Time-dependent Vehicle Routing Problem Research Articles

Uncrewed Aerial Vehicle Routing Problem for Integrated Crewed and Uncrewed Aircraft Operations

Last-mile delivery systems are one of the numerous industrial applications of uncrewed aircraft systems (UAS). This study analyzes UAS-based package delivery systems in the context of the integrated operations of crewed and uncrewed aircraft. It introduces a multi-trip, time-dependent vehicle routing problem aimed at optimizing UAS delivery routes within integrated operations. By conducting numerical simulations of helicopters (crewed aircraft) and UAS missions sharing the same airspace, the optimal UAS routes for resolving potential conflicts in the shared airspace can be effectively determined without any adverse impact on helicopter operations. Knowing the helicopter mission times in advance enables the efficient planning of UAS missions. The results of this study provide valuable insights regarding the implementation of integrated crewed and uncrewed aircraft operations.

Collaboration and resource sharing in the multidepot time-dependent vehicle routing problem with time windows

Concerns about energy conservation and emission reduction have highlighted the importance of environmentally sound logistics networks in urban areas. These networks are deeply intertwined with urban traffic systems, where variations in transit speeds can significantly increase the energy consumption and carbon emissions of delivery vehicles, compromising the environmental sustainability of urban deliveries. To address this, we propose a multidepot time-dependent vehicle routing problem with time windows, enhancing route planning flexibility and resource configuration. Our approach begins with a route spatiotemporal decomposition method to estimate vehicle travel times and emissions based on varying vehicle speeds. We then develop a multiobjective mixed integer linear programming model that aims to minimize total operating costs, the number of vehicles, and carbon dioxide emissions. A hybrid heuristic algorithm combining spectral clustering, multiobjective ant colony optimization, and variable neighborhood search is proposed to solve the model. This algorithm incorporates collaboration and resource sharing strategies, a pheromone initialization mechanism, a novel heuristic operator that accounts for time dependency, and a self-adaptive update mechanism, enhancing both solution quality and algorithm convergence. We compare the performance of our algorithm with that of the CPLEX solver, multiobjective ant colony optimization, non-dominated sorting genetic algorithm-Ⅲ, and multiobjective particle swarm optimization. The results demonstrate the superior convergence, uniformity, and spread of our proposed algorithm. Furthermore, we apply our model and algorithm to a real-world case in Chongqing, China, analyzing optimized results for different time intervals and vehicle speeds. This study offers robust methodologies for theoretically and practically addressing the multidepot time-dependent vehicle routing problem with time windows, contributing to the development of economical, efficient, collaborative, and sustainable urban logistics networks.

A branch-and-cut algorithm for the time-dependent vehicle routing problem with time windows and combinatorial auctions

The urban logistics industry typically faces traffic congestion problems that result in variation of travel times across the day and impose a great challenge in routing design. While companies may either use a private fleet of delivery vehicles or outsource the tasks to the third-party logistics (3PL) providers to fulfill their logistics demand, some companies might employ a combination of both when their resource is unable to cope with demand or if demand fluctuates significantly over time. To tackle both challenges, we focus on a new variant of the vehicle routing problem known as the time-dependent vehicle routing problem with time windows and combinatorial auctions (TD-VRPTWCA), which considers time-dependent travel times in the routing design of the private fleet while selecting competitive bids from the 3PLs to serve a subset of the customers economically. The goal is to minimize the sum of the travel cost incurred by the private fleet and the outsourcing cost charged by the 3PLs for the chosen bids. To solve this problem, we present an arc-flow model with nine families of valid inequalities to strengthen the linear relaxation of the model. Based on this, a branch-and-cut approach is developed and evaluated on instances adapted from the well-known Solomon’s benchmark data. Extensive computational results demonstrate the effectiveness of the proposed method.

Correction to the Paper “Branch and Price for the Time-Dependent Vehicle Routing Problem with Time Windows”

The aim of this erratum is to correct an error in the computer implementation of the algorithm proposed by Dabia, Ropke, and van Woensel [Dabia S, Ropke S, van Woensel T (2013) Branch and price for the time-dependent vehicle routing problem with time windows. Transportation Sci. 47(3):295–454]. Section 6 , “Computational Results,” from the original paper is rewritten to reflect the corrected implementation, the new computational setup, and the updated results.

