Abstract In this paper, we address a discrete facility location problem where a retailer aims at locating new facilities with possibly different characteristics. Customers visit the facilities based on their preferences which are represented as probabilities. These probabilities are determined in a novel way by using a fuzzy clustering algorithm. It is assumed that the sum of the probabilities with which customers at a given demand zone patronize different types of facilities is equal to one. However, among the same type of facilities they choose the closest facility, and the strength at which this facility covers the customer is based on two distances referred to as full coverage distance and gradual (partial) coverage distance. If the distance between the customer location and the closest facility is smaller (larger) than the full (partial) coverage distance, this customer is fully (not) covered, whereas for all distance values between full and partial coverage, the customer is partially covered. Both distance values depend on both the customer attributes and the type of the facility. Furthermore, facilities can only be opened if their revenue exceeds a certain threshold value. A final restriction is incorporated into the model by defining a minimum separation distance between the same facility types. This restriction is also extended to the case where a minimum threshold distance exists among facilities of different types. The objective of the retailer is to find the optimal locations and types of the new facilities in order to maximize its profit. Two versions of the problem are formulated using integer linear programming, which differ according to whether the minimum separation distance applies to the same facility type or different facility types. The resulting integer linear programming models are solved by three approaches: commercial solver CPLEX, heuristics based on Lagrangean relaxation, and local search implemented with 1-Add and 1-Swap moves. Apart from experimentally assessing the accuracy and the efficiency of the solution methods on a set of randomly generated test instances, we also carry out sensitivity analysis using a real-world problem instance.

中文翻译：

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考虑客户偏好的多类型设施的渐进覆盖位置问题

摘要在本文中，我们解决了一个离散设施位置问题，其中零售商旨在定位具有可能不同特征的新设施。客户根据以概率表示的偏好访问设施。这些概率是通过使用模糊聚类算法以一种新颖的方式确定的。假设给定需求区的顾客光顾不同类型设施的概率之和等于 1。但是，在同类设施中，他们选择最近的设施，该设施覆盖客户的强度基于两个距离，称为全覆盖距离和渐进（部分）覆盖距离。如果客户位置和最近设施之间的距离小于（大于）完全（部分）覆盖距离，则该客户被完全（未）覆盖，而对于完全覆盖和部分覆盖之间的所有距离值，该客户被部分覆盖. 两个距离值都取决于客户属性和设施类型。此外，设施只有在收入超过某个阈值时才能开放。通过定义相同设施类型之间的最小间隔距离，将最终限制合并到模型中。此限制也扩展到不同类型设施之间存在最小阈值距离的情况。零售商的目标是找到新设施的最佳位置和类型，以实现利润最大化。该问题的两个版本使用整数线性规划来表述，它们根据最小间隔距离是适用于相同设施类型还是不同设施类型而有所不同。生成的整数线性规划模型通过三种方法求解：商业求解器 CPLEX、基于拉格朗日松弛的启发式算法以及使用 1-Add 和 1-Swap 移动实现的局部搜索。除了在一组随机生成的测试实例上通过实验评估解决方法的准确性和效率之外，我们还使用现实世界的问题实例进行了敏感性分析。生成的整数线性规划模型通过三种方法求解：商业求解器 CPLEX、基于拉格朗日松弛的启发式算法以及使用 1-Add 和 1-Swap 移动实现的局部搜索。除了在一组随机生成的测试实例上通过实验评估解决方法的准确性和效率之外，我们还使用现实世界的问题实例进行了敏感性分析。生成的整数线性规划模型通过三种方法求解：商业求解器 CPLEX、基于拉格朗日松弛的启发式算法以及使用 1-Add 和 1-Swap 移动实现的局部搜索。除了在一组随机生成的测试实例上通过实验评估解决方法的准确性和效率之外，我们还使用现实世界的问题实例进行敏感性分析。