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March  2022, 12(1): 109-120. doi: 10.3934/naco.2021054

## Optimal pre-sale policy for deteriorating items

 1 School of Management, Shanghai University, Shanghai 200444, China 2 School of Engineering, Open University of China, Beijing 100039, China

* Corresponding author: Qi An

Received  May 2020 Revised  July 2021 Published  March 2022 Early access  November 2021

Fund Project: This work is supported by Humanities and Social Sciences Foundation of the Chinese Ministry of Education (20YJAZH135)and National Natural Science Foundation of China (71502100)

Pre-sale policy is a frequently-used sales approach for deteriorating products, e.g, fruits, vegetables, seafood, etc. In this paper, we consider an EOQ inventory model under pre-sale policy for deteriorating products, in which the demand of pre-sale period depends on price and pre-sale horizon, and the demand of spot-sale period depends on the price and stock level. Optimal pricing decisions and economic order quantity are also provided. We compare pre-sale model with a benchmark inventory model in which all the products are sold in spot-sale period. Theoretical results are derived to show the existence and uniqueness of the optimal solution. Numerical experiments are carried out to to illustrate the theoretical results. And sensitivity analysis is conducted to identify conditions under which the pre-sale policy is better off than the spot-sale only policy.

Citation: Lianxia Zhao, Hui Qiao, Qi An. Optimal pre-sale policy for deteriorating items. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 109-120. doi: 10.3934/naco.2021054
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The graphic of the sale model during a cycle
Comparison of the profits and prices at different pre-sale market potential
Comparison of the profits and prices at different deteriorating rate
Comparison of the profits and prices at different price sensitivity
Comparison of the profits and prices at different planning horizon
Summary of notations
 Symbol Description $\epsilon$ The market demand in pre-sale period $\delta$ The price sensitivity of the demand $T$ The fixed spot-sale planning horizon $p_0$ The selling price of a unit in pre-sale period under pre-sale policy $p$ The selling price of a unit in spot-sale period under pre-sale policy $p_b$ The selling price of a unit under spot-sale only policy $w$ The purchase price of a unit $\theta$ The deterioration rate of the item $c$ The sensitivity to the advertising policy, $0\leq\delta\leq1$ $c_h$ The holding cost per unit per unit time $c_d$ The deterioration cost per unit per unit time $L$ The length of the pre-sale horizon, and suppose $L=\epsilon T (0\leq\epsilon\leq\frac{1}{2})$ $I(t)$ The instantaneous inventory level on hand at time $[0, T].$
 Symbol Description $\epsilon$ The market demand in pre-sale period $\delta$ The price sensitivity of the demand $T$ The fixed spot-sale planning horizon $p_0$ The selling price of a unit in pre-sale period under pre-sale policy $p$ The selling price of a unit in spot-sale period under pre-sale policy $p_b$ The selling price of a unit under spot-sale only policy $w$ The purchase price of a unit $\theta$ The deterioration rate of the item $c$ The sensitivity to the advertising policy, $0\leq\delta\leq1$ $c_h$ The holding cost per unit per unit time $c_d$ The deterioration cost per unit per unit time $L$ The length of the pre-sale horizon, and suppose $L=\epsilon T (0\leq\epsilon\leq\frac{1}{2})$ $I(t)$ The instantaneous inventory level on hand at time $[0, T].$

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