Routing Problem Research Articles

Research on vehicle path planning of automated guided vehicle with simultaneous pickup and delivery with mixed time windows

AbstractThe authors investigate new Automated Guided Vehicle (AGV) Routing Problem with Simultaneous Pickup and Delivery with Mixed Time Windows (VRPSPDMTW) in smart workshops, a variation of the classic Vehicle Routing Problem (VRP). A mixed time window vehicle routing model was developed for simultaneous deliveries. This model reduces the cost of AGVs used and distribution cost, along with time window penalties. To address this complex challenge, a Hybrid Adaptive Genetic Algorithm using Variable Neighbourhood Search (AGA‐VNS) is proposed. This algorithm enhances the genetic algorithm's local search capabilities while preserving solution diversity, thereby improving both efficiency and quality of solutions. Comprehensive computational experiments are conducted, which include both VRPSPDTW test benchmark and real‐world smart factory instance studies. The outcomes reveal that the AGA‐VNS algorithm outperforms both professional solver software and advanced heuristic methods significantly. Moreover, the newly developed mixed time window model is more aligned with the requirements of real‐world production processes compared to the traditional time window model. Thus, this research not only presents novel insights into the domain of vehicle routing problems but also demonstrates its significant applicability and potential in the background of intelligent workshops.

Genetic Algorithm Optimization with Selection Operator Decider

AbstractGenetic Algorithm (GA) is a powerful and flexible meta-heuristic tool to deal with the complexity of optimization problems, as they are directly related to real-life situations. The primary goal of an optimization problem could be to obtain a solution with less effort and near-optimal rather than slow, improbable optimal. GAs serve this purpose by broadly exploring the possible solution space and using genetic operators. The performance of GAs can vary significantly depending on the genetic operators. Although each operator type has upsides and downsides, the selection operator greatly influences the GA’s performance. Conventional GAs initialize with predetermined genetic operators and continue with the same throughout all iterations. In this paper, dynamically adjusting the selection operators to the current progress of the iteration will be shown to be a crucial strategy to improve the performance of the GA. This study aims to propose a novel GA capable of harnessing multiple selection operators by a self-deciding operator structure, which is more advantageous at the current iteration. For this, Technique for Order Preference by Similarity to Ideal Solution (TOPSIS), which is known as a simple and effective multi-criteria decision-making method, will be integrated into the GA by a proposed dynamic decision matrix. The proposed Selection Operator Decider Genetic Algorithm (SODGA) has unique properties with varying selection processes and is capable of using TOPSIS as a decider of the operator inside the iterations. The effectiveness of the presented SODGA framework will be analyzed by a Capacitated Vehicle Routing Problems (CVRPs) benchmark set.

Mathematical programming models for order picking in warehouses with decoupling of picker and cart

As a special Travelling Salesman Problem (TSP), the Single-Picker Routing Problem (SPRP) in warehouses is of important theoretical and practical significance. In manual order picking, the items are usually picked up by a picker with a cart. To speed up the picking process, the picker is allowed to stop the cart to pick up the items individually and return to the location of the cart. This paper proposes a mixed integer linear programming formulation for an order picking problem with the decoupling of the picker and the cart in a multi-block warehouse, where the picker capacity is at most four. And based on this model, order batching problem is also analysed. Through numerical experiments, the cost of above decoupling problems is computed under different settings of the speed ratio and the capacity of the picker while he/she is travelling alone. The results indicate how the decoupling of picker and cart leads to cost reduction. Our model shows good performance for orders of small and medium scale and hence has great potential to improve the order picking operation which is executed by decoupling containers and robots/vehicles in modern digital warehouse systems.

Two-echelon multi-commodity multimodal vehicle routing problem considering user heterogeneity in city logistics

Two-echelon multi-commodity multimodal vehicle routing problem considering user heterogeneity in city logistics

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.

