Large-Scale Transit Network Optimization

This article proposes a methodology for developing optimal transit networks (route structures and headways) that minimizes transit transfers and total user cost while maximizing service coverage, given information on transit demand, transit fleet size, and the street network of the transit service area. The research provides an effective mathematical computational tool with minimal reliance on heuristics. The methodology includes representation of transit route networks and solution search spaces, objective functions representing total user cost and unwillingness of users to make transfers, and a global search scheme based on simulated annealing. The methodology has been tested with published solutions to benchmark problems and has been applied to a large-scale realistic network optimization problem in Miami-Dade County, Florida. Introduction As congestion in large urban areas continues to worsen and gas prices began to rise in the recent years, the attractiveness of public transit as an alternative to private cars has also been growing. However, for a public transit system to help meet the growing travel demand and alleviate the congestion problem, it must be able to provide reasonable travel time and convenience relative to private vehicles. Travel time and convenience are affected directly by the configuration of a transit netJournal of Public Transportation, Vol. 9, No. 2, 2006 108 work (TN) and service frequency, although other service and traffic characteristics and pedestrian environment will also have an impact on the willingness of the public to use transit. The quality of a TN may be evaluated in terms of a number of parameters including route directness, service coverage, operator cost, transit user cost (including waiting, in-vehicle, and transfer times), and the average number of transfers required to accomplish a trip. Route directness may be measured by the additional travel time incurred to a transit user when a bus does not follow the most direct route between the user’s origin and destination. Service coverage refers to the percentage of total estimated demand (i.e., transit trips) that may be potentially satisfied by the transit services provided, based on a given transit route network. Operator cost is the cost to a transit property to provide transit services within a given network. Transfers are a result of not being able to provide direct services between all pairs of origins and destinations. Transfers are known to discourage transit use. According to a survey conducted by Stern (1996) of various transit agencies in the United States, about 58 percent of the respondents believed that transit riders were willing to transfer only once per trip. Reducing transfers, therefore, has great potential in increasing the attractiveness of public transit and ridership. Transfers may be reduced by optimizing transit network configuration, or optimally laying out transit routes such that the services are as direct as possible and transfers are minimized. Improvements of TN configuration may also lead to lower transit operating cost and more services provided, which, in turn, help increase transit use. In TN optimization, route network layouts and route headways are sought that minimize the overall cost of providing transit services, which is generally considered to have two components: user cost and operator cost. Unfortunately, TN design optimization processes that attempt to find global optimal solutions from a search space with reasonable completeness suffer from combinatorial intractability. Newell (1979) observes the difficulty in developing efficient TN optimization methods with traditional mathematical programming techniques and points out that TN design optimization “is generally a nonconvex (even concave) optimization problem for which no simple procedure exists short of direct comparisons of the various local minima.” Furthermore, the resultant system for a TN problem is usually a NP-hard, mixed combinatorial optimization problem that is unlikely to be solved with traditional mathematical optimization techniques. The NP-hard problem (the hard problem in nondeterministic polynomial problem/ algorithm class) refers to a problem for which the number of elementary numeriLarge-Scale Transit Network Optimization 109 cal operations is not likely to be expressed or bounded by a function of polynomial form where the variable(s) of the function reflect(s) the size of the problem. The NP-hard intractability is due to the need to search for optimal solutions from a large search space made up by all possible solutions. A mixed problem refers to a problem that involves both continuous and discrete variables; a combinatorial problem usually refers to an integer optimization problem where the unknown variable set (called combinatorial set) consists of all feasible integer subsets of a larger base integer set. In TN optimization, the base set is the set of all street nodes that are suitable to serve as transit stops, and the combinatorial set consists of all street paths (subsets or integer vectors of the base street node set) in the street network that are suitable for transit vehicle operations. Even for a small street node set, the corresponding combinatorial set (i.e., the set that includes all possible paths) may be huge. Baaj and Mahmassani (1991) observe that large-scale TN optimization problems tend to suffer from several forms of difficulties with traditional mathematical approaches, such as nonlinearity, nonconvexity, multiobjectives, and combinatorial intractability due to the discrete nature of the problems. Similar observations are also made by Ceder and Wilson (1986), Charkroborty and Dwivedi (2002), and Zhao and Ubaka (2004), among others. These seem to be why the solutions to most TN optimization problems in practice are either relying on certain heuristic assumptions or are limited to relatively small or idealized networks. To date, the solutions to large-scale transit network problems that include both route network and headway as design components have been mostly limited to the use of various heuristic approaches where the solution search schemes are based on a collection of design guidelines, criteria established from past experiences, and cost and feasibility constraints. In recent years, genetic algorithm (GA) has been applied to various TN optimization problems. GA is a stochastic algorithm based on natural evolution principle (i.e., genetic inheritance and the Darwinian strife for survival process). Mathematically, GAs may be categorized as weak solution search schemes that make few assumptions about problem domains and function properties, such as the smoothness, uniqueness, or compatibility of the objective functions, design parameters, and constraints. While this makes GAs attractive and popular for complex problems, it also causes GAs to suffer from combinatorial explosive solution costs due to huge solution search spaces often associated with large-scale problems. In the current TN literature, most GA applications are limited to smallor medium-sized network problems. Recently, Agrawal and Mathew (2004) applied a GA approach to a large-scale transit network. However, the travel demand (about 900 origin-destiJournal of Public Transportation, Vol. 9, No. 2, 2006 110 nation pairs) was relatively small, and the search method required multiprocessor parallel processing due to intensive computation involved. Table 1 summarizes the main features of some of the approaches reported in the literature. In the table, H&M indicates a combination of both mathematical programming methods and heuristic search schemes; MATH stands for mathematical optimization; H&M/AI means a combination of H&M and artificial intelligence techniques; and multiconstraints indicates use of multiple constraints such as maximum/minimum route length, maximum number of routes, minimum frequency, etc. Due to space limitations, the merits, solution strategies, and applicability to practical problems of the individual approaches are not discussed. Detailed information about various optimization approaches may be found in Fan and Machemehl (2004), which provides an extensive review and comparison of various optimization methods for TN design, and Zhao and Gan (2003), among others. Table 1. Main Features of Some Approaches Used in Transit Network Design Large-Scale Transit Network Optimization 111 The development of the combined simulated annealing and fast descent (SAFD) method in this study has been motivated by the lack of optimization procedures that are capable of tackling large-scale TN problems and finding global optimal solutions in terms of both user and operator costs. Unlike other search algorithms such as various heuristic methods and genetic algorithms, which do not theoretically guarantee good performance to ensure a global optimum, simulated annealing is supported by a solid theory. Under fairly general conditions, it has been shown that a global optimal will be obtained with probability 1 (Hajek 1988). The simulated annealing search scheme used in this study is based on the integrated simulated annealing, tabu, and greedy search method developed by Zhao and Gan (2003), originally designed for finding optimal TN route layouts to minimize passenger transfers. Solution Methodology For simplicity, the following assumptions were made in this study: 1. The demand pattern, expressed in a transit origin-destination (OD) matrix, remains the same during the period of study. 2. Passengers’ choices of routes are based on the shortest travel time. Terminal times are not included, although may be added easily. 3. Transit vehicles have the same seating capacity. 4. Passengers arrive at transit stops randomly (uniform distribution); therefore, the average waiting time to board a vehicle (twait) is one half of the headway (h), i.e.,