Vehicle routing with stochastic time-dependent travel times

Assigning and scheduling vehicle routes in a stochastic time-dependent environment is a crucial management problem. The assumption that in a real-life environment everything goes according to an a priori determined static schedule is unrealistic. Our methodology builds on earlier work in which the traffic congestion is captured in an analytical way using queueing theory. The congestion is then applied to the VRP problem. In this paper, we introduce the variability in traffic flows into the model. This allows for an evaluation of the routes based on the uncertainty involved. Different experiments show that the risk taking behavior of the planner can be taken into account during optimization. As more weight is given to the variability component, the resulting optimal route will take a slightly longer travel time, but will be more reliable. We propose a powerful objective function that is easily implemented and that captures the trade-off between the average travel time and its variance. The evaluation of the solution is done in terms of the 95th-percentile of the travel time distribution (assumed to be lognormal), which reflects well the quality of the solution in this stochastic time-dependent environment.     

Parking buses in a depot with stochastic arrival times

Given buses of different types arriving at a depot during the evening, the bus parking problem consists of assigning these buses to parking slots in such a way that they can be dispatched adequately to the next morning routes without moving them between their arrivals and departures. In practice, the bus arrival times deviate stochastically from the planned schedule. In this paper, we introduce for this problem two solution approaches that produce solutions which are robust to variations in the arrival times. The first approach considers that each arrival can deviate from its planned arrival order (sooner or later) by at most k positions, where k is a predefined parameter. In the second approach, the objective aims at minimizing the expectation of a function positively correlated with the number of buses that make the planned solution infeasible because they arrive too late or too early. In both approaches, the problem is modeled as an integer linear program that can be solved by a commercial mip solver. Computational results obtained on instances derived from a real-world dataset are reported.     

The orienteering problem with stochastic travel and service times

In this paper, we introduce a variant of the orienteering problem in which travel and service times are stochastic. If a delivery commitment is made to a customer and is completed by the end of the day, a reward is received, but if a commitment is made and not completed, a penalty is incurred. This problem reflects the challenges of a company who, on a given day, may have more customers than it can serve. In this paper, we discuss special cases of the problem that we can solve exactly and heuristics for general problem instances. We present computational results for a variety of parameter settings and discuss characteristics of the solution structure.     

A framework for user equilibrium dynamic traffic assignment

Several analytic approaches have been developed to describe or predict traffic flows on networks with time-varying (dynamic) travel demands, flows and travel times. A key component of these models lies in modelling the flows and/or travel times on the individual links, but as this is made more realistic or accurate it tends to make the overall model less computationally tractable. To help overcome this, and for other reasons, we develop a bi-level user equilibrium (UE) framework that separates the assignment or loading of flows on the time–space network from the modelling of flows and trip times within individual links. We show that this model or framework satisfies appropriate definitions of UE satisfies a first-in-first-out (FIFO) property of road traffic, and has other desirable properties. The model can be solved by iterating between (a) a linear network-loading model that takes the lengths of time–space links as fixed (within narrow ranges), and (b) a set of link flow sub-models which update the link trip times to construct a new time–space network. This allows links to be processed sequentially or in parallel and avoids having to enumerate paths and compute path flows or travel times. We test and demonstrate the model and algorithms using example networks and find that the algorithm converges quickly and the solutions behave as expected. We show how to extend the model to handle elastic demands, multiple destinations and multiple traffic types, and traffic spillback within links and from link to link.     

Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

Solving the logit-based stochastic user equilibrium problem with elastic demand based on the extended traffic network model

This paper proposes a novel extended traffic network model to solve the logit-based stochastic user equilibrium (SUE) problem with elastic demand. In this model, an extended traffic network is established by properly adding dummy nodes and links to the original traffic network. Based on the extended traffic network, the logit-based SUE problem with elastic demand is transformed to the SUE problem with fixed demand. Such problem is then further converted to a linearly constrained convex programming and addressed by a predictor–corrector interior point algorithm with polynomial complexity. A numerical example is provided to compare the proposed model with the method of successive averages (MSA). The numerical results indicate that the proposed model is more efficient and has a better convergence than the MSA.     

A stochastic network with mobile users in heavy traffic

We consider a stochastic network with mobile users in a heavy traffic regime. We derive the scaling limit of the multidimensional queue length process and prove a form of spatial state space collapse. The proof exploits a recent result by Lambert and Simatos (preprint, 2012), which provides a general principle to establish scaling limits of regenerative processes based on the convergence of their excursions. We also prove weak convergence of the sequences of stationary joint queue length distributions and stationary sojourn times.     

System optimal and user equilibrium time-dependent traffic assignment in congested networks

This paper formulates two dynamic network traffic assignment models in which O-D desires for the planning horizon are assumed known a priori: the system optimal (SO) and the user equilibrium (UE) time-dependent traffic assignment formulations. Solution algorithms developed and implemented for these models incorporate a traffic simulation model within an overall iterative search framework. Experiments conducted on a test network provide the basis for a comparative analysis of system performance under the SO and UE models.     

