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doi: 10.3934/jimo.2021042
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## Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain

 1 School of Economics and Management, Hebei University of Technology, Tianjin 300401, China 2 The Center for Enterprise Informatization and Management Innovation, Tianjin Humanities and Social Science Key Research Base, Tianjin 300401, China

* Corresponding author: Jing Zhang

Received  April 2020 Revised  December 2020 Early access March 2021

Fund Project: The first author is supported by NSF grant 14BGL055, 18BGL012

By applying Stackelberg game theory, this paper investigates the supply chain with a risk-neutral retailer and a risk-averse supplier, measuring risk-averse behavior by using conditional value-at-risk (CVaR). The equilibrium solutions of the supplier's wholesale price and the retailer's order quantity are obtained under two financing strategies: supplier financing (SF) and supplier investment (SI). It is found that the supplier's risk aversion is a crucial factor affecting both parties' financing decisions, and the supplier should offer different financing strategies to the retailer based on his risk attitude and the profit-sharing coefficient. However, the retailer prefers SF regardless of the supplier's risk aversion. Taking bank credit financing as a basic model, the advantages of SF and SI have been investigated. A Pareto improvement region for the two finance strategies has been identified and some suggestions are provided for the supplier's optimal utility. Then we extend to the situation that both parties are risk-averse and use the financing cost-sharing mechanism to achieve centralized decision-making.

Citation: Kai Kang, Taotao Lu, Jing Zhang. Financing strategy selection and coordination considering risk aversion in a capital-constrained supply chain. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021042
##### References:

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##### References:
Relationship between each participant and financing mechanism
The effect of $\eta$ and $R$ on $w^{F*}$
The effect of $\eta$ and $c$ on $q^{F*}$
Changes of $w^{I*}$ with $c$, $\eta$, $\beta$ and $\alpha$
Changes of $q^{I*}$ with $c$, $\eta$, $\beta$ and $\alpha$
Pareto improvement region under dominant of SF
Pareto improvement region under dominant of SI
Range of the transfer payments that can be assigned to a supplier under SF
Range of the transfer payments that can be assigned to a supplier under SI
Summary of notations
 Parameters and variables Description $w$ Supplier's wholesale price for each product $q$ Retailer's order quantity of the product Parameters $\quad$ $p$ Unit market price $c$ Unit production cost of the product $y$ Market demand which is a random variable with F(.) and f(.) as the $\quad$ cumulative distribution function and the density function $R$ The interest rate under SF $\alpha$ Supplier's sharing coefficient of retailer's profit $\beta$ Supplier investment as a percentage of retailers capital needs $\psi$ Supplier's financing cost-sharing coefficient $\eta$ Supplier's degree of risk aversion $T$ Transfer payment $\delta$ Influence of the retailer Index $\quad$ $r$ Parameter's related to the retailer $s$ Parameter's related to the supplier $i$ Superscript, retailer's financing scheme, where $i=\{F,I,B\}$ $C$ The status related to centralized decision $a$ Risk-averse scenario $n$ Risk-neutral scenario
 Parameters and variables Description $w$ Supplier's wholesale price for each product $q$ Retailer's order quantity of the product Parameters $\quad$ $p$ Unit market price $c$ Unit production cost of the product $y$ Market demand which is a random variable with F(.) and f(.) as the $\quad$ cumulative distribution function and the density function $R$ The interest rate under SF $\alpha$ Supplier's sharing coefficient of retailer's profit $\beta$ Supplier investment as a percentage of retailers capital needs $\psi$ Supplier's financing cost-sharing coefficient $\eta$ Supplier's degree of risk aversion $T$ Transfer payment $\delta$ Influence of the retailer Index $\quad$ $r$ Parameter's related to the retailer $s$ Parameter's related to the supplier $i$ Superscript, retailer's financing scheme, where $i=\{F,I,B\}$ $C$ The status related to centralized decision $a$ Risk-averse scenario $n$ Risk-neutral scenario
Impact of production cost
 $c$ $w^{F*}$ $q^{F*}$ $\pi_{r}^{F*}$ $CVaR_{s}^{F*}$ $w^{I*}$ $q^{I*}$ $\pi_{r}^{I*}$ $CVaR_{s}^{I*}$ 0.30 1.1500 6.6400 9.3151 1.7040 1.8398 6.6000 3.6000 3.6899 0.35 1.1812 6.5800 9.0218 1.3735 1.8950 6.5333 3.4500 3.6683 0.40 1.2130 6.5200 8.7248 1.0460 1.9513 6.4666 3.3000 3.6433 0.45 1.2454 6.4600 8.4241 0.7214 2.0089 6.4000 3.1499 3.6150
 $c$ $w^{F*}$ $q^{F*}$ $\pi_{r}^{F*}$ $CVaR_{s}^{F*}$ $w^{I*}$ $q^{I*}$ $\pi_{r}^{I*}$ $CVaR_{s}^{I*}$ 0.30 1.1500 6.6400 9.3151 1.7040 1.8398 6.6000 3.6000 3.6899 0.35 1.1812 6.5800 9.0218 1.3735 1.8950 6.5333 3.4500 3.6683 0.40 1.2130 6.5200 8.7248 1.0460 1.9513 6.4666 3.3000 3.6433 0.45 1.2454 6.4600 8.4241 0.7214 2.0089 6.4000 3.1499 3.6150
Impact of risk aversion
 $\eta$ $w^{F*}$ $q^{F*}$ $\pi_{r}^{F*}$ $CVaR_{s}^{F*}$ $w^{I*}$ $q^{I*}$ $\pi_{r}^{I*}$ $CVaR_{s}^{I*}$ 0.25 0.9282 7.1000 11.4450 0.2049 1.4904 7.0555 4.6250 3.0361 0.30 1.2130 6.5200 8.7248 1.0460 1.9513 6.4666 3.3000 3.6433 0.35 1.5534 5.9400 5.6682 1.8870 2.5047 5.8777 1.9750 4.2505 0.40 1.9674 5.3600 2.2752 2.7279 3.1812 5.2888 0.6499 4.8577
 $\eta$ $w^{F*}$ $q^{F*}$ $\pi_{r}^{F*}$ $CVaR_{s}^{F*}$ $w^{I*}$ $q^{I*}$ $\pi_{r}^{I*}$ $CVaR_{s}^{I*}$ 0.25 0.9282 7.1000 11.4450 0.2049 1.4904 7.0555 4.6250 3.0361 0.30 1.2130 6.5200 8.7248 1.0460 1.9513 6.4666 3.3000 3.6433 0.35 1.5534 5.9400 5.6682 1.8870 2.5047 5.8777 1.9750 4.2505 0.40 1.9674 5.3600 2.2752 2.7279 3.1812 5.2888 0.6499 4.8577
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