Multi-depot Vehicle Routing Problem Research Articles

The multidepot vehicle routing problem with intelligent recycling prices and transportation resource sharing

The increasing focus on environmental regulations and the economic advantages of recycling has spurred interest in the design of multidepot reverse logistics networks (MDRLNs). In these networks, the growing use of intelligent recycling bins (IRBs) has been beneficial for both product recycling and standardizing recycling product pricing. Furthermore, collaboration and resource sharing enhance the efficiency of resource utilization and recycling. This study proposes a multidepot vehicle routing problem with time windows that incorporates intelligent recycling prices (IRPs) and transportation resource sharing (MDVRPTW-IRPTRS). Initially, a linear function is developed to define the relationship between the volume of returned products and IRPs. Subsequently, the problem is expressed as a mathematical model aiming to minimize total operating costs and maximize total recycling profits. Additionally, a hybrid algorithm that combines a three-dimensional (3D) k-means clustering algorithm with a self-adapting genetic algorithm-particle swarm optimization (SGA-PSO) is devised to determine the optimal solution for MDVRPTW-IRPTRS. The 3D k-means clustering algorithm is utilized to categorize IRBs within an MDRLN. The SGA-PSO algorithm incorporates elite preservation and self-adaptive update mechanisms to enhance the solution quality and algorithm convergence. A transportation resource sharing (TRS) strategy is integrated into the SGA-PSO, facilitating the allocation of shared vehicles to alternative recycling routes. A comparative analysis of SGA-PSO against other algorithms, including a hybrid genetic algorithm, an improved particle swarm optimization algorithm, and a hybrid genetic algorithm with variable neighborhood search, demonstrates its superiority in solving the MDVRPTW-IRPTRS. The model and algorithm are applied in a real-world case study in Chongqing, China, and the study discusses the optimized results under varying TRS strategies and IRP schemes, contributing to the development of an efficient and synergistic urban reverse logistics network. Moreover, the superior performance of the proposed approach is validated through the ablation experiments. This study offers valuable decision-making support for fostering an environmentally sustainable and resource-efficient city.

Logistics Distribution Route Optimization With Time Windows Based on Multi-Agent Deep Reinforcement Learning

Multi-depot vehicle routing problem with time windows (MDVRPTW) is a valuable practical issue in urban logistics. However, heuristic methods may fail to generate high-quality solutions for massive problems instantly. Thus, this article presents a novel reinforcement learning algorithm integrated with a multi-head attention mechanism and a local search strategy to solve the problem efficiently. The routing optimization was regarded as a vehicle tour generation process and an encoder-decoder was used to generate routes for vehicles departing from different depots iteratively. A multi-head attention strategy was employed for mining complex spatiotemporal correlations within time windows in the encoder. Then, a decoder with multi-agent was designed to generate solutions by optimizing reward and observing transition state. Meanwhile, a local search strategy was employed to improve the quality of solutions. The experiments results demonstrate that the proposed method can significantly outperform traditional methods in effectiveness and robustness.

Tailoring Evolutionary Algorithms to Solve the Multiobjective Location-Routing Problem for Biomass Waste Collection

Location-routing problems widely exist in logistics activities. For the biomass waste collection, there is a recognized need for novel models to locate the collection facilities and plan the vehicle routes. So far most location-routing models fall into the cost-driven-only category. However, comprehensive objectives are required in the specific context, such as time-dependent pollution and speed-and load-related emission. Furthermore, location-routing problems are hierarchical by nature, containing the facility location problems (strategic level) and the vehicle routing problems (tactical level). Existing studies in this field usually adopt computational intelligence methods directly without decomposing the problem. This can be inefficient especially when multiple objectives are applied. Motivated by these, we develop a novel multi-objective optimization model for the location-routing problem for biomass waste collection. To solve this model, we explore the way to tailor evolutionary algorithms to the hierarchical structure. We develop adapted versions of two commonly used evolutionary algorithms: the genetic algorithm and the ant colony optimization algorithm. For the genetic algorithm, we divide the population by the strategic level decisions, so that each subpopulation has a fixed location plan, breaking the location-routing problem down into many multi-depot vehicle routing problems. For the ant colony optimization, we use an additional pheromone vector to track the good decisions on the location level, and segregate the pheromones related to different satellite depots to avoid misleading information. Thus, the problem degenerates into vehicle routing problem. Experimental results show that our proposed methods have better performances on the location routing problem for biomass waste collection.

