Two-dimensional Loading Constraints Research Articles

A Multi-Objective Learning Whale Optimization Algorithm for Open Vehicle Routing Problem with Two-Dimensional Loading Constraints

With the rapid development of the sharing economy, the distribution in third-party logistics (3PL) can be modeled as a variant of the open vehicle routing problem (OVRP). However, very few papers have studied 3PL with loading constraints. In this work, a two-dimensional loading open vehicle routing problem with time windows (2L-OVRPTW) is described, and a multi-objective learning whale optimization algorithm (MLWOA) is proposed to solve it. As the 2L-OVRPTW is integrated by the routing subproblem and the loading subproblem, the MLWOA is designed as a two-phase algorithm to deal with these subproblems. In the routing phase, the exploration mechanisms and learning strategy in the MLWOA are used to search the population globally. Then, a local search method based on four neighborhood operations is designed for the exploitation of the non-dominant solutions. In the loading phase, in order to avoid discarding non-dominant solutions due to loading failure, a skyline-based loading strategy with a scoring method is designed to reasonably adjust the loading scheme. From the simulation analysis of different instances, it can be seen that the MLWOA algorithm has an absolute advantage in comparison with the standard WOA and other heuristic algorithms, regardless of the running results at the scale of 25, 50, or 100 datasets.

A variable neighborhood search for the green vehicle routing problem with two-dimensional loading constraints and split delivery

We address the Green Vehicle Routing Problem with Two-Dimensional Loading Constraints and Split Delivery (G2L-SDVRP), which extends the split delivery vehicle routing problem to include customer demands represented by two-dimensional, rectangular items. We aim to minimize carbon dioxide (CO2) emissions instead of travel distance, a critical issue in contemporary logistics activities. The CO2 emission rate is proportional to fuel consumption and measured in terms of the vehicle’s total weight and traveled distance. We propose the first metaheuristic for the G2L-SDVRP, based on a variable neighborhood search approach that designs effective routes and guarantees the feasibility of loading constraints using various strategies, such as lower bound procedures, the open space heuristic, and a constraint programming model. We evaluate the performance of our approach through computational experiments using benchmark and newly created instances. The results indicate that the proposed approach is effective. It achieves improved solutions for 21 out of 60 instances in relatively short computing times when compared to existing methods for the G2L-SDVRP. Furthermore, our approach is competitive on benchmark instances of a related variant, namely the Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints, improving the best-known solutions for 50 out of 180 instances.

An Improved Chemical Reaction Optimisation for the Dynamic CVRP with Two-Dimensional Loading Constraints

An Improved Chemical Reaction Optimisation for the Dynamic CVRP with Two-Dimensional Loading Constraints

Modelling and heuristically solving many-to-many heterogeneous vehicle routing problem with cross-docking and two-dimensional loading constraints

In light of the existing practical applications of the two-dimensional loading on vehicle scheduling and many-to-many supply-demand relationships between suppliers and customers, we address a many-to-many heterogeneous vehicle routing problem with cross-docking and two-dimensional loading constraints (or the heterogeneous 2L-MVRPCD). The newly proposed problem can be regarded as a generalized problem of the many-to-many vehicle routing with cross-docking (MVRPCD) and capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP). To solve small-scale 2L-MVRPCDs, a mixed integer linear programming (MILP) model is developed, whereas two hybrid optimization heuristic algorithms are proposed to solve large-scale 2L-MVRPCDs. The first heuristic incorporates a basic adaptive large neighborhood search (ALNS) algorithm and a new best-fit-skyline (BFS) packing heuristic, while the second heuristic, a Tabu-based ALNS (ALNS/TS), extends the first one by embedding an insert-tabu strategy to enhance the intensity and diversity of search. Wide-ranging instances with various many-to-many supply-demand scenarios and different loading configurations are employed to verify the efficacy of the proposed MILP model and two heuristics. Numerical results show that small-scale heterogeneous 2L-MVRPCD instances with and without the rotation constraints can be solved to optimality by commercial solvers, and the proposed heuristics can achieve high-quality solutions within a reasonable computational time for large-scale instances. Meanwhile, the comparison between the proposed heuristics and existing methods verifies the effectiveness and applicability of the proposed heuristics for solving MVRPCD and 2L-CVRP.

