Two-dimensional Loading Research Articles

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.

Nonlinear dynamic analysis of the deep-water bridge piers under combined earthquakes and wave actions

When the Stokes fifth-order wave is utilized to study the wave force of large-diameter bridge piers, the iterative method is usually utilized to solve the transcendental equations and then solve the wave parameter wavelength, coefficient and wave number. The solving process is particularly complicated, and the ratio of water depth and wavelength is generally assumed to be 0.1–0.15, which is suitable for shallow water depths and has not been investigated for deep water depths. In this study, the wave force and action point of the large-diameter rectangular section bridge pier under Stokes fifth-order wave action are derived, and the range of Stokes wave number is derived when the deep water level is constant. Combined with the functional analysis of transcendental equations, a program for calculating the Stokes fifth-order wave parameters is derived, and the approximate solution of the coefficient and wavelength of the Stokes wave is accurately and quickly solved. The dynamic nonlinear analysis of super deep-water large-diameter bridge pier under combined multi-dimensional earthquake and nonlinear wave action is investigated. It is found that nonlinear waves can significantly increase the dynamic response of deep-water bridge pier structure, but under combined wave and earthquake loadings on the deep-water bridge pier structure, and earthquakes play a dominant role. The influence of resonant waves and common linear waves on the dynamic response of large-diameter bridge piers in deep water depth is quite different. Resonant waves have a significant influence on the displacement response and stress response of deep-water bridge piers. When under two-dimensional earthquake and resonant wave independent loadings, the influence on the dynamic response of deep-water bridge pier structures is insignificant. When under combined earthquakes and waves loadings on the deep-water bridge pier structure, earthquakes play a significant role.

Horizontal bearing capacity of composite bucket foundation in clay: A case study

Failure of offshore wind power structural system is an important research topic in many countries. Most previous researches focus on the tower failures and blade failures rather than foundation failure. Taking the 3 MW of the composite bucket foundation (CBF) in China as a case study, a comprehensive study of the horizontal bearing response of CBF were carried out by a centrifuge model test and a series of numerical simulations. Results show that the horizontal bearing capacity of CBF increase at first and then decrease with the increase of the existing vertical load in two-dimensional loading space. The relationship between bending moment and horizontal load in two-dimensional loading space is approximate linear. The aspect ratio has influence on the capacity envelopes peak in horizontal-vertical load loading space. Three-dimensional envelope surface is obtained, the results provide a basis for the design of CBF of offshore wind turbine.

Load Spectrum Compilation Method of Hybrid Electric Vehicle Reducers Based on Multi-Criteria Decision Making

This article proposes a method for compiling the load spectra of reducers for hybrid electric vehicles. Selecting typical working conditions for real vehicle data collection, the load data under each typical working condition were divided into five categories according to the state of the power source and the data were preprocessed. The optimal sample loads for compiling load spectra were obtained based on a multi-criteria decision-making method, rainflow counting for optimal sample loads was performed according to different power source output patterns, non-parametric extrapolation was performed to obtain the full-life two-dimensional load spectrum after dimensionality reduction, and a full-life eight-level programmed load spectrum that could be used for bench tests was obtained. Using the programmed load spectrum and the extracted sample load as the load input, a fatigue life prediction simulation of the reducer gear of a hybrid electric vehicle was carried out. The reducer gear fatigue life from the programmed load spectrum was compared to the gear fatigue life under actual load. The fatigue life of the reducer gear when the programmed load spectrum was used as the input was 1.412 × 103. When the actual load was used as the input load, the fatigue life of the reducer gear was 1.933 × 103. The relative error between the two is only 26%, which is in the normal range. The results show that the programmed load spectrum is effective and reliable and that the load spectrum compilation method provides a basis for accurately evaluating the reliability of the hybrid electric vehicle reducer.

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.

Mathematical Model and Simulated Annealing Algorithm for the Two-Dimensional Loading Heterogeneous Fixed Fleet Vehicle Routing Problem

This paper addresses the two-dimensional loading heterogeneous fixed fleet vehicle routing problem, which is a complex and unstudied variant of the classical vehicle routing problem and has a wide range of applications in transportation and logistics fields. In this problem, each customer demands a set of rectangular two-dimensional items, and the objective is to find the minimum cost delivery routes for a limited set of vehicles with different capacities, fixed and variable operating costs, and rectangular two-dimensional loading surfaces. We formulate a mixed integer linear programming model to obtain optimal solutions for small-scale problems. To obtain solutions for large-scale problems, we develop an algorithm based on simulated annealing and local search, which uses a collection of packing heuristics to address the loading constraints, and we also propose three new heuristics. We conduct experiments on benchmark instances derived from the two-dimensional loading heterogeneous fleet vehicle routing problem. The results indicate that the proposed model correctly describes the problem and can solve small-scale problems, that the new packing heuristics are effective in improving the collection of packing heuristics, and that the proposed simulated annealing algorithm can find good solutions to large-scale problems within an acceptable computational time. Hence, it can be used by logistic companies using a heterogeneous fixed fleet in the integrated planning of vehicle loading and routing.

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.

