Hierarchical Reinforcement Learning for Vehicle Routing Problems with Time Windows

‘Route balance’ is the difference between the longest and the shortest among all the route lengths. The route balance is frequently considered in a vehicle routing problem (VRP) for balancing the distance travelled among delivery vehicles. This is a common practice since in VRPs there are no time elements. However, in recent years, some research works have considered route balance in a vehicle routing problem with time windows (VRPTW). As there are more time elements in VRPTW, this paper refutes that route balance is sufficient for VRPTW by simultaneously optimising makespan and workload imbalance using ‘total time balance’ instead of route balance. On the other hand, ‘total time balance’ is the difference between the longest total time taken and the shortest total time taken among delivery vehicles. As such, makespan is the longest total time taken among vehicles. Total time taken for a vehicle is the sum of vehicle run time, waiting time and service time. In order to demonstrate the importance of using total time balance instead of route balance on VRPTW, three different multi-objective VRPTW models, namely, (1) only with general VRPTW objectives (i.e. ‘total distance travelled’ by all vehicles and ‘total number of vehicles’ used), (2) general VRPTW objectives with route balance and (3) general VRPTW objectives with total time balance are developed and solved by fitness aggregated genetic algorithm (FAGA) for 36 Solomon’s benchmark instances. By comparison of the makespan produced by the FAGA between the three cases, the importance of using total time balance instead of route balance on multi-objective VRPTW is demonstrated. Also, the makespan produced by the FAGA for the third case, i.e. general VRPTW objectives with total time balance is compared with the makespan produced by fitness aggregated differential evolution (FADE). By comparing makespan with statistical testing between FAGA and FADE, the outperformance of FAGA over FADE is verified. To check the practicality of the total time balance on multi-objective VRPTW, an instance with real time windows is also solved for the three cases by the FAGA and its makespan are compared and reported.

Vehicle Routing Problem with Time Windows (VRPTW) is a constrained NP-hard problem that designs the least cost routes from one depot to a set of geographically scattered points. Most of traditional techniques are difficult to solve this kind of problem. In this paper, we explored the validity of another swarm-intelligence-based model-Bacterial Foraging Optimization (BFO) for VRPTW solving. Original BFO, BFO with linear decreasing chemotaxis step (BFO-LDC) and BFO with non-linear decreasing chemotaxis step (BFO-NDC) are used to obtain the best solutions of a given VRPTW problem, respectively. The experimental results demonstrated that the proposed BFO algorithms have the potential to solve Vehicle Routing Problem with Time Windows (VRPTW) with rapid convergence rate and the good result accuracy.

In the field of operations research, the vehicle routing problem with time windows (VRPTW) has been widely studied because it is extensively used in practical applications. Real-life situations discussed in the relevant research include time windows and vehicle capabilities. Among the constraints in a VRPTW, the practical consideration of the fairness of drivers’ performance bonuses has seldom been discussed in the literature. However, the shortest routes and balanced performance bonuses for all sales drivers are usually in conflict. To balance the bonuses awarded to all drivers, an auction bidding approach was developed to address this practical consideration. The fairness of performance bonuses was considered in the proposed mathematical model. The nearest urgent candidate heuristic used in the auction bidding approach determined the auction price of the sales drivers. The proposed algorithm both achieved a performance bonus balance and planned the shortest route for each driver. To evaluate the performance of the auction bidding approach, several test instances were generated based on VRPTW benchmark data instances. This study also involved sensitivity and scenario analyses to assess the effect of the algorithm’s parameters on the solutions. The results show that the proposed approach efficiently obtained the optimal routes and satisfied the practical concerns in the VRPTW.

The Vehicle Routing Problem with Time Windows (VRPTW) consists of a homogenous set of vehicles and a set of customer located in a city. In VRPTW all the vehicles starts from the depot visit the customer and end at the depot. Each customer is visited exactly by one vehicle within the specified time window. The objective is to minimize the number of vehicles and total distance travelled simultaneously. This represents the multiobjective Vehicle Routing Problem with Time Windows. The proposed work consists of Hybrid Genetic Search with Diversity Control using the Genetic Algorithm for solving the VRPTW. The Pareto approach is used for finding the set of optimal solutions for achieving the multiobjective. The crossover operator is used for exchanging the best routes, which have shortest distance. Two mutation operators such as relocation mutation operator and split mutation operator were used in this application. In this, it accounts penalty for an infeasible solutions with respect to time-window and duration constrains. The computations are performed using the instances which are obtained from the VRPLIB.

