- Email: [email protected]
Cell formation with alternative process plans and machine capacity constraints: A new combined approach
Int. J. Production Economics 64 (2000) 279}284
Cell formation with alternative process plans and machine capacity constraints: A new combined approach C. Caux*, R. Bruniaux, H. Pierreval Equipe de Recherche en Syste% mes de Production de l'IFMA, Laboratoire d'Informatique, de Mode& lisation et d'Optimisation des Syste% mes, Campus des Ce& zeaux-BP 265, F-63175 Aubie% re, France
Abstract This paper addresses the problem of manufacturing cell formation with alternative process plans and machine capacity constraints. Given routings, capacities of machines and quantities of parts to produce, the problem consists in grouping machines into manufacturing cells and in selecting one process plan for each part. The objective is to minimize the inter-cell tra$c, respecting machine capacity constraints. A new approach combining the simulated annealing method for the cell formation and a branch-and-bound method for the routing selection is presented. An illustrative example and numerical results are given. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Cellular manufacturing; Alternative process plan; Branch and bound; Simulated annealing
1. Introduction Group technology is a principle, which decomposes a global system into several subsystems, which are easier to manage than the entire system. Applied to manufacturing, this principle is the base of the design of production cells. According to WemmerloK v and Hyer [1], the main improvements that can be expected from cellular manufacturing are reductions in throughput time, in material handling, in setup time and improvement of part quality. The cell formation problem consists in grouping machines (and eventually tools, storages, operators,
* Corresponding author. Tel.: #33-473-28-81-08; fax: #33473-28-81-00. E-mail address: [email protected] (C. Caux)
etc.) in manufacturing cells as independent as possible in order to satisfy a performance criterion, for example the inter-cell tra$c or the number of inter-cell movements. During the decades, the cell formation problem has been addressed in numerous research papers. A "rst category of authors only considers one possible way to produce each type of part (i.e. a part is assigned to one process plan or routing). In this case, the manufacturing process of parts is generally given by an incidence part}machine matrix or by a #ow machine}machine matrix [2}9] or by a sequence of manufacturing operations [10}12]. In order to increase the #exibility, a second category of authors takes into account several possible ways to manufacture the same part type. Some works consider that each operation can be achieved by a set of machines [6,13}20] and some others consider that each part has a set of several possible
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process. In association with these two problem formulations, machine capacity constraints are usually added (i.e. the loading of each machine must be lower than its capacity). This type of problem is tackled in [21}23]. In this paper, we focus on the cell formation problem with alternative process plans and machine capacity constraints in order to minimize the inter-cell tra$c. In this case, several process plans are available for the process of one part. Gupta [22] proposes a two step solution to solve this problem. At the "rst step, one routing is de"nitely determined for each part, respecting machine capacity constraints. At the second step, cell formation is achieved. The problem of this method is that routing selection is performed once and that the #exibility given by alternative routings is not used to minimize inter-cell tra$c. Nagi et al. propose in [23] an iterative method solving the two distinct subproblems: cell formation solved by a heuristic [10] and routing selection solved by the Simplex method. At each iteration, one problem of the two is solved using the solutions of the other problem. Unfortunately, their method is limited by the Simplex method because of the number of constraints to deal with which becomes very large as the size of the problem increases. Moreover, due to the use of a heuristic, this method does not guaranty the convergence toward an optimum. In this paper, we propose a new combined approach to solve this problem. This approach is based both on the simulated annealing method and on a branch-and-bound method in order to solve simultaneously the problem of selecting a routing and the problem of minimizing the inter-cell tra$c. The problem addressed is more formally described in the following section. Our approach is then presented in the next section. Computational results are "nally presented.
