Abstract: With the introduction of a low-carbon economy and green GDP, green supply chain management has been received increasingly attention by the scholars and the market administrators in present times. In real world, especially for some new green products, the relevant precise dates are difficult to obtain due to lack of historical data. In this situation, the market base and production cost can usually be predicted by some experts.
In this paper, in a two-stage green supply chain composed of one manufacturer and one retailer, the optimal price and greening level decisions in an uncertainty environment is researched, where the costs of the manufacturer and the retailer, and the parameters of the market demand are all regarded as nonnegative and mutually independent uncertain variables. Three non-cooperative games including two Stackelberg games of the manufacturer and the retailer respectively dominating supply chain, and one Vertical-Nash game of the manufacturer and the retailer having the equal market power are analyzed. In the Manufacturer-Stackelberg game, the greening level and the wholesale price are firstly set by the manufacturer, and then the profit margin is decided by the retailer. In the Retailer-Stackelberg game, the profit margin is firstly set by the retailer, and then the greening level and the wholesale price are decided by the manufacturer. In the Vertical-Nash game, the greening level, the wholesale price and the profit margin are set simultaneously and independently. Their optimal solutions are derived by the method of uncertainty theory.
Owing to the complicated forms of the solutions, a numerical example is given to compare the optimal solutions under three non-cooperative games. Based on the discussions, four main findings can be obtained. First, the profits of the manufacturer and the retailer are higher than those with crisp parameters. Second, the Manufacturer-Stackelberg game and the Retailer-Stackelberg game are the optimal strategy for the manufacturer and the retailer respectively, while the Vertical-Nash game is the optimal strategy for the customer. Third, with the uncertain degree of the coefficient of the greening level decreasing, the profit of the retailer are decreased in the three games, while, the profit of the manufacturer are decreased in the Manufacturer-Stackelberg game and the Retailer-Stackelberg game, and the profit of the manufacturer are increased in the Vertical-Nash game. Fourth, the profits of the manufacturer and the retailer increase as the uncertain degree of the green investment parameter decreases in the three games.

Key words: green supply chain, green level, uncertainty theory, uncertain variables