Mikko Linnala
email: mikko.linnala@uef.fi.

Elina Madetoja
email: elina.madetoja@iki.fi.

Henri Ruotsalainen
email: henrimatias.ruotsalainen@gmail.com.

Jari Hämäläinen
email: jari.hamalainen@lut.fi.

Bi-level optimization for a dynamic multiobjective problem

Keywords: Bi-level optimization, Dynamic multiobjective optimization, Dynamic process simulation Bi-level optimization, dynamic multiobjective optimization, dynamic process simulation

In this paper, a bi-level optimization problem covering upper (design) and lower (operation) levels is defined and a solution procedure for bi-level optimization problems is presented. This is devised as a dynamic multiobjective optimization problem, i.e. the values of the control and state variables change over a predefined time horizon and several competing criteria are optimized simultaneously. Moreover, the interaction between the upper and lower levels is analysed. The benefits of bi-level dynamic multiobjective optimization are illustrated in detail by examining an industrial case in which the design of a paper mill (upper level) and the mill operation (lower level) are optimized at the same time. However, the problem definition and the solution procedure are not limited to any specific application but can be exploited in many different industrial areas.

Introduction

A bi-level optimization makes possible the simultaneous optimization of design and operation. A bi-level optimization problem has two levels where the upper level (e.g. process design) problem is the leader and the lower level (e.g. process operation) problem is the follower. Thus, the upper level solution affects the lower level solution and vice versa. For example, the optimization variables of the upper level are used as constants in the lower lever. If the objective function on one or both of the levels is vector-valued, the problem is called a bi-level multiobjective optimization problem [START_REF] Eichfelder | Multiobjective bilevel optimization[END_REF]). Bilevel multiobjective optimization makes it possible to conduct an efficient optimization of complicated practical applications, such as a papermaking process.

Bi-level optimization can be applied in many fields and for different kinds of problems. For example, in chemical engineering, bi-level optimization is exploited to optimize the design and control of processes (Mohideen et al. 1996b, Bansal et al. 2000a,b), in electricity markets, it is applied to strategic pricing [START_REF] Fampa | Bilevel optimization applied to strategic pricing in competitive electricity markets[END_REF], and in supply chain problems, production-distribution interactions are studied [START_REF] Calvete | Bilevel model for production-distribution planning solved using ant colony optimization[END_REF]). Bi-level optimization methodology was developed by Mohideen et al. (1996b) who proposed an algorithm for solving integrated design and operation problems including dynamic mathematical models, uncertainty parameters, timevarying disturbances and robust stability criteria. Subsequently, Bansal et al. (2000b) described both simultaneous and sequential solution procedures for bi-level problems. In the sequential procedure, the process design and operation are optimized in turn until optimal design with optimal operation is achieved. Multiobjective bi-level problems have been studied by [START_REF] Fliege | Multicriteria approach to bilevel optimization[END_REF] as well as [START_REF] Deb | Solving bilevel multi-objective optimization problems using evolutionary algorithms[END_REF] and Li et al. (2010) solved multiobjective bilevel problems by using evolutionary algorithms. Different cases and useful background to bi-level optimization were reviewed by [START_REF] Colson | Bilevel optimization: A survey[END_REF].

Practical optimization problems in industrial processes can usually be considered as being both dynamic and multiobjective. In addition, the solution methods may be based on the process models [START_REF] Kameswaran | Simultaneous dynamic optimization strategies: Recent advances and challenges[END_REF] in which case they are referred to as model-based problems. As well as bi-level optimization, dynamic multiobjective optimization has been widely applied in chemical engineering. For example, [START_REF] Cervantes | Large-scale dynamic optimization for grade transitions in a low density polyethylene plant[END_REF] have considered optimal control strategies of an industrial low-density polyethylene plant, and [START_REF] Barakat | Multi-objective optimisation of batch separation processes[END_REF] have presented dynamic multiobjective optimization as applied in batch separation processes.

In the paper industry, although dynamic process modelling has been used for a long time [START_REF] Barber | Dynamic modeling of a paper machine, part I: programming and software development[END_REF]Scott 2007, Niemenmaa et al. 1998) the process optimization has mainly been simply single objective [START_REF] Höfferl | A dynamic programming extension to the steady state refinery-LP[END_REF]Steinschorn 2009, Ropponen andRitala 2010) or multiobjective with steady-state models [START_REF] Hämäläinen | Simulation-based optimization and decision support for conflicting objectives in papermaking[END_REF]Tarvainen 2008, Hämäläinen et al. 2010). However, there are examples of dynamic multiobjective approaches being used in single level optimization (Linnala et al. 2009(Linnala et al. , 2010)). Bi-level multiobjective optimization has already been exploited so that the design and operation of one sub-process (broke system) are optimized simultaneously (Ropponen et al. 2010). In this paper bilevel multiobjective optimization is applied for the broke system similarly to previous studies, but as far as it is known the process model used is more exact, realistic and larger than those used in the previous studies. In this way the results of the optimization are more reliable for implementation in real processes, which is one important goal of this kind of research.

In this study, dynamic and multiobjective properties were taken into account in both levels of a bi-level optimization problem unlike in previous studies of bi-level optimization.

Engineering Optimization 3

That is because several of the practical optimization problems in the process industry are dynamic and multiobjective, and those problems have been not discussed extensively from a bi-level optimization point of view. Here, definition of the bi-level problem as well as the solution procedure is generalized, i.e. they are not limited to any specific application, algorithm or solver. Since the optimization problem is formulated in the bi-level multiobjective mode, it is possible to handle both long-term and short-term objectives simultaneously and efficiently. In the long-term, one can decrease capital expenditure in the process as well as raw material costs, and in the short-term one can assure optimal operation during different production tasks. In addition to handling different problems, a similar approach can be exploited for the design of both new processes and for the revision of existing processes.

The paper is organized as follows: the next section handles theoretical background and presents a solution procedure for a bi-level optimization problem. Then, there are numerical experiments and the results are presented in Section 3. Finally, conclusions sum up the results in Section 4.

Problem setting and solution procedure

Here a bi-level optimization problem in which the upper (design) and lower (operation) levels are taken into account will be defined. There are several papers on this topic [START_REF] Deb | Multi-Objective Optimization using Evolutionary Algorithms[END_REF]Sinha 2008, Mohideen et al. 1996b). However, certain aspects presented in this paper have not been considered in previously, where the emphasis of the research may have been different. [START_REF] Deb | Solving bilevel multi-objective optimization problems using evolutionary algorithms[END_REF] have described an algorithm that uses a multiobjective evolutionary algorithm on both levels of a bi-level optimization problem. Thus, there are some similarities with this solution procedure. For instance, Deb and Sinha emphasize that the upper level control variables do not change in the lower level optimization. The same assumption is also used in this approach. In addition, both levels have separate objective functions and control variables. Moreover, Deb and Sinha highlighted one important issue: Pareto optimal solutions of the lower level optimization problem become feasible solutions to the upper level optimization problem. Pareto optimal solutions are mathematically equally good solutions of a multiobjective optimization problem (Sawaragi et al. 1985). However, this approach can use different optimization methods and algorithms whereas their algorithm was developed as a multiobjective evolutionary algorithm.

Similarly to this approach and Deb and Sinha's algorithm, also Mohideen et al. (1996b) attempted to find an optimal design on the upper level with feasible and efficient operation on the lower level. They proposed a framework for solving integrated design and control problems that included dynamic mathematical models similar to this approach. Moreover their framework also takes into account uncertainty parameters and time-varying disturbances as well as robust stability criteria. These two last features are not taken into account to the same extent in this approach, though this optimization can also cope with some time-varying disturbances (Linnala et al. 2010). The main difference between this approach and the algorithm of Mohideen et al. (1996b) is that their solution procedure can only manage a single objective function.

Since there are two important aspects in this approach; multiobjectivity and dynamics, next a dynamic optimization problem and its solution procedure are defined, and then a dynamic multiobjective optimization problem is studied.

Dynamic optimization

A dynamic optimization problem involves a dynamic process model where dynamic variable values change with time [START_REF] Biegler | Retrospective on optimization[END_REF]). Thus, its solution differs from the steady state problem. The dynamic optimization problem can be formulated as follows:

optimize

u u u(t) f (x x x(t f ), u u u(t f ), p p p, t f ) subject to      x x x ∈ S x u u u ∈ S u h dx x x(t) dt , x x x(t), u u u(t), ω ω ω(t), t = 0 for all t ∈ [t o , t f ],
(2.1)

where f is an objective function, h is a system of the differential and algebraic equation constraints, which can also be divided into two separate systems (Biegler andGrossmann 2004, Cervantes and[START_REF] Cervantes | A stable elemental decomposition for dynamic process optimization[END_REF], x x x denotes the differential and algebraic state vectors (these can also be presented separately), u u u is the vector of optimization variables, also known as a control vector, p p p is a steady parameter vector, and t f is a length of time horizon. Sometimes the steady parameter vectors can be ignored, because they are constants from the optimization point of view. Since dynamic optimization has been studied for years, there are several different solution approaches for (2.1). Typically some of the solutions require at least partial discretization of the originally continuous variables (Biegler andGrossmann 2004, Grossmann andBiegler 2004).

In the case when several objectives need to be optimized simultaneously and objectives and/or variables change with time, the optimization problem is called a dynamic multiobjective optimization problem and it can be formulated as follows:

optimize u u u(t) (f 1 (x x x(t f ), u u u(t f ), p p p, t f ), . . . , f n (x x x(t f ), u u u(t f ), p p p, t f )) subject to      x x x ∈ S x u u u ∈ S u h dx x x(t)
dt , x x x(t), u u u(t), p p p(t), t = 0 for all t ∈ [t o , t f ],

(2.2)

where f f f = (f 1 , . . . , f n ) T is a vector-valued objective function. In this case, the solution process is different compared to (2.1) because all the objective functions f i , for all i = 1, . . . , n, need to be optimized at the same time. Thus, there does not typically exist a unique solution, but instead, there can be a set of solutions that are mathematically equally good. These solutions are called Pareto optimal or non-dominated or efficient solutions (Sawaragi et al. 1985).

