Online optimization in routing and scheduling

Abstract

In this thesis we study online optimization problems in routing and scheduling. An online problem is one where the problem instance is revealed incrementally. Decisions can (and sometimes must) be made before all information is available. We design and analyze (polynomial-time) online algorithms for a variety of problems. We utilize worst-case competitive ratio (and relaxations thereof), asymptotic and Monte Carlo simulation analyses in our study of these algorithms. The focus of this thesis is on online routing problems in arbitrary metric spaces. We begin our study with online versions of the Traveling Salesman Problem (TSP) and the Traveling Repairman Problem (TRP). We then generalize these basic problems to allow for precedence constraints, capacity constraints and multiple vehicles. We give the first competitive ratio results for many new online routing problems. We then consider resource augmentation, where we give the online algorithm additional resources: faster servers, larger capacities, more servers, less restrictive constraints and advanced information. We derive new worst-case bounds that are relaxations of the competitive ratio.

(cont.) We also study the (stochastic) asymptotic properties of these algorithms - introducing stochastic structure to the problem data, unknown and unused by the online algorithm. In a variety of situations we show that many online routing algorithms are (quickly) asymptotically optimal, almost surely, and we characterize the rates of convergence. We also study classic machine sequencing problems in an online setting. Specifically, we look at deterministic and randomized algorithms for the problems of scheduling jobs with release dates on single and parallel machines, with and without preemption, to minimize the sum of weighted completion times. We derive improved competitive ratio bounds and we show that many well-known machine scheduling algorithms are almost surely asymptotically optimal under general stochastic assumptions. For both routing and sequencing problems, we complement these theoretical derivations with Monte Carlo simulation results.