Time-dependent Vehicle Routing Problem Research Articles

Simplified Swarm Optimization for the Heterogeneous Fleet Vehicle Routing Problem with Time-Varying Continuous Speed Function

Transportation planning has been established as a key topic in the literature and practices of social production, especially in urban contexts. To consider traffic environment factors, more and more researchers are taking time-varying factors into account when scheduling their logistic activities. The time-dependent vehicle routing problem (TDVRP) is an extension of the classical Vehicle Routing Problem with Time Windows (VRPTW) by determining a set of optimal routes serving a set of customers within specific time windows. However, few of them use the continuous speed function to express the time-varying. In practice, many vehicle routing problems are addressed by a fleet of heterogeneous vehicles with different capacities and travel costs including fix costs and variable costs. In this study, a Heterogeneous Fleet Vehicle Routing Problem (HFPRP) Time-Varying Continuous Speed Function has been proposed. The objective is to minimize distribution costs, which contained fixed costs of acquiring and variable fuel costs. To address this problem, our research developed a mathematical model and proposed a Simplified Swarm Optimization (SSO) heuristic for HFVRP with Time-Varying Continuous Speed Function.

On path ranking in time-dependent graphs

In this paper we study a property of time-dependent graphs, dubbed “path ranking invariance”. Broadly speaking, a time-dependent graph is “path ranking invariant” if the ordering of its paths (w.r.t. travel time) is independent of the start time. In this paper we show that, if a graph is path ranking invariant, the solution of a large class of time-dependent vehicle routing problems can be obtained by solving suitably defined (and simpler) time-independent routing problems. We also show how this property can be checked by solving a linear program. If the check fails, the solution of the linear program can be used to determine a tight lower bound. In order to assess the value of these insights, the lower bounds have been embedded into an enumerative scheme. Computational results on the time-dependent versions of the Travelling Salesman Problem and the Rural Postman Problem show that the new findings enable to outperform state-of-the-art algorithms.

A model for the time dependent vehicle routing problem with time windows under traffic conditions with intelligent travel times

A new mixed-integer nonlinear programming model is presented for the time-dependent vehicle routing problem with time windows and intelligent travel times. The aim is to minimize fixed and variable costs, with the assumption that the travel time between any two nodes depends on traffic conditions and is considered to be a function of vehicle departure time. Depending on working hours, the route between any two nodes has a unique traffic parameter. We consider each working day to be divided into several equal and large intervals, termed as a time interval of traffic. Here, allowing for long distances between some of the nodes, travel time may take more than one time interval of traffic, resulting in resetting the time interval of traffic at the start of each large interval. This repetition of time interval of traffics has been used in modeling and calculating travel time. A tabu search optimization algorithm is devised for solving large problems. Also, after linearization, a number of random instances are generated and solved by the CPLEX solver of GAMS to assess the effectiveness of our proposed algorithm. Results indicate that the initial travel time is estimated appropriately and updated properly in accordance with to the repeating traffic conditions.

The secure time-dependent vehicle routing problem with uncertain demands

This paper addresses the transportation of valuable goods in which security carriers are interested to optimize the operating costs and the vehicle routes’ security. In real operations, route planning and scheduling are normally performed manually based on individual experience. Hereupon, an optimizing approach should be developed to target the robbery risk reduction and the distribution network’ operational costs. Facing the dangerous nature of operations, some considerations play an important role, including demand fluctuations and traffic congestion of the urban environment. To handle these issues, we respectively benefit from the robust optimization theory and considering time-dependent travel speeds with satisfying the “first-in-first-out” property. This paper proposes a rich vehicle routing problem denoted by the secure time-dependent vehicle routing problem with time windows including pickup and delivery with uncertain demands (S-TD-VRPTWPD-UD). A mathematical formulation and an efficient solution approach combining the greedy randomized adaptive search procedure (GRASP) and the iterated local search (ILS) are developed to minimize the predictability of route plans using an integrated risk index, besides the travel costs. Extensive computational experiments on this problem are performed to analyze the impact of the demand uncertainty and the speed time-dependency and to show the efficiency of the GRASP×ILS implementation. The results show the significant improvements due to the time-dependency as well as the extra cost of protecting the model against the worst-case scenario of demand requests by deriving the robust counterpart of the S-TD-VRPTWPD-UD.

