Modeling techniques for periodic vehicle routing problems

This paper presents a continuous approximation model for the period vehicle routing problem with service choice (PVRP-SC). The PVRP-SC is a variant of the period vehicle routing problem in which the visit frequency to nodes is a decision of the model. This variation can result in more efficient vehicle tours and/or greater service benefit to customers. We present a continuous approximation model to facilitate strategic and tactical planning of periodic distribution systems and evaluate the value of service choice. Further, results from the continuous model can provide guidelines for constructing solutions to the discrete PVRP-SC.     

Continuous approximation for skip-stop operation in rail transit

Operating speed of a transit corridor is a key characteristic and has many consequences on its performance. It is generally accepted that an increased operating speed for a given fleet leads to reduced operating costs (per kilometer), travel and waiting times (three changes that can be computed precisely), an improved comfort and level of service, which can attract new passengers who are diverted from automobile (items harder to estimate precisely). That is why several operation schemes which aim to increase the operating speed are studied in the literature, such as deadheading, express services, and stop skipping.A novel category of solutions to this problem for one-way single-track rail transit is to perform accelerated transit operations with fixed stopping schedules. The concept is quite simple: as the time required for stopping at each station is an important part of travel time, reducing it would be a great achievement. Particular operations that take advantage of this idea already exist. This paper focuses on one of them: the skip-stop operation for rail transit lines using a single one-way track. It consists in defining three types of stations: AB stations where all the trains stop, and A and B stations where only half of the trains stop (stations type A and B are allocated interchangeably). This mode of operation is already described in the literature (Vuchic, 1973, Vuchic, 1976, Vuchic, 2005) and has been successfully implemented in the Metro system of Santiago, Chile.This work tackles the problem with a continuous approximation approach. The problem is described with a set of geographically dependent continuous parameters like the density of stations for a given line. Cost functions are built for a traditional (all-stop) operation and for skip-stop operation as described above. A simple example is presented to support this discussion. Finally, a discussion about the type of scenarios in which skip-stop operations are more beneficial is presented.     

Estimating the inefficiency in the Norwegian bus industry from stochastic cost frontier models

A stochastic cost frontier function based on data from 170 of the 175 Norwegian subsidized bus companies is estimated under two alternative presumptions regarding the distribution of the inefficency among the bus operators. When the inefficiency is assumed to be half-normally distributed, the average inefficiency in the industry is estimated to be 13.7 per cent. This calculated value is nearly halved (7.2 per cent) when the exponential distribution is applied, while the ranking of the companies according to inefficiency is unchanged. By regressing the estimated inefficiency values for each company on some exogenous variables describing its ownership structure and the subsidy policy which it faces, it is seen that inefficiency of the companies which negotiate with the public authorities over the subsidy amounts is slightly higher than the inefficiency of the companies which face a subsidy policy based on cost norms. Our analysis gives, however, no significant differences in the efficiency between privately owned bus companies and publicly owned bus operators, and shows only minor economies of scale.     

Strategic evacuation planning with pedestrian guidance and bus routing: a mixed integer programming model and heuristic solution

This paper presents a mathematical model to plan emergencies in a densely populated urban zone where a certain numbers of pedestrians depend on transit for evacuation. The proposed model features an integrated operational framework, which simultaneously guides evacuees through urban streets and crosswalks (referred to as “the pedestrian network”) to designated pickup points (e.g., bus stops), and routes a fleet of buses at different depots to those pick‐up points and transports evacuees to their destinations or safe places. In this level, the buses are routed through the so‐called “vehicular network.” An integrated mixed integer linear program that can effectively take into account the interactions between the aforementioned two networks is formulated to find the maximal evacuation efficiency in two networks. Because the large instances of the proposed model are mathematically difficult to solve to optimality, a two‐stage heuristic is developed to solve larger instances of the model. Results from hundreds of numerical examples analysis indicate that proposed heuristic works well in providing (near) optimal or feasibly good solutions for medium‐scale to large‐scale instances that may arise in real transit‐based evacuation situations in a much shorter amount of computational time compared with cplex (can find optimal/feasible solutions for only five instances within 3 hours of running). Copyright © 2016 John Wiley & Sons, Ltd.     

