Lifted and Local Reachability Cuts for the Vehicle Routing Problem with Time Windows

The Vehicle Routing Problem with Time Windows (VRPTW) requires to design minimum cost routes for a fleet of vehicles with identical capacities to serve a set of customers within given time windows. Each customer must be visited exactly once and the load of a vehicle must not exceed its capacity.     

Heuristic methods for the fleet size and mix vehicle routing problem with time windows and split deliveries

This paper proposes a scatter-search (SS) approach to solve the Fleet Size and Mixed Vehicle Routing Problem with Time Windows and Split Deliveries (FSMVRPTWSD). In the Vehicle Routing Problem with Split Deliveries (VRPSD), each customer can be served by more than one vehicle, as opposed to the classical VRP in which each customer is served only once. In the FSMVRPTW, the customers must be serviced within their time windows with minimal costs using a heterogeneous fleet. Experimental testing and benchmark examples are used to assess the merit of our proposed procedure. The results show that the proposed heuristics are competitive with the best results found in the literature.     

The min–max split delivery multi-depot vehicle routing problem with minimum service time requirement

The min–max Split Delivery Multi-Depot Vehicle Routing Problem with Minimum Service Time Requirement (min–max SDMDVRP-MSTR) is a variant of the Multi-Depot Vehicle Routing Problem. Each customer requires a specified amount of service time. The service time can be split among vehicles as long as each vehicle spends a minimum amount of service time at a customer. The objective is to minimize the duration of the longest route (where duration is the sum of travel and service times).We develop a heuristic (denoted by MDS) that solves the min–max SDMDVRP-MSTR in three stages: (1) initialize a feasible solution without splits; (2) improve the longest routes by splitting service times; (3) ensure all minimum service time requirements are satisfied. The first stage of MDS is compared to an existing heuristic to solve the min–max Multi-Depot Vehicle Routing Problem on 43 benchmark instances. MDS produces 37 best-known solutions. We also demonstrate the effectiveness of MDS on 21 new instances whose (near) optimal solutions can be estimated based on geometry. Finally, we investigate the savings from split service and the split patterns as we vary the required service times, the average number of customers per route, and the minimum service time requirement.     

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

This paper addresses a recently practical combinatorial problem named Three-Dimensional Loading Capacitated Vehicle Routing Problem, which combines three-dimensional loading problem and vehicle routing problem in distribution logistics. The problem requires a combinatorial optimization of a feasible loading and successive routing of vehicles to satisfy customer demands, where all vehicles must start and finish at a central depot. The goal of this combinatorial problem is to minimize the total transportation cost while serving customers. Despite its clearly practical significance in the real world distribution management, for its high combinatorial complexity, published papers on this problem in literature are very limited.     

Particle Swarm Optimization for the Vehicle Routing Problem with Stochastic Demands

This paper introduces a new hybrid algorithmic approach based on Particle Swarm Optimization (PSO) for successfully solving one of the most popular supply chain management problems, the Vehicle Routing Problem with Stochastic Demands (VRPSD). The VRPSD is a well known NP-hard problem in which a vehicle with finite capacity leaves from the depot with full load and has to serve a set of customers whose demands are known only when the vehicle arrives to them. A number of different variants of the PSO are tested and the one that performs better is used for solving benchmark instances from the literature.     

Vehicle Routing Problem with elementary shortest path based column generation

The usual column generation model for a Vehicle Routing Problem involves an elementary shortest-path subproblem. The worst-case complexity of the known algorithms for this problem being too high, the elementary-path constraint is usually relaxed. Indeed, as each customer must be visited exactly once, the two problems with and without the elementary-path constraint have the same optimal integer solutions. In this article, we propose one theoretical and several practical improvements to the algorithm for elementary paths. We obtain better lower bounds and pruning of the search tree, and these improvements allowed us to find an exact solution to 17 instances of the Solomon benchmark suite which were previously open.     