Bi-level optimization

Based on the dynamic multiobjective optimization problem (2.2), the problem setting can be extended such that there are two separate levels in the optimization problem; the upper level and the lower level. Assumption is that both levels have multiple objectives Engineering Optimization and the upper level includes the lower level optimization problem. In other words, optimization of the upper level requires that the lower level is optimized. The optimization problem considered on the upper level can be given as follows:

optimize

a a a {F 1 (a a a, x x x(t f ), u u u(t f ), ω ω ω) . . . , F k (a a a, x x x(t f ), u u u(t f ), ω ω ω)} subject to a a a ∈ S a (2.4), (2.3)
where F i , for all i = 1, . . . , k, are the upper level objective functions, a a a is a vector of the optimization variables on the upper level (design variables), x x x is a vector of the state variables, u u u is a vector of the optimization variables on the lower level (control variables), ω ω ω is a vector of the operational tasks, t f is the length of the optimization horizon and S a is a feasible set of a a a defined by all the constraints including box constraints as well as linear and nonlinear equality and inequality constraints. In (2.3), the operation tasks (ω ω ω) are related to the lower level optimization problem and they typically present some assignment or modification of the system state. One should note that compared to some other formulations in (2.3), steady parameters are ignored.

The problem (2.3) becomes dynamic because the lower level optimization problem includes a system of transient differential and algebraic equations that is a dynamic system. The optimization problem on the lower level can be given as:

optimize u u u {f 1 (x x x(t f ), u u u(t f ), ω ω ω(a a a, t)), . . . , f l (x x x(t f ), u u u(t f ), ω ω ω(a a a, t))} subject to      x x x ∈ S x (a a a) u u u ∈ S u (a a a) h dx x x(t)
dt , x x x(t), u u u(t), ω ω ω(a a a, t), t = 0 for all t ∈ [t o , t f ],

(2.4)

where f j for all j = 1 . . . , l are the lower level objective functions, x x x, a a a, u u u and ω ω ω are the same as above, S x and S u are the feasible sets of x x x and u u u, respectively, and h is the system of dynamic differential and algebraic equations, similar to (2.1) and (2.2). As (2.3) and (2.4) show, in this problem formulation, the current optimization variable values on the upper level affect the lower level problem setting. They appear not only in the operational tasks but also in defining the feasible sets of the state variables and the optimization variables. On the other hand, the solution of the lower level optimization problem (optimal values of the optimization variables) affects the objective functions on the upper level. Thus, the optimization problems on the upper and lower levels have two-way coupling which is presented in Figure 1. Here assumption is that the objective functions and optimization variables are different between the levels, and the optimal variable values are denoted by *.

Solution procedure

Next a solution procedure for the bi-level optimization problem given in (2.3) will be presented. As described, this approach has a few similar features to simultaneous and sequential optimization procedures presented by Mohideen et al. (1996a) and the algorithm of [START_REF] Deb | Solving bilevel multi-objective optimization problems using evolutionary algorithms[END_REF]. In practice, this approach is able to handle dynamic and multiobjective optimization problems on both levels. The main emphasis is given to definition of an abstract solution procedure that can be applied and modified easily for different problems. In the following section, an example of how it is applied for solving a bi-level optimization problem from paper industry will be presented.

Algorithm 2.1 (Bi-level optimization)

Let the bi-level optimization problem be defined as in (2.3) and (2.4). Then the solution procedure for such a problem can be defined as follows.

1. Initialize the optimization method selected for the upper level optimization. 2. Define objective functions, a vector of optimization variables and constraints for the upper level optimization problem (2.3) (including the lower level optimization problem parameterized by the upper level optimization variables) and perform optimization as follows. Do until some stopping criterion is fulfilled: a) Let vector ã a a contain the current values for the optimization variables on the upper level. Define the corresponding lower level optimization problem (2.4) at ã a a. b) Find the optimal state for lower level with the selected optimization method, i.e. perform the optimization based on Algorithm 2.2. c) Let u u u * be the optimal solution of the lower level optimization problem and x x x * the corresponding vector of state variables. In the multiobjective case u u u * is selected from a set of Pareto optimal solutions using even human decision maker or some predefined information. d) Based on u u u * and x x x * , evaluate objective functions F 1 , . . . , F k on the upper level and provide the objective function values to the optimization method. End Do 3. The optimal solutions for the bi-level optimization problem are F 1 , . . . , F k at a a a * with the corresponding optimal lower level optimization solutions f 1 , . . . , f l at u u u * .

Algorithm 2.2 (Lower level optimization) Let a lower level optimization problem be defined as in (2.4). Then the solution procedure is as follows:

1. Initialize the optimization method. 2. Define the objective functions, a vector of the optimization variables and constraints in (2.4) for the given parameters ã a a. Start optimization procedure as follows. Do until some stopping criterion is fulfilled: In this approach, the bi-level optimization problem is multiobjective in both optimization levels. If the objectives are conflicting, a set of mathematically equally good solutions (Pareto optimal solutions) is obtained on both levels. In the work of [START_REF] Deb | Solving bilevel multi-objective optimization problems using evolutionary algorithms[END_REF] bi-level Pareto optimality was analysed and their algorithm could handle the Pareto optimal solutions produced on the lower level. However, in many situations only one solution has to be chosen as the final one. This choice can be done by a decision maker who is capable of comparing the Pareto optimal solutions based on her/his expert knowledge of the current problem. Sometimes computational time becomes unacceptably long and that precludes participation of the decision maker. Then some kind of predefined information with classical scalarizing functions can be used to choose the best solution during the bi-level optimization process (Miettinen 1999).

Numerical experiment

Here a numerical experiment where a bi-level optimization was applied to the process of papermaking is presented. To be precise, a broke system in which the rejected paper is being collected and re-circulated back into the process as a raw material is considered. Briefly, the aim was to optimize the capital costs and the operating costs simultaneously. The capital costs were described by the broke tower volumes that were minimized, and the operating costs were minimized by maximizing the broke dosage (in order to minimize the need for virgin raw material) and the net production. In order to highlight the special features of this problem, the application will be introduced.

Dynamic multiobjective optimization related to papermaking

In this experiment a dynamic process model of a supercalendered (SC) papermaking line was used. The model consists of a TMP (thermomechanical pulp) mill where the mechanical pulp is produced, an approach system where the raw materials are mixed and diluted, a broke system where the rejected paper is being collected, a paper machine where the paper web is formed and dried, and a reel where the ready paper is packaged for post processing. The dynamic process model was conducted with Apros software (VTT 2010) in co-operation with VTT Technical Research Centre of Finland.

The problem became bi-level one because the design and operation of the broke system had to be considered. The broke system had its own subsystems for a wet broke (paper rejected before drying having approximately 50 % moisture content) and a dry broke (ready paper being rejected). Hence, the broke system consisted of the elements presented in Figure 2.

When the broke is recycled back into the process, its properties differ from those of the virgin raw materials and therefore this affects the paper properties. For example, when the dosage of broke increases, the strength properties of the paper web decrease and furthermore, the risk of production failure, referred to as a break, increases significantly. Should a break occur, all the paper needs to be rejected and fed back to the broke system (that creates a need for larger broke tower volume). This phenomenon was modelled such that when the broke dosage increased, a larger part of the paper web was rejected, but no actual break occurred. This illustrated the growing risk of break caused by the increased broke dosage over the long-term. On the other hand, broke is a much cheaper raw material than virgin raw materials such as TMP or chemical pulp. Therefore the dosage of broke should be kept as high as possible. Hence, on the lower level (operation of the broke system) there were conflicting objectives: maximize both dosage of broke and net production. On the upper level (process design) investment costs (broke tower volumes) which depend on the operations of the lower level were tried to minimize. Overall, there were multiple objectives at both levels which had very strong interactions with each other.

Optimization problem setting

At the operational level, there were two continuous control variables; dosage of broke and the wet broke proportion of total broke dosaged. Instead, at the design level different tower volumes as control variables were considered and they were defined as integer variables in order to limit the search space. The objective functions optimized on the operational level consisted of production loss (f 1 ), variation of fill percentage of the dry broke tower (f 2 ), variation of fill percentage of the wet broke tower (f 3 ), and broke dosage (f 4 ). From these functions, f 1 , f 2 and f 3 were minimized, and f 4 was maximized. The objective functions on the operational level were defined as follows:

f j =

Engineering Optimization 9

On the design level, the objective functions consisted of the fill percentages of wet broke (F 1 ) and dry broke (F 2 ) towers and both functions were minimized. The objective functions were defined as follows:

F i = max t∈[t0,tf ] |x(t) -x target |, for all i = 1, 2. (3.2)
In this way, the tower volumes could be minimized, but runnability and stability were able to be maintained with buffer volume needed. This can be seen in the objective functions f 2 , f 3 , F 1 and F 2 which were based on the same state variables. At the operational level (f 2 and f 3 ), the process stability was the most important parameter whereas at the design level (F 1 and F 2 ) maximum values were the most important. Numerical values for the design, control and state variables are presented in Table 1. The upper and lower limits of the design and control variables were defined based on the expert knowledge. The initial values of the variables represented a stable process situation (reference point). The optimization problem on the operational level was multiobjective, and thus, a set of Pareto optimal solutions was obtained. Here only one solution was wanted to be brought to the design level. To achieve that goal, a scalarizing function was used to select a single solution. The method of global criteria was applied as a scalarizing function [START_REF] Deb | Multi-Objective Optimization using Evolutionary Algorithms[END_REF]:

S(f f f , z z z) = l m=1 |f m (u u u) -z m | p 1/p , (3.3)
where f f f = (f 1 , . . . , f l ) T is a vector of the objective functions, z z z = (z 1 , . . . , z l ) is the reference point defined close to the ideal point based on the expert knowledge and here p was 2. After the scalarizing, a solution which corresponded to the smallest value of the scalarizing function was brought to the design level. Since the optimization problem on the operational level was dynamic, a receding horizon prediction principle (model predictive control) was applied (Rawlings 2000). In this example, the same dynamic process model was used for both prediction and simulation. Therefore two different time horizons were defined. The longer time horizon, T pred =3000 s, was used for prediction of the control sequence, and the shorter time horizon, T sim =600 s, was used to simulate one time stage forward with the first controls predicted. This loop was repeated until the end of total time horizon, 4800 s, was achieved (T end = T 8 ). The principle of receding horizon prediction is presented in Figure 3, and an example of this has been presented previously (Linnala et al. 2010). In this example, the optimization problem on the both levels was solved by using the differential evolution algorithm (DE) implemented in MATLAB. DE belongs to the group of evolution strategies i.e. it simulates natural evolution. The simulation is based on generations and populations as in other evolution algorithms (Price et al. 2005[START_REF] Lampinen | Multi-Constrained Nonlinear Optimization by the Differential[END_REF][START_REF] Coello Coello | An updated survey of GA-based multiobjective optimization techniques[END_REF]. DE algorithm can be formulated to solve multiobjective problems with linear and nonlinear equality and inequality constraints. The solution of the problem begins with the definition of initial vector population (target vectors) from the permitted search space. After the initialization, differential mutation produces a population of mutation vectors, such that, the scaled difference of two randomly sampled vectors is added to the third vector. The scale factor is used to control the rate at which the population evolves. After a mutation, DE performs a uniform crossover or discrete recombination. Subsequently, trial vectors are built by copying parameter values from two different vectors, i.e. each target vector is crossed with a corresponding mutant vector. The crossover probability is used to control the proportion of parameters copied from the mutant vector. The defined crossover probability is compared to the output of random number generator and if the probability is greater than the random number, then this parameter is copied from the mutant vector. Otherwise, the parameter value is copied from the target vector. Still, at least one parameter value is copied from the mutant vector in order to avoid duplication of the target vector. Finally, DE selects the vectors for the new population. If the trial vector is equally good or better from the objective function point of view, it will replace the target vector in the new generation. In the multiobjective case, the nondominated vector is selected to the new generation. In this way, the search moves toward the optimal solution in a step by step manner (Price et al. 2005).

Even if the same solver was used on both levels, some parameter values of DE differed. The mutation rate was 0.3 and the crossover rate was 0.6 on both levels. The number of generations and the population size were five at the design level. At the operational level, the number of generations was five also, but the population size was eight in order Engineering Optimization 11 to increase variation in the set of optimal solutions.

Results

After the optimization problem was solved, several different design candidates with the optimal operation were obtained. Since the design optimization problem was multiobjective and the operational level affected the design level, design candidates differed from each other. As an example, the objective function values of two chosen design candidates are presented in Table 2. Here the conflicts between the objectives can be appreciated: At the design level, if F 2 decreased, then F 1 increased and vice versa. At the operational level, conflicts were more complicated because several variables affected several objectives. Nonetheless, it could be considered as a conflict between the dry broke and wet broke line operations as can be seen in Table 2. If f 3 (wet broke tower variations) decreased, f 2 (dry broke tower variations) increased (Candidate 1) and vice versa, if f 2 (dry broke tower variations) decreased, f 3 (wet broke tower variations) increased (Candidate 2). 24.98 25.00

Each optimal process design had its own optimal control sequence defined by the operational optimization over the time horizon T 1 -T 8 . Table 3 presents the previous two design candidates including the design variable values (tower volumes) and control variable values (broke dosage and wet broke proportion of total broke).

Design variables [m

3 ] Control variables [%] T 1 T 2 T 3 T 4 T 5 T 6 T 7
T 8 Candidate 1, a 1 4000 u 1 15.0 7.6 2.3 18.7 3.9 38.6 8.5 27.5 a 2 1000 u 2 60.0 57.4 23.7 80.8 95.7 95.4 39.0 65.2 Candidate 2, a 1 1000 u 1 0.5 61.0 7.2 7.7 3.9 40.7 25.7 12.4 a 2 2750 u 2 37.5 58.5 38.5 38.4 50.4 42.2 46.3 2.8 a 1 : Wet broke tower volume, a 2 : Dry broke tower volume. u 1 : Broke dosage, u 2 : Wet broke proportion of the total broke.

The optimization results can be visualized with the time series of different state variables used in the objective functions as presented in Figures 4 and5. Here, five optimal solutions are shown in order to illustrate the variation between the different designs and operations. The dotted lines mark the target levels, i.e. the objective function values are better the closer the state variable values are to the target level. For the sake of clarification, design candidates 1 and 2 presented above are marked with arrows in Figure 4. Figure 6 presents variation in basis weight and filler content which are very important paper quality parameters. These parameters were not included in the objective functions but produce interesting additional information for the decision maker because the quality parameters should stay within the target interval (marked with dotted lines).

As the results show, there was variation between the solutions both in design and in operation. At the operational level, variations in the broke tower liquid levels (Figure 4) differed rather markedly between the design candidates. For example, designs 2 and 4 show that too strong operations lead to loss of the process stability. The same phenomenon can be seen in Figure 5 in which too high a broke dosage leads to production losses (design 2). In addition, some solutions, such as designs 2 and 4, could be discounted in practice based on the expert knowledge because the paper quality would not stay at the target interval, as can be seen in Figure 6. Since the paper quality was not included in the objective functions, all the solutions presented are equally good from the optimization point of view. Therefore, a decision maker is the best individual to compare the various solution candidates and choose the best one to be realized.

Above, only one control sequence for each design is presented. Nonetheless, depending on the method used at the operational level, different control sequences are also possible. Here, the scalarizing function selected the best controls for eight time stages, one after another. When the first controls were implemented, the next control possibilities were fixed, because the next stage depended on the previous stage. Note that if the scalarizing function or the value of reference point was changed, the optimal control sequence could be different. This is the challenge of dynamics in a bi-level case which has not been considered previously in the literature. Overall, this numerical experiment shows that dynamic and multiobjective bi-level optimization can be applied in practical optimization problems, but it requires expert knowledge to understand the relationships between different optimization levels, objectives and variables.

Conclusions

In this paper a real life optimization problem being dynamic, multiobjective and bi-level by nature was discussed. All these special features set requirements for the solution process as well as the problem formulation. Thus, the generalized formulation for a dynamic multiobjective bi-level optimization problem was defined, and a solution procedure for such problems was presented. Subsequently, the approach was illustrated with a real life example in which the process design and operation were optimized simultaneously in place of theoretical test case examination. The results show that this approach was successful and it could be implemented in practical optimization problems. Moreover, this complicated problem cannot be solved efficiently as a single level optimization problem. Here, the bi-level optimization problem was solved by using differential evolution algorithm, but other techniques can be implemented as well. It would be interesting to apply this approach with different optimization problems and algorithms. Furthermore, this generalized approach could be implemented in some industrial, even more complicated and challenging, optimization problems in order to demonstrate the real advantages of this approach.

F

Figure 1 .

 1 Figure 1. Interactions between the optimization levels and a dynamic process model.

  the dynamic process model with the current optimization variables ũ u u. b) Evaluate objectives f 1 , . . . , f l at ũ u u based on the state variables x x x. End Do 3. Save the optimal values of the optimization variables u u u * and the corresponding state variable values x x x * as well as f 1 , . . . , f l which are needed in Algorithm 2.1.

Figure 2 .

 2 Figure 2. Broke system of the papermaking line modeled.

  )x target |, for all j = 1, . . . ,

Figure 3 .

 3 Figure 3. Receding horizon prediction principle.

Figure 4 .Figure 5 .

 45 Figure 4. Fill percentages of wet broke (left) and dry broke (right) towers during the simulation. The dotted line represents the target value.

FillerFigure 6 .

 6 Figure 6. Basis weight (left) and filler content (right) during the simulation. The dotted lines represent the target intervals.

  Fill percentages of wet broke (left) and dry broke (right) towers during the simulation. The dotted line represents the target value. 253x200mm (600 x 600 DPI) Broke dosage for each of the eight time stages (left) and production (right) during the simulation. The dotted line represents the target value. 190x150mm (600 x 600 DPI) Basis weight (left) and filler content (right) during the simulation. The dotted lines represent the target intervals. 190x150mm (600 x 600 DPI)

Table 1 .

 1 Numerical values of the design, control and state variables. Lower and upper limits or step size were not needed for the state variables and target values were not needed for the control variables. Thus, they are marked with '-'.

	Operational Operational Operational	Control variables Control variables Control variables	State variables State variables State variables
	level level level Wet broke prop. [%] Broke dos. Initial value 60 15	51.39	50
	Lower limit	0	0	-	-
	Upper limit	100	75	-	-
	Target value	-	75	53.69	50
	Design Design Design	Design variables Design variables Design variables	State variables State variables State variables
	level level level	V wet broke [m 3 ]	V dry broke [m 3 ]	Fill-% [%]
	Initial value	2000	4000	50
	Lower limit	250	250	-
	Upper limit	6000	6000	-
	Step size	250	250	-
	Target value	-	-	75

*

[%] Prod. [t/h] Fill-% [%] * Broke dosage was both a control and a state variable. Its target value was used only in the objective function formulation.

Table 2 .

 2 The objective function values of two different solution candidates.

	Candidate 1 Candidate 2

Table 3 .