In this paper, the vehicle routing problem with time windows (VRPTW) was considered, and a mixed integer programming mathematic model of VRPTW was proposed in detail. Meantime, an improved ant colony algorithm (IACA) was proposed to overcome the shortcomings of premature convergence and slow convergence of conventional ant colony algorithm (ACA). The experiment results showed that the IACA can solve VRPTW effectively.

In this study, Vehicle Routing Problem with Time Windows (VRPTW) with known customer demands, a single depot and identical vehicles, is considered. Minimizing the total distance and the total waiting time of the vehicles are determined as objective functions for VRPTW which is capable to serve the customers in a prespecified time interval. A hybridized version of genetic algorithm which is a metaheuristic solution technique with constructive heuristic methods is proposed to produce effective solutions for VRPTW. By using sweep algorithm in initial population generation phase of genetic algorithm, it is planned to begin the search with high quality solution sets and in this way, get more feasible solutions faster. A benchmark problem in the literature is solved and obtained results are compared with the results of genetic algorithm with the nearest neighbor algorithm based algorithm. It is observed that the proposed genetic algorithm beginning with sweep based initial population generation algorithm reaches more effective solutions.

In this paper, we discuss the vehicle routing problem with time windows (VRPTW) where stochastic travel times are assumed. In a first step, a variable neighborhood search (VNS) approach is proposed for the VRPTW where deterministic data is assumed. This approach is extended in a second step by adding a sampling approach to deal with the stochastic travel times. Problem instances are proposed that are based on the well-known Solomon instances from the literature. The resulting routes are executed in a stochastic environment to assess the performance of the proposed heuristic. Computational experiments demonstrate that it is reasonable to incorporate stochastic knowledge into VRPTW algorithms.

This paper investigates a well-known complex combinatorial problem known as the vehicle routing problem with time windows (VRPTW). Unlike the standard vehicle routing problem, each customer in the VRPTW is served within a given time constraint. This paper solves the VRPTW using an improved artificial bee colony (IABC) algorithm. The performance of this algorithm is improved by a local optimization based on a crossover operation and a scanning strategy. Finally, the effectiveness of the IABC is evaluated on some well-known benchmarks. The results demonstrate the power of IABC algorithm in solving the VRPTW.

The balanced cargo vehicle routing problem with time windows

In order to improve the accuracy of the artificial bee colony algorithm (ABC) on vehicle routing problem with time window (VRPTW), This paper makes the following improvements to the ABC. Three kinds of neighborhood search methods are introduced. In the leader bee and follower bee search stage, the single search mode is changed into a three-way search method, which improves the optimization depth of the algorithm. Multiple neighborhood searches of new food sources generated by the scouter bee is conducted and proceeding to the next iteration has enhanced the survival of new food sources and increased the diversity of populations. The global optimal solution is recorded by setting and updating the bulletin board. Simulation experiments show that the improved discrete ABC algorithm has obvious advantages in solving large-scale VRPTW. Therefore, the improved discrete ABC algorithm has great potential and application value in solving VRPTW.

In this paper, an optimization algorithm based on lagrangian relaxation is proposed to solve the vehicle routing problem with time windows (VRPTW). The VRPTW is the generalization of the Vehicle Routing Problem, with a service time interval constraint available with every customer node. Here, an approach is used which relaxes the constraint set that justify that each customer should be serviced. The problem consists of introducing optimal lagrangian multipliers and then the subgradient method is used to find the appropriate multipliers. Here the problem focuses on, compared to the problems in the literature, not only the minimization of the transportation cost but also the used number of vehicles. The method has been implemented and tested on a series of well known benchmark test problems. The algorithm results to be very competitive and the results are better than the optimal solution.

Path optimization is a critical issue in vehicle routing problems (VRPs). This study tends to solve a VRP with time windows (VRPTW), which is an important version of VRP. Its principal objective is to find out the minimum cost routes to serve the customers within service time intervals, by a fleet of vehicles having confined capacity. An improved ant colony optimization (ACO) is applied to the problem, which uses a new pheromone reset and update function to enhance the route searching and a 2-opt method to improve the optimized path. The proposed ACO is tested on standard benchmarks of VRPTW. The results verifies the effectiveness of the algorithm and confirms it as better path optimizing method for VRPTW, in comparison to the traditional ACO and other heuristics.

