Stochastic Travel Research Articles

Scheduled service network design with quality targets and stochastic travel times

We address the stochastic scheduled service network design problem with quality targets and uncertainty on travel times. This important problem, raising in the tactical planning process of consolidation-based freight carriers, has been little studied up to now. We define the problem considering quality targets for on-time operation of services and delivery of demand loads to destinations. We introduce a two-stage mixed-integer stochastic model defined over a space-time network, with quality targets modeled through penalties. We also propose an effective progressive-hedging-based meta-heuristic, based on a partial-decomposition concept aiming to address the challenges raised by the presence of flow-distribution decisions in the first-stage problem and by the flow-related degeneracy particular to network design. The results of an extensive numerical experimentation emphasize the worthiness of the formulation, as well as the very good performance of the proposed meta-heuristic when compared to a well-known commercial solver.

Column generation-based stochastic school bell time and bus scheduling optimization

Given a trip plan (a sequence of bus stops), the integrated school bell time and bus scheduling problem simultaneously optimizes the school bell time and bus routes (the assignment of buses to serve trips at a specific time) while maintaining the minimum level-of-service requirements. The objective is to minimize the total number of buses and the total vehicle time. The previous efforts focused on the deterministic problem. However, uncertainty is unavoidable in a real-world problem. Thus, we consider this integrated problem with stochastic travel times. Two Mixed Integer Programming (MIP) models and a Column Generation-based algorithm are proposed. The algorithm incorporates the merits of the Column Generation, Simulated Annealing, Insertion Algorithm, and Greedy Randomized Adaptive Search Procedure and gains the computational power that the existing methods do not have. The experiments on a real-world problem show that, after the bell time optimization, up to 41% of current buses can be saved with even better service in terms of the higher punctuality and shorter student ride time.

A simheuristic for routing electric vehicles with limited driving ranges and stochastic travel times

Green transportation is becoming relevant in the context of smart cities, where the use of electric vehicles represents a promising strategy to support sustainability policies. However the use of electric vehicles shows some drawbacks as well, such as their limited driving-range capacity. This paper analyses a realistic vehicle routing problem in which both driving-range constraints and stochastic travel times are considered. Thus, the main goal is to minimize the expected time-based cost required to complete the freight distribution plan. In order to design reliable Routing plans, a simheuristic algorithm is proposed. It combines Monte Carlo simulation with a multi-start metaheuristic, which also employs biased-randomization techniques. By including simulation, simheuristics extend the capabilities of metaheuristics to deal with stochastic problems. A series of computational experiments are performed to test our solving approach as well as to analyse the effect of uncertainty on the routing plans.

A Distributed Assignment Method for Dynamic Traffic Assignment Using Heterogeneous-Adviser Based Multi-Agent Reinforcement Learning

The Dynamic Traffic Assignment (DTA) is one of the important measures to alleviate urban network traffic congestion. The congestions are usually caused by stochastic traffic demands, which are generally unassignable from time dimension in the real-world but are assumed to be assignable in existing DTA methods (i.e. real-time travel demands). In this paper, a distributed DTA method for preventing urban network traffic congestion caused by stochastic real-time travel demands by improving Multi-Agent Reinforcement Learning (MARL). A team structure, which consists of decision-makers and advisers, is designed to learn parallelly in realistic DTA tasks. To reduce the size of the solution space adaptively, the dynamic critical values advised by adviser agents are adopted as constraints for the strategy space of decision-makers (i.e. main agents). A collaborative heterogeneous-adviser mechanism is designed to avoid deviation of guidance. To enhance the adaptability of DTA to the changeable external environment, the mixed strategy concept is introduced to improve the decision-making process of main agents. The respective mapping mechanisms are designed to define adaptive learning rates to improve the sensitivity of MARL. The Sioux Falls (SF) network is established as a test platform via a Dynamic Network Loading (DNL). The effectiveness of the suggested DTA method is assessed through numerical simulations SF network. Under the influence of the scenario with stochastic real-time travel demands, the results show that the proposed method outperforms in terms of the throughput of the network and the individual average travel time among the overall network. Additionally, the ability of the proposed method in response to the external environment rapidly has also been demonstrated. Adopting the suggested method can improve the state of the art to assign stochastic real-time travel demands dynamically and to avoid potential traffic congestion fundamentally.