 3 Design and control variable values of two design candidates.

  EvolutionAlgorithm., 305-316, London, England. Li, M., Lin, D., and Wang, S., 2010. Solving a type of biobjective bilevel programming problem using NSGA-II. Computers and Mathematics with Applications, 59 (2), 706-715. Linnala, M., et al., 2009. Dynamic multiobjective optimization in papermaking process simulation. In: E. Madetoja, H. Niskanen and J. Hämäläinen, eds. Perkins, J., and Pistikopoulos, E., 1996a. A framework for process design and control. In: Proceedings of UKACC International Conference on CONTROL96, Exeter, United Kingdom. Mohideen, M., Perkins, J., and Pistikopoulos, E., 1996b. Optimal design of dynamic systems under uncertainty. AlChE Journal, 42 (8), 2251-2272. Niemenmaa, A., et al., 1998. A multi-purpose tool for dynamic simulation of paper mills. Simulation Practice and Theory, 6 (3), 297-304. Price, K., Storn, R., and Lampinen, J., 2005. Differential Evolution -A Practical Approach to Global Optimization. Berlin: Springer. Rawlings, J., 2000. Tutorial Overview of Model Predictive Control. Control Systems Magazine, IEEE, 20 (3), 38-52. Ropponen, A. and Ritala, R., 2010. Optimizing the design and operation of measurements for control: Dynamic optimization approach. Measurement, 43 (1), 9-20. Ropponen, A., Ritala, R., and Pistikopoulos, E., 2010. Broke management optimization in design of paper production systems. In: S. Pierucci and G. Buzzi Ferraris, eds. 20th European Symposium on Computer Aided Process Engineering, ESCAPE20, Naples, Italy. Sawaragi, Y., Nakayama, H., and Tanino, T., 1985. Theory of Multiobjective Optimization. Orlando: Academic Press, Inc. VTT, Chapter title. Apros Overview [online], Technical Research Centre of Finland Available from: http://www.apros.fi [Accessed 27 October 2010], 2010. .

	February 16, 2011 13:2 February 16, 2011 13:2 Engineering Optimization main
	o r F o r Papermaking Research Symposium 2009 (PRS2009), Kuopio, Finland. Engineering Optimization Proceedings of 15 Linnala, M., et al., 2010. Dynamic simulation and optimization of an SC papermaking line -illustrated with case studies. Nordic Pulp and Paper Research Journal, 25 (2), 213-220. Madetoja, E. and Tarvainen, P., 2008. Multiobjective process line optimization under uncertainty applied to papermaking. Structural and Multidisciplinary Optimization, 35, 461-472. Miettinen, K., 1999. Nonlinear Multiobjective Optimization. Boston: Kluwer Academic Publishers. Mohideen, M., Page 18 of 27
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URL: http:/mc.manuscriptcentral.com/geno Email: A.B.Templeman@liverpool.ac.uk Engineering Optimization

URL: http:/mc.manuscriptcentral.com/geno Email: A.B.Templeman@liverpool.ac.uk Engineering Optimization

Acknowledgements

The authors wish to thank M.Sc. Jouni Savolainen from VTT Technical Research Centre of Finland for their co-operation, and Finnish Forestcluster Ltd. and Finnish Funding Agency for Technology and Innovation (TEKES) for financial support (EffTech and


Ph.D Paolo Renna
email: paolo.renna@unibas.it

Ph.D Michele Ambrico
email: michele.ambrico@unibas.it

Loadingpolicies in cellular manufacturing systems with remainder cell

Keywords: CELLULAR MANUFACTURE, SCHEDULING, Cellular manufacture, scheduling, dynamic environments, discrete event simulation, remainder cell cellular manufacture, scheduling, dynamic environments, discrete event simulation, remainder cell

published or not. The documents may come

Loading policies in cellular manufacturing systems with remainder cell

Introduction and motivations

Cellular manufacturing systems (CMSs)are the more appropriate configuration to obtain the better performance when the production variety and volume are at medium level. The CMSs are characterized by the following benefits: simplification and reduction in material handling, decreasing the work in process, reduction in set-up time, increment in flexibility, better production control, and shorter lead time (Askinand Estrada, 1999). These benefits can be obtained when the demand volume and mix are rather constant respect to the design data of the manufacturing cells.In order to remain efficient, the CMSs need to be reconfigured when the manufacturing conditions change [START_REF] Chen | Amathematical programming model for system reconfiguration in a dynamic cellular manufacturing environment[END_REF]. Nowadays, competition is characterized by short life-cycle of the products, introduction of new products, demand and mix fluctuations. These issues lead to reduce drastically the "life cycle" of a CMS configuration with numerous re-configuration activities. Each reconfiguration activity causes costs of re-design of manufacturing cells and set-up times that reduce the availability of the manufacturing system. Tompkins et al. (2003) estimatedthat$250billionisannuallyspentin USA forplanningandre-planningbecauseofthesechanges,andthis huge costcanbereducedby10-30%viaeffectiveplanning.In literature, several alternatives were proposed as: virtual manufacturing cells (McLeanetal., 1982), fractal cells, (Vektadari et al., 1997;Monteruil et al., 1999),holonic cells [START_REF] Montreuil | Holographic layout of manufacturing systems operating in chaotic environments[END_REF] and remainder cells introduction in the design of CMSs [START_REF] Maddisetty | Design of shared cells in a probabilistic demand environment[END_REF]. The focus of this paper is on the operation management of remainder cells in a CMS environment. The aim of the research concerns the development and analysis of loading approaches to integrate the remainder cell within the classical CMS. In particular, there are proposed three loading policies to decide when a generic part will be loaded in remainder cell instead of the manufacturing cell designed for the family of the part. A simulation environment has been developed in order to test the proposed policies when market fluctuations occur in terms of product mix and demand. Therefore, the benefits of the proposed approaches are tested in several dynamic conditions compared with a classical cellular manufacturing configuration.The rest of the paper is structured as follows: Section 2 provides an overview of the literature of remainder cells in CMSs, while in Section 3 the problem context is formulated. The loading approaches proposed are described in Section 4. Section 5 presents the simulation environment and the case study. The numerical results are discussed in Section 6, while the conclusions and future research paths are drawn in Section 7.

Literature review

Several authors addressed the problem of adaptability of CMSs when some factors change such as: uncertain in demand, mix fluctuations and new product introduction. [START_REF] Mak | Product Grouping and Resource Allocation in Multiline Manufacturing Systems[END_REF] investigated the problem of resource allocation in a context of multiple production lines, each line is able to manufacture a range of products. A genetic algorithm was developed for product grouping and allocates the group to a production line. The uncertain considered is the volume of product and the demand. [START_REF] Saad | A new integrated system for loading and scheduling in cellular manufacturing[END_REF] discussed a multiple objective simulation optimization model for loading flexible cells. The approach developed is composed by three main modules: generic process planning module, a multiobjective algorithm based on tabu search method to generate and evaluate candidate part to cell scenarios; the performancemeasures are determined by simulation module. [START_REF] Huang | Fast Algorithm for Evaluating the Similarity of Manufacturing Processes within Dynamic Production Environment[END_REF]developed an algorithm to optimize decision on capacity exchange between rush orders and prescheduled orders. Therefore, it can be considered as a mix production variation on the CMS. [START_REF] Maddisetty | Design of shared cells in a probabilistic demand environment[END_REF] integrated the design and operational control of CMS when the product demand is stochastic. The configurations investigated are: classical, shared and remainder cells. The performance measures evaluated are the work in process and average flow time. [START_REF] Süer | Design of dedicated, shared and remainder cells in a probabilistic demand environment[END_REF] proposed a new layered cellular manufacturing system to form dedicated, shared and remainder cells to deal with the probabilistic demand, and its performance is compared with the classical cellular manufacturing system.The performance of work in process and average flow time are betterthan the classical cellular system when high demand fluctuation was observed. Safei et al. (2007) proposed a fuzzy programming based approach to design a cellular manufacturing system under dynamic and uncertain conditions. The dynamic conditions are characterized by multi-period planning horizon, in which the product mix and demand in each period can be different. [START_REF] Bhandwale | A methodology to incorporate product mix variations in cellular manufacturing[END_REF] proposed a methodology to incorporate new parts, production mix changes and machines into an existing cellular manufacturing system. The objective is to fit the new parts and machines into an existing cellular manufacturing system thereby increasing machine utilizationand reducing investment in new equipment. [START_REF] Viguier | An approach to the design of a hybrid organization of workshops into functional layout and group technology cells[END_REF] proposed an evolutionary programming algorithm to design a hybrid cellular manufacturing system. The hybrid cellular manufacturing system is composed by classical cells and functional cells (i.e. cells composed of machines of the same type). The proposed algorithm is illustrated on a test example with a known optimum. [START_REF] Mak | An ant colony optimization algorithm for scheduling virtual cellular manufacturing systems[END_REF] presented a methodology to solve the manufacturing cell creation and the production scheduling problems for designing virtual cellular manufacturing systems. The methodology is based on ant colony optimization algorithm and two simple heuristics are developed to assign workstations to the operations of the jobs, and to construct the final schedule. Numerical experiments showed that the proposed algorithm generates excellent final solutions in a much shorter computation time when compared with the genetic algorithm. [START_REF] Kesen | A mixed integer programming formulation for scheduling of virtual manufacturing cells (VMCs)[END_REF] developeda multi-objective mixed integer programming formulation for job scheduling in virtual manufacturing cells. The objective function is to minimize the sum of the makespanand total traveling distance/cost. [START_REF] Balakrishnan | Multi-period planning and uncertainty issues in cellular manufacturing: A review and future directions[END_REF] presented a review research that has been done to address cellular manufacturing of multi period planning horizons, with demand and resources uncertainties. The authors identify, among future areas for research the following: "Comparison of dynamic cells (in which physical [START_REF] Hoeck | A workload control procedure for an FMC integrated in a job shop[END_REF] proposed an approach for order release and loading problem in a flexible manufacturing cell environment. The workload control approach involves three steps: lead orders are identified; transfer batches of the lead part types are calculated; workload of the machining centres is determined. A simulation environment was used to test the proposed approach. [START_REF] Drolet | Simulation-based performance comparison between dynamic cells, classical cells and job shops: a case study[END_REF] discussed the results of a simulation-based performance comparison between dynamic cellular manufacturing systems and two other well-known systems as classical cells and job shop systems. The experiments conducted regard 13 independent variables related to sources of turbulence and 17 independent variables related to performance measures. The research highlighted in what conditions one system is better than the others. Renna and [START_REF] Padalino | Adaptive Dynamic Scheduling in Agent Based Manufacturing Environment: A Budget Approach[END_REF] and [START_REF] Padalino | A Multi-Agent Architecture for Budget-based Scheduling in Dynamic Environments[END_REF] proposed an innovative decision making strategy for autonomous agents in a cellular manufacturing environment by a budget assigned to each job in order to purchase manufacturing cell services. The budget manages as a market like approach among agents to coordination the multi agent system. Moreover, a fuzzy tool has been proposed to assign the budget to each typology job. A simulation environment is developed in order to test the proposed approach. The simulations show that the proposed approach is robust and a scheduling approach able to select the jobs that have been the better performance. From the analysis of the literature, the following issues can be drawn:

• few papers investigated the performance comparison between cellular manufacturing systems with remainder cells and other configurations. In thesepapers, the remainder cell is used as a secondary cell that can be used when the manufacturing cells are affected by machine breakdowns. • most of the researches consider one uncertainty between mix and volume products. Moreover, a research on the rapidity change of volume and mix products was not discussed in literature. The discussion in the literature concerns the amplitude of variability of the uncertain (as the standard deviation of stochastic demand). The aim of this paper is to overcome the above limitations, developing loading approaches in hybrid cellular manufacturing systems composed by classical cells and remainder cells. The use of an opportune loading approach allows to response tomarket changes (volume and mix) avoiding the reconfiguration of the manufacturing cells.Then, a simulation environment has been developed in order to test the proposed approaches introducing market fluctuations (mix and inter-arrival demand) and evaluate the effect of rapidity change of the conditions. A wide range of performance measures are investigated using a classical CMS as a benchmark. The number of machines is the same for all configurations; therefore, the comparison is evaluated in a mid-term horizon in which only the different configuration is evaluated.

Manufacturing system context

The configurations of the manufacturing system investigated are: classical cellular manufacturing and hybrid cellular with a remainder cell. The objective of this research is the comparison of the performance in different conditions; therefore, the composition of the manufacturing cells is known. It has been considered three part families that are manufactured by three manufacturing cells. Then, the configuration with remainder cell is composed by the same number of machines (N machines) re-arranged to include in the manufacturing system the remainder cell. Therefore, the comparison is performed with the same machines changing only the configuration. The demand is not known at priori, and it follows an exponential distribution. Each cell is assumed to be independent, i.e. each family partperforms all the operations in only one cell.Set-up times are assumed to be zero, because the machines of the manufacturing cells are configured for the particular part family (dedicated machines).The machines of the remainder cell are general purpose machines; therefore, they have a processing timegreater than the machines in the manufacturing cells. This increment of processing time is due to the machines that are able to manufacture all part families. The processing time of machine i-th in remainder cell rpt ir is greater than the processing time of the machine i-th in the cell j-th(pt ij ). The processing times are correlated by the following expression:

, 1

ir ij pt pt α α = ≥ (1)
The cellular manufacturing system has been designed for specific conditions; then, several disturbances have been introduced. These disturbances can be external (demand volatility) or internal (machine breakdowns). In this paper, it has been introduced the following disturbances:

• demand volatility; the volatility has been considered in term of mix fluctuation that changes the volume of each part family. Therefore, the workload of the manufacturing cells changes dynamically. Moreover, it has been considered volume fluctuations requested (inter-arrival time) by the market. • machine breakdowns; each machine can breakdown randomly with a reduction of the productivity of the manufacturing cells. Figure 1 shows the classical cellular manufacturing system, where each part family is assigned to the related manufacturing cell designed.

[Insert figure 1

here]

Figure 2 shows the hybrid configuration with the remainder cell. In this case, each part of a family has two possible routings: the manufacturing cell assigned and the remainder cell (the arrows of the figure 2 show the possible routing). The "cell loading policy" controller implements the strategy in order to decide the routing between the two possibilities when a part enters in the manufacturing system. The machines assigned to the reminder cellare able to perform any manufacturing operations required by all part families. The proposed strategies are deeply described in the following paragraph.

[Insert figure 2 here]

Loading approaches

The loading approach regards the hybrid manufacturing system with remainder cell; the problem is to decide when a part of a generic family can be manufactured by the remainder cell instead of the manufacturing cell. In literature, the remainder cell is used when an exception occurssuch as machine breakdowns. In this paper, three methodologies are proposed to use the remainder cell to keep a high performance level of the entire manufacturing system.The loading approaches proposed are based on the evaluation of the Work InProcess (WIP) of the manufacturing cells: local approach, global and global exclusive approaches. The loading policy is based on evaluation of the state of each manufacturing cell compared to the state of the remainder cell. The controller of the generic cellj-th (WIP j ) computes the work in process as the sum of the parts waiting in queuesof the machinesthat compose the cell.

Local approach

j i j WIP NQi ∈ = ∑ (2)
whereNQ i is the number of parts waiting in the machine's queuei-th computed for the machines of the cell j-th.

When a part arrives in the manufacturing system, the controller will decide if the part enters to the manufacturing cell designedfor its family or the part entersto the remainder cell (rsubscript). The decision is based on the following conditions computed for all manufacturing cells.

r P j WIP WI ≥ (3) r P j WIP WI < (4)
The manufacturing cells that verified the condition (3) re-route the parts to the reminder cell. While, the manufacturing cells that verify the condition (4) don't re-route the parts. Therefore, the generic part of a family is loaded in j-th manufacturing cell(designed for the part family), if the condition ( 4) is verified, otherwise it is loaded in remainder cell. This approach allows to limit the level ofthe work in process in the manufacturing cells and, therefore, reducing throughput time and delay of the parts.The main drawback is the possibility toincrease the work in process in the remainder cell reducing the performance of the parts worked in this cell.This increment is due to the decision strategy; the expressions 3 and 4 don't take into account the global state of the manufacturing system. Each manufacturing cell decides the re-routing of the parts independently.The advantage of this approach is the reduction of communication; each manufacturing cell to take the decision needs to know only the state of the remainder cell.

Global approach

This approach is proposed to avoid the drawback of the local approach evaluating the global state of the manufacturing system. In this case, each manufacturing cell computes a congestion level (cong j ) related to the WIP of the manufacturing system by the following expression:

[ ] 1 0,1 j j M i WIP cong NQi = = ∈ ∑ (5)
whereWIP j is the same value computed in expression (2) and the denominator is the sum of the parts in the queue for all machines (M) of the manufacturing system (the work in process of the entire manufacturing system is the global information).

The part assigned to the generic j-th manufacturing cell has to be re-routed to the remainder cell, if two conditions are verified. The first is the following:

[ ] 1 2 , ,....., j N cong MAX cong cong cong = (6)
The expression ( 6) means that the manufacturing cellj-th is the cell with maximum value of the work in process compared to other manufacturing cells (N is the number of manufacturing cells). The second condition is the following:

r P 1 j WIP WI - > (7)
Expression ( 7) means that the work in process of the manufacturing cell (j)has to be greater than the remainder cell (r).The difference between the two cells has to be two parts. It is chosen this strategy in order to obtain a work in process of the remainder cell always lower than other manufacturing cells. In fact, if the second part of the expression ( 7) is zero, when the part enters the remainder cell the two cells have the same work in process. Applying this methodology, the remainder cell plays a role in supporting the main manufacturing cells. Following this policy, only, the manufacturing cell with higher WIP re-route the parts. However, if the WIP of some manufacturing cells are equal, many parts can be re-route to the remainder cell. For this reason, a modification of this approach is proposed in the following sub-section.

Global approach exclusive

This approach is simply an adjustment of the previous approach. In some cases, the conditions ( 6) and ( 7) can be verified for more than one manufacturing cell. In this approach only one manufacturing cell can reroute the part, if more cells can re-route the part at the same time. In particular, the first manufacturing cell thatverifies the expressions ( 6) and ( 7) can re-route the part.This approach leads to reduce the utilization of the remainder cell.

Simulation environment

The objective of the simulation experiments is to measure the performance of proposed approaches benchmarked to a classical cellular manufacturing system in a very dynamic environment. The authors selected the Arena® discrete event simulation platform by Rockwell Software Inc. it was used to develop the simulation model of the presented approaches.Discrete event simulation -in many commercial tools and simulation packages, nowadays the simulation model is automatically created from high level modeling languages and notations -allows to validate and optimize dynamic and discrete systems such as production systems, but also workflows such as negotiation mechanisms. These models facilitate evaluating different coordination scenarios and maximizing their potential output and benefits. Arena® -based on the known SIMAN simulation language -is well suited for modeling shop floors of production systems in which each entity (part) follows a manufacturing route through production resources (servers, material handling systems, buffers, and so forth), (Kelton and Sadowski, 2009).The manufacturing system consists of ten machines (M=10), and it has been considered three part families. The configurations of the manufacturing system have to implement three manufacturing cells (N=3). Table 1 reports the mix for each part family.

[Insert table1 here]

In the caseof the cellular manufacturing system (used as a benchmark), it has been designed two manufacturing cells with four machines for the part 1 and 2 (the two families have the same part mix); the third manufacturing cell is composed by twomachines for the part 3 (lower value of mix). The configuration with the remainder cell consists of three manufacturing cells, but some machines need to be used in the remainder cell and subtracted from the manufacturing cells of the cellular manufacturing system. In this case,the configuration is the following: three machines are assigned for each manufacturing cell dedicated to part 1 and 2; two machines are assigned for the manufacturing cell dedicated to the part 3; two machines are assignedto the remainder cell. The machines assigned to the reminder cell perform the manufacturing operations with higher processing time (see equation 1),because these machinesare general purpose in order to perform any kind of operations. Figure 3 shows the cellular manufacturing system used for the simulation experiments with the assignment of the machines to the cells.