A Hybrid Adaptive Simulated Annealing and Tempering Algorithm for Solving the Half-Open Multi-Depot Vehicle Routing Problem

The half-open multi-depot vehicle routing problem (HOMDVRP) is a typical decision optimization problem in the field of collaborative logistics that considers resource sharing. This study aims to develop an effective meta-heuristic algorithm for solving the HOMDVRP. Firstly, a mixed-integer programming model of HOMDVRP is established to minimize the total travel distance of the vehicles. After that, a novel hybrid adaptive simulated annealing and tempering algorithm (HASATA) is proposed based on the features of HOMDVRP. The proposed algorithm combines the strengths of the simulated annealing algorithm and the large-neighborhood search algorithm to balance the algorithm’s searching capabilities in both breadth and depth. Meanwhile, an adaptive Markov chain length mechanism and a tempering mechanism are designed to improve the algorithm’s computational efficiency and convergence ability. Finally, simulation experiments are conducted to verify the effectiveness of the proposed model and the computational performance of the proposed algorithm. Four comparison algorithms are selected and analyzed using 24 groups of problem instances. The comparison results show that the proposed HASATA can solve the HOMDVRP more efficiently and obtain a solution with better optimization performance and satisfactory stability.

Variable neighbourhood search embedded perturbation mechanism for multi-depot vehicle routing problem with simultaneous delivery & pickup, and time limit

Given its complex combinatorial features, the multi-depot vehicle routing problem with simultaneous delivery & pickup, and time limits (MDVRPSDPTL) is a well-known and challenging real-world problem faced by manufacturing enterprises. Vehicles depart from different depots to simultaneously deliver new products and pick up packaging or used products within a prescribed time limit. In this study, we developed a mixed-integer linear programming model to assign customers to depots and determine a group of vehicle routes to minimise total travel cost. A variable neighbourhood search algorithm embedded with a novel perturbation mechanism (including a route-selection strategy and a retention strategy for the current best solution) was proposed to solve the model efficiently. The perturbation mechanism helps the algorithm quickly escape local optima to improve computational precision and reduce computation time. The numerical results indicate that the proposed algorithm outperforms existing state-of-the-art algorithms in terms of solution quality and computation time for well-known MDVRPSDPTL benchmark instances.

A hybrid heuristic approach for the multi-objective multi depot vehicle routing problem

Efficiency in logistics is often affected by the fair distribution of the customers along the routes and the available depots for goods delivery. From this perspective, in this study, the Multi-depot Vehicle Routing Problem (MDVRP), by considering two objectives, is addressed. The two objectives in conflict for MDVRP are the distance traveled by vehicles and the standard deviation of the routes’ length. A significant standard deviation value provides a small distance traveled by vehicles, translated into unbalanced routes. We have used a weighted average objective function involving the two objectives. A Variable Neighborhood Search algorithm within a Chu-Beasley Genetic Algorithm has been proposed to solve the problem. For decision-making purposes, several values are chosen for the weight factors multiplying the terms at the objective function to build up a non-dominated front of solutions. The methodology is tested in large-size instances for the MDVRP, reporting noticeable results for managerial insights.