hypothesis [23]:
2. The cell formation problem
2.2. Formulation
2.1. Description
We propose a formulation of this problem using two decision variables: xr "1 if the part i is processed using the routing ik r , 0 otherwise. ik
We are interested in "nding solutions of a cell formation problem, which respect the following
f Each machine is considered as unique: even if two machines are functionally similar, they are considered as di!erent in the model. f One and only one routing has to be selected for each part type. f At least one routing selection exists to produce the required part quantities, respecting machine capacity constraints. We consider a set M"Mm , m ,2, m N of m ma1 2 m chines in a given manufacturing system. Each machine m has a "nite time capacity of tc time units j j in the chosen horizon H. We also consider a set p , p ,2, p of p part 1 2 p types. Let n denote the required part quantity for i part type p . This information is calculated by the i projected production forecasts (in the case of new facilities), or by historical production information (in the case of existing facilities). Each part type p is associated with nr possible i i routings. The kth possible routing for the part type p is denoted by r (1)k)nr ). A routing r is i ik i ik a sequence of nbo operations. Because each operaik tion is performed on one given machine, we can de"ne a routing r as ik r "Mm (1), m (2),2, m (nbo )N ik ik ik ik ik where m (x)3M, 1)x)nbo . ik ik The processing time required to perform an operation of the part i on the machine m (j) is denoted ik by pt (j). These processing times can be deduced ik from the process plans, the bills of materials and the production forecasts by the production management system. As in most production management systems, we will consider that processing times include waiting times and setup times. The machines must be grouped in c cells. The number of machines a cell can contain is nm.
C. Caux et al. / Int. J. Production Economics 64 (2000) 279}284
281
xc "1 if the machine i and the machine j are in the ij same cell, 0 otherwise (xc "0, 1)j)m). jj The objective is p nri nboik~1 minimize + + + n .xr xc ik ik i ik m (q)m (q`1) i/1 k/1 q/1 subject to
(1)
m nri + + xc )nm, (2) ik i/1 k/1 p nri + + xr "1, (3) ik i/1 k/1 m p nri nboik + + + + n .pt .xr )tc . (4) i ik ik j j/1 i/1 k/1 t/1 mik(t)/j The objective function (1) to be minimized is the inter-cell tra$c (expressed in number of parts). Constraint (2) restricts the number of machines in a cell to the value nm. Constraint (3) indicates that one and only one routing is assigned to a part. Constraint (4) limits the sum of the processing times of a machine to its capacity. This problem is more complex (on a combinatorial aspect) than the problem which consists in minimizing the inter-cell tra$c because in addition, the assignment of parts to routings has to be determined in order to evaluate the objective function. The approach we propose is based on two interconnected methods and is described in the next section.
3. Proposed approach 3.1. General principle We propose in this paper a combined iterative approach able to solve simultaneously the two subproblems mentioned above as shown in Fig. 1. This method combines the simulated annealing (SA) algorithm and a branch-and-bound (BB) algorithm in a reiterated way. The SA algorithm proposes and selects partitions, respecting the maximum number of machines authorized in a cell. Then, for a given partition, the BB algorithm optimally assigns one
Fig. 1. The proposed approach.
routing to each part, respecting machine capacity constraints. The advantage of this approach is twofold. First, one of the two subproblems is solved with an exact method, instead of considering the whole problem as a combinatorial problem [24]. Second, as it will be explained in the following, the implemented SA algorithm satis"es the convergence conditions, so that it can reach the optimal solution. Moreover, the use of the SA algorithm will permit us to deal with other objective functions (handling costs, intra-cell tra$c, etc.). The two next sections, respectively, describe the SA algorithm and the BB algorithm we implement to solve these two linked problems. 3.2. The partition of the set of machines problem This problem is solved by using the simulated annealing algorithm [25]. This algorithm is an optimization method based on ideas from statistical mechanics and motivated by an analogy to the behavior of physical systems in the presence of a heat bath. The simulated annealing method can obtain an optimum or near optimum solution by using stochastic principles to avoid to be trapped in local minima. Indeed, general deterministic iterative improvement methods accept those solutions that decrease cost value, so that the search fails in certain local optima. To avoid this, the simulated annealing is based on the following principles. The state transition by which the value of the cost function is increased is also accepted with a certain probability which is given by exp(!*f/¹) where ¹ is a parameter called temperature and *f denotes the di!erence in cost between the candidate solution and the current solution. Decreasing the temperature ¹ at a low rate permits the control of the acceptance probability. A classical simulated
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annealing method is the following: Generate an initial solution S. Generate a neighbor S@ of S. *"cost(S@)!cost(S). If *)0 then S"S@. If *'0 then S"S@ with probability exp(!*/¹). Step 6: Go to step 2 until a stopping criterion is reached.
