Algoritmos de partição e geração de colunas para dimensionamento de lotes de produção

 

Carina Maria Oliveira Pimentel ^†

Filipe Pereira e Alvelos ^†‡

José Manuel Valério de Carvalho ^†‡

 

^† Centro Algoritmi, Universidade do Minho

carina@dps.uminho.pt

falvelos@dps.uminho.pt

vc@dps.uminho.pt

^‡ Departamento de Produção e Sistemas, Universidade do Minho

 

 

 

Title: Branch and Price Algorithms for Production Lot Sizing

 

Abstract

In this paper, we present two algorithms for the multi-item capacitated lot-sizing problem with setup times. In this problem we aim at finding a production plan for several items over a number of time periods that minimizes the production, inventory and setup costs and satisfies all demand requirements without exceeding capacity limits.

Both of the algorithms are based on the application of the Dantzig-Wolfe principle to a classical model of Mixed Integer Programming. In one case, we apply item decomposition and in the other case we apply a period decomposition. In both cases the reformulated models are stronger than the original one. These reformulated models are solved by branch-and-price, which is a combination of column generation and branch-and-bound methods.

We present computational results for a set of instances with different characteristics, to establish comparisons between the two decomposition models. These results are then compared with the classic Mixed Integer Programming formulation solved by the commercial solver Cplex 8.1.

Keywords: Production Planning, Mixed Integer Programming, Branch and Price

 

 

Resumo

Neste artigo apresentam-se dois algoritmos para o problema de lotes de produção multi-artigo capacitado com tempos de preparação. Neste problema pretende-se determinar um plano de produção para vários artigos ao longo de um determinado horizonte temporal, que minimize os custos de produção, de armazenagem e de preparação e respeite restrições de procura e de capacidade.

Os algoritmos baseiam-se na aplicação do princípio de decomposição de Dantzig-Wolfe a um modelo clássico de Programação Inteira Mista. Num dos casos é efectuada uma decomposição por artigo e no outro uma decomposição por período. Em ambos os casos os modelos reformulados são mais fortes do que o modelo original. Os modelos reformulados são resolvidos através do método de partição e geração de colunas, que resulta da combinação do método de geração de colunas com o método de partição e avaliação ("branch-and-bound").

São apresentados resultados de testes computacionais para um conjunto de instâncias com diferentes características, que permitem estabelecer comparações entre os dois modelos de decomposição. Esses resultados computacionais são ainda comparados com a formulação clássica resolvida através do Cplex 8.1, um software comercial para Programação Inteira Mista.

 

Texto completo apenas disponível em PDF.

Full text only in PDF.

 

 

Referências

 

Alvelos, F. (2005) Branch-and-Price and Multicommodity Flows, PhD Thesis, Universidade do Minho.        [ Links ]

Barany, I., Roy, T. J. V. and Wolsey, L. A. (1984) Strong Formulations for Multi-Item Capacitated Lot Sizing, Management Science, Vol. 30, No. 10, pp.1255-1261.

Barnhart, C., Johnson, E. L., Nemhauser, G. L., Savelsbergh, M. W. P. and Vance, P. H. (1998) Branch-and-Price: Column Generation for Solving Huge Integer Programs, Operations Research, Vol. 46, No. 3, pp.316-329.

Belvaux, G. and Wolsey, L. A. (2000) bc-prod: A Specialized Branch-and-Cut System for Lot-Sizing Problems, Management Science, Vol. 46, pp.724-738.

Chen, W. H. and Thizy, J. M. (1990) Analysis of Relaxations for the Multi-Item Capacitated Lot-Sizing Problem, Annals of Operations Research, Vol. 26, pp.29-72.

Dantzig, G. B. and Wolfe, P. (1960) Decomposition Principle for Linear Programs, Operations Research, Vol. 8, pp.101-111.

Diaby, M., Bahl, H. C., Karwan, M. H. and Zionts, S. (1992a) Capacitated Lot-Sizing and Scheduling by Lagrangean Relaxation, European Journal of Operational Research, Vol. 59, pp.444-458.

Diaby, M., Bahl, H. C., Karwan, M. H. and Zionts, S. (1992b) A Lagrangean Relaxation Approach for Very-Large-Scale Capacitated Lot-Sizing, Management Science, Vol. 38, No. 9, pp.1329-1340.

Drexl, A. and Kimms, A. (1997) Lot Sizing and Scheduling - Survey and Extensions, European Journal of Operational Research, Vol. 99, pp.221-235.

Dzielinski, B. P. and Gomory, R. E. (1965) Optimal Programming of Lot Sizes, Inventory and Labor Allocations, Management Science, Series A, Vol. 11, No. 9, pp.874-890.

Eppen, G. D. and Martin, R. K. (1987) Solving Multi-Item Capacitated Lot-Sizing Problems using Variable Redefinition, Operations Research, Vol. 35, No. 6, pp.832-848.

Geoffrion, A. M. (1974) Lagrangean Relaxation for Integer Programming, Mathematical Programming Study, Vol. 2, pp.82-114.

ILOG (2002) CPLEX 8.1, User's Manual.

Jans, R. (2002) Capacitated Lot Sizing Problems: New Applications, Formulations and Algorithms, PhD Thesis, Katholieke Universiteit.

Karimi, B., Ghomi, S. and Wilson, J. (2003) The Capacitated Lot Sizing Problem: a Review of Models and Algorithms, Omega, Vol. 31, pp.365-378.

Kuik, R., Salomon, M. and Wassenhove, L. v. (1994) Batching Decisions: Structure and Models, European Journal of Operational Research, Vol. 75, pp.243-263.

Lasdon, L. S. and Terjung, R. C. (1971) An Efficient Algorithm for Multi-Item Scheduling, Operations Research, Vol. 19, No. 4, pp.946-969.

Manne, A. S. (1958) Programming of Economic Lot Sizes, Management Science, Vol. 4, No. 2, pp.115-135.

Merle, O. d., Goffin, J.-L., Trouiller, C. and Vial, J.-P. (1999) A Lagrangian Relaxation of the Capacitated Multi-Item Lot Sizing Problem Solved with an Interior Point Cutting Plane Algorithm, Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems, Pardalos, P. M. (eds.), Kluwer Academic Publishers.

Miller, A. J., Nemhauser, G. L. and Savelsbergh, M. W. P. (2000) Solving Multi-Item Capacitated Lot-Sizing Problems with Setup Times by Branch-and-Cut, Technical report CORE Discussion Paper 2000/39.

Perrot, N. and Vanderbeck, F. (2004) Knapsack Problems with Setups, Technical report U-04.03.

Pochet, Y. (2001) Mathematical Programming Models and Formulations for Deterministic Production Planning Problems, Computational Combinatorial Optimization, Jünger, M. and D. Naddef (eds.), Springer-Verlag.

Pochet, Y. and Wolsey, L. A. (1991) Solving Multi-Item Lot-Sizing Problems using Strong Cutting Planes, Management Science, Vol. 37, pp.53-67.

Thizy, J. M. and Wassenhove, L. N. V. (1985) Lagrangean Relaxation for the Multi-Item Capacitated Lot Sizing Problem: a Heuristic Implementation, IIE Transactions, Vol. 17, pp.308-313.

Trigeiro, W. W., Thomas, L. J. and McClain, J. O. (1989) Capacitated Lot Sizing with Setup Times, Management Science, Vol. 35, No. 3, pp.353-366.

Wagner, H. and Whitin, T. (1958) Dynamic Version of the Economic Lot Size Model, Management Science, Vol. 5, No. 1, pp. 89-96.