Replenishment Lot Size and Number of Shipments for EPQ Model Derived Without Derivatives

Kuang-Ku Chen,Singa Chiu

doi:10.3390/mca16030753

Abstract

This paper investigates optimal replenishment lot size and optimal number of shipments for economic production quantity (EPQ) model with rework process using an algebraic approach. The classic EPQ model assumes a perfect quality production for all items produced and a continuous issuing policy for satisfying customer‟s demand. However, in a real life vendor-buyer integrated-production-inventory system, a multi- delivery policy is commonly used in lieu of a continuous issuing policy and generation of defective items during a production run seems to be inevitable. A recent published work examined such an identical problem using mathematical modeling and differential calculus to derive the optimal replenishment lot size and optimal number of deliveries that minimizes overall production-inventory-shipment costs. This paper proposes a straightforward algebraic approach to replace the use of calculus on the cost function for determining optimal production-shipment policies. It also presents a simpler form for calculating the long-run average cost for such an imperfect EPQ problem.

Highlights

Two fundamental questions that need to be answered by inventory controllers for items they routinely replenish are “when should a replenishment lot be initiated?” and “how many to be refilled in a lot?” in order to minimize the long-run average cost [1]
The economic production quantity (EPQ) model is often used by production and inventory managers to assist them in addressing the aforementioned issues [2]
This paper re-examines the EPQ model with rework and multi-delivery policy using algebraic approach, to demonstrate that the optimal replenishment production-shipments policies as well as the long-run average system costs can all be derived without derivatives

Summary

INTRODUCTION

Two fundamental questions that need to be answered by inventory controllers for items they routinely replenish are “when should a replenishment lot be initiated?” and “how many to be refilled in a lot?” in order to minimize the long-run average cost [1]. In a real life vendor-buyer integrated production-inventory system, multiple or periodic deliveries of finished products are commonly adopted As a result, another issue to be addressed is “how many shipments should a replenishment lot be broken down to?” so that the overall costs can be minimized. Siajadi et al [8] presented a methodology to obtain the joint economic lot size in the case where multiple buyers are demanding one type of item from a single vendor They proposed a model to minimize the joint total relevant cost (JTRC) for both vendor and buyer(s). Chiu et al [22] investigated the optimal replenishment lot size and optimal number of shipments for economic production quantity (EPQ) model with rework They used the mathematical modeling together with the conventional derivatives on the cost function of the proposed system, to prove its optimality and derived the optimal replenishment production-shipments decisions respectively. This paper re-examines the EPQ model (considered by Chiu et al [22]) with rework and multi-delivery policy using algebraic approach, to demonstrate that the optimal replenishment production-shipments policies as well as the long-run average system costs can all be derived without derivatives

PROBLEM STATEMENT AND MATHEMATICAL MODELLING

OPTIMAL SOLUTIONS DERIVED WITHOUT DERIVATIVES

Z1 Z2 2 Z3 Z4

NUMERICAL EXAMPLE

CONCLUSION