This paper addresses retail store location in a duopoly. One firm, the so-called leader, has already located several stores, while a second firm, the follower that offers a different product, considers locating its stores in the same area. Both products have some degree of complementarity. Marianov et al. (2018) studied this problem considering a customer decision rule that is binary (winner-take-all) and deterministic (there are no uncertainties related to customers’ behavior). In that work, they concluded that multipurpose shopping has a strong influence on optimal store location. This paper extends that work by introducing random utility customer choice rules based on the Multinomial Logit model. Along these lines, we propose the new Partially Binary Logit rule, applicable to the case in which all stores of a firm have identical attributes, except for their location. In this case, a customer always chooses the least cost trip store of the firm to purchase its product. We propose one nonlinear and three linear formulations for the follower location problem, and a Branch and Cut method for one of them. Two of the formulations have a good performance for instances of up to at least 200 nodes. We compare the binary, Multinomial Logit, and Partially Binary Logit rules by analyzing the captured markets and location patterns for a large set of randomly generated 100-node instances.

本文讨论了双头垄断中的零售店位置。一家公司，即所谓的领导者，已经找到了几家门店，而另一家公司，即提供不同产品的追随者，考虑将其门店设在同一地区。两种产品都有一定程度的互补性。马里亚诺夫等人。(2018) 研究了这个问题，考虑了一个二元（赢家通吃）和确定性（不存在与客户行为相关的不确定性）的客户决策规则。在那项工作中，他们得出结论，多功能购物对最佳商店位置有很大影响。本文通过引入基于多项 Logit 模型的随机效用客户选择规则来扩展这项工作。沿着这些思路，我们提出了新的 Partially Binary Logit 规则，适用于公司所有商店具有相同属性的情况，除了他们的位置。在这种情况下，客户总是选择公司成本最低的旅行商店来购买其产品。我们为跟随者位置问题提出了一种非线性和三种线性公式，并为其中之一提出了一种分支和切割方法。其中两个公式对于多达至少 200 个节点的实例具有良好的性能。我们通过分析大量随机生成的 100 节点实例的捕获市场和位置模式来比较二进制、多项式 Logit 和部分二进制 Logit 规则。其中两个公式对于多达至少 200 个节点的实例具有良好的性能。我们通过分析大量随机生成的 100 节点实例的捕获市场和位置模式来比较二进制、多项式 Logit 和部分二进制 Logit 规则。其中两个公式对于多达至少 200 个节点的实例具有良好的性能。我们通过分析大量随机生成的 100 节点实例的捕获市场和位置模式来比较二进制、多项式 Logit 和部分二进制 Logit 规则。