R.F. Jans (Raf)
http://repub.eur.nl/ppl/3219/
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RePub, Erasmus University RepositoryEconomic lot-sizing with remanufacturing: Complexity and efficient formulations
http://repub.eur.nl/pub/76484/
Thu, 02 Jan 2014 00:00:01 GMT<div>M.R. Helmrich</div><div>R.F. Jans</div><div>W. van den Heuvel</div><div>A.P.M. Wagelmans</div>
Within the framework of reverse logistics, the classic economic lot-sizing problem has been extended with a remanufacturing option. In this extended problem, known quantities of used products are returned from customers in each period. These returned products can be remanufactured so that they are as good as new. Customer demand can then be fulfilled from both newly produced and remanufactured items. In each period, one can choose to set up a process to remanufacture returned products or produce new items. These processes can have separate or joint setup costs. In this article, it is shown that both variants are NP-hard. Furthermore, several alternative mixed-integer programming (MIP) formulations of both problems are proposed and compared. Because "natural" lot-sizing formulations provide weak lower bounds, tighter formulations are proposed, namely, shortest path formulations, a partial shortest path formulation, and an adaptation of the (l, S,WW) inequalities used in the classic problem with Wagner-Whitin costs. Their efficiency is tested on a large number of test data sets and it is found that, for both problem variants, a (partial) shortest path-type formulation performs better than the natural formulation, in terms of both the linear programming relaxation and MIP computation times. Moreover, this improvement can be substantial. CopyrightThe Economic Lot-Sizing Problem with an Emission Constraint
http://repub.eur.nl/pub/37650/
Thu, 03 May 2012 00:00:01 GMT<div>M. Retel Helmrich</div><div>R.F. Jans</div><div>W. van den Heuvel</div><div>A.P.M. Wagelmans</div>
We consider a generalisation of the lot-sizing problem that includes an emission constraint. Besides the usual financial costs, there are emissions associated with production, keeping inventory and setting up the production process. Because the constraint on the emissions can be seen as a constraint on an alternative cost function, there is also a clear link with bi-objective optimisation. We show that lot-sizing with an emission constraint is NP-hard and propose several solution methods. First, we present a Lagrangian heuristic to provide a feasible solution and lower bound for the problem. For costs and emissions for which the zero inventory property is satisfied, we give a pseudo-polynomial algorithm, which can also be used to identify the complete Pareto frontier of the bi-objective lot-sizing problem. Furthermore, we present a fully polynomial time approximation scheme (FPTAS) for such costs and emissions and extend it to deal with general costs and emissions. Special attention is paid to an efficient implementation with an improved rounding technique to reduce the a posteriori gap, and a combination of the FPTASes and a heuristic lower bound. Extensive computational tests show that the Lagrangian heuristic gives solutions that are very close to the optimum. Moreover, the FPTASes have a much better performance in terms of their gap than the a priori imposed performance, and, especially if the heuristic’s lower bound is used, they are very fast.Economic lot-sizing with remanufacturing: complexity and efficient formulations
http://repub.eur.nl/pub/21936/
Wed, 22 Dec 2010 00:00:01 GMT<div>M. Retel Helmrich</div><div>R.F. Jans</div><div>W.J. van den Heuvel</div><div>A.P.M. Wagelmans</div>
Within the framework of reverse logistics, the classic economic lot-sizing problem has been extended with a remanufacturing option. In this extended problem, known quantities of used products are returned from customers in each period. These returned products can be remanufactured, so that they are as good as new. Customer demand can then be fulfilled both from newly produced and remanufactured items. In each period, we can choose to set up a process to remanufacture returned products or produce new items. These processes can have separate or joint set-up costs. In this paper, we show that both variants are NP-hard. Furthermore, we propose and compare several alternative MIP formulations of both problems. Because ‘natural’ lot-sizing formulations provide weak lower bounds, we propose tighter formulations, namely shortest path formulations, a partial shortest path formulation and an adaptation of the (l, S, WW)-inequalities for the classic problem with Wagner-Whitin costs. We test their efficiency on a large number of test data sets and find that, for both problem variants, a (partial) shortest path type formulation performs better than the natural formulation, in terms of both the LP relaxation and MIP computation times. Moreover, this improvement can be substantial.Analysis of an Industrial Component Commonality Problem
http://repub.eur.nl/pub/13588/
Wed, 16 Apr 2008 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div><div>L. Schepens</div>
We discuss a case study of an industrial production-marketing coordination problem involving component commonality. For the product line considered, the strategic goal of the company is to move from the current low volume market to a high volume market. The marketing department believes that this can be achieved by substantially lowering the end products’ prices. However, this requires a product redesign to lower production costs in order to maintain profit margins. The redesign decision involves grouping end products into families. All products within one family use the same version of some components. This paper fits in the stream of recent literature on component commonality where the focus has shifted from inventory cost savings to production and development cost savings. Further, we consider both costs and revenues, leading to a profit maximization approach. The price elasticity of demand determines the relationship between the price level and number of units sold. Consequently, we integrate information from different functional areas such as production, marketing and accounting. We formulate the problem as a net-present-value investment decision. We propose a mixed integer nonlinear optimization model to find the optimal commonality decision. The recommendation based on our analysis has been implemented in the company. In addition, the application allows us to experimentally validate some claims made in the literature and obtain managerial insights into the trade-offs.A note on a symmetrical set covering problem: The lottery problem