The Flexible Periodic Vehicle Routing Problem

This paper introduces the Flexible Periodic Vehicle Routing Problem (FPVRP) where a carrier has to establish a distribution plan to serve his customers over a planning horizon. Each customer has a total demand that must be served within the horizon and a limit on the maximum quantity that can be delivered at each visit. A fleet of homogeneous capacitated vehicles is available to perform the services and the objective is to minimize the total routing cost. The FPVRP can be seen as a generalization of the Periodic Vehicle Routing Problem (PVRP) which instead has fixed service frequencies and schedules and where the quantity delivered at each visit is fixed. Moreover, the FPVRP shares some common characteristics with the Inventory Routing Problem (IRP) where inventory levels are considered at each time period and, typically, an inventory cost is involved in the objective function. We present a worst-case analysis which shows the advantages of the FPVRP with respect to both PVRP and IRP. Moreover, we propose a mathematical formulation for the problem, together with some valid inequalities. Computational results show that adding flexibility improves meaningfully the routing costs in comparison with both PVRP and IRP.     

The vehicle routing problem with multiple prioritized time windows: A case study

This paper addresses Multi-objective Vehicle Routing Problem with Multiple Prioritized Time Windows (VRPMPTW) in which the distributer proposes a set of all non-overlapping time windows with equal or different lengths and the customers prioritize these delivery time windows. VRPMPTW aims to find a set of routes of minimal total traveling cost and maximal customer satisfaction (with regard to the prioritized time windows), starting and ending at the depot, in such a way that each customer is visited by one vehicle given the capacity of the vehicle to satisfy a specific demand. This problem is inspired from a real life application. The contribution of this paper lies in its addressing the VRPMPTW from a problem definition, modeling and methodological point of view. We developed a mathematical model for this problem. This model can simply be used for a wide range of applications where the customers have multiple flexible time windows and violation of time windows may drop the satisfaction levels of customers and lead to profit loss in the long term. A Cooperative Coevolutionary Multi-objective Quantum-Genetic Algorithm (CCMQGA) is also proposed to solve this problem. A new local search is designed and used in CCMQGA to reach an appropriate pareto front. Finally, the proposed approach is employed in a real case study and the results of the proposed CCMQGA are compared with the current solution obtained from managerial experience, the results of NSGA-II and the multi-objective quantum-inspired evolutionary algorithm.     

A branch-price-and-cut algorithm for the commodity constrained split delivery vehicle routing problem

We consider the Commodity constrained Split Delivery Vehicle Routing Problem (C-SDVRP), a routing problem where customers may request multiple commodities. The vehicles can deliver any set of commodities and multiple visits to a customer are allowed only if the customer requests multiple commodities. If the customer is visited more than once, the different vehicles will deliver different sets of commodities. Allowing the splitting of the demand of a customer only for different commodities may be more costly than allowing also the splitting of each individual commodity, but at the same time it is easier to organize and more acceptable to customers. We model the C-SDVRP by means of a set partitioning formulation and present a branch-price-and-cut algorithm. In the pricing phase, the ng-path relaxation of a constrained elementary shortest path problem is solved with a label setting dynamic programming algorithm. Capacity cuts are added in order to strengthen the lower bound. We solve to optimality within 2 h instances with up to 40 customers and 3 commodities per customer.     

Hybrid metaheuristics for the Clustered Vehicle Routing Problem

The Clustered Vehicle Routing Problem (CluVRP) is a variant of the Capacitated Vehicle Routing Problem in which customers are grouped into clusters. Each cluster has to be visited once, and a vehicle entering a cluster cannot leave it until all customers have been visited. This paper presents two alternative hybrid metaheuristic algorithms for the CluVRP. The first algorithm is based on an Iterated Local Search algorithm, in which only feasible solutions are explored and problem-specific local search moves are utilized. The second algorithm is a hybrid genetic search, for which the shortest Hamiltonian path between each pair of vertices within each cluster should be precomputed. Using this information, a sequence of clusters can be used as a solution representation and large neighborhoods can be efficiently explored, by means of bi-directional dynamic programming, sequence concatenation, and appropriate data structures. Extensive computational experiments are performed on benchmark instances from the literature, as well as new large scale instances. Recommendations on the choice of algorithm are provided, based on average cluster size.     

