Production/inventory competition between firms with fixed-proportions co-production systems

We study the production/inventory competition problem between two manufacturers each adopting the co-production system to process a raw material into two products in fixed proportions that are substitutable between the manufacturers. We first consider the one-period problem where the demands for the two manufacturers’ products are uncertain and independent. The unmet demand of one manufacturer's product can be met by the other manufacturer's leftover stock of the same product, if available, and is lost otherwise. The manufacturers compete for the substitute demands by choosing their own purchase/processing quantities. We show the existence of the unique Nash equilibrium processing quantity, and examine the impact of competition by comparing the total equilibrium quantity of the game with that of the centralized-decision case. We then study the multi-period production/inventory competition problem in which the manufacturers make decisions in each period according to the initial inventory levels of the products and the rivals’ decisions. We show the existence and uniqueness of the Nash equilibrium strategy and that the equilibrium strategy has the simple base-stock structure. We then consider the case where the manufacturers’ demands are correlated. In addition, studying the case where the manufacturers are capacitated, we show that the unique Nash equilibrium exists and has a modified base-stock structure. We further generalize the results to the infinite-period case and the case with more than two products. Finally, numerical studies are conducted to investigate the impacts of the model parameters on the equilibrium outcomes and get some managerial insights from the analytical findings.