A multiclass,multicriteria logit-based traffic equilibrium assignment model under ATIS

This paper presents a multiclass, multicriteria (cost versus time) logit-based traffic equilibrium assignment model in road networks served by advanced traveler information systems (ATIS). All users are differentiated by their own value of time (VOT) that follows some probability distribution. Users of each class, having their own VOT, are further divided into two groups, equipped and unequipped with ATIS respectively. The travel disutility received by each user is defined as a linear bi-criteria combination of travel time and monetary travel cost. It is assumed that all users make their route choices in a logit-based stochastic manner, but the equipped users have lower perception variation on the travel disutility than the unequipped due to the ATIS service. The model is formulated as a fixed-point problem and solved by the method of successive averages in conjunction with logit assignment. Numerical results show that the traditional single-class and/or single-criterion models may overestimate or underestimate the benefit from ATIS services.     

A modified Physarum-inspired model for the user equilibrium traffic assignment problem

The user equilibrium traffic assignment principle is very important in the traffic assignment problem. Mathematical programming models are designed to solve the user equilibrium problem in traditional algorithms. Recently, the Physarum shows the ability to address the user equilibrium and system optimization traffic assignment problems. However, the Physarum model are not efficient in real traffic networks with two-way traffic characteristics and multiple origin–destination pairs. In this article, a modified Physarum-inspired model for the user equilibrium problem is proposed. By decomposing traffic flux based on origin nodes, the traffic flux from different origin–destination pairs can be distinguished in the proposed model. The Physarum can obtain the equilibrium traffic flux when no shorter path can be discovered between each origin–destination pair. Finally, numerical examples demonstrate the rationality and convergence properties of the proposed model.     

A dimension-reduced method of sensitivity analysis for stochastic user equilibrium assignment model

This paper gives a new dimension-reduced method of sensitivity analysis for perturbed stochastic user equilibrium assignment (SUEA) model based on the relation between its Lagrange function and logarithmic barrier function combined with a Courant quadratic penalty term. The advantage of this method is of smaller dimension than general sensitivity analysis and reducing complexity. Firstly, it presents the dimension-reduced sensitivity results of the general nonlinear programming perturbation problem and the improved results when the objective or constraint functions are not twice continuously differentiable. Then it proves the corresponding conclusion of SUEA with smooth or non-smooth cost functions by the method of converting constraint conditions and decision variables. Finally, two corresponding examples (smooth and non-smooth) are given to illustrate the feasibility of this method.     

On the stochastic vehicle routing problem with time windows,correlated travel times,and time dependency

4OR - Most state-of-the-art algorithms for the Vehicle Routing Problem, such as Branch-and-Price algorithms or meta heuristics, rely on a fast feasibility test for a given route. We devise the...     

Using fuzzy integral for evaluating subjectively perceived travel costs in a traffic assignment model

A good traffic assignment model can be a powerful tool to describe the characteristics of traffic behavior in a road network. The traffic assignment results often play an important role in transportation planning, e.g., an optimal and economical network design. Many traditional traffic assignment models rely heavily on the travel cost function established by Wardrop’s principles; however, the Wardrop’s travel cost function has been proven to be weak for explaining the uncertainty and interactivity of traffic among links. This study tries to construct a traffic assignment model that is different from Wardrop’s in many aspects. First, it considers the cross-effect among the links. Second, a fuzzy travel cost function is established based on the possibility concept instead of precise calculation of traffic volumes. Third, the techniques of fuzzy measure and fuzzy integral are applied to calculate the subjectively perceived travel costs during traffic assignment. Furthermore, in order to validate our model, a detailed network with 22 nodes and 36 links is used to illustrate it. Study results show that our model explains more interactivity and uncertainty of traffic among links when compared with the traditional model of Wardrop’s.     

Vehicle routing with soft time windows and stochastic travel times: A column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported.     

Applications of sensitivity analysis for probit stochastic network equilibrium

Network equilibrium models are widely used by traffic practitioners to aid them in making decisions concerning the operation and management of traffic networks. The common practice is to test a prescribed range of hypothetical changes or policy measures through adjustments to the input data, namely the trip demands, the arc performance (travel time) functions, and policy variables such as tolls or signal timings. Relatively little use is made, however, of the full implicit relationship between model inputs and outputs inherent in these models. By exploiting the representation of such models as an equivalent optimisation problem, classical results on the sensitivity analysis of non-linear programs may be applied, to produce linear relationships between input data perturbations and model outputs. We specifically focus on recent results relating to the probit Stochastic User Equilibrium (PSUE) model, which has the advantage of greater behavioural realism and flexibility relative to the conventional Wardrop user equilibrium and logit SUE models. The paper goes on to explore four applications of these sensitivity expressions in gaining insight into the operation of road traffic networks. These applications are namely: identification of sensitive, ‘critical’ parameters; computation of approximate, re-equilibrated solutions following a change (post-optimisation); robustness analysis of model forecasts to input data errors, in the form of confidence interval estimation; and the solution of problems of the bi-level, optimal network design variety. Finally, numerical experiments applying these methods are reported.     