Step Step Step Step Step
1: 2: 3: 4: 5:
The temperature ¹ is given by C/Ln(n#1) where C is a parameter called initial temperature and n is the iteration number. The use of simulated annealing for a particular problem requires the de"nition of a coding of the solutions and the de"nition of a method to obtain a neighbor from a solution. In our case, a solution is represented by a vector l where l indicates the number of the cell containi ing the machine i. A neighbor of a solution is obtained by applying an insertion or a permutation to the solution. The insertion consists in moving a machine from one cell to another one. In order to meet the cell capacity constraints, the destination cell must contain at least one free place. The permutation consists in swapping two machines from two di!erent cells. These two functions ensure that the search space is connected (all the solutions are reachable from a particular solution). This is a condition to ful"ll, to obtain the optimal solution. The implementation of the SA algorithm also implies the setting of some parameters (such as the initial temperature) which will be described in the computational results section.
Fig. 2. Structure of the enumeration tree.
3.3.1. Using constraints to prune the enumeration tree The tree as shown in Fig. 2 becomes huge when the number of parts and routings increases. But some nodes do not meet the machine capacity constraints. It is easy to check, for each new node, if the sum of the processing times of each machine exceeds the capacity of this machine (i.e. if the constraint 4 is not met). In this case, the node is not created and the tree is cut. Depending on the characteristics of the routings, this cut method may reduce the number of solutions to consider.
3.3. The part-routing assignment problem
3.3.2. Lower bound computation The lower bound of a node indicates the minimum (but not necessarily reachable) value of the objective function for a given node of the tree. This bound is computed for a node as follows. Let p and r the part and the routing associated i j with the considered node. The inter-cell tra$c induced by parts from p to p is directly computed 1 i (the tree indicates the assignment of parts to routings). The inter-cell tra$c for the parts p to p is i`1 p evaluated after relaxation of constraint (4), that is to say the machine capacities are no more taken into account. Then, for each part p to p , the i`1 p chosen routing is the one leading to the smallest inter-cell tra$c value. The following algorithm details the lower bound computation.
As described previously, the BB procedure is aimed to assign, for a given partition of machines, one routing to each part. This assignment must meet the machine capacity constraints and must minimize the objective function. The solutions are enumerated using a tree where a part represents one level and where the possible routings for these parts represents the nodes of this level. Fig. 2 illustrates the tree structure.
Set the lower bound ¸B to the inter-cell tra$c for parts p to p 1 i For each part p (i#1)k)p) k For each possible routing r (1)m)n ) for p km rk k Compute the inter-cell tra$c for the part p to k the routing r km Choose the routing corresponding to the smallest inter-cell tra$c Add the minimal inter-cell tra$c value to ¸B
C. Caux et al. / Int. J. Production Economics 64 (2000) 279}284
The branching rule used consists in partitioning the node with the smallest bound (best bound rule). In the case where two or more nodes have the same bound, we choose the deepest one. 3.3.3. Improvement of branch-and-bound performances In order to speed up the branch-and-bound method, parts are sorted before the tree construction. The idea of this improvement is to avoid backtracks to the top of the tree which are costly in terms of computing time. The principle is to have at the top of the tree the parts such that the inter-cell tra$c is di!erent depending on the selected routing and at the bottom of the tree the parts whose tra$c is identical whatever the selected routing is. Such a strategy allows backtrack to occur most often at the bottom of the tree. The following sort algorithm has been implemented. For each part p (1)k)p) k For each possible routing r (1)m)n ) for km rk the part p k Compute the inter-cell tra$c for the part p and the routing r k km Set index(k) to the di!erence between the two smallest values of the inter-cell tra$c Sort parts descending on index( )
4. Computational results The method has been implemented on a HP/UNIX workstation with C language. We tested it on the example published in [23]. In this example, 20 parts, 51 routings and 20 machines have to be grouped into 5 cells. In order to set the SA parameters, 25 runs of the calculation have been performed, changing at each time the two parameters of the simulated annealing: initial temperature C and permutation probability pp (i.e. a neighbor is created from a solution by using permutation with probability pp and by using insertion with probability 1!pp). The optimal solution, given in [23] has been found in 218 iterations (CPU time"10 s) in the best case (C"10 and permutation probability pp"0.5). Table 1 presents the number of iterations
283
Table 1 Number of iterations required to "nd the optimal solution pp
0 0.25 0.50 0.75 1.00
C 1
5
10
15
20
2000 1031 775 764 2000
397 556 319 280 2000
450 321 218 671 2000
743 308 949 1861 2000
1538 1622 1100 2000 2000
required to reach the solution (2000 means that the solution has not been reached after 2000 iterations). For this example, our method gives the optimal solution in most of the cases. In the case where pp"1 (i.e. neighbors are created only by permutations), the method cannot "nd the optimal solution. Indeed, because of permutations, the number of machines per cell cannot be changed and then the "nal solution depends on the initial solution.
