Arc Travel Times Research Articles

A general variable neighborhood search for the traveling salesman problem with time windows under various objectives

The traveling salesman problem with time windows (TSPTW) has wide practical applications in transportation and scheduling operations. We study the TSPTW in the deterministic case as well as under travel time uncertainty. We consider the three classical TSPTW variants with the objectives of minimizing cost, completion time, and tour duration. In addition, a new TSPTW variant that maximizes the minimum slack of a tour, i.e., the smallest time buffer between the arrival time and the end of the time window over all nodes visited on the tour, is introduced. We further address a variant of the TSPTW in which the arc travel times are uncertain. This problem is modeled by means of an uncertainty set, and the goal is to determine a tour that remains feasible in the worst case within a budget stipulating the sum of all travel time deviations from their nominal values. To solve all targeted problem variants, we develop a two-phase general variable neighborhood search (GVNS) that applies an efficient move evaluation approach within the local search. Extensive numerical experiments on benchmark instances from the literature show that our GVNS finds high-quality solutions that are competitive with those obtained by the state-of-the-art heuristics for all problem variants.

Charging Network Design and Service Pricing for Electric Vehicles With User-Equilibrium Decisions

This paper aims to investigate the electric vehicle (EV) charging network design and utilization management considering user-centric decisions. A hierarchical formulation is developed with the EV charging network design and demand-driven pricing scheme in the upper level and users’ charging decisions to minimize their own travel costs and charging expenses in the lower level. The model aims to minimize the facility deployment cost and maximize the charging income of the network operator while minimizing the user-centric costs. We have converted the proposed bi-level formulation into an equivalent single-level model using the lower-level objective function as complementary equations. Then, we have developed an iterative active-set based solution technique to determine the strategic decisions on charging network design. To partially overcome the computational burden, the arc travel times are estimated using a macroscopic fundamental diagram concept. The proposed integrated methodology is applied to a hypothetical and an empirical case study to evaluate its performance and solution quality. The numerical results indicate that the proposed algorithm can solve the problem efficiently and outperform a system-level bi-level optimization benchmark. Our experiments show a CPU time of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.3~hr$ </tex-math></inline-formula> for the proposed approach compared to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$173.1~hr$ </tex-math></inline-formula> of the benchmark. Finally, a series of sensitivity analyses has been conducted to study the impact of input parameters on the solutions and draw managerial insights.

Joint power distribution and charging network design for electrified mobility with user equilibrium decisions

AbstractRapid adoption of electric vehicles (EVs) requires the development of a highly flexible charging network. The design and management of the charging infrastructure for EV‐dominated transportation systems are intertwined with power grid operations both economically and technically. High penetration of EVs in the future can increase the charging loads and cause a wide range of operational issues in power distribution networks (PDNs). This paper aims to design an EV charging network with an embedded PDN layout to account for energy dispatch and underlying traffic flows in urban transportation networks supporting electric mobility in the near future. A mixed‐integer bilevel model is proposed with the EV charging facility location and PDN energy decisions in the upper level and user equilibrium traffic assignment in the lower level considering an uncertain charging demand. The objective is to minimize the cost of PDN operations, charging facility deployments, and transportation. The proposed problem is solved using a column and constraint generation (C&amp;CG ) algorithm, while a macroscopic fundamental diagram concept is implemented to estimate the arc travel times. The methodology is applied to a hypothetical and two real‐world case study networks, and the solutions are compared to a Benders decomposition benchmark. The east‐coast analysis results indicate a 77.3% reduction in the computational time. Additionally, the benchmark technique obtains an optimality gap of 1.15%, while the C&amp;CG algorithm yields a 0.61% gap. The numerical experiments show the robustness of the proposed methodology. Besides, a series of sensitivity analyses has been conducted to study the impact of input parameters on the proposed methodology and draw managerial insights.

Dynamic Discretization Discovery Algorithms for Time-Dependent Shortest Path Problems