http://repub.eur.nl/pub/13587/
Tue, 01 Apr 2008 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
In a lottery, n numbers are drawn from a set of m numbers. On a lottery ticket we fill out n numbers. Consider the following problem: what is the minimum number of tickets so that there is at least one ticket with at least p matching numbers? We provide a set-covering formulation for this problem and characterize its LP solution. The existence of many symmetrical alternative solutions, makes this a very difficult problem to solve, as our computational results indicate.Modeling Industrial Lot Sizing Problems: A Review
http://repub.eur.nl/pub/13589/
Sat, 01 Mar 2008 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
In this paper we give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems. The focus of this paper is on the modeling of 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.A new Dantzig-Wolfe reformulation and branch-and-price algorithm for the capacitated lot-sizing problem with setup times
http://repub.eur.nl/pub/75191/
Sat, 01 Sep 2007 00:00:01 GMT<div>Z. Degraeve</div><div>R.F. Jans</div>
Although the textbook Dantzig-Wolfe decomposition reformulation for the capacitated lot-sizing problem, as already proposed by Manne [Manne, A. S. 1958. Programming of economic lot sizes. Management Sci. 4(2) 115-135], provides a strong lower bound, it also has an important structural deficiency. Imposing integrality constraints on the columns in the master program will not necessarily give the optimal integer programming solution. Manne's model contains only production plans that satisfy the Wagner-Whitin property, and it is well known that the optimal solution to a capacitated lot-sizing problem will not necessarily satisfy this property. The first contribution of this paper answers the following question, unsolved for almost 50 years: If Manne's formulation is not equivalent to the original problem, what is then a correct reformulation? We develop an equivalent mixed-integer programming (MIP) formulation to the original problem and show how this results from applying the Dantzig-Wolfe decomposition to the original MIP formulation. The set of extreme points of the lot-size polytope that are needed for this MIP Dantzig-Wolfe reformulation is much larger than the set of dominant plans used by Manne. We further show how the integrality restrictions on the original setup variables translate into integrality restrictions on the new master variables by separating the setup and production decisions. Our new formulation gives the same lower bound as Manne's reformulation. Second, we develop a branch-and-price algorithm for the problem. Computational experiments are presented on data sets available from the literature. Column generation is accelerated by a combination of simplex and subgradient optimization for finding the dual prices. The results show that branch-and-price is computationally tractable and competitive with other state-of-the-art approaches found in the literature.Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches
http://repub.eur.nl/pub/66124/
Fri, 16 Mar 2007 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
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 combinatorial 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.Solving Lotsizing Problems on Parallel Identical Machines Using Symmetry Breaking Constraints
http://repub.eur.nl/pub/7985/
Wed, 20 Sep 2006 00:00:01 GMT<div>R.F. Jans</div>
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.Modeling Industrial Lot Sizing Problems: A Review
http://repub.eur.nl/pub/6912/
Fri, 09 Sep 2005 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
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.Meta-Heuristics for Dynamic Lot Sizing: a review and comparison of solution approaches
http://repub.eur.nl/pub/1336/
Thu, 24 Jun 2004 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
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.Improved lower bounds for the capacitated lot sizing problem with setup times
http://repub.eur.nl/pub/76437/
Mon, 01 Mar 2004 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
We present new lower bounds for the capacitated lot sizing problem, applying decomposition to the network reformulation. The demand constraints are the linking constraints and the problem decomposes into subproblems per period containing the capacity and setup constraints. Computational results and a comparison to other lower bounds are presented.An industrial extension of the discrete lot-sizing and scheduling problem
http://repub.eur.nl/pub/76322/
Thu, 01 Jan 2004 00:00:01 GMT<div>R.F. Jans</div><div>Z. Degraeve</div>
Improved Lower Bounds For The Capacitated Lot Sizing Problem With Set Up Times
http://repub.eur.nl/pub/326/
Tue, 06 May 2003 00:00:01 GMT<div>Z. Degraeve</div><div>R.F. Jans</div>
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.A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times
http://repub.eur.nl/pub/275/
Tue, 04 Mar 2003 00:00:01 GMT<div>Z. Degraeve</div><div>R.F. Jans</div>
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.Combining Column Generation and Lagrangian Relaxation
http://repub.eur.nl/pub/1098/
Wed, 01 Jan 2003 00:00:01 GMT<div>D. Huisman</div><div>R.F. Jans</div><div>M. Peeters</div><div>A.P.M. Wagelmans</div>
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.