[Insert figure 3 here]

Figure 4 shows the re-configuration of the machines 4 and 8 to form the remainder cell. These machines are configured to perform any kind of operations required by the three part families.

[Insert figure 4 here] Table 2 reports the processing time of the machines in the two configurations investigated.

[Insert table2 here]

The total time to manufacture the parts is 40 equal for all part families; in this way, the performance comparison is due only to the different configurations.The total processing time is distributed to the machines uniformly. In the case of cellular manufacturing configuration, the machines from 1 to 4 (cell 1) have a processing time of 10 unit times in order to perform all the processing time required (40 unit times); the same processing times are assigned to the machines from 5 to 8 (cell 2); while the machines 9 and 10 (cell 3) have 20 unit times. In case of remainder cell introduction, the cells 1 and 2 have three machines with a processing time of 13.33 unit times. The machines 4 and 8 assigned to the remainder cell have a processing time of 24 unit times. It is assigned a value of α=1.2 (see equation 1); the processing time of general purpose machines of remainder cell isgrater of 20% than the machines of a cell that are specifically configured for a part family. This choice overestimates the difference of processing time between dedicated and general purpose machines. However, the focus of this paper is the evaluation of the behavior of loading approaches; the study of α will be associated with the economic evaluation of the investment in machines. Parts enter the system following an exponential arrival stream whose inter-arrival times are reported intable 3. The simulations are performed for four congestion levels of the manufacturing system.

[Insert table3 here]

Several experiments have been conducted in order to set the inter-arrival time; the values reported in table 3 allow to obtain an order of magnitude of average utilization of the manufacturing system.These interarrival times allow to investigate several degrees of congestion of the manufacturing system. A due date is assigned to the parts by the following expression:

40 = • index due date duedate (8)
The due date is obtained by the technological processing time (40 unit times) multiplied with an index; this index is 1.5 for parts 1 and 2, while it is one for part 3 (part 3 has a low part mix). However, the objective of the research is the comparison of the performance; therefore, the choice of thedue date index does not affect the analysis. Concerning the machine breakdowns, it has been assumed that all the manufacturing machines are subject to faults;Mean Time Between Failures (MTBF) is distributed according to thenormally distribution, with mean 2000 unit times and variance 200 unit times. Mean Time To Repair (MTTR) follows a normal distribution with mean 40unit times and variance 6 unit times (equal for all machines).

Simulation results

The proposed approaches are tested in static and dynamic situations; the dynamicity of the manufacturing system is characterized by the stage length andthe simulation length is fixed to 43200 time units.In order to emulate a market dynamic environment the demand characteristics (inter-arrival time and product mix) changing during the production run consisting of several alternating stages. Four stage lengths have been considered; table 4 reports the stage length and the number of changes (demand characteristics) that occur over the entire simulation horizon.

[Insert table4 here] Table 5 describes the design of the simulation experiments conducted for the two configurations of the manufacturing system. Combining the four inter-arrival times, four stage length and the two demand changes (mix and inter-arrival) and the static condition (without any changes) it has been obtained 28 experimental classes.

[Insert table5 here]

The two demand changes are obtained by the following expressions:

0 9 1 1 = • * [ . , . ], p p mix product mix product UNIFORM
for product p=1,2,3;

Expression ( 9) means that the mix of the products (for which the cellular manufacturing systems were designed, mix product p ) is affected by a variation of 20% (1.1-0.9) extracted by a uniform distribution.

5 5 0 9 1 1 - = • int * .
[ . , . ] er arrival UNIFORM (10) Expression (10) computes the inter-arrival time of the parts starting from 5.5 multiplied for an uncertain of20% extracted by a uniform distribution. The value 5.5 of inter-arrival time assures a medium level of average utilization of the manufacturing system about 70%.The inter-arrival and mix changes are computed for each alternation between two consequently stages. The uniform distribution is used to simulate demand and mix random fluctuations. For each experiment class, a number of replications able to assure a 5% confidence interval and 95 % of confidence level for each performance measure have been conducted. The performance measures investigated are the following:

• Throughput time for each part j (thr. Time j);

• Average throughput time (average thr. Time);

• Throughput (thr.);

• Work In Process (WIP);

• Average utilization of the manufacturing system (av. utilization);

• Total tardiness time of the parts (tardiness). The results reported in the following tables and figures are an elaboration of the simulation results reported in the appendix. Table 6 reports the percentage difference among the proposed approaches and cellular manufacturing system (used as the base for percentage computation), when the environmental conditions are static and without machine breakdowns. Table 6 reports the average value and the standard deviation of the performance over the congestion levels. The average values show the difference among the approaches if the congestion levels have the same probability to occur. The standard deviation (dev.st)is an index of the performance variation whenthe congestion level changes (robustness).

[Insert table6 here]

The results of the table 6 show that in a static condition and without machine breakdowns, the cellular manufacturing system leads to the better performance level. Only, the throughput time of the part 3 improves, because the remainder cell can support the related manufacturing cell when the peakof WIP 3 occurs. The influence of congestion levels is very low except for tardiness and throughput time of the part 3 (see dev.st of table 6).Table 7 reports the same analysis of table 6 withmachine breakdowns.

[Insert table7 here]

From the analysis of table 7 the following issues can be drawn:

• The local approach leads to performance very close to the cellular manufacturing system, except for the tardiness performance. In fact, the main benefit of this approach is the reduction of the tardiness (average of 16%).

• The global approach leads to worst performance for all measures, except the throughput time of the part 3. However, the average throughput time getsworse than the cellular manufacturing system. The global approach exclusive reduces the deterioration of the performance, but the performance measures are worse than the cellular manufacturing system. • Generally, the performance measures that are influenced by the congestion level are the tardiness and the throughput time of the part 3 (for these performance measuresthe standard deviation is relevant). Table 8 reports the percentage difference among the proposed approaches and the cellular manufacturing system for different congestion levels, when mix fluctuations are present. The values reported are the average over the different stage lengths; the performance measures have a low dependence on stage length (as showed in appendix with the numerical results).

[Insert table8 here]

In these environmental conditions, the proposed approaches lead to better results for all performance measures of the manufacturing system: reduction of average throughput time, reduction of work in process and reduction of tardiness. The global approach and global approach exclusive increase the benefit compared tothe cellular manufacturing system. It can be noticed that the reduction of throughput time is obtained for all family parts, whilethe local approach reduces only the throughput time of the part 3. Moreover, the performance measures of the global and global exclusive have a greaterdependence on the congestion levels than the local (see dev.st). In particular, the benefits of two global approaches are relevant incases of medium and high congestion levels (inter-arrival 4.5 and 5), while in case of low congestion level (inter arrival 7) the performance measures areworse than the cellular manufacturing system. Table 9 reports the simulation results when the inter-arrival fluctuation is present. Also, in this case the values are the average over the stage lengths.

[Insert table9 here]

The results show how the proposed approaches lead to better results when the inter-arrival fluctuation occurs. The improvement of the performance is better for the global and global exclusive approaches (very similar between them).The stage length has a low influence on the results' comparison, because the standard deviation values are very low. The above comments are valid when the inter-arrival and mix fluctuations are present together (see table 10). In this case, the improvements are greater than the case of only inter-arrival fluctuations.

[Insert table10 here]

In summary, the three performance measures with higher improvement are: average throughput time, work in process and tardiness. Figures 5a,5b approaches and the cellular manufacturing system over the different unforeseen events considered. As the reader can notice, when the conditions are static the approach with remainder cell leads to worst performance while, the loading approaches with remainder cell lead to better performance measures with the introduction of exceptions (mix and inter-arrival fluctuations and machine breakdowns).

[Insert figure 5 here]

Conclusions and future development

The paper proposesloading policies for cellular manufacturing systems with a general purpose cell defined in literature reminder cell. A simulation environment has been developed to test the proposed approaches compared to a classical cellular manufacturing system. The simulations have been conducted in static conditions and in a very dynamic environment with market changes (mix and inter-arrival time fluctuations) and machine breakdowns. The results of this research can be summarized as it follows: -The cellular manufacturing configuration is better when the market conditions are static, this validates the benchmark developed; -The proposed loading policies allow to obtain relevant improvements of the performance when mix fluctuations occur. The benefits are reduced when the inter-arrival fluctuations occur, but however, the proposed approaches are better than the cellular manufacturing configuration. The performance measurewith better improvement is the due date (tardiness).The performance of work in process and average throughput time are also improved.

-The two approaches with the better performance (global and global exclusive)have higherdependence on the congestion levels, whilethelocal approach is more robust.

-The better performance of the global approaches than the local approach underlines that the decision taken on global information can lead to relevant improvements of the performance measures.

-Finally, the low values of standard deviation when external and internal exceptions occur (see table 10) show that the proposed approaches are robust to the rapidity change of the manufacturing system conditions. In briefly, the remainder cell was used in literature as a cell to support the manufacturing cells when machine breakdowns occur; the loading policies proposed show how the remainder cell can be used to keep a high level of performance when market fluctuations happen. The benefits of the remainder cell are evaluated when the processing time of the remainder cell is greater than the dedicated cell. This assumption is made to introduce the effect of the efficiency; general purpose machines can be less efficient than task-specific dedicated machines. This strategy can avoid the re-configuration of cellular manufacturing systems when market conditions change, reducing costs and set-up times. Future research paths concern the following issues. The performance measures of the hybrid cellular manufacturing system for different amplitude of market fluctuations. Another future research path is the evaluation of the increment of processing time for the machines assigned to the remainder cell. This analysis needs to be conducted with the related investment cost in machines, because the parameter α is strength related to the cost of machines.

  periodically) versus robust cells (in which cells stay static and uncertainties are managed through strategies such as VCMS) is needed to identify the conditions under which one would be favoured over the other"

  and 5cshow the percentage difference among the proposed

  Figure 1. Cellular manufacturing system

Table 1

 1

	. Partmix

Table 6 .