Adaptive Ant Colony Optimization With Node Clustering for the Multidepot Vehicle Routing Problem

This article deals with the novel metaheuristic algorithm based on the Ant Colony Optimization (ACO) principle. It implements several novel mechanisms that improve its overall performance, lower the optimization time, and reduce the negative behavior which is typically connected with ACO-based algorithms (such as prematurely falling into local optima, or the impact of setting of control parameters on the convergence for different problem configurations). The most significant novel techniques, implemented for the first time to solve the Multi-Depot Vehicle Routing Problem (MDVRP), are as follows: (a) node clustering where transition vertices are organized into a set of candidate lists called clusters; (b) adaptive pheromone evaporation which is adapted during optimization according to the diversity of the population of ant solutions (measured by information entropy). Moreover, a new termination condition, based also on the population diversity, is formulated. The effectiveness of the proposed algorithm for the MDVRP is evaluated via a set of experiments on 23 well-known benchmark instances. Performance is compared with several state-of-the-art metaheuristic methods; the results show that the proposed algorithm outperforms these methods in most cases. Furthermore, the novel mechanisms are analyzed and discussed from points of view of performance, optimization time, and convergence. The findings achieved in this article bring new contributions to the very popular ACO-based algorithms; they can be applied to solve not only the MDVRP, but also, if adapted, to related complex NP-hard problems.

Sustainable collaborative strategy in pharmaceutical refrigerated logistics routing problem

The rapid growth of pharmaceutical refrigerated logistics poses sustainability challenges, including elevated costs, energy consumption, and resource inefficiency. Collaborating multiple depots can enhance logistics efficiency when standalone distribution centres have limited transport resources, i.e. refrigerated vehicles. However, the sustainable benefits and performance across different strategies remain unexplored. This study fills this research gap by addressing a refrigerated pharmaceutical routing problem. While many collaborative strategies prioritise economic and environmental benefits, our approach highlights a vital social indicator: maintaining vehicle flow equilibrium at each depot during collaboration. This ensures the stability of transport resources for all stakeholders, promoting sustainable collaborative logistics. The problem is formulated as a multi-depot vehicle routing problem with time windows (MDVRPTW). Three collaborative strategies using Clustering VRP (CLUVRP) and improved Open VRP (OVRP) are proposed and compared. We develop two approaches to address traditional OVRP limitations in ensuring vehicle flow equilibrium at each depot. Our models consider perishable pharmaceuticals and time-dependent travel speeds. Three hybrid heuristics based on Simulated Annealing and Variable Neighborhood Search (SAVNS) are proposed and evaluated for efficacy. Computational experiments and a case study demonstrate distinct sustainable benefits across various strategies, offering valuable insights for decision-makers in the refrigerated logistics market.

Multi-Depot Vehicle Routing Problem with Drones: Mathematical formulation, solution algorithm and experiments

The use of Unmanned Aerial Vehicles (UAVs) is expected to grow rapidly in the coming years, driven by technological advancements, cost-effectiveness, and the increasing demand for faster and more efficient delivery solutions. This article deals with the mathematical formulation of the Multi-Depot Vehicle Routing Problem with Drones (MDVRP-D), whereby a set of heterogeneous trucks, each paired with a UAV, are located in different depots. Both types of vehicles deliver goods to customers; UAVs are dispatched from trucks while en route to make the last-mile delivery. A metaheuristic algorithm based on the Ant Colony Optimization (ACO) principle is proposed as the solution. This algorithm has been adapted for this newly proposed problem; the novel mechanics include the probabilistic decision to dispatch an UAV, the selection of a customer to be served, and local search optimization. Extensive computational experiments are performed to verify the proposed algorithm. First, its performance is compared with Adaptive Large Neighborhood Search (ALNS) metaheuristics on a set of Vehicle Routing Problem with Drones (VRP-D) benchmarks. A set of various benchmark instances are subsequently proposed for the newly formulated MDVRP-D (differing in complexity and graph topology). Finally, the behavior of the proposed algorithm is thoroughly analyzed, especially in respect of features connected with UAVs. The findings presented in this article provide valuable contributions to the NP-hard models related to the Travelling Salesman Problem (TSP) and to the very popular ACO-based algorithms.

