A stochastic model for the cell formation problem considering machine reliability
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Abstract
This paper presents a new mathematical model to solve cell formation problem in cellular manufacturing systems, where interarrival time, processing time, and machine breakdown time are probabilistic. The objective function maximizes the number of operations of each part with more arrival rate within one cell. Because a queue behind each machine; queuing theory is used to formulate the model. To solve the model, two metaheurstic algorithms such as modified particle swarm optimization and genetic algorithm are proposed. For the generation of initial solutions in these algorithms, a new heuristic method is developed, which always creates feasible solutions. Both metaheurstic algorithms are compared against global solutions obtained from Lingo software’s branch and bound (B&B). Also, a statistical method will be used for comparison of solutions of two metaheurstic algorithms. The results of numerical examples indicate that considering the machine breakdown has significant effect on block structures of machinepart matrixes.
Keywords
Cell formation Queuing theory Metaheurstic algorithm ReliabilityIntroduction
The concept of group technology (GT) has emerged to reduce setups, batch sizes, and travel distances. In essence, GT tries to retain the flexibility of a job shop with the high productivity of a flow shop. Cellular manufacturing (CM) is an application of GT in a manufacturing system. CM involves processing a collection of similar parts (part families) on a dedicated cluster of machines or manufacturing processes (cells). The cell formation (CF) problem in CM systems is the decomposition of the manufacturing systems into cells (Singh and Rajamani 1996). Cellular manufacturing problems can be under static or dynamic conditions. In static conditions, CF is done for a singleperiod planning horizon. In real problems, some input parameters such as costs, demands, processing times, and setup times are uncertain and this uncertainty can affect the results. In the static stochastic problems, it is assumed that our information on model parameters is incomplete. In other words, the exact value of the parameters is unknown. It can only be predicted with probability; however, parameters are uncertain, static and do not change during time. In the following, a review of previous studies about static stochastic cell formation problem is presented in four parts such as processing time, the mix product, the demand, and the reliability, respectively.
SaidiMehrabad and Ghezavati (2009) assumed the processing time and the time between two successive arrivals to cell described by exponential distribution in the CF problem. For analyzing this problem, they used queuing theory in which machine has been considered as a server and the part as a customer. The aim of this model is to minimize the summation of three costs: (1) the idleness costs for machines (2) total cost of subcontracting for exceptional elements, and (3) the cost of resource underutilization. Ghezavati and SaidiMehrabad (2010) proposed a mathematical model for the CM problem integrated with group scheduling in an uncertain space. Within this model, CF and scheduling decisions are optimized concurrently. It is assumed that processing time of parts on machines is stochastic and described by discrete scenarios. Their model minimizes the expected total cost including maximum tardiness cost among all parts, the cost of subcontracting for exceptional elements, and the cost of resource underutilization. Egilmez and Suer (2011a, b) presented a mathematical model for CF which minimized the number of tardy jobs and total probability of tardiness. They assumed that processing time of each job has a normal distribution. Ghezavati and SaidiMehrabad (2011) assumed that each machine works as a server and each part is a customer where servers should provide service to customers. Accordingly, they defined formed cells as a queue system which can be optimized by queuing theory. The optimal cells and part families were formed by maximizing the probability that a server is busy. Ghezavati (2011) evaluated the CF problem, scheduling, and layout decisions, concurrently. Also, he considered processing time as stochastic with discrete scenarios under supply chain characteristics. This model minimized holding cost and the costs with respect to the suppliers network in a supply chain in order to outsource exceptional operations. Fardis et al. (2013) examined the CF problem considering that the arrival rate of parts into cells and machine service rate are stochastic parameters, which have described by an exponential distribution. The objective function of presented model minimized summation of the idleness cost of machines, the subcontracting cost for exceptional parts, nonutilizing machine cost, and the holding cost of parts in the cells.
The mix product is defined as the set of part types in a factory which can be produced and each factory looks for the best mix product. The mix product and value of demand for each product in the mix product due to customized products, shorter product life cycles, and unpredictable patterns of demand, are not known exactly at the time of designing the manufacturing cells. The composition of the product mix is determined by demand and is probabilistic in nature. Seifoddini (1990) proposed a stochastic CF model in which for each mix product a probability had been attributed. He calculated the expected intercellular material handling cost for each machine cell arrangement under all possible product mixes. Madhusudanan Pillai and Chandrasekharan (2010) evaluated the CF problem under probabilistic product mix. Each product mix is specified with a scenario and to each scenario has been attributed probability of occurrence. They minimized intercell material handling. Jayakumar and Raju (2011) presented a mathematic model for the CF problem in which for each scenario probability of occurrence has been attributed. The objective function of this model is to minimize the total machine constant (investment) cost, the operating cost, the intercell material handling cost, and the intracell material handling cost for a particular product mix.