An Improved Tabu Search Algorithm for the Stochastic Vehicle Routing Problem With Soft Time Windows

In real-world environments, vehicle travel and service time will be affected by unpredictable factors and present a random state. Because of this situation, this article proposes the vehicle routing problem with soft time windows and stochastic travel and service time (SVRP-STW). The probability distribution of vehicle travel and service time are introduced into the model, and a stochastic programming model with modification is established to minimize the distribution cost. An Improved Tabu Search algorithm (I-TS) based on greedy algorithm is proposed, in which adaptive tabu length and neighborhood structure are introduced; the greedy algorithm is used instead of the random methods to generate the initial solution. Experiments on different scale instances prove the effectiveness and superiority of the proposed algorithm.

Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions.

Uncovering Spatial Invasion on Metapopulation Networks with SIR Epidemics

Understanding how infectious diseases spatially diffuse is critical to predict and control the epidemic prevalence. However, uncovering epidemic spatial invasion is challenging due to stochastic travel of hosts and insufficient data availability. In this study, we develop a methodology for inferring global invasion pathways on metapopulation networks with the susceptible-infected-recovered (SIR) epidemics. To solve the inference problem, we infer the invasion pathways of each invasion case as a subgraph containing a part of global invasion pathways at each epidemic arrival time. We first develop a reduction approach to decrease the sizes of the dominant invasion cases, and propose a local optimization method aiming at the SIR epidemics based on epidemic maximum diffusion (EMD) to infer spatial invasion pathways of the reduced invasion cases, and reconstruct the global invasion pathways. Compared with the previous work, we reduce the computational complexity of invasion cases, and improve the calculation of epidemic diffusion likelihood and effectiveness of the algorithm for epidemic recovery. Simulations on real and synthetic metapopulation networks verify the validity of our algorithm. Finally, an empirical example of the 2009 A (H1N1) in the USA is presented to uncover the spatial invasion pathways and identify the superinvader s.

A distribution strategy using a two-step optimization to maximize blood services considering stochastic travel times

The blood service is a health service that utilizes human blood as basic material with humanitarian purposes, not for commercial one. Indonesian hospital ability in blood transfusions is generally still low, especially in terms of blood supply adequacy. In fact, there are still some provinces that experience excess blood supply while many other provinces experience a shortage of blood supply. The Blood Bank in Jakarta has the highest excess blood supply. Therefore, the blood can be transferred evenly from one province to another nearby province. The aim of this paper is to determine the allocation and the route of blood distribution to achieve the minimum travel times. Some variations in travel time are difficult to predict, so we take into account the stochastic properties of them. The effectiveness of blood distribution is very dependent on the accuracy of the target number of beneficiaries and the accuracy of the number of blood bags received in distribution activities. Meanwhile, the efficiency of blood bag distribution is measured by distribution routes that are directly related to transportation costs. This study uses a two-step optimization model to reach optimality. The first step is utilizing the transportation model to make sure the destination points are only the fastest to arrive. The second step is making use of the capacitated vehicle routing problem to ensure the routing is global optimal. This model successfully creates better blood demand fulfillment while minimizing transportation cost.

Vehicle Routing with Space- and Time-Correlated Stochastic Travel Times: Evaluating the Objective Function

We study how to model and handle correlated travel times in two-stage stochastic vehicle-routing problems. We allow these travel times to be correlated in time and space; that is, the travel time on one link in one period can be correlated to travel times on the same link in the next and previous periods as well as travel times on neighboring links (links sharing a node) in both the same and the following periods. Hence, we are handling a very high-dimensional dependent random vector. We discuss how such vehicle-routing problems should be modeled in time and space, how the random vector can be represented, and how scenarios (discretizations) can meaningfully be generated to be used in a stochastic program. We assume that the stochastic vehicle-routing problem is being solved by a search heuristic and focus on the objective function evaluation for any given solution. Numerical procedures are given and tested. As an example, our largest case has 142 nodes, 418 road links, and 60 time periods, leading to 25,080 dependent random variables. To achieve an objective function evaluation stability of 1%, we need only 15 scenarios for problem instances with 64 customer nodes and nine vehicles.

RISK-AVERSE STOCHASTIC ORIENTEERING PROBLEMS

In this study, we consider risk-averse orienteering problems with stochastic travel times or stochastic rewards. In risk-neutral orienteering problems, the objective is generally to maximize the expected total reward of visited notes. However, due to uncertain travel times or uncertain rewards, the dispersion in total reward collected may be large, which necessitates an approach that minimizes the dispersion (risk) in addition to maximizing the expected total reward. To handle this, for the orienteering problems with stochastic travel times or stochastic rewards, we suggest two different formulations with an objective of coherent measures of risk. For both problems, we conduct an experimental study using two different coherent measures of risk, which have been extensively used in the literature, and compare the results. The computational results show that, in both models suggested and under both risk measures used, the decision maker is able to obtain a tour with expected total reward being close to the expected total reward of risk-neutral solution, however with a significant decrease in the standard deviation of total reward.