A Branch-and-Price-and-Cut Algorithm for the Vehicle Routing Problem with Two-Dimensional Loading Constraints

The vehicle routing problem with two-dimensional loading constraints (2L-CVRP) is a practical variant of the classic capacitated vehicle routing problem. A number of algorithms have been developed for the problem, but it is very difficult for the existing exact methods to optimally solve instances featuring with large rectangular items. To address this issue, a branch-and-price-and-cut (BPC) algorithm is proposed in this study. A novel data structure and a new dominance rule are developed to build an exact pricing algorithm that takes the loading constraints into account. Several valid inequalities are used to strengthen the linear relaxation. Extensive computational experiments were conducted on the benchmark instances of the 2L-CVRP, showing that the BPC algorithm outperforms all the existing exact methods for the problem in terms of the solution quality. Fourteen instances are solved to optimality for the first time. In particular, the size of solvable instances with large items is nearly doubled. Moreover, managerial insights about the impact of respecting the last-in-first-out constraint are also obtained.

A simulated annealing based heuristic for a location-routing problem with two-dimensional loading constraints

A hybrid heuristic is proposed to solve a location-routing problem with two-dimensional loading constraints. This problem appears in military situations and natural disasters in such a way that decisions are taken in a short time horizon. The proposed heuristic combines the simulated annealing method and the artificial algae algorithm. Simulated annealing is used to handle the location-routing problem, while the artificial algae algorithm is used to determine the sequence in which the items will be packed. Therefore, we apply the Skyline technique to find a feasible packing of such items onto the vehicle’s rectangular surface. As there is no other work in the literature handling the location-routing problem with two-dimensional loading constraints that we can compare the results, we evaluate the heuristic performance on its subproblems: the location-routing problem and the vehicle routing problem with two-dimensional loading constraints. Although the heuristic is not designed for these subproblems, it still obtains competitive results, with an average relative difference of 1.26% and equal or better solutions for more than 90 instances. Regarding the problem under study, the heuristic obtains solutions close to an estimated lower bound for instances having more items per customer.

A branch-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints

In this study, we develop a branch-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints. A separation algorithm is proposed to simultaneously identify infeasible set inequalities and weak capacity inequalities for fractional solutions. Classic capacitated vehicle routing problem (CVRP) inequalities are adopted as well. A branch-and-cut (B&C) algorithm is built to solve the problem. Experimental results illustrate that the algorithm is competitive. In particular, we solve 6 instances to optimality for the first time. Extensive computational analysis is also conducted to reveal the impact of infeasible set inequalities, the branching strategy, and the packing algorithms on the B&C algorithm.

Solving the capacitated vehicle routing problem with two-dimensional loading constraints using a parallel VNS approach

Solving the capacitated vehicle routing problem with two-dimensional loading constraints using a parallel VNS approach

Solving the capacitated vehicle routing problem with two-dimensional loading constraints using a parallel VNS approach

Solving the capacitated vehicle routing problem with two-dimensional loading constraints using a parallel VNS approach

INTEGER FORMULATIONS FOR THE INTEGRATED VEHICLE ROUTING PROBLEM WITH TWO-DIMENSIONAL PACKING CONSTRAINTS

Two integer linear programming models are developed for the unrestricted vehicle routing problem with two-dimensional loading constraints. The first one is a complete model, and the other uses valid inequalities to guarantee that routes are connected and respect the two-dimensional loading constraints. The models are solved with a branch-and-cut algorithm. Computational experiments on benchmark instances showed the complete model has allowed optimal solutions for 5% of the instances, while the second model optimally solved 64% of the instances. Given the superior performance of the second model, we adapted it to handle the sequential variant of the problem, which is harder, and then optimal solutions were obtained for 46% of the instances within the given time limit. The second model compared with a branch-and-cut algorithm from the literature found identical or better solutions for all the instances.