A review of studies done in the demand area is provided in the following. Cao and Chen (2005) offered CF with scenarios for product demand. In this model, an occurrence probability had been assigned for each scenario. The objective function of this model minimized machine cost and the expected intercell material handling cost. TavakkoliMoghaddam et al. (2007) examined a mathematical model to solve a facility layout problem in CM systems with stochastic demands. The main purpose of their study is to minimize the total costs of intercell and intracell movements in both machine and cell layout problems in CM system simultaneously. They considered part demands as an independent variable with the normal probability distribution. Egilmez and Suer (2011a, b) proposed a twophase hierarchical methodology to find the optimal manpower assignment and cell loads simultaneously. In the first phase, the manufacture cells are formed with the objective function of the production rate maximization. Then, the manpower is assigned to the manufacture cells to minimize number of labors. In both the models, the processing time and demand have the normal distribution. Ariafar et al. (2011) purposed the model for cell layout in the shop and machines in the machine cells. The demand has been considered as stochastic and with a uniform distribution. This model minimizes the intercell and intracell material handling costs. Egilmez et al. (2012) considered processing times and customer demand uncertain with the normal distribution. The objective is to design a CM system with product families that are formed with the most similar products and minimum number of cells and machines for a specified risk level. Ariafar et al. (2012) examined the effect of demand fluctuation on cell layout in shop and machine layout in cell. This model minimizes the intercell and intracell material handling costs. They assumed which demand has the normal distribution. Rabbani et al. (2012) proposed a biobjective CF problem with demand of products expressed in a number of probabilistic scenarios. Their model in the first objective minimizes the sum of machine constant cost, expected machine variable cost, cell fixedcharge cost, and expected intercell movement cost and in the second objective minimizes the expected total cell loading variation. Egilmez and Suer (2014) offered two models for analyzing the interaction between CF stage and cell scheduling stage in terms of the risk taken by decisionmakers. The first model formed manufacturing cells with the objective of maximizing total pairwise similarity among products assigned to cells and minimizing the total number of cells. The second model maximizes the number of early jobs. The demand and the processing time in both models are random variables with the normal distribution.
Jabal Ameli et al. (2008) investigated the effects of machine breakdowns in the CF problem with a new perspective. The results of their study showed that although considering machine reliability can increase the movement costs, it significantly reduces the total costs and total time for CM system. Jabal Ameli and Arkat (2008) have conducted a study on the configuration of machine cells considering production volumes and process sequences of parts. Further, they studied on alternative process routings for part types and machine reliability considerations. They found out that the reliability consideration has significant impacts on the final block diagonal form of machinepart matrixes. Chung et al. (2011) found that machine reliability has meaningful effects on reducing the total system cost in the CF problem. Arkat et al. (2011) presented the CF problem in general state and considering the reliability. The generalized CF problem follows selecting the best process plan for each part and assigning machines to the cells. In this model, it has been assumed that the number of breakdowns for each machine follows a Poisson distribution with a known failure rate. Because of the probabilistic nature of the machine breakdowns, a set of chance constraints have been introduced. These constraints guarantee that the number of breakdowns for each machine never exceeds a predefined percentile. The objective function of this model minimized intercellular and intracellular movement costs and machine breakdown costs.
Many articles investigate CM system problems in certain conditions, demand, machine availability, processing time, raw materials, and etc., but they are uncertain in real world and are changed randomly during the time horizon. Therefore, cellular manufacturing in uncertainty condition is an important area for investigation for making more accurate decisions. In this paper, uncertainty in processing time, time between two successive arrivals to cell, and reliability are considered to fill this gap in literature. The structure of this paper is as follows. In Sect. “Problem formulation”, the problem formulation is described. The MPSO algorithm and GA are described in Sect. “Notation”. The computational results and conclusion are reported in Sects. “Mathematical model” and “The proposed algorithms”, respectively.
Problem formulation
Notation
Indexing sets