 6 Simulation results -static environment without machine breakdowns

	Inter-	Thr.	Thr.	Thr.	Average	Thr.	WIP	Av.	tardiness
	arrival	Time 1	Time 2	Time 3	Thr. time			utilization
					Local appraoch
	Average	5.08%	4.60%	-24.04%	-2.04%	-0.08% -2.12%	4.25%	-16.00%
	Dev.st	6.43%	5.53%	12.16%	3.16%	0.05%	3.19%	0.44%	10.27%
				Global approach
	Average	17.79%	18.96%	-20.17%	9.96%	-0.16%	9.86%	4.26%	44.88%
	Dev.st	2.69%	3.47%	12.32%	3.67%	0.10%	3.63%	0.38%	41.22%
				Global approach exclusive
	Average	12.69%	12.11%	-21.45%	4.40%	-0.14%	4.27%	0.85%	12.97%
	Dev.st	5.69%	4.66%	11.84%	2.94%	0.23%	2.78%	4.84%	15.00%
		F o r
			P
			e
				e r
				R
					e
					v i e
					w
						O n l
							y

Table 7 .

 7 Simulation results -static environment with machine breakdowns

	Inter-	Thr.	Thr.	Thr.	Average	Thr.	WIP	Av.	tardiness
	arrival	Time 1	Time 2	Time 3	Thr. time			utilization
					Local approach
	4.5	5.88%	5.52%	-43.67%	-6.83%	-0.02%	-6.87%	3.66%	-13.08%
	5	-0.25%	-0.78%	-31.34%	-7.91%	0.04%	-7.88%	4.14%	-25.73%
	6	1.18%	1.46%	-18.11%	-3.09%	-0.10%	-3.16%	4.43%	-25.26%
	7	2.80%	2.87%	-12.21%	-0.48%	-0.32%	-0.82%	4.14%	-21.83%
	Average	2.40%	2.27%	-26.33%	-4.58%	-0.10%	-4.68%	4.09%	-21.48%
	Dev.st	2.63%	2.64%	14.06%	3.42%	0.16%	3.28%	0.32%	5.86%
					Global approach
	4.5	-18.20% -18.61% -51.97%	-26.96%	0.00%	-27.05%	4.09%	-46.21%
	5	-6.82%	-7.30%	-34.29%	-13.60%	-0.04%	-13.67%	3.76%	-39.39%
	6 7 Average Dev.st 4.5	2.98% 6.34% F -3.93% 11.03% 11.25% 2.86% 6.42% -4.16% o r -18.51% -18.84% -51.81% -18.57% -12.13% -29.24% 17.78% Global approach exclusive -1.94% -0.05% 2.32% -0.06% -10.05% -0.04% -10.12% -2.00% 2.23% 13.13% 0.03% 13.14% -27.12% 0.06% -27.17%	3.63% 2.91% 3.60% 0.50% 4.09%	-18.02% 1.35% -25.57% 21.59% -46.50%
	5 6 7 Average Dev.st	-7.36% 2.83% 6.47% -4.14% 11.23% 11.41% -8.38% 2.78% P -34.32% -18.55% 6.31% -12.02% -4.53% -29.18% e r -14.23% -2.03% 2.35% -10.26% e 17.76% 13.25%	-0.08% -0.02% -0.03% -0.02% -10.33% -14.33% -2.08% 2.26% 0.06% 13.24%	3.79% 3.63% 3.21% 3.68% 0.37%	-41.23% -18.77% 1.29% -26.30% 21.98%
				R
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Table 8 .

 8 Simulation results -mix changes

	Exp. no.	Thr.	Thr.	Thr.	Average	Thr.	WIP	Av.	tardiness
		Time 1	Time 2	Time 3	Thr. time			utilization
				Local approach
	Average	1.31%	1.10%	-23.05%	-4.36%	-0.02% -4.42%	4.61%	-22.70%
	Dev.st	0.16%	0.17%	0.39%	0.22%	0.14%	0.37%	0.22%	0.90%
				Global approach
	Average	-0.79%	-1.01%	-24.59%	-6.34%	0.07% -6.36%	4.09%	-28.56%
	Dev.st	0.34%	0.40%	0.23%	0.25%	0.50%	0.69%	0.14%	1.18%
				Global approach exclusive
	Average Dev.st	r o -0.95% -1.05% F 0.08% 0.17%	-24.37% 0.39%	-6.36% 0.15%	0.00% -6.42% 0.22% 0.36%	3.85% 0.29%	-28.85% 0.74%
			P
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Table 9 .

 9 Simulation results -inter-arrival changes

	Exp. no.	Thr.	Thr.	Thr.	Average	Thr.	WIP	Av.	tardiness
		Time 1	Time 2	Time 3	Thr. time			utilization
				Local approach
	Average	0.18%	-0.11%	-24.33%	-5.61%	-0.14% -5.76%	3.73%	-26.18%
	Dev.st	0.25%	0.12%	0.52%	0.22%	0.26%	0.39%	1.03%	0.77%
				Global approach
	Average	-1.63%	-2.17%	-26.04%	-7.50%	-0.32% -7.81%	3.63%	-31.50%
	Dev.st	0.85%	0.66%	0.38%	0.58%	0.26%	0.74%	0.29%	1.73%
				Global approach exclusive
	Average	-1.71%	-2.20%	-25.78%	-7.49%	-0.09% -7.56%	3.63%	-31.52%
	Dev.st	r o F 0.57% 0.43%	0.25%	0.38%	0.17%	0.40%	0.35%	1.18%
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Table 10 .

 10 Simulation results -inter-arrival and mix changes

	APPENDIX
	WIP [parts]
	Thr. Time [unit time]
	Average Thr. Time [unit time]
	Tardiness [unit time]
	Inter-	WIP	Thr.	Thr.	Thr.	Average	Tardiness	Thr.	Av.
	arrival		Time 1	Time 2	Time 3	Thr. time			utilization
				Cellular manufacturing system
	4.5	19.350	79.250	79.150	118.560	87.160	339000	13.294	0.880
	5	12.800	60.040	59.840	80.040	64.000	128000	11.980	0.790
	6 7 4.5 5 6	8.690 6.820 F 49.890 46.570 19.580 92.080 o 12.60 64.070 r 8.700 52.770	50.130 46.630 91.770 64.150 52.810	60.320 53.158 Local approach 52.090 47.920 72.970 88.160 58.250 62.940 50.610 52.350	42335 20479 333000 108000 36006	10.010 8.530 13.300 11.990 9.960	0.660 0.566 0.916 0.827 0.686
	7 4.5 5	6.990 21.400 14.120	49.400 P 49.580 101.260 101.170 e 72.950 72.590	47.990 Global approach 49.190 77.170 96.430 61.800 70.580	19339 411000 170000	8.520 13.280 11.980	0.592 0.914 0.828
	6 7 4.5	9.920 7.980 20.621	61.030 57.610 96.850	61.470 57.430 e Global approach exclusive 53.040 59.610 49.700 55.960 r 96.980 76.340 92.830	81686 54160 376000	9.980 8.550 13.290	0.692 0.596 0.915
	5 6 7	13.270 9.290 7.440	68.100 56.730 53.020	67.910 56.810 53.060	60.070 51.960 R e 49.020	66.430 55.810 52.250	134000 55324 33884	11.970 9.980 8.540	0.822 0.684 0.588
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Table A1 .

 A1 Numerical results -Static environment without machine breakdowns

	Inter-	WIP	Thr.		Thr.	Thr.	Average	tardiness	Thr.	Av.
	arrival		Time 1		Time 2	Time 3	Thr. time			utilization
					Cellular manufacturing system
	4.5	29.750	123.450		124.840	168.680	133.760	769000	13.290	0.880
	5	15.660	74.220		74.440	93.480	78.330	237000	11.980	0.790
	6	9.530	55.230		55.420	64.230	57.160	69743	9.990	0.662
	7	7.320	50.130		49.970	55.650	51.190	34091	8.570	0.570
						Local approach
	4.5	26.590	126.990		127.120	89.930	119.720	624000	13.280	0.910
	5	14.300	73.490		73.700	63.180	71.540	172000	11.980	0.824
	6	9.180	55.900		55.980	52.350	55.240	51233	9.970	0.689
	7	7.200	51.460		51.240	48.750	50.840	26054	8.490	0.590
	4.5 5 6 7	20.840 13.370 F 98.280 68.430 9.310 57.020 7.460 53.040 o r	Global approach 76.410 93.870 60.320 67.000 51.960 56.000 49.020 52.330 Global approach exclusive 97.990 68.820 56.920 53.240	386000 139000 56732 34375	13.290 11.960 9.960 8.550	0.916 0.820 0.684 0.583
	4.5	20.830	98.100		98.000	76.230	93.780	385000	13.300	0.916
	5 6	13.260 9.290	68.330 P 67.710 56.730 56.790	60.040 52.090	66.460 55.840	134000 55651	11.950 9.980	0.822 0.684
	7	7.470	53.120	e 53.050	49.140	52.310	34345	8.570	0.587
					e r
						R
						e
	Inter-arrival 4.5	WIP 26.000	Thr. Time 1 108.790		Thr. Time 2 Cellular manufacturing system Thr. Time 3 Average Thr. time v i e 110.910 144.650 116.980 w	tardiness 609000	Thr. 13.300	Av. utilization 0.880
	5 6 7 4.5 5 6 7	15.100 9.440 7.280 26.158 14.120 9.210 7.270	72.300 54.950 49.700 124.640 72.670 55.980 51.370		72.300 55.100 49.850 125.040 72.700 56.150 51.440	88.340 63.100 55.430 Local approach 75.570 56.660 50.910 88.550 117.620 62.900 70.730 O 213000 66439 32775 604000 165000 n l 11.970 9.990 8.570 13.290 11.970 52.470 55.350 51813 9.980 48.940 50.910 26673 8.560 y	0.790 0.660 0.570 0.915 0.824 0.692 0.592
						Global approach
	4.5	28.290	135.750		135.930	93.170	127.330	697000	13.280	0.913
	5	15.960	83.280		82.940	67.030	79.900	242000	11.960	0.823
	6	10.430	64.630		64.860	54.800	62.760	99858	9.960	0.690
	7	8.330	59.920		60.240	51.100	58.290	65920	8.560	0.596
					Global approach exclusive
	4.5	27.540	131.590		131.800	92.48	123.890	664000	13.290	0.912
	5	15.132	78.100		78.230	65.65	75.680	205000	11.980	0.824
	6	9.890	60.700		60.710	54.04	59.380	75866	9.980	0.684
	7	7.730	55.330		55.440	50.07	54.330	43396	8.530	0.584
			Table A2. Numerical results -Static environment

Table A3 .

