Two-dimensional loading in vehicle routing problem with release and due dates

A meta-heuristic algorithm for heterogeneous fleet vehicle routing problems with two-dimensional loading constraints

In this paper we consider a single machine preemptive scheduling problem to minimize the weighted number of late jobs. There are given n jobs and for each job we have a release date, a processing time and a due date. It is assumed that certain specified jobs have to be completed on time. The due dates for these jobs are considered as deadlines. while for the other jobs due dates may be violated. For the case of similarly ordered release and due dates (when there is no advantage to preemption) as well as for the case of embedded release and due date intervals and preemption allowed, we transform the initial problem into a reduced problem, where all jobs with deadlines are eliminated. It gives an opportunity to propose O(n log n) algorithms for some special cases of the considered problems.

Multi-compartment vehicles (MCVs) can deliver several product segments jointly. Separate compartments are necessary as each product segment has its own specific characteristics and segments cannot be mixed during transportation. The size and position of the compartments can be adjusted for each tour with the use of flexible compartments. However, this requires that the compartments can be accessed for loading/unloading. The layout of the compartments is defined by the customer and segment sequence, and it needs to be organized in a way that no blocking occurs during loading/unloading processes. Routing and loading layouts are interdependent for MCVs. This paper addresses such loading/unloading issues raised in the distribution planning when using MCVs with flexible compartments, loading from the rear, and standardized transportation units. The problem can therefore be described as a two-dimensional loading and multi-compartment vehicle routing problem (2L-MCVRP). We address the problem of obtaining feasible MCV loading with minimal routing, loading and unloading costs. We define the loading problem that configures the compartment setup. Consequently, we develop a branch-and-cut (B&C) algorithm as an exact approach and extend a large neighborhood search (LNS) as a heuristic approach. In both cases, we use the loading model in order to verify the feasibility of the tours and to assess the problem as a routing and loading problem. The loading model dictates the cuts to be performed in the B&C, and it is used as a repair mechanism in the LNS. Numerical studies show that the heuristic reaches the optimal solution for small instances and can be applied efficiently to larger problems. Additionally, further tests on large instances enable us to derive general rules regarding the influence of loading constraints. Our results were validated in a case study with a European retailer. We identified that loading constraints matter even for small instances. Feasible loading can often be achieved only through minor changes to the routing solution and therefore with limited additional costs. Further, the importance to integrate loading constraints grows as the problem size increases, especially when a heterogeneous mix of segments is ordered.

Traditional vehicle routing problems implicitly assume that only one crew operates a vehicle for the entirety of its journey. However, this assumption is violated in many applications arising in humanitarian and military logistics. This paper considers a joint vehicle and crew routing and scheduling problem in which crews are able to interchange vehicles, resulting in space and time interdependencies between vehicle routes and crew routes. The problem is formulated as a mixed integer programming (MIP) model and a constraint programming (CP) model that overlay crew routing constraints over a standard vehicle routing problem. The constraint program uses a novel optimization constraint to detect infeasibility and to bound crew objectives. This paper also explores methods using large neighborhood search over the MIP and CP models. Experimental results indicate that modeling the vehicle and crew routing problems jointly and supporting vehicle interchanges for crews may bring significant benefits in cost reduction compared with a method that sequentializes these decisions.

In this paper, the vehicle routing optimization problem in the distribution process of express logistics joint distribution alliance is studied. According to the actual situation that client nodes have with regard to time windows, the delivery goods arrive at the distribution center in batches, and the vehicle can be repeatedly used in these actual situations, and based on the purpose of saving the transportation cost. We put forward with the Multi Trip Split Delivery Vehicle Routing Problem with Time Window and Release Dates and establish the problem's mathematical model with the minimizing of distribution costs as object function. Considering the difficulty of solving the problem, this paper designs a modern heuristic algorithm, the discrete particle swarm optimization algorithm, and takes the instance data as the input to solve the algorithm programmatically. The experimental results show that the vehicle routing optimization method proposed in this paper can bring more cost savings compared with the separate distribution of vehicles of different enterprises in the non-joint distribution mode, and this method is more suitable for the joint distribution mode.

Addressed in this study is a vehicle routing problem with two‐dimensional loading constraints and time windows (2L‐CVRPTW), aiming to minimize the transportation cost while satisfying the two‐dimensional loading and routing constraints with time windows. To solve this problem, for the first time a mixed‐integer linear programming model is formulated with considering practical last‐in‐first‐out loading constraints, and a branch‐and‐price‐based (BP‐based) heuristic is proposed based on a set partitioning formulation. In the heuristic, a modified labeling algorithm is proposed for the complex pricing problem, which is a relaxation of the elementary shortest path problem with resource constraints and two‐dimensional loading constraints. Therein, an effective Tabu‐maximum open space packing heuristic is proposed to verify the feasibility of the two‐dimensional packing problem of each route generated by the labeling algorithm. In addition, effective accelerating and branching strategies are introduced to improve the solving efficiency of the heuristic. To evaluate the effectiveness and the advantages of the proposed heuristic, extensive computational experiments are performed based on the generated instances. The computational results show that the proposed BP‐based heuristic can effectively solve the 2L‐CVRPTW, in which the optimal solutions can be achieved much faster than CPLEX in small‐scale problems. Relationships between the transportation cost and the characteristics of the instances are analyzed. The stability of the algorithm and the effectiveness of the accelerating strategies are verified and discussed.

