A GRASP×ELS for the vehicle routing problem with basic three-dimensional loading constraints

This paper addresses an extension of the capacitated vehicle routing problem where the client demand consists of three-dimensional weighted items (3L-CVRP). The objective is to design a set of trips for a homogeneous fleet of vehicles based at a depot node which minimizes the total transportation cost. Items in each vehicle trip must satisfy the three-dimensional orthogonal packing constraints. This problem is strongly connected to real-life transportation systems where the packing of items to be delivered by each vehicle can have a significant impact on the routes. We propose a new way to solve the packing sub-problem. It consists of a two-step procedure in which the z-constraints are first relaxed to get a (x,y) positioning of the items. Then, a compatible z-coordinate is computed to get a packing solution. Items can be rotated but additional constraints such as item fragility, support and LIFO are not considered. This method is included in a GRASP×ELS hybrid algorithm dedicated to the computation of VRP routes. The route optimization alternates between two search spaces: the space of VRP routes and the space of giant trips. The projection from one to the other is done by dedicated procedures (namely the Split and the concatenation algorithms). Moreover, a Local Search is defined on each search space. Furthermore, hash tables are used to store the result of the packing checks and thus save a substantial amount of CPU time. The effectiveness of our approach is illustrated by computational experiments on 3L-CVRP instances from the literature. A new set of realistic instances based on the 96 French districts are also proposed. They range from 19 nodes for the small instances to 255 nodes for the large instances and they can be stated as realistic since they are based on true travel distances in kilometers between French cities. The impact of the hash tables is illustrated as well.     

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

This paper addresses an extension of the capacitated vehicle routing problem where customer demand is composed of two-dimensional weighted items (2L-CVRP). The objective consists in designing a set of trips minimizing the total transportation cost with a homogenous fleet of vehicles based on a depot node. Items in each vehicle trip must satisfy the two-dimensional orthogonal packing constraints. A GRASP×ELS algorithm is proposed to compute solutions of a simpler problem in which the loading constraints are transformed into resource constrained project scheduling problem (RCPSP) constraints. We denote this relaxed problem RCPSP-CVRP. The optimization framework deals with RCPSP-CVRP and lastly RCPSP-CVRP solutions are transformed into 2L-CVRP solutions by solving a dedicated packing problem. The effectiveness of our approach is demonstrated through computational experiments including both classical CVRP and 2L-CVRP instances. Numerical experiments show that the GRASP×ELS approach outperforms all previously published methods.     

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

The capacitated vehicle routing problem with three-dimensional loading constraints combines capacitated vehicle routing and three-dimensional loading with additional packing constraints concerning, for example, unloading operations. An efficient hybrid algorithm including a tabu search algorithm for routing and a tree search algorithm for loading is introduced. Computational results are presented for all publicly available test instances. Most of the best solutions previously reported in literature have been improved while the computational effort is drastically reduced compared to other methods.     

Bilevel model for production–distribution planning solved by using ant colony optimization

This paper addresses a hierarchical production–distribution planning problem. There are two different decision makers controlling the production and the distribution processes, respectively, that do not cooperate because of different optimization strategies. The distribution company, which is the leader of the hierarchical process, controls the allocation of retailers to each depot and the routes which serve them. In order to supply items to retailers, the distribution company orders from the manufacturing company the items which have to be available at the depots. The manufacturing company, which is the follower of the hierarchical process, reacts to these orders deciding which manufacturing plants will produce them. A bilevel program is proposed to model the problem and an ant colony optimization based approach is developed to solve the bilevel model. In order to construct a feasible solution, the procedure uses ants to compute the routes of a feasible solution of the associated multi-depot vehicle route problem. Then, under the given data on depot needs, the corresponding production problem of the manufacturing company is solved. Global pheromone trail updating is based on the leader objective function, which involves costs of sending items from depots to retailers and costs of acquiring items from manufacturing plants and unloading them into depots. A computational experiment is carried out to analyze the performance of the algorithm.     

A two-stage tabu search algorithm with enhanced packing heuristics for the 3L-CVRP and M3L-CVRP

The Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) addresses practical constraints frequently encountered in the freight transportation industry. In this problem, the task is to serve all customers using a homogeneous fleet of vehicles at minimum traveling cost. The constraints imposed by the three-dimensional shape of the goods, the unloading order, item fragility, and the stability of the loading plan of each vehicle are explicitly considered. We improved two well-known packing heuristics, namely the Deepest-Bottom-Left-Fill heuristic and the Maximum Touching Area heuristic, for the three-dimensional loading sub-problem and provided efficient implementations. Based on these two new heuristics, an effective tabu search algorithm is given to address the overall problem. Computational experiments on publicly available test instances show our new approach outperforms the current best algorithms for 20 out of 27 instances. Our approach is also superior to the existing algorithm on benchmark data for the closely related problem variant M3L-CVRP (which uses a slightly different unloading order constraint compared to 3L-CVRP).     