Collaborative Orchard Pesticide Spraying Routing Problem with Multi-Vehicles Supported Multi-UAVs

Collaborative Orchard Pesticide Spraying Routing Problem with Multi-Vehicles Supported Multi-UAVs

Integrated Optimisation Model of Daily Freight Train Scheduling and Dynamic Railcar Routing Based on a Two-Layer Space-Time Network

This paper focuses on daily freight train scheduling and dynamic railcar routing problems for rail freight transportation at the operational level.  Two mixed integer linear programming models that adopted different strategies were formulated based on a continuous two-layer time-space network. We simultaneously considered the benefits of railroad company and service quality when setting the objective function. By solving the  models, we can distribute the dynamic railcar flows to the train paths in the basic train timetable to obtain the daily train operation plan over a short time horizon (e.g. a day), which will be helpful for dispatchers to make decisions such as the empty railcar distribution and car routing (trip planning). Finally, we compared two models on a part of the Chinese railroad network. The results show that the second model can effectively improve the efficiency of railroad freight transportation.

Construction of line Layout Model Based on Geometric Deduction and Constrained Heuristic Search

Based on problem B of the National Mathematical Contest in Modeling for College Students in 2023, this paper establishes a model through geometric deductive deduction, and establishes and solves an optimal model of survey line layout through constrained heuristic algorithm. In problem 1, a mathematical model of coverage width and overlap rate is established based on triangle relation and triangle sine theorem by finding the mathematical relationship between target quantity and known quantity. In problem 2, the angle between the known direction of the survey line and the projection of the normal direction of the submarine slope on the horizontal plane is . We can imagine a surface perpendicular to the direction of the measurement line, and then find the slope of the imaginary surface according to the geometric relationship, and replace the slope Angle of problem 1 with the imaginary slope Angle to obtain the mathematical model of the coverage width. Based on this, the constraints of the Angle and overlap rate of the line are added, and the problem 3 is simplified into the optimization problem of multiple routes in problem 2.  Finally, by changing the parameter setting of slope Angle in the model, the sensitivity analysis is carried out, and it is found that the model is stable and reliable.To sum up, the problems from one to four are gradually deepened and progressively advanced, and the development of Marine survey field is facilitated by establishing a model of depth survey line layout.

Глобальное планирование маршрута мобильного робота на основе графовых методов

The problem of global route planning for a mobile robot between two given points in a known area with static obstacles is considered. To solve the problem of constructing a route in an area with a large number of obstacles of complex shape, an integrated approach based on graph theory methods is proposed. It includes the Voronoi diagram, visibility graph and the Dijkstra's algorithm. At the first stage, the study area is represented as a polygonal object, the space outside the object is considered as obstacles. Next, to get a safe distance from obstacles, an internal buffer of the polygonal object is built using the Minkowski difference. Then the vertices of the polygon are compacted, and the Voronoi polygons are constructed from the resulting vertices. The median axis of the polygon is calculated from the Voronoi polygons. Then the Dijkstra's algorithm is applied to calculate the shortest path. The resulting path is used to construct a visibility graph, and the Dijkstra's algorithm is reapplied to the resulting graph. The proposed approach allows to build a route that is optimal in terms of length and distance to obstacles. It significantly reduces the computational complexity of constructing a visibility graph. The approach was implemented in the freely distributed QGIS geographic information system for planning the route of a mobile robot in an aquatic environment. The results of the experiment showed that the Voronoi diagram reduced the number of vertices required to construct the visibility graph by 8.3 times, while the visibility graph improved the path obtained from the Voronoi diagram by 8%. The proposed approach can be used for global planning of routes for mobile robots in various environments.

Many-to-many locomotive routing problem for the steel industry

The locomotive routing problem (LRP) considers the transportation of non-motorised containers called torpedo ladle cars (TPCs) within a steelworks. Locomotives are responsible for picking up and delivering TPCs to fulfil requests for various facilities, including steelmaking, ironmaking, heating, and repair facilities. This study introduces the many-to-many locomotive routing problem (M2MLRP), specifically tailored for the steel industry. M2MLRP was formulated using pattern- and routing-based models, and a logic-based Benders-decomposition-based matheuristic algorithm was developed. The pattern-based model successfully solved 20 small randomly generated instances, while the proposed matheuristic algorithm demonstrated the rapid generation of near-optimal solutions. Notably, unlike the pattern- and routing-based models, the proposed algorithm extended its capability to generate good feasible solutions for medium- and large-sized instances. Across all instances, the matheuristic algorithm outperformed both the routing-based model and an adaptive large neighbourhood search algorithm, showcasing its superior solution quality. This study provides a significant contribution to the field by being the first to address M2MLRP in the context of the steel industry.

