Abstract This paper addresses a generalized version of the facility location problem with customer preferences which includes an additional constraint on the number of customers which can be allocated to each facility. The model aims to minimize the total cost due to opening facilities and allocating customers while taking into account both customer preferences for the facilities and these cardinality constraints. First, two approaches to deal with this problem are proposed, which extend the single level and bilevel formulations of the problem in which customers are free to select their most preferred open facility. After analyzing the implications of assuming any of the two approaches, in this research, we adopt the approach based on the hierarchical character of the model which leads to the formulation of a bilevel optimization problem. Then, taking advantage of the characteristics of the lower level problem, a single level reformulation of the bilevel optimization model is developed based on duality theory which does not require the inclusion of additional binary variables. Finally, we develop a simple but effective matheuristic for solving the bilevel optimization problem whose general framework follows that of an evolutionary algorithm and exploits the bilevel structure of the model. The chromosome encoding pays attention to the upper level variables and controls the facilities which are open. Then, an optimization model is solved to allocate customers in accordance with their preferences and the availability of the open facilities. A computational experiment shows the effectiveness of the matheuristic in terms of the quality of the solutions yielded and the computing time.

中文翻译：

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用于解决具有基数约束和偏好的设施位置问题的双层方法的数学方法

摘要 本文解决了具有客户偏好的设施位置问题的广义版本，其中包括对可以分配给每个设施的客户数量的附加约束。该模型旨在最大限度地减少由于开放设施和分配客户而导致的总成本，同时考虑到客户对设施的偏好和这些基数约束。首先，提出了两种处理这个问题的方法，它们扩展了问题的单层和双层公式，客户可以自由选择他们最喜欢的开放设施。在分析了假设这两种方法中的任何一种的含义之后，在本研究中，我们采用基于模型的分层特征的方法，这导致了双层优化问题的制定。然后，利用低层问题的特点，基于对偶理论开发了双层优化模型的单层重构，不需要包含额外的二元变量。最后，我们开发了一种简单但有效的数学算法来解决双层优化问题，其一般框架遵循进化算法的框架并利用模型的双层结构。染色体编码关注上层变量并控制开放的设施。然后，求解优化模型以根据客户的偏好和开放设施的可用性来分配客户。计算实验显示了数学在产生的解决方案的质量和计算时间方面的有效性。