A branch-and-cut algorithm for the hub location and routing problem

We study the hub location and routing problem where we decide on the location of hubs, the allocation of nodes to hubs, and the routing among the nodes allocated to the same hubs, with the aim of minimizing the total transportation cost. Each hub has one vehicle that visits all the nodes assigned to it on a cycle. We propose a mixed integer programming formulation for this problem and strengthen it with valid inequalities. We devise separation routines for these inequalities and develop a branch-and-cut algorithm which is tested on CAB and AP instances from the literature. The results show that the formulation is strong and the branch-and-cut algorithm is able to solve instances with up to 50 nodes.     

A tabu search algorithm for the vehicle routing problem with simultaneous pick-up and delivery service

The vehicle routing problem with simultaneous pick-up and delivery (VRP_SPD) is a variant of the classical vehicle routing problem (VRP) where clients require simultaneous pick-up and delivery service. Deliveries are supplied from a single depot at the beginning of the vehicle's service, while pick-up loads are taken to the same depot at the conclusion of the service. One important characteristic of this problem is that a vehicle's load in any given route is a mix of pick-up and delivery loads.     

Exact algorithms for the double vehicle routing problem with multiple stacks

The Double Traveling Salesman Problem with Multiple Stacks (DTSPMS) is a one-to-one pickup-and-delivery single-vehicle routing problem with backhaul deliveries. The vehicle carries a container divided into stacks of fixed height, each following a Last-In-First-Out policy, and the aim is to perform pickups and deliveries by minimizing the total routing cost and ensuring a feasible loading/unloading of the vehicle.A realistic generalization of the DTSPMS arises when a single vehicle is not enough to collect all products, and therefore multiple, and possibly heterogeneous vehicles are needed to perform the transportation operations. This paper introduces and formulates this generalization, that we refer as the Double Vehicle Routing Problem with Multiple Stacks. It proposes three models, the first one based on a three-index formulation and solved by a branch-and-cut algorithm, and the other two based on two set partitioning formulations using different families of columns and solved by a branch-and-price and a branch-and-price-and-cut algorithm, respectively.The performance of these algorithms has been studied on a wide family of benchmark test instances, observing that, although the branch-and-cut algorithm shows a better performance on instances with a small number of vehicles, the performance of the branch-and-price and the branch-and-price-and-cut algorithms improves as the number of vehicles grows. Additionally, the first set partitioning formulation yields tighter lower bounds, but the second formulation, because of its simplicity, provides better convergence properties, solving instances with up to fifty vertices to proven optimality.     

Heuristic and exact algorithms for a min–max selective vehicle routing problem

In this work, we investigate a vehicle routing problem where not all clients need to be visited and the goal is to minimize the longest vehicle route. We propose two exact solution approaches for solving the problem: a Branch-and-cut (BC) algorithm and a Local Branching (LB) method that uses BC as its inner solver. Our computational experience indicates that, in practice, the problem is difficult to solve, mainly when the number of vehicles grows. In addition to the exact methods, we present a heuristic that relies on GRASP and on the resolution of a restricted integer program based on a set covering reformulation for the problem. The heuristic was capable of significantly improving the best solutions provided by BC and LB, in one tenth of the times taken by them to achieve their best upper bounds.     

用于求解混合车辆路径问题的混合进化算法

文中研究了具有NP难度的混合车辆路径问题(Mixed Capacitated General Routing Problem,MCGRP),其是在基本车辆路径问题(Vehicle Routing Problem,VRP)的基础上通过添加限载容量约束及弧上的用户需求而衍生的。给定一列车辆数不限的车队,使车辆从站点出发向用户提供服务,服务完用户需求后仍返回站点;规定每辆车的总载重不能超过其载重量,且每个需求只能被一辆车服务且仅服务一次。MCGRP旨在求解每辆车的服务路线,使得在满足以上约束条件的情况下所有车辆的旅行消耗之和最小。混合车辆路径问题具有较高的理论价值和实际应用价值,针对该问题提出了一种高效的混合进化算法。该算法采用基于5种邻域算符的变邻域禁忌搜索来提高解的质量,并通过一种基于路径的交叉算符来继承解的优异性,从而有效地加速算法的收敛。在一组共计23个经典算例上的实验结果表明,该混合进化算法在求解混合车辆路径问题时是非常高效的。     

