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Journal ArticleDOI

Capacitated lot sizing with setup times

01 Mar 1989-Management Science (INFORMS)-Vol. 35, Iss: 3, pp 353-366
TL;DR: In this paper, a Lagrangian relaxation of the capacity constraints of CLSP allows it to be decomposed into a set of uncapacitated single product lot sizing problems, which are solved by dynamic programming.
Abstract: This research focuses on the effect of setup time on lot sizing. The setting is the Capacitated Lot Sizing Problem (the single-machine lot sizing problem) with nonstationary costs, demands, and setup times. A Lagrangian relaxation of the capacity constraints of CLSP allows it to be decomposed into a set of uncapacitated single product lot sizing problems. The Lagrangian dual costs are updated by subgradient optimization, and the single-item problems are solved by dynamic programming. A heuristic smoothing procedure constructs feasible solutions (production plans) which do not require overtime. The algorithm solves problems with setup time or setup cost. Problems with extremely tightly binding capacity constraints were much more difficult to solve than anticipated. Solutions without overtime could not always be found for them. The most significant results are that (1) the tightness of the capacity constraint is a good indicator of problem difficulty for problems with setup time; and (2) the algorithm solve...
Citations
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Journal ArticleDOI
TL;DR: This chapter presents the basic schemes of VNS and some of its extensions, and presents five families of applications in which VNS has proven to be very successful.

3,572 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider single-level lot sizing problems, their variants and solution approaches, together with exact and heuristic approaches for their solution, and conclude with some suggestions for future research.
Abstract: Lot sizing is one of the most important and also one of the most difficult problems in production planning. This subject has been studied extensively in the literature. In this article, we consider single-level lot sizing problems, their variants and solution approaches. After introducing factors affecting formulation and the complexity of production planning problems, and introducing different variants of lot sizing and scheduling problems, we discuss single-level lot sizing problems, together with exact and heuristic approaches for their solution. We conclude with some suggestions for future research.

670 citations


Cites background or methods from "Capacitated lot sizing with setup t..."

  • ...[53] remark that the set-partitioning approaches tend to under account for the setup costs and times, because these are charged only once when a lot is split....

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  • ...The LP relaxation of this problem Table 3 Summary of mathematical based heuristics Algorithm Methods used in algorithm Newson [48] Lagrangian relaxation, shortest path, Wagner–Whitin Thizy and Van Wassenhove [50] Lagrangian relaxation, transportation, Wagner–Whitin, subgradient optimisation Billington et al. [49] Lagrangian relaxation, transportation, Wagner–Whitin, subgradient optimisation Trigeiro [52] Lagrangian relaxation, Wagner–Whitin, subgradient optimisation, smoothing Trigeiro et al. [53] Lagrangian relaxation, transportation, Wagner–Whitin, subgradient optimisation Diaby et al. [58] Lagrangian relaxation, transportation, Wagner–Whitin, subgradient optimisation Bitran and Matsuo [51] Various relaxations Millar and Yang [56] Network formulation, Lagrangian relaxation, Lagrangian decomposition Chen and Thizy [31] Relaxation methods Thizy [54] Lagrangian decomposition Gelders et al. [57] Branch and bound, Lagrangian relaxation Diaby et al. [58] Branch and bound, Lagrangian relaxation, subgradient optimisation Hindi [61] Branch and bound, shortest path, column generation, minimum cost network Tow Armentano et al. [62] Branch and bound, minimum cost network Tow Chung et al. [59] Branch and bound, dynamic programming LotD and Yoon [60] Branch and bound Manne [63] Set partitioning, linear programming Cattrysse et al. [64] Set partitioning, column generation Lozano et al. [65] Primal–dual, relaxation Hindi [66] Tabu search, transshipment Hindi [67] Shortest path, column generation, minimum cost network Tow, tabu search Hung and Hu [68] Shadow prices is then solved by column generation....

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  • ...Trigeiro [47] developed a heuristic algorithm for CLSP with setup times....

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  • ...Trigeiro used a feedback mechanism for ensuring feasibility....