A vehicle routing problem with time windows (VRPTW) has been studied widely in the literature. Most effective methods to find a solution are still heuristics, which are constructed from the inventor’s ideas of how a good solution should be obtained, and their efficiency is quantified only by empirical data. However, these ideas may or may not yield a near-optimal solution. It would be better if we can identify some characteristics of a solution to VRPTW to help guide the heuristic in the right direction. Arnold and Sörensen had proposed a statistical learning model to determine a characteristic of a near-optimal solution for a vehicle routing problem (VRP). They used that knowledge to create a guided heuristic for a local search and the results were impressive. Therefore, we wish to extend this work to VRPTW by defining new solution metrics that involve time windows constraints and using a statistical tool to obtain characteristic(s) of a good solution to VRPTW. By applying a machine learning model, we examined 14,000 VRPTW instances to identify the key characteristics that differentiate high-quality solutions. Our experiment revealed that the average width per route is the most distinguishing characteristic, which also holds true for the standard VRP according to Arnold and Sorensen's findings. Moreover, the runner-up characteristics are the average compactness per route, measured by width, and the average total travel time and service time per route. These insights contribute to a deeper understanding of the VRPTW solution structures and offer a meaningful direction for the development of more effective heuristics.

The Vehicle Routing Problem (VRP) is a well-known combinatorial optimization problem where a number of customers must be served with a fleet of vehicles. The classical variation of the VRP is the Capacicated Vehicle Routing Problem (CVRP) with the additional constraint that each vehicle must have uniform capacity. Many Decision Support Systems (DSS) have been implemented to solve real life problems of the VRP and its’ variants, but they do not allow multiple decision makers to explore several scenarios of a given problem simultaneously and collaborate with each other in order to find the best possible solution. In this paper, we extend our previous work [35] and incorporate in our spatial DSS four variants of the CVRP: (i) Distance Constrained Vehicle Routing Problem (DVRP), (ii) Vehicle Routing Problem with Time Windows (VRPTW), (iii) Vehicle Routing Problem with Backhauls (VRPB), and (iv) Vehicle Routing Problem With Pickup and Delivery (VRPPD). This extension allows decision makers to solve specific routing problems according to their needs. The proposed collaborative spatial DSS allows two decision makers to collaborate with each other in order to find the best possible solution through a tangible interface. The locations are added through interactive Google Maps and other parameters through user-friendly forms that can be manipulated via tangible interfaces. The proposed DSS has been implemented using Java, TUIO protocol, jsprit, and Google Maps.

The Vehicle Routing Problem with Time Windows (VRPTW) is a critical combinatorial optimization problem in modern logistics, where finding optimal routes is essential for minimizing operational costs and enhancing service reliability. While Genetic Algorithms (GAs) are a powerful tool for solving VRPTW, their effectiveness is often undermined by premature convergence, a phenomenon in which the algorithm stagnates at suboptimal solutions, thus failing to achieve maximum efficiency. This study directly addresses this challenge by systematically evaluating how different reproduction schemes impact GA performance. The primary objective is to identify operator combinations that mitigate premature convergence to achieve superior solution quality, measured by total travel time, while also analyzing the trade-off with computational cost. We investigate combinations of conventional operators, such as Tournament Selection (TS) and Order Crossover (OX), against more advanced schemes, including Split Rank Selection (SRS) and Multi-Parent Order Crossover (MPOX), as well as different mutation methods, namely Scramble Mutation (SM) and Inversion Mutation (IM). Results demonstrate that advanced schemes, particularly the combination of SRS, MPOX, and IM, yield the most robust convergence and the lowest average total travel time of 48,422.8 minutes. However, this superior performance requires the longest computation time at 30.9 h. In contrast, conventional operator combinations are significantly faster, with execution times as low as 8.7 h, but they produce lower-quality solutions and exhibit unstable convergence. This study highlights the crucial role of the reproduction scheme in balancing the trade-off between solution quality and computational efficiency, confirming that a synergistic combination of advanced operators is essential for solving complex VRPTW instances effectively.