Three essays on sequencing and routing problems

Abstract

In this thesis we study different combinatorial optimization problems. These problems arise in many practical settings where there is a need for finding good solutions fast. The first class of problems we study are vehicle routing problems, and the second type of problems are sequencing problems. We study approximation algorithms and local search heuristics for these problems. First, we analyze the Vehicle Routing Problem (VRP) with and without split deliveries. In this problem, we have to route vehicles from the depot to deliver the demand to the customers while minimizing the total traveling cost. We present a lower bound for this problem, improving a previous bound of Haimovich and Rinnooy Kan. This bound is then utilized to improve the worst-case approximation algorithm of the Iterated Tour Partitioning (ITP) heuristic when the capacity of the vehicles is constant. Second, we analyze a particular case of the VRP, when the customers are uniformly distributed i.i.d. points on the unit square of the plane, and have unit demand. We prove that there exists a constant c > 0 such that the ITP heuristic is a 2 - c approximation algorithm with probability arbitrarily close to one as the number of customers goes to infinity. This result improves the approximation factor of the ITP heuristic under the worst-case analysis, which is 2. We also generalize this result and previous ones to the multi-depot case. Third, we study a language to generate Very Large Scale Neighborhoods for sequencing problems. Local search heuristics are among the most popular approaches to solve hard optimization problems.

(cont.) Among them, Very Large Scale Neighborhood Search techniques present a good balance between the quality of local optima and the time to search a neighborhood. We develop a language to generate exponentially large neighborhoods for sequencing problems using grammars. We develop efficient generic dynamic programming solvers that determine the optimal neighbor in a neighborhood generated by a grammar for a list of sequencing problems, including the Traveling Salesman Problem and the Linear Ordering Problem. This framework unifies a variety of previous results on exponentially large neighborhoods for the Traveling Salesman Problem.