Formulations and exact algorithms for the vehicle routing problem with time windows

In this paper we review the exact algorithms proposed in the last three decades for the solution of the vehicle routing problem with time windows (VRPTW). The exact algorithms for the VRPTW are in many aspects inherited from work on the traveling salesman problem (TSP). In recognition of this fact this paper is structured relative to four seminal papers concerning the formulation and exact solution of the TSP, i.e. the arc formulation, the arc-node formulation, the spanning tree formulation, and the path formulation. We give a detailed analysis of the formulations of the VRPTW and a review of the literature related to the different formulations. There are two main lines of development in relation to the exact algorithms for the VRPTW. One is concerned with the general decomposition approach and the solution to certain dual problems associated with the VRPTW. Another more recent direction is concerned with the analysis of the polyhedral structure of the VRPTW. We conclude by examining possible future lines of research in the area of the VRPTW.     

基于粒子碰撞的粒子群算法求解带时间窗车辆调度问题*

带时间窗车辆调度问题属于离散NP-hard组合优化问题,传统的粒子群算法在离散域上表现了一定的劣性,对此提出了一种基于粒子碰撞的离散PSO算法来求解该问题。受物体相互碰撞之后物体的速度和位置会发生改变的现象启发,使当前粒子与个体最优和全局最优粒子发生碰撞来更新粒子的位置,以避免传统更新操作中的取整,保证种群的进化能力。采用Solomon’s VRP标准问题集的实例来对算法进行测试,实验结果数据表明了该算法的有效性。     

Analysis of OpenMP and MPI implementations of meta-heuristics for vehicle routing problems

The parallelization of heuristic methods allows the researchers both to explore the solution space more extensively and to accelerate the search process. Nowadays, there is an increasing interest on developing parallel algorithms using standard software components that take advantage of modern microprocessors including several processing cores with local and shared cache memories. The aim of this paper is to show it is possible to parallelize algorithms included in computational software using standard software libraries in low-cost multi-core systems, instead of using expensive high-performance systems or supercomputers. In particular, it is analyzed the benefits provided by master-worker and island parallel models, implemented with MPI and OpenMP software libraries, to parallelize population-based meta-heuristics. The capacitated vehicle routing problem with hard time windows (VRPTW) has been used to evaluate the performance of these parallel strategies. The empirical results for a set of Solomon's benchmarks show that the parallel approaches executed on a multi-core processor produce better solutions than the sequential algorithm with respect to both the quality of the solutions obtained and the runtime required to get them. Both MPI and OpenMP parallel implementations are able to obtain better or at least equal solutions (in terms of distance traveled) than the best known ones for the considered benchmark instances.     

蜂群算法在带时间窗的车辆路径问题中的应用*

根据带时间窗车辆路径问题的实际情况,通过考察车辆数和总行程两个目标函数,给出了该问题的一种新的算法——蜂群算法。通过计算若干benchmark问题,并将结果与其他算法相比较分析,验证了算法的有效性。目前关于蜂群算法的文献较少,故不仅是拓宽蜂群算法应用范围的有效尝试,同时也给带时间窗车辆路径问题提供了一种新的解决方法。     

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.     

求解带容量和时间窗约束车辆路径问题的改进蝙蝠算法

带时间窗和容量约束的车辆路径问题是车辆路径问题重要的扩展之一,属于NP难题,精确算法的求解效率较低,且对于较大规模问题难以在有限时间内给出最优解.为了满足企业和客户快速有效的配送需求,使用智能优化算法可以在有限的时间内给出相对较优解.研究了求解带容量和时间窗约束车辆路径问题的改进离散蝙蝠算法,为增加扰动机制,提高搜索速...     

Arcs-states models for the vehicle routing problem with time windows and related problems

This paper presents several Arcs-States models that can be applied to numerous vehicle routing problems, one of which is the well-known vehicle routing problem with capacities and time windows. In these models, the variables correspond to the states (i.e. the resource quantities) of the vehicles when they travel through an arc. The LP relaxation of the problem provides a lower bound that can be embedded in a branch and bound algorithm that solves the problem exactly. However, for the pseudo-polynomial number of variables and constraints of our models, column and row generations have to be used. Generally, in a branch and bound algorithm, the lower bound needs to be very efficient to minimize the size of the branch and bound trees as far as possible. One of the models we present, the time-only, relies on a relaxation of the Arcs-States model where a resource is removed from the states in the variables. Although the quality of the bounds decreases, the global resolution time is greatly improved, as illustrated on instances of Solomon's benchmark.     