\( i \) index for parts \( i = 1,. . .,P \)

\( j \) index for machines \( j = 1,. . .,M \)

\( k \) index for cells \( k = 1,. . .,C \)
Parameters

\( \lambda_{i} \) mean arrival rate for part \( i \) (mean number of parts entered per unit time)

\( \mu_{j} \) mean service rate for machine j (mean number of customers served per unit time by machine j)

M _{max} The maximum number of machines per cell

MTBFj Mean time between failures for machine j

MTTRj Mean time to repair for machine j
Decision variables
Mathematical model
As might be expected, the value of production rate is reduced by considering reliability. As said by above contents, the reliability affects only on the production rate.
The objective function (5) maximizes the average effective arrival rate. Maximizing the average effective arrival rate increases processing operations of the part with more arrival rate within one cell. Hence, the number of intercellular movement becomes lower. The main point in the objective function is that the arrival rate for each part is added to the effective arrival rate for each machine when part needs to be operated on the machine, and the part and the machine allowed in each cell. Constraint (6) guarantees that each part must be allocated to one cell only. Constraint (7) guarantees that each machine must be allocated to one cell only. Constraint (8) guarantees that the number of machines to be allocated to each cell should be less than the maximum number of machines allowed in each cell. Constraint (9) avoids instability of the queuing system, that is, the effective arrival rate will be necessarily less than the service rate. Constraint (10) specifies the type of decision variables.
The proposed algorithms
The CF problem is NPhard problem (King and Nakornchai 1982). Therefore, precise solution procedures and commercial optimization softwares are unable to reach global optimum in an acceptable amount of time for medium and largescale problems. To deal with this deficiency, two algorithms based on MPSO and GA metheuristics have been developed in this paper.
Particle or chromosome structure
The proposed generating initial population
The MPSO algorithm
Particle swarm optimization (PSO) algorithm by Kennedy and Eberhart (1995), (Eberhart and Kennedy 1995) has been presented for problems which have continuous solution space. PSO is a naturebased evolutionary algorithm and starts with an initial population of random solutions. Each potential solution is called particle (\( {\vec{\text{x}}} \)). Particles move around in a multidimensional search space, and during movement, each particle adjusts its position based on its own past and the experience of neighbor particles. Particle’s fitness is compared with its \( {\text{pbest}}_{i} \) (value of the best function result so far, for particle \( i \)). If existing value is better than \( {\text{pbest}}_{i} \), then set \( {\text{pbest}}_{i} \) equal to the current value, and \( p_{i} \) equal to the current location \( {\vec{\text{x}}}_{\text{i}} \) in multidimensional space. Value of the best function result so far for all particles is called \( {\text{gbest}} \), and its location is assigned to \( p_{\text{g}} \).
In the original PSO process, the velocity of each particle is iteratively adjusted so that the particle stochastically oscillates around \( {\vec{\text{p}}}_{\text{i}} \) and \( {\vec{\text{p}}}_{\text{g}} \) locations. In fact, the velocity of a particle must be understood as an ordered set of transformations that operate on a solution. The MPSO algorithm uses this concept for optimizing.
 1.
Initial population is generated using the proposed heuristic algorithm (Fig. 3).
 2.
The fitness value of all particles is calculated by the linearization objective function.
 3.
The fitness value of each particle is assigned to \( {\text{pbest}}_{i} \) and its location to \( {\vec{\text{p}}}_{\text{i}} \). Identification of the particle in whole swarm with the best success so far, and assignment of its fitness value to \( {\text{gbest}} \) and its location to \( {\vec{\text{p}}}_{\text{g}} \).
 4.Producing a new population is based on the repetition of the following steps:
 4.1.
A new vector P is generated to record the positions where the \( {\vec{\text{x}}}_{\text{i}} \) and \( {\vec{\text{p}}}_{\text{i}} \) elements are not equal. A vector Q is defined with the same length with the vector P. Binary elements for the vector Q are generated randomly.