Providing an integrated multi-depot vehicle routing problem model with simultaneous pickup and delivery and package layout under uncertainty with fuzzy-robust box optimization method

This paper modeled and solved an integrated multi-depot vehicle routing problem (MDVRP) with simultaneous pickup and delivery (SPD) with package layout under unpredictable pickup, delivery, and transfer costs. The model described in this paper is divided into two stages. In the first stage, the SCA algorithm is used to optimize the package dimensions (a collection of commodities consumers need). The NSGA II and MOALO algorithms are used in the second stage to optimize the three objective functions of 1 simultaneously) minimizing total costs, 2) minimizing co2 emissions, and 3) minimizing the maximum working hours of drivers based on the optimal dimensions (length, width, and height) obtained from solving the first stage model. Determining the quantity and ideal location of possible warehouses, the best route for trucks to take to deliver and collect customer items, and the distribution of customers to warehouses are the key goals of the second stage. Since the model is unclear, the problem's uncertainty parameters are controlled using a novel fuzzy-robust box optimization (FRBO) technique. This technique, which combines the advantages of fuzzy programming with robust box-based optimization, produces excellent results when used to optimize objective functions. The numerical calculations in the numerical example show that the total network costs and CO2 emissions increased in the second stage in the presented model with an increasing uncertainty rate. At the same time, the maximum working hours of drivers decreased due to the shortened communication route and the number of vehicles increasing. Finally, the MOALO algorithm was used to resolve a case study at Safir Broadcasting Company because of its excellent efficiency in resolving the created model, the findings of which revealed the presence of 13 potential effective solutions. The quantity of greenhouse gas emissions rose by 1.11%, the overall expenditures climbed by 1.72%, and the number of hours that drivers worked fell by 11.98% when the uncertainty rate was raised from 0.5 to 0.7, according to research on the FRBO.

Multiobjective Mission Planning for Multiple Geosynchronous Spacecraft Refueling

This paper studies the multiple geosynchronous spacecraft refueling problem (MGSRP) with multiple servicing spacecraft (Ssc) and fuel depots (FDs). In the mission scenario, multiple Ssc and FDs are parked in the geosynchronous Earth orbit (GEO) initially. Ssc start from FDs and maneuver to visit and refuel multiple GEO targets with known demands. These capacitated Ssc are expected to rendezvous with fuel-deficient GEO targets and FDs for the purpose of delivering the fuel stored in FDs to GEO targets. The objective is to find a set of Pareto-optimal solutions with minimum fuel cost and mission duration. The MGSRP is a much more complex variant of multidepot vehicle routing problems mixing discrete and continuous variables. A two-nested optimization model is built. We propose a new multiobjective hybrid particle swarm optimization to solve the outer-loop problem, and the design variables are the refueling sequence, task assignment, time distribution, and locations of FDs. In the inner-loop problem, branch and bound method is used to find the optimal decision variable for a given outer-loop solution. Finally, numerical simulations are presented to illustrate the effectiveness and validity of the proposed approach.

Green vehicle routing using mixed fleets for cold chain distribution

Introducing clean energy into urban cold chain distribution to achieve a green transition is crucial for reducing transportation and logistics emissions and promoting sustainable urban development. This study focuses on the Cold Chain Green Multi-Depot Vehicle Routing Problem with Time Windows and Mixed Fleets (CC-GMD-VRPTW-MF) for city logistics distribution, utilizing both electric vehicles (EVs) and gasoline and diesel vehicles (GDVs). To accurately assess energy consumption, a realistic energy consumption model is employed. The CC-GMD-VRPTW-MF aims to minimize total costs (including fixed, energy consumption, damage, and environmental costs). To address the problem, we propose an improved Variable Neighborhood Search (VNS) algorithm that introduces a new balanced mechanism for perturbation and a new memory-based mechanism for local search to enhance computational performance. Numerical studies on newly designed CC-GMD-VRPTW-MF instances are conducted to investigate the effect of incorporating EVs for joint delivery, considering different carbon prices, and adjusting time windows. Furthermore, we evaluate the effectiveness of the proposed improvement strategies in VNS and demonstrate the algorithm’s performance on benchmarks of related problems.