The dynamic multiperiod vehicle routing problem with probabilistic information

This paper introduces the Dynamic Multiperiod Vehicle Routing Problem with Probabilistic Information, an extension of the Dynamic Multiperiod Vehicle Routing Problem in which, at each time period, the set of customers requiring a service in later time periods is unknown, but its probability distribution is available. Requests for service must be satisfied within a given time window that comprises several time periods of the planning horizon. We propose an adaptive service policy that aims at estimating the best time period to serve each request within its associated time window in order to reduce distribution costs. The effectiveness of this policy is compared with that of two alternative basic policies through a series of computational experiments.     

求解需求可拆分车辆路径问题的聚类算法

针对需求可拆分车辆路径问题（SDVRP）,提出一种先分组后路径的聚类算法。该算法考虑车辆载重的均衡性和可行解的特征,优先安排载重大于等于车辆限载的客户;然后结合客户间的距离和载重,设定一个拆分阈值限定车辆载重范围,按照就近原则对客户进行聚类分组,当组内客户载重未达到车辆载重最小值而加入新客户后超出限载时,对新加入客户进行拆分和调整,最终完成对所有客户的分组;最后采用蚁群优化算法对各组内客户进行线路规划。实验结果表明,所提算法在求解需求可拆分车辆路径问题时,具有更高的稳定性,得到的结果更优。     

A cooperative coevolutionary algorithm for the Multi-Depot Vehicle Routing Problem

The Multi-Depot Vehicle Routing Problem (MDVRP) is an important variant of the classical Vehicle Routing Problem (VRP), where the customers can be served from a number of depots. This paper introduces a cooperative coevolutionary algorithm to minimize the total route cost of the MDVRP. Coevolutionary algorithms are inspired by the simultaneous evolution process involving two or more species. In this approach, the problem is decomposed into smaller subproblems and individuals from different populations are combined to create a complete solution to the original problem. This paper presents a problem decomposition approach for the MDVRP in which each subproblem becomes a single depot VRP and evolves independently in its domain space. Customers are distributed among the depots based on their distance from the depots and their distance from their closest neighbor. A population is associated with each depot where the individuals represent partial solutions to the problem, that is, sets of routes over customers assigned to the corresponding depot. The fitness of a partial solution depends on its ability to cooperate with partial solutions from other populations to form a complete solution to the MDVRP. As the problem is decomposed and each part evolves separately, this approach is strongly suitable to parallel environments. Therefore, a parallel evolution strategy environment with a variable length genotype coupled with local search operators is proposed. A large number of experiments have been conducted to assess the performance of this approach. The results suggest that the proposed coevolutionary algorithm in a parallel environment is able to produce high-quality solutions to the MDVRP in low computational time.     

An ILP improvement procedure for the Open Vehicle Routing Problem

We address the Open Vehicle Routing Problem (OVRP), a variant of the “classical” (capacitated and distance constrained) Vehicle Routing Problem (VRP) in which the vehicles are not required to return to the depot after completing their service. We present a heuristic improvement procedure for OVRP based on Integer Linear Programming (ILP) techniques. Given an initial feasible solution to be possibly improved, the method follows a destruct-and-repair paradigm, where the given solution is randomly destroyed (i.e., customers are removed in a random way) and repaired by solving an ILP model, in the attempt of finding a new improved feasible solution. The overall procedure can be considered as a general framework which could be extended to cover other variants of Vehicle Routing Problems. We report computational results on benchmark instances from the literature. In several cases, the proposed algorithm is able to find the new best known solution for the considered instances.     

