A dual algorithm for the economic lot-sizing problem (original) (raw)
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Sensitivity Analysis of the Economic Lot-Sizing Problem
Discrete Applied Mathematics, 1993
Van Hoesel, S. and A. Wagelmans, Sensitivity analysis of the economic lot-sizing problem, Discrete Applied Mathematics 45 (1993) 291-312. In this paper we study sensitivity analysis of the uncapacitated single level economic lot-sizing problem, which was introduced by Wagner and Whitin about thirty years ago. In particular we are concerned with the computation of the maximal ranges in which the
Dynamic-programming-based inequalities for the capacitated lot-sizing problem
IIE Transactions, 2010
Iterative solutions of forward dynamic programming formulations for the capacitated lot-sizing problem are used to generate inequalities for an equivalent integer programming formulation. The inequalities capture convex and concave envelopes of intermediate-stage value functions, and can be lifted by examining potential state information at future stages. We test several possible implementations that employ these inequalities, and demonstrate that our approach is more efficient than alternative integer programming based algorithms. For certain datasets, our algorithm also outperforms a pure dynamic programming algorithm for the problem.
A note on “The economic lot sizing problem with inventory bounds”
European Journal of Operational Research, 2012
In a recent paper, Liu (2008) considers the lot-sizing problem with lower and upper bounds on the inventory levels. He proposes an O(n 2) algorithm for the general problem, and an O(n) algorithm for the special case with non-speculative motives. We show that neither of the algorithms provides an optimal solution in general. Furthermore, we propose a fix for the former algorithm that maintains the O(n 2) complexity.
Analysis of bounds for a capacitated single-item lot-sizing problem
Computers & Operations Research, 2007
Lot-sizing problems are cornerstone optimization problems for production planning with time varying demand. We analyze the quality of bounds, both lower and upper, provided by a range of fast algorithms. Special attention is given to LP-based rounding algorithms. † Corresponding author program to solve it. Polyhedral results for these problems are especially plentiful. Notable results are found in, among others, [9], , and [13] (see for a more comprehensive review). The abundance of polyhedral results has resulted in more effective integer programming based solution approaches, but many instances still require prohibitive amounts of computation time. Van Hoesel & Wagelmans [17] approach the problem differently and present a fully polynomial approximation scheme for single-item capacitated lot-sizing, with the time required to find a solution depending on the desired nearness of the value of the resulting solution to the optimal value. Unfortunately, the practical value of this approach is limited as the computational requirements are high even for moderate approximation factors.
European Journal of Operational Research, 2010
We present a fully polynomial time approximation scheme (FPTAS) for a capacitated economic lot-sizing problem with a monotone cost structure. An FPTAS delivers a solution with a given relative error ε in time polynomial in the problem size and in 1/ε. Such a scheme was developed by van Hoesel and Wagelmans [8] for a capacitated economic lot-sizing problem with monotone concave (convex) production and backlogging cost functions. We omit concavity and convexity restrictions. Furthermore, we take advantage of a straightforward dynamic programming algorithm applied to a rounded problem.
Primal-dual approach to the single level capacitated lot-sizing problem
European Journal of Operational Research, 1991
The Lagrangean relaxation of the single level capacitated dynamic lot-sizing problem can be solved using the primal-dual method. The algorithm has monotone and finite convergence properties. It works as a steepest ascent method. A variant of this approach is also studied. A heuristic routine used to obtain a feasible solution in each iteration is presented. Computational experiences show that this method usually yields better solutions than the subgradient method although it requires greater CPU times.
An efficient algorithm for the dynamic economic lot size problem
Computers & Operations Research, 1992
Abstract This paper addresses the question of production/procurement planning for finite horizon, deterministic, dynamic demand process, known as the “Dynamic Economic Lot size Problem”. A new algorithm is presented and compared to existing exact procedures. The algorithm first decomposes the problem into much smaller sequences (planning horizons). It then applies an exact method to schedule the production in each sequence (this stage is illustrated here with the “classical” dynamic programming algorithm of Zangwill [8]). Finally, it combines the partial solutions to an overall optimal solution. Camputational results which demonstrate the effectiveness of the proposed algorithm are provided.
A note on solving the concave cost dynamic lot-sizing problem in almost linear time
Journal of Operations Management, 1989
The heavilydebated Wagner-Whitin algorithm is known to produce optimal ordering policies for minimal-cost dynamic lot--sizing problems. In an earlier paper in this journal, Evans showed that the Wagner-Whitin algorithm is essentially a shortest path computation on an acyclic network, and presented a simple O(n2) computer implementation with low storage requirements.
Efficient post-optimization analysis procedure for the dynamic lot-sizing problem
European Journal of Operational Research, 1993
In this paper, we develop an efficient post-optimization analysis procedure for the Wagner-Whitin solution to the Dynamic Lot-Sizing Problem (DLSP). The proposed procedure can be used in the context of a branch-and-bound algorithm, or in smoothing heuristic approaches for solving the Capacitated Lot-Sizing Problem (CLSP).
A global constraint for the capacitated single-item lot-sizing problem
ArXiv, 2019
The goal of this paper is to set a constraint programming framework to solve lot-sizing problems. More specifically, we consider a single-item lot-sizing problem with time-varying lower and upper bounds for production and inventory. The cost structure includes time-varying holding costs, unitary production costs and setup costs. We establish a new lower bound for this problem by using a subtle time decomposition. We formulate this NP-hard problem as a global constraint and show that bound consistency can be achieved in pseudo-polynomial time and when not including the costs, in polynomial time. We develop filtering rules based on existing dynamic programming algorithms, exploiting the above mentioned time decomposition for difficult instances. In a numerical study, we compare several formulations of the problem: mixed integer linear programming, constraint programming and dynamic programming. We show that our global constraint is able to find solutions, unlike the decomposed constra...