Finding a shortest path in a network is a fundamental optimization problem. We focus on settings in which the travel time on an arc in the network depends on the time at which traversal of the arc begins. In such settings, reaching the destination as early as possible is not the only objective of interest. Minimizing the duration of the path, that is, the difference between the arrival time at the destination and the departure from the origin, and minimizing the travel time along the path from origin to destination, are also of interest. We introduce dynamic discretization discovery algorithms to efficiently solve such time-dependent shortest path problems with piecewise linear arc travel time functions. The algorithms operate on partially time-expanded networks in which arc costs represent lower bounds on the arc travel time over the subsequent time interval. A shortest path in this partially time-expanded network yields a lower bound on the value of an optimal path. Upper bounds are easily obtained as by-products of the lower bound calculations. The algorithms iteratively refine the discretization by exploiting breakpoints of the arc travel time functions. In addition to time discretization refinement, the algorithms permit time intervals to be eliminated, improving lower and upper bounds, until, in a finite number of iterations, optimality is proved. Computational experiments show that only a small fraction of breakpoints must be explored and that the fraction decreases as the length of the time horizon and the size of the network increases, making the algorithms highly efficient and scalable. Summary of Contribution: New data collection techniques have increased the availability and fidelity of time-dependent travel time information, making the time-dependent variant of the classic shortest path problem an extremely relevant problem in the field of operations research. This paper provides novel algorithms for the time-dependent shortest path problem with both the minimum duration and minimum travel time objectives, which aims to address the computational challenges faced by existing algorithms. A computational study shows that our new algorithm is indeed significantly more efficient than existing approaches.

One-sided arc averaging geometries in time–distance local helioseismology

Context. The study of solar oscillations (helioseismology) has been a very successful method of researching the Sun. Helioseismology teaches us about the structure and mean properties of the Sun. Together with mid-resolution data, the local properties were uncovered in quiet-Sun regions. However, magnetic fields affect the oscillations and prevent us from studying the properties of magnetically active regions with helioseismology. Aims. We aim to create a new methodology to suppress the negative effects of magnetic fields on solar oscillations and measure plasma properties close to active regions. Methods. The methodology consists of new averaging geometries, a non-linear approach to travel-time measurements, and a consistent inversion method that combines plasma flows and sound-speed perturbations. Results. We constructed the one-sided arc averaging geometries and applied them to the non-linear approach of travel-time measurements. Using the one-sided arc travel times, we reconstructed the annulus travel times in a quiet-Sun region. We tested the methodology against the validated helioseismic inversion pipeline. We applied the new methodology for an inversion for surface horizontal flows in a region with a circular H-type sunspot. The inverted surface horizontal flows are comparable with the output of the coherent structure tracking, which is not strongly affected by the presence of the magnetic field. We show that the new methodology suppresses the negative effects of magnetic fields up to outer penumbra. We measure divergent flows with properties comparable to the moat flow. Conclusions. The new methodology can teach us about the depth structure of active regions and physical conditions that contribute to the evolution of the active regions.

On the shortest $$\alpha$$-reliable path problem

In this variant of the constrained shortest path problem, the time of traversing an arc is given by a non-negative continuous random variable. The problem is to find a minimum cost path from an origin to a destination, ensuring that the probability of reaching the destination within a time limit meets a certain reliability threshold. To solve this problem, we extend the pulse algorithm, a solution framework for shortest path problems with side constraints. To allow arbitrary non-negative continuous travel-time distributions, we model the random variables of the travel times using Phase-type distributions and Monte Carlo simulation. We conducted a set of experiments over small- and medium-size stochastic transportation networks with and without spatially-correlated travel times. As an alternative to handling correlations, we present a scenario-based approach in which the distributions of the arc travel times are conditioned to a given scenario (e.g., variable weather conditions). Our methodology and experiments highlight the relevance of considering on-time arrival probabilities and correlations when solving shortest path problems over stochastic transportation networks.

Benders' decomposition for robust travel time-constrained inventory routing problem

The time-constrained inventory routing problem (TCIRP) is a variant of the standard inventory routing problem (IRP), in which each vehicle's trip duration must not exceed a predefined maximum duration limit. In this paper, we consider the TCIRP on a network in which arc travel times are uncertain. In particular, we investigate the case where the only available information on the arc travel times is that they take values in symmetric and bounded intervals around their mean values. To solve this problem, a robust optimisation approach that generates solutions with a controlled level of conservatism is implemented. To cope with the complexity of the resulting robust counterpart, we develop a Benders' decomposition-based heuristic. Furthermore, we describe how the valid inequalities developed for the IRP can be strengthened so that the Benders' decomposition master problem can be solved effectively. We compare the proposed approach with two standard approaches for finding robust solutions. The analysis of the results show that the proposed method is able to find robust solutions that are not too conservative in reasonable time.