 4.2.In each position of the vector Q, if the element is 0, the change is not made, but if the element is 1, the element of the same position of vector P is selected. This element in the vector P shows the position of vector \( {\vec{\text{p}}}_{\text{i}} \) which should be copied in the vector \( {\vec{\text{x}}}_{\text{i}} \). Then, the feasibility of constraints (8) and (9) is evaluated. The procedure continues, if it is true, otherwise, the made changes return and the next element of vector P will be tested, which is specified by the vector Q (see Fig. 4).
 4.3.
For the new location (\( \overrightarrow {\text{x'}}_{\text{i}} \)) a new vector P is generated to record the positions where the \( \overrightarrow {\text{x'}}_{\text{i}} \) and \( {\vec{\text{p}}}_{\text{g}} \) elements are not equal. A new vector Q, is defined with the same length with vector P. Binary elements for vector Q are generated randomly.
 4.4.
In each position of vector Q, if the element is 0, the change is not made, but if the element is 1, the element of the same position of vector P is selected. This element in the vector P shows the position of vector \( {\vec{\text{p}}}_{\text{g}} \) which should be copied in vector \( \overrightarrow {\text{x'}}_{\text{i}} \). Then, the feasibility of constraints (8) and (9) is evaluated. The procedure continues, if it is true, otherwise, the made changes return and the next element of vector P will be tested, which is specified by vector Q.
 4.1.
 5.
Comparing particle’s fitness value with its \( {\text{pbest}}_{i} \). If current value is better than \( {\text{pbest}}_{i} \), then set \( {\text{pbest}}_{\text{i}} \) equal to the current value, and set \( {\vec{\text{p}}}_{\text{i}} \) equal to the current location \( {\vec{\text{x}}}_{\text{i}} \). Comparing previous \( {\text{gbest}} \) with current \( {\text{gbest}} \). If current value is better than previous \( {\text{gbest}} \), then set \( {\text{gbest}} \) equal to the current value, and assign its location to \( {\vec{\text{p}}}_{\text{g}} \).
 6.
Check stopping criteria (number of iterations).
 7.
If the stopping condition is not met, go to step four.
The proposed genetic algorithm
Genetic algorithms (GAs) are search algorithms based on mechanics of the natural selection and the natural genetics. GA exploits the idea of the survival of the fittest and the interbreeding population to create a novel and innovative search strategy. A population of the strings representing solution to the specified problem is maintained by GA, which then iteratively creates the new population from the old by ranking the strings and interbreeding the fittest to create the new strings, which are closer to the optimum solution to a specified problem (Venkata 2011).
 1.
Initial population is generated using the proposed heuristic algorithm (see Fig. 3).
 2.
The fitness value of a chromosome is calculated by the linearization of objective function.
 3.Producing a new population is based on the repetition of the following steps:
 3.1.Crossover operator:
 3.1.1.
Selection of two parent chromosomes in one population is based on the tournament selection method. Tournament selection involves running several “tournaments” among a few individuals chosen (two or three) at random from the population. The winner of each tournament (the one with the best fitness) is selected for crossover.
 3.1.2.
Two parents are selected from the selection population. Then a number between 1 and M + P (M is the number of machines and P is the number of parts) is selected. A single crossover point on both parents’ chromosome is selected. All data beyond that point in either chromosome are swapped between the two parent chromosomes. The resulting combinations are the children. After crossover, the feasibility of constraints (8) and (9) are evaluated. The procedure continues, if it is true, otherwise, the made change returns.
 3.1.1.
 3.2.
The fraction of the initial population is selected with a probability and then mutations are performed on them. Used mutation alters one array value in a chromosome from its initial state. A number between 1 and M + P is selected. Then, mutation operator of the source (Mahdavi et al. 2009) is used for the mutation. After mutation, the feasibility of constraints (8) and (9) are evaluated. The procedure continues, if it is true, otherwise, the made change returns.
 3.1.
 4.
The size of the next population is the same as the previous one, that is derived from selecting the best solutions by comparing the previous generations and the solutions generated by mutation and crossover operators.
 5.
Check stopping criteria (number of iterations).
 6.
If the stopping condition is not met, go to step two.
Computational results
The obtained values for MPSO parameters
Size parameter  8 × 11  9 × 18  16 × 30 