Learning-Based Branch-and-Price Algorithms for the Vehicle Routing Problem with Time Windows and Two-Dimensional Loading Constraints

A capacitated vehicle routing problem with two-dimensional loading constraints is addressed. Associated with each customer are a set of rectangular items, the total weight of the items, and a time window. Designing exact algorithms for the problem is very challenging because the problem is a combination of two NP-hard problems. An exact branch-and-price algorithm and an approximate counterpart are proposed to solve the problem. We introduce an exact dominance rule and an approximate dominance rule. To cope with the difficulty brought by the loading constraints, a new column generation mechanism boosted by a supervised learning model is proposed. Extensive experiments demonstrate the superiority of integrating the learning model in terms of CPU time and calls of the feasibility checker. Moreover, the branch-and-price algorithms are able to significantly improve the solutions of the existing instances from literature and solve instances with up to 50 customers and 103 items. Summary of Contribution: We wish to submit an original research article entitled “Learning-based branch-and-price algorithms for a vehicle routing problem with time windows and two-dimensional loading constraints” for consideration by IJOC. We confirm that this work is original and has not been published elsewhere, nor is it currently under for publication elsewhere. In this paper, we report a study in which we develop two branch-and-price algorithms with a machine learning model injected to solve a vehicle routing problem integrated the two-dimensional packing. Due to the complexity brought by the integration, studies on exact algorithms in this field are very limited. Our study is important to the field, because we develop an effective method to significantly mitigate computational burden brought by the packing problem so that exactness turns to be achievable within reasonable time budget. The approach can be generalized to the three-dimensional case by simply replacing the packing algorithm. It can also be adapted for other VRPs when high-dimensional loading constraints are concerned. Broadly speaking, the study is a typical example of adopting supervised learning to achieve acceleration for operations research algorithms, which expands the envelop of computing and operations research. Hence, we believe this manuscript is appropriate for publication by IJOC.

A Novel Adaptive Genetic Algorithm for Dynamic Vehicle Routing Problem With Backhaul and Two-Dimensional Loading Constraints

In this paper, we consider an extension of the Dynamic Vehicle Routing Problem with Backhauls integrated with two-dimensional loading problem called DVRPB with 2D loading constraints (2L-DVRPB). In the VRPB, a vehicle can deliver (Linehaul) then collect goods from customers (backhaul) and bring back to the depot. Once customer demand is formed by a set of two-dimensional items the problem will be treat as a 2L-VRPB. The 2L-VRPB has been studied on the static case. However, in most real-life application, new customer requests can be happen over time of backhaul and thus perturb the optimal routing schedule that was originally invented. This problem has not been analysed sofar in the literature. The 2L-DVRPB is an NP-Hard problem, so, we propose to use a Genetic algorithm for routing and a packing problems. We applied our approach in a real case study of the Regional Post Office of the city of Jendouba in the North of Tunisia. Results indicate that the AGA approach is considered as the best approach in terms of solutions quality for a real world routing system.

An enhanced neighborhood search algorithm for solving the split delivery vehicle routing problem with two-dimensional loading constraints

The split delivery vehicle routing problem with two-dimensional loading constraints (2L-SDVRP) is a complex practical problem in the field of logistics. It aims at finding the optimal vehicle routes and two-dimensional packing layouts of items in each vehicle to minimize the transportation cost. In this work, we present a mathematical model of 2L-SDVRP with wide-ranging practical considerations. Then we propose an enhanced neighborhood search algorithm (ENS) incorporated with the maximum-space-utilization-based tabu search packing algorithm (MSUTS) to solve the 2L-SDVRP, where the MSUTS examines the feasibility of packing in each route. To accelerate the solving process, a sequence-optimization and random-splitting-based initialization strategy is further proposed. Meanwhile, three classical neighborhood operators are modified to adapt to the 2L-SDVRP with split delivery and two-dimensional packing. The proposed algorithms are tested on a large number of instances varying with scales and properties, which verifies the effectiveness of the proposed algorithm for solving the 2L-SDVRP. Observation of the results also infers the split delivery shows an economic advantage over the non-split case and the advantage becomes significant when the total area of items of each customer increases.

