163 research outputs found
Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints
Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems.Mixed Integer Programming;Formulations;Symmetry;Lotsizing
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
Improved Lower Bounds For The Capacitated Lot Sizing Problem With Set Up Times
We present new lower bounds for the Capacitated Lot Sizing Problem with Set Up Times. We improve the lower bound obtained by the textbook Dantzig-Wolfe decomposition where the capacity constraints are the linking constraints. In our approach, Dantzig-Wolfe decomposition is applied to the network reformulation of the problem. The demand constraints are the linking constraints and the problem decomposes into subproblems per period containing the capacity and set up constraints. We propose a customized branch-and-bound algorithm for solving the subproblem based on its similarities with the Linear Multiple Choice Knapsack Problem. Further we present a Lagrange Relaxation algorithm for finding this lower bound. To the best of our knowledge, this is the first time that computational results are presented for this decomposition and a comparison of our lower bound to other lower bounds proposed in the literature indicates its high quality.Lagrange relaxation;Dantzig-Wolfe decomposition;capacitated lot sizing;lower bounds
Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints
Production planning on multiple parallel machines is an interesting problem, both from a theoretical and practical point of view. The parallel machine lotsizing problem consists of finding the optimal timing and level of production and the best allocation of products to machines. In this paper we look at how to incorporate parallel machines in a Mixed Integer Programming model when using commercial optimization software. More specifically, we look at the issue of symmetry. When multiple identical machines are available, many alternative optimal solutions can be created by renumbering the machines. These alternative solutions lead to difficulties in the branch-and-bound algorithm. We propose new constraints to break this symmetry. We tested our approach on the parallel machine lotsizing problem with setup costs and times, using a network reformulation for this problem. Computational tests indicate that several of the proposed symmetry breaking constraints substantially improve the solution time, except when used for solving the very easy problems. The results highlight the importance of creative modeling in solving Mixed Integer Programming problems
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 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
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
Modeling Industrial Lot Sizing Problems: A Review
In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research
Improved Lower Bounds For The Capacitated Lot Sizing Problem With Set Up Times
We present new lower bounds for the Capacitated Lot Sizing Problem with Set Up Times. We improve the lower bound obtained by the textbook Dantzig-Wolfe decomposition where the capacity constraints are the linking constraints. In our approach, Dantzig-Wolfe decomposition is applied to the network reformulation of the problem. The demand constraints are the linking constraints and the problem decomposes into subproblems per period containing the capacity and set up constraints. We propose a customized branch-and-bound algorithm for solving the subproblem based on its similarities with the Linear Multiple Choice Knapsack Problem. Further we present a Lagrange Relaxation algorithm for finding this lower bound. To the best of our knowledge, this is the first time that computational results are presented for this decomposition and a comparison of our lower bound to other lower bounds proposed in the literature indicates its high quality
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
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