Population  450  1050  4000 
Iteration  10  60  50 
The obtained values for GA parameters
Size parameter  8 × 11  9 × 18  16 × 30 

Population  450  1100  4000 
Iteration  20  60  70 
Probability of crossover  0.7  0.7  0.6 
Probability of mutation  0.4  0.3  0.1 
Number of members competing in the tournament  3  2  3 
Comparison of B&B, MPSO, and GA results for state of ignoring machine reliability
Problem No.  No. of parts  No. of machines  No. of cells  Mmax  B&B  MPSO  

F _{best}  F _{bound}  TB&B(s)  Z _{ave}  Z _{best}  
1  4  4  2  3  15.25  15.25  0  15.25  15.25 
2  5  5  2  3  21.00  21.00  0  21.00  21.00 
3  7  6  2  3  20.50  20.50  0  20.50  20.50 
4  8  6  2  4  25.17  25.17  1  24.83  24.83 
5  9  7  3  4  22.71  22.71  3  21.74  22.71 
6  11  8  3  4  22.00  22.00  13  21.13  22.00 
7  12  9  3  4  27.44  27.44  70  25.47  26.33 
8  18  8  3  5  28.00  28.00  588  26.11  26.63 
9  17  10  3  5  26.80  27.30  5400  26.40  26.60 
10  18  9  3  5  28.11  28.11  1766  25.79  26.44 
11  19  9  3  5  28.78  28.78  997  27.10  27.33 
12  20  9  3  5  28.44  28.44  3560  27.21  27.56 
13  19  10  3  5  27.70  28.20  5400  26.25  26.60 
14  24  11  4  5  27.91  29.64  5400  26.43  27.36 
15  24  14  4  5  25.14  27.57  5400  23.84  24.29 
16  30  16  4  5  25.56  29.77  5400  26.81  27.50 
17  35  20  4  7  21.60  29.60  5400  26.02  27.05 
18  37  20  5  7  25.40  30.10  5400  26.86  27.70 
19  43  22  5  7  22.18  27.68  5400  22.89  23.68 
Average 
GA  MPSO & GA comparison (%)  

