Stochastic Routing Research Articles

Stochastic single vehicle routing problem with ordered customers and partial fulfilment of demands

ABSTRACTWe consider the problem of finding the optimal routing of a single vehicle that starts its route from a depot and delivers a product to N customers that are served according to a particular order. The vehicle during its route can return to the depot for replenishment. It is assumed a stochastic demand for each customer. The actual demand of each customer becomes known upon the vehicle's arrival at the customer's site. It is permissible to satisfy fully or to satisfy partially or not to satisfy the demand of a customer. The cost structure includes travel costs between consecutive customers, travel costs between the customers and the depot and penalty costs if a customer's demand is not satisfied or if it is satisfied partially. A dynamic programming algorithm is developed for the determination of the optimal routing policy. It is shown that the optimal routing policy has a specific threshold-type structure. Furthermore, if we consider the same problem without the assumption that the customers are ordered, numerical experiments indicate that the optimal routing strategy can be computed for N smaller or equal to nine.

Stochastic route choice models based on VMS information and hierarchy for urban transport network

This paper firstly establishes four variational inequalities for stochastic traffic assignment by quantifying information utility of Variable Message Signs (VMS) and hierarchy of transport network. When VMS systems are operated in the higher level of roads such as arterials, the attenuation factor of information utility is integrated with information quantity to define a new formulation of the impact coefficient of VMS on traffic flow, which appears in the definitions of link and path lengths, and allocation parameters. Furthermore, equivalent Variational Inequalities (VIs) of Logit models considering these elements are developed and proved. A case study was conducted using the VI models on a hierarchical road network. We get some practically meaningful results, for example, VMS information can improve hierarchical network performance by interacting with traffic flow in all models except the C-Logit model and hierarchical level can improve network performance. Among the results, the most important is that the C-Logit or path-size Logit will produce a flow pattern with less fluctuation of traffic densities among links and is less affected by stochastic factors than the multinomial Logit and cross-nested Logit in a stochastic hierarchical road network.

Solving stochastic multiobjective vehicle routing problem using probabilistic metaheuristic

In this paper we propose a new metaheuristic algorithm for solving stochastic multiobjective combinatorial optimization(SMOCO) problems. Indeed, we find that the various initiatives that have been launched recently on this subject, they propose the classical metaheuristics to solve a stochastic multi-objective problems,but when the stochasticity effects is not taken into account, the choice of an arbitrary value leading to a particular configuration and a high loss of information. To conserve the stochastic nature of SMOCO problems,the pareto ranking should be defined on the random objectives functions directly rather than converted deterministic objectives. From these considerations, the scope of this research should consist of: (i)Proposed novel methodology of stochastic optimality for ranking objective functions characterized by non-continuous and no closed form expression. This novel approach is based on combinatorial probability and can be incorporated in a multiobjective evolutionary algorithm. (ii)Provide probabilistic approaches to elitism and diversification in multiobjective evolutionary algorithms. Finally, The behavior of the resulting Probabilistic Multi-objective Evolutionary Algorithms (PrMOEAs) is empirically investigated on the multi-objective stochastic VRP problem.

Complex vehicle transport problems: taxonomy, new variants, challenges and solution methodology

Due to appeared of several and new alternatives of vehicle routing problems (VRPs) in the last years, the contributions focus on design of traffic flow and transportation systems, vehicle routing, and other important aspects of supply chain management such as dynamic routing, multi commodity problems, inventory management, lot sizing, and lot scheduling. In this paper, we are going to present an extensive review of the general vehicle routing problem then we are going to focus on the problematic approaches in this field. For each one of these problematic, a positioning compared to this state of the art will be developed. In the first part a synthesis of literature review devoted to the complex and logistic transport problems in general and a transport problem with limited fleet and common carrier in particular with stochastic and dynamic vehicle routing problem. In the second part, a description of the vehicle routing problem with mathematical formulations will be presented. The different variants of this problem will be approached.