Solving school bus routing using the multiple vehicle traveling purchaser problem: A branch-and-cut approach

School bus routing problems, combining bus stop selection and bus route generation, look simultaneously for a set of bus stops to pick up students from among a group of potential locations, and for bus routes to visit the selected stops and carry the students to their school. These problems, classified as Location-Routing problems, are of interest in densely populated urban areas.This article introduces a generalization of the vehicle routing problem called the multi-vehicle traveling purchaser problem, modeling a family of routing problems combining stop selection and bus route generation. It discusses a Mixed Integer Programming formulation extending previous studies on the classical single vehicle traveling purchaser problem. The proposed model is based on a single commodity flow formulation combining continuous variables with binary variables by means of coupling constraints. Additional valid inequalities are proposed with the purpose of strengthening its Linear Programming relaxation. These valid inequalities are obtained by projecting out the flow variables.We develop a branch-and-cut algorithm that makes use of the proposed model and valid inequalities. This cutting plane algorithm is implemented and tested on a large family of symmetric and asymmetric instances derived from randomly generated problems, showing the usefulness of the proposed valid inequalities.     

A generalized exchange heuristic for the capacitated vehicle routing problem

We consider the problem of dispatching the minimum number of vehicles from a central depot to make deliveries to a set of clients with known demands. The objective is to minimize the total distance travelled, subject to vehicle capacity requirements. We present a new heuristic algorithm for solving this problem. The algorithm is based on generalized edge-exchange search procedures, and relaxation of the capacity requirements. Computational results, based upon standard test problems with up to 249 customers, indicate that our heuristic compares favourably with known heuristics in terms of solution quality.     

Variable neighbourhood search for the variable sized bin packing problem

The variable sized bin packing problem is a generalisation of the one-dimensional bin packing problem. Given is a set of weighted items, which must be packed into a minimum-cost set of bins of variable sizes and costs. This problem has practical applications, for example, in packing, transportation planning, and cutting. In this work we propose a variable neighbourhood search metaheuristic for tackling the variable sized bin packing problem. The presented algorithm can be seen as a hybrid metaheuristic, because it makes use of lower bounding techniques and dynamic programming in various algorithmic components. An extensive experimentation on a diverse set of problem instances shows that the proposed algorithm is very competitive with current state-of-the-art approaches.     

Agile optimization of a two‐echelon vehicle routing problem with pickup and delivery

In this paper, we consider a vehicle routing problem in which a fleet of homogeneous vehicles, initially located at a depot, has to satisfy customers' demands in a two‐echelon network: first, the vehicles have to visit intermediate nodes (e.g., a retail center or a consolidation center), where they deliver raw materials or bulk products and collect a number of processed items requested by the customers in their route; then, the vehicles proceed to complete their assigned routes, thus delivering the processed items to the final customers before returning to the depot. During this stage, vehicles might visit other intermediate nodes for reloading new items. In some real‐life scenarios, this problem needs to be solved in just a few seconds or even milliseconds, which leads to the concept of “agile optimization.” This might be the case in some rescue operations using drones in humanitarian logistics, where every second can be decisive to save lives. In order to deal with this real‐time two‐echelon vehicle routing problem with pickup and delivery, an original constructive heuristic is proposed. This heuristic is able to provide a feasible and reasonably good solution in just a few milliseconds. The constructive heuristic is extended into a biased‐randomized algorithm using a skewed probability distribution to modify its greedy behavior. This way, parallel runs of the algorithm are able to generate even better results without violating the real‐time constraint. Results show that the proposed methodology generates competitive results in milliseconds, being able to outperform other heuristics from the literature.     

Artificial intelligence heuristics in solving vehicle routing problems with time window constraints

This paper describes the authors’ research on various heuristics in solving vehicle routing problem with time window constraints (VRPTW) to near optimal solutions. VRPTW is NP-hard problem and best solved to near optimum by heuristics. In the vehicle routing problem, a set of geographically dispersed customers with known demands and predefined time windows are to be served by a fleet of vehicles with limited capacity. The optimized routines for each vehicle are scheduled as to achieve the minimal total cost without violating the capacity and time window constraints. In this paper, we explore different hybridizations of artificial intelligence based techniques including simulated annealing, tabu search and genetic algorithm for better performance in VRPTW. All the implemented hybrid heuristics are applied to solve the Solomon's 56 VRPTW with 100-customer instances, and yield 23 solutions competitive to the best solutions published in literature according to the authors’ best knowledge.     

