in 16 cases (Serial number: 1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21 from Table 2) out of 21 cases (76%), than that of any of the other methods. It yields equal results in 5 cells (Sl. No.: 3,8,9,16,17) to the best one of other methods. The cumulative cell load variation provides better outcome in 11 cases (Sl. No.: 4, 5, 10, 11, 12, 13, 14, 15, 17, 19, 20 from Table 2) out of 21 cases (52%) than that of any of the other methods. It produces equal results in 9 cells (Sl. No.: 1, 2, 3, 6, 7, 8,9,16,18) to the best of the other methods.  From Table 2, it has been inferred that there are 5 different matrix-sized problems and 21 cells. The cumulative intercellular moves provide better results in 16 cases (Serial number : 1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21 from Table 2) out of 21 cases (76%), than that of any of the other methods. It yields equal results in 5 cells (Sl. No. : 3, 8, 9, 16, 17) to the best one of other methods. The cumulative cell load variation provides better outcome in 11 cases (Sl. No. : 4, 5, 10, 11, 12, 13, 14, 15, 17, 19, 20 from Table 2) out of 21 cases (52%) than that of any of the other methods. It produces equal results in 9 cells (Sl. No.: 1, 2, 3, 6,7,8,9,16,18) to the best of the other methods.    From the analysis of Table 3, it has been known that there are 10 different matrix-sized problems and 70 cells are considered for results comparison. The cumulative intercellular moves provide better results in 40 cases out of 70 cases (57%) than that of any other method, and yield equal results in 18 cells compared to the best of the other methods. The cumulative cell load variation provides better outcome in 44 cases out of 70 cases (63%) than that of any of the method, and produces equal results in 18 cells to the best one of the other methods. Table 4 indicates that there are 5 different matrix-sized problems and 55 cases. The cumulative intercellular moves provide better results in 41 cases out of 55 cases (75%) than that of any one of the other methods, and yields equal result in 7 cells compared to the best of the other methods. The cumulative cell load variation provides better outcome in 42 cases out of 55 cases (76%) than that of any one of the other methods, and produces equal results in 4 cells compared to the best of the other methods. Based on the analysis and results, the bar charts are plotted for clarity of results for the intercellular moves and cell load variations for the problem sizes 8 × 20, 10 × 12, and 10 × 15, to compare the results of MABC with GA and ACO. Figure 7 indicates that in cell numbers 2 and 3, MABC intercellular moves are considered to be at a minimum, as there is a better convergence compared to GA and ACO. In cell number 4, MABC produces results that are on par with GA and ACO, and there is no appreciable difference observed when using MABC.
From the analysis of Table 3, it has been known that there are 10 different matrix-sized problems and 70 cells are considered for results comparison. The cumulative intercellular moves provide better results in 40 cases out of 70 cases (57%) than that of any other method, and yield equal results in 18 cells compared to the best of the other methods. The cumulative cell load variation provides better outcome in 44 cases out of 70 cases (63%) than that of any of the method, and produces equal results in 18 cells to the best one of the other methods. Table 4 indicates that there are 5 different matrix-sized problems and 55 cases. The cumulative intercellular moves provide better results in 41 cases out of 55 cases (75%) than that of any one of the other methods, and yields equal result in 7 cells compared to the best of the other methods. The cumulative cell load variation provides better outcome in 42 cases out of 55 cases (76%) than that of any one of the other methods, and produces equal results in 4 cells compared to the best of the other methods. Based on the analysis and results, the bar charts are plotted for clarity of results for the intercellular moves and cell load variations for the problem sizes 8 × 20, 10 × 12, and 10 × 15, to compare the results of MABC with GA and ACO. Figure 7 indicates that in cell numbers 2 and 3, MABC intercellular moves are considered to be at a minimum, as there is a better convergence compared to GA and ACO. In cell number 4, MABC produces results that are on par with GA and ACO, and there is no appreciable difference observed when using MABC.

Conclusions
The main purpose of this paper is to compare the performance of the MABC approach with the other two heuristic algorithms in the literature GA [16] and ACO [21] to minimize the cumulative intercellular moves and the cumulative cell load variation. From the results obtained, it has been indicated that, in most of the problems discussed in the literature, cumulative intercellular moves and cumulative cell load variations obtained by the proposed method, MABC, is either better (minimized) than the other methods, or it is equal to the best one. The outcome obtained in the computational experiences carried out show that the proposed algorithm MABC deserves to be well placed among the three approaches in studies on machine cell formation. The work can be further extended in the future, incorporating parameters like demand of parts, cost factor, etc., enhancing it to a more generalized sustainability in manufacturing environments.

Conclusions
The main purpose of this paper is to compare the performance of the MABC approach with the other two heuristic algorithms in the literature GA [16] and ACO [21] to minimize the cumulative intercellular moves and the cumulative cell load variation. From the results obtained, it has been indicated that, in most of the problems discussed in the literature, cumulative intercellular moves and cumulative cell load variations obtained by the proposed method, MABC, is either better (minimized) than the other methods, or it is equal to the best one. The outcome obtained in the computational experiences carried out show that the proposed algorithm MABC deserves to be well placed among the three approaches in studies on machine cell formation. The work can be further extended in the future, incorporating parameters like demand of parts, cost factor, etc., enhancing it to a more generalized sustainability in manufacturing environments.

Conclusions
The main purpose of this paper is to compare the performance of the MABC approach with the other two heuristic algorithms in the literature GA [16] and ACO [21] to minimize the cumulative intercellular moves and the cumulative cell load variation. From the results obtained, it has been indicated that, in most of the problems discussed in the literature, cumulative intercellular moves and cumulative cell load variations obtained by the proposed method, MABC, is either better (minimized) than the other methods, or it is equal to the best one. The outcome obtained in the computational experiences carried out show that the proposed algorithm MABC deserves to be well placed among the three approaches in studies on machine cell formation. The work can be further extended in the future, incorporating parameters like demand of parts, cost factor, etc., enhancing it to a more generalized sustainability in manufacturing environments.