th
19 International Conference on Production Research
VIRTUAL MANUFACTURING CELL FORMATION PROBLEM (VMCFP) IN A DISTRIBUTED LAYOUT A.R. Xambre, P.M. Vilarinho Departamento de Economia, Gestão e Engenharia Industrial - Universidade de Aveiro Campus de Santiago - 3810-193 Aveiro, Portugal Phone: +351-234 370361, Fax: +351-234 370315, E-mail:
[email protected]
Abstract There are essentially two ways to implement a Cellular Manufacturing System (CMS): physical and logical. In the first case, the machines are laid out in the floor space in dedicated manufacturing cells to exploit flow shop efficiency in processing part families, whereas in the second case machines are dedicated to process part families in a process layout. Logical oriented CMS are known as Virtual Cellular Manufacturing Systems (VCMS). A VCMS presents a higher degree of flexibility, useful in a volatile environment where demand changes occur frequently. A novel approach to this problem is to consider that machines are spread in the shop floor in a distributed layout. In this work an algorithm is present that includes two stages: (i) the part family formation problem and (ii) the definition of the virtual cells configuration. The algorithm uses a metaheuristic to improve the exploration of the solution space. Keywords: Facilities Layout, Virtual Cells, Metaheuristics.
1 INTRODUCTION In manufacturing companies characterized by a high variety and low to medium volume product mix the traditional layout approach is process oriented: machines are grouped in departments according to their functionality. In this type of systems, each part visits the different departments according to its process requirements. Process oriented manufacturing systems are very flexible and induce high equipment utilization rates. However, they are difficult to manage efficiently and can lead to long and uncertain throughput times, high machine setup and material handling costs, and high work in process inventories. An alternative to this traditional layout is the use of Cellular Manufacturing Systems (CMS) that can collect the advantages of both product and process oriented systems for a high variety and medium volume product mix [2]. The physical implementation of CMS results in a high degree of set-up efficiency [4], but the use of dedicated equipment has some drawbacks, namely high investment costs, low capacity utilization and low routing flexibility [7]. The concept of virtual cells allows maintaining high set-up efficiency without loosing the flexibility inherent to a process layout [1]. As with CMS’s, in a Virtual Cellular Manufacturing System (VCMS) machines and part families are assigned to cells, but because VCMS are based on process layouts, the cells can be reconfigured when changes in demand occur, without any physical changes of the plant layout. The advantages of this type of cells are that, by dedicating machines to part families, reduction in setup times and the simplification of material flows are accomplished and the flexibility of the process based manufacturing system is maintained. There is, however, alternative ways to implement VCMS’s as, for example, the one suggested by [8]: the use of a distributed layout as a basis to create the virtual cells. In a distributed layout, functional departments are fragmented and distributed strategically throughout the plant floor [12].
It is stated in [10] that forming virtual cells over a process layout may adversely affect the performance of the VCMS and a Simulated Annealing procedure, for developing distributed layouts based on machine capabilities, is explained. It is also presented an example to illustrate the superiority of a distributed layout as a basis for the formation of virtual cells. In view of these new developments it was considered relevant the study and development of a procedure that could help solving the Virtual Manufacturing Cell Formation Problem (VMCFP) in a distributed layout. The VMCFP can be divided into two sub problems: the part family formation and the machine cells formation. These two issues were consider within an algorithm that has been developed in previous work [9], in which the second problem was simplified since the virtual cells were formed taking into account a process configuration. If a distributed layout is used, as suggested by [8], the problem becomes much more complex and includes, not only the assigning of families to machines and the balancing of workload between machines of the same type, but also the distance traveled by each part and the flow complexity between machines assigned to the same cell. The proposed algorithm, presented in the next sections, is a Genetic Algorithm based procedure and addresses the Virtual Manufacturing Cell Formation Problem (VMCFP) in a distributed layout considering all those issues: reducing setup times by grouping parts with similar operational requirements, balancing the workload between machines of the same type, reducing the distance traveled by parts and simplifying the existing flow. 2 PROBLEM DEFINITION The VMCFP will be solved using as a starting point the procedure presented in [9]. That procedure determined the composition of the part families, so that setup times were reduced, and allocated part families to specific machines, so that workload was balanced. A relevant aspect of the problem was that multiple part families could be processed
