Combining genetic local search into a multi-population Imperialist Competitive Algorithm for the Capacitated Vehicle Routing Problem

A hybrid genetic search and dynamic programming-based split algorithm for the multi-trip time-dependent vehicle routing problem

Particle swarm optimization (PSO) is adopted to solve capacitated vehicle routing problems (CVRPs). A nonlinear S decrease curve is employed to adjust the inertia weight of PSO to balance global and local searches. Meanwhile, local search and path relink are also involved to improve the quality of solutions. Experiments were conducted on Auger at problems in the OR Library. Experimental results will were compared with those from other studies. Experimental results prove the algorithm designed in this article can effectively solve capacitated type vehicle routing problems.

Vehicle Routing Problem (VRP) is a classical NP-hard problem in combinational optimization with great practicality. A branch of VRP is the Capacitated Vehicle Routing Problem (CVRP), where vehicles have capacity constraints. In this paper, a Hybrid Chaotic Particle Swarm Optimization (HCPSO) is proposed to solve CVRP. To make this algorithm successful, three main components play an important role. Firstly, we introduce a new mutual mapping method to encode and decode between decimal numbers and integer solutions. Secondly, the ergodicity and sensitivity on initial conditions of chaos theory are utilized to achieve chaotic initialization and renewing. Thirdly, various local search strategies like neighbor change strategy, move strategy are employed to improve the local search ability. In the end, benchmarks have been tested. It is clearly shown that for small and medium sized CVRP, the algorithm is efficient and effective and the results are better than those of some recent papers.

Design and development of a hybrid artificial bee colony algorithm for the environmental vehicle routing problem

Crow Search Algorithm is a novel Metaheuristic method based on the intelligent behavior of crows. It has been used to solve some optimization problems like engineering design problems, feature extraction and classification problems but has not been applied to vehicle routing problem. In this paper, crow search algorithm has been utilized to solve capacitated vehicle routing problem and the performance of it on various size-capacitated vehicle routing problem instances is analyzed. The factorial design ANOVA is used to determine the performance of crow search algorithm under different parameter settings on capacitated vehicle routing problem instances.

The hybrid genetic search (HGS) metaheuristic has produced outstanding results for several variants of the vehicle routing problem. A recent implementation of HGS specialized to the capacitated vehicle routing problem (CVRP) is a state‐of‐the‐art method for this variant. This paper proposes an improved HGS for the CVRP obtained by incorporating a new solution generation method into its (re‐)initialization process to guide the search more efficiently and effectively. The solution generation method introduced in this work combines an approach based on frequent patterns extracted from good solutions by a data mining process and a randomized version of the Clarke and Wright savings heuristic. As observed in our experimental comparison, the proposed method significantly outperforms the original algorithm regarding the final gap to the best known solutions and the primal integral.

Capacitated vehicle routing problem is a vehicle routing problem that has constraints in the form of vehicle capacity. It is a matter of optimization to find the optimal route for some vehicles with a specific capacity and homogeneous vehicles that serve some agents with several known requests before the distribution process takes place. LPG distribution is the application of capacitated vehicle routing problems. This research aims to determine the LPG distribution route based on the Clarke and Wright Algorithm and LINGO. The capacitated vehicle routing problem model will be solved in two phases. The first stage completes the capacitated vehicle routing problem model using the Clarke and Wright algorithm. The second stage is completing the results of the first stage in the form of LINGO-based sub tours to obtain the optimal solution. The solution of the capacitated vehicle routing problem in distribution LPG using the Clarke and Wright algorithm has a total distance of 151.94 km. The solution of the capacitated vehicle routing problem in distribution LPG using LINGO has a total distance of 161.59 km.

