Metaheuristics for vehicle routing problems with three-dimensional loading constraints

A hybrid approach for the vehicle routing problem with three-dimensional loading constraints

In this paper, we consider the Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) which combines the routing of a fleet of vehicles and the loading of three dimensional shaped goods into vehicles while minimizing total travel distance. Compared with previous works which mainly focus on improving the performance of solving the routing problem, we propose a new least waste packing heuristic based approach for solving the loading subproblem, which is iteratively invoked by a simple tabu search algorithm for the routing. The proposed algorithm illustrates the state-of-the-art performance in terms of the solution quality and computational efficiency of our approach through the numerical experiments on the same test instances from literature.

This study addresses the Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP), a highly complex NP-hard problem that combines vehicle routing with spatially constrained three-dimensional bin packing. To tackle this challenge, we propose an enhanced Artificial Immune System (En-AIS) that integrates a novel local search heuristic called “Bring-i-to-j,” designed to improve routing feasibility and loading efficiency. The En-AIS algorithm is further refined through rigorous parameter tuning using a full factorial design and ANOVA analysis. Comparative experiments were conducted against conventional AIS and the Firefly Algorithm (FA) across 27 benchmark instances. Results demonstrate that En-AIS consistently outperforms both baseline methods in terms of solution quality, achieving an average improvement of 15–20% while maintaining competitive computational times. These findings highlight the algorithm’s robustness and its practical potential for application in logistics and supply chain optimization tasks involving joint routing and loading decisions.

This paper proposes an abstraction of emerging vehicle routing problems, the Vehicle Routing Problem with Black Box Feasibility. In this problem the routes of a basic VRP need to satisfy an unknown set of constraints. A black box function to test the feasibility of a route is provided. This function is considered of non-linear complexity (in the length of the route). Practical examples of such problems are combinations of VRP with Loading problems or VRP with Scheduling problems. The difficulty in addressing the VRP with Black Box Feasibility lies in the unknown problem structure and the costly feasibility check. We propose a column generation-based approach to locally optimize this problem. Columns are heuristically generated by so-called Collector ants, executing a construction heuristic while guided by pheromones. To find an integer solution we solve an integer Set Partitioning Problem defined on the set of columns generated by the ants. We test the proposed approach on two applications from the literature, the Three-Dimensional Loading Capacitated Vehicle Routing Problem and the Multi-Pile Vehicle Routing Problem, showing the applicability of our approach and its good behavior compared to dedicated approaches.

The paper addresses the heterogeneous fleet vehicle routing problem with three-dimensional (3D) loading constraints (3L-HFVRP), a new practical variant of the combined routing and loading problem. In this problem, the loads consist of a set of three-dimensional, rectangular shaped items. The fleet is composed of heterogeneous vehicles with different weight and space capacities. The objective is to serve all customers by selecting a set of vehicles such that the total transportation cost is minimized. The cost consists of the fixed cost of the selected vehicles and their travel cost. In addition, loading sequence related constraints frequently encountered in realistic applications are respected when loading and unloading the items. To solve this challenging problem, we develop an adaptive variable neighborhood search (AVNS) which utilizes an extreme point based first fit heuristic to find a feasible loading pattern for each route. We design two strategies to accelerate the loading and routing processes. The Trie data structure is used to record the loading information of routes already visited and to control the computational effort spent for each route. The Fibonacci heap data structure is used to maintain all of the possible moves and vehicle type assignments, which avoids the duplicated evaluation of some moves and unnecessary loading check of unpromising solutions. The robustness and effectiveness of the proposed algorithm is validated by computational tests performed both on some newly generated 3L-HFVRP instances and well-known benchmark instances from the literature for two simplified VRP variants: the capacitated vehicle routing problem with 3D loading constraints (3L-CVRP) and the pure heterogeneous fleet vehicle routing problem (HFVRP). The numerical experiments show that the proposed AVNS outperforms other algorithms in 3L-CVRP and improves several best known solutions reported in the literature. The results obtained for the pure HFVRP are very close to the best known solutions.