The Time-Dependent Vehicle Routing Problem with Time Windows and Road-Network Information

Most time-dependent vehicle routing problems are based on a similar modeling paradigm: travel time information is represented by travel time functions between pairs of points of interest (e.g., depot or customers). Only a few papers investigate how these functions can be computed using the available travel time information. Furthermore, most of them neglect the possibility that different paths could be selected in the road network depending on the compromises they offer between cost (distance) and travel time. In this paper, we propose a new setting where travel time functions are defined on road-network arcs. We consider the Time-Dependent Vehicle Routing Problem with Time Windows and solve it with a branch-and-price algorithm. As far as we know, this is the first exact approach for a time-dependent vehicle routing problem when travel time functions are initially defined on the segments of a road-network.

Time-dependent vehicle routing problem with backhaul with FIFO assumption: Variable neighborhood search and mat-heuristic variable neighborhood search algorithms

This paper presents a mathematical model for a single depot, time-dependent vehicle routing problem with backhaul considering the first in first out (FIFO) assumption. As the nature of the problem is NP-hard, variable neighborhood search (VNS) meta-heuristic and mat-heuristic algorithms have been designed. For test problems with large scales, obtained results highlight the superior performance of the mat-heuristic algorithm compared with that of the other algorithm. Finally a case study at the post office of Khomeini-Shahr town, Iran, was considered. Study results show a reduction of roughly 19% (almost 45 min) in the travel time of the vehicle.

A hybrid algorithm for time-dependent vehicle routing problem with time windows

In this paper, we study the duration-minimizing time-dependent vehicle routing problem with time windows (DM-TDVRPTW), where time-dependent travel times represent different levels of road congestion throughout the day. The departure time from depot becomes an important decision to reduce the route duration. We provide an alternative arc-based mixed-integer programming model with explicit arc time zone index. For larger scale instances, we extend the segment-based evaluation method to speed up the feasibility check of a given vehicle route. It is thereafter incorporated to implement an efficient hybrid adaptive large neighborhood search with tabu search (ALNS-TS) algorithm, to explore both feasible and infeasible solution spaces. The ALNS-TS exploits the strength of diversification in ALNS through adaptive control of neighborhood size, as well as the strength of intensification in TS by limiting the maximum number of steps allowed without improvement. Related parameters are tuned using an automatic configuration tool to determine the best set of configurations. The computational results demonstrate its excellent performance on the benchmark instances proposed by Dabia et al. (2013), obtaining optimal solutions for all instances with 25 customers and improving best-known solutions for 36 instances with 50 and 100 customers. Computational experiments are also conducted to evaluate the effectiveness of the algorithm on an extra set of large scale instances and on the simplified VRPTW instances.

Multi-trip time-dependent vehicle routing problem with time windows

In this study, we investigate a routing problem in urban transportation which considers time-dependent travel time, multiple trips per vehicle, and loading time at the depot simultaneously. Its objective is to minimize the total travel distance while satisfying the time windows, vehicle capacity, and maximum trip duration constraints. We model the problem as a multi-trip time-dependent vehicle routing problem with time windows (MT-TDVRPTW). We formulate the time-dependent ready time function and duration function for any segment of consecutive nodes as piecewise linear functions and develop an iterative algorithm to derive them efficiently. Then, these two functions are embedded in the segment-based evaluation scheme to accelerate the local search operators. Based on them, we design a hybrid meta-heuristic algorithm to solve the problem, leveraging the adaptive large neighborhood search (ALNS) for guided exploration and the variable neighborhood descend (VND) for intensive exploitation. Moreover, we propose problem-specific local search operators and removal operators to enhance the effectiveness of the algorithm. Extensive experiments are conducted to assess the performance of the algorithm on instances of varied sizes. The algorithm is shown to be robust and efficient under different speed profiles and maximum trip duration limits. Finally, we evaluate the performance of the algorithm on a special case: the multi-trip vehicle routing problem with time windows.