Simulation-based robust optimization of limited-stop bus service with vehicle overtaking and dynamics: A response surface methodology

We propose a robust optimization model for limited-stop bus service with vehicle overtaking and demand dynamics. Time-dependent stochastic travel time is also considered. The objective is to minimize the total cost (user cost and operation cost) at the planning phase given a target reliability imperative. We further propose a simulation-based optimization framework incorporating response surface methodology to solve the problem efficiently. A real-world application result shows that vehicle overtaking and demand dynamics have significant impacts on the performance of limited-stop service and that the stop patterns are quite distinct when overtaking and demand dynamics are considered.

Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.

The value of foresight in the drybulk freight market

We analyze the value of foresight in the drybulk freight market when repositioning a vessel through space and time. In order to do that, we apply an optimization model on a network with dynamic regional freight rate differences and stochastic travel times. We evaluate the value of the geographical switching option for three cases: the upper bound based on having perfect foresight, the baseline set by a random strategy, and the case of perfect foresight but only for a limited time horizon. By combining a neural network with optimization, we can assess the impact of varying foresight horizons on economic performance. In a simple but realistic two-region case, we evaluate empirically that the upper bound for large vessels can be as high as 25% cumulative outperformance, and that a significant portion of this theoretical value can be captured with limited foresight of a few weeks. Our research sheds light on the important issue of spatial efficiency in global ocean freight markets and provides a useful upper bound for the value of investing in predictive analysis.

A general method for lower bounds on fluctuations of random variables

There are many ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. This paper introduces a general method for lower bounds on fluctuations. The method is used to obtain new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington–Kirkpatrick model of spin glasses, first-passage percolation and random matrices. A long list of open problems is provided at the end.

Comparing Route Choice Models for Managed Lane Networks with Multiple Entrances and Exits

Developing an appropriate route choice model for managed lanes with multiple entrances and exits is critical for the success of managed lane planning and operations. This research focuses on route choice models for managed lane networks with stochastic and time-varying tolls and travel times. In the model, a traveler receives real-time information about the tolls and travel times upon arrival at each diverge node and makes a dynamic lane choice decision that minimizes the total expected cost. The online route choice model is formulated as a Markov decision process and solved using a backward recursion algorithm. The model is compared against four other routing models: a binary logit model, a model based on decision routes, a model that chooses paths a priori, and a model with routes chosen randomly. The study also models irrational driver behavior with parameters like driver’s inclination toward making optimal lane choices and preference for certain lanes. Findings show that the expected costs from the routes chosen using the decision route model from the literature are close to the optimal cost with an average percentage error of 0.93%. The binary logit model is shown to have a high average error of 50% in the expected cost when a driver is assumed to behave rationally, but the same model shows optimal prediction for certain irrational driver behaviors. The proposed routing model forms a basis for future work in the area of managed lane pricing and planning.

A latent-class adaptive routing choice model in stochastic time-dependent networks

Transportation networks are inherently uncertain due to random disruptions; meanwhile, real-time information potentially helps travelers adapt to realized traffic conditions and make better route choices under such disruptions. Modeling adaptive route choice behavior is essential in evaluating real-time traveler information systems and related policies. This research contributes to the state of the art by developing a latent-class routing policy choice model in a stochastic time-dependent network with revealed preference data. A routing policy is defined as a decision rule applied at each link that maps possible realized traffic conditions to decisions on the link to take next. It represents a traveler’s ability to look ahead in order to incorporate real-time information not yet available at the time of decision.A case study is conducted in Stockholm, Sweden and data for the stochastic time-dependent network are generated from hired taxi Global Positioning System (GPS) readings. A latent-class Policy Size Logit model is specified, with routing policy users who follow routing policies and path users who follow fixed paths. Two additional layers of latency in the measurement equation are accounted for: 1) the choice of a routing policy is latent and only its realized path on a given day can be observed; and 2) when GPS readings have relatively long gaps, the realized path cannot be uniquely identified, and the likelihood of observing vehicle traces with non-consecutive links is instead maximized.Routing policy choice set generation is based on the generalization of path choice set generation methods. The generated choice sets achieve 95% coverage for 100% overlap threshold after correcting GPS mistakes and breaking up trips with intermediate stops, and further achieve 100% coverage for 90% overlap threshold.Estimation results show that the routing policy user class probability increases with trip length, and the latent-class routing policy choice model fits the data better than a single-class path choice or routing policy choice model. This suggests that travelers are heterogeneous in terms of their ability and/or willingness to plan ahead and utilize real-time information, and an appropriate route choice model for uncertain networks should take into account the underlying stochastic travel times and structured traveler heterogeneity in terms of real-time information utilization.

