Abstract:

Substitute competition is a common phenomenon in economy and business that affects inventory management significantly. The competition problem between two firms is studied with the initial inventory level and the capacity constraints. Two substitutable products are operated by two firms respectively. The unmet demand of one product can be met by the other product’s leftover stock， if available， and is lost otherwise. The firms compete for the substitute demand by choosing their own order quantities in multi-period. The problem is formulated as a dynamic game under independent stochastic demands and it is shown that the inventory game is submodular and the response functions are contraction maps， so the existence of a unique Nash equilibrium is guaranteed. It is found that the best response function is strictly decreasing in the rival’s decisions and decreasing in the initial inventory levels of the two products. It is found that the base-stock policy is the equilibrium policy. The capacitated case is considered where each firm can hold no more than a certain quantity of the product in each period and find that the equilibrium policy is a modified base-stock strategy. Finally， numerical studies are conducted to compare our strategy with the myopic policy and to generate managerial insights from the analytical findings. Compared with the existing literature， it contributes to the fields of multi-period inventory management and dynamic game in this paper.

Key words: initial inventory level, capacity constraint, dynamic inventory game, base-stock policy