An Enhanced Network-Consistent Travel Speed Generation Scheme on Time-Dependent Shortest Path and Routing Problems

The time-dependent shortest path and vehicle routing literature depends on realistic and reasonable test data for demonstration and performance evaluation. Despite the advancements in GPS and tracking technologies there is still lack and inaccessibility of publicly available real-world road networks with time-dependent arc costs and speeds. Since most of the time-dependent travel time layer generation models proposed for vehicle routing problems (VRPs) are mainly developed for synthetic networks, they cannot capture some realistic features of the real road networks and cannot be used in time-dependent shortest path problems (TDSPPs). In this paper, we first exploit spatial and temporal behavior of travel times using real life road network and speed data, and discuss the cases where the existing methods in the literature are not applicable. Then, we propose an enhanced method that is best fitted for TDSPP and time-dependent VRP (TDVRP). The proposed method can be implemented on both synthetic and real road networks. Finally, we apply our method to generate realistic speed data on Istanbul road network and demonstrate the applicability in TDSPP and TDVRP.

A Fissile Ripple Spreading Algorithm to Solve Time-Dependent Vehicle Routing Problem via Coevolutionary Path Optimization

The time-dependent vehicle routing problems have lately received great attention for logistics companies due to their crucial roles in reducing the time and economic costs, as well as fuel consumption and carbon emissions. However, the dynamic routing environment and traffic congestions have made it challenging to make the actual travelling trajectory optimal during the delivery process. To overcome this challenge, this study proposed an unconventional path optimization approach, fissile ripple spreading algorithm (FRSA), which is based on the advanced structure of coevolutionary path optimization (CEPO). The objective of the proposed model is to minimize the travelling time and path length of the vehicle, which are the popular indicators in path optimization. Some significant factors usually ignored in other research are considered in this study, such as congestion evolution, routing environment dynamics, signal control, and the complicated correlation between delivery sequence and the shortest path. The effectiveness of the proposed approach was demonstrated well in two sets of simulated experiments. The results prove that the proposed FRSA can scientifically find out the optimal delivery trajectory in a single run via global research, effectively avoid traffic congestion, and decrease the total delivery costs. This finding paves a new way to explore a promising methodology for addressing the delivery sequence and the shortest path problems at the same time. This study can provide theoretical support for the practical application in logistics delivery.

Adaptive Cultural Algorithm-Based Cuckoo Search for Time-Dependent Vehicle Routing Problem with Stochastic Customers Using Adaptive Fractional Kalman Speed Prediction

For the Time-Dependent Vehicle Routing Problem with Stochastic Customers (TDVRPSC), an adaptive Cultural Algorithm-Based Cuckoo Search (CACS) has been proposed in this paper. The convergence of the new algorithm is proved. An adaptive fractional Kalman filter (AFKF) for traffic speed prediction is proposed. An adaptive mechanism for choosing the covariance of state noise is designed. Its mathematical process is proved. Several benchmark instances with different scales are tested, and new solutions are discovered, which are better than the published solutions. The effects of the parameters on the convergence and the results are studied. According to cargo weight of customers to be delivered, the customers can be divided into large, small, and retail customers. The algorithm is tested with fixed demand probability and also different customer types with stochastic demand. The traffic speeds in different business districts in Xiamen at different times are predicted by AFKF. The results show that AFKF has smaller prediction error and better prediction accuracy than fractional Kalman filter and Kalman filter. The effect of different fractional orders on prediction error is compared. The performance of the new algorithm is compared with that of the cultural algorithm and the Cuckoo Search. The result shows that the new algorithm can efficiently and effectively solve DTVRPSC and improve the accuracy of vehicle routing planning of time-varying actual urban traffic road.

Tabu search for the time-dependent vehicle routing problem with time windows on a road network

Travel times inside cities often vary quite a lot during a day and significantly impact the duration of commercial delivery routes. Several authors have suggested time-dependent variants of the most commonly encountered vehicle routing problems. In these papers, however, time-dependency is usually defined on customer-based graphs. Thus, one major impact of travel time variations is missed: in an urban environment, not only do travel times change, but also the paths used to travel from one customer to another. In fact, during a day, different paths may be used at different points in time. To address this issue, one possible approach is to work directly with the road network and consider travel time (or travel speed) variations on each road segment.In this paper, we propose a solution approach for a time-dependent vehicle routing problem with time windows in which travel speeds are associated with road segments in the road network. This solution approach involves a tabu search heuristic that considers different shortest paths between any two customers at different times of the day. A major contribution of this work is the development of techniques to evaluate the feasibility and the approximate cost of a solution in constant time, which allows the solution approach to handle problem instances with up to 200 nodes and 580 arcs in very reasonable computing times. The performance of our algorithm is assessed by comparing it to an exact method on a set of benchmark instances. The results show that solutions of high quality are produced.