in the same machine, subject to capacity availability. Furthermore, the procedure was then enhanced by the inclusion, in the objective function, of an improved similarity coefficient; use to evaluate the likeness, in terms of processing sequence, between parts within the several families [11]. Because the VMCFP is quite complex it is essential to assume a set of assumptions. Also, the amount of required information, in particular when a distributed layout is use, increases. It is assumed the following: • The production throughput is stable over the time frame considered. • The overall production throughput can be adequately represented by a limited set of parts produced in predetermined quantities within a given planning horizon. • The operation sequence for each part is known. • Each of the O operations (i=1,…,O) can be performed on different parts (p=1,…,P) however the setup is the same no matter in which part the operation is performed. • The production will be processed through equipment classified into a set of machine types (t=1,...,T). • Each operation can be performed on a set of functional identical machines (m=1,…,Mt). • The usage rate of each operation i performed on part p is expressed as the percentage of the capacity available on each machine of type t and is denoted by uipt. • The usage rate of each processing operation includes the time used for the corresponding setup operation. • The distributed layout is predefined and the location of each machine is known. • The coordinates of the geometric center of each machine is defined using a common referential and are denoted by xk and yk. • A part generates flow when it is transported between two machines (k and g) and that flow corresponds to its demand during the planning horizon (Dp).
Generate initial population
Current population
• The totality of available machines are classified between 1 and K, regardless of their type. 3 A GENETIC ALGORITHM FOR THE VMCFP The previously proposed heuristic [9] included two stages: (i) part family formation (f=1,…,F) and (ii) family allocation, using a greedy allocation heuristic. Although the algorithm took into account the relevant aspects in forming virtual cells it was developed considering that the configuration of the layout was process based. However, when that assumption changes and a distributed layout is used new considerations must be well thought-out. In this work a brief description of the procedure is presented highlighting the changes that were introduced due to those new considerations, namely: (i) the greedy allocation heuristic was changed (ii) the objective or fitness function has two new elements in order to reflect the distance traveled by each part and the complexity of the flow. As stated previously the procedure uses a well known metaheuristic, Genetic Algorithm (GA). GA’s are iterative search procedures, based on the biological process of natural selection and genetic inheritance. A set of individuals is created and then they suffer transformations over many generations. The idea is that the “best” characteristics are maintained and the individuals are perfected until the optimal or near-optimal individual is obtained. The GA procedure that was used is depicted in the flowchart shown in Figure 1. The steps that are signaled with grey are the ones that were altered in order to improve the algorithm and include the distributed layout configuration. The first steps in the algorithm include the encoding of the individuals used to represent a solution and the initialization of the procedure by generating an initial population.
Apply greedy allocation heuristic to individuals
Evaluate fitness value of each individual
Replace current solution Apply mutation operator
Generate future population (reproduction and crossover)
Rank individuals using tournament selection
No
Stoping criteria?
Yes Stop Figure 1: Genetic Algorithm.
As for the encoding it was maintained the same structure where each individual is represented in an array of length P (number of parts), where each element represents a part and the corresponding value represents the part family.
On the other hand, the initial population composed of P individuals, is generated randomly using the same heuristic as before.
