Designing robust layout in cellular manufacturing systems with uncertain demands

Article history: Received October 2


Introduction
Group technology (GT) is a manufacturing theory, which attempts to increase production efficiency by grouping products having similar attributes and manufacturing processes.Similar parts are classified into part families and manufactured by clusters of dissimilar machines.Cellular Manufacturing system (CMS) is a practical application of GT, which determines part families and machine cells known as cell formation (CF) problem, to attain some manufacturing characteristics.In CMSs, the flexibility of job shop manufacturing for producing different products is considered to the efficiency of flow line manufacturing to increase production outputs (Mungwattana, 2000).According to Wemmerlöv and Hyer (1989), the main improvements derived from cellular manufacturing (CM) are reductions in throughput time, in material handling, in setup time and improvement in the production quality.
Facilities layout is one of the key areas in a manufacturing system, which has a direct influence on the operational performance, as measured by manufacturing lead time, throughput rate and work-in-process (Benjaafar, 2002).Tompkins et al. (2003) estimated that 20-50% of the manufacturing costs are due to the handling parts and an efficient arrangement of facilities could reduce expenditures up to 30%.The interaction among various cells due to exceptional elements, parts require processing on machines in two or more cells, is a major barrier to maintain the benefits of CMS (Arıkan & Güngör, 2009).Therefore, the layout problem in CMSs should be designed, efficiency.In the context of layout problem in designing CMSs, Aktürk and Turkcan (2000) developed a heuristic algorithm with three main stages to form machine cells and obtain the layout of machines.The first two stages are to form independent machine cells, completely and to obtain intra-cell layout.In the third stage, we decrease additional machine investment expenditures by allowing inter-cell movements of parts.Chiang and Lee (2004) addressed the joint problem of manufacturing CF and its layout assignment, where machine cells are located along the bi-directional linear layout.The primary objective is to minimize the inter-cell flow cost, instead of minimizing the number of inter-cell movements.Simulated annealing (SA) (Haddad et al., 2012;Kazemipoor et al., 2012) approach is combined with dynamic programming algorithm to solve the resulted problem.Chan et al. (2006) proposed a two-stage method to solve CF problem as well as cell layout problem.The first stage was to detect machine cells and part families and the second stage is to arrange the layout sequence of machine cells, formed in the first stage.Both problems, presented in the first and second stages were solved by genetic algorithm (GA).Wu et al. (2007) presented an approach to concurrently make the CF, group layout and group scheduling decisions.A hierarchical GA was developed to solve the proposed problem.Mahdavi et al. (2008) proposed a matrix based heuristic to design an efficient CF and layout.Ahi et al. (2009) implemented the multiple attribute decision making concept and proposed a two-stage technique to determine CF, inter-and intra-cell layouts.Khaksar-Haghani et al. (2011) presented a mathematical model to integrate CF, group layout and group scheduling decisions in designing multi-floor CM.
Shorter product life-cycle, higher product variety, unpredictable demand, and shorter delivery times have caused manufacturing systems to operate under dynamic and uncertain environments (Mungwattana, 2000).In practice, costs, demands, processing times, set-up times and other inputs for manufacturing systems may be highly uncertain so that these could significantly influence the production processes in the manufacturing systems.Manufacturing systems ought to be able to adapt/respond to such changes and uncertainties quickly with reasonable investment and operating costs.Addressing data uncertainty in designing CMSs is relatively new class of CMS problems.In this way, Mungwattana (2000) developed a model for designing CMSs by assuming dynamic and stochastic production requirements and employing routing flexibility and simulated annealing (SA) was used to solve the proposed model.Tavakkoli-Moghaddam et al. (2007) presented a mathematical model to solve a facility layout problem in CMSs with stochastic demands by assuming that CF is first completed and known a priori.In this model, depending on the level of uncertainty, the optimal layout in CMSs can change, significantly.Ghezavati and Saidi-Mehrabad (2009) addressed a mathematical model for CMS integrated with group scheduling in an uncertain space.Discrete scenarios were used to describe stochastic processing time of parts on machines by assuming that all parts in a part family could be processed in the same cell and no inter-cellular transfer is required.This model aims to minimize the total expected cost consisting maximum tardiness cost among all parts, cost of subcontracting for exceptional elements and the cost of resource underutilization.A hybrid method based on combination of GA and SA was implemented to solve such a stochastic model.Arıkan and Güngör (2009) presented a multi-objective fuzzy mathematical model with fuzzy part demand, machine capacity and the EEs' elimination costs.The objective functions were minimization of the cost of EE elimination, minimization of the number of inter-cell operations and maximization of the utilized machine capacity.The proposed model gives the decision maker alternative decision plans for various grades of precision.Kia et al. (2011) presented a mathematical model for the layout design of dynamic CMSs with fuzzy uncertain demands and machine time-capacities.The disadvantage of this method was that it gives only one decision plan for the fuzzy parameters in various risk levels.Ghezavati and Saidi-Mehrabad (2011) implemented queuing system theory to formulate data uncertainty in designing CMSs with exponentially distributed inter-arrival and processing times.A hybrid method based on GA and SA was employed to solve the problem.
As was mentioned, the efficiently of CMSs depends on the structure of cells and arrangement of machines within the cells, significantly.In this paper, a new mathematical model is developed to form machine cells and determine inter-and intra-cell layouts under an uncertain environment while designing CMSs, concurrently.The uncertainty stems from part demands.Unlike the scenario based approaches, which use predefined scenarios to represent uncertainty, the proposed model of this paper uses an interval approach to address data uncertainty for part demands, which is more realistic and practical.The problem is formulated as a mixed-integer non-linear programming model to minimize the total inter-and intra-cell material handling cost, and then linearization methods are used to transform the resulted problem into a linear mathematical model.Sim's (2004) approach is applied to handle the uncertainty for the problem formulation and control the robustness of the layout against the level of conservatism.Finally, an illustrative example is solved by CPLEX 10 to demonstrate the efficiency of the proposed approach.