 A3 Numerical results -Mix changes (stage length: 8640 )

	Inter-	WIP	Thr.		Thr.	Thr.	Average	tardiness	Thr.	Av.
	arrival		Time 1		Time 2	Time 3	Thr. time		utilization
					Cellular manufacturing system
	4.5	28.130	118.970		119.020	155.050	126.510	699000	13.290	0.880
	5	15.550	73.690		75.250	90.820	77.820		232000	11.970	0.794
	6	9.500	55.580		55.220	63.600	57.070		68734	9.980	0.660
	7	7.300	49.830		50.010	56.120	51.180		34309	8.550	0.566
						Local approach
	4.5	26.630	127.210		127.470	89.460	119.820	625000	13.290	0.913
	5	14.280	73.420		73.810	63.010	71.510		171000	11.970	0.823
	6	9.240	56.190		56.220	52.410	55.450		52426	9.980	0.692
	7	7.250	51.300		51.470	49.070	50.920		26794	8.540	0.592
	4.5 5 6 7	20.870 13.430 F 98.370 68.770 9.320 56.890 7.460 53.220 o r	Global approach 76.400 93.980 60.340 67.250 52.060 56.080 49.010 52.280 Global approach exclusive 98.290 69.140 57.240 52.930		388000 141000 57341 34236	13.290 11.970 9.970 8.550	0.916 0.822 0.684 0.586
	4.5	20.760	97.710		97.580	76.720	93.500		383000	13.290	0.916
	5 6	13.280 9.290	68.270 P 68.040 56.840 56.870	60.370 52.140	66.610 55.920		136000 56172	11.950 9.960	0.820 0.684
	7	7.460	53.010	e 53.100	49.100	52.270		34139	8.560	0.588
					e r
						R
	Inter-arrival	WIP	Thr. Time 1		Thr. Time 2	Thr. Time 3 e	Average Thr. time	tardiness	Thr.	Av. utilization
	4.5 5 6 7 4.5	28.713 15.610 9.4700 7.2900 26.640	120.480 73.680 55.230 49.810 127.250		Cellular manufacturing system 120.660 161.370 129.150 73.940 94.410 78.040 55.310 63.860 57.030 50.060 55.690 51.110 v i e Local approach 127.430 89.450 119.840 w	725000 234000 68844 33848 625000	13.290 11.980 9.960 8.550 13.290	0.880 0.792 0.660 0.569 0.913
	5	14.290	73.690		73.620	62.720	71.500		171000	11.970	0.823
	6 7 4.5 5 6 7	9.2100 7.2500 20.900 13.400 9.300 7.470	55.880 51.210 98.520 68.730 56.920 53.110		56.250 51.420 98.300 68.930 56.640 53.340	52.460 49.020 Global approach 55.350 50.860 76.410 94.070 60.210 67.130 52.180 55.870 48.920 52.380	51827 26573 388000 140000 56155 O n l 9.970 8.540 13.300 11.960 9.980 34736 8.550 y	0.690 0.592 0.916 0.820 0.686 0.583
					Global approach exclusive
	4.5	21.020	98.780		98.720	77.380	94.520		393000	13.310	0.916
	5	13.310	68.080		68.380	60.450	66.690		136000	11.960	0.822
	6	9.300	56.860		56.840	52.060	55.910		56135	9.970	0.684
	7	7.440	53.070		53.150	48.930	52.290		34081	8.530	0.584
			Table A4. Numerical results -Mix changes (stage length: 4320 )

Table A5 .

 A5 Numerical results -Mix changes (stage length: 2880 )

	Stage	WIP	Thr.		Thr.	Thr.	Average	tardiness	Thr.	Av.
	Length		Time 1		Time 2	Time 3	Thr. time		utilization
					Cellular manufacturing system
	8640	11.710	62.000		62.050	73.100	64.270		120000	10.910	0.720
	4320	11.710	61.830		62.100	73.300	64.270		120000	10.910	0.720
	2880	11.740	61.870		62.100	73.790	64.370		121000	10.920	0.723
	2160	11.720	61.860		62.110	73.090	64.230		120000	10.930	0.720
						Local approach
	8640	11.250	62.890		62.870	56.550	61.620		94149	10.930	0.754
	4320	11.170	62.580		62.690	56.290	61.380		92256	10.900	0.754
	2880	11.180	62.580		62.710	56.430	61.410		92398	10.910	0.754
	2160	11.210	62.760		62.810	56.400	61.520		92984	10.920	0.754
	8640 4320 2880 2160	10.970 11.010 F 61.440 61.460 11.060 61.600 10.860 61.110 o r	Global approach 55.250 60.180 55.440 60.250 55.450 60.460 55.030 59.950 Global approach exclusive 61.370 61.440 61.810 61.220		85759 86338 87777 83760	10.920 10.950 10.970 10.860	0.748 0.750 0.753 0.750
	8640	11.010	61.390		61.520	55.590	60.290		86458	10.940	0.750
	4320 2880	10.970 10.940	61.310 P 61.480 61.230 61.400	55.390 55.430	60.210 60.150		85675 85101	10.920 10.900	0.748 0.748
	2160	10.950	61.290	e 61.340	55.380	60.140		85009	10.910	0.748
					e r
	Inter-arrival 4.5	WIP 27.680	Thr. Time 1 117.170		Thr. Time 2 Cellular manufacturing system Thr. Time 3 Average Thr. time R 117.420 152.450 124.590 e	tardiness 681000	Thr. 13.290	Av. utilization 0.880
	5 6 7 4.5 5 6	15.250 9.500 7.270 26.490 14.300 9.170	73.110 55.200 49.790 126.660 73.360 55.870		73.590 55.390 49.810 126.300 73.760 56.130	88.390 64.140 55.460 Local approach 76.430 57.100 50.950 89.760 119.180 v i e 62.980 71.460 52.280 55.260 w	220000 69468 33092 619000 171000 51358	11.950 9.970 8.550 13.290 11.980 9.940	0.790 0.660 0.567 0.913 0.827 0.688
	7	7.240	51.180		51.450	48.870	50.830		26352	8.540	0.592
	4.5 5 6 7 4.5	20.700 13.380 9.310 7.440 20.560	97.400 68.670 57.010 52.840 96.500		Global approach 76.500 93.280 60.200 66.980 52.110 55.990 48.920 52.190 Global approach exclusive 97.480 68.630 56.880 53.170 96.700 76.420 92.600	381000 139000 56678 33801 O n l 13.280 11.970 9.970 8.550 374000 13.290 y	0.916 0.823 0.684 0.586 0.916
	5	13.320	68.330		68.160	60.100	66.630		136000	11.980	0.822
	6	9.330	57.070		57.000	52.070	56.060		56864	9.980	0.686
	7	7.470	53.270		53.150	48.950	52.370		34494	8.550	0.586
			Table A6. Numerical results -Mix changes (stage length: 2160)

Table A7 .

 A7 Numerical results -interarrival changes

	Stage	WIP	Thr.	Thr.	Thr.	Average	tardiness	Thr.	Av.
	Length		Time 1	Time 2	Time 3	Thr. time			utilization
				Cellular manufacturing system
	8640	11.920	62.910	63.060	74.330	65.330	128000	10.920	0.724
	4320	11.990	63.140	63.290	75.180	65.670	130000	10.940	0.723
	2880	11.900	62.500	62.710	75.150	65.170	127000	10.940	0.725
	2160	11.920	62.600	63.090	74.810	65.290	127000	10.940	0.722
					Local approach
	8640	11.290	63.020	63.100	56.790	61.830	95598	10.940	0.740
	4320	11.270	63.040	63.170	56.620	61.830	95425	10.920	0.754
	2880	11.170	62.680	62.620	56.570	61.440	92676	10.890	0.754
	2160	11.250	62.860	62.990	56.630	61.680	94243	10.930	0.754
					Global approach
	8640	10.920	61.510	61.480	55.180	60.250	86315	10.850	0.748
	4320	10.970	61.610	61.470	55.320	60.310	86632	10.900	0.748
	2880	11.050	62.120	61.860	55.370	60.690	89043	10.910	0.753
	2160	11.060	61.820	61.860	55.600	60.600	88684	10.940	0.750
				Global approach exclusive
	8640	10.970	61.500	61.480	55.360	60.280	86289	10.900	0.750
	4320	11.060	61.790	61.630	55.620	60.500	88044	10.950	0.752
	2880	11.010	61.620	61.610	55.630	60.430	87314	10.910	0.748
	2160	11.080	61.940	61.890	55.650	60.670	88933	10.940	0.749

Table A8 .

 A8 Numerical results -interarrival and mix changes

URL: http://mc.manuscriptcentral.com/tandf/tcim Email:ijcim@bath.ac.ukInternational Journal of Computer Integrated Manufacturing

http://mc.manuscriptcentral.com/tandf/tcim Email:ijcim@bath.ac.ukInternational Journal of Computer Integrated Manufacturing