A practical vehicle routing problem in small and medium cities for fuel consumption minimization

Today, reducing fossil fuel consumption in the transport sector is one of the important requirements to preserve the ecological environment. In this paper, we investigated a logistics problem in small and medium cities. We proposes the time-dependent vehicle routing problem model in small and medium cities considering heterogeneous fleets, time windows, multi-trip, time dependency, and road network. We conducted a numerical analysis on fuel consumption minimizing path selection between two customer nodes and found that the path selection is independent of the payload in most cases. The model is formulated as a mathematical integer linear programming model, and small problems are solved using a commercial optimization solver. For large-scale instances, we proposed a heuristic algorithm based on the variable neighborhood search and evaluated its performance. The proposed algorithm can decrease fuel consumption by 25 % compared to manual routing and scheduling. Furthermore, for provide useful managerial insights, sensitivity analyses were performed considering different vehicle capacities.

An accelerated benders decomposition algorithm for the solution of the multi-trip time-dependent vehicle routing problem with time windows

The logistics companies executing the last mile delivery of goods in urban areas deal every day with the problem of routing their vehicles, while taking into account multiple trips per vehicle, time-dependent travel time, customers’ time windows and loading time at the depot simultaneously. This paper addresses this problem, known as Multi-Trip Time-Dependent Vehicle Routing Problem with Time Windows, aiming at its exact solution, minimizing the total travelled distance of a company's fleet. Based on a literature model, a new reformulation with reduced size is suggested. This reformulation is decomposed by applying the Benders method in an effective way, resulting in a subproblem with no duality gap. By exploiting the special features of the problem and the particular structure of the decomposition made, several novel valid inequalities are introduced, in order to both tighten the non-decomposed formulations and warm start the relaxed master problem to achieve less infeasible solutions and higher lower bounds. For the solution of the problem, an innovative algorithm is proposed, including suboptimal master solutions and a multi-cut generation procedure, which is based on the careful observation of the values of the Benders dual subproblem variables. The impact of the valid inequalities as well as two variants of the suggested algorithm are tested on benchmark data and they are compared with the non-decomposed models and a heuristic introduced in the literature. The computational results indicate improved efficiency and stronger bounds for the proposed algorithm.

A hybrid genetic search and dynamic programming-based split algorithm for the multi-trip time-dependent vehicle routing problem

We design a hybrid algorithm for the multi-trip time-dependent vehicle routing problem (MT-TD-VRP). One of its components is the Time-Dependent SPlit Algorithm (TD-SPA), which is a dynamic programming-based algorithm specifically designed to handle both the multi-trip per vehicle and the time-dependent aspects of the problem. The hybrid algorithm combines the proposed TD-SPA, designed to efficiently split a giant tour into complete vehicle routes, with a genetic algorithm for generating these tours. We introduce a monotone queue optimization (MQO) technique to accelerate the TD-SPA. The effectiveness of MQO is evaluated by comparing computation times between the original split algorithm for the capacitated vehicle routing problem (CVRP) and its MQO-enhanced counterpart. Extensive numerical experiments with a real-world dataset from a Singapore food and beverage company are conducted to assess our algorithm’s performance on various MT-TD-VRP instances. The results indicate that our algorithm surpasses the performance of the commercial solver Gurobi, with an average improvement of 25.13% on the best solutions found within a prescribed duration. Our numerical simulations further reveal the algorithm’s ability to efficiently solve both the capacitated vehicle routing problem (CVRP) and the multi-trip vehicle routing problem (MTVRP), consistently producing competitive solutions. Moreover, to highlight the importance of incorporating the time-dependent (TD) factor into our model and algorithm, we demonstrate a notable enhancement in performance–averaging at 7.47% for the best solutions under TD conditions for an MTVRP dataset.

Two-echelon time-dependent vehicle routing problem with simultaneous pickup and delivery and satellite synchronization

The multi-echelon distribution strategy has been widely applied in city logistics systems. This study develops a mixed-integer linear programming model for a new variant of vehicle routing problem, namely, a two-echelon time-dependent vehicle routing problem with simultaneous pickup and delivery and satellite synchronization, in which customers are divided into multiple customer regions according to different geographical characteristics. The pickup and delivery activities with hard time windows are performed simultaneously by the same vehicles from depots to satellites in the first echelon and from satellites to customers in different customer regions in the second echelon. A vehicle speed function is developed for each echelon based on traffic conditions. The function depends on commuter traffic and urbanization level. Satellites with storage buffer capacities for split and consolidation are allowed. Thus, demand unloading, storage, and reloading are synchronized. Costs of transportation, loading/unloading, inventory, and environment during one complete distribution process are considered. A memetic algorithm (MA) is proposed including a split algorithm for chromosome decoding and a three-phase construction heuristic for initial population generation. Self-adaptive operators of crossover, mutation, and local search are designed, and a neighborhood route improvement strategy is devised that includes distance-related destroy, route-based destroy, distance-based greedy repair, and time-distance cluster repair operators. Results of different scales of numerical experiments and sensitivity analysis show the effectiveness of the model and MA by comparing it with an exact algorithm, CPLEX, and adaptive large neighborhood search.