Robust optimization
In recent years, a body of literature has been developed under the name of robust optimization, in which we seek a solution, which remains feasible and near-optimal under the perturbation of parameters for the optimization problem.To address data uncertainity, Mulvey et al. (1995) presented an approach, which integrates goal optimization formulations with scenario-based description of the problem data.Kouvelis and Yu (1997) proposed a framework for robust discrete optimization, which seeks to find a solution, which minimizes the worst case performance under a set of scenarios for the data.Unfortunately, under their approach, the robust counterpart of many polynomially solvable discrete optimization problems becomes NP-hard; moreover, the scenario-based approaches may not be paractial for some real-world cases.Soyster (1973) proposeed a linear optimization model to obtain a solution, which is feasible for all data belonging to a convex set.The Soyster's method produces solutions, which are too conservative in the sense that we give up too much optimality for the nominal problem to ensure robustness.To overcome such problem, Ben-Tal and Nemirovski (2000) presented a non-linear robust model under ellipsoidal uncertainty sets, that is less conservative than the Soyster's model.However, A practial drawback of Ben-Tal and Nemirovski's approach is that it leads to nonlinear, although convex, models and consequently the difficulty of the robust problem increases.Sim (2004) proposed methodologies in robust optimization, which cover a broad range of mathematical optimization problems, including linear optimization, quadratic constrained quadratic optimization, general conic optimization including second order cone programming and semidefinite optimization, mixed integer optimization, network flows and 0-1 discrete optimization.Sim's approach retains the same complexity class as the original model and allows the decision maker to vary the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations, while keeping the problem tractable.
In this section, we present a robust approach developed by Sim (2004) for discrete optimization problems with uncertain parameters in the objective function.This approach allows to control the degree of conservatism of the solution.Let be n-vector and be the feasible space.Without loss of generality, it is assumed that data uncertainty influences only the elements on the matrix .Each entity , ∈ independently takes values in , + , where represents the deviation from the nominal coefficient .Consider the following nominal mixed-integer programming (MIP) problem on a set of variables: In practice, it is unlikely that all of the , ∈ will change.The goal is to be protected against all cases in which up to Γ of these coefficients are allowed to change.Therefore, we are interested in finding an optimal solution, which optimizes against all scenarios under which a number Γ of the , ∈ can vary in such a way that maximally influences the objective function.Let = > 0 ; therefore the robust counterpart of problem (1) can be written as follows: In model ( 2), parameter Γ takes value in the interval 0, | | and adjusts the level of robustness in the objective function.If Γ = 0, the influence of the deviations are completely ignored and the problem becomes the same as the nominal problem, while if Γ = | |, all possible deviations are considered, which is indeed more conservative.Sim (2004) showed that problem (2) is equivalent to the following MIP model: As it can be seen, no auxiliary integral variable has been used in problem (3); thus, if the nominal problem can be solved in a polynomial time, its robust problem can also be solved, polynomially.This robust framework is used as the basis of this research to formulate our problem and it has been used in other areas, significantly (Roghanian & Foroughi, 2010;Moghadam, & Seyedhosseini, 2010)

Problem statement and formulation
The single line (flow line) facility layout problem is considered when multi-products with various production volume and different process routings need to be manufactured (El-Baz, 2004).It is a common layout type implemented in the layout design of CMSs; see Chiang and Lee (2004), Chan et al. (2006) and Jolai et al. (2012).In this paper, we assume that machines, within each cell, are arranged along a linear flow line on the equally spaced locations by considering cell size limit.In addition, machine cells are sequentially placed in multiple rows.Furthermore, it is assumed that cell locations are approximately equally spaced.The proposed layout approach is demonstrated in Fig. 1.In this approach, each row represents a machine cell and each column represents machine locations.Parts are transferred among machines based on their operation sequences.
Sets parts index = 1, … , ( is the total number of parts) operations index = 1, … , ( is the number of operations of part ) , ′ machines index , ′ = 1, … , ( is the total number of machines) , ′ cells index = 1, … , ( is the maximum number of cells allowed) location index = 1, … , ( is the maximum number of machines permissible in a cell) In order to approximate and ℎ for the problem we can use the following equations:

Parameters
where and ℎ are the average width and height of machines respectively, is the spacing between machines and is the spacing between cells.