Stochastic Route Planning in Public Transport

Embodiments relate to generating a route plan. A method of generating a route plan is provided. The method receives a route planning request that includes a starting location, a destination location, a desired arrival time, and a set of user preferences. The method obtains transport service information that includes schedules of a plurality of transport services provided for a geographic region including the starting location and the destination location. The method determines a set of transport services for each of a plurality of possible intermediate locations between the starting and destination locations. The sets of transportation services provide a maximum probability of arriving at the destination location before the desired arrival time as long as a user takes a first-arriving transportation among the set of transport services at each intermediate location while traveling from the starting location to the destination location.

Integrated Analysis of Inventory Management and Transportation Systems for the Single-Period Problem

In a two-stage supply chain system, vendor managed inventory policies (VMI) is an integrating decisions between a supplier and his customers.The supplier assumes the responsibility of maintaining inventory at its customers and ensuring that they will not run out of stock at any moment.The objective of this paper is to optimize the inventory and the transportation costs of the customers for a two-stage supply chain. The problem is to determine the delivery quantities as well as the times and routes to the customers, while ensuring feasibility regardless of the realized demands.This paper considers the single-period stochastic inventory routing problem (SP-SIRP) system. The problem formulated as a linear mixed-integer stochastic program for the solutions of the SP-SIRP problem. An illustrative example for the SP-SIRP is presented to explain the proposed model. The finding shows that the vehicle capacity is optimized efficiently with the average capacity utilization for all tours is 100%.

Finding the Shortest Path in Stochastic Vehicle Routing: A Cardinality Minimization Approach

This paper aims at solving the stochastic shortest path problem in vehicle routing, the objective of which is to determine an optimal path that maximizes the probability of arriving at the destination before a given deadline. To solve this problem, we propose a data-driven approach, which directly explores the big data generated in traffic. Specifically, we first reformulate the original shortest path problem as a cardinality minimization problem directly based on samples of travel time on each road link, which can be obtained from the GPS trajectory of vehicles. Then, we apply an $\ell_{1}$ - norm minimization technique and its variants to solve the cardinality problem. Finally, we transform this problem into a mixed-integer linear programming problem, which can be solved using standard solvers. The proposed approach has three advantages over traditional methods. First, it can handle various or even unknown travel time probability distributions, while traditional stochastic routing methods can only work on specified probability distributions. Second, it does not rely on the assumption that travel time on different road segments is independent of each other, which is usually the case in traditional stochastic routing methods. Third, unlike other existing methods which require that deadlines must be larger than certain values, the proposed approach supports more flexible deadlines. We further analyze the influence of important parameters to the performances, i.e., accuracy and time complexity. Finally, we implement the proposed approach and evaluate its performance based on a real road network of Munich city. With real traffic data, the results show that it outperforms traditional methods.

Distributed stochastic multi-vehicle routing in the Euclidean plane with no communications

ABSTRACTThis paper presents an algorithm for the multi-vehicle routing problem with no communications among the vehicles. The scenario consists in a convex Euclidean mission space, where targets are generated according to a Poisson distribution in time and to a generic continuous spatial distribution. The targets must be visited by the vehicles, which, therefore, must act in coordination. Even if no communications are required, the proposed routing strategy succeeds in effectively partitioning the mission space among the vehicles: at low target generation rates, the algorithm leads to the well-known centroidal Voronoi tessellation, whereas at high target generation rates, simulation results show that it has better performances with respect to a reference algorithm with no communications among vehicles.

Travelers’ Risk-Taking Behaviors in Day-to-Day Dynamic Evolution Model

Uncertainties often exist in traffic systems because of interday route flow swapping and intraday road capacity degradation. These uncertainties obviously affect the traveler’s individual route adjustment behavior and the corresponding network traffic dynamic evolution at an aggregate level. Seeking to give a more realistic description of traffic dynamics, this study extends the existing day-to-day (DTD) model by considering both travelers’ study of uncertain travel time and travelers’ choice of risky routes. In the proposed model, an exponential-smoothing filter is introduced to describe travelers’ study of uncertain travel time. Meanwhile, on-time arrival probability is used to model travelers’ dynamic adjustment to their reference travel time (RTT). Under the framework of cumulative prospect theory, the perceived stochastic route travel time and the RTT are both integrated into a unified prospect value function to reflect travelers’ perception of route attractiveness. On this basis, a logit model is applied to describe travelers’ stochastic route choice behaviors. The proposed model’s several mathematical properties, including fixed-point existence, uniqueness, and stability, are also investigated through theoretical analyses. Numerical examples are used to illustrate the applications of the DTD model and verify some important dynamic system properties.