5. Conclusion A new method to solve cell formation problem with alternative routings and machine capacity constraints has been presented. This method permits to solve simultaneously the cell formation problem and the part-routing assignment problem whereas other methods are based on two heuristics or algorithms: one of the two problems is then solved from the solutions of the second one. Although exact methods, like branch-andbound, often lead to large computational times, our method provides us with solutions very quickly. Applied to the example in [23], it returns the optimal solution in 10 seconds. Moreover, the branch-and-bound algorithm we propose can be considered independent of the rest of the method. Indeed in the case where process plans are modi"ed (e.g. when short-term variations of production are considered), the BB algorithm is able to optimally select routings in order to minimize inter-cell tra$c, respecting machine capacity constraints. This feature makes our method more robust to variations of production. Although acceleration processes have been introduced in the branch-and-bound, our method can be
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limited with large-sized problems or unconstrained problems due to calculation time. Further work involves applying the bene"ts of this method to other criteria.
Acknowledgements Romain Bruniaux is very grateful to Pr. Andrew Kusiak since a part of this work has been carried out in the Intelligent Systems Laboratory of the Industrial Engineering Department of the University of Iowa (USA). This work has been partially supported by the French Ministry of Education and Research (DSPT 8).
References [1] U. WemmerloK v, N.L. Hyer, Cellular manufacturing in the US industry: A survey of users, International Journal of Production Research 27 (9) (1989) 1511}1530. [2] C. Caux, H. Pierreval, R. Bruniaux, Algorithmes eH volutionnistes pour la technologie de groupe: Application a` la formation d'm( lots de fabrication, Revue d'automatique et de productique appliqueH es 8 (2}3) (1995) 485}490. [3] C.-H. Cheng, M.S. Madan, J. Motwani, Designing cellular manufacturing systems by a truncated tree search, International Journal of Production Research 34 (2) (1996) 349}361. [4] Y. Gupta, M. Gupta, A. Kumar, C. Sundaram, A genetic algorithm-based approach to cell composition and layout design problems, International Journal of Production Research 34 (2) (1996) 447}482. [5] S. Kaparthi, N.C. Suresh, R.P. Cerveny, An improved neural network leader algorithm for part-machine grouping in group technology, European Journal of Operational Research 69 (1993) 342}356. [6] A. Kusiak, The generalized group technology concept, International Journal of Production Research 25 (4) (1987) 561}569. [7] Ch.V.R. Murthy, G. Srinivasan, Fractional cell formation in group technology, International Journal of Production Research 33 (5) (1995) 1323}1337. [8] J.C. Wei, G.M. Fern, Commonality analysis: A linear cell clustering algorithm for group technlogy, International Journal of Production Research 27 (12) (1989) 2053}2062. [9] A. Kusiak, W.S. Chow, Decomposition of manufacturing systems, IEEE Journal of Robotics and Automation 4 (5) (1988) 457}471.
