In this paper, we study a facility location problem within a competitive market context, where customer demand is predicted by a random utility choice model. Unlike prior research, which primarily focuses on simple constraints such as a cardinality constraint on the number of selected locations, we introduce routing constraints that necessitate the selection of locations in a manner that guarantees the existence of a tour visiting all chosen locations while adhering to a specified tour length upper bound. Such routing constraints find crucial applications in various real-world scenarios. The problem at hand features a non-linear objective function, resulting from the utilization of random utilities, together with complex routing constraints, making it computationally challenging. To tackle this problem, we explore three types of valid cuts, namely, outer-approximation and submodular cuts to handle the nonlinear objective function, as well as sub-tour elimination cuts to address the complex routing constraints. These lead to the development of two exact solution methods: a nested cutting plane and nested branch-and-cut algorithms, where these valid cuts are iteratively added to a master problem through two nested loops. We also prove that our nested cutting plane method always converges to optimality after a finite number of iterations. Furthermore, we develop a local search-based metaheuristic tailored for solving large-scale instances and show its pros and cons compared to exact methods. Extensive experiments are conducted on problem instances of varying sizes, demonstrating that our approach excels in terms of solution quality and computation time when compared to other baseline approaches.

翻译：本文研究了竞争市场背景下的设施选址问题，其中客户需求由随机效用选择模型预测。与现有研究主要关注简单的约束（如所选设施数量的基数约束）不同，我们引入了路由约束，要求在选择设施时必须保证存在一条能够访问所有选定设施且满足指定路径长度上限的路线。这类路由约束在多种实际场景中具有关键应用。该问题因采用随机效用而具有非线性目标函数，加之复杂的路由约束，导致计算极具挑战性。为解决此问题，我们探索了三种有效割平面：外逼近割与子模割用于处理非线性目标函数，以及子环消除割用于应对复杂路由约束。由此开发出两种精确求解方法：嵌套割平面算法与嵌套分支切割算法，通过两个嵌套循环将有效割平面迭代添加到主问题中。我们证明了嵌套割平面算法在有限次迭代后必然收敛至最优解。此外，我们设计了一种基于局部搜索的元启发式算法以求解大规模实例，并分析了其相较于精确方法的优劣。通过在不同规模的实例上进行大量实验，结果表明我们的方法在求解质量与计算时间上均优于其他基准方法。