Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees

• Published in: Operations research Vol. 42; no. 4; pp. 626 - 642
• Main Author: Fisher, Marshall L
• Format: Journal Article
• Language: English
• Published: Baltimore, Md INFORMS 01.07.1994; Operations Research Society of America; Institute for Operations Research and the Management Sciences
• Subjects: Algorithms; Cost efficiency; Heuristics; integer; Lagrangian function; Linear programming; Marketing strategies; Operations research; Optimal solutions; optimization algorithm; programming; relaxation/subgradient: k-tree based; Routing; Scheduling; Studies; Time windows; transportation; Travel expenses; Vehicle capacity; vehicle routing; Vehicles
• ISSN: 0030-364X; 1526-5463
• DOI: 10.1287/opre.42.4.626

We consider the problem of optimally scheduling a fleet of K vehicles to make deliveries to n customers subject to vehicle capacity constraints. Given a graph with n + 1 nodes, a K-tree is defined to be a set of n + K edges that span the graph. We show that the vehicle routing problem can be modeled as the problem of finding a minimum cost K-tree with two K edges incident on the depot and subject to some side constraints that impose vehicle capacity and the requirement that each customer be visited exactly once. The side constraints are dualized to obtain a Lagrangian problem that provides lower bounds in a branch-and-bound algorithm. This algorithm has produced proven optimal solutions for a number of difficult problems, including a well-known problem with 100 customers and several real problems with 25–71 customers.