Linear edge costs and labeling algorithms: The case of the time‐dependent vehicle routing problem with time windows

AbstractIn this paper we implement a branch‐price and cut algorithm for a time dependent vehicle routing problem with time windows in which the goal is to minimize the total route duration. The travel time between two customers is given by a piecewise linear function on the departure time and, thus, it need not remain fixed along the planning horizon. We discuss different alternatives for the implementation of these linear functions within the labeling algorithm applied to solve the pricing problem. We also provide a tailored implementation for one of these alternatives, relying on efficient data structures for storing the labels, and show several strategies to accelerate the algorithm. Computational results show that the proposed techniques are effective and improve the column generation step, solving all instances with 25 customers, 49 of 56 with 50 customers, and many instances with 100 customers. Furthermore, heuristic adaptations are able to find good quality solutions in reasonable computation times.

Using Congestion Zones for Solving the Time Dependent Vehicle Routing Problem

This paper provides a framework for solving the Time Dependent Vehicle Routing Problem (TDVRP) by using historical data. The data are used to predict travel times during certain times of the day and derive zones of congestion that can be used by optimization algorithms. A combination of well-known algorithms was adapted to the time dependent setting and used to solve the real-world problems. The adapted algorithm outperforms the best-known results for TDVRP benchmarks. The proposed framework was applied to a real-world problem and results show a reduction in time delays in serving customers compared to the time independent case.

Multi-Trip Time-Dependent Vehicle Routing Problem with Soft Time Windows and Overtime Constraints

The multi-trip time-dependent vehicle routing problem with soft time windows and overtime constraints (MT-TDVRPSTW-OT) is considered in this paper. The modified hierarchical multi-objective formulation and the equivalent single-objective formulation are proposed. The iterative multi-trip tour construction and improvement (IMTTCI) procedure and the single-trip tour counterpart procedure with post-processing greedy heuristic (ISTTCI-GH) are proposed to solve the problem. These procedures are based on the ruin and recreate principle, and consider trade-offs among cost components (vehicle usage, number of early/late soft time window occurrences, transport distance, transport time, overtime and early/late soft time window penalty costs) in the search process. From the computational experiment, the IMTTCI procedure outperforms the existing efficient insertion heuristic with 42.09% improvement in number of vehicles and 24.30% improvement in travel time for the constant-speed multi-trip vehicle routing problem with hard time windows and shift time limits, a special case of MT-TDVRPSTW-OT problem, on all problem instances. For the MT-TDVRPSTW-OT problem, the ISTTCI-GH and the IMTTCI outperform the ISTTCI on all problem groups by 43.21% and 69.44% in the number of vehicles (the primary objective), respectively. The IMTTCI outperforms the ISTTCI-GH in number of vehicles by 51.07%, but takes 42.83% longer CPU time than the ISTTCI-GH. The performance of IMTTCI in terms of the primary objective improves with the increase of mean speed as well as the increase of time window width and planning horizon across all customer configuration types, tighter/looser time windows, shorter/longer planning horizon and various time-dependent travel speed profiles. The sensitivity analysis of different problem parameters is performed. The complexity analysis of the proposed procedures shows that the proposed procedures are solvable in polynomial time, and from the computational results the relationships between the CPU time and problem size confirm this. For the MT-VRPSTW-OT, a mixed integer program and the special case of MT-TDVRPSTW-OT, on 12-customer instances, the proposed IMTTCI algorithm yields the average optimality gap of 1.35% with the average CPU time of 0.66 seconds, whereas the GAMS/CPLEX solver yields the average optimality gap of 0.83% with the average CPU time of 256.39 seconds. The proposed IMTTCI algorithm yields only 0.52% greater optimality gap but 388 times faster CPU time than the commercial solver.

Performance Comparison of Two Recent Heuristics for Green Time Dependent Vehicle Routing Problem

This article concerns a green Time Dependent Capacitated Vehicle Routing Problem (TDCVRP) which is confronted in urban distribution planning. The problem is formulated as a Markovian Decision Process and a dynamic programming (DP) approach has been used for solving the problem. The article presents a performance comparison of two recent heuristics for the green TDCVRP that explicitly accounts for time dependent vehicle speeds and fuel consumption (emissions). These heuristics are the classical Restricted Dynamic Programming (RDP) algorithm, and the Simulation Based RDP that consists of weighted random sampling, RDP heuristic and simulation. The numerical experiments show that the Simulation Based Restricted Dynamic Programming heuristic can provide promising results within relatively short computational times compared to the classical Restricted Dynamic Programming for the green TDCVRP.