An iterated local search for the multi-commodity multi-trip vehicle routing problem with time windows

The Multi-Commodity Multi-Trip Vehicle Routing Problem with Time Windows calls for the determination of a routing planning to serve a set of customers that require products belonging to incompatible commodities. Two commodities are incompatible if they cannot be transported together into the same vehicle. Vehicles are allowed to perform several trips during the working day. The objective is to minimize the number of used vehicles.We propose an Iterated Local Search that outperforms the previous algorithm designed for the problem. Moreover, we conduct an analysis on the benefit that can be obtained introducing the multi-trip aspect at the fleet dimensioning level. Results on classical VRPTW instances show that, in some cases, the fleet can be halved.     

A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem

This paper presents a generic template-based solution framework and its application to the so-called Consistent Vehicle Routing Problem (ConVRP). The ConVRP is an NP-hard combinatorial optimization problem and involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over multiple days. Customers may receive service either once or with a predefined frequency; however frequent customers must receive consistent service, i.e., must be visited by the same driver over approximately the same time throughout the planning period. The proposed solution framework adopts a two-level master-slave decomposition scheme. Initially, a master template route schedule is constructed in an effort to determine the service sequence and assignment of frequent customers to vehicles. On return, the master template is used as the basis to design the actual vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search improvement method is employed that operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Computational experiments on benchmark data sets illustrate the competitiveness of the proposed approach compared to existing results.     

需求可拆分车辆路径问题的禁忌搜索算法

为解决实际配送运输中的车辆路径问题（Vehicle Routing Problem,VRP）,通过改进传统的数学模型,解除每个客户需求只能由l辆车配送的约束,建立改进的可拆分车辆路径问题（Split Delivery VRP,SDVRP）数学模型,并利用禁忌搜索算法（Taboo Search Algorithm,TSA）进行求解．在TSA的设计中,根据SDVRP模型的特点对初始解、邻域搜索和解的评价等进行特殊处理．算例表明,该模型不仅可以解决VRP模型中不允许配送点需求量超出装载量的限制,而且通过相应配送点需求量的拆分和重新组合,可节省车辆数目、缩短路线长度、提高车辆装载率．     

多物流中心共同配送的车辆路径问题研究

研究多物流中心共同配送的车辆路径问题。首先考虑客户服务关系变化与客户需求的异质性情况,设计一种共享客户需求、配送车辆与物流中心的共享物流模式;再综合考虑车辆容量、油耗、碳排放、最长行驶时间、客户需求量与服务时间等因素,以总成本最小为目标构建多物流中心共同配送的车辆路径规划模型,并设计一种改进蚁群算法进行求解;最后采用多类型算例进行仿真实验,结果表明共享物流模式能有效避免交叉配送与迂回运输等不合理现象,降低物流成本,缩短车辆行驶距离,减少车辆碳排放,促进物流与环境的和谐发展。     

有数量限制的开放式车辆路径加速算法

设计有数量限制的开放式车辆路径加速禁忌搜索算法,将所有点(包括客户和仓库)做Delaunay三角剖分后,限制问题的解的大多数边与Delaunay三角剖分的边重合。实验结果表明,该算法在保证寻求到相对较优解的前提下,执行速度得到大幅度的提升,解与上界关联紧密,可以应用到其他启发式搜索问题的求解中。     

An iterated local search heuristic for the split delivery vehicle routing problem

This paper concerns the Split Delivery Vehicle Routing Problem (SDVRP). This problem is a relaxation of the Capacitated Vehicle Routing Problem (CVRP) since the customers׳ demands are allowed to be split. We deal with the cases where the fleet is unlimited (SDVRP-UF) and limited (SDVRP-LF). In order to solve them, we implemented a multi-start Iterated Local Search (ILS) based heuristic that includes a novel perturbation mechanism. Extensive computational experiments were carried out on benchmark instances available in the literature. The results obtained are highly competitive, more precisely, 55 best known solutions were equaled and new improved solutions were found for 243 out of 324 instances, with an average and maximum improvement of 1.15% and 2.81%, respectively.