Approximating earliest arrival flows with flow-dependent transit times

For the earliest arrival flow problem one is given a network G=(V,A) with capacities u(a) and transit times τ(a) on its arcs a∈A, together with a source and a sink vertex s,t∈V. The objective is to send flow from s to t that moves through the network over time, such that for each time θ∈[0,T) the maximum possible amount of flow up to this time reaches t. If, for each θ∈[0,T), this flow is a maximum flow for time horizon θ, then it is called earliest arrival flow. In practical applications a higher congestion of an arc in the network often implies a considerable increase in transit time. Therefore, in this paper we study the earliest arrival problem for the case that the transit time of each arc in the network at each time θ depends on the flow on this particular arc at that time θ.For constant transit times it has been shown by Gale that earliest arrival flows exist for any network. We give examples, showing that this is no longer true for flow-dependent transit times. For that reason we define a relaxed version of this problem where the objective is to find flows that are almost earliest arrival flows. In particular, we are interested in flows that, for each θ∈[0,T), need only α-times longer to send the maximum flow to the sink. We give both constant lower and upper bounds on α; furthermore, we present a constant factor approximation algorithm for this problem.     

A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium

Suppose that a road network model is given, together with some given demand for travel by (say) car and that the demand for travel varies with time of day but not from day to day. Suppose that this demand is given in the form of specified total outflow rates from each origin headed towards each destination, for each origin-destination pair and for each time of day, and that some initial time-dependent routeinflow rates, meeting the given demand, are given. Finally, suppose that within-day time is represented by a continuous variable. This paper specifies a natural smooth day-to-day route-swapping procedure wherein drivers swap toward less expensive routes as day succeeds day, and shows that under reasonable conditions there is an equilibrium state of this dynamical system. If such a collection of route-inflows has arisen today, say, then there is no incentive for any route-inflow to change tomorrow, in the sense that at each moment of today each of today's route-inflows isalready on a route which today yielded the smallest travel cost. Such a set of no-incentive-to-change route-inflows is called adynamic equilibrium, or adynamic user-equilibrium, and may be regarded as a solution of the dynamic equilibrium traffic assignment problem. Thus, the paper introduces a smooth day-to-day dynamic assignment model and, using this model, shows that there is a dynamic user-equilibrium in a continuous time setting. The paper briefly considers the day-to-day stability of the route-swapping process, also in a continuous setting. Finally, the paper gives a simple dynamical example illustrating the stability of the route-swapping process in a simple two-route network when there is deterministic queueing at bottlenecks.     

Reconstructing freeway travel times with a simplified network flow model alternating the adopted fundamental diagram

The present study summarises the travel time reconstruction performance of a network flow model by explicitly analysing the adopted fundamental diagram relation under congested and un-congested traffic patterns. The incorporated network flow model uses a discrete meso-simulation approach in which the anisotropic property of traffic flow and the uniform acceleration of vehicle packets are explicitly considered. The flow performances on link-route dynamics have been derived by reasonably alternating the adopted two-phase, i.e., congested and un-congested, fundamental relation of traffic flow. The linear speed–density relation with the creeping speed assumption is substituted with the triangular flow–density relation in order to investigate the performance of the network flow model in varying flow patterns. Applying the anisotropic mesoscopic model, the measure of travel time is obtained as a link performance from a simplified dynamic network loading process. Travel time reconstruction performance of the network flow model is sought considering the actual measures that are obtained by a probe vehicle, in addition to reconstructions by a macroscopic network flow model. The main improvements on travel time reconstruction process are encountered in terms of the computation load within the explicit analyses by the alternation of adopted two-phase fundamental diagram. Although the accuracies of the flow model with the adoption of two different fundamental diagrams are hard to differentiate, the computational burden of the simulation process by the triangular fundamental diagram is found to be considerably different.     

Fixed-point fast sweeping weighted essentially non-oscillatory method for multi-commodity continuum traffic equilibrium assignment problem

This work presents a fixed-point fast sweeping weighted essentially non-oscillatory method for the multi-commodity continuum traffic equilibrium assignment problem with elastic travel demand. The commuters’ origins (i.e. home locations) are continuously dispersed over the whole city with several highly compact central business districts. The traffic flows from origins to the same central business district are considered as one commodity. The continuum traffic equilibrium assignment model is formulated as a static conservation law equation coupled with an Eikonal equation for each commodity. To solve the model, a pseudo-time-marching approach and a third order finite volume weighted essentially non-oscillatory scheme with Lax–Friedrichs flux splitting are adopted to solve the conservation law equation, coupled with a third order fast sweeping numerical method for the Eikonal equation on rectangular grids. A fixed-point fast sweeping method that utilizes Gauss–Seidel iterations and alternating sweeping strategy is designed to improve the convergence for steady state computations of the problem. A numerical example is given to show the feasibility of the model and the effectiveness of the solution algorithm.