Lagrangian Relaxation Methods for Solving the Minimum Fleet Size Multiple Traveling Salesman Problem with Time Windows

We consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints, for a homogeneous fleet of vehicles located at a common depot. This problem can be formulated as a network flow problem with additional time constraints. The pa...

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Bibliographic Details
Published inManagement science Vol. 34; no. 8; pp. 1005 - 1022
Main Authors Desrosiers, Jacques, Sauve, Michel, Soumis, Francois
Format Journal Article
LanguageEnglish
Published Linthicum, MD INFORMS 01.08.1988
Institute of Management Sciences
Institute for Operations Research and the Management Sciences
SeriesManagement Science
Subjects
Algorithms
Applied sciences
Commercial workers
Exact sciences and technology
fleet size
Flows in networks. Combinatorial problems
Integers
Lagrangian function
Lagrangian relaxation methods
multiple traveling salesman
Objective functions
Operational research and scientific management
Operational research. Management science
Scheduling
School busing
Theorie
Time windows
Tourenplanung
Travel expenses
Vehicle fleets
Vehicles
Integer programming
Travelling salesman problem
Relaxation method
Transportation problem
Combinatory problem
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Summary:We consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints, for a homogeneous fleet of vehicles located at a common depot. This problem can be formulated as a network flow problem with additional time constraints. The paper presents an optimal solution approach using the augmented Lagrangian method. Two Lagrangian relaxations are studied. In the first one, the time constraints are relaxed producing network subproblems which are easy to solve, but the bound obtained is weak. In the second relaxation, constraints requiring that each node be visited are relaxed producing shortest path subproblems with time window constraints and integrality conditions. The bound produced is always excellent. Numerical results for several actual school busing problems with up to 223 nodes are discussed. Comparisons with a set partitioning formulation solved by column generation are given.
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ISSN:0025-1909
1526-5501
DOI:10.1287/mnsc.34.8.1005