th
19 International Conference on Production Research
For each individual of the population a solution for the part family formation problem is obtained, however it is necessary to allocate those families to the available machines. The greedy allocation heuristic starts by determining the usage rate of each family for each machine type, the families are then arranged in accordance to the total usage rate for every machine type used by each part family. Afterwards they are, by that order, allocated to the specific machines (m=1,…,Mt) of each type choosing the ones with less workload. Once again it should be noticed that several part families can be processed on an individual machine. If there is a part family that does not fit completely in a specific machine of a certain type, then it is partially allocated to the machine with less workload, then to the next machine in those conditions and so on until it is completely allocated [9]. By applying this heuristic to each individual a solution to the VCMFP is found but, in order to improve that solution, a random local search procedure is then employed. Each family uses several machines, for each family one of those machines is randomly chosen and the family is transferred from that machine to another one (of the same type) also chosen randomly between those with available capacity. By doing this the solution can improve because a better workload balance is obtained or because the distance traveled by the parts in that family is reduced or even because there is a simplification of the flow. Every individual and every solution created around that individual by using the greedy random procedure has to be evaluated in accordance to the defined fitness function. That function is quite complex since it involves several different objectives: (i) the setup efficiency, (ii) the workload balance, (iii) the transportation costs and (iv) the flow simplification. The fitness value is given by: F= a*((part similarity coefficient + family dissimilarity coefficient)/2)) + b*(workload balance coefficient) + c*(distance coefficient) + d*(flow coefficient). The parameters a, b, c and d are user defined in order to allow the decision maker the possibility of attributing different weights to different objectives. Naturally the sum of these parameters should be one, to reflect the weight of each coefficient in terms of a percentage. It is important to notice that each of those coefficients will vary between 0 and 1 which helps in the definition of unequal weights for the separate objectives. The part similarity coefficient measures the level of similarity between parts within the several families [9] and [11]. That similarity takes into account the common operations between parts and the correspondence in its processing sequences. It was however necessary to include in the first term of the fitness function a family dissimilarity coefficient in order to prevent that solutions have a large number of families with a small number of very similar elements. That coefficient helps the algorithm converge to a solution with a smaller number of families where the degree of dissimilarity between families is, hopefully, quite high. As for the second term of the equation it measures the balance between the workload of the different machines within the same machine type t. The third term (distance coefficient – DC) identifies the distance that each part travels and tries to minimize that value. It can be calculated in the following manner:
P
K
K
∑∑ ∑ z pkg ⋅ D p ⋅ d kg DC = 1 −
p =1 k =1 g =k +1
(1)
P
∑ D p ⋅ ∑ ∑ d kg p =1
k =1 g =k +1
Where dkg=|xk-xg|+|yk-yg| is the rectilinear distance between the geometric centers of machine k and machine g, and zpkg assumes the value one if part p uses machine k and machine g, and also machine k immediately precedes machine g in part p processing sequence. If not zpkg value is zero. Finally the flow coefficient (FC) is a measure of the complexity found in the movement of the parts between machines. The analysis is done in both axis (x and y), if a part changes direction in one of this axis, the objective function is penalized. By doing so solutions where parts move in a more linear manner are valued. This coefficient can be measured in the following way: P
P
P
P
∑ GM px + ∑ GM py ∑ BM px + ∑ BM py FC =
p =1
p =1
2⋅
P
∑ TM p − 1 p =1
−
p =1
p =1
2⋅
P
(2)
∑TM p − 1 p =1
TM is the total number of movements a part performs, typically is the number of processing operations minus one, unless more than one sequential operation is performed in the same machine. All those movements are divided into good (GM) and bad (BM) movements, both in the x and y axis. The first movement is not classified, the second one is classified as good if it is in the same direction (in one axis) as the first and bad if not. The third movement is compared to the second one, the fourth to the third and so on, and this is done for both axis. By observing the coefficients we can conclude that the best values for F are those nearer to 1, whereas the values closer to 0 represent solutions that are undesirable. Therefore the procedure seeks to maximize the fitness value of the individuals. As for the other aspects of the GA they were maintained, namely: the technique to select the individuals, the method for replacing the current population, the crossover and mutation operators and the stopping criteria. The tournament technique for population selection was chosen because it reflects more adequately the natural behavior of individuals, by simulating mutual ‘competition’ during random ‘encounters’ [3]. As for the replacement of the populations, basically two populations are kept in, what is called, a generational GA: the current and the future one [5]. This means that at each generation a new population is constructed and replaces the current population. The individuals of the future population are drawn from two sources: reproduction (individuals from the current population are copied into the next generation) and recombination or crossover (individuals that result from the recombination of two parents from the current population). After the new population is created and before it replaces the current one, the mutation operator is applied. The reproduction rate used is of 0.5, which means that 50% of the best individuals in the population are carried over into the next generation. The remaining individuals of the future population result from the application of the crossover operator.
A traditional partial match crossover operator was chosen [6], by which the two parents produce two children and are replaced by them. As for the mutation operator, it is applied to 50% of the individuals of the future generation, before it becomes the current one. The individuals are chosen randomly and part i (where i is a random number between 1 and P) is changed to another family f or creates a new one since f is also chosen randomly between 1 and P. Finally the GA stops when one of two conditions is met: (i) the fitness value of the best individual does not improve after P iterations, or (ii) the total number of iterations exceeds a maximum number (PxP). The use of the number of parts (P) in the stopping criteria draws from the fact that the size of the search space is directly dependent of that number. 4 CONCLUSIONS AND FUTURE DEVELOPMENTS The problem presented in this work has not, in our knowledge, been address in this manner. The combination of two fairly recent concepts, virtual cells and distributed layouts is an innovative idea and although the algorithm that was explained can be improved, it was a first attempt at solving this problem. It is also an objective of this work to bring to the attention of both researchers and practitioners this possibility as a new way to organize a production layout. The work also intends to contribute to the further advancement of the research in this area and to separate the VMCFP in a distributed layout from the considerations used to solve the manufacturing cells formation problem and the virtual cells formation problem in a process organized environment. The proposed procedure accounts for several important issues, namely: (i) setup reductions, (ii) workload balancing, (iii) the use of the equipment for more than one family, (iv) generated flow and (v) flow simplification. Also using GA, a proven efficient search technique helps to explore the solution space in order to escape local minima and obtain a good solution. The procedure is, however, a first effort of solving the VCMFP in this type of layout so there are a lot of issues to improve. Future developments of this work include: (i) the improvement of the greedy random procedure in order to create and evaluate more solutions, (ii) the measurement of the flow in terms of number of travels completed and not number of transported parts, (iii) the use of the pick up and drop off points of the machines instead of the geometric center to determine the distance between them, (iv) the improvement of the flow coefficient so it can reflect in a superior way the complexity of the existing flow, (v) the refinement of the parameters used and (vi) the test of the procedure to randomly generated problems and preferably through its application to a real life problem. 5 ACKNOWLEDGMENTS This research has been partially supported by Fundação para a Ciência e Tecnologia and Fundo Social Europeu (III Quadro Comunitário de Apoio PRAXIS XXI / BD / 1338 / 97). 6 REFERENCES [1] Drolet J.R., Moodie C.L., Montreuil B., 1989, Scheduling factories of the future, Journal of Mechanical Working Technology, 20, 183-194. [2] Burbidge J.L., 1992, Change to group technology: process organization is obsolete, International Journal of Production Research, 30 (5), 1209-1219.
[3]
Michalewicz Z., 1996, Genetic Algorithms + Data Structures = Evolution Programs, Springer-Verlag, Berlin and Heidelberg, Germany. [4] Kannan V.R., 1997, A simulation analysis of the impact of family configuration on virtual cellular manufacturing, Production Planning & Control, 8 (1), 14-24. [5] Falkenaeur E., 1999, Genetic Algorithms and Grouping Problems, John Wiley & Sons, Chichester, UK. [6] Hamamoto S., Yih Y., Salvendy G., 1999, Development and validation of genetic algorithmbased facility layout – a case study in the pharmaceutical industry, International Journal of Production Research, 37 (4), 749-768. [7] Mertins K., Friedland R., Rabe M., 2000, Capacity assignment of virtual manufacturing cells by applying lot size harmonization, International Journal of Production Research, 38 (17), 4385-4391. [8] Benjaafar S., Heragu S.S., Irani S.A., 2002, Next generation factory layouts: research challenges and recent progress, Interfaces, 32 (6), 58-76. [9] Xambre A.R., Vilarinho P.M., 2002, An algorithm for the virtual manufacturing cell formation problem (VMCFP), Proceedings of the 30th International Conference on Computers & Industrial Engineering – volume II, Tinos Island (Greece), 1005-1010. [10] Baykasoğlu A., 2003, Capability-based distributed layout approach for virtual manufacturing cells, International Journal of Production Research, 41 (11), 2597-2618. [11] Xambre A.R., Vilarinho P.M., 2003, A new similarity coefficient for the part family formation in a virtual cellular manufacturing system, Proceedings from the 8th International Conference on Industrial Engineering Theory, Applications and Practice, Las Vegas (EUA), 1122-1128. [12] Lahmar M., Benjaafar S., 2005, Design of distributed layouts, IIE Transactions, 37, 303-318.