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) and the last-in-first-out (LIFO) rule presents significant practical and algorithmic challenges. While numerous heuristic approaches have been proposed to address its complexity, stemming from two NP-hard problems: the vehicle routing problem (VRP) and the two-dimensional bin packing problem (2D-BPP), less attention has been paid to developing exact algorithms. Bridging this gap, this article presents an exact algorithm that integrates advanced machine learning techniques, specifically a novel combination of attention and recurrence mechanisms. This integration accelerates the state-of-the-art exact algorithm by a median of 29.79% across various problem instances. Moreover, the proposed algorithm successfully resolves an open instance in the standard test-bed, demonstrating significant improvements brought about by the incorporation of machine learning models. Code is available at https://github.com/xyfffff/NCG-for-2L-CVRP.

This paper proposes an efficient algorithm, with a reduced number of parameters, for solving the two‐dimensional loading‐capacitated vehicle routing problem (2L‐CVRP). This problem combines two of the most important issues in logistics, that is, vehicle routing and packing problems. Our approach contemplates unrestricted loading including the possibility of applying 90° rotations to each rectangular‐shaped item while loading it into the vehicle, which is a realistic assumption seldom considered in the existing literature. The algorithm uses a multistart approach that is designed to avoid local minima and also to make the algorithm an easily parallelizable one. At each restart, a biased randomization of a savings‐based routing algorithm is combined with an enhanced version of a classical packing heuristic to produce feasible good solutions for the 2L‐CVRP. The proposed algorithm has been compared with the classical benchmarks for two different 2L‐CVRP variants, that is, with and without item rotations. Experimental results show that our approach outperforms several best‐known solutions from previous work, both in terms of quality and the computational time needed to obtain them.

The chapter by Teodor Crainic in this book discusses the long haul truck routing problem. Some of the key differences between the long haul truck routing problem and the vehicle routing problems considered in this chapter are as follows: 1. In the long haul truck routing problem, the routes can extend over several days whereas, in the vehicle routing problem described herein, the routes are of one-day duration. 2. In the long haul truck routing problem, the locations can be scattered over a wide region, even the entire United States, whereas in the vehicle routing problems considered herein, the locations requiring service are packed in a small region.

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.

In this paper the authors present a review on a quite recent topic: the integrated resolution of the vehicle routing problem and the vehicle/container loading problem, or, as the authors name it, the routing problem with loading constraints. In the Capacitated Vehicle Routing Problem loading constraints have always existed, but just as a scalar measure, like volume or weight. In this new form of the problem we are dealing with geometric (length, width and height of each item) and other cargo related constraints, as loading order and cargo stability

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

The field of grain logistics transport vehicle route has seen the development of a variety of mathematical techniques, and the practical application of these techniques has depended on the use of spatial information, such as regional allocation information of road network. Vehicle Routing Problem (VRP) is regarded as the key of grain logistics distribution. In this paper, firstly, focus on the grain logistics VRP with time windows, then VRP model is successful constructed. In order to reduce the cost of grain logistics distribution, to combine the VRP model to geographic information system (GIS) by analyzing the VRP model. GIS takes into account for grain logistics, and the Particle Swarm Optimization (PSO) is used for the identification of optimal routes. And the optimal route for grain distribution is urgent since it has been calculated that, of the total cost for transportation. It improves the efficiency of grain logistics distribution and gets a better effect.

This paper presents an overview of recent advances in the field of the vehicle routing problem (VRP), based on papers published in high-quality journals during the period from January 2015 to July 2017. A distinctive feature of the presented survey is its focus on new versions of the VRP, which have recently emerged or gained momentum, and the corresponding new solution methods, with particular emphasis on computational intelligence (CI) approaches. The list of newly proposed or currently popular VRP formulations include last mile and same day delivery, crowdshipping, bike sharing systems, post-disaster response plans, local routing in large production or cargo plants, customer-centric VRP, autonomous delivery, unnamed aerial vehicle delivery, green VRP, waste collection VRP, rich VRP, or VRP with backhauls. Simultaneously, an adequate increase of interest in the application of traditional CI methods (e.g., genetic, memetic, ant colony or particle swarm optimization, simulated annealing, or their various hybrid versions) can be observed in the VRP domain. At the same time, approaches proven efficient in other optimization areas (e.g., hyperheuristics, methods based on Monte Carlo simulations, algorithms rooted in game theory and bi-level optimization-Stackelberg games, or cognitively motivated methods) have lately entered the VRP field and become a viable alternative to more traditional techniques. Since VRP is one of the fastest growing fields in the operations research area, we believe that an analysis of the recently published VRP papers from the perspective of their novelty in problem formulation and/or applied solution method can provide a true value for the CI community, especially young researchers entering the field and seeking challenges in this interesting and fast developing research area.

Since increasing oil prices and vehicle costs increase transportation costs in order delivery systems, the optimal vehicle loading and routing is very crucial for the companies in competitive conditions. Although there are many studies related to optimal vehicle loading and routing by using linear programming and heuristic algorithms, there is not enough practical web service-based application in the literature. In this study, we propose a hybrid model to solve the problem of two-dimensional vehicle pallet loading with routing for a real-world data by combining Knapsack Problem solver algorithms such as MaxRects, Skyline and Guillotine with Dijkstra's algorithm for loading and routing respectively as a web service-based application.