TMPSO(s)  G _{ave}(%)  G _{best}(%)  Z _{ave}  Z _{best}  T _{GA}(s)  G _{ave}(%)  G _{best}(%)  Gaave  Gabest  R 
0  0  0  15.25  15.25  0  0  0  0  0  – 
0  0  0  21.00  21.00  0  0  0  0  0  – 
0  0  0  20.50  20.50  0  0  0  0  0  – 
0  −1.32  −1.32  24.83  24.83  0  −1.32  −1.32  0  0  – 
0  −4.28  0  22.34  22.71  0  −1.64  0  −2.76  0  – 
0  −3.98  0  21.16  21.63  0  −3.81  −1.7  −0.18  1.7  – 
4.3  −7.21  −4.05  25.53  26.33  7  −6.96  −4.05  −0.26  0  −38.57 
4.7  −6.74  −4.91  26.20  26.50  7  −6.43  −5.36  −0.34  0.47  −32.86 
18.3  −1.49  −0.75  26.39  26.60  80.6  −1.53  −0.75  0.04  0  −77.3 
4.8  −8.26  −5.93  26.14  26.67  7  −7  −5.14  −1.38  −0.84  −31.43 
6.1  −5.83  −5.02  26.87  27.22  7  −6.64  −5.41  0.86  0.41  −12.86 
19  −4.34  −3.13  26.93  27.56  7  −5.31  −3.13  1.02  0  171.43 
30.2  −5.23  −3.97  26.41  26.50  77.2  −4.66  −4.33  −0.61  0.38  −60.88 
21.9  −5.31  −1.95  26.27  26.82  84.1  −5.86  −3.91  0.58  1.99  −73.96 
23.6  −5.2  −3.41  23.65  24.07  90.4  −5.94  −4.26  0.78  0.88  −73.89 
34.5  4.89  7.58  26.29  27.19  89.1  2.84  6.36  1.96  1.14  −61.28 
40.7  20.46  25.23  25.55  26.25  96.9  18.29  21.53  1.81  2.96  −58 
40.9  5.73  9.06  26.47  27.10  101.75  4.21  6.69  1.43  2.17  −59.8 
42.7  3.18  6.76  22.53  23.27  113.8  1.56  4.92  1.57  1.73  −62.48 
−1.31  0.75  −1.59  0.01  0.24  0.68  −36.3 
Comparison of B&B, MPSO, and GA results for state of considering machine reliability
Problem No.  No. of parts  No. of machines  No. of cells  Mmax  B&B  MPSO  

F _{best}  F _{bound}  TB&B (s)  Z _{ave}  Z _{best}  
1  4  4  2  3  15.25  15.25  0  15.25  15.25 
2  5  5  2  3  19.20  19.20  0  19.20  19.20 
3  7  6  2  3  20.50  20.50  0  20.50  20.50 
4  8  6  2  4  22.67  22.67  1  22.67  22.67 
5  9  7  3  4  20.86  20.86  5  20.23  20.86 
6  11  8  3  4  22.00  22.00  12  20.59  22.00 
7  12  9  3  4  24.89  24.89  24  23.88  24.89 
8  18  8  3  5  26.25  26.25  250  25.14  25.75 
9  17  10  3  5  25.50  25.50  1517  24.77  25.00 
10  18  9  3  5  25.67  25.67  2154  25.08  25.56 
11  19  9  3  5  26.11  26.11  1811  25.53  26.11 
12  20  9  3  5  27.11  27.11  3855  26.27  26.78 
13  19  10  3  5  26.20  26.20  3870  25.74  26.20 
14  24  11  4  5  25.64  26.82  5400  25.04  25.64 
15  24  14  4  5  22.57  26.00  5400  22.02  22.57 
16  30  16  4  5  23.44  27.31  5400  25.91  26.69 
17  35  20  4  7  23.55  26.55  5400  24.39  25.00 
18  37  20  5  7  24.05  27.85  5400  25.46  26.55 
19  43  22  5  7  20.00  25.32  5400  22.15  23.32 
Average 
GA  MPSO & GA comparison (%)  