A metaheuristic for a time-dependent vehicle routing problem with time windows, two vehicle fleets and synchronization on a road network

In this work, we extend the time-dependent vehicle routing problem with time windows on a road network by considering two types of vehicles, large and small, to serve customers. Motivated from city logistics applications, large vehicles are forbidden from the downtown area. Accordingly, goods must be transferred from large to small vehicles to serve downtown customers. This leads to synchronization issues at transfer points, which are special locations without storage capacity. The problem is not a pure two-echelon vehicle routing problem, since customers outside of the downtown area can be served directly by large vehicles. The problem is further compounded by the presence of time-dependent travel times that are defined on the arcs of the road network and are used to model congestion periods. To solve this difficult problem, we propose an adaptation of the Slack Induction by String Removals metaheuristic, which is state-of-the-art for the classical capacitated vehicle routing problem. Computational results on a set of test instances with different characteristics empirically demonstrate the optimization capabilities of this new metaheuristic on a problem which is much more complicated than the capacitated vehicle routing problem.

Optimization algorithm for the time-dependent vehicle routing problem with softtime windows: A research based on mixed integer model and mixed tabu search

Optimization algorithm for the time-dependent vehicle routing problem with softtime windows: A research based on mixed integer model and mixed tabu search

Data-driven preference-based routing and scheduling for activity-based freight transport modelling

Understanding preferences and behaviours in road freight transport is valuable for planning and analysis. This paper proposes a data-driven vehicle routing and scheduling approach for use as a descriptive tool to study road freight transport activities. The model developed seeks to capture planners’ or drivers’ preferences in order to reproduce observed road freight activities. The model is based on a parametrized time-dependent vehicle routing problem whose parameters can be estimated from a set of observed planned tours. We propose a Bayesian optimization technique for parameter estimation of the model. Empirical results show that the model can fit real-world data accurately and synthesize tour flows close to reality.

A bi-objective time-dependent vehicle routing problem with delivery failure probabilities

This work presents a bi-objective time-dependent vehicle routing problem with delivery failure probabilities (TDVRPDFP). Two objectives are jointly minimized: operational costs and delivery failure rates. Both travel times and costs, as well as the probabilities of delivery failure and service times and costs, are considered time-dependent. A mixed-integer linear programming model is proposed to obtain sets of non-dominated solutions for small-size instances, while a multi-objective genetic algorithm, NSGA-II, is implemented to obtain approximate sets of non-dominated for large-size instances. Five sets of instances are proposed and used to evaluate the solution approaches. Our results indicate that the implemented NSGA-II algorithm can optimally solve small and large instances and find large Pareto fronts for instances with more than 25 nodes and 40-time intervals.

Pareto Frontier Approach to Determining the Optimal Path on Multi-Objectives

Every issue we face in daily life can be resolved through mathematical modeling. The use of mathematical modeling to generate solutions frequently produces value that serves a single purpose. Sometimes a single-purpose function's solution does not offer the best solution value. In this study, the author models the multiobjective time-dependent vehicle routing problem using the Ant Colony Optimization (ACO) metaheuristic algorithm. The author then applies the pareto optimization principle to the determination of the optimal starting point for the route. An optimal Pareto frontier principle solution on a multi-objective model under control of the Ant Colony Optimization algorithm is the outcome of this study.

Deep attention models with dimension-reduction and gate mechanisms for solving practical time-dependent vehicle routing problems

