Derivation of the Optimal Solution for the Economic Production Quantity Model with Planned Shortages without Derivatives

Cenk Çalışkan

doi:10.3390/modelling3010004

Abstract

In this paper, we study a reformulation of the Economic Production Quantity (EPQ) problem. We study a more general version of the problem first and derive the conditions for an optimal solution, as well as the optimal solution itself, all without using derivatives. Then, we apply the approach to the reformulated EPQ problem. This version of the EPQ problem has been tackled by a number of researchers, wherein they have derived the conditions for the optimal solution and proposed algebraic derivations. However, their derivations for the conditions, as well as the optimal solution, have been shown to be questionable. Other than being questionable, the existing approaches are so complicated that they defeat the purpose of simplifying the optimization by using a derivative-free approach. We propose a correct and more succinct, much less complicated approach to derive the conditions and the optimal solution without using derivatives.

Highlights

The basic EOQ model was developed by Harris [1]
Cárdenas-Barrón [3] extends the same approach to the Economic Production Quantity (EPQ) model with backorders, and Wee et al [4] extend it to the EOQ problem with a one-time discount offer
We answer the open question posed in Chang et al [7]: deriving the optimal solution for a reformulation of the EPQ problem with backorders using an approach that does not require derivatives

Summary

Introduction

The basic EOQ model was developed by Harris [1]. In this pioneering work, he states that “the solution to this problem requires higher mathematics.” Grübbstrom and Erdem [2]. Cárdenas-Barrón [19] uses analytic geometry in conjunction with this approach for the EOQ and EPQ problems. Minner [24] proposes the cost comparisons approach to find the optimal order interval in the EOQ and EPQ problems (and in their backordering extensions). Cárdenas-Barrón [26] compares the earlier algebraic approaches and the cost comparison approach and derives the optimal backorder quantity in the EOQ and EPQ models. Open question: deriving the optimal solution for a reformulation of the EPQ problem with backorders. The conditions for the existence and uniqueness of the optimal solution for the posed problem Analysis of both Lau et al [29] and Chiu et al [30], a purely algebraic approach as opposed to the former. Answering the open question in Chang et al [7]; correction of the former approaches; analysis of the general problem in Lau et al [29]

Description of the Problem

The Optimal Solution and Its Conditions

The Original Inventory Problem

Limitations and Further

Conclusions