TMPSO (s)  G _{ave} (%)  G _{best} (%)  Z _{ave}  Z _{best}  TGA (s)  G _{ave} (%)  G _{best} (%)  Gaave  Gabest  R 
0  0  0  15.25  15.25  0  0  0  0  0  – 
0  0  0  19.20  19.20  0  0  0  0  0  – 
0  0  0  20.50  20.50  0  0  0  0  0  – 
0  0  0  22.67  22.67  0  0  0  0  0  – 
0  −3.01  0  20.70  20.86  0  −0.75  0  −2.33  0  – 
0  −6.42  0  21.10  22.00  0  −4.09  0  −2.49  0  – 
0  −4.06  0  24.44  24.89  7  −1.79  0  −2.37  0  −100 
0.4  −4.24  −1.9  25.58  26.00  7  −2.57  −0.95  −1.74  −0.97  −94.29 
6.3  −2.86  −1.96  24.71  25.00  7  −3.1  −1.96  0.24  0  −10 
18.7  −2.29  −0.43  25.03  25.33  7  −2.47  −1.3  0.18  0.87  167.14 
7.2  −2.21  0  25.71  26.11  7  −1.53  0  −0.7  0  2.86 
24.2  −3.11  −1.23  25.84  26.33  7  −4.67  −2.87  1.61  1.66  245.71 
29.5  −1.76  0  25.19  26.10  7  −3.85  −0.38  2.14  0.38  321.43 
22  −2.34  0  25.09  25.45  85.08  −2.13  −0.71  −0.21  0.71  −74.14 
23.2  −2.44  0  22.00  22.50  89.2  −2.53  −0.32  0.1  0.32  −73.99 
37.7  10.56  13.87  25.38  25.69  88.6  8.27  9.6  2.07  3.75  −57.45 
40.9  3.55  6.16  23.86  24.35  94.9  1.3  3.4  2.17  2.6  −56.9 
42.5  5.86  10.4  24.73  25.60  100.4  2.83  6.44  2.87  3.58  −57.67 
42  10.73  16.59  21.55  22.77  117.2  7.73  13.86  2.71  2.34  −64.16 
−0.21  2.18  −0.49  1.31  0.22  0.8  11.43 
Detailed statistics of paired t test for state of ignoring machine reliability
Paired differences  t  df  Sig. (2tailed)  

Mean  Std. deviation  Std. error Mean  95 % Confidence interval of the difference  
Lower  Upper  
Pair1 MPSOGA  0.06947  0.27840  0.06387  −0.06471  0.20366  1.088  18  0.291 
Detailed statistics of paired t test for state of considering machine reliability
Paired Differences  t  df  Sig. (2tailed)  

Mean  Std. deviation  Std. error mean  95 % confidence interval of the difference  
Lower  Upper  
Pair1 MPSOGA  0.06789  0.40029  0.09183  −0.12504  0.26083  0.739  18  0.469 
Comparison of effect of machine failure for two states of ignoring reliability and considering reliability
Problem number  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19 

Ignoring machine reliability  15.25  21.00  20.50  25.17  22.71  22.00  27.44  28.00  26.80  28.11  28.78  28.44  27.70  27.91  25.14  27.50  27.05  27.70  23.68 
Considering machine reliability  15.25  19.20  20.50  22.67  20.86  22.00  24.89  26.25  25.50  25.67  26.11  27.11  26.20  25.64  22.57  26.69  25.00  26.55  23.32 
Reduction percent  0.00  8.57  0.00  9.93  8.18  0.00  9.31  6.25  4.85  8.70  9.27  4.69  5.42  8.14  10.23  2.95  7.58  4.15  1.54 
Conclusion and future work
In this paper, a new stochastic nonlinear model to solve CF problem within the queuing theory framework with random variables such as time between two successive arrival parts, processing time, and machine availability has been presented. To find out the optimal solution in a reasonable time, the proposed nonlinear model was linearized using auxiliary variable. The time between two successive arrival customers had exponential distribution and service time is distributed generally. Numerical examples showed that the reliability consideration has meaningful effects on the final block diagonal form of machinepart matrixes. Because of complexity class of this problem that was categorized as NPhard, two metaheurstic algorithms based on genetic and MPSO algorithms were developed to solve problems. Also, since the efficiency of metaheurstic algorithms depends strongly on the operators and the parameters, design of experiment was done for set parameters. Deterministic method of the Lingo software’s B&B algorithm was used to evaluate the results of both metaheurstic algorithms. The results indicated that proposed metaheurstic algorithms have better performance in quality of final answer and solving time against the method of Lingo software’s B&B. For future research, considering machine capacity and costs in stochastic CF problem are offered.
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