Receding Horizon Control for Station Inventory Management in a Bike-Sharing System

A docking bike-sharing system (BSS) is modeled as a network representing the underlying transportation network. Mobile agents (replenishment trucks) traverse the network making routing decisions and deciding how and when to replenish station inventories so as to prevent imbalances due to users’ one-way rides as well as time-varying demand. This load balancing process entails selecting both optimal routes for the agents and the number of bikes to load/unload at a station with an objective of minimizing a user dissatisfaction metric. First, we establish a time-dependent replenishment fill-to level policy for each station based on the demand rates and station capacities. Next, we focus on developing a receding horizon controller (RHC) to find optimal routes. The controller proceeds in an event-driven manner to determine after each event the optimal routes for a fleet of agents over a finite planning horizon, with the control applied over a shorter action horizon. The proposed controller is applied to a simulated BSS with station and demand parameters taken from the public data sets of Bluebikes, the BSS in Boston, MA, USA, and a cost-benefit analysis is performed on agent shift hours. In order to demonstrate the robustness of the RHC, sensitivity analysis is also performed on the arc travel times and the demand processes. Note to Practitioners—This paper is motivated by the load balancing problem faced by BSS with finite-capacity docking stations and time-varying demand; stations become empty or full as their popularity as an origin or destination varies over the course of a day. The proposed RHC creates dynamic cooperative routes for a fleet of load balancing trucks to move bikes between stations by considering the current inventory, demand rates, and proximity of all stations and trucks. The strength of this controller is that it achieves optimality over a specified planning horizon and reacts quickly to random inventory changes which update the planning horizon on a rolling basis. Thus, the optimal routes found for a planning horizon are updated at the next decision point, i.e., an intersection. This event-driven RHC decreases the complexity of finding optimal routes so that it may be used in real time. Insights obtained from the application of this approach to the Boston BSS are that extending the length of the receding horizon provides marginal benefits beyond a certain value and that the controller is robust with respect to the stochastic behavior of the user demand and the truck travel time processes. The approach is amenable to extensions that can include incentivizing users so as to enhance load balancing beyond external truck-based interventions.

A trade-off between average and maximum arc congestion minimization in traffic assignment with user constraints

In system optimal traffic assignment of traffic flows with user constraints the total travel time is minimized on a set of paths with bounded length ensuring a certain level of fairness for users. Minimizing the total travel time may lead to experience large travel times on some arcs. On the other hand, when minimizing the maximum arc travel time, the total travel time cannot be controlled. The increase of arc travel time with respect to arc free-flow travel time is related to the arc congestion that is one of the main issues in road networks. In this paper we propose a new model that is a compromise between minimizing the average arc congestion and the worst arc congestion on the road network. The model is a linear program that minimizes the average arc congestion over a given percentage of the most congested arcs. A heuristic algorithm aimed at reducing the number of paths that are considered for the model is also proposed. A computational study is performed that shows the flexibility of the model and the quality of the heuristic.

Reliable vehicle routing problem in stochastic networks with correlated travel times

This study is motivated by the fact that travel times on road networks are correlated. However, all existing studies on the vehicle routing problem share a common simplifying assumption that arc travel times are independently distributed. In this paper, we address a variant of the vehicle routing problem with soft time windows in which travel times are treated as correlated random variables. To this end, correlations among arc travel times are modeled by a variance–covariance matrix. We use a mathematical model in which penalties are incurred for early and late arrival at each customer (violation of time window constraints). A Max–Min ant colony system is hybridized with a tabu search algorithm to solve the model. Results show that ignorance of correlations among arc travel times can significantly lead to inefficient solutions to the vehicle routing problem in stochastic networks. We also conduct an exploratory analysis of real travel time data. The results of the analysis demonstrate that travel times are significantly correlated and the shifted log-normal distribution is an appropriate candidate for modeling travel time uncertainty.

A robust optimization approach for itinerary planning with deadline

We propose a robust optimization approach to address the itinerary planning problem with deadline in public transit networks. Given departure times at origins and deadlines at destinations, we help the travelers meet the deadlines as much as possible. Our model maximizes the size of the uncertainty set of arc travel times, while guaranteeing that the corresponding worst-case arrival time of itinerary would not exceed the deadline. We exploit the model’s structure and develop efficient solution algorithms. We demonstrate in numerical studies that our approach can effectively mitigate the lateness and can solve real-world instances within one second.