A model for taxi pooling with stochastic vehicle travel times

Taxi pooling means the sharing of a taxi by more than one passenger with at least a semi-common route. Currently, a trial-and-error process is adopted, taking into account expected travel times, for taxi pooling, a method which is neither effective nor efficient. This is because stochastic disturbances arising from variations in the taxi travel times that occur in actual traveling are neglected. In the worst case scenario, where vehicle travel times fluctuate wildly during operations, the planned schedule can be disturbed enough to lose its optimality. Therefore, in this study, a network flow technique is used to construct a stochastic taxi pooling model incorporating stochastic vehicle travel times. A solution algorithm is also proposed to efficiently solve the problem. The performance of the proposed models and the use of the solution method in practice are evaluated by executing numerical tests using real data with suitable assumptions. The test results show that the stochastic model and the solution method could be useful references for practice.

Capacitated vehicle routing problem with soft time windows and stochastic travel times

A full multiobjective approach is employed in this paper to deal with a stochastic multiobjective capacitated vehicle routing problem (CVRP). In this version of the problem, the demand is considered to be deterministic, but the travel times are assumed to be stochastic. A soft time window is tied to every customer and there is a penalty for starting the service outside the time window. Two objectives are minimized, the total length and the time window penalty. The suggested solution method includes a non-dominated sorting genetic algorithm (NSGA) together with a variable neighborhood search (VNS) heuristic. It was tested on instances from the literature and compared to a previous solution approach. The suggested method is able to find solutions that dominate some of the previously best known stochastic multiobjective CVRP solutions.

Last Mile Delivery With Stochastic Travel Times Considering Dual Services

Nowadays there are two prevailing delivery modes in the last mile delivery; one is the home delivery (HD) that vehicles deliver parcels to customers' homes; and the other is the customers' pickup (CP) that vehicle deliver parcels to some kind of intelligent express boxes where customers can pick up their parcels with free time. This article studies a green vehicle routing problem considering dual services (HD and CP) with stochastic travel times (GVRP-DS-STT) to provide customers with sustainable and diversified delivery services. The GVRP-DS-STT problem is formulated as a two-stage stochastic optimization model with recourse strategy. The purpose of the model is to minimize the total operational cost under stochastic environment. In addition, a two-stage heuristic algorithm integrating with a sampling strategy is developed to solve approximately the problem, the first of which is the greedy-based initial feasible solution generation, and the second of which is an improvement heuristic with late acceptance to explore the solution space. The computational results show that there are increasing benefit in terms of total operational cost, number of the vehicles used and loading rate with the increase of percentage of CP customers. The time windows have great effect on the operational cost, but more CP customers can reduce their impact. The stochastic model outperforms the deterministic model in terms of total operational cost.

An EV Charging Demand Model for the Distribution System Using Traffic Property

This paper proposes a mathematical model for the spatial-temporal charging demand for electric vehicle (EV). The determination of spatial-temporal charging demand is a key step for the planning of distribution systems with a scalable application of EV. The spatial-temporal allocation of EV is conventionally obtained through a simulation procedure using traffic topology data, which is not suitable for the regions lacking such information. This model converts the problem of travel distance to travel duration so that the requirement of network geographic information can be avoided. Static EV parameters, EV spatial-temporal moving parameters and system charging model parameters are treated as the deterministic factors for the charging demand allocation. A stochastic travel route simulation procedure, which relies purely on statistical data of traffic flow, is also developed to obtain the EV moving parameters by adopting the traffic property information. The designed procedure derives travel time parameters from the vehicle dynamic-location-property (DLP) model and the travel time probability distribution. The DLP model is established using the traffic property matrix and the regional origin-destination matrix. A simple case is presented to illustrate the result of stochastic travel route simulation. Then, a modified eastern China system is used as an example to analyze the EV charging demand under multiple scenarios. The feasibility and versatility of the proposed model in the large complex system are verified by the test results.