Exact and heuristic algorithms for the vehicle routing problem with multiple interdependent time windows

In this paper a model and several solution procedures for a novel type of vehicle routing problems where time windows for the pickup of perishable goods depend on the dispatching policy used in the solution process are presented. This problem is referred to as Vehicle Routing Problem with multiple interdependent time windows (VRPmiTW) and is motivated by a project carried out with the Austrian Red Cross blood program to assist their logistics department. Several variants of a heuristic constructive procedure as well as a branch-and-bound based algorithm for this problem were developed and implemented. Besides finding the expected reduction in costs when compared with the current procedures of the Austrian Red Cross, the results show that the heuristic algorithms find solutions reasonably close to the optimum in fractions of a second. Another important finding is that increasing the number of pickups at selected customers beyond the theoretical minimum number of pickups yields significantly greater potential for cost reductions.     

A planning model and solution algorithm for multi-trip split-delivery vehicle routing and scheduling problems with time windows

This study proposes a daily vehicle routing model for minimizing the total cost of replenishing inventory within a supply chain. The first major contribution of this research is to allow multiple use of vehicles in a split delivery vehicle routing problem with time windows (SDVRPTW), which is more realistic for various real-life applications. The multi-trip SDVRPTW (MTSDVRPTW) is formulated using the time–space network technique, which provides greater flexibility for formulating the complicated interactions between vehicles and products when multi-trip, split delivery, and delivery time windows are simultaneously considered. The resulting formulation of the MTSDVRPTW can be categorized as an integer multi-commodity network flow problem with side constraints. A two-step solution algorithm is proposed to solve this NP-hard problem, which is the second major contribution of this research. Finally, a real-world scale numerical example is performed to demonstrate and to test the methodology. The results indicate that these vehicle routing problems can be solved effectively and efficiently and that the proposed methodology has great potential for inventory replenishment scheduling where split deliveries and multiple trips for a single vehicle are allowed and time window constraints are imposed.     

A genetic algorithm for vehicle routing with backhauling

In this paper, a greedy route construction heuristic for a vehicle routing problem with backhauling is described. This heuristic inserts customers one by one into the routes using a fixed a priori ordering of customers. Then, a genetic algorithm is used to identify an ordering that produces good routes. Numerical comparisons are provided with an exact algorithm and with other heuristic approaches.     

基于Memetic算法的带时间窗车辆路径问题研究*

提出一种模拟文化进化的Memetic算法求解带时间窗的车辆路径问题。设计了一种实数编码方案,将离散的问题转为连续优化问题。采用邻域搜索帮助具备一定学习能力的个体提高寻优速度;采用禁忌搜索帮助部分个体跳出局部最优点,增强全局寻优性能。实验结果表明,该算法可以更有效地求出优化解,是带时间窗车辆路径问题的一种有效求解算法。     

Waste collection vehicle routing problem with time windows

In this paper, we address a real life waste collection vehicle routing problem with time windows (VRPTW) with consideration of multiple disposal trips and drivers’ lunch breaks. Solomon's well-known insertion algorithm is extended for the problem. While minimizing the number of vehicles and total traveling time is the major objective of vehicle routing problems in the literature, here we also consider the route compactness and workload balancing of a solution since they are very important aspects in practical applications. In order to improve the route compactness and workload balancing, a capacitated clustering-based waste collection VRPTW algorithm is developed. The proposed algorithms have been successfully implemented and deployed for the real life waste collection problems at Waste Management, Inc. A set of waste collection VRPTW benchmark problems is also presented in this paper.Waste collection problems are frequently considered as arc routing problems without time windows. However, that point of view can be applied only to residential waste collection problems. In the waste collection industry, there are three major areas: commercial waste collection, residential waste collection and roll-on-roll-off. In this paper, we mainly focus on the commercial waste collection problem. The problem can be characterized as a variant of VRPTW since commercial waste collection stops may have time windows. The major variation from a standard VRPTW is due to disposal operations and driver's lunch break. When a vehicle is full, it needs to go to one of the disposal facilities (landfill or transfer station). Each vehicle can, and typically does, make multiple disposal trips per day. The purpose of this paper is to introduce the waste collection VRPTW, benchmark problem sets, and a solution approach for the problem. The proposed algorithms have been successfully implemented and deployed for the real life waste collection problems of Waste Management, the leading provider of comprehensive waste management services in North America with nearly 26,000 collection and transfer vehicles.     

