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Posted 2023-03-16

Vehicle scheduling is one important step in the hierarchical planning process
in public transportation. The Multiple-Depot Vehicle Scheduling Problem
(MDVSP) is to assign a fleet of vehicles, possibly stationed at several garages, to a given set of passenger trips such that operational, company-specific, technical, and further side constraints are satisfied and the available resources are employed as efficiently as possible. In the last three decades considerable research has gone into the development of academic as well as practice-oriented solution techniques for the A/'P-hard MDVSP and special often polynomially solvable cases of it. Review articles on this topic are, for instance,
Desrosiers/Dumas/Solomon/Soumis (1995), Daduna/Paixäo (1995) and Bussieck/Winter/Zimmermann (1997).
The most successful solution approaches for the MDVSP are based on network flow models and their integer programming analogues. In the liter ature, there are two basic mathematical models of this type: irst, a direct arcoriented model leading to a multicommodity flow problem and, second, a path-oriented model leading to a set partitioning problem. The latter can also be derived from Dantzig-Wolfe decomposition applied to the first. Both approaches lead to large-scale integer programs, and column generation techniques
are required to solve their LP relaxations. We shall explicitly discuss the differences between these two models in Section 3.
We investigate in this paper the solution of the multicommodity flow for mulation. Solution techniques for models of this flavour have been discussed in various articles: Carpaneto/Dell'Amico/Fischetti/Toth (1989) describe an integer LP (ILP) formulation based on an arc-oriented assignment problem with additional path-oriented flow conservation constraints. They apply a so-called "additive lower bounding" procedure to obtain a lower bound for their ILP formulation. Ribeiro/Soumis (1994) show that this additive lower bounding is a special case of Lagrangean relaxation and its corresponding subgradient method. Forbes/Holt/Watts (1994) solve the integer linear programming
formulation of the multicommodity flow model by branch-and-bound. The sizes of the problems that have been solved to optimality in these publications are relatively small involving up to 600 timetabled trips and 3 depots.
The contribution of this paper is the efficient solution of the ILP (de rived from the multicommodity flow formulation) by means of LP column generation techniques. We use a new technique, called Lagrangean pricing, that is based on Lagrangean relaxations of the multicommodity flow model Embedded within a branch-and-cut frame, this method makes it possible to solve problems from practice to proven optimality. Lagrangean pricing has been developed independently at the same time by Fischetti/Vigo (1996)
and Fischetti/Toth (1996)
for solving the Asymmetric Travelling Saleman Problem and the Resource-Constrained Arborescence Problem.

参考技术A My god. 赏我200分都要考虑下