An exact algorithm for a Vehicle-and-Driver Scheduling Problem

This article introduces a combinatorial optimization problem that consists of assigning tasks to machines and operators, and sequencing the tasks assigned to each one. Two configurations exist. Machines alternate configurations, while the operators must start and finish the process in the same configuration. Moreover, machines and operator have limited capacities. The sequencing of the tasks must guarantee that each one is performed by a machine and an operator at the same time, and it is determined in order to minimize an overall cost function. Two critical aspects of the problem are the need of synchronizing the machine and the operator performing each task, and the need of minimizing the changeovers, which are pairs of tasks done consecutively by the same machine but by different operators. The problem is modeled as a vehicle routing problem with two types of vehicles and with two depots. We propose a mixed integer programming formulation, and introduce valid inequalities to strengthen its linear programming relaxation. We describe separation routines for these inequalities and design a branch-and-cut algorithm for the problem. The algorithm is tested on a set of benchmark instances showing that it is able to solve to optimality instances with up to 50 customers.     

A hybrid genetic algorithm for multi-depot homogenous locomotive assignment with time windows

This paper presents a hybrid genetic algorithm to solve a multi-depot homogenous locomotive assignment problem with time windows. The locomotive assignment problem is to assign a set of homogeneous locomotives locating in a set of dispersed depots to a set of pre-schedules trains that are supposed to be serviced in pre-specified hard/soft time windows. A mathematical model is presented, using vehicle routing problem with time windows (VRPTW) for formulation of the problem. A cluster-first, route-second approach is used to inform the multi-depot locomotive assignment to a set of single depot problems and after that we solve each problem independently. Each single depot problem is solved heuristically by a hybrid genetic algorithm that in which Push Forward Insertion Heuristic (PFIH) is used to determine the initial solution and λ-interchange mechanism is used for neighborhood search and improving method. A medium sized numerical example with different scenarios is presented and examined to more clarification of the approach as well as to check capabilities of the model and algorithm. Also some of the results are compared with the solutions produced by branch & bound technique to determine validity and quality of the model. The experiments with a set of 15 completely random generated instance problems indicate that this algorithm is efficient and solves the problem in a polynomial time.     

Savings based ant colony optimization for the capacitated minimum spanning tree problem

The problem of connecting a set of client nodes with known demands to a root node through a minimum cost tree network, subject to capacity constraints on all links is known as the capacitated minimum spanning tree (CMST) problem. As the problem is NP-hard, we propose a hybrid ant colony optimization (ACO) algorithm to tackle it heuristically. The algorithm exploits two important problem characteristics: (i) the CMST problem is closely related to the capacitated vehicle routing problem (CVRP), and (ii) given a clustering of client nodes that satisfies capacity constraints, the solution is to find a MST for each cluster, which can be done exactly in polynomial time. Our ACO exploits these two characteristics of the CMST by a solution construction originally developed for the CVRP. Given the CVRP solution, we then apply an implementation of Prim's algorithm to each cluster to obtain a feasible CMST solution. Results from a comprehensive computational study indicate the efficiency and effectiveness of the proposed approach.     

考虑卸载顺序约束的成品油二次配送车辆路径问题

研究考虑卸载顺序约束的成品油二次配送车辆路径问题,已知油库使用容量有限的多隔舱运输槽车为加油站配送多种类型的成品油,每个隔舱只能装载一种特定的油品,且装载到各个隔舱中的油品具有固定的卸载顺序.已知加油站对各种油品的需求量,假设每个加油站对同一种油品的需求只能由一辆车配送,考虑配送车辆的固定动用成本和运输成本,以总配送成本极小化为目标建立该问题的混合整数规划模型,并设计求解模型的自适应大邻域搜索启发式算法.利用不同规模的算例进行模拟计算与分析,从而验证算法的有效性.实验结果显示:对于小规模算例,大邻域搜索启发式算法能够以较高的概率得到全局最优解;对于中、大规模算例,大邻域搜索启发式算法可以在短时间内得到近似最优解,近似比不超过1.2.所提出的模型和算法可为石油公司制定成品油二次配送计划提供理论依据和决策支持.     

Scalable cache invalidation algorithms for mobile data access

In this paper, we address the problem of cache invalidation in mobile and wireless client/server environments. We present cache invalidation techniques that can scale not only to a large number of mobile clients, but also to a large number of data items that can be cached in the mobile clients. We propose two scalable algorithms: the Multidimensional Bit-Sequence (MD-BS) algorithm and the Multilevel Bit-Sequence (ML-BS) algorithm. Both algorithms are based on our prior work on the Basic Bit-Sequences (BS) algorithm. Our study shows that the proposed algorithms are effective for a large number of cached data items with low update rates. The study also illustrates that the algorithms ran be used with other complementary techniques to address the problem of cache invalidation for data items with varied update and access rates.