A dual algorithm for the economic lot-sizing problem (original) (raw)
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...
Facets and algorithms for capacitated lot sizing
Mathematical Programming, 1989
The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use
Hybrid Formulation of the Multi-Item Capacitated Dynamic Lot Sizing Problem
American Journal of Operations Research, 2015
It is shown that when backorders, setup times and dynamic demand are included in capacitated lot sizing problem, the resulting classical formulation and one of the transportation formulations of the problem (referred to as CLSP_BS) are equivalent. And it is shown that both the formulations are "weak" formulations (as opposed to "strong" formulation). The other transportation version is a strong formulation of CLSP_BS. Extensive computational studies are presented for medium and large sized problems. In case of medium-sized problems, strong formulation produces better LP bounds, and takes lesser number of branch-and-bound (B&B) nodes and less CPU time to solve the problem optimally. However for large-sized problems strong formulation takes more time to solve the problem optimally, defeating the benefit of strength of bounds. This essentially is because of excessive increase in the number of constraints for the large sized problems. Hybrid formulations are proposed where only few most promising strong constraints are added to the weak formulation. Hybrid formulation emerges as the best performer against the strong and weak formulations. This concept of hybrid formulation can efficiently solve a variety of complex real life large-sized problems.
2021
Efficient algorithms for the economic lot-sizing problem with storage capacity are proposed. On the one hand, for the cost structure consisting of general linear holding and ordering costs and fixed setup costs, an OT2 dynamic programming algorithm is introduced, where T is the number of time periods. The new approach induces an accurate partition of the planning horizon, discarding most of the infeasible solutions. Moreover, although there are several algorithms based on dynamic programming in the literature also running in quadratic time, even considering more general cost structures and assumptions, the new solution uses a geometric technique to speed up the algorithm for a class of subproblems generated by dynamic programming, which can now be solved in linearithmic time. To be precise, the computational results show that the average occurrence percentage of this class of subproblems ranges between 13% and 45%, depending on both the total number of periods and the percentage of ...
Polynomial cases of the economic lot sizing problem with cost discounts
European Journal of Operational Research, 2014
In this paper we study the economic lot sizing problem with cost discounts. In the economic lot sizing problem a facility faces known demands over a discrete finite horizon. At each period, the ordering cost function and the holding cost function are given and they can be different from period to period. There are no constraints on the quantity ordered in each period and backlogging is not allowed. The objective is to decide when and how much to order so as to minimize the total ordering and holding costs over the finite horizon without any shortages. We study two different cost discount functions. The modified allunit discount cost function alternates increasing and flat sections, starting with a flat section that indicates a minimum charge for small quantities. While in general the economic lot sizing problem with modified all-unit discount cost function is known to be NP-hard, we assume that the cost functions do not vary from period to period and identify a polynomial case. Then we study the incremental discount cost function which is an increasing piecewise linear function with no flat sections. The efficiency of the solution algorithms follows from properties of the optimal solution. We computationally test the polynomial algorithms against the use of CPLEX.
Lecture Notes in Computer Science, 2007
We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a time-dependent fixed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to find a feasible solution with minimum overall ordering and holding costs.
Mathematics of Operations Research, 2008
We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a time-dependent fixed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to find a feasible solution with minimum overall ordering and holding costs.
Single item lot-sizing with non-decreasing capacities
Mathematical Programming, 2010
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 setup 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 nondecreasing and arbitrary capacities over time.
In this paper we study the problem of multi-item capacitated lot-sizing from the point of commercial enterprises. We consider profit as the main criteria. This dynamic problem belongs to the class of discrete optimization and contains boolean variables, algorithmic objective function, where various types of constraints such as analytical functions, algorithmic and simulation models can be used. We present model and direct search algorithm that consists of an intelligent iterative search and upper bound set construction, and allows finding exact solution in reasonable time. We carry out computational investigation and solve a real task with using developed computational tool to show the efficiency and practical application of the proposed model and algorithm.
The joint economic lot sizing problem: Review and extensions
European Journal of Operational Research, 2008
With the growing focus on supply chain management, firms realize that inventories across the entire supply chain can be more efficiently managed through greater cooperation and better coordination. This paper presents a comprehensive and up-todate review of the joint economic lot sizing problem (JELP) and also provides some extensions of this important problem. In particular, a detailed mathematical description of, and a unified framework for, the main JELP models are given. Additionally, a comparative empirical study of the main policies proposed for JELP is conducted. The focus of this study is on assessing the deviation of these policies from the optimal solution. Studying the performance of different models provides additional insights that will help in justifying their use in more complex supply chain models that involve more stages or other practical considerations of interest. (C) 2007 Elsevier B.V. All rights reserved.