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.

我们研究了两个制造商之间的生产/库存竞争问题，每个制造商都采用联合生产系统，将一种原材料按固定比例加工成两种制造商之间可替代的产品。我们首先考虑两个制造商的产品需求不确定且独立的单期问题。一个制造商产品的未满足需求可以通过其他制造商的相同产品的剩余库存（如果可用）来满足，否则就会丢失。制造商通过选择他们自己的采购/加工数量来竞争替代需求。我们证明了唯一纳什均衡处理量的存在，并通过比较博弈的总均衡量与集中决策情况的均衡量来检验竞争的影响。然后我们研究了多时期生产/库存竞争问题，其中制造商根据产品的初始库存水平和竞争对手的决策在每个时期做出决策。我们证明了纳什均衡策略的存在性和唯一性，并且该均衡策略具有简单的基本股结构。然后我们考虑制造商的需求相关的情况。此外，研究制造商有能力的情况，我们表明存在独特的纳什均衡并且具有修改的基础油结构。我们进一步将结果推广到无限期情况和两个以上产品的情况。最后，进行数值研究以调查模型参数对平衡结果的影响，并从分析结果中获得一些管理见解。