Vehicle routing problems (VRPs) are a fundamental class of combinatorial optimization problems (COPs) in computer science and operations research, with diverse real-world applications in logistics, transportation, and manufacturing. The intrinsic NP-hard nature makes VRPs exponentially expensive to be solved by exact solvers. As an alternative, heuristic solvers deliver suboptimal solutions within reasonable time but require extensive hand-crafted rules and domain expertise tailored to each specific problem. Recently, neural combinatorial optimization (NCO), which leverages machine learning to learn heuristics in a data-driven manner, has gained significant attention. These neural solvers demonstrate strong performance while reducing computational overhead and reliance on domain expertise compared to traditional solvers. However, they face significant generalization challenges. For example, their performance may degrade substantially when applied to instances with different data distributions, problem scales, or constraints than those encountered during training. This generalization issue severely limits their practical applicability. This thesis aims to systematically address these challenges and advance the development of generalizable neural solvers for VRPs. The thesis begins by enhancing the cross-distribution generalization of neural VRP solvers through an adversarial training approach. We introduce an ensemble-based Collaborative Neural Framework (CNF) that adversarially trains multiple models in a collaborative manner, promoting robustness against adversarial attacks while also boosting performance on clean instances. In this framework, an attacker generates hard instances by perturbing the data distribution, challenging the models to adapt and thereby strengthening cross-distribution generalization. Additionally, a neural router is designed to efficiently distribute training instances among the models, thereby improving load balancing and collaborative efficacy. Extensive experiments on two classic routing problems -- the travelling salesman problem (TSP) and the capacitated vehicle routing problem (CVRP) -- demonstrate the effectiveness and versatility of CNF in enhancing the cross-distribution generalization. Next, the thesis further expands neural solvers to simultaneously address both cross-size and cross-distribution generalization challenges. We propose Omni-VRP, a generic meta-learning framework that enables effective training of an initial model with the capacity for rapid adaptation to new tasks during inference, where each task corresponds to a set of instances with the same problem scale and data distribution. Additionally, we introduce a simple yet efficient approximation method to reduce the training overhead associated with second-order derivative calculations. Extensive experiments on both synthetic and benchmark instances of TSP and CVRP validate the effectiveness of our approach. Finally, the thesis pioneers the study of cross-problem generalization, pushing the boundaries of neural VRP solvers in learning generalizable representations across diverse constraints. We propose MVMoE, a unified neural solver capable of solving 16 VRP variants simultaneously in a zero-shot manner through attribute composition. To efficiently enhance model capacity, we employ a mixture-of-experts architecture with a hierarchical gating mechanism, striking a balance between empirical performance and computational complexity. Experimentally, our method significantly promotes zero-shot generalization on 10 unseen VRP variants, and showcases decent results on the few-shot setting and real-world benchmark instances. Overall, this thesis represents a pioneering exploration and substantial advancement in developing generalizable neural solvers for VRPs. It investigates innovative designs across various components, including model architectures, training algorithms, and problem settings, to enhance the generalization of neural solvers across diverse contexts such as varying data distributions, problem scales, and constraints. These contributions encourage neural solvers to learn more robust and generalizable representations, paving the way for the first generation of foundation models capable of addressing a wide range of COPs. Ultimately, the insights gained here not only propel the field of NCO forward but also enrich the broader landscape of learning-based optimization methods.

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.

Capacitated vehicle routing problem is extension of vehicle routing Problem where the purpose is to determine a set of routes to fulfil all the transportation requests with a given fleet of vehicles in such a way that the travelling cost comes out to be a minimum. In this paper, the demands of the customers will be very well known in advance, but they are known imprecisely. To deal with such kind of impreciseness, interval type-2 fuzzy number (IT2FN) can be used. Therefore, a capacitated vehicle routing problem with interval type-2 fuzzy number has been modelled. A procedure to solve the model has been introduced using Clark and Wright algorithm.

Vehicle Routing Problem (VRP) is concerned with the optimal design of routes to be used by a fleet of vehicles to serve a set of customers. In this study we analyze constructive heuristics for a subcase of VRP, where the vehicles have a limited capacity – Capacitated Vehicle Routing Problem (CVRP). The problem is NP-hard, therefore heuristic algorithms which provide near-optimal polynomial-time solutions are still actual. The aim of this work is to make a comparison of constructive heuristics as there were not found any such classification. Finally, the leader by a criterion of quality is admitted being a Clarke and Wright Savings heuristic; however, this algorithm cannot find the solution for all used instances. This fact and other ones are discussed in the paper. Our future goal is to make an experimental comparison of the most common and state-of-the-art metaheuristics using well suited constructive heuristic to build a suboptimal solution.

