Summary of “Optimal Solution of Vehicle Routing Problems Using Minimum K-Trees” by Marshall L. Fisher (1994)

The paper [1] has three main contributions: • Modeling VRP (Vehicle Routing Problem) into the minimum cost Ktree problem. • Dualizing the side constraints to obtain a Lagrangian problem which provides lower-bounds in B&B. • Performing the algorithm on real problems. 1. Problem Definition and Lagrangian Relaxation The vehicle routing problem has the following parameters: • K = the number of vehicles, • n = the number of customers, • b = vehicle capacity, • ai = delivery size to customer i, • cij = travel cost between points i and j, and the problem is to assign a path for each vehicle to start from the depot and deliver some items to the customers satisfying the size and capacity constraints and minimizing the cost. Discrete Optimization June 19, 2015 Let N0 = N ∪{0} denote the set of customers plus the depot. xij is a 0-1 variable denoting whether the edge (i, j) is selected in the solution or not. In this part the author models the VRP into K-tree instance in which 2K of edges connected to depot are selected. So define