Analyses of Advanced Iterated Tour Partitioning Heuristics for Generalized Vehicle Routing Problems Anupam Seth , Diego Klabjan , Placid M. Ferreira Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL June 26, 2011 Abstract: Theoretical analyses of a set of iterated-tour partitioning vehicle routing algorithms applicable to a wide variety of commonly-used vehicle routing problem variants are presented. We analyze the worst- case performance of the algorithms and establish tightness of the derived bounds. Among other variants we capture the cases of pick-up and delivery, and multiple depots. We also introduce brand new concepts such as mobile depots, partitioning of customer nodes into groups, and potential opportunistic under-utilization of vehicle capacity by only partially loading the vehicle, among others, which arise from a printed circuit board application. The problems studied are of critical importance in many practical applications. Keywords: vehicle routing, worst-case analysis, printed circuit board, approximation algorithms 1 Introduction Logistics in today’s increasingly global economies has become a more important field than ever before. Even though logistics in itself has a very significant business impact, nevertheless, the breadth of applications of vehicle routing goes well beyond the traditional trucking industry. Vehicle routing algorithms play an im- portant role in vehicle routing related information systems. Theoretical analyses of vehicle routing problems have led to significant insights into the nature and complexity of the proposed algorithms and they help in better understanding the potentials and limitations of such algorithms. A novel and recent area of applications of vehicle routing is production planning of printed circuit board assembly on an automated (primarily surface-mount type) equipment, for example, on the collect-and-place or pick-and-place type machines [27]. More specifically, collect-and-place machines are typically used for high- volume, high-flexibility production and are becoming increasingly popular in industrial settings because of their inherently advantageous characteristics and capabilities. The placement sequencing problem, i.e., the problem of planning the pick-up and delivery of electronic components or chips from feeder trays on the side of the machine onto a bare printed circuit board (PCB) in the center on such machines is an NP-hard problem that turns out to be essentially a generalization of the vehicle routing problem (VRP) in several different dimensions. Several interesting variants appear based on the manufacturing scenario and configuration of the machine, which are generalizations of many well-known and standard vehicle routing problem variants that have been studied in the past. Thus, existing algorithms and research do not directly apply to the problems motivated by the PCB context, but they are interesting generalizations of many important standard VRP variants, and hence a study of these problems is of great value. PCB manufacturing plays a very important role in today’s economy. Global revenues for the PCB industry exceeded 50 billion in 2007 and are expected to reach more than 76 billion in 2012 [22]. A PCB is a flat board that carries the chips and various other electronic components. This board is made up of alternating layers of copper and plastic, with the etching process performed on the copper layers to provide interconnects. These boards are capable of holding several components depending on the required specifications and produce complex interconnections. PCBs can be of different sizes and varying densities and are manufactured in automated assembly lines where high-speed placement machines put components 1