Abstract
Clustering parts and machines into part families and machine cells is a major decision in the design of cellular manufacturing systems which is defined as cell formation. This paper presents a nonlinear mixed integer programming model to design cellular manufacturing systems which assumes that the arrival rate of parts into cells and machine service rate are stochastic parameters and described by exponential distribution. Uncertain situations may create a queue behind each machine; therefore, we will consider the average waiting time of parts behind each machine in order to have an efficient system. The objective function will minimize summation of idleness cost of machines, subcontracting cost for exceptional parts, nonutilizing machine cost, and holding cost of parts in the cells. Finally, the linearized model will be solved by the Cplex solver of GAMS, and sensitivity analysis will be performed to illustrate the effectiveness of the parameters.
Introduction
Cellular manufacturing system (CMS) is an application of the group technology concept, which classifies parts with closest features and processes into the part families and assigns machines into the cells, with the goal of increasing production efficiency while decreasing the unit cost. Some advantages of CMS such as simplification of material flow, reduction of transportation and queuing times, reduction of material handling cost and setup times, and the increase of machine utilization and throughput rates are declared in literatures (Muruganandam et al. 2005; Olorunniwo and Udo 2002; Wemmerlov and Hyer 1989). The four major decisions in the implementation of cellular manufacturing systems are the following:

1.
Cell formation: grouping parts with the similar processes and features into part families and allocating machines to the cells (Mahdavi et al. 2007; Muruganandam et al. 2005; Yasuda et al. 2005)

2.
Group layout: laying out cells and machines within each cell (Mahdavi and Mahadevan 2008; TavakkoliMoghaddam et al. 2007; Wu et al. 2007a)

3.
Group scheduling: operating and managing the cell operation (Mak and Wang 2002; TavakkoliMoghaddam et al. 2010)

4.
Resource allocation: assigning resources, such as tools, materials, and human resources, to the cells Cesani and Steudel (2005; Mahdavi et al. 2010)
Wu et al. (2007b) considered cell formation, group layout, and group scheduling decisions simultaneously in their model, which minimize the makespan. They presented a hierarchical genetic algorithm to solve it. Logendran (1993) developed a mathematical programming model to minimize part intercell and intracell movements and proposed a heuristic algorithm to solve it. Chen (1998) proposed an integer programming model to minimize material handling and machine cost and reconfiguration cost to design a sustainable cellular manufacturing system in a dynamic environment. Most of the developed models in cellular manufacturing systems are costbased, but there are some models in which machine reliability is considered simultaneously with different cost types. A multiobjective mixed integer programming model was presented by Das et al. (2006) that minimizes variable cost of machine and penalty cost of nonutilizing machine as well as intercell material handling cost, and maximizes system reliability with minimizing failure rate. Machine breakdown cost (JabalAmeli and Arkat 2007) and inverse of the reliability of the system (Das et al. 2007) are two more objective components in order to maximize machine reliability, which will develop cell performance.
Cellular manufacturing problems can be under static or dynamic conditions. In static conditions, cell formation is done for a singleperiod planning horizon, where product mix and demand rate are constant. However, in dynamic conditions, the planning horizon is considered as a multiperiod planning horizon where the product mix and demand rate differ from one period to another. In order to reach the best efficiency, there will be different cell formations for each period. Some recent researches are under dynamic conditions (Kioon et al. 2009; Mahdavi et al. 2010; Schaller 2007).
Inputs of classical cellular manufacturing system problems are certain, but in real problems, some input parameters such as costs, demands, processing times, and setup times are uncertain so that this uncertainty can affect on the results. Some approaches such as stochastic programming, queuing theory, and robust optimization can be applied for uncertain models (SaidiMehrabad and Ghezavati 2009). Ariafar et al. (2011) proposed a mathematical model for layout problem in cellular manufacturing systems that the demand of each product was described by uniform distribution. The objective function of their model minimizes total cost of intercell and intracell material handling, concurrently. It was solved by the Lingo software. SaidiMehrabad and Ghezavati (2009) presented a stochastic CMS problem applying queuing theory approach. They assumed parts as customers and machines as servers, in which the arrival rate of parts and service rate of machines were identified by the exponential distribution. Their objective function components are the following: nonutilization cost, machine idleness cost, and subcontracting cost. They computed the utilization factor for each machine which shows the probability that each machine is busy.
Table 1 illustrates the summarization of more reviewed literatures consisting objectives, model definitions, solution methods, and outputs. This helps to compare our work with previous researches. Table 2 describes the abbreviations used in Table 1.
In this paper, we will develop a CMS problem as a queue system considering the average waiting time of parts behind each machine and holding cost of parts in cells. The goal of this model is to minimize summation of four cost types: (1) the idleness costs for machines, (2) total cost of subcontracting, (3) nonutilization cost of machines in cells, and (4) holding cost of parts in cells. The rest of the paper is organized as follows: The proposed model is described in the ‘Mathematical modeling’ section. In the ‘Experimental results’ section, the experimental results and sensitivity analysis are presented. The last section is the ‘Conclusions and future directions.’
Mathematical modeling
Problem description
In this section, a stochastic cellular manufacturing system problem will be formulated as a queue system, which considers parts and machines as customers and servers with the arrival and service rate of λ and μ, respectively. Also, we consider that at each time, only one part can be processed by a machine; thus, when a machine is processing a part, the others should wait, and a queue will be created behind the machine. The population of this queue as a new part arrives (birth) can be increased, or it can be decreased (death) by service completion.
In a steady system, to avoid infinite growth of queue, the service rate must be greater than the arrival rate, so the utilization factor (probability of being busy) of each machine (ρ = λ / μ) will be less than 1. Also, as each machine processes different parts with different arrival rates, according to this property, the minimum of some independent exponential random variables with the arrival rate of λ_{1}, λ_{2}, …, λ_{ m } is also exponential with the arrival rate of \lambda ={\displaystyle \sum}_{i=1}^{m}{\lambda}_{i\phantom{\rule{0.25em}{0ex}}}. Hence, the utilization factor is ρ = ∑ λ_{ i } / μ.
Our queue system is formulated as M / M / 1, where the arrival and processing time of parts are uncertain and described by exponential distribution, and as it was mentioned earlier, each machine can process at most one part at a time. In this problem, the decision maker needs to allocate parts and machines to cells in order to minimize objective function value. In the previous work of SaidiMehrabad and Ghezavati (2009), only the impact of utilization factor in designing CMS is considered, but our study shows the impact of utilization factor and maximum waiting time of parts simultaneously. Due to the uncertainty of arrival and service time of parts, the time that each part spends in the cell is uncertain, and as the time passes, the parts will be broken. Thus, in order to avoid long waiting time, a chance constraint is considered to show that the probability that the average waiting time of a part behind each machine exceeds the critical time is less than the service level (α). By knowing that, this probability affects on the utilization factor.
Notation
Indexes
The following are the indexes:

i Part index, i = 1, …, p

j Machine index, j = 1, …, m

k Cell index, k = 1, …, c

{a}_{\mathit{ij}}=\left\{\begin{array}{c}\hfill 1\phantom{\rule{0.5em}{0ex}}\mathit{if}\phantom{\rule{0.25em}{0ex}}\mathit{machine}\phantom{\rule{0.25em}{0ex}}j\phantom{\rule{0.25em}{0ex}}\mathit{processes}\phantom{\rule{0.25em}{0ex}}\mathit{part}\phantom{\rule{0.25em}{0ex}}i\phantom{\rule{0.25em}{0ex}}\hfill \\ \hfill 0\phantom{\rule{0.5em}{0ex}}\mathit{otherwise}\hfill \end{array}\right.

C_{ i } Subcontracting cost per unit for part i

u_{ j } Idleness cost of machine j

M_{ m } Maximum number of machines permitted in a cell

α Maximum allowed probability that the waiting time behind each machine can be more than the critical time

t_{ qj } Average time parts spend behind machine j

t Critical waiting time

np_{ i } Total number of part i

C_{ u } Maximum number of cells

λ_{ i } Mean arrival rate for part i

μ_{ j } Mean service rate for machine j

U_{ ij } Cost of part i not utilizing machine j

h_{ i } Holding cost per unit for part i
Decision variables
The following are the decision variables:

{X}_{\mathit{ik}}=\left\{\begin{array}{l}1\phantom{\rule{0.25em}{0ex}}\mathrm{if}\phantom{\rule{0.25em}{0ex}}\mathrm{part}\phantom{\rule{0.25em}{0ex}}i\phantom{\rule{0.25em}{0ex}}\mathrm{is}\phantom{\rule{0.25em}{0ex}}\mathrm{assigned}\phantom{\rule{0.25em}{0ex}}\mathrm{to}\phantom{\rule{0.25em}{0ex}}\mathrm{cell}\phantom{\rule{0.25em}{0ex}}k\\ 0\phantom{\rule{0.25em}{0ex}}\mathrm{otherwise}\end{array}\right.

{Y}_{\mathit{jk}}=\left\{\begin{array}{l}1\phantom{\rule{0.25em}{0ex}}\mathrm{if}\phantom{\rule{0.25em}{0ex}}\mathrm{machine}\phantom{\rule{0.5em}{0ex}}j\phantom{\rule{0.25em}{0ex}}\mathrm{is}\phantom{\rule{0.25em}{0ex}}\mathrm{assigned}\phantom{\rule{0.25em}{0ex}}\mathrm{to}\phantom{\rule{0.25em}{0ex}}\mathrm{cell}\phantom{\rule{0.25em}{0ex}}k\\ 0\phantom{\rule{0.25em}{0ex}}\mathrm{otherwise}\end{array}\right.

