optimizing wholesale and retail prices, quality and marketing improvement efforts in a closed-loop supply chain using a game theory

this paper develops a closed-loop supply chain including one manufacturer, one retailer and a third party, in which the manufacturer produces a product under appropriate quality and sells it through a retailer in the market. the retailer pays the expense of marketing effort and a third party collects the used products from consumers and sends to the manufacturer. the manufacturer remanufactures the recived second hand products such that the quality of the remanufactured items bacome as same as the new ones. we use two different stackelberg games, and the decision vraibles of the model are the quality level, marketting expenditure, whole sale and retail prices. after investigating the concavity of the profit functions, the optimal closed-form solutions for all decision varaibales are derived. subsequently, to show the applicability of the proposed models, a numerical example is presented and a sensitivity anslysis is then conducted to investigate the effects of key parameters on the values of decision variables.