Implementation Issues for the Reliable a Priori Shortest Path Problem

Xing Wu, Yu (Marco) Nie
2009 Transportation Research Record  
The reliable a priori shortest path problem (RASP) studied in this research aims to find a priori paths that are shortest to ensure a specified probability of on-time arrival. The authors (28) have shown that the RASP belongs to a class of multiple-criteria shortest path problems that rely on a dominance relationship to obtain Paretooptimal solutions (7, 16, 29) ; that is, no further travel time improvements associated with any on-time arrival probability can be made without worsening those
more » ... ciated with other probability levels. Because the dominance relationship in RASP is defined with respect to the cumulative distribution function (CDF) of path travel times, it is effectively equivalent to the FSD rule considered by and Bard and Bennett (27). The RASP formulation proposed by Nie and Wu (28) is continuous and solved with a label-correcting algorithm similar to that of . This paper proposes and tests several implementation strategies intended to improve the computational performance of the solution algorithms for RASP. Because the dominance relationship is determined on the basis of CDFs, how to calculate and store them is critical to the efficiency of solution algorithms. These operations often involve discretizing continuous probability density functions and numerically evaluating convolution integrals. A challenge in the conventional discretization scheme (e.g., 28, 30) is that the length of the analysis period T has to be set so large that trips on most paths can be completed with a probability of 1.0. For one thing, it is difficult to determine T with analytical methods. More important, T is problem-specific in the sense that it increases with network size and depends on travel time distributions. Because computational cost increases rapidly with T for the same resolution, the existing discretization scheme is not suitable for large networks. An alternative scheme is proposed to overcome this drawback, using the inverse of a CDF. A procedure to evaluate the convolution integral using this discretization scheme is also proposed. It is well known that multicriteria shortest path problems are intractable because of the nondeterministic polynomial (NP) bound of Pareto-optimal solutions. The RASP is no exception. Typical heuristic strategies attempt to overcome the difficulty by limiting the size of nondominant paths. For example, Nie and Wu recently proposed the extreme-dominance approximation (EDA) strategy (28). EDA ignores nondominant paths that do not contribute directly to the Pareto frontier, thereby effectively restricting the number of these paths. Preliminary results demonstrated the satisfactory performance of EDA (28). However, like other heuristics, this strategy may not yield correct Pareto-optimal solutions. In the worst case, it may not even identify a subset of nondominant paths. Therefore, a comprehensive computational study is needed to evaluate EDA on networks of different sizes and densities and with the newly proposed discretization scheme. Nie and Wu proved the acyclicity of nondominant paths and suggest that preventing cyclic paths from temporarily entering the non-Solution techniques are studied for the problem of finding a priori paths that are shortest to ensure a specified probability of on-time arrival in a stochastic network. A new discretization scheme called ␣-discrete is proposed. The scheme is well suited to large-scale applications because it does not depend on problem-specific parameters. A procedure for evaluating convolution integrals based on the new scheme is given, and its complexity is analyzed. Other implementation strategies also are discussed to improve the computational performance of the exact yet nondeterministic polynomial label-correcting algorithm. These include an approximate method based on extreme dominance and two cycle-avoidance strategies. Comprehensive numerical experiments are conducted to test the effects of the proposed implementation strategies using different networks and different distribution types. Optimal path problems in a stochastic network have been intensively studied. Conventionally, a path is considered optimal if it incurs the least-expected travel time (1-12). To address the reliability of path travel times, Frank (13) and Mirchandani (14) define the optimal path as the one that maximizes the probability of realizing a travel time equal to or less than a given threshold. Sigal et al. suggest using the maximum probability of being the shortest path as an optimality index (15). Using expected utility theory, Loui shows that, for polynomial functions, utility maximization is reduced to a class of bi-criteria shortest path problems that trade off mean and variance of random travel times (16). This result is consistent with the mean-variance rule that has long been used in portfolio selection (17 ) . Similar routing problems have been studied elsewhere (18-20). Stochastic optimal-path problems also have been approached using robust optimization, which usually implies that a path is optimal if its worst-case travel time is the minimum (21-23). Miller-Hooks and Mahmassani define the optimal path as the one that realizes the least possible travel time in stochastic and time-varying networks (24). Later, other definitions of optimality based on various dominance relationships also were explored, namely, deterministic dominance, first-order stochastic dominance (FSD), and expected value dominance (25, 26). Label-correcting algorithms were proposed to solve nondominant paths corresponding to each definition of dominance rules. Bard and Bennett also used FSD to determine optimal paths in a stochastic network and proposed a network reduction algorithm for acyclic networks (27 ).
doi:10.3141/2091-06 fatcat:nkdfgadzeza4zbqziojmbvsvym