ρ_{ j } Utilization factor for machine j (the value of ρ indicates the probability in which machine j is busy)
Mathematical model
In this section, details of the mathematical formulation which we are interested will be described. For this purpose, the following formulations are presented:
subject to
Equation 1 indicates the objective function which can compute the total idleness cost for machines in cells, subcontracting cost, resource underutilization cost, and holding cost of parts in cells. Set constraint (2) restricts that each part is allocated to only one cell. Set constraint (3) ensures that each machine is allocated to only one cell. Set constraint (4) computes the utilization factor. Set constraint (5) guarantees that the number of machines in each cell will not exceed the maximum number. Equation (6) is a chance constraint and ensures that the probability that the average waiting time of parts behind each machine exceeds the critical time is less than the service level (α). Set constraint (7) says that the proportion of time that the machine is processing a part must be less than or equal to 1. Set constraint (8) specifies binary and nonnegative variables.
Linearization of the model
As the objective function has nonlinear terms, we can change it to a mixed integer linear programming. For this purpose, we replace a new binary variable Z_{ ijk } instead of the multiple of X_{ ik } and Y_{ jk }. We reformulate the model, and the three auxiliary constraints (10), (11), and (12) are added to the model to guarantee the correctness of the replacement. Constraint (6) is equal to {e}^{{\mu}_{j}\left(1{\rho}_{j}\right)t}\le \alpha and contains a nonlinear equation; therefore, Equation 14 denotes the linear form of this constraint.
Subject to constraints (2), (3), (5), (7), and (8),
Constraints (4) and (6) are changed as follows:
Experimental results
Consider a manufacturing system consisting of ten machines to process parts, wherein the decision maker should allocate machines and parts to three cells. Also, the maximum number of machines permitted to be located is four. In this section, we present some numerical examples which have been generated randomly, to illustrate the effect of changing the main parameters (α, t, u_{ j }), on the number of subcontracting movements and utilization factors. The proposed mixed integer model was performed by GAMS and Cplex solver and was run on a processor Intel Core 2 Duo CPU running at 2 GHz with 2GB RAM.
In Table 3, the results associated to solve two sets of examples for ten times for each is shown, where only idleness cost is not fix, and the impact of its changes on the average utilization factor and the number of subcontracting movements are illustrated. As the utilization factor of the machines is directly related to the idleness cost, it can be found that the more the average of idleness cost leads to the better design of cellular manufacturing system. It means more costs lead to a higher utilization factor. Also, more utilization factor may lead to less number of subcontracting movements. Results of both sets show the same changes.
Figure 1 indicates the relation between the average utilization factor and number of subcontracting movements of set problem 1. It can be found that by increasing idleness cost, the average utilization factor will be increased, too. As the term ∑(1  ρ_{ j })u_{ j } indicates, the direct relation between idleness cost and utilization factor is established. Therefore, in order to minimize this term, by increasing idleness cost, idleness rate of machine must be decreased. This means that the probability that each machine is busy increases. Therefore, the total number of subcontracting movements must be decreased in order to decrease idleness of each machine. The less total number of subcontracting movements makes the queue system be more populated. Therefore, the total objective function value will be minimized.
The effect of service level's changes (the maximum allowed probability that the waiting time behind each machine can be more than the critical time) on the subcontracting movements is shown in Figure 2. This figure illustrates that for a fixed critical waiting time, if the service level (α) increases, the upper bound of utilization factor (ρ_{ j }) will increase, where this growth may cause decreasing in the total number of subcontracting movements.
Figure 3 demonstrates the relation between changes of critical waiting time and number of subcontracting movements. If we assume all the parameters to be fixed except the service level (α), increasing the critical waiting time may lead to the reduction of the number of subcontracting movements, which is due to the increase of the upper bound of the utilization factor.
Conclusions and future directions
We have developed a stochastic CMS model that considers the arrival rate of parts into cells and machine service rate as uncertain parameters. The proposed nonlinear mixed integer programming model was linearized using auxiliary variables. Then, the linearized model was solved using the Cplex solver of GAMS. As the CMS problem is NPhard, by increasing the size of the problem, GAMS stops solving it; due to the increase of computational time, the branchandcut algorithm is unable to give good solutions. Therefore, it is necessary to present a heuristic or metaheuristic approach to solve this model for largescale problems. Also, the following directions can be applied for further considerations:

1.
Developing the proposed model under new stochastic parameters such as capacities, lead times, and machine failures.

2.
Analyzing the defined problem under scenariobased planning approach and robust optimization theory.

3.
Incorporating our objectives with productivity and production planning aspects in uncertain situations.
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VG carried out the modeling, drafted the manuscript, involved in revising the manuscript, and gave final approval of the version to be published. FF was involved in performing the model and analyzing the results, participated in the sequence alignment, and helped draft the manuscript. AZ carried out the literature study and participated in the design of the study and the sequence alignment. All authors read and approved the final manuscript.
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Fardis, F., Zandi, A. & Ghezavati, V. Stochastic extension of cellular manufacturing systems: a queuingbased analysis. J Ind Eng Int 9, 20 (2013). https://doi.org/10.1186/2251712X920
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DOI: https://doi.org/10.1186/2251712X920
Keywords
 Cellular manufacturing system
 Stochastic arrival rate and service rate
 Average waiting time
 Queuing theory