Single item lot-sizing with non-decreasing capacities

We consider the single item lot-sizing problem with capacities that are non-decreasing over time. When the cost function is i) non-speculative or Wagner-Whitin (for instance, constant unit production costs and non-negative unit holding costs), and ii) the production set-up costs are non-increasing over time, it is known that the minimum cost lot-sizing problem is polynomially solvable using dynamic programming. When the capacities are non-decreasing, we derive a compact mixed integer programming reformulation whose linear programming relaxation solves the lot-sizing problem to optimality when the objective function satisfies i) and ii). The formulation is based on mixing set relaxations and reduces to the (known) convex hull of solutions when the capacities are constant over time. We illustrate the use and effectiveness of this improved LP formulation on a new test instances, including instances with and without Wagner-Whitin costs, and with both non-decreasing and arbitrary capacities over time.lot-sizing, mixing set relaxation, compact reformulation, production planning, mixed integer programming

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

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 Lot Sizing Problem (CLSP),as already proposed by Manne in 1958, has an important structural deficiency. Imposingintegrality constraints on the variables in the full blown master will not necessarily give theoptimal IP solution as only production plans which satisfy the Wagner-Whitin condition canbe selected. It is well known that the optimal solution to a capacitated lot sizing problem willnot necessarily have this Wagner-Whitin property. The columns of the traditionaldecomposition model include both the integer set up and continuous production quantitydecisions. Choosing a specific set up schedule implies also taking the associated Wagner-Whitin production quantities. We propose the correct Dantzig-Wolfe decompositionreformulation separating the set up and production decisions. This formulation gives the samelower bound as Manne's reformulation and allows for branch-and-price. We use theCapacitated Lot Sizing Problem with Set Up Times to illustrate our approach. Computationalexperiments are presented on data sets available from the literature. Column generation isspeeded up by a combination of simplex and subgradient optimization for finding the dualprices. The results show that branch-and-price is computationally tractable and competitivewith other approaches. Finally, we briefly discuss how this new Dantzig-Wolfe reformulationcan be generalized to other mixed integer programming problems, whereas in the literature,branch-and-price algorithms are almost exclusively developed for pure integer programmingproblems

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