An Adaptive Routing Protocol with Balanced Stochastic Route Exploration and Stabilization Based on Short-Term Memory

An Adaptive Routing Protocol with Balanced Stochastic Route Exploration and Stabilization Based on Short-Term Memory

Dynamic modelling of traffic incident impacts on network reliability

A traffic incident is one of the major sources for degrading network capacity, inducing traffic congestion, and decreasing network reliability. The impacts of a traffic incident on network reliability have been extensively studied with the use of static network equilibrium or dynamic simulation models. In this paper, an analytical reliability-based dynamic traffic assignment (DTA) model is proposed for assessing the temporal and spatial impacts of a traffic incident on network reliability. The proposed DTA model can be used to estimate the stochastic link flow pattern and route travel time distribution for examining the impacts of a traffic incident on the on-time arrival probability with and without dynamic speed limit (SL) control. It is shown that a traffic incident on a congested road during peak period will greatly decrease the on-time arrival probability, particularly when the incident has greater effects on link capacity degradation with longer duration. Under certain circumstances, SL control can be employed to reduce total network delay during the time intervals with a traffic incident.

Modeling Stochastic Route Choice Behaviors with Equivalent Impedance

A Logit-based route choice model is proposed to address the overlapping and scaling problems in the traditional multinomial Logit model. The nonoverlapping links are defined as a subnetwork, and its equivalent impedance is explicitly calculated in order to simply network analyzing. The overlapping links are repeatedly merged into subnetworks with Logit-based equivalent travel costs. The choice set at each intersection comprises only the virtual equivalent route without overlapping. In order to capture heterogeneity in perception errors of different sizes of networks, different scale parameters are assigned to subnetworks and they are linked to the topological relationships to avoid estimation burden. The proposed model provides an alternative method to model the stochastic route choice behaviors without the overlapping and scaling problems, and it still maintains the simple and closed-form expression from the MNL model. A link-based loading algorithm based on Dial’s algorithm is proposed to obviate route enumeration and it is suitable to be applied on large-scale networks. Finally a comparison between the proposed model and other route choice models is given by numerical examples.

Modelling Stochastic Route Choice Behaviours with a Closed-Form Mixed Logit Model

A closed-form mixed Logit approach is proposed to model the stochastic route choice behaviours. It combines both the advantages of Probit and Logit to provide a flexible form in alternatives correlation and a tractable form in expression; besides, the heterogeneity in alternative variance can also be addressed. Paths are compared by pairs where the superiority of the binary Probit can be fully used. The Probit-based aggregation is also used for a nested Logit structure. Case studies on both numerical and empirical examples demonstrate that the new method is valid and practical. This paper thus provides an operational solution to incorporate the normal distribution in route choice with an analytical expression.

Estimating Route Choice Models from Stochastically Generated Choice Sets on Large-Scale Networks

Route choice is one of the most complex decision-making contexts to represent mathematically, and the most frequently used approach to model route choice consists of generating alternative routes and modeling the preferences of utility-maximizing travelers. The main drawback of this approach is the dependency of the parameter estimates from the choice set generation technique. Bias introduced in model estimation has been corrected only for the random walk algorithm, which has problematic applicability to large-scale networks. This study proposes a correction term for the sampling probability of routes extracted with stochastic route generation. The term is easily applicable to large-scale networks and various environments, given its dependence only on a random number generator and the Dijkstra shortest path algorithm. The implementation for revealed preferences data, which consist of actual route choices collected in Cagliari, Italy, shows the feasibility of generating routes stochastically in a high-resolution network and calculating the correction factor. The model estimation with and without correction illustrates how the correction not only improves the goodness of fit but also turns illogical signs for parameter estimates to logical signs.