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  • ...The problems solved are from various families including big-bucket and small-bucket variants of sizes up Table 1 Summary of CLSP algorithms Barany et al. [33] Exact methods Methods for solving CLSP Eppen and Martin [34] Leung et al. [35] Belvaux and Wolsey [36] Belvaux and Wolsey [37] Fatemi Ghomi and Hashemin [38] Eisenhut [39] Period-by-period Common-sense Lambrecht and Vanderveken [40] heuristics or Dixon and Silver [41] specialised Maes and Van Wassenhove [42] heuristics Kirca and Kokten [15] Dogramaci et al. [43] Improvement Karni and Roll [44] heuristics Gunther [45] Selen and Heuts [46] Trigeiro [47] Newson [48] Relaxation Mathematical Billington et al. [49] heuristics Programming Thizy and Van Wassenhove [50] Based Bitran and Matsuo [51] Heuristics Trigeiro [52] Trigeiro et al. [53] Chen and Thizy [31] Thizy [54] Diaby et al. [55] Millar and Yang [56] Gelders et al. [57] Branch-and-bound Diaby et al. [58] heuristics Chung et al. [59] LotD and Yoon [60] Hindi [61] Armentano et al. [62] Manne [63] Set partitioning and Cattrysse et al. [64] column generation heuristics Lozano et al. [65] Other heuristics Hindi [66] Hindi [67] Hung and Hu [68] to 10 machines, up to 20 items and up to three periods....

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Journal ArticleDOI
TL;DR: Simpson and Vakharia as discussed by the authors proposed a method to identify the most important parts of the human brain in order to determine whether a person's brain is composed of neurons or not.

556 citations

Journal ArticleDOI
TL;DR: In this article, the authors give an overview of recent developments in the field of modeling deterministic single-level dynamic lot sizing problems, focusing on the modeling of various industrial extensions and not on the solution approaches.
Abstract: 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.

350 citations

References
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Book
01 Jan 1979
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Abstract: This is the second edition of a quarterly column the purpose of which is to provide a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’’ W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed. Readers having results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this journal.

40,020 citations

Journal ArticleDOI
TL;DR: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations.
Abstract: A technique is presented for the decomposition of a linear program that permits the problem to be solved by alternate solutions of linear sub-programs representing its several parts and a coordinating program that is obtained from the parts by linear transformations. The coordinating program generates at each cycle new objective forms for each part, and each part generates in turn from its optimal basic feasible solutions new activities columns for the interconnecting program. Viewed as an instance of a “generalized programming problem” whose columns are drawn freely from given convex sets, such a problem can be studied by an appropriate generalization of the duality theorem for linear programming, which permits a sharp distinction to be made between those constraints that pertain only to a part of the problem and those that connect its parts. This leads to a generalization of the Simplex Algorithm, for which the decomposition procedure becomes a special case. Besides holding promise for the efficient computation of large-scale systems, the principle yields a certain rationale for the “decentralized decision process” in the theory of the firm. Formally the prices generated by the coordinating program cause the manager of each part to look for a “pure” sub-program analogue of pure strategy in game theory, which he proposes to the coordinator as best he can do. The coordinator finds the optimum “mix” of pure sub-programs using new proposals and earlier ones consistent with over-all demands and supply, and thereby generates new prices that again generates new proposals by each of the parts, etc. The iterative process is finite.

2,281 citations

Journal ArticleDOI
TL;DR: Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods and desire a minimum total cost inventory management scheme which satisfies known demand in every period.
Abstract: (This article originally appeared in Management Science, October 1958, Volume 5, Number 1, pp. 89-96, published by The Institute of Management Sciences.) A forward algorithm for a solution to the following dynamic version of the economic lot size model is given: allowing the possibility of demands for a single item, inventory holding charges, and setup costs to vary over N periods, we desire a minimum total cost inventory management scheme which satisfies known demand in every period. Disjoint planning horizons are shown to be possible which eliminate the necessity of having data for the full N periods.

2,114 citations

Journal ArticleDOI
TL;DR: It is concluded that the “relaxation” procedure for approximately solving a large linear programming problem related to the traveling-salesman problem shows promise for large-scale linear programming.
Abstract: The "relaxation" procedure introduced by Held and Karp for approximately solving a large linear programming problem related to the traveling-salesman problem is refined and studied experimentally on several classes of specially structured large-scale linear programming problems, and results on the use of the procedure for obtaining exact solutions are given It is concluded that the method shows promise for large-scale linear programming

1,339 citations