[10] G. Harhalakis, R. Nagi, J.M. Proth, An e$cient heuristic in manufacturing cell formation for group technology applications, International Journal of Production Research 28 (1) (1990) 185}198. [11] S. Kamal, L.I. Burke, FACT: A new neural network-based clustering algorithm for group technology, International Journal of Production Research 34 (4) (1996) 919}946. [12] R. Logendran, Impact of sequence of operations and layout of cells in cellular manufacturing, International Journal of Production Research 29 (2) (1991) 375}390. [13] F.F. Boctor, The minimum-cost, machine-part cell formation problem, International Journal of Production Research 34 (4) (1996) 1045}1063. [14] N.-E. Dahel, S.B. Smith, Designing #exibility into cellular manufacturing systems, International Journal of Production Research 31 (4) (1993) 933}945. [15] J.F. Ferreira Ribeiro, B. Pradin, A methodology for cellular manufacturing design, International Journal of Production Research 31 (1) (1993) 235}250. [16] T.W. Liao, Design of line-type cellular manufacturing systems for minimum operating and material-handling costs, International Journal of Production Research 32 (2) (1994) 387}397. [17] R. Logendran, P. Ramakrishna, C. Sriskandarajah, Tabu search-based heuristics for cellular manufacturing systems in the presence of alternative process plans, International Journal of Production Research 32 (2) (1994) 273}297. [18] R. Logendran, P. Ramakrishna, Manufacturing cell formation in the presence of lot splitting and multiple units of the same machine, International Journal of Production research 33 (3) (1995) 675}693. [19] S. Viswanathan, Con"guring cellular manufacturing systems: A quadratic integer programming formulation and a simple interchange heuristic, International Journal of Production Research 33 (2) (1995) 361}376. [20] N. Wu, G. Salvendy, A modi"ed network approach for the design of cellular manufacturing systems, International Journal of Production Research 31 (6) (1993) 1409}1421. [21] A. Atmani, R.S. Lashkari, R.J. Caron, A mathematical programming approach to joint cell formation and operation allocation in cellular manufacturing, International Journal of Production Research 33 (1) (1995) 1}15. [22] T. Gupta, Design of manufacturing cells for #exible environment considering alternative routeing, International Journal of Production Research 31 (6) (1993) 1259}1273. [23] R. Nagi, G. Harhalakis, J.-M. Proth, Multiple routeings and capacity considerations in group technology applications, International Journal of Production Research 28 (12) (1990) 2243}2257. [24] A. Kusiak, W.J. Boe, C. Cheng Chunhung, Designing cellular manufacturing systems: Branch-and-bound and AH approaches, IIE Transactions 25 (4) (1993) 46}56. [25] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Optimization by simulated annealing, Science 220 (1983) 671}680.
Cell formation with alternative process plans and machine capacity constraints: A new combined approach C. Caux*, R. Bruniaux, H. Pierreval Equipe de Recherche en Syste% mes de Production de l'IFMA, Laboratoire d'Informatique, de Mode& lisation et d'Optimisation des Syste% mes, Campus des Ce& zeaux-BP 265, F-63175 Aubie% re, France
Abstract This paper addresses the problem of manufacturing cell formation with alternative process plans and machine capacity constraints. Given routings, capacities of machines and quantities of parts to produce, the problem consists in grouping machines into manufacturing cells and in selecting one process plan for each part. The objective is to minimize the inter-cell tra$c, respecting machine capacity constraints. A new approach combining the simulated annealing method for the cell formation and a branch-and-bound method for the routing selection is presented. An illustrative example and numerical results are given. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Cellular manufacturing; Alternative process plan; Branch and bound; Simulated annealing
1. Introduction Group technology is a principle, which decomposes a global system into several subsystems, which are easier to manage than the entire system. Applied to manufacturing, this principle is the base of the design of production cells. According to WemmerloK v and Hyer [1], the main improvements that can be expected from cellular manufacturing are reductions in throughput time, in material handling, in setup time and improvement of part quality. The cell formation problem consists in grouping machines (and eventually tools, storages, operators,
* Corresponding author. Tel.: #33-473-28-81-08; fax: #33473-28-81-00. E-mail address: [email protected] (C. Caux)
etc.) in manufacturing cells as independent as possible in order to satisfy a performance criterion, for example the inter-cell tra$c or the number of inter-cell movements. During the decades, the cell formation problem has been addressed in numerous research papers. A "rst category of authors only considers one possible way to produce each type of part (i.e. a part is assigned to one process plan or routing). In this case, the manufacturing process of parts is generally given by an incidence part}machine matrix or by a #ow machine}machine matrix [2}9] or by a sequence of manufacturing operations [10}12]. In order to increase the #exibility, a second category of authors takes into account several possible ways to manufacture the same part type. Some works consider that each operation can be achieved by a set of machines [6,13}20] and some others consider that each part has a set of several possible