A multi-agent deep reinforcement learning approach for solving the multi-depot vehicle routing problem

The multi-depot vehicle routing problem (MDVRP) is one of the most essential and useful variants of the traditional vehicle routing problem (VRP) in supply chain management (SCM) and logistics studies. Many supply chains (SC) choose the joint distribution of multiple depots to cut transportation costs and delivery times. However, the ability to deliver quality and fast solutions for MDVRP remains a challenging task. Traditional optimization approaches in operation research (OR) may not be practical to solve MDVRP in real-time. With the latest developments in artificial intelligence (AI), it becomes feasible to apply deep reinforcement learning (DRL) for solving combinatorial routing problems. This paper proposes a new multi-agent deep reinforcement learning (MADRL) model to solve MDVRP. Extensive experiments are conducted to evaluate the performance of the proposed approach. Results show that the developed MADRL model can rapidly capture relative information embedded in graphs and effectively produce quality solutions in real-time.

Development of Machine Learning Based State-of-Charge Prediction Model for Plug-in Electric Vehicle's Relocation in Sharing System

This paper presents a Plug-in electric vehicle (PEV) relocation strategy in a sharing system considering machine learning based state-of-charge (SOC) prediction model. The proposed work involves the machine learning approaches like the Adaboost algorithm and XGBoost algorithm for PEVs SOC prediction in relocation problems for the first time. Also, the low-capacity batteries of PEVs are considered as it is affordable by the common people with less charging time. It is also believed that the large number of low capacities PEVs based taxi owners in metropolitan cities will be attracted towards relocation-based sharing system considering machine learning based SOC prediction model. The customer request for sharing PEVs will be reflected in the management system at the beginning. Assuming all the PEVs with a certain level of SOC during pre-relocation are connected to the management system. The customer request for PEVs service will be assigned to the nearest PEVs depending on the predicted value of SOC by machine learning approaches. The machine learning-based predicted value of PEVs SOC will help the management system to identify the nearest PEVs for the customer sharing system. The present work steps are listed here. Firstly, the PEVs with higher predicted SOC value by machine learning have been shared with the customer from the nearest location considering descending order of their SOC. Further, the rest of the PEVs will be assigned to the nearest charging station by the PEV sharing management system. Secondly, the customer request is assigned from the station to relocate PEVs with higher SOC values in descending order. Different case studies are assumed including one practical scenario of Mumbai, India. The optimization problem of multi-depot vehicle routing problem (MDVRP) with the minimization of relocation cost has been solved by IBM CPLEX optimizer in this work. Finally, the proposed work efficacy has been demonstrated with some validated results including less relocation costs for all the case studies under consideration with lower capacity based PEVs.

A multi-depot vehicle routing problem with time windows, split pickup and split delivery for surplus food recovery and redistribution

The problem faced by the food banks operating on the front-end model is the collection of surplus food from donation points and re-distribution of it to the demand points using a fleet of vehicles available at different volunteer locations/depots within specified time windows. Further, due to the limited vehicle capacity, demand splitting is allowed at the pickup as well and delivery points. Therefore, this problem can be seen as a rich variant of the vehicle routing problem (VRP) and named as a multi-depot VRP with time windows, split pickup and split delivery (MDVRP-TW-SP-SD). The problem is to determine the optimal number of vehicles from each depot to be hired and their optimal routes (sequence for visiting pickup and demand nodes) such that the total cost of hiring the vehicles, routing cost and penalty for unmet demand are minimized and it is formulated as a mixed-integer programming problem. To the best of our knowledge, this is the first attempt to formulate the MDVRP-TW-SP-SD. To efficiently and effectively solve the problem, an elitist GA and a GA hybridized with the local search are proposed. The performance of the proposed algorithms is tested through extensive computational experiments carried out on 180 modified benchmark problem instances from the literature. The results of computational experiments show the superiority of the GA hybridized with 3-opt local search. A case of the Robin Hood Army operating in the Lucknow city of India is also presented to demonstrate the applicability of the proposed model to practice.

