# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020031

## Coordination contracts for a dual-channel supply chain under capital constraints

 1 School of Management, Nanjing University of Posts and Telecommunications, Nanjing, 210003, Jiangsu, China 2 School of Economics & Managements, Southeast University, Nanjing, 210089, Jiangsu, China

* Corresponding author: Chong Zhang

Received  May 2019 Revised  July 2019 Published  February 2020

Fund Project: The paper is supported by the Humanity and Social Science Youth Foundation of Ministry of Education of China (18YJC630235), the National Natural Science Foundation of China(71531004, 71803088) and the Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY219117)

Manufacturers often face capital constraints when opening up online channel, at this time external financing and internal financing are usually considered. Previous literature has shown that internal financing, turns out to be a better option. To figure out how trade credit financing discount contract affects operations and performances of supply chain, this paper studies the pricing decision of a retailer-dominant dual-channel supply chain with manufacturer's capital constraints. The Stackelberg game models under centralized decision and decentralized decision are constructed. Moreover, this paper conducts research about the effects of revenue-sharing (RS) contract, direct channel price discount (DP) contract and retail channel price discount (RP) contract on the performance of supply chain. Numerical examples are provided to explore the comparison of the optimal pricing strategies and total profits under different contracts. The results show that the retailer prefers RS and DP contracts to RP contract. Among them, RS contract has a broader scope of coordination, while DP contract can achieve a higher profit. The results can serve as insights for decision-makers to choose the most appropriate financial discount contract.

Citation: Chong Zhang, Yaxian Wang, Ying Liu, Haiyan Wang. Coordination contracts for a dual-channel supply chain under capital constraints. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020031
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##### References:
Model structure of the supply chain
Sequences of event and decisions
Effects of on optimal solutions
Effects of ${I_m}$ on optimal solutions
Effects of $\xi$ on optimal solutions
Effects of $\mu$ on optimal solutions
Effects of $\kappa$ on optimal solutions
Comparison of the total profit of the three contracts
Notations and descriptions used in the paper
 Parameters $c$ The unit production cost of products, ＄/unit. $w$ The unit wholesale price of products, ＄/unit. $I_i$ The interest charged per dollar per year, ＄/year, $i = m, r$ refer to the manufacturer and the retailer. $N$ The length of advance payment period offered by the retailer in years. $a$ Market primary demand, $a > 0$ $\lambda$ The cross price sensitivity and $0 < \lambda < 1$, which can reveal that the effect of ownership price is greater than that of cross-price. $s$ The degree of customer loyalty to the retail channel and $0 < s < 1$, and $1 - s$ represents the degree of customer loyalty to the direct channel. $L$ Retailer's advance payment to manufacturer as a function of >$c$, ${D_r}$, ${D_d}$ and $B$. $L = c({D_r}{{ + }}{D_d}) - B$, where $B \le c({D_r} + {D_d})$ $\le w{D_r}{ + }B$. ${\pi _i}$ The supply chain member's annual profit, $i = m, r, sc$ refer to the manufacturer, the retailer and the supply chain separately. $*$ Represents the optimal value of a decision variable. Decision variables ${p_d}$ Direct price decided by the manufacturer, ＄/unit, with ${p_d} > w > c$. ${p_r}$ Retail price decided by the retailer, ＄/unit, with ${p_r} > w > c$. Superscript $S$ The Stackelberg game with the retailer as the leader $C$ The centralized decision $RS$ The revenue-sharing contract $DP$ The retail channel price discount contract $PR$ The case with green products in the dual-channel supply chain Subscript $m$ Manufacturer $r$ Retailer $sc$ Supply chain
 Parameters $c$ The unit production cost of products, ＄/unit. $w$ The unit wholesale price of products, ＄/unit. $I_i$ The interest charged per dollar per year, ＄/year, $i = m, r$ refer to the manufacturer and the retailer. $N$ The length of advance payment period offered by the retailer in years. $a$ Market primary demand, $a > 0$ $\lambda$ The cross price sensitivity and $0 < \lambda < 1$, which can reveal that the effect of ownership price is greater than that of cross-price. $s$ The degree of customer loyalty to the retail channel and $0 < s < 1$, and $1 - s$ represents the degree of customer loyalty to the direct channel. $L$ Retailer's advance payment to manufacturer as a function of >$c$, ${D_r}$, ${D_d}$ and $B$. $L = c({D_r}{{ + }}{D_d}) - B$, where $B \le c({D_r} + {D_d})$ $\le w{D_r}{ + }B$. ${\pi _i}$ The supply chain member's annual profit, $i = m, r, sc$ refer to the manufacturer, the retailer and the supply chain separately. $*$ Represents the optimal value of a decision variable. Decision variables ${p_d}$ Direct price decided by the manufacturer, ＄/unit, with ${p_d} > w > c$. ${p_r}$ Retail price decided by the retailer, ＄/unit, with ${p_r} > w > c$. Superscript $S$ The Stackelberg game with the retailer as the leader $C$ The centralized decision $RS$ The revenue-sharing contract $DP$ The retail channel price discount contract $PR$ The case with green products in the dual-channel supply chain Subscript $m$ Manufacturer $r$ Retailer $sc$ Supply chain
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