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 0 6 5 - 1
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process. In association with these two problem formulations, machine capacity constraints are usually added (i.e. the loading of each machine must be lower than its capacity). This type of problem is tackled in [21}23]. In this paper, we focus on the cell formation problem with alternative process plans and machine capacity constraints in order to minimize the inter-cell tra$c. In this case, several process plans are available for the process of one part. Gupta [22] proposes a two step solution to solve this problem. At the "rst step, one routing is de"nitely determined for each part, respecting machine capacity constraints. At the second step, cell formation is achieved. The problem of this method is that routing selection is performed once and that the #exibility given by alternative routings is not used to minimize inter-cell tra$c. Nagi et al. propose in [23] an iterative method solving the two distinct subproblems: cell formation solved by a heuristic [10] and routing selection solved by the Simplex method. At each iteration, one problem of the two is solved using the solutions of the other problem. Unfortunately, their method is limited by the Simplex method because of the number of constraints to deal with which becomes very large as the size of the problem increases. Moreover, due to the use of a heuristic, this method does not guaranty the convergence toward an optimum. In this paper, we propose a new combined approach to solve this problem. This approach is based both on the simulated annealing method and on a branch-and-bound method in order to solve simultaneously the problem of selecting a routing and the problem of minimizing the inter-cell tra$c. The problem addressed is more formally described in the following section. Our approach is then presented in the next section. Computational results are "nally presented.
hypothesis [23]:
2. The cell formation problem
2.2. Formulation
2.1. Description
We propose a formulation of this problem using two decision variables: xr "1 if the part i is processed using the routing ik r , 0 otherwise. ik
We are interested in "nding solutions of a cell formation problem, which respect the following
f Each machine is considered as unique: even if two machines are functionally similar, they are considered as di!erent in the model. f One and only one routing has to be selected for each part type. f At least one routing selection exists to produce the required part quantities, respecting machine capacity constraints. We consider a set M"Mm , m ,2, m N of m ma1 2 m chines in a given manufacturing system. Each machine m has a "nite time capacity of tc time units j j in the chosen horizon H. We also consider a set p , p ,2, p of p part 1 2 p types. Let n denote the required part quantity for i part type p . This information is calculated by the i projected production forecasts (in the case of new facilities), or by historical production information (in the case of existing facilities). Each part type p is associated with nr possible i i routings. The kth possible routing for the part type p is denoted by r (1)k)nr ). A routing r is i ik i ik a sequence of nbo operations. Because each operaik tion is performed on one given machine, we can de"ne a routing r as ik r "Mm (1), m (2),2, m (nbo )N ik ik ik ik ik where m (x)3M, 1)x)nbo . ik ik The processing time required to perform an operation of the part i on the machine m (j) is denoted ik by pt (j). These processing times can be deduced ik from the process plans, the bills of materials and the production forecasts by the production management system. As in most production management systems, we will consider that processing times include waiting times and setup times. The machines must be grouped in c cells. The number of machines a cell can contain is nm.
C. Caux et al. / Int. J. Production Economics 64 (2000) 279}284
281
xc "1 if the machine i and the machine j are in the ij same cell, 0 otherwise (xc "0, 1)j)m). jj The objective is p nri nboik~1 minimize + + + n .xr xc ik ik i ik m (q)m (q`1) i/1 k/1 q/1 subject to
(1)
m nri + + xc )nm, (2) ik i/1 k/1 p nri + + xr "1, (3) ik i/1 k/1 m p nri nboik + + + + n .pt .xr )tc . (4) i ik ik j j/1 i/1 k/1 t/1 mik(t)/j The objective function (1) to be minimized is the inter-cell tra$c (expressed in number of parts). Constraint (2) restricts the number of machines in a cell to the value nm. Constraint (3) indicates that one and only one routing is assigned to a part. Constraint (4) limits the sum of the processing times of a machine to its capacity. This problem is more complex (on a combinatorial aspect) than the problem which consists in minimizing the inter-cell tra$c because in addition, the assignment of parts to routings has to be determined in order to evaluate the objective function. The approach we propose is based on two interconnected methods and is described in the next section.
