Contents lists available at ScienceDirect Computers, Environment and Urban Systems journal homepage: www.elsevier.com/locate/ceus TRANSMax II: Designing a flexible model for transit route optimization Richard L. Church ⁎ , Timothy J. Niblett Department of Geography, University of California, Santa Barbara, Santa Barbara, CA 93106-4060, United States of America ARTICLE INFO Keywords: Covering path problems Covering tour problems Transit design Traveling salesman problem ABSTRACT Covering path problems date from the pioneering work of Current et al. (1984, 1985). Two basic forms were defined in their work: the shortest covering path problem and the maximal covering shortest path problem. These two problems differ in that one requires complete coverage by the defined path and the other involves determining path alternatives which cover as much as possible while keeping the path length as short as possible. The latter of these two problems, the maximal covering shortest path problem, embodies the two major goals in transit planning: that is, finding efficient paths which serve as many people as possible. Often transit routes are restricted to major road segments, and when that occurs, routes do not compete with one another unless they overlap along a street segment or at an intersection. In addition, coverage distances can be quite small, barely extending to other streets. Given this type of situation, Curtin and Biba (2011) developed a model called TRANSMax (Transit Route Arc-Node Service Maximization), which maximizes node and arc service, where service coverage is defined for only those street and node segments that are part of a route. They based their model on a structure first proposed by Vajda (1961) in formulating and solving the traveling salesman problem. Because of this structure, we demonstrate that it is possible that a route generated by their original TRANSMax model may not be Pareto optimal with respect to both distance and access. In this paper, we develop a flexible TRANSMax model formulation that finds Pareto Optimal solutions when the original form does not. We also present computational experience in solving this new model on the same street network of Curtin and Biba involving Richardson, Texas. This application allows us to make comparisons between this work and the original work of Curtin and Biba. Overall, we show that this new model can identify new, improved routes over the existing TRANSMax model. 1. Introduction The problem class of covering path location was first defined in the pioneering work of Current, ReVelle, and Cohon (1984) when they proposed the shortest covering path (SCP) problem. This original pro- blem involved finding the shortest path starting at a designated origin and ending at a pre-specified destination such that all nodes must be covered within a maximal service distance. The SCP problem represents a special form of the location set covering problem which involves lo- cating a minimum number of needed facilities that covers all nodes. That is, in both problems all nodes need to be covered; one by a small set of facilities and the other by the shortest possible path. The devel- opment of the SCP was followed by that of the Maximal Covering/ Shortest Path (MCSP) problem which was formulated by Current, ReVelle, & Cohon, 1985. This problem involves relaxing the require- ment for total coverage and represents finding those solutions which simultaneously maximize coverage and minimize path length. The backbone in structuring and solving the SCP/MCSP problems is based on tour-breaking constraints that are borrowed from the Traveling Salesman Problem (TSP) literature. For example, Current et al. (1984 & 1985 based their formulation and solution process on TSP constraints developed by Dantzig, Fulkerson, and Johnson (1954). The basic idea is that without such constraints, solutions will have disconnected cycles and independent elements in a solution in addition to a covering path. 1 Because of the difficulty in solving problems with TSP constraints, a number of different formulations to the TSP have been developed and tested (see for example, Miller, Tucker, and Zemlin (1960); Vajda (1961); and Gavish and Graves (1978)). An excellent review can be found in Orman and Williams (2006). Virtually all work on solving the SCP/MCSP problems have been based on the approach of Current et al. (1984 & 1985. There are several exceptions to this: Niblett and Church (2016), Curtin and Biba (2011), and Capar, Kuby, Leon, and Tsai https://doi.org/10.1016/j.compenvurbsys.2019.101395 Received 28 March 2019; Received in revised form 15 July 2019; Accepted 23 August 2019 ⁎ Corresponding author. E-mail address: rickchurch@ucsb.edu (R.L. Church). 1 If the desired path involves a pre-specified order in which nodes are visited, then the problem can be solved without needing tour-breaking constraints (Matisziw & Demir, 2012). However, most routing problems are more general and the order in which nodes are visited is not fixed. Computers, Environment and Urban Systems 79 (2020) 101395 0198-9715/ © 2019 Elsevier Ltd. All rights reserved. T