Itinerary planning with time budget for risk-averse travelers

Itinerary planning problem (IPP) in public transport networks has witnessed to be a fundamental and important problem in public transport operations management. Although empirical studies show that time budgets are usually prescribed by travelers, the time budget is rarely considered in literature related to IPP under stochastic travel times. In this paper, we study the IPP with time budget (IPPB) in public transport networks with stochastic travel times, which consists of planning an itinerary from an origin to a destination that helps risk-averse travelers mitigate uncertainty and effectively meet their time budgets. A mathematical model (MRHA-IPPB) is developed for the IPPB, in which we use the utility theory to characterize travelers’ risk-averse behaviors. Since minimizing the lateness probability or the expected lateness duration is intractable, we propose a new decision criterion, maximizing risk-hedging ability (short for MRHA hereafter) while guaranteeing that the corresponding certainty equivalent of itinerary travel time would not exceed the time budget. The model MRHA-IPPB is shown theoretically and numerically to consider both the lateness probability and the lateness duration. We show NP-completeness of the IPPB with fully correlated arc travel times and study two tractable scenarios, i.e., the IPPB with independent arc travel times and that with partially correlated arc travel times. We decompose an IPPB as a two-stage problem and efficiently solve it through a binary search scheme with a label-setting algorithm embedded. We demonstrate the effectiveness of the MRHA-IPPB model through an illustrative example and show that the MRHA-based decomposition approach requires only less than one second per query over a real-world public transport network.

Multicriteria Stochastic Shortest Path Problem for Electric Vehicles

This paper formulates the reliable routing of electric vehicles in stochastic networks as a multicriteria shortest path problem with travel time and charging cost components. The reliability term is defined as the probability of finishing the trip without running out of charge. The arc travel times are represented as stochastic variables, and arc energy consumption is modeled as a linear function of arc length and arc travel time. The traveler aims to minimize the generalized cost, formulated as a linear function of travel time and charging cost, subject to a minimum reliability threshold, representing the level of risk a traveler is willing to take in favor of routes with lower cost. We propose a solution algorithm based on generalized dynamic programming and show that the optimal solution may include cycles that visit at least one charging station. The properties of the proposed multicriteria shortest path problem are mathematically proved. The simulation results on randomly-generated networks show that cyclic paths are very rare, and that the generalized cost of travel is a monotone increasing function of minimum reliability threshold.

An Optimal Path Model for the Risk-Averse Traveler

In this paper we discuss stochastic optimal path problems where the goal is to find a path that has minimal expected cost and at the same time is less risky (in terms of travel time) than a given benchmark path. The model is suitable for a risk-averse traveler who prefers a path with a more guaranteed travel time to another path that could be faster but could also be slower. Such risk attitude is incorporated using the concept of second order stochastic dominance constraints. A recently developed theory for optimization problems with stochastic dominance constraints ensures that the resulting problem can be written as a large linear integer program with binary variables; for networks of realistic size, however, such a direct approach is not practical because of the size of the resulting optimization problem. Moreover, the solution returned by the model may contain cycles, which is clearly undesirable from a practical perspective. A number of strategies are explored to solve the problem. First, we prove that cycles can be prevented by a simple modification of the model if the arc travel times are mutually independent. We then propose a sample average approximation approach to the problem using samples from the distribution of travel times. Because of the randomness resulting from sampling, it is important that statistical guarantees for the solution returned by the algorithm be given, and we provide heuristic procedures to deal with stochastic constraints. We also incorporate a branch-and-cut approach that exploits the structure of the problem to deal with the integrality constraints more efficiently. We present some numerical experiments for a 1,522-arc system that corresponds to a large portion of the Chicago area network. The results show that our approach can solve the problem very effectively, producing solutions with statistical guarantees of optimality within a reasonable computational time.

Adaptive Transit Routing in Stochastic Time-Dependent Networks

We define an adaptive routing problem in a stochastic time-dependent transit network in which transit arc travel times are discrete random variables with known probability distributions. We formulate it as a finite horizon Markov decision process. Routing strategies are conditioned on the arrival time of the traveler at intermediate nodes and real-time information on arrival times of buses at stops along their routes. The objective is to find a strategy that minimizes the expected travel time, subject to a constraint that guarantees that the destination is reached within a certain threshold. Although this framework proves to be advantageous over a priori routing, it inherits the curse of dimensionality, and state space reduction through preprocessing is achieved by solving variants of the time-dependent shortest path problem. Numerical results on a network representing a part of the Austin, Texas, transit system indicate a promising reduction in the state space size and improved tractability of the dynamic program.

A bi-objective model for uncertain multi-modal shortest path problems

Abstract This paper employs uncertain programming to investigate the uncertain multi-modal shortest path problem, in which the arc weights (arc travel time, arc travel costs) associated with different transport modes are characterized by uncertain variables. By using the chance-constrained programming approach, we firstly formulate a bi-objective optimization model to minimize the total travel time and travel costs simultaneously with the given confidence levels. Moreover, using the basic concepts and properties in the uncertainty theory, we transform the proposed model into its deterministic crisp equivalent with an explicit proof. Finally, some numerical experiments are implemented to show the performance of the proposed approaches on a multi-modal transportation network with three specific modes.

