This article comprises the first theoretical and computational study on mixed integer programming (MIP) models for the connected facility location problem (ConFL). ConFL combines facility location and Steiner trees: given a set of customers, a set of potential facility locations and some inter-connection nodes, ConFL searches for the minimum-cost way of assigning each customer to exactly one open facility, and connecting the open facilities via a Steiner tree. The costs needed for building the Steiner tree, facility opening costs and the assignment costs need to be minimized. We model ConFL using seven compact and three mixed integer programming formulations of exponential size. We also show how to transform ConFL into the Steiner arborescence problem. A full hierarchy between the models is provided. For two exponential size models we develop a branch-and-cut algorithm. An extensive computational study is based on two benchmark sets of randomly generated instances with up to 1300 nodes and 115,000 edges. We empirically compare the presented models with respect to the quality of obtained bounds and the corresponding running time. We report optimal values for all but 16 instances for which the obtained gaps are below 0.6%.

中文翻译：

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用于连接设施位置的 MIP 模型：理论和计算研究

本文包括对用于连接设施位置问题 (ConFL) 的混合整数规划 (MIP) 模型的首次理论和计算研究。ConFL 结合了设施位置和 Steiner 树：给定一组客户、一组潜在设施位置和一些互连节点，ConFL 搜索将每个客户准确分配到一个开放设施并连接开放设施的最低成本方式通过斯坦纳树。需要最小化构建 Steiner 树所需的成本、设施开放成本和分配成本。我们使用指数大小的七个紧凑型和三个混合整数规划公式对 ConFL 进行建模。我们还展示了如何将 ConFL 转换为 Steiner 树状光问题。提供了模型之间的完整层次结构。对于两个指数大小的模型，我们开发了一种分支切割算法。一项广泛的计算研究基于两个基准集随机生成的实例，最多 1300 个节点和 115,000 个边。我们根据获得的边界的质量和相应的运行时间，凭经验比较了所提出的模型。我们报告了除 16 个实例之外的所有实例的最佳值，其中获得的差距低于 0.6%。