Production, Manufacturing and Logistics
On “An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging” by Dye and Ouyang

https://doi.org/10.1016/j.ejor.2005.04.024Get rights and content

Abstract

In 2005, Dye and Ouyang proposed an EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging, and then established the unique optimal solution to the problem when building up inventory is not profitable. However, they did not provide the optimal solution to the problem when building up inventory is profitable. In this note, we establish an appropriate model in which building up inventory is profitable, and then provide an algorithm to find the optimal solution to the problem. A numerical example is used to illustrate the proposed model.

Introduction

Padmanabhan and Vrat [4] considered an economic order quantity (i.e., EOQ) model for perishable items with stock-dependent demand, and presented three shortage models: without backlogging, with backlogging, and partial backlogging. They assumed that the backlogging rate depends on the amount of unsatisfied demand. The more the amount of unsatisfied demand, the smaller the backlogging rate would be. In contrast, Abad [1] reflected to the fact that the willingness of a customer to wait for backlogging during a shortage period is diminishing with the length of the waiting time, and proposed several distinct backlogging rates related to the length of the waiting time. Recently, Dye and Ouyang [2] amended Padmanabhan and Vrat’s model by using Abad’s linear time-proportional backlogging rate, and established a unique optimal solution to the problem in which building up inventory has a negative effect on the profit. However, an increase in shelf space for an item induces more consumers to buy it. This occurs because of its visibility, popularity or variety. Conversely, low stocks of certain baked goods (e.g., donuts) might raise the perception that they are not fresh. Therefore, building up inventory often has a positive impact on the sales, as well as the profit. As a result, in this note, we complement Dye and Ouyang’s paper by establishing an appropriate model in which building up inventory is profitable, and then provide an algorithm to find the optimal solution to the problem.

Section snippets

Assumptions and notations

In this note, the following notations are similar to those in Dye and Ouyang [2], except the ending inventory.

    t1

    the length of time interval with positive or zero inventory in a replenishment cycle, where t1  0

    t2

    the length of time interval with negative inventory (or shortages) in a cycle, where t2  0

    T

    the duration of a replenishment = t1 + t2

    θ

    the constant deterioration rate, where 0  θ < 1

    I(t)

    the inventory level at time t

    E

    the ending inventory level (a decision variable) if t2 = 0, where 0  E = I(t1)

    D(t)

    α + βI(t

The model and theoretical results

In this section, we first obtain the conditions under which the shortage is profitable. We then provide the necessary and sufficient conditions to solve the problem in which shortage is profitable. Finally, we discuss the case in which the shortage is not optimal to the problem.

Numerical example

We consider the data as the same as in Dye and Ouyang [2]: α = 600, A = 250, θ = 0.05, R = 5, δ = 50, C2 = 3, C = 5 and i = 0.35. In order to get βS  (i + β + θ)C > 0, we adopt S = 16, β = {0.2, 0.3, 0.4, 0.5, 0.6} and U = 500. If β = 0.2 and 0.3, then we obtain t1 from (3). Since α[βS-(i+β+θ)C][e(β+θ)t1-1-(β+θ)t1]/(β+θ)2<A, by solving (4), we obtain the unique positive optimal shortage period t2. If β = 0.4, 0.5 and 0.6, then α[βS-(i+β+θ)C][e(β+θ)t1-1-(β+θ)t1]/(β+θ)2>A. Then we can obtain the unique E* and t1 by solving (12)

Acknowledgements

We would like to thank the referees for their constructive suggestions. This research was partially supported by the Assigned Released Time for Research and a Summer Research Funding from the William Paterson University of New Jersey.

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