Using the General Link Transmission Model in a Dynamic Traffic Assignment to Simulate Congestion on Urban Networks

This article presents two new models of Dynamic User Equilibrium that are particularly suited for ITS applications, where the evolution of vehicle flows and travel times must be simulated on large road networks, possibly in real-time. The key feature of the proposed models is the detail representation of the main congestion phenomena occurring at nodes of urban networks, such as vehicle queues and their spillback, as well as flow conflicts in mergins and diversions. Compared to the simple word of static assignment, where only the congestion along the arc is typically reproduced through a separable relation between vehicle flow and travel time, this type of DTA models are much more complex, as the above relation becomes non-separable, both in time and space.Traffic simulation is here attained through a macroscopic flow model, that extends the theory of kinematic waves to urban networks and non-linear fundamental diagrams: the General Link Transmission Model. The sub-models of the GLTM, namely the Node Intersection Model, the Forward Propagation Model of vehicles and the Backward Propagation Model of spaces, can be combined in two different ways to produce arc travel times starting from turn flows. The first approach is to consider short time intervals of a few seconds and process all nodes for each temporal layer in chronological order. The second approach allows to consider long time intervals of a few minutes and for each sub-model requires to process the whole temporal profile of involved variables. The two resulting DTA models are here analyzed and compared with the aim of identifying their possible use cases.A rigorous mathematical formulation is out of the scope of this paper, as well as a detailed explanation of the solution algorithm.The dynamic equilibrium is anyhow sought through a new method based on Gradient Projection, which is capable to solve both proposed models with any desired precision in a reasonable number of iterations. Its fast convergence is essential to show that the two proposed models for network congestion actually converge at equilibrium to nearly identical solutions in terms of arc flows and travel times, despite their two diametrical approaches wrt the dynamic nature of the problem, as shown in the numerical tests presented here.

Floyd-A∗Algorithm Solving the Least-Time Itinerary Planning Problem in Urban Scheduled Public Transport Network

We consider an ad hoc Floyd-A∗algorithm to determine the a priori least-time itinerary from an origin to a destination given an initial time in an urban scheduled public transport (USPT) network. The network is bimodal (i.e., USPT lines and walking) and time dependent. The modified USPT network model results in more reasonable itinerary results. An itinerary is connected through a sequence of time-label arcs. The proposed Floyd-A∗algorithm is composed of two procedures designated as Itinerary Finder and Cost Estimator. The A∗-based Itinerary Finder determines the time-dependent, least-time itinerary in real time, aided by the heuristic information precomputed by the Floyd-based Cost Estimator, where a strategy is formed to preestimate the time-dependent arc travel time as an associated static lower bound. The Floyd-A∗algorithm is proven to guarantee optimality in theory and, demonstrated through a real-world example in Shenyang City USPT network to be more efficient than previous procedures. The computational experiments also reveal the time-dependent nature of the least-time itinerary. In the premise that lines run punctually, “just boarding” and “just missing” cases are identified.

Online Stochastic Routing Incorporating Real-Time Traffic Information

In this paper, online stochastic routing policies are developed to identify the optimal next action (path choice) at the current decision node (intersection) for travelers on the basis of their preference for future paths with the shortest travel time, the lowest travel time variability, or a combination, given the network conditions. A modified label-correcting algorithm is provided to solve for the shortest path that results from the proposed routing policies. Its running time is bounded by O(mn2), where m and n are the number of arcs and nodes, respectively, in the network. Since real-time traffic information is usually available with a certain level of accuracy, the proposed online routing policy integrates an existing information fusion model, which provides real-time short-term arc travel time distributions by considering information accuracy. Numerical experiments are used to demonstrate the performance of the proposed routing policies and algorithms, as well as the impacts of real-time information accuracy on the online stochastic routing.

The general traffic equilibrium problem in a stochastic network with travellers' risk aversion and inaccurate perceptions

We study the general traffic equilibrium problem through a stochastic network in which travellers perceive inaccurately probabilistic arc travel times and use disutility functions to evaluate path perceived travel times. The objective is to determine the arc equilibrium flows when each trip-maker travels along the perceived optimal path which minimises the perceived expected disutility. A fixed point formulation for this general problem is presented in which flows are defined through paths. The problem is difficult to solve exactly due to the non-additive path cost structure of the perceived expected disutility function. However, we introduce a solution strategy that, without requiring a complete path enumeration, can approximate the arc flows. Extensive computational experiments are performed to investigate the effects of various factors on travellers’ route choice decisions, and to demonstrate that the proposed approach provides good quality flows in a very reasonable amount of time.