Expectation and fractile models for decentralised distribution systems under demand uncertainty and their computational methods

In this study, we deal with the expectation and the fractile models for obtaining a Nash equilibrium point of the two-stage game for describing the competition and cooperation in decentralised distribution systems with stochastic demands, and develop computational methods. In the first stage of the equilibrium problem, each retailer independently determines the inventory level, and in the second stage for the coordination of retailers, the addition profit arising from the transshipment of the leftover inventories of all the retailers is maximised. Formulating the transshipment of the leftover inventories as a two-stage programming problem with simple recourse, we define an allocation rule based on the optimal dual solution of the transshipment problem which belongs to the core of the cooperative game. Using numerical examples, we demonstrate the effectiveness of the expectation and the fractile models, and examine the validity of their computational methods.