Cost allocation in some routing problems : a game theoretic approach

Abstract: In many situations a set of decision makers have the opportunity to cooperate. In this way they may reduce the total cost for satisfying their objectives. However, the reduction of cost is often not enough to motivate cooperation. The problem of how to divide the total cost (or gain) among the decision makers must also be solved.This thesis include the modeling of cost allocation problems related to some routing problems. The cost allocation problems are formulated as cooperative games and we compute cost allocations based on concepts from cooperative game theory, such as the core and the nucleolus. 'We illustrate the problems using real-life data from Norsk Hydro. We consider how to divide the cost of an optimal traveling salesman tour among the customers that were served on a tour, and how to divide the cost of au optimal solution to a vehicle routing problem with a heterogeneous fleet among the customers served. These problems are formulated as a traveling salesman game and a vehicle routing game, respectively. For these games, we present procedures based on constraint generation to decide if the core is empty or not, and to compute the nucleolus.One subproblem in the constraint generation procedures is a generalized multiple tour problem. In this problem each customer has a demand that either can be satisfied or not. If it is satisfied, revenue is collected and there are also travel costs to pay. A tabu search heuristic is used to identify which customers to serve and 011 which routes, with the objective to maximize the collected revenue minus the cost of traveling. We present numerical results based on a set of test instances.We also consider the node weighted Steiner tree problem. In this problem there are costs associated with connecting the nodes of a network to a tree. In addition, there is a potential revenue to collect at each node, if it is connected. The problem is to decide which nodes to connect, and how, so as to maximize the revenue collected minus the connecting costs. For this problem, we suggest a Lagrangean-based heuristic to produce strong lower bounds and to obtain feasible solutions. We present numerical results based on a set of test instances.