We study the maximum capture problem in facility location under random utility models, i.e., the problem of seeking to locate new facilities in a competitive market such that the captured user demand is maximized, assuming that each customer chooses among all available facilities according to a random utility maximization model. We employ the generalized extreme value (GEV) family of discrete choice models and show that the objective function in this context is monotonic and submodular. This finding implies that a simple greedy heuristic can always guarantee a $(1-1/e)$ approximation solution. We further develop a new algorithm combining a greedy heuristic, a gradient-based local search, and an exchanging procedure to efficiently solve the problem. We conduct experiments using instances of different sizes and under different discrete choice models, and we show that our approach significantly outperforms prior approaches in terms of both returned objective value and CPU time. Our algorithm and theoretical findings can be applied to the maximum capture problems under various random utility models in the literature, including the popular multinomial logit, nested logit, cross nested logit, and mixed logit models.

我们研究了随机效用模型下设施选址的最大捕获问题，即在竞争市场中寻找新设施以使捕获的用户需求最大化的问题，假设每个客户根据随机效用最大化模型。我们采用离散选择模型的广义极值 (GEV) 系列，并表明在这种情况下的目标函数是单调和子模的。这一发现意味着一个简单的贪婪启发式总是可以保证$(1-1/e)$近似解。我们进一步开发了一种结合贪婪启发式、基于梯度的局部搜索和交换过程的新算法，以有效地解决问题。我们使用不同大小的实例和不同的离散选择模型进行实验，我们表明我们的方法在返回的目标值和 CPU 时间方面都显着优于先前的方法。我们的算法和理论发现可以应用于文献中各种随机效用模型下的最大捕获问题，包括流行的多项式 logit、嵌套 logit、交叉嵌套 logit 和混合 logit 模型。