Optimizing two-dimensional vehicle loading and dispatching decisions in freight logistics

This paper introduces a multi-period, two-dimensional vehicle loading and dispatching problem, called Two-Dimensional Vehicle Loading and Dispatching Problem with Incompatibility Constraints (VLDP). The problem concerns preparing a single-origin single-destination transportation plan of loading required orders to vehicles at the origin and dispatching the vehicles to deliver the orders to the destination within their due dates. The decision maker uses their own fleet of vehicles, with each vehicle having a fixed transportation cost per trip, and may outsource additional vehicles at a higher cost. VLDP involves constraints regarding the due dates of the orders, pairwise incompatibility of orders packed in the same vehicle, incompatibility of orders and vehicles, as well as area and weight capacity of the vehicles. An order can be delivered earlier than its due date, incurring an earliness penalty due to storage requirements at the destination. The objective is to minimize the total vehicle usage and earliness penalty costs. A Mixed-Integer Linear Programming model (MILP) is provided, as well as an Adaptive Large Neighbourhood Search (ALNS) algorithm. Results of computational experiments on instances derived from real-world data show the effectiveness of the ALNS algorithm.

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.

Learning Whale Optimization Algorithm for Open Vehicle Routing Problem with Loading Constraints

This paper addresses the two-dimensional loading open vehicle routing problem with time window (2L-OVRPTW). We propose a learning whale optimization algorithm (LWOA) to minimize the total distance; an improved skyline filling algorithm (ISFA) is designed to solve the two-dimensional loading problem. In LWOA, the whale optimization algorithm is used to search the solution space and get the high-quality solution. Then, by learning and accumulating the block structure and customer location information in the high-quality solution individuals, a three-dimensional matrix is designed to guide the updating of the population. Finally, according to the problem characteristics, the local search method based on fleet and vehicle is designed and performed on the high-quality solution region. IFSA is used to optimize the optimal individual. The computational results show that the proposed algorithm can effectively solve 2L-OVRPTW.

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 peridynamic approach for modeling of two dimensional functionally graded plates

This study investigates the crack initiation and its progression in two-dimensional functionally graded (FG) plates under dynamic and quasi-static loading conditions by using an ordinary state-based PeriDynamic (OSB PD) theory. Functionally gradient materials (FGMs) are a new class of advanced materials that provide a smooth transition among the layers of materials to meet the desired requirements in engineering structures. The effective material properties in the FG plate were functionally tailored in two directions by employing a rule of mixture. The present OSB PD theory is very suitable for the failure analysis of materials without the use of surface and volume correction factors. The robustness of the present approach on monitoring the mixed-mode fractures is established with the experimental results under various loading conditions. The PD predictions successfully captured the critical load levels and damage evolutions. Subsequently, numerical analyses were carried out to assess the influence of the one- and two-dimensional material variations, boundary, and loading conditions on the fracture behavior of the FG plates. The compositional gradient exponent played a major role on the reaction force levels and crack trajectories. It was observed that two-dimensional material variations paved the way for increasing the plate strength and fracture resistance.

One novel dynamical calibration method to identify two-dimensional distributed load

For dynamic load identification, the identification of distributed load can be converted to that of the orthogonal polynomial coefficients by introducing the generalized orthogonal polynomial. Thus, the dynamic calibration technology is constructed to study the relationship between the system responses and the orthogonal polynomial coefficients to identify the load. The traditional dynamic calibration method, that is simulation dynamic calibration, is applied based on the finite element model, which has the model error that leads to the low identification accuracy of the dynamic load. Therefore, in this paper, we propose a novel dynamical calibration method based on the orthogonal polynomials and the Gauss-Legendre integral to reconstruct the distributed load in frequency-domain. This method is introduced in detail by determining the Gauss-points appropriately and establishing the dynamic calibration matrix of elastic thin plate structure. Thus, the influence of the model error on load identification accuracy can be eliminated to improve the identification accuracy. To verify the effectiveness and accuracy, two simulation examples are studied to demonstrate the advantage of the proposed method compared with the simulation dynamic calibration. Moreover, the experiment is also performed to further validate the feasibility, and the results show that this method is effective and can achieve high accuracy for distributed load identification.

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.

Sexual dimorphism in bat wing morphology — variation among foraging styles

Sexual dimorphism can lead to differences in foraging style among conspecifics due to morphological differences. Within bats, maneuverability and speed of flight are influenced by wing shape and size, which may differ between sexes. Female bats gain about 30% of their body mass during pregnancy, affecting their agility and flight efficiency. To fill the same foraging niche as males, pregnant female bats would require wing size and (or) shape modifications to maintain maneuverability. We investigated sexual dimorphism in bat wing morphology and how it varies among foraging guilds. Wing photos of male and female adult bats (19 species) in Canada, Belize, and Dominica were analyzed using two-dimensional geometric morphometrics, wing loading, and aspect ratios. Nonpregnant female bats had higher wing loading than males, suggesting that they are less maneuverable than males. Additionally, mass increases during pregnancy may not permit female bats to forage as male conspecifics do. Wing shape differed minimally among foraging guilds with only frugivores differing significantly from all other guilds. Further studies should investigate how female bats forage during their reproductive cycle and determine how frugivore wings differ and whether there are individual differences in wing shape that are not consistent among bat species.