This paper introduced the Bin Packing Capacitated Vehicle Routing Problem. It introduced the constraints and differences between algorithms of two-dimensional and three-dimensional loading capacitated vehicle routing problem. It gave a review of models and algorithms for Bin Packing Capacitated VRP. Finally, it prospected future research orientations and possible improvement in this area.

To improve the efficiency of logistics distribution and balance the workload of drivers, this study introduces the standard deviation of workload time as an equilibrium measure and constructs a dual-objective mixed-integer programming model based on three-dimensional loading constraints to achieve the minimum total distance and workload balance. To solve the above model, a multi-stage hybrid algorithm is designed in this paper. Firstly, the clustering algorithm is used to effectively divide the distribution area, and then the non-dominated sorting genetic algorithm is used to solve the vehicle routing problem considering workload balance. At the same time, the constructive heuristic three-dimensional loading algorithm based on block loading is invoked to ensure the feasibility of cargo loading. Based on the historical data of a material flow distribution company, the feasibility of the constructed model and algorithm is verified. The research shows that the standard deviation of workload time can be reduced by 35.86%, and the average loading rate can be increased by 9.19% under the premise of increasing the distribution distance by only 1.6%. In addition, the proposed algorithm is compared with three other multi-objective optimization algorithms by constructing examples of different scales and using a series of standard example sets to verify its effectiveness and superiority.

In order to promote the cost reduction and efficiency improvement of the logistics distribution process and to guarantee the safety of goods transportation, this paper studies the portfolio optimization of goods loading and the problem of simultaneous pickup and delivery vehicle routing. A balanced loading constraint was introduced to restrict loading through two aspects of axle weight bearing and lateral center-of-gravity offset. With the shortest total route length as the objective, this paper constructs a simultaneous pickup and delivery vehicle routing model with three-dimensional (3D) balanced loading constraints (3BL-VRPSPD). Additionally, a hybrid tabu search (TS) algorithm embedded loading test was proposed to solve this problem. Firstly, a heuristic insertion method was applied to determine the initial routing scheme, and the node swapping and relocation operators were designed to construct the tabu neighborhood scheme for routing optimization. On this basis, the 3D balanced loading was incorporated into the routing iteration process. A balanced loading algorithm, combining multiple-indicator ordering and maximum space division strategies (MOMD), was formulated to develop a 3D-balanced loading plan for goods with a pickup and delivery vehicle routing scheme. Finally, standard instances verified the effectiveness of the method. The results show that the proposed method can effectively optimize 3BL-VRPSPD and outperform other algorithms.

An Australian company is faced with the logistics problem of distributing small quantities of fibre boards to hundreds of customers every day. The resulting Heterogeneous Fleet Vehicle Routing Problem with Time Windows and Three-Dimensional Loading Constraints has to be solved within a single hour, hence the use of a heuristic instead of an exact method. In previous work, the loading was performed after optimising the routes, which in some cases generated infeasible solutions in need of a repair mechanism. In this work, the feasibility of the loading constraints is maintained during the route optimisation. Iterated Local Search has proved very effective at solving vehicle routing problems. Its success is mainly due to its biased sampling of locl optima. However, its performance heavily depends on the perturbation procedure. We trialled different perturbation procedures where the first one perturbs the given solution by moving deliveries that incur the highest cost on the objective function, whilst the second one moves deliveries that have been shifted less frequently by the local search in previous iterations. Our industry partner provided six sets of daily orders which have varied characteristics in terms of the number of customers, customer distribution, number of fibre boards and fibre boards’ sizes. Our investigations show that an instance becomes more constrained when the customer order contains many different board sizes, which makes it harder to find feasible solutions. The results show that the proposed perturbation procedures significantly enhances the performance of iterated local search specifically on such constrained problems.

