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**The multiple depot, multiple traveling salesmen facility-location problem: Vehicle range, service frequency, and heuristic implementations.**
*(English)*
Zbl 1082.90058

Summary: At the core of many delivery-logistics problems is the multiple depot, multiple traveling salesmen facility-location problem (MDMTSFLP). This paper addresses a mixed integer MDMTSFLP formulation, expanded to include vehicle range and multiple service-frequency requirements. Here, vehicle range can be interpreted as limitation on tour length, often due to crew duty day restriction. Once defined, we use this optimization formulation to validate a heuristic solution for location and routing. Our heuristic employs a combination of the minimum spanning forest (MSF) and a modified Clarke-Wright (CW) procedure. The spanning forest is used for geographic partitioning and for depot location, while the CW routes the multiple aircraft fleets. Generalizing from a minimum \(K\)-tree, the MSF is an exact \(O(|I|^3)\) method for partitioning the demands into service regions and in locating regional depots, where \(|I|\) is the number of nodes.

The Defense Courier Service (DCS) aerial network provides a huge instance of the hierarchical multiple-frequency, range-restricted MDMTSFLP. For this network, we find our MSF/CW procedure to be suboptimal by 3.86 percent on average over 16 validation runs, with no run worse than 20.27 percent. We show that several depots of the DCS may be closed without a large increase of routing cost. The model and solution procedure presented here have implications upon many other logistics problems, which are typically characterized by multiple vehicular visits and shortage of crew duty days.

The Defense Courier Service (DCS) aerial network provides a huge instance of the hierarchical multiple-frequency, range-restricted MDMTSFLP. For this network, we find our MSF/CW procedure to be suboptimal by 3.86 percent on average over 16 validation runs, with no run worse than 20.27 percent. We show that several depots of the DCS may be closed without a large increase of routing cost. The model and solution procedure presented here have implications upon many other logistics problems, which are typically characterized by multiple vehicular visits and shortage of crew duty days.

### MSC:

90B85 | Continuous location |

90C27 | Combinatorial optimization |

90C59 | Approximation methods and heuristics in mathematical programming |

90C11 | Mixed integer programming |

90B06 | Transportation, logistics and supply chain management |

### Keywords:

Combined facility location and routing; Minimum K-tree; Multiple traveling salesmen; Multiple service frequency; Split delivery; Minimum spanning forest; Clarke-Wright heuristic
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\textit{Y. Chan} and \textit{S. F. Baker}, Math. Comput. Modelling 41, No. 8--9, 1035--1053 (2005; Zbl 1082.90058)

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