有时间窗车辆路径问题的捕食搜索算法

有时间窗车辆路径问题是当前物流配送系统研究中的热点问题,该问题具有NP难性质。难以求得最优解或满意解,在建立有时间窗车辆路径问题数学模型的基础上。设计了一种模仿动物捕食策略的捕食搜索算法．该算法利用控制搜索空间的限制大小来实现算法的局域搜索和全局搜索,具有良好的局部集中搜索和跳出局部最优的能力．通过实例计算,并与相关启发式算法比较．取得了满意的结果．     

A hybrid meta-heuristic for multi-objective vehicle routing problems with time windows

The Capacitated Vehicle Routing Problem with Time Windows is an important combinatorial optimization problem consisting in the determination of the set of routes of minimum distance to deliver goods, using a fleet of identical vehicles with restricted capacity, so that vehicles must visit customers within a time frame. A large number of algorithms have been proposed to solve single-objective formulations of this problem, including meta-heuristic approaches, which provide high quality solutions in reasonable runtimes. Nevertheless, in recent years some authors have analyzed multi-objective variants that consider additional objectives to the distance travelled. This paper considers not only the minimum distance required to deliver goods, but also the workload imbalance in terms of the distances travelled by the used vehicles and their loads. Thus, MMOEASA, a Pareto-based hybrid algorithm that combines evolutionary computation and simulated annealing, is here proposed and analyzed for solving these multi-objective formulations of the VRPTW. The results obtained when solving a subset of Solomon’s benchmark problems show the good performance of this hybrid approach.     

An adaptive guidance approach for the heuristic solution of a minimum multiple trip vehicle routing problem

One of the most important problems in combinatorial optimization is the well-known vehicle routing problem (VRP), which calls for the determination of the optimal routes to be performed by a fleet of vehicles to serve a given set of customers. Recently, there has been an increasing interest towards extensions of VRP arising from real-world applications. In this paper we consider a variant in which time windows for service at the customers are given, and vehicles may perform more than one route within a working shift. We call the resulting problem the minimum multiple trip VRP (MMTVRP), where a “multiple trip” is a sequence of routes corresponding to a working shift for a vehicle. The problem objective is to minimize the overall number of the multiple trips (hence the size of the required fleet), breaking ties in favor of the minimum routing cost.     

带时间窗车辆路径问题的分布式多Agent蚁群算法

带时间窗车辆路径问题(VRPTW)多年来一直受到人们关注。针对以往研究中求解效率有限、求解复杂度有限、难以求解较大规模问题的不足,本文以提高精度和速度为目标,在传统蚁群算法的基础上,改进了状态转移规则,结合了邻域搜索算法;同时将本算法设计为分布式结构。利用多分布式agent系统实现了分布式求解VRPTW问题。针对国际标准算例设计了四个实验,结果表明：本算法在精确度、速度、可靠性以及求解大规模问题方面具有明显优势。     

带时间窗口动态车辆路径规划模型及其求解算法

基于事件触发,把带时间窗口动态车辆路径规划问题（DVRPTW）分解成一系列延迟快照,在快照基础上建立相应的动态数学模型,并提出双缓冲区改进大邻域搜索算法进行求解。利用算法的特点,实现新请求无缝插入。采用Solomon设计的56个100节点范例和Lackner相应的动态测试数据,经不同类型动态实例的实验表明,所建立的模型和给出的算法是有效的。     

时间窗约束下的配送车辆调度问题研究

为解决时间窗约束下的物流配送车辆的多目标调度优化问题,给出了一种基于免疫计算的配送车辆调度优化方案。设计了配送车辆调度问题的数学模型和一种基于非劣邻域支配的多目标调度优化算法,在仿真环境下进行了实验。实验结果表明,算法能够有效地解决物流配送车辆调度问题,具有较好的应用价值。     

带时间窗车辆路径问题的多目标文化基因算法

针对物流配送中带时间窗的车辆路径问题,以最小化车辆使用数和行驶距离为目标,建立了多目标数学模型,提出了一种求解该问题的多目标文化基因算法。种群搜索采用遗传算法的进化模式和Pareto排序的选择方式,局部搜索采用禁忌搜索机制和存储池的结构,协调两者得到的Pareto非占优解的关系。与不带局部搜索的多目标遗传算法和单目标文化基因算法的对比实验表明,本文算法的求解质量较高。     

A hybrid evolutionary algorithm for heterogeneous fleet vehicle routing problems with time windows

This paper presents a hybrid evolutionary algorithm (HEA) to solve heterogeneous fleet vehicle routing problems with time windows. There are two main types of such problems, namely the fleet size and mix vehicle routing problem with time windows (F) and the heterogeneous fixed fleet vehicle routing problem with time windows (H), where the latter, in contrast to the former, assumes a limited availability of vehicles. The main objective is to minimize the fixed vehicle cost and the distribution cost, where the latter can be defined with respect to en-route time (T) or distance (D). The proposed unified algorithm is able to solve the four variants of heterogeneous fleet routing problem, called FT, FD, HT and HD, where the last variant is new. The HEA successfully combines several metaheuristics and offers a number of new advanced efficient procedures tailored to handle the heterogeneous fleet dimension. Extensive computational experiments on benchmark instances have shown that the HEA is highly effective on FT, FD and HT. In particular, out of the 360 instances we obtained 75 new best solutions and matched 102 within reasonable computational times. New benchmark results on HD are also presented.