Customized bus service design for jointly optimizing passenger-to-vehicle assignment and vehicle routing

Emerging transportation network services, such as customized buses, hold the promise of expanding overall traveler accessibility in congested metropolitan areas. A number of internet-based customized bus services have been planned and deployed for major origin-destination (OD) pairs to/from inner cities with limited physical road infrastructure. In this research, we aim to develop a joint optimization model for addressing a number of practical challenges for providing flexible public transportation services. First, how to maintain minimum loading rate requirements and increase the number of customers per bus for the bus operators to reach long-term profitability. Second, how to optimize detailed bus routing and timetabling plans to satisfy a wide range of specific user constraints, such as passengers’ pickup and delivery locations with preferred time windows, through flexible decision for matching passengers to bus routes. From a space-time network modeling perspective, this paper develops a multi-commodity network flow-based optimization model to formulate a customized bus service network design problem so as to optimize the utilization of the vehicle capacity while satisfying individual demand requests defined through space-time windows. We further develop a solution algorithm based on the Lagrangian decomposition for the primal problem and a space-time prism based method to reduce the solution search space. Case studies using both the illustrative and real-world large-scale transportation networks are conducted to demonstrate the effectiveness of the proposed algorithm and its sensitivity under different practical operating conditions.     

A reinforcement learning framework for the adaptive routing problem in stochastic time-dependent network

Most previous work in addressing the adaptive routing problem in stochastic and time-dependent (STD) network has been focusing on developing parametric models to reflect the network dynamics and designing efficient algorithms to solve these models. However, strong assumptions need to be made in the models and some algorithms also suffer from the curse of dimensionality. In this paper, we examine the application of Reinforcement Learning as a non-parametric model-free method to solve the problem. Both the online Q learning method for discrete state space and the offline fitted Q iteration algorithm for continuous state space are discussed. With a small case study on a mid-sized network, we demonstrate the significant advantages of using Reinforcement Learning to solve for the optimal routing policy over traditional stochastic dynamic programming method. And the fitted Q iteration algorithm combined with tree-based function approximation is shown to outperform other methods especially during peak demand periods.     

A simulation evaluation of the maximum approximate composite marginal likelihood (MACML) estimator for mixed multinomial probit models

This paper evaluates the ability of the maximum approximate composite marginal likelihood (MACML) estimation approach to recover parameters from finite samples in mixed cross-sectional and panel multinomial probit models. Comparisons with the maximum simulated likelihood (MSL) estimation approach are also undertaken. The results indicate that the MACML approach recovers parameters much more accurately than the MSL approach in all model structures and covariance specifications. The MACML inference approach also estimates the parameters efficiently, with the asymptotic standard errors being, in general, only a small proportion of the true values. As importantly, the MACML inference approach takes only a very small fraction of the time needed for MSL estimation. In particular, the results suggest that, for the case of five random coefficients, the MACML approach is about 50 times faster than the MSL for the cross-sectional random coefficients case, about 15 times faster than the MSL for the panel inter-individual random coefficients case, and about 350 times or more faster than the MSL for the panel intra- and inter-individual random coefficients case. As the number of alternatives in the unordered-response model increases, one can expect even higher computational efficiency factors for the MACML over the MSL approach. Further, as should be evident in the panel intra- and inter-individual random coefficients case, the MSL is all but practically infeasible when the mixing structure leads to an explosion in the dimensionality of integration in the likelihood function, but these situations are handled with ease in the MACML approach. It is hoped that the MACML procedure will spawn empirical research into rich model specifications within the context of unordered multinomial choice modeling, including autoregressive random coefficients, dynamics in coefficients, space-time effects, and spatial/social interactions.