Mathematical model
According to the descriptions given above, the problem can be formulated as the following mixedinteger non-linear programming model: = 1, ∀ , . ≤ 1, ∀ , , ∈ 0,1 , ∀ , , where ′ is calculated as follows: Objective function (6) minimizes the total material handling cost, where the first term is the total intracell handling cost and the remaining terms are correspond to the total inter-cell handling cost.Constraints ( 7) and ( 8) enforce that each machine is assigned to one cell and one location respectively.In addition, constraint (9) ensures that each location is occupied by at most one machine.Finally, the binary restriction on the decision variables is given in constraints ( 10) and (11).

Model linearization
Since the proposed model is non-linear (due to the presence of the absolute and product terms in objective function (6) and constraint ( 9)), and non-linear models are usually much harder to solve for optimality than linear models, thus in this section we reformulate the problem as a MIP model.In order to linearize objective function ( 6) we introduce new sets of variables ′ to ′ to replace as ∑ − ′ and .′ , respectively, also constraints ( 17)-( 21) are added to the model.So, objective function (6) can be written as follows: Now, we introduce new sets of variables ′ and ′ ′ to replace the . ′ ′ + ′ .′ product terms respectively; also, constraints ( 15), ( 16) and ( 21)-( 23) are added to the model, in order to linearize Eq. ( 13).Finally, constraint ( 9) is linearized by introducing a new set of variables to replace the .product term, and adding constraints ( 24) and ( 25) to the model.The linearized model is shown below.

Robust model
In this section, we present a mathematical model, which is robust to demand changes.It is assumed that the uncertainty only affects the demand of parts in which the uncertain demand of each part takes values in the interval , + .According to the robust framework presented in Section 2, the robust counterpart of the problem presented in Subsection 3.2, is as follows: subject to: (7), ( 9), ( 10), ( 11) and ( 15)-( 30) + ≥ , ∀ , , ≥ 0 In the above model, parameter takes values in the interval 0, | | = and adjusts the level of robustness in the objective function.In general, a higher values of increases the level of robustness at the expense of higher nominal cost.

Illustrative example
In this section, to validate the proposed model and illustrate its various features, an example is solved by CPLEX 10.0 on a PC with 2.4 GHz CPU and 1 GB RAM.The example consists of 20 part types and 10 machine types.The main inputs, including operation sequences, uncertain demand of parts and unit inter-and intra-cell handling costs have been included in Table 1.The machines are to be grouped into 3 cells with the capacity of 4 machines in each cell (i.e. = 3 and = 4).The average width and height of machines ( and ℎ) are assumed to be 2 and 1.5 units respectively.Also the spacing between machines in the same cell ( ) and the spacing between cells ( ) are assumed to be 1.5 and 3 units respectively.Eq. ( 4) and Eq. ( 5) are used to calculate and ℎ for the problem.This example was investigated by considering various values of conservatism (i.e.Γ ∈ 0,20 ), to verify the behavior of the model.The problem was solved optimally within 14,348 seconds for all values of Γ (on average 683 seconds for each level of Γ ).The relationship between the total material handling cost and level of conservatism has been plotted in Fig. 2. In addition, the machine cells as well as arrangement of machines within the cells for various values of Γ has been illustrated in Fig. 3.The results show that 7 different solutions can be produced for this problem when we change the level of conservatism.In addition, from Fig. 3 we figure out that depending on the level of conservation, the optimal solution is changed, significantly.

Conclusion
In this study, a new approach was presented to design CMSs with uncertain demands.The primary objective of this paper was to form machine cells and obtain inter-and intra-cell layouts, in such a way that the decision maker was able to control the robustness of the solution against the level of conservatism.The problem was formulated as a mixed-integer non-linear programming model to minimize total material handling costs, and then the linearization procedures were applied to linearize it.To illustrate the proposed approach, a numerical example was solved for various grades of conservatism.The results revealed that depending on the attitude of the decision maker towards conservatism the optimal solution can change, significantly.For the proposed approach of this paper, it was assumed that the uncertainty only arises from part demands; however, in real-world problems, other parameters such as processing times, capacity of machines, etc. may be uncertain and can be considered along alternative processing routings in the problem formulation.In addition, employing metaheuristics such as GA, SA, etc. for efficiently solving large-scale problems in a reasonable computational time is a suitable research area for interested researchers.

Fig. 1 .
Fig. 1.Proposed approach for arrangement of machines within the cells

Fig. 2 .Fig. 3 .
Fig. 2. Behavior of the objective function in terms of various values of Γ

Table 1
Input parameter for numerical example