A Robust Optimizing Reverse Logistics Model for Beef Products Using Multi Depot Vehicle Routing Problem

A Robust Optimizing Reverse Logistics Model for Beef Products Using Multi Depot Vehicle Routing Problem

Overview Discourse on Inherent Distinction of Multiobjective Optimization in Routing Heuristics for Multi-Depot Vehicle Instances

This paper reviews the research methodologies used in earlier years on the benefits and traits reflected by multi-depot vehicle routing problem (MDVRP) instances and assesses the efficacy of various improvised techniques to improve the current recurrent problems in routing procedures. Management of logistics involves moving finished goods from depots to end-user clients. Routing and scheduling systems that are improved will be able to serve a more significant number of customers in a shorter amount of time while also increasing customer satisfaction. To thoroughly discuss the current state of MDVRP implementation in routing heuristics, an analysis of the selected approaches involving multi-depot task distribution under VRP incorporations is further extrapolated. These approaches address the most common routing issues involving constraints like cost optimality, time window impositions, and load capacity flexibility. Recent research focuses on the advantages, proficiency, problem magnitude and adaptability in MDVRP. The MDVRP framework can still be significantly improved by reducing routing costs with efficient heuristics to generate optimized solutions.

Multi-Depot Vehicle Routing Problem with Drones: Mathematical Formulation, Solution Algorithm and Experiments

Multi-Depot Vehicle Routing Problem with Drones: Mathematical Formulation, Solution Algorithm and Experiments

Energy Saving-Oriented Multi-Depot Vehicle Routing Problem with Time Windows in Disaster Relief

This paper studies the distribution of emergency relief for electric vehicles (EVs), which considers energy saving, multi-depot, and vehicle routing problems with time windows, and the named energy saving-oriented multi-depot vehicle routing problem with time windows (ESMDVRPTW). Our aim is to find routes for EVs such that all the shelter demands are fulfilled during their time windows and the total cost traveled by the fleet is minimized. To this end, we formulate the ESMDVRPTW as a mixed-integer linear programming model. Since the post-disaster transportation network contains a large number of vertices and arcs composed of vertices, we propose a two-stage approach to solve the ESMDVRPTW. The first stage is to obtain the minimal travel cost between any two vertices in real-time on a post-disaster transportation network using the proposed Floyd algorithm combined with the neighboring list (Floyd-NL algorithm). In the second stage, we develop the genetic algorithm (GA) incorporating large neighborhood search (GA-LNS), which determines the delivery scheme of shelters. Simulation results of the MDVRPTW benchmark illustrate that the performance of the GA-LNS is better than GA, simulated annealing (SA) and tabu search (TS). Finally, case studies are constructed on two real cases acquired from the OpenStreetMap (OSM) generated by the Quantum Geographic Information System (QGIS) in Ichihara city, Japan, and the test results of case studies show the effectiveness of the proposed two-stage approach.

Graph attention reinforcement learning with flexible matching policies for multi-depot vehicle routing problems

Multi-depot vehicle routing problem with soft time windows (MD-VRPSTW) is a valuable practical issue in urban logistics. However, heuristic methods may fail to generate high-quality solutions for massive problems instantly. Thus, this paper presents a novel reinforcement learning algorithm integrated with graph attention network (GAT-RL) to efficiently solve the problem. This method utilizes the encoder–decoder architecture to produce routes for vehicles starting from different depots iteratively. The encoder architecture employs graph attention network to mine the complex spatial–temporal correlations within time windows. Then, the decoder architecture designs fixed-order and full-pair matching policies to generate solutions. After off-line training, experiments show that this approach consistently outperforms Google OR-Tools with negligible computational time. Particularly, the robustness of the pre-trained model is validated under multiple sources of variations and uncertainties, including customer/depot numbers, vehicle capacities, and en-route traffic conditions.