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May  2020, 16(3): 1389-1414. doi: 10.3934/jimo.2019008

## Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns

 1 School of Business, Central South University, The Collaborative Innovation Center for Resource-conserving & Environment-friendly Society and Ecological Civilization, Changsha 410083, China 2 Hunan University of Commerce, Changsha 410205, China 3 School of Business, Central South University, Changsha 410083, China

* Corresponding author: Fuqiang Wang

Received  February 2018 Revised  October 2018 Published  May 2020 Early access  March 2019

This paper studies an original equipment manufacturer's (OEM's) optimal production and pricing decisions and the governments optimal subsidy level when the number of used products returning to the OEM is uncertain. The government aims to minimize its total expenditures but also attempt to achieve a given target collection level. We model the problem as an extended price-setting newsvendor model, which simultaneously incorporates supply uncertainty and external government influence. Moreover, we consider separately the cases of stochastic supplies with additive and multiplicative return uncertainty. We show that under the above settings, the governments optimal strategy is to provide only sufficient subsidies that cause its target to be met exactly. The government subsidies will mitigate the cost of remanufacturing and increase the total collection efforts of the government and the manufacturer. Moreover, the return uncertainty lowers both the manufacturers profits and selling price, whereas its effects on the governments optimal subsidies and the manufacturers optimal return efforts are insignificant. Therefore, the manufacturer is worse off but consumers are better off under the conditions of uncertain returns. By comparing the optimal decisions when the government is a central planner with the case of decentralized decision making, or comparing the arrangement in which the government provides subsidies directly to the manufacturer rather than to consumers, we find that the government subsidies would coordinate the supply chain only when its target collection level is high. Moreover, no essential differences exist between providing subsidies directly to the manufacturer and to consumers. Our results are robust under both the additive and multiplicative uncertainty models.

Citation: Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1389-1414. doi: 10.3934/jimo.2019008
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Order of events: (1) subsidies; (2) price, quantity and return price; (3) sales
Notations
 $d(p)$ Demand for new products $p$ Selling price $M$ Market size $a$ Price sensitivity factor $q_{n}$ Number of new products produced from raw materials $q_{r}$ Number of new products produced from used products $q$ Total number of products produced $c_{n}$ Unit cost of producing a new product from raw materials $c_{r}$ Unit cost of producing a new product from used products $r$ Price of the returned products $\eta$ Per unit subsidy for returned products the government provides for consumers $\eta^\prime$ Per unit subsidy for returned products the government provides for the manufacturers $z$ Effective price of the returned products $\Gamma$ Target collection level $s$ Salvage value for each unit of unsold product $f_{\epsilon}( \cdot )$ Probability density function of random variable $\epsilon$ $F_{\epsilon}( \cdot )$ Cumulative distribution function of random variable $\epsilon$ $\Pi$ Manufacturer's profit $Exp$ Minimal expected subsidy expenditure
 $d(p)$ Demand for new products $p$ Selling price $M$ Market size $a$ Price sensitivity factor $q_{n}$ Number of new products produced from raw materials $q_{r}$ Number of new products produced from used products $q$ Total number of products produced $c_{n}$ Unit cost of producing a new product from raw materials $c_{r}$ Unit cost of producing a new product from used products $r$ Price of the returned products $\eta$ Per unit subsidy for returned products the government provides for consumers $\eta^\prime$ Per unit subsidy for returned products the government provides for the manufacturers $z$ Effective price of the returned products $\Gamma$ Target collection level $s$ Salvage value for each unit of unsold product $f_{\epsilon}( \cdot )$ Probability density function of random variable $\epsilon$ $F_{\epsilon}( \cdot )$ Cumulative distribution function of random variable $\epsilon$ $\Pi$ Manufacturer's profit $Exp$ Minimal expected subsidy expenditure
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