Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models

Fully Polynomial Approximation Schemes for Single-Item Capacitated Economic Lot-Sizing Problems

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known which produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist, by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models.single-item capacitated lot-sizing;fully polynomial approximation schemes;lot-sizing models;suboptimal algorithms

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

Approximating the Nonlinear Newsvendor and Single-Item Stochastic Lot-Sizing Problems When Data Is Given by an Oracle

The single-item stochastic lot-sizing problem is to find an inventory replenishment policy in the presence of discrete stochastic demands under periodic review and finite time horizon. A closely related problem is the single-period newsvendor model. It is well known that the newsvendor problem admits a closed formula for the optimal order quantity whenever the revenue and salvage values are linear increasing functions and the procurement (ordering) cost is fixed plus linear. The optimal policy for the single-item lot-sizing model is also well known under similar assumptions. In this paper we show that the classical (single-period) newsvendor model with fixed plus linear ordering cost cannot be approximated to any degree of accuracy when either the demand distribution or the cost functions are given by an oracle. We provide a fully polynomial time approximation scheme for the nonlinear single-item stochastic lot-sizing problem, when demand distribution is given by an oracle, procurement costs are provided as nondecreasing oracles, holding/backlogging/disposal costs are linear, and lead time is positive. Similar results exist for the nonlinear newsvendor problem. These approximation schemes are designed by extending the technique of K-approximation sets and functions.National Science Foundation (U.S.) (Contract CMMI-0758069)United States. Office of Naval Research (Grant N000141110056

A dynamic lot-sizing model with demand time windows

One of the basic assumptions of the classical dynamic lot-sizing model is that the aggregate demand of a given period must be satisfied in that period. Under this assumption, if backlogging is not allowed then the demand of a given period cannot be delivered earlier or later than the period. If backlogging is allowed, the demand of a given period cannot be delivered earlier than the period, but can be delivered later at the expense of a backordering cost. Like most mathematical models, the classical dynamic lot-sizing model is a simplified paraphrase of what might actually happen in real life. In most real life applications, the customer offers a grace period - we call it a demand time window - during which a particular demand can be satisfied with no penalty. That is, in association with each demand, the customer specifies an earliest and a latest delivery time. The time interval characterized by the earliest and latest delivery dates of a demand represents the corresponding time window. This paper studies the dynamic lot-sizing problem with demand time windows and provides polynomial time algorithms for computing its solution. If shortages are not allowed, the complexity of the proposed algorithm is of the order T square. When backlogging is allowed, the complexity of the proposed algorithm is of the order T cube.dynamic programming;lot-sizing;time windows

Fast approximation schemes for multi-criteria flow, knapsack, and scheduling problems

Includes bibliographical references (p. 40-42).Supported by an NSF Presidential Young Investigator grant. Supported by the Air Force Office of Scientific Research. AFOSR-88-0088 Supported by the NSF. DDM-8921835by Hershel M. Safer, James B. Orlin

An algorithm for the serial capacitated economic lot-sizing problem with non-speculative costs and stationary capacities

We address the serial capacitated economic lot-sizing problem under particular assumptions on the costs and the capacity pattern. We prove that when the involved costs are non-speculative with respect to the transfer to future periods and the capacity pattern is stationary for all levels, the optimal plan for each level can be obtained independently in O(T 3) time. This leads to an O(T 3L) algorithm for the problem with L levels. and the capacity pattern is stationary for all levels, the optimal plan for each level can be obtained independently in O(T 3) time. This leads to an O(T 3L) algorithm for the problem with L levels

Green Lot-Sizing

The lot-sizing problem concerns a manufacturer that needs to solve a production planning problem. The producer must decide at which points in time to set up a production process, and when he/she does, how much to produce. There is a trade-off bet