In this paper we consider a two-echelon supply chain with one supplier that controls a limited resource and a finite set of manufacturers who need to purchase this resource. We analyze the effect of the limited resource on the horizontal cooperation of manufacturers. To this end, we use cooperative game theory and the existence of stable distributions of the total profit among the manufacturers as a measure of the possibilities of cooperation. The game theoretical model that describes the horizontal cooperation involves externalities, which arise because of the possible scarcity of the limited resource and the possible coalition structures that can be formed. Furthermore, manufacturers do not know how the supplier will allocate the limited resource, therefore, how much of this resource they will obtain is uncertain for all concerned. Nevertheless, when the limited resource is not scarce for the grand coalition, the existence of stable distributions of the total profit is guaranteed and consequently the collaboration among the manufacturers is profitable for them all. In the event that the limited resource is insufficient for the grand coalition, we introduce a new cooperative game that assesses the expectations of each coalition of manufacturers regarding the amount of the limited resource they can obtain. We analyze two extreme expectations: the optimistic and the pessimistic. In the optimistic case, we cannot reach a conclusion regarding the full cooperation of the manufacturers. In the pessimistic case, with one reasonable assumption, the existence of stable distributions of the total profit is guaranteed and as a result the collaboration among manufacturers is a win–win deal.

中文翻译：

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在线性生产过程中与控制有限资源的供应商进行横向合作

在本文中，我们考虑一个两级供应链，其中一个供应商控制着有限的资源，而有限的一组制造商需要购买这种资源。我们分析了有限资源对制造商横向合作的影响。为此，我们使用合作博弈理论和制造商之间总利润稳定分配的存在来衡量合作可能性。描述横向合作的博弈论模型涉及外部性，这是由于有限资源的可能稀缺和可能形成的联盟结构而产生的。此外，制造商不知道供应商将如何分配有限的资源，因此，对于所有相关人员来说，他们将获得多少资源是不确定的。不过，当有限的资源对于大联盟来说并不稀缺时，总利润的稳定分配就可以得到保证，因此制造商之间的合作对他们所有人都是有利的。如果有限的资源不足以容纳大型联盟，我们将引入一个新的合作博弈，该博弈评估每个制造商联盟对他们可以获得的有限资源的期望。我们分析了两个极端期望：乐观和悲观。在乐观的情况下，我们无法得出有关制造商充分合作的结论。在悲观的情况下，只要有一个合理的假设，就可以保证总利润的稳定分配，因此制造商之间的合作是双赢的。