A Multi-Start Algorithm for Solving the Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints

This work presents a multistart algorithm for solving the capacitated vehicle routing problem with 2D loading constraints (2L-CVRP) allowing for the rotation of goods. Research dedicated to graph theory and symmetry considered the vehicle routing problem as a classical application. This problem has complex aspects that stimulate the use of advanced algorithms and symmetry in graphs. The use of graph modeling of the 2L-CVRP problem by undirected graph allowed the high performance of the algorithm. The developed algorithm is based on metaheuristics, such as the Constructive Genetic Algorithm (CGA) to construct promising initial solutions; a Tabu Search (TS) to improve the initial solutions on the routing problem, and a Large Neighborhood Search (LNS) for the loading subproblem. Although each one of these algorithms allowed to solve parts of the 2L-CVRP, the combination of these three algorithms to solve this problem was unprecedented in the scientific literature. In our approach, a parallel mechanism for checking the loading feasibility of routes was implemented using multithreading programming to improve the performance. Additionally, memory structures such as hash-tables were implemented to save time by storing and querying previously evaluated results for the loading feasibility of routes. For benchmarks, tests were done on well-known instances available in the literature. The results proved that the framework matched or outperformed most of the previous approaches. As the main contribution, this work brings higher quality solutions for large-size instances of the pure CVRP. This paper involves themes related to the symmetry journal, mainly complex algorithms, graphs, search strategies, complexity, graph modeling, and genetic algorithms. In addition, the paper especially focuses on topic-related aspects of special interest to the community involved in symmetry studies, such as graph algorithms and graph theory.

An exact approach for the green vehicle routing problem with two-dimensional loading constraints and split delivery

This paper presents a study about the Capacitated Vehicle Routing Problem with Two-Dimensional Loading Constraints (2L-CVRP) and its three variants: allowing split delivery (2L-SDVRP), with green requirements (G2L-CVRP), and integrating split delivery with green requirements (G2L-SDVRP). When considering split delivery, a customer can be served by more than one vehicle. The green variant takes into consideration the CO2 emission. Our objective is to analyze the cost benefits obtained with the aggregation of split delivery and the reduction of CO2 emission. Mathematical models are presented for each variant, and instances are solved with a branch-and-cut approach. We develop a tailored procedure to address the packing subproblem, including the computation of lower bounds, a constructive-based heuristic, and a constraint programming formulation. Computational experiments performed on literature instances and newly created ones show that the proposed approach can outperform previous results. Besides that, the green variant has solutions with low emissions of CO2. The variant with split delivery has solutions with lower cost but at the expense of higher computing time.

An Adaptive Genetic Algorithm for a New Variant of the Gas Cylinders Open Split Delivery and Pickup with Two-dimensional Loading Constraints

This paper studies a combination of two well-known problems in distribution logistics, which are the truck loading problem and the vehicle routing problem. In our context, a customer daily demand exceeds the truck capacity. As a result, the demand has to be split into several routes. In addition, it is required to assign customers to depots, which means that each customer is visited just once by any truck in the fleet. Moreover, we take into consideration a customer time windows. The studied problem can be defined as a Multi-depots open split delivery and pickup vehicle routing problem with two-dimensional loading constraints and time windows (2L-MD-OSPDTW). A mathemat-ical formulation of the problem is proposed as a mixed-integer linear programming model. Then, a set of four class instances is used in a way that reflects the real-life case study. Furthermore, a genetic algorithm is proposed to solve a large scale dataset. Finally, preliminary results are reported and show that the MILP performs very well for small test instances while the genetic algorithm can be efficiently used to solve the problem for a wide-reaching test instances.

A specialized genetic algorithm for the fuel consumption heterogeneous fleet vehicle routing problem with bidimensional packing constraints