On risk attitude and optimal yacht racing tactics

When the future wind direction is uncertain, the tactical decisions of a yacht skipper involve a stochastic routing problem. The objective of this problem is to maximise the probability of reaching the next mark ahead of all the other competitors. This paper describes some numerical experiments that explore the effect of the skipper׳s risk attitude on their policy when match racing another boat. The tidal current at any location is assumed to be negligible, while the wind direction is modelled by a Markov chain. Boat performance in different wind conditions is defined by the output of a velocity prediction program, and we assume a known speed loss for tacking and gybing. We compare strategies that minimise the average time to sail the leg with those that seek to maximise the probability of winning, and show that by adopting different attitudes to risk when leading or trailing the competitor, a skipper can improve their chances of winning.

A stochastic route to simulate the growth of porous anodic alumina

Porous anodic alumina (PAA) film is composed of highly ordered and controllable structures, and their extensive application requires the understanding of their growing mechanism. Herein, we present a localized oxidation model to unravel the phenomena of PAA growth, showing that random processes converge into an ordered formation due to the unique characteristics of ion transport confinement in alumina. The anodizing voltage shows a quadratic relationship with barrier layer depth. In addition, we predict the furcate conditions of PAA and the voltage threshold to produce a PAA by our model.

Efficient computation of evacuation routes on a three-dimensional geometric network

We consider a real-time emergency evacuation problem that seeks to compute a set of rapid evacuation routes in a building. Given a three-dimensional geometric structure of the evacuation network, an emergency evacuation route is a sequence of movements of people away from the threat or actual occurrence of a hazard (such as a fire, a hidden bomb) to a safe exit in the network. In such a network each room/crossing/exit in the building is designated as a node and the corridors/staircases/links between the rooms are edges. The evacuation times assigned to the edges are normally distributed random variables. This stochastic routing problem subject to deadline constraints is NP-hard. We provide a new pseudo-polynomial-time dynamic programming algorithm to solve this problem. Based on this algorithm, we construct two types of approximation algorithm, namely a fully polynomial-time approximation scheme providing “almost-optimal” solutions and a fully polynomial-time approximately feasible scheme yielding a best “almost feasible” solution. We present a case study and results of computational experiments to illustrate the working and efficacy of the proposed solution methods, respectively.

Reliability-based traffic network design with advanced traveler information systems

As the demands on urban transportation networks grow rapidly, problems of network design have attracted a great deal of interest because of the need to effectively handle urban transport planning using information technology. A bi-level continuous network design model is proposed in this paper to address the optimal road capacity expansion of existing links. Based on the fact that every origin–destination demand is random and affected by traffic travel information, the network is subject to relatively minimal day-to-day events of stochastic link capacity variations. Therefore, the primary objective is to maximize the reliability of the total travel time, while the lower level model, utilizing the behaviors of stochastic route choice, is aimed at reducing drivers’ travel time uncertainty through traffic information provided by advanced traveler information systems. The Particle Swarm Optimization algorithm is used to solve the suggested model, and a numerical example using the Sioux Falls network is provided. The computation results show that travel time reliability is improved by system optimization using traffic information.

General mechanism of two-state protein folding kinetics.

We describe here a general model of the kinetic mechanism of protein folding. In the Foldon Funnel Model, proteins fold in units of secondary structures, which form sequentially along the folding pathway, stabilized by tertiary interactions. The model predicts that the free energy landscape has a volcano shape, rather than a simple funnel, that folding is two-state (single-exponential) when secondary structures are intrinsically unstable, and that each structure along the folding path is a transition state for the previous structure. It shows how sequential pathways are consistent with multiple stochastic routes on funnel landscapes, and it gives good agreement with the 9 order of magnitude dependence of folding rates on protein size for a set of 93 proteins, at the same time it is consistent with the near independence of folding equilibrium constant on size. This model gives estimates of folding rates of proteomes, leading to a median folding time in Escherichia coli of about 5 s.

A Maximum Entropy Fixed-Point Route Choice Model for Route Correlation

In this paper we present a stochastic route choice model for transit networks that explicitly addresses route correlation due to overlapping alternatives. The model is based on a multi-objective mathematical programming problem, the optimality conditions of which generate an extension to the Multinomial Logit models. The proposed model considers a fixed point problem for treating correlations between routes, which can be solved iteratively. We estimated the new model on the Santiago (Chile) Metro network and compared the results with other route choice models that can be found in the literature. The new model has better explanatory and predictive power that many other alternative models, correctly capturing the correlation factor. Our methodology can be extended to private transport networks.