238 research outputs found
Facets and Algorithms for Capacitated Lot Sizing
The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use these results to develop solution methods for multi-item applications. In particular, we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron. Computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane/branch and bound procedure. Moreover, our results show that in many instances adding certain of these inequalities a priori to the problem formulation, and avoiding the generation of cutting planes, can be equally effective
Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs
This paper presents a new class of valid inequalities for the single-item capacitated lotsizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithmis proposed for these three classes of valid inequalities. The exact separation algorithmproposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities.single-item capacitated lot sizing problem, flow cover inequalities, cutting plane algorithm
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 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
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
The multi-item capacitated lot-sizing problem with setup times and shortage costs
International audienceWe address a multi-item capacitated lot-sizing problem with setup times and shortage costs that arises in real-world production planning problems. Demand cannot be backlogged, but can be totally or partially lost. The problem is NP-hard. A mixed integer mathematical formulation is presented. Our approach in this paper is to propose some classes of valid inequalities based on a generalization of Miller et al. [A.J. Miller, G.L. Nemhauser, M.W.P. Savelsbergh, On the polyhedral structure of a multi-item production planning model with setup times, Mathematical Programming 94 (2003) 375–405] and Marchand and Wolsey [H. Marchand, L.A. Wolsey, The 0–1 knapsack problem with a single continuous variable, Mathematical Programming 85 (1999) 15–33] results. We also describe fast combinatorial separation algorithms for these new inequalities. We use them in a branch-and-cut framework to solve the problem. Some experimental results showing the effectiveness of the approach are reported
Polyhedral techniques in combinatorial optimization II: applications and computations
The polyhedral approach is one of the most powerful techniques available for solving hard combinatorial optimization problems. The main idea behind the technique is to consider the linear relaxation of the integer combinatorial optimization problem, and try to iteratively strengthen the linear formulation by adding violated strong valid inequalities, i.e., inequalities that are violated by the current fractional solution but satisfied by all feasible solutions, and that define high-dimensional faces, preferably facets, of the convex hull of feasible solutions. If we have the complete description of the convex hull of feasible solutions at hand all extreme points of this formulation are integral, which means that we can solve the problem as a linear programming problem. Linear programming problems are known to be computationally easy. In Part 1 of this article we discuss theoretical aspects of polyhedral techniques. Here we will mainly concentrate on the computational aspects. In particular we discuss how polyhedral results are used in cutting plane algorithms. We also consider a few theoretical issues not treated in Part 1, such as techniques for proving that a certain inequality is facet defining, and that a certain linear formulation gives a complete description of the convex hull of feasible solutions. We conclude the article by briefly mentioning some alternative techniques for solving combinatorial optimization problems
Polyhedral techniques in combinatorial optimization II: computations
Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience
Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation
This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;
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