eCommerce, postal and logistics’ planners require to solve large-scale capacitated vehicle routing problems (CVRPs) on a daily basis. CVRP problems are NP-Hard and cannot be easily solved for large problem instances. Given their complexity, we propose a methodology to reduce the size of CVRP problems that can be later solved with state-of-the-art optimization solvers. Our method is an efficient version of clustering that considers the constraints of the original problem to transform it into a more tractable version. We call this approach Constrained Clustering Capacitated Vehicle Routing Solver (CC-CVRS) because it produces a soft-clustered vehicle routing problem with reduced decision variables. We demonstrate how this method reduces the computational complexity associated with the solution of the original CVRP and how the computed solution can be transformed back into the original space. Extensive numerical experiments show that our method allows to solve very large CVRP instances within seconds with optimality gaps of less than 16%. Therefore, our method has the following benefits: it can compute improved solutions with small optimality gaps in near real-time, and it can be used as a warm-up solver to compute an improved solution that can be used as an initial solution guess by an exact solver.

We consider the asymmetric capacitated vehicle routing problem (CVRP), a particular case of the standard asymmetric vehicle routing problem in which only the vehicle capacity constraints are imposed. CVRP is known to be NP-hard and finds practical applications in distribution and scheduling. We describe two new bounding procedures for CVRP, based on the so-called additive approach. Each procedure computes a sequence of nondecreasing lower bounds, obtained by solving different relaxations of CVRP. Effective implementations of the procedures are also outlined which considerably reduce the computational effort. The two procedures are combined into an overall bounding algorithm. A branch-and-bound exact algorithm is then proposed, whose performance is enhanced by means of reduction procedures, dominance criteria, and feasibility checks. Extensive computational results on both real-world and random test problems are presented, showing that the proposed approach favorably compares with previous algorithms from the literature.

This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.

The last-mile delivery problem is one of the most complex and resource-intensive aspects of modern logistics, especially within the growing e-commerce sector. As online shopping continues to expand, companies are under immense pressure to deliver goods more quickly, efficiently, and at lower costs, all while meeting the demands of increasingly time-sensitive customers. This has created a need for innovative solutions that can tackle challenges related to dynamic traffic patterns, fluctuating customer preferences, and operational constraints such as vehicle capacities and delivery windows. In response to these challenges, this paper explores the application of predictive analytics as a tool for optimizing last-mile delivery routes in real-time.The study begins by identifying the core challenges inherent in last-mile logistics, particularly in the U.S. e-commerce landscape, where the cost of last-mile delivery can represent up to 53% of total shipping costs. With traffic congestion, unpredictable customer availability, and delivery time constraints posing significant hurdles, conventional static route planning models often fall short. In this paper, predictive analytics is proposed as a solution to these challenges, utilizing real-time data to inform more efficient routing decisions. By processing vast amounts of real-time traffic data, customer preferences, and delivery constraints, predictive models can offer a more flexible and responsive approach to last-mile delivery.The research then presents a comprehensive literature review of existing route optimization methods, such as the traditional Vehicle Routing Problem (VRP) and its extensions, including VRP with Time Windows (VRPTW), Dynamic VRP (DVRP), and Capacitated VRP (CVRP). While these models have proven useful, their limitations are exposed when faced with real-time operational complexities in the e-commerce sector. Therefore, this study introduces an advanced dynamic routing model that integrates machine learning algorithms—such as decision trees and neural networks—with traditional VRP frameworks. These machine learning models, trained on historical data, are capable of predicting future traffic patterns, customer behavior, and delivery time windows.A case study is conducted using data from U.S.-based e-commerce companies to demonstrate the practical application of predictive analytics in optimizing last-mile delivery. The case study outlines how predictive models are used to dynamically adjust delivery routes based on real-time conditions, leading to significant improvements in efficiency, cost savings, and customer satisfaction. Key performance indicators such as delivery times, fuel consumption, and vehicle utilization are examined before and after the implementation of the predictive models, with the results showing a reduction in delivery time by 20% and fuel costs by 15%, alongside improved on-time delivery rates.The paper concludes by presenting the proposed dynamic route optimization model as a solution that combines the flexibility and responsiveness of predictive analytics with the robust framework of traditional VRP models. Through the integration of machine learning, real-time data processing, and dynamic routing, the model is shown to significantly improve last-mile delivery efficiency. This study's findings highlight the potential for predictive analytics to revolutionize the logistics industry, particularly in the high-demand e-commerce sector, where quick and reliable delivery is paramount. The research suggests that as e-commerce continues to grow, predictive analytics will play an increasingly critical role in ensuring that last-mile delivery is both cost-effective and responsive to the evolving needs of consumers.