Time dependencies of travel speeds in time-dependent vehicle routing problems (TDVRPs) are usually accounted for by discretizing the planning horizon into several time periods. However, travel speeds usually change frequently in real road networks, so many time periods are needed to evaluate candidate solutions accurately in model and solution construction for practical TDVRPs, which increases substantially the computational complexity of TDVRPs. We develop two deep attention models with dimension-reduction and gate mechanisms to solve practical TDVRPs in real urban road networks. In the two models, a multi-head attention-based dimension-reduction mechanism is proposed to reduce the dimension of model inputs and obtain enhanced node representation, whereas a gate mechanism is introduced to obtain better information representation. On the basis of a travel speed dataset from an urban road network, we conduct extensive experiments to validate the effectiveness of the proposed models on practical TDVRPs with or without consideration of time windows. Experimental results show that our models can solve TDVRPs with 240 time periods and up to 250 customers effectively and efficiently and provide significantly superior overall performances over two representative heuristics and two state-of-the-art deep reinforcement learning models. Especially, compared with a recent tabu search method, our models can reduce the computation time by up to 3,540 times and improve the solution performance by up to 23%. Moreover, our models have an outstanding generalization performance. The model trained for the 30-customer TDVRP with time windows can be used directly to solve problems with up to 250 customers effectively by generating superior solutions over those generated by benchmarking methods.

Research on the Time-Dependent Vehicle Routing Problem for Fresh Agricultural Products Based on Customer Value

With continuous improvements in people’s consumption levels, consumers’ demands for safe and fresh agricultural products increase. The increase in the number of vehicles and serious congestion on roads has led to problems, such as the weak timeliness of urban cold chain logistics, high carbon emissions, low customer value and reduced customer satisfaction. In this study, carbon emissions, customer satisfaction, customer value and cost are considered, and an optimization algorithm is established to solve the time-dependent vehicle routing problem in urban cold chain logistics. For road congestion at different time periods during the cold chain distribution process, the segment function is used to express the vehicle speed. According to the characteristics of the model, considering the constraints of the time window and vehicle capacity, an improved NSGA-II algorithm with the local optimization characteristics of the greedy algorithm (G-NSGA-II) is proposed, and the sorting fitness strategy is optimized. In addition, we carry out a series of experiments on existing vehicle routing problem examples and analyze them in a real background to evaluate and prove the effectiveness of the proposed model and algorithm. The experiment results show that the proposed approach effectively reduces the total cost, enhances customer value and promotes the long-term development of logistics companies.

Time-dependent vehicle routing problem of perishable product delivery considering the differences among paths on the congested road

Time-dependent vehicle routing problem of perishable product delivery considering the differences among paths on the congested road

Mixed Integer Programming Formulation for Time-Dependent Petrol Station Replenishment Problem: A Real-Life Case in İstanbul

With the increase in the consumption of petroleum and petroleum products, these limited resources must be provided efficiently, accurately, and with minimal damage. Therefore, the accurate and effective distribution of petroleum, and related problems with petroleum distribution have attracted much attention among the practitioners and optimization working researchers over the years. The petroleum distribution problem, as a version of the Vehicle Routing Problem (VRP), deals with the planning of petroleum distribution from the depot(s) to the petrol stations safely and quickly. In this study, the petrol station replenishment problem (PSRP) is handled and a case study is presented for a public company located in İstanbul. The problem is considered as a time-dependent VRP with time windows. A novel mixed integer mathematical model is proposed for the problem. In order to handle the proposed time-dependent problem in a more realistic way, variable tanker speeds are considered based on traffic density. The optimum route is determined in which risks such as environment and marine pollution may occur in case of possible accidents, and these risks are minimized by the proposed mathematical model considering the factors as traffic, vehicle speed, road structure, the road's proximity to the sea and living areas

Multi-Trip Time-Dependent Vehicle Routing Problem with Split Delivery

Motivated by some practical applications of post-disaster supply delivery, we study a multi-trip time-dependent vehicle routing problem with split delivery (MTTDVRP-SD) with an unmanned aerial vehicle (UAV). This is a variant of the VRP that allows the UAV to travel multiple times; the task nodes’ demands are splittable, and the information is time-dependent. We propose a mathematical formulation of the MTTDVRP-SD and analyze the pattern of the solution, including the delivery routing and delivery quantity. We developed an algorithm based on the simulation anneal (SA) framework. First, the initial solution is generated by an improved intelligent auction algorithm; then, the stochastic neighborhood of the delivery route is generated based on the SA algorithm. Based on this, the model is simplified to a mixed-integer linear programming model (MILP), and the CPLEX optimizer is used to solve for the delivery quantity. The proposed algorithm is compared with random–simulation anneal–CPLEX (R-SA-CPLEX), auction–genetic algorithm–CPLEX (A-GA-CPLEX), and auction–simulation anneal–CPLEX (A-SA) on 30 instances at three scales, and its effectiveness and efficiency are statistically verified. The proposed algorithm significantly differs from R-SA-CPLEX at a 99% confidence level and outperforms R-SA-CPLEX by about 30%. In the large-scale case, the computation time of the proposed algorithm is about 30 min shorter than that of A-SA. Compared to the A-GA-CPLEX algorithm, the performance and efficiency of the proposed algorithm are improved. Furthermore, compared to a model that does not allow split delivery, the objective function values of the solution of the MTTDVRP-SD model are reduced by 52.67%, 48.22%, and 34.11% for the three scaled instances, respectively.

