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

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

Meta-Heuristics for Dynamic Lot Sizing: a review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinational optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig-Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples

A technique for speeding up the solution of the Lagrangean dual

"April 10, 1991."Includes bibliographical references (p. 26-29).Research supported by the National Science Foundation. DDM-9010332 DDM-9014751 DDM-8921835 Research supported by the Air Force Office of Scientific Research. AFOSR-88-0088 Research supported by grants from UPS, Prime Computer Corp. and Draper Laboratories.Dimitris Bertsimas, James B. Orlin

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

Combining Column Generation and Lagrangian Relaxation

Although the possibility to combine column generation and Lagrangian relaxation has been known for quite some time, it has only recently been exploited in algorithms. In this paper, we discuss ways of combining these techniques. We focus on solving the LP relaxation of the Dantzig-Wolfe master problem. In a first approach we apply Lagrangian relaxation directly to this extended formulation, i.e. no simplex method is used. In a second one, we use Lagrangian relaxation to generate new columns, that is Lagrangian relaxation is applied to the compact for-mulation. We will illustrate the ideas behind these algorithms with an application in Lot-sizing. To show the wide applicability of these techniques, we also discuss applications in integrated vehicle and crew scheduling, plant location and cutting stock problems.column generation;Lagrangean relaxation;cutting stock problem;lotsizing;vehicle and crew scheduling

A technique for speeding up the solution of the Lagrangean dual

"April 10, 1991."Includes bibliographical references (p. 26-29).Research supported by the National Science Foundation. DDM-9010332 DDM-9014751 DDM-8921835 Research supported by the Air Force Office of Scientific Research. AFOSR-88-0088 Research supported by grants from UPS, Prime Computer Corp. and Draper Laboratories.Dimitris Bertsimas, James B. Orlin

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

On the single level capacitated lot sizing problem.

Yip Ka-yun.Thesis (M.Phil.)--Chinese University of Hong Kong, 1998.Includes bibliographical references (leaves 107-113).Abstract also in Chinese.Chapter Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Overview --- p.1Chapter 1.2 --- Our Contributions --- p.2Chapter 1.3 --- Organization of the Thesis --- p.4Chapter Chapter 2 --- Literature Review --- p.5Chapter 2.1 --- Overview --- p.5Chapter 2.2 --- Research in Capacitated Lot Sizing Problem without significant setup times --- p.5Chapter 2.3 --- Research in Capacitated Lot Sizing Problem with setup time consideration --- p.12Chapter 2.4 --- Summary --- p.15Chapter Chapter 3 --- Capacitated Lot Sizing Problem with Setup Times --- p.16Chapter 3.1 --- Overview --- p.16Chapter 3.2 --- Problem Description and Formulation --- p.20Chapter 3.2.1 --- Our problem formulationChapter 3.2.2 --- Comparison between our problem formulation and traditional problem formulationChapter 3.3 --- Description of the Algorithm --- p.26Chapter 3.3.1 --- Wagner-Whitin algorithmChapter 3.3.2 --- Transportation problemChapter 3.3.3 --- Consistence testChapter 3.3.4 --- Subgradient optimizationChapter 3.3.5 --- Computation of lower boundChapter 3.4 --- Design of Experiment --- p.43Chapter 3.4.1 --- Product demandsChapter 3.4.2 --- Setup costsChapter 3.4.3 --- Setup timesChapter 3.4.4 --- Capacity costsChapter 3.4.5 --- Inventory holding costsChapter 3.4.6 --- Quantity of capacity available for productionChapter 3.4.7 --- Capacity absorption rateChapter 3.4.8 --- Generation of larger problemsChapter 3.4.9 --- Initialization of Lagrangean multipliersChapter 3.4.10 --- Close testChapter 3.5 --- Open test --- p.58Chapter 3.6 --- Managerial Implications --- p.61Chapter 3.7 --- Summary --- p.61Chapter Chapter 4 --- Capacitated Lot Sizing Problem without Setup Times --- p.63Chapter 4.1 --- Overview --- p.63Chapter 4.2 --- Problem Description and Formulation --- p.64Chapter 4.3 --- Description of the Algorithm --- p.67Chapter 4.3.1 --- Decomposition schemeChapter 4.3.2 --- Wagner-Whitin algorithmChapter 4.3.3 --- Transportation problemChapter 4.3.4 --- Subgradient optimizationChapter 4.3.5 --- Computation of lower boundChapter 4.4 --- Design of Experiment --- p.80Chapter 4.4.1 --- Product demandsChapter 4.4.2 --- Setup costsChapter 4.4.3 --- Capacity costsChapter 4.4.4 --- Inventory holding costsChapter 4.4.5 --- Quantity of capacity available for productionChapter 4.4.6 --- Capacity absorption rateChapter 4.4.7 --- Generation of larger problemsChapter 4.4.8 --- Initialization of Lagrangean multipliersChapter 4.4.9 --- Selection of the extent of geometrical reduction and exponential smoothingChapter 4.4.10 --- Close testChapter 4.5 --- Open test --- p.92Chapter 4.6 --- Managerial Implications --- p.95Chapter 4.7 --- Comparison with other approaches --- p.96Chapter 4.7.1 --- Gilbert and Madan's approachChapter 4.7.2 --- Our algorithm for CLS problem with setup time considerationChapter 4.8 --- Summary --- p.102Chapter Chapter 5 --- Conclusion --- p.104Appendix A Vogel's approximation method --- p.106Bibliography --- p.10