The vehicle routing problem combined with loading of goods, considering the reduction of fuel consumption, aims at finding the set of routes that will serve the demands of the customers, arguing that the fuel consumption is directly related to the weight of the load in the paths that compose the routes. This study integrates the Fuel Consumption Heterogeneous Vehicle Routing Problem with Two-Dimensional Loading Constraints (2L-FHFVRP). To reduce fuel consumption taking the associated environmental impact into account is a classical VRP variant that has gained increasing attention in the last decade. The objective of this problem is to design the delivery routes to satisfy the customers’ demands with the lowest possible fuel consumption, which depends on the distances of the paths, the assigned vehicles, the loading/unloading pattern and the load weight. In the vehicle routing problem literature, the approximate algorithms have had great success, especially the evolutionary ones, which appear in previous works with quite a sophisticated structure, obtaining excellent results, but that are difficult to implement and adapt to other variants such as the one proposed here. In this study, we present a specialized genetic algorithm to solve the design of routes, keeping its main characteristic: the easy implementation. By contrast, the loading of goods restriction is validated by means of a GRASP algorithm, which has been widely employed for solving packing problems. With a view of confirming the performance of the proposed methodology, we provide a computational study that uses all the available benchmark instances, allowing to illustrate the savings achieved in fuel consumption. In addition, the methodology suggested can be adapted to the version of solely minimizing the total distance traveled for serving the customers (without the fuel consumption) and it is compared to the best works presented in the literature. The computational results show that the methodology manages to be adequately adapted to this version and it is capable of finding improved solutions for some benchmark instances. As for future work, we propose to adjust the methodology to consider the three-dimensional loading problem so that it adapts to more real-life conditions of the industry.

An Adaptive Memetic Approach for Heterogeneous Vehicle Routing Problems with two-dimensional loading constraints

Abstract The heterogeneous fleet vehicle routing problem with two-dimensional loading constraints (2L- HFVRP) is a complex variant of the classical vehicle routing problem. 2L-HFVRP seeks for minimal cost set of routes to serve a set of customers using a fleet of vehicles of different capacities, fixed and variable operating costs, different dimensions, and restricted loading constraints. To effectively deal with the 2L-HFVRP, we propose a two-stage method that successively calls the routing stage and the packing stage. For the routing stage, we propose an adaptive memetic approach that integrates new multi-parent crossover operators with multi-local search algorithms in an adaptive manner. A time-varying fitness function is proposed to avoid prematurity and improve search performance. An adaptive quality-and-diversity selection mechanism is devised to control the application of the memetic operators and the local search algorithms. In the packing stage, five heuristics are adopted and hybridised to perform the packing process. Experiments on a set of 36 2L-HFVRP benchmark instances demonstrate that the proposed method provides highly competitive results in comparison with state-of-the-art algorithms. In particular, the proposed method obtains the best results for several instances.

A simulated annealing algorithm for the capacitated vehicle routing problem with two-dimensional loading constraints

This paper studies the well-known capacitated vehicle routing problem with two-dimensional loading constraints (2L-CVRP). It requires designing a set of min-cost routes, starting and terminating at the central depot, to satisfy customer demands which involve a set of two-dimensional, rectangular, weighted items. A simulated annealing algorithm with a mechanism of repeatedly cooling and rising the temperature is proposed to solve the four versions of this problem, with or without the LIFO constraint, and allowing rotation of goods or not. An open space based heuristic is employed to identify the feasible loading patterns. In addition, the data structure Trie is used to accelerate the procedure by keeping track of the packing feasibility information of routes examined, and also by controlling the effort spent on different routes. The proposed algorithm is tested on the widely used instances of 2L-CVRP. The results show that our approach outperforms all existing algorithms on the four problem versions, and reaches or improves the best-known solutions for most instances. Furthermore, we compared the impact of different loading constraints, and observed some interesting results.

Uma revisão sistemática da literatura sobre o Problema de Roteirização de Veículos Capacitados com Restrições de Carregamento Bidimensional (2L-CVRP)

Roteirização e carregamento são dois aspectos centrais na gestão de distribuição logística. Recentemente, o pro-blema integrado de roteirização e carregamento tem atraído a atenção da comunidade de pesquisa operacional. Esse interesse se deve não só a seu enorme potencial de aplicação em situações reais, mas também aos desafios metodológicos para sua solução, pois se trata da combinação de dois problemas de elevada complexidade computacional. Este artigo tem por objetivo realizar uma revisão sistemática das principais publicações relacionadas ao Problema de Roteirização de Veículos Capacita-dos com Restrições de Carregamento Bidimensional (Capacitated Vehicle Routing Problem with Two-dimensional Loading Constraints – 2L-CVRP). O intuito é analisar o que foi publicado até o momento e identificar abordagens de solução, carac-terísticas incorporadas ao problema e lacunas de pesquisa que ainda não foram tratadas.