Time-dependent Vehicle Routing Problem with Departure Time and Speed Optimization for Shared Autonomous Electric Vehicle Service

Shared autonomous electric vehicles (SAEVs) are expected to become a popular choice for urban transportation in the future. A solution to a vehicle routing problem that considers congestion and energy consumption of SAEVs is proposed to assist green transportation under poor traffic conditions. In addition to routing the fleet of SAEVs to serve customers, the proposed method also determines the vehicle speed in each arc and the departure time at each node by minimizing the cost considering the travel distance, energy consumption, and travel time. In the time-dependent vehicle routing problem with departure time and speed optimization for SAEV service (SAEV-TD-VRP-DSO), the electric vehicles (EVs) exchange batteries at battery swapping stations (BSSs), and the corresponding vehicle routing method also includes recharge scheduling. We develop a mixed-integer linear model (MILP) to formulate the SAEV-TD-VRP-DSO and show that the state-of-the-art commercial optimization solver (CPLEX) can only solve a few instances (no more than 8 requests). Thus, an adaptive large neighborhood search (ALNS) algorithm is proposed to find near-optimal solutions for larger-size instances. The removal and insertion operators of ALNS help optimize the node sequences. A departure time and speed optimization procedure (DSOP) is employed to optimize the speed and departure time in each arc. The good performance of the proposed algorithm is demonstrated using instance sets. The optimization results of a case study in the Yanta Administrative District in Xi'an, China, show that speed and departure time optimization result in 24.99% cost savings. The sensitivity analysis results provide the SAEV operators with alternative cost-saving solutions. Empty vehicles traveling at a slower speed can reduce energy cost. The total cost of operating vehicles with passengers can be reduced by avoiding congestion and utilizing larger capacity batteries (32 kWh or more) and a lower minimum battery level (0.2-0.3).

Branch-Cut-and-Price for the Time-Dependent Green Vehicle Routing Problem with Time Windows

Motivated by rising concerns regarding global warming and traffic congestion effects, we study the time-dependent green vehicle routing problem with time windows (TDGVRPTW), aiming to minimize carbon emissions. The TDGVRPTW is a variant of the time-dependent vehicle routing problem (TDVRP) in which, in addition to the time window constraints, the minimization of carbon emissions requires determination of the optimal departure times for vehicles, from both the depot and customer location(s). Accordingly, the first exact method based on a branch-cut-and-price (BCP) algorithm is proposed for solving the TDGVRPTW. We introduce the notation of a time-dependent (TD) arc and describe how to identify the nondominated TD arcs in terms of arc departure times. In this way, we reduce infinitely many TD arcs to a finite set of nondominated TD arcs. We design a state-of-the-art BCP algorithm for the TDGVRPTW with labeling and limited memory subset row cuts, together with effective dominance rules for eliminating dominated TD arcs. The exact method is tested on a set of test instances derived from benchmark instances proposed in the literature. The results show the effectiveness of the proposed exact method in solving TDGVRPTW instances involving up to 100 customers. Summary of Contribution: Due to the environmental situation, green vehicle routing problems (GVRPs) aim to consider greenhouse gas emissions reduction, while routing the vehicles, and play a key role in transportation and logistics. Vehicle greenhouse gas emissions strongly depend on the vehicle speeds and traffic conditions which in real life vary continuously over time. To tackle these challenges, we address the time-dependent green vehicle routing problem with time windows (TDGVRPTW) aimed at reducing total carbon emissions under time-dependent travel times and time window constraints. We design an effective exact method for the TDGVRPTW based on a state-of-the-art branch-cut-and-price algorithm. The paper is both of methodological value for researchers and of interest for practitioners. For researchers, the presented algorithm is amenable for various routing constraints and provides a ground for further studies and research. For practitioners, the paper suggests insights on how the carbon emissions change based on different vehicle speed profiles. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This research is supported by the National Natural Science Foundation of China [Grants 71831003, 71831006, 72171043, and 71901180] and the Fundamental Research Funds for the Central Universities [Grants N170405005 and N180704015]. Supplemental Material: The electronic companion is available at https://doi.org/10.1287/ijoc.2022.1195 .