Effective Algorithms for the Economic Lot-Sizing Problem with Bounded Inventory and Linear Fixed-Charge Cost Structure

José M Gutiérrez,Beatriz Abdul-Jalbar,Inmaculada Rodríguez-Martín,Joaquín Sicilia

doi:10.3390/math9060689

José M Gutiérrez, Beatriz Abdul-Jalbar + Show 2 more

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https://doi.org/10.3390/math9060689

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Journal: Mathematics	Publication Date: Mar 23, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: University of La Laguna

Abstract

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 storage capacity availability. Furthermore, this percentage significantly increases from 13% to 35% as the capacity availability decreases. This reveals that the usage of the geometric technique is predominant under restrictive storage capacities. Specifically, when the percentage of capacity availability is below 50%, the average running times are on average 100 times faster than those when this percentage is above 50%. On the other hand, an OT on-line array searching method in Monge arrays can be used when the costs are non-speculative costs.

Highlights

IntroductionAs an extension of the economic order quantity (EOQ) model to cases with time-varying parameters
The economic lot-sizing problem is a classical deterministic inventory problem which was independently introduced by Wagner and Whitin [1] and Manne [2]as an extension of the economic order quantity (EOQ) model to cases with time-varying parameters
For the general linear cost structure, the new O T approach focused on the evaluation of the feasible solution set by conveniently decomposing the original problem into subproblems

Summary

Introduction

As an extension of the economic order quantity (EOQ) model to cases with time-varying parameters. The cost structure defined in [1] is the sum of the fixed charge replenishment costs and the linear holding costs for each period. It is a well-known result that among the optimal solutions for the dynamic version of the EOQ, there is at least one satisfying the zero inventory ordering (ZIO) policy (i.e., an order is placed only if the inventory is depleted, which can be used to derive polynomially efficient solutions). Veinott [3] proved that ZIO policies could be likewise used to derive optimal plans when the cost structure was concave in general. A significant number of contributions have been published in the literature considering diverse extensions of this simple model (refer to Brahimi et al [4] and Wolsey [5] for more comprehensive surveys and to [6,7,8,9,10] for more recent contributions)

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