A dynamic vehicle routing problem with time-dependent

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车辆调度算法研究及其应用文献综述

文献综述车辆调度算法研究及其应用一、前言局部车辆调度问题是现代物流系统优化中关键的一环，也是开展电子商务不可缺少的内容。

对车辆调度优化理论与算法进展系统研究是构建综合物流系统、建立现代调度指挥系统、发展智能交通运输系统和开展电子商务的根底[1]。

车辆调度问题是运筹学与组合优化领域的研究热点。

有效的调度车辆，不仅可以提高物流工作效率，而且能够为及时生产模式的企业提供运输上的保障，从而实现物流管理科学化。

由于该问题的理论涉及很多学科，很多实际问题的理论抽象都可归结为这一类问题，研究该问题具有很重要的理论意义和实际意义。

1 . VRP〔Vehicle Routing Problem〕问题描述及其分类VRP问题一般可定义为：对一系列的装货点或卸货点，组织适当的行车路线，使车辆有序地通过它们，在满足一定的约束条件(货物需求量、发送量、车辆容量限制、行驶里程限制、时间限制)下，到达一定的目标(路程最短、时间最小、费用最省、车辆数目最少等)。

由于该问题研究范围非常广，根据其网络性能大致可以分为两类：一类为静态 VRP (StaticVRP, SVRP)，一类为动态VRP (dynamic VRP, DVRP)。

(1)静态VRP问题描述SVRP 问题是VRP 中较简单的一类问题，是大局部研究者研究的热点。

该问题具有一个很重要的特征：在安排初始路线时，和路线相关的所有信息，并且在安排路线以后其相关信息始终保持改变[2]。

以以下举了一些常见的SVRP 问题:仅考虑车辆容量限制的VRP(CVRP)、带时间窗的VRP(VRPTW)、带有回收的VRP(VRP with backhauls)、带有集派的VRP(VRPPD)。

除此以外，还有许多其它CVRP 的延伸问题，如顾客有优先权，考虑卸货时间、装卸时间、等待时间等，甚至综合了以上不同的特征。

这些问题的相关信息均且保持不变[3]。

(2)动态VRP问题描述所谓DVRP，是指在安排初始路线时，并不是和路线相关的所有信息都为，并且初始路线安排以后，其相关信息可能发生改变。

车辆路径问题专题—VehicleRoutingProblem

Periodic VRP (PVRP)
• In classical VRPs, typically the planning period is a single day. In the case of the Period Vehicle Routing Problem (PVRP), the classical VRP is generalized by extending the planning period to M days. • We define the problem as follows: Objective: The objective is to minimize the vehicle fleet and the sum of travel time needed to supply all customers. Feasibility: A solution is feasible