Sustainable last-mile distribution with autonomous delivery robots and public transportation

In recent years, the rapid growth of e-commerce and the need to make last-mile logistics more sustainable have stimulated the development of new distribution paradigms, based on air drones and autonomous delivery robots, which are less sensitive to traffic congestion. In this article we deal with a routing problem in which a fleet of autonomous delivery robots can travel not only on the road network but also on the public transportation system (or part of it) to extend their range of action with a given battery capacity. The problem entails building delivery robot routes synchronized with the rides of the public transportation lines, enabling robots to drop on/off public vehicles, where they use dedicated compartments, to reach customers that would be otherwise out of reach. To this purpose, we develop a tailored destroy-and-repair mechanism that, embedded into a neighborhood search algorithm, allows to effectively explore the feasibility region of large-scale instances. Extensive computational results on instances resembling the distribution of drugs to pharmacies in Rome (Italy) show that the proposed algorithmic mechanism allows to obtain a cost reduction up to about 7.5% with respect to a more traditional approach. Moreover, from a managerial point of view, our experiments show that autonomous delivery robots combined with public transportation can provide huge benefits in terms of costs and emissions reduction, when compared to both traditional and electric vans.

Determination of optimal depot location for a capacitated vehicle routing problem (CVRP) based on gross vehicle weight

The fuel consumption of a vehicle depends on its gross weight, which varies as goods are successively unloaded at each wholesaler site or distribution centre within a transportation network. There exist multiple routes and many wholesalers with different demand quantities within the network. Again, transportation cost varies according to the route and the depot location (DL) from which the vehicle departs. Hence, this article aims to determine the optimal DL for Ly-Foods Ltd.'s transportation network, utilising a capacitated vehicle to fulfil wholesalers’ demands, based on gross vehicle weight (GVW). First, we develop some analytical and mathematical models (mixed-integer programming-MIP) to determine the minimum transportation cost for each location. Second, we verify our models for other instances and compare our results with some heuristic results. We find the optimal location for the depot for the transportation network of Ly-Foods, which saves a maximum of 10.43 litres of fuel in a trip. This research holds significance for the manufacturing or distribution companies aiming to set up depots within a new transportation network. Furthermore, our models are applicable to transportation networks featuring multiple routes and numerous wholesalers, where the management is planning to shift DL with the changes in wholesalers’ demands.

The joint maintenance operation selection and technician routing problem

Numerous studies have explored the challenge of optimizing maintenance grouping, aiming to efficiently manage maintenance resources and reduce costs associated with various maintenance opportunities. Maintenance grouping is even more important when the systems to maintain are geographically distributed because it significantly reduces travel costs. In this research, we tackle a novel problem that integrates condition-based maintenance with the selection of maintenance operations and technician routing. The problem involves the selection of machines requiring maintenance for each time period, determining the nature of required operations (based on the uncertain degradation state of the machines), selecting suitable technicians based on their skills and availability, and planning their routes. We formulate this problem as a mixed-integer program and propose a heuristic approach for its solution. Our method constructs a solution by iteratively adding maintenance operations to technician routes. To assess the method’s performance, we conduct experiments that evaluate both running times and solution quality.

A planar graph cluster‐routing approach for optimizing medical waste collection based on spatial constraint