Time-dependent vehicle routing problem with time windows of city logistics with a congestion avoidance approach

The time-dependent vehicle routing problem with time windows (TDVRPTW) is examined in this study. A mathematical model for the TDVRPTW is formulated by considering the time-dependent vehicle speeds, capacity, travel time, wait time, customer demand, service time, time window, impact of dynamic loads, and travel speed effect on vehicle carbon emissions. The objective of the TDVRPTW model is to minimize the sum of the fixed costs of the vehicle used, as well as the costs of drivers, fuel consumption, and carbon emissions. A calculation method for the road travel time across time periods with time-dependent speeds is presented. A congestion avoidance approach is proposed to avoid peak hour traffic congestion and temporal traffic congestion. An improved ant colony algorithm with a congestion avoiding approach (IACACAA) is elucidated according to the characteristics of the TDVRPTW model. The effectiveness of the proposed approaches was demonstrated in several computational experiments. The results show that the proposed IACACAA can scientifically plan the departure time and driving route for each vehicle, effectively avoid traffic congestion, and reduce the total distribution costs. The congestion avoidance approach effectively reduces vehicle fuel consumption and carbon emissions and protects the environment.

A metaheuristic for the time-dependent vehicle routing problem considering driving hours regulations – An application in city logistics

Abstract New retail trends show the increasing importance of providing cost efficient deliveries in cities, where congestion and compliance with driving hours regulations should be incorporated into routing software. This paper introduces a large neighbourhood search algorithm that substantially improves the benchmark solutions (in terms of the number of vehicles required, travel distance and duty time) for the vehicle routing problem variant considering time windows, time-dependent travel times and driving hours regulations (EC) 561/2006 that apply to vehicles over 3.5 tons in European cities. Additionally, instances for The Road Transport (Working Time) Regulation 2005 that applies to drivers in the United Kingdom are introduced. The proposed algorithm is also used in scenarios that represent home delivery conditions to evaluate the impacts of the length of time windows, customer density, congestion and regulations in terms of cost and environmental impact.

Efficient solution algorithms for a time-critical reliable transportation problem in multigraph networks with FIFO property

Relief distribution in urban environments is one of the major activities in emergency logistics management. The effective and time-saving dispatching process in affected areas is pivotal in rescue operations. In this study, we formulate a reliable time-dependent vehicle routing problem with time windows in a multigraph based network. In such networks, there exist parallel arcs with multiple attributes between nodes. The purpose of the provided model is to minimize delays in delivering prioritized items in disaster response operations. It also controls the minimum reliability of each route. Controlling the reliability in relief distribution gives this assurance that emergency packages on vehicles can reach their destinations safely and in a timely manner. In order to solve the problem, a novel restricted dynamic programming is applied to the problem through the giant-tour representation. The proposed algorithm can reach the optimal solution when utilized in an unrestricted way. In addition, a modified caching genetic algorithm and a three-phase optimization method based on the tabu search heuristic are provided to deal with larger instances in reasonable computation times. Finally, a real transportation case is presented to illustrate the potential applicability of the model in urban environments. The results accentuate the efficiency of the proposed methods and show the significance of multigraph to accelerate the distribution operations for reliable emergency logistics planning.

Integer programming formulations for the time-dependent elementary shortest path problem with resource constraints

The impact of congestion in transportation has become one of the main concerns regarding urban planing in large cities. Time-Dependent Vehicle Routing Problems (TDVRPs) is the name given to a broad family of VRPs that explicitly incorporate the congestion by considering variable travel times. In this paper we study the Time-Dependent Elementary Shortest Path Problem with Resource Constraints (TDESPPRC), that appears as the pricing sub-problem in column generation-based approaches for TDVRPs. We consider two integer programming formulations which exploit the characteristics of the time-dependent travel time function and are evaluated on benchmark instances. On preliminary computational experiments, the approach is able to effectively solve instances with up to 25 vertices in reasonable times, showing its potential to be used within a Branch and Price algorithm.