# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020109

## Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering

 1 School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China, Springfield, MO 65801-2604, USA 2 School of Economics and Management, Xi'an University of Posts and Telecommunications, Xi'an Shaanxi, 710121, China, Springfield, MO 65810, USA

* Corresponding author: Fujun Hou

Received  October 2019 Revised  February 2020 Published  June 2020

Fund Project: This work is supported by the Natural Science Foundation of China (71571019)

This paper studies ordering and pricing issues under multiple times ordering. A manufacturer and a retailer are involved in our discussion. The definition of a reasonable price is given based on the practical requirement. First, we construct a Stackelberg model in which the manufacturer and the retailer make their decisions respectively. During the process of derivation, both ordering time-points and optimal prices are expressed as functions of number of times of ordering. By solving a quadratic programming model with an undetermined parameter, we demonstrate that the optimal ordering time-points of the retailer are equidistant time points on the given selling period. Second, a cooperative model is developed in which the manufacturer and the retailer jointly make decisions. It is shown that the optimal retail price is lower and the number of times of ordering is more in the cooperative situation than the noncooperative one. Further, an allocation method based on revenue proportions is proposed.

Citation: Zhenkai Lou, Fujun Hou, Xuming Lou. Optimal ordering and pricing models of a two-echelon supply chain under multipletimes ordering. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020109
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Model parameters
 Parameters Definition [0, $T$] The given selling period $t_{i }$ The ordering time-point, where $t_{i} \in$[0, $T$] and $i \in M$, $M$ = {$1, \ldots, m$} $p_{b}$ The wholesale price determined by the manufacturer $p_{c}$ The retail price determined by the retailer $q$ The total procurement volume of the retailer $e$ The production cost per item of the manufacturer $m$ The number of times of ordering of the retailer $k$ The fixed ordering cost of each order $\lambda$ The linear price-sensitive coefficient of the demand rate $r(p_{c})$ The demand rate under price $p_{c}$: $r(p_{c}) = a$ – $\lambda p_{c}$, $a$ ¿ 0 $h$ The stock-holding cost per item per unit time $W$ The revenue of the manufacturer $Z$ The revenue of the retailer $S$ The total revenue incurred by centralized decision-making
 Parameters Definition [0, $T$] The given selling period $t_{i }$ The ordering time-point, where $t_{i} \in$[0, $T$] and $i \in M$, $M$ = {$1, \ldots, m$} $p_{b}$ The wholesale price determined by the manufacturer $p_{c}$ The retail price determined by the retailer $q$ The total procurement volume of the retailer $e$ The production cost per item of the manufacturer $m$ The number of times of ordering of the retailer $k$ The fixed ordering cost of each order $\lambda$ The linear price-sensitive coefficient of the demand rate $r(p_{c})$ The demand rate under price $p_{c}$: $r(p_{c}) = a$ – $\lambda p_{c}$, $a$ ¿ 0 $h$ The stock-holding cost per item per unit time $W$ The revenue of the manufacturer $Z$ The revenue of the retailer $S$ The total revenue incurred by centralized decision-making
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