AbstractMedical Solid Wastes (MSWs) are major hazardous materials containing harmful biological or chemical compounds that present public and environmental health risks. The collection and transportation of waste are usually informed by optimized work‐balanced routing based on comprehensive spatial data in urban traffic networks, called a Vehicle Routing Problem (VRP). This may be unsuitable for MSWs as their special category means they impose additional complexity. The present article develops a planar graph‐based cluster‐routing approach for the optimal collection of MSWs informed by a Geospatial Information System (GIS). The problem is first formulated as a mixed integer linear program in road network spatial data, in the context of Tehran city. The work has two key aims: (i) to minimize the total routing cost of MSW collection and transfer to waste landfills; (ii) to balance workload across waste collectors. There are three main contributions of the proposed approach: (i) to simplify the large search space area by converting the road network to a planar graph based on graph theory, spatial parameters, and topological rules; (ii) to use a modified K‐means algorithm for clustering; (iii) to consider average traffic impacts in the clustering stage and momentary traffic in the route planning stage. A planar graph extraction procedure is applied to capture the network sketch (i.e., a directed graph) from the traffic roadway network. An iterative cluster‐first‐route‐second heuristic is employed to solve the proposed routing problem. This heuristic customizes a K‐means algorithm to determine the optimal number and size of clusters (i.e., routes). A Traveling Salesman Problem (TSP) algorithm is applied to regulate the optimal sequence of visits to medical centers. The experimental results show improvements in balancing collectors' workload (i.e., ~4 min reduction in the standard deviation of average travel time) with reductions in travel time (i.e., an average ~1 h for the entire fleet and ~4 min per route). These findings confirm that the proposed methodology can be considered as an approach for optimizing waste collection routes.

A Branch-and-Price Algorithm for an Integrated Online and Offline Retailing Distribution System with Product Return

This study identifies critical inefficiencies within a dual-channel operation model employed by a fast fashion company, particularly the independent operation of three logistics distribution systems. These systems result in high operational costs and low resource utilization, primarily due to redundant vehicle dispatches to meet the distinct demands of retail store replenishment, online customer orders, and customer return demands, as well as random and scattered return requests leading to vehicle underutilization. To address these challenges, we propose a novel integrated logistics distribution system design and management method tailored for dual-channel sales and distribution businesses. The approach consolidates the three distribution systems into one cohesive framework, thus streamlining the delivery process and reducing vehicle trips by combining retail and customer visits. An optimization algorithm is introduced to factor in inventory and distribution distance, aiming to achieve global optimization in pairing retail store inventory with online customer orders and unifying the distribution of replenishment products, online products, and returned products. The paper contributes to the field by introducing a new variation of the Vehicle Routing Problem (VRP) that arises from an integrated distribution system, combining common VRP issues with more complex challenges. A custom Branch-and-Price (B&P) algorithm is developed to efficiently find optimal routes. Furthermore, we demonstrate the benefits of the integrated system over traditional, segregated systems through real-world data analysis and assess various factors including return rates and inventory conditions. The study also enhances the model by allowing inventory transfers between retail stores, improving inventory distribution balance, and offering solutions for scenarios with critically low inventory levels. Our findings highlight a significant reduction in total operating cost savings of up to 49.9% and vehicle usage when using the integrated distribution system compared to independent two-stage and three-stage systems. The integrated approach enables the utilization of vacant vehicle space and the dynamic selection and combination of tasks, preventing unnecessary mileage and space wastage. Notably, the integration of inventory sharing among retail stores has proven to be a key factor in generating feasible solutions under tight inventory conditions and reducing operational costs and vehicle numbers, with the benefits amplified in large-scale problem instances.

Application of Lexisearch Algorithm to Vehicle Routing Problem with Time Windows

Vehicle Routing Problem with Time Windows (VRPTW) is an NP hard combinatorial scheduling optimization problem in which a minimum number of routes have to be determined so as to serve each of the destinations within their specified time windows. In this paper, the mail delivery of the Nigerian Postal Services (NIPOST) is modelled as a VRPTW in order to address the problem of delay in mail delivery occurring regularly in NIPOST. The Abuja Post Office is used as a case study and the Model of a related literature is applied with modifications to solve the problem. The problem is solved by applying Lexisearch algorithm using data that was obtained from Abuja Post Office and computational results on Solomon’s 100 instances were used to validate the algorithm.

An Integrated Framework for Dynamic Vehicle Routing Problems with Pick-up and Delivery Time Windows and Shared Fleet Capacity Planning

This paper proposes a novel route optimization framework to solve the problem of instant pick-up and delivery for e-grocery orders. The proposed framework extends the traditional time-windowed package delivery problem. We demonstrate the effectiveness of our approach for this integrated problem using actual delivery data from HepsiJet, a leading e-commerce logistics provider in Turkey. We first employ several machine learning algorithms and simulations to investigate the capacity of the courier. Subsequently, a dynamic route planning workflow is executed with a highly specialized and novel routing algorithm. Our proposed heuristic approach considers combined fleet operations for delivering regular packages originating from a central depot and dynamic e-grocery orders picked up at local supermarkets and delivered to the customers. The heuristic algorithm constitutes k-opt and node transfer operation variations customized for this integrated problem. We report the performance of our approach in problem instances from the literature and instances from HepsiJet’s daily operations, which we also publicly share as new route optimization problem instances. Our results suggest that, despite the more complex nature of the integrated problem, our proposed algorithm and solution framework produce more efficient and cost-effective solutions that offer additional business opportunities for companies such as HepsiJet. The computational analyses reveal that implementing our proposed approach yields significant efficiency gains and cost reductions for the company, with a distance reduction of over 30%, underscoring our approach’s effectiveness in achieving substantial cost savings and enhanced efficiency through integrating two distinct delivery operations.

Qubit Efficient Quantum Algorithms for the Vehicle Routing Problem on Noisy Intermediate‐Scale Quantum Processors

AbstractThe vehicle routing problem with time windows (VRPTW) is a common optimization problem faced within the logistics industry. In this work, the use of a previously‐introduced qubit encoding scheme is explored to reduce the number of qubits, to evaluate the effectiveness of Noisy Intermediate‐Scale Quantum (NISQ) devices when applied to industry relevant optimization problems. A quantum variational approach is applied to a testbed of multiple VRPTW instances ranging from 11 to 3964 routes. These intances are formulated as quadratic unconstrained binary optimization (QUBO) problems based on realistic shipping scenarios. The results are compared with standard binary‐to‐qubit mappings after executing on simulators as well as various quantum hardware platforms, including IBMQ, AWS (Rigetti), and IonQ. These results are benchmarked against the classical solver, Gurobi. The approach can find approximate solutions to the VRPTW comparable to those obtained from quantum algorithms using the full encoding, despite the reduction in qubits required. These results suggest that using the encoding scheme to fit larger problem sizes into fewer qubits is a promising step in using NISQ devices to find approximate solutions for industry‐based optimization problems, although additional resources are still required to eke out the performance from larger problem sizes.

A three-phase algorithm for the pollution traveling Salesman problem

This paper studies a variant of the Pollution Traveling Salesman Problem (PTSP) focused on fuel consumption and pollution emissions (PTSPC). The PTSPC generalizes the well-known Traveling Salesman Problem (TSP), classified as NP-Hard. In the PTSPC, a vehicle must deliver a load to each customer through a Hamiltonian cycle, minimizing an objective function that considers the speed of each edge, the mass of the truck, the mass of the load pending delivery, and the distance traveled. We have proposed a three-phase algorithm for the PTSPC. The first phase solves the Traveling Salesman Problem (TSP) exactly with a time limit and heuristically using a Nearest Neighborhood Search approach. This phase considers the constraints associated with the PTSPC by using commercial software. In the second phase, both the obtained solutions and their inverse sequences from the initial phase undergo enhancement utilizing metaheuristic algorithms tailored for the PTSPC. These algorithms include Variable Neighborhood Search (VNS), Tabu Search (TS), and Simulated Annealing (SA). Subsequently, for the third phase, the best solution identified in the second phase—determined by having the minimum value by the PTSPC objective function—is subjected to resolution by a mathematical model designed for the PTSPC, considering the heuristic emphasis of commercial software. The efficiency of the former algorithm has been validated through experimentation involving the adaptation of instances from the Pollution Routing Problem (PRP) to the PTSPC. This approach demonstrates the capacity to yield high-quality solutions within acceptable computing times.

Electric Vehicle Routing Problem with States of Charging Stations

This paper proposes an electric vehicle routing problem, considers the states of charging stations and suggests solution strategies. The charging of electric vehicles is a main issue in the field of electric vehicle routing. There are many studies that find the locations of charging stations, recharging functions for the batteries of vehicles, and so on. However, the state of charging stations significantly affects the routes of electric vehicles, which is not much explored. The states may include open or closed charging stations, occupied or empty charging slots, and so on. This paper investigates how the states of charging stations are estimated and how routing strategies are determined. We formulate a mixed integer programming model and suggest how to solve the problem with an exact method. Numerical examples provide the optimal routing strategies of electric vehicles for the changing environments regarding the states of charging stations.