The optimization of auto parts supply chain logistics plays a decisive role in the development of the automotive industry. To reduce logistics costs and improve transportation efficiency, this paper addresses the joint optimization problem of multi-vehicle pickup and delivery transportation paths under time window constraints, coupled with the three-dimensional loading of goods. The model considers mixed time windows, three-dimensional loading constraints, cyclic pickup and delivery paths, varying vehicle loads and volumes, flow balance, and time window constraints. Evaluation rules for the three-dimensional loading test of goods are also set, resulting in constructing a comprehensive optimization model for the inbound logistics of auto parts and components. In this study, a Greedy-Tabu Dual-Heuristic algorithm is proposed, which integrates an Improved Greedy Algorithm with an Enhanced Tabu Search Algorithm based on the -sampling strategy. The overall problem-solving process for the Improved Greedy Algorithm and the Tabu Search Algorithm is outlined. The superiority, efficiency, and stability of the improved algorithm are verified by solving cases of various sizes and analyzing the algorithm’s results before and after improvement. A case study involving the third-party logistics company R Enterprise compares the pickup and delivery-separated Milk-Run mode with the simultaneous delivery and pickup Milk-Run mode. The proposed method shows a 26.67% reduction in total distance traveled and a 46.60% decrease in waiting time compared to the traditional Milk-Run approach. Additionally, when evaluated against the standard 3D loading inspection method, the proposed approach improves average vehicle load utilization by 17% and vehicle volume utilization by 15%. These findings verify the applicability and superiority of the proposed algorithms and models in practical scenarios.

A multi-start evolutionary local search for the two-dimensional loading capacitated vehicle routing problem

In this work, a Prognosis-Decision-Making (PDM) methodology to calculate the best route of an Electric Vehicle (EV) in order to reach a destination in a street network, incorporating stochastic traffic information, is presented. The problem is formulated as an optimization problem that seeks to minimize the expected value of an objective function, which includes the time and energy spent to complete the route. The proposed methodology first selects a number of the best routes using standard routing optimization algorithms. Then, information about the traffic state, state of charge (SOC) of the lithium-ion (Li-ion) battery, as well as the elevation and distance profiles of the routes are used as inputs for a prognosis algorithm, which allows to evaluate the different possible paths. Finally the route with minimal expected travel cost is selected according to the optimization criterion. Simulation results show that the inclusion of the prognosis algorithm, incorporating traffic, battery and route information, effectively helps finding the best route in terms of the travel cost.

In this article we present a branch‐and‐price‐and‐cut method for the solution of the split delivery vehicle routing problem (SDVRP). The SDVRP is the problem to serve customers with a fleet of capacitated vehicles at minimum traveling cost. With respect to the classical vehicle routing problem, where each customer is visited exactly once, in the SDVRP a customer may be visited any number of times. The exact method we propose is based on a decomposition of the problem where the possible routes, with the delivery quantities, are generated in the subproblem. The generated routes are also used to find a heuristic solution to the problem. We consider both the case where the fleet of vehicles is unlimited and the case where the fleet is limited to the minimum possible number of vehicles. We solve to optimality instances with larger size with respect to previous approaches, find new best solutions to several benchmark instances and reduce the optimality gap on most of the benchmark instances. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(4), 241–254 2011

The Capacitated Vehicle Routing Problem (CVRP) is a combinatorial optimization problem widely applied in logistics and supply chain management. It involves determining the optimal routes for a fleet of vehicles with limited capacity to serve a set of customers with specific demands while minimizing travel costs. This study compares the performance of two popular metaheuristic algorithms, Ant Colony Optimization (ACO) and Artificial Bee Colony (ABC), in solving the CVRP. The research implements both algorithms on standard benchmark datasets, evaluating solution accuracy and computational efficiency. Simulation results indicate that ACO tends to excel in finding high-quality solutions, particularly for problems with high complexity, whereas ABC demonstrates superior computational efficiency on small- to medium-scale datasets. A detailed analysis of algorithm parameters was also conducted to understand their impact on the performance of both methods. This study provides valuable insights into the strengths and limitations of each algorithm in the context of CVRP and paves the way for the development of hybrid approaches in the future.

A vehicle routing problem involves finding a set of optimal route for a fleet of capacitated vehicles which are available at a location to service the demands of a set of customers. In its simplest form, a customer is required to be visited once and the capacity of a vehicle must not be exceeded. In this paper, a review of four common and applicable variants of the vehicle routing problem, namely, capacitated vehicle routing problem, vehicle routing problem with time windows, periodic vehicle routing problem and the dynamic vehicle routing problem, are considered based on formulation techniques, methods of solution and areas of application. A summary table is presented for each variant to emphasis some key features that represent direction of current research.