3. Proposed approach 3.1. General principle We propose in this paper a combined iterative approach able to solve simultaneously the two subproblems mentioned above as shown in Fig. 1. This method combines the simulated annealing (SA) algorithm and a branch-and-bound (BB) algorithm in a reiterated way. The SA algorithm proposes and selects partitions, respecting the maximum number of machines authorized in a cell. Then, for a given partition, the BB algorithm optimally assigns one
Fig. 1. The proposed approach.
routing to each part, respecting machine capacity constraints. The advantage of this approach is twofold. First, one of the two subproblems is solved with an exact method, instead of considering the whole problem as a combinatorial problem [24]. Second, as it will be explained in the following, the implemented SA algorithm satis"es the convergence conditions, so that it can reach the optimal solution. Moreover, the use of the SA algorithm will permit us to deal with other objective functions (handling costs, intra-cell tra$c, etc.). The two next sections, respectively, describe the SA algorithm and the BB algorithm we implement to solve these two linked problems. 3.2. The partition of the set of machines problem This problem is solved by using the simulated annealing algorithm [25]. This algorithm is an optimization method based on ideas from statistical mechanics and motivated by an analogy to the behavior of physical systems in the presence of a heat bath. The simulated annealing method can obtain an optimum or near optimum solution by using stochastic principles to avoid to be trapped in local minima. Indeed, general deterministic iterative improvement methods accept those solutions that decrease cost value, so that the search fails in certain local optima. To avoid this, the simulated annealing is based on the following principles. The state transition by which the value of the cost function is increased is also accepted with a certain probability which is given by exp(!*f/¹) where ¹ is a parameter called temperature and *f denotes the di!erence in cost between the candidate solution and the current solution. Decreasing the temperature ¹ at a low rate permits the control of the acceptance probability. A classical simulated
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C. Caux et al. / Int. J. Production Economics 64 (2000) 279}284
annealing method is the following: Generate an initial solution S. Generate a neighbor S@ of S. *"cost(S@)!cost(S). If *)0 then S"S@. If *'0 then S"S@ with probability exp(!*/¹). Step 6: Go to step 2 until a stopping criterion is reached.
Step Step Step Step Step
1: 2: 3: 4: 5:
The temperature ¹ is given by C/Ln(n#1) where C is a parameter called initial temperature and n is the iteration number. The use of simulated annealing for a particular problem requires the de"nition of a coding of the solutions and the de"nition of a method to obtain a neighbor from a solution. In our case, a solution is represented by a vector l where l indicates the number of the cell containi ing the machine i. A neighbor of a solution is obtained by applying an insertion or a permutation to the solution. The insertion consists in moving a machine from one cell to another one. In order to meet the cell capacity constraints, the destination cell must contain at least one free place. The permutation consists in swapping two machines from two di!erent cells. These two functions ensure that the search space is connected (all the solutions are reachable from a particular solution). This is a condition to ful"ll, to obtain the optimal solution. The implementation of the SA algorithm also implies the setting of some parameters (such as the initial temperature) which will be described in the computational results section.
Fig. 2. Structure of the enumeration tree.
3.3.1. Using constraints to prune the enumeration tree The tree as shown in Fig. 2 becomes huge when the number of parts and routings increases. But some nodes do not meet the machine capacity constraints. It is easy to check, for each new node, if the sum of the processing times of each machine exceeds the capacity of this machine (i.e. if the constraint 4 is not met). In this case, the node is not created and the tree is cut. Depending on the characteristics of the routings, this cut method may reduce the number of solutions to consider.
3.3. The part-routing assignment problem
3.3.2. Lower bound computation The lower bound of a node indicates the minimum (but not necessarily reachable) value of the objective function for a given node of the tree. This bound is computed for a node as follows. Let p and r the part and the routing associated i j with the considered node. The inter-cell tra$c induced by parts from p to p is directly computed 1 i (the tree indicates the assignment of parts to routings). The inter-cell tra$c for the parts p to p is i`1 p evaluated after relaxation of constraint (4), that is to say the machine capacities are no more taken into account. Then, for each part p to p , the i`1 p chosen routing is the one leading to the smallest inter-cell tra$c value. The following algorithm details the lower bound computation.
As described previously, the BB procedure is aimed to assign, for a given partition of machines, one routing to each part. This assignment must meet the machine capacity constraints and must minimize the objective function. The solutions are enumerated using a tree where a part represents one level and where the possible routings for these parts represents the nodes of this level. Fig. 2 illustrates the tree structure.
Set the lower bound ¸B to the inter-cell tra$c for parts p to p 1 i For each part p (i#1)k)p) k For each possible routing r (1)m)n ) for p km rk k Compute the inter-cell tra$c for the part p to k the routing r km Choose the routing corresponding to the smallest inter-cell tra$c Add the minimal inter-cell tra$c value to ¸B
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The branching rule used consists in partitioning the node with the smallest bound (best bound rule). In the case where two or more nodes have the same bound, we choose the deepest one. 3.3.3. Improvement of branch-and-bound performances In order to speed up the branch-and-bound method, parts are sorted before the tree construction. The idea of this improvement is to avoid backtracks to the top of the tree which are costly in terms of computing time. The principle is to have at the top of the tree the parts such that the inter-cell tra$c is di!erent depending on the selected routing and at the bottom of the tree the parts whose tra$c is identical whatever the selected routing is. Such a strategy allows backtrack to occur most often at the bottom of the tree. The following sort algorithm has been implemented. For each part p (1)k)p) k For each possible routing r (1)m)n ) for km rk the part p k Compute the inter-cell tra$c for the part p and the routing r k km Set index(k) to the di!erence between the two smallest values of the inter-cell tra$c Sort parts descending on index( )
4. Computational results The method has been implemented on a HP/UNIX workstation with C language. We tested it on the example published in [23]. In this example, 20 parts, 51 routings and 20 machines have to be grouped into 5 cells. In order to set the SA parameters, 25 runs of the calculation have been performed, changing at each time the two parameters of the simulated annealing: initial temperature C and permutation probability pp (i.e. a neighbor is created from a solution by using permutation with probability pp and by using insertion with probability 1!pp). The optimal solution, given in [23] has been found in 218 iterations (CPU time"10 s) in the best case (C"10 and permutation probability pp"0.5). Table 1 presents the number of iterations
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Table 1 Number of iterations required to "nd the optimal solution pp
0 0.25 0.50 0.75 1.00
C 1
5
10
15
20
2000 1031 775 764 2000
397 556 319 280 2000
450 321 218 671 2000
743 308 949 1861 2000
1538 1622 1100 2000 2000
required to reach the solution (2000 means that the solution has not been reached after 2000 iterations). For this example, our method gives the optimal solution in most of the cases. In the case where pp"1 (i.e. neighbors are created only by permutations), the method cannot "nd the optimal solution. Indeed, because of permutations, the number of machines per cell cannot be changed and then the "nal solution depends on the initial solution.
5. Conclusion A new method to solve cell formation problem with alternative routings and machine capacity constraints has been presented. This method permits to solve simultaneously the cell formation problem and the part-routing assignment problem whereas other methods are based on two heuristics or algorithms: one of the two problems is then solved from the solutions of the second one. Although exact methods, like branch-andbound, often lead to large computational times, our method provides us with solutions very quickly. Applied to the example in [23], it returns the optimal solution in 10 seconds. Moreover, the branch-and-bound algorithm we propose can be considered independent of the rest of the method. Indeed in the case where process plans are modi"ed (e.g. when short-term variations of production are considered), the BB algorithm is able to optimally select routings in order to minimize inter-cell tra$c, respecting machine capacity constraints. This feature makes our method more robust to variations of production. Although acceleration processes have been introduced in the branch-and-bound, our method can be
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limited with large-sized problems or unconstrained problems due to calculation time. Further work involves applying the bene"ts of this method to other criteria.
Acknowledgements Romain Bruniaux is very grateful to Pr. Andrew Kusiak since a part of this work has been carried out in the Intelligent Systems Laboratory of the Industrial Engineering Department of the University of Iowa (USA). This work has been partially supported by the French Ministry of Education and Research (DSPT 8).
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