An economic order quantity model with partial backordering and a special sale price

doi:10.1016/j.ejor.2012.03.032

European Journal of Operational Research

Volume 221, Issue 3, 16 September 2012, Pages 571-583

https://doi.org/10.1016/j.ejor.2012.03.032 Get rights and content

Abstract

A constant unit purchase cost is one of the main assumptions in the classic economic order quantity model. In practice, suppliers sometimes offer special sale prices to stimulate sales or decrease inventories of certain items. In this paper we develop an EOQ model with a special sale price and partial backordering. We prove the convexity of the cost-reduction function if a special order is placed at the special sale price. A solution method is proposed and numerical examples are presented.

Highlights

► We combine two of the extensions of the basic EOQ found in the literature. ► Partial backordering of demand during a stockout period and responding to a short-term one-time opportunity for a discount. ► We develop and prove optimality of the solution.

Section snippets

Introduction and literature review

Since the basic economic order quantity model (EOQ) was introduced by Harris (1913), there have been many extensions that have relaxed one or more of its assumptions or added additional features. In this paper we combine two of those extensions: (1) determining whether to place a special order if there is an opportunity to buy at a temporary sale price and, if so, the order quantity and (2) partial backordering at a constant backordering rate.

A constant unit purchase cost is one of the main

Problem definition and assumptions

Consider a situation in which a company stocks a particular item which, since the demand rate is constant and the other usual conditions for doing so are met, it orders regularly using an EOQ model. Assume that it is discovered that a temporary discount is available for a short time, during which the regular price of the item, C, is reduced to C_S = C − C′. After the temporary sale the price of the item will return to C. The logical reaction to finding an item on sale at the time of a regular

Model development

We introduce the parameters and the variables of the model in Section 3.1. We next review some of the previous work on the two basic components of our model: the special sale problem without backordering (Section 3.2) and the EOQ with partial backordering (Section 3.3). Our models for the problem under different assumptions about when the sale price is available are developed in Sections 3.4.1 Scenario 1: The sale price coincides with the normal time to place an order, 3.4.2 Scenario 2: The

Solution method

To justify ordering a special quantity during the sale period, the profit from ordering the optimal special quantity must be more than the profit from using regular EOQ order quantities for the same amount of time T_S (CTP_S > CTP_n). Another way of saying this is to say that the optimal value of G₁, G₂, or G₃, whichever is relevant for the particular scenario being faced, must be positive to justify placing a special order at the sale price.

We noted above that the coefficients of the variable terms

Numerical examples

To illustrate the application of the solution procedure given above, we will use the numerical example from Pentico and Drake (2009), adding the new parameters which are used in this paper. The fixed parameters in all examples are D = 200 units/year, A = $50/order, h = $3/unit/year, π = $1/unit/year, π′ = $2/unit lost, C = $11/unit, C′ = $3/unit, P = $11/unit, and i = 0.3. Following we introduce four examples based on those parameter values.

Conclusion

We have developed EOQ models for three scenarios with a special sale price and partial backordering: the sale price is available at the normal time to place an order, it is available only when there is still existing inventory, or it is available only when there is a stockout. We consider that the purchaser either orders a special quantity Q_s to take advantage of the discount price or sticks to the regular EOQ model and orders Q. Since ordering a special quantity during the sale period requires

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Production, Manufacturing and LogisticsAn economic order quantity model with partial backordering and a special sale price

Abstract

Highlights

Section snippets

Introduction and literature review

Problem definition and assumptions

Model development

Solution method

Numerical examples

Conclusion

Computers & Operations Research

European Journal of Operational Research

Omega

Applied Mathematical Modeling

International Journal of Production Economics

Computers & Industrial Engineering

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Computers & Industrial Engineering

International Journal of Production Economics

Computers & Industrial Engineering

European Journal of Operational Research

European Journal of Operational Research

Production, Manufacturing and Logistics
An economic order quantity model with partial backordering and a special sale price