An exact solution framework for the multiple gradual cover location problem.

*(English)*Zbl 1458.90408Summary: Facility and covering location models are key elements in many decision aid tools in logistics, supply chain design, telecommunications, public infrastructure planning, and many other industrial and public sectors. In many applications, it is likely that customers are not dichotomously covered by facilities, but gradually covered according to, e.g., the distance to the open facilities. Moreover, customers are not served by a single facility, but by a collection of them, which jointly serve them. In this paper we study the recently introduced multiple gradual cover location problem (MGCLP). The MGCLP addresses both of the issues described above.

We propose the first exact approach to solve the MGCLP. In particular, we provide four different mixed-integer programming formulations for the MGCLP, all of them exploiting the submodularity of the objective function and developed a branch-and-cut framework based on these formulations. The framework is further enhanced by starting and primal heuristics and preprocessing procedures based on domination between facilities.

The computational results show that our approach allows to effectively address different sets of instances. We provide equal or better solution values for 33 out of 40 instances from the literature, and provide proof of optimality for 20 of them (15 instances more with respect to previously published results). Many of these instances can be solved within a minute. We also introduce new instances with different characteristics and analyze the dependence of the structure of the obtained solutions with respect to such characteristics.

We propose the first exact approach to solve the MGCLP. In particular, we provide four different mixed-integer programming formulations for the MGCLP, all of them exploiting the submodularity of the objective function and developed a branch-and-cut framework based on these formulations. The framework is further enhanced by starting and primal heuristics and preprocessing procedures based on domination between facilities.

The computational results show that our approach allows to effectively address different sets of instances. We provide equal or better solution values for 33 out of 40 instances from the literature, and provide proof of optimality for 20 of them (15 instances more with respect to previously published results). Many of these instances can be solved within a minute. We also introduce new instances with different characteristics and analyze the dependence of the structure of the obtained solutions with respect to such characteristics.

##### MSC:

90B80 | Discrete location and assignment |

90C11 | Mixed integer programming |

90C57 | Polyhedral combinatorics, branch-and-bound, branch-and-cut |

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