Abstract: Product quality is fundamental factor for supply chain to obtain a competitive advantage and to build stable supply chain relationship. Retailers are the link between manufacturers and consumers. The effective product quality control by retailers not only can encourage manufacturers to improve the quality of products but also can ensure the quality of products purchased by consumers. In reality, the supply chain network in which the competition and cooperation coexist is complex. The research on the equilibrium state of the supply chain network has become a hot topic in recent years. In addition, fierce competition brings more uncertainty to the market demand, and supply chain members will show the characteristics of risk aversion which could affect them in making quality decisions. Therefore, this paper attempts to find the supply chain network equilibrium under quality control by retailers with risk-averse manufacturers and risk-averse retailers.
In order to control the product quality, the retailer inspects the product purchased from the manufacturer. If unqualified products are found, the retailer claims compensation from the manufacturer. Meanwhile, conditional Value-at-Risk is used to evaluate the risk aversion of manufacturers and retailers. The effect of risk aversion on the quality decision of supply chain members is considered. Firstly, under quality control, conditional Value-at-Risk model of manufacturers and retailers are established respectively. According to the principle of maximizing the conditional Value-at-Risk, the optimal decisions of manufacturers and retailers are solved. Secondly, The optimal economic behavior of players in the supply chain network is portrayed by way of variational inequalities theory and supply chain network equilibrium model is constructed. Then, the equilibrium pattern in terms of existence and uniqueness results is provided. Finally, the Levenberg-Marquardt method is built to solve the equilibrium model. The impacts of the changes of the parameters in the model on the equilibrium solutions are presented through the sensitivity analysis of the parameters. Through the analysis, the impact of quality control and risk aversion on the solution of equilibrium model and player's conditional Value-at-Risk can be revealed.
Several conclusions are drawn:first, with the risk aversion of manufacturers or retailers increasing, the product quality will improve and the manufacturers' conditional Value-at-Risk will increase more then retailers. In addition, if the risk aversion of manufacturers or retailers increases, their own conditional Value-at-Risk will reduce, and the manufacturers' conditional Value-at-Risk is very sensitive to changes in retailers' risk aversion. Further, the increase in the product qualification rate will increase the conditional risk value of both manufacturers and retailers. The manufacturers' benefits due to the increase in the qualification rate will be greater than those obtained by the retailers. Additionally when the manufacturers' qualification rate is too low, the retailers will refuse to order their products. Finally, increased selling price can promote the optimization of product quality.

Key words: quality control, conditional Value-at-Risk, network equilibrium, variational inequalities