Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models.single-item capacitated lot-sizing;fully polynomial approximation schemes;lot-sizing models;suboptimal algorithms

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

A heuristic approach for big bucket multi-level production planning problems

Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper, we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other heuristics, it generates high quality lower bounds using strong formulations, and its simple scheme allows it to be easily implemented in the Xpress-Mosel modeling language. Extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic, particularly for challenging problems

A computational analysis of lower bounds for big bucket production planning problems

In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research

A Tabu List-Based Algorithm for Capacitated Multilevel Lot-Sizing with Alternate Bills of Materials and Co-Production Environments

[EN] The definition of lot sizes represents one of the most important decisions in production planning. Lot-sizing turns into an increasingly complex set of decisions that requires efficient solution approaches, in response to the time-consuming exact methods (LP, MIP). This paper aims to propose a Tabu list-based algorithm (TLBA) as an alternative to the Generic Materials and Operations Planning (GMOP) model. The algorithm considers a multi-level, multi-item planning structure. It is initialized using a lot-for-lot (LxL) method and candidate solutions are evaluated through an iterative Material Requirements Planning (MRP) procedure. Three different sizes of test instances are defined and better results are obtained in the large and medium-size problems, with minimum average gaps close to 10.5%.This paper shows the results of the project entitled "Algoritmo heuristico basado en listas tabu para la planificacion de la produccion en sistemas multinivel con listas de materiales alternativas y entornos de coproduccion" supported by Universidad de la Costa and Universitat Politecnica de Valencia.Romero-Conrado, AR.; Coronado-Hernandez, J.; Rius-Sorolla, G.; García Sabater, JP. (2019). A Tabu List-Based Algorithm for Capacitated Multilevel Lot-Sizing with Alternate Bills of Materials and Co-Production Environments. Applied Sciences. 9(7):1-17. https://doi.org/10.3390/app9071464S11797Karimi, B., Fatemi Ghomi, S. M. T., & Wilson, J. M. (2003). The capacitated lot sizing problem: a review of models and algorithms. Omega, 31(5), 365-378. doi:10.1016/s0305-0483(03)00059-8Martí, R., & Reinelt, G. (2010). Heuristic Methods. 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Journal of the Operational Research Society, 51(11), 1309-1318. doi:10.1057/palgrave.jors.2601026Kang, Y., Albey, E., & Uzsoy, R. (2018). Rounding heuristics for multiple product dynamic lot-sizing in the presence of queueing behavior. Computers & Operations Research, 100, 54-65. doi:10.1016/j.cor.2018.07.019BERRETTA, R., FRANÇA, P. M., & ARMENTANO, V. A. (2005). METAHEURISTIC APPROACHES FOR THE MULTILEVEL RESOURCE-CONSTRAINED LOT-SIZING PROBLEM WITH SETUP AND LEAD TIMES. Asia-Pacific Journal of Operational Research, 22(02), 261-286. doi:10.1142/s0217595905000510KIMMS, A. (1996). Competitive methods for multi-level lot sizing and scheduling: tabu search and randomized regrets. International Journal of Production Research, 34(8), 2279-2298. doi:10.1080/00207549608905025Sabater, J. P. G., Maheut, J., & Garcia, J. A. M. (2013). A new formulation technique to model materials and operations planning: the generic materials and operations planning (GMOP) problem. European J. of Industrial Engineering, 7(2), 119. doi:10.1504/ejie.2013.052572Maheut, J., & Sabater, J. P. G. (2013). Algorithm for complete enumeration based on a stroke graph to solve the supply network configuration and operations scheduling problem. Journal of Industrial Engineering and Management, 6(3). doi:10.3926/jiem.550Rius-Sorolla, G., Maheut, J., Coronado-Hernandez, J. R., & Garcia-Sabater, J. P. (2018). Lagrangian relaxation of the generic materials and operations planning model. Central European Journal of Operations Research, 28(1), 105-123. doi:10.1007/s10100-018-0593-0Maheut, J., Garcia-Sabater, J. P., & Mula, J. (2012). The Generic Materials and Operations Planning (GMOP) Problem Solved Iteratively: A Case Study in Multi-site Context. IFIP Advances in Information and Communication Technology, 66-73. doi:10.1007/978-3-642-33980-6_8Maheut, J. P. D. (s. f.). Modelos y Algoritmos Basados en el Concepto Stroke para la Planificación y Programación de Operaciones con Alternativas en Redes de Suministro. doi:10.4995/thesis/10251/29290Glover, F. (1989). Tabu Search—Part I. ORSA Journal on Computing, 1(3), 190-206. doi:10.1287/ijoc.1.3.190Glover, F., Taillard, E., & Taillard, E. (1993). A user’s guide to tabu search. Annals of Operations Research, 41(1), 1-28. doi:10.1007/bf02078647Chelouah, R., & Siarry, P. (2000). Tabu Search applied to global optimization. European Journal of Operational Research, 123(2), 256-270. doi:10.1016/s0377-2217(99)00255-6Raza, S. A., Akgunduz, A., & Chen, M. Y. (2006). A tabu search algorithm for solving economic lot scheduling problem. Journal of Heuristics, 12(6), 413-426. doi:10.1007/s10732-006-6017-7Cesaret, B., Oğuz, C., & Sibel Salman, F. (2012). A tabu search algorithm for order acceptance and scheduling. Computers & Operations Research, 39(6), 1197-1205. doi:10.1016/j.cor.2010.09.018Li, X., Baki, F., Tian, P., & Chaouch, B. A. (2014). A robust block-chain based tabu search algorithm for the dynamic lot sizing problem with product returns and remanufacturing. Omega, 42(1), 75-87. doi:10.1016/j.omega.2013.03.003Li, J., & Pan, Q. (2015). Solving the large-scale hybrid flow shop scheduling problem with limited buffers by a hybrid artificial bee colony algorithm. Information Sciences, 316, 487-502. doi:10.1016/j.ins.2014.10.009Hindi, K. S. (1995). Solving the single-item, capacitated dynamic lot-sizing problem with startup and reservation costs by tabu search. Computers & Industrial Engineering, 28(4), 701-707. doi:10.1016/0360-8352(95)00027-xHindi, K. S. (1996). Solving the CLSP by a Tabu Search Heuristic. Journal of the Operational Research Society, 47(1), 151-161. doi:10.1057/jors.1996.13Gopalakrishnan, M., Ding, K., Bourjolly, J.-M., & Mohan, S. (2001). 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Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times

The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

On alternative mixed integer programming formulations and LP-based heuristics for lot-sizing with setup times

We address the multi-item, capacitated lot-sizing problem (CLSP) encountered in environments where demand is dynamic and to be met on time. Items compete for a limited capacity resource, which requires a setup for each lot of items to be produced causing unproductive time but no direct costs. The problem belongs to a class of problems that are difcult to solve. Even the feasibility problem becomes combinatorial when setup times are considered. This difculty in reaching optimality and the practical relevance of CLSP make it important to design and analyse heuristics to nd good solutions that can be implemented in practice. We consider certain mixed integer programming formulations of the problem and develop heuristics including a curtailed branch and bound, for rounding the setup variables in the LP solution of the tighter formulations. We report our computational results for a class of instances taken from literature