The channel members are following Nash game to decide their decisions that is both of the manufacturer and the retailer move simultaneously. The necessary conditions of optimization of the minimum expected profit of the retailer yields, Moreover, concavity condition is found from , ; ; and . Solving and for and , we get optimal of and and are presented in the The Proposition-11. necessary conditions of optimization of the minimum expected profit of the manufacturer yields, Optimal value of the decision variables (, , ) of the manufacturer are obtained from solving the above necessary conditions and are given in the Proposition-12. The manufacturer makes its decisions through a Nash game, then the manufacturer’s decision is independent from the retailer's decision. Note that that is, is a concave function of . Thus, for given and , optimal selling price () of the manufacturer in online channel is given by Also note that, that is, is a concave function of . for given , optimal delivery lead time () of the manufacturer in online channel is given by Now, consider the situation when none of the decision variables are given. To check the concavity of expected profit function of the manufacturer we compute its corresponding Hessian matrix, as follows If the principal minors are alternatively negative and positive, i.e., the kth order leading principal minor follows the sign of , then the minimum expected profit function of the manufacturer () for a is concave. Here , and Hence, the minimum expected profit function of the manufacturer () for any distribution will be concave if and at the stationary points. S4. Special case: deterministic demand in decentralized case Consider the special case where the demand is deterministic in both channels. That is, and normalize both and to be 0. Thus, the profit functions for the retailer and manufacturer are respectively
The channel members are following Nash game to decide their decisions that is both of the manufacturer and the retailer move simultaneously. The necessary conditions of optimization of the minimum expected profit of the retailer yields, Moreover, concavity condition is found from , ; ; and . Solving and for and , we get optimal of and and are presented in the Proposition-11. The necessary conditions of optimization of the minimum expected profit of the manufacturer yields, Optimal value of the decision variables (, , ) of the manufacturer are obtained from solving the above necessary conditions and are given in the Proposition-12. The manufacturer makes its decisions through a Nash game, then the manufacturer’s decision is independent from the retailer's decision. Note that that is, is a concave function of . Thus, for given and , optimal selling price () of the manufacturer in online channel is given by Also note that, that is, is a concave function of . for given , optimal delivery lead time () of the manufacturer in online channel is given by Now, consider the situation when none of the decision variables are given. To check the concavity of expected profit function of the manufacturer we compute its corresponding Hessian matrix, as follows If the principal minors are alternatively negative and positive, i.e., the kth order leading principal minor follows the sign of , then the minimum expected profit function of the manufacturer () for a is concave. Here , and Hence, the minimum expected profit function of the manufacturer () for any distribution will be concave if and at the stationary points. S4. Special case: deterministic demand in decentralized case Consider the special case where the demand is deterministic in both channels. That is, and normalize both and to be 0. Thus, the profit functions for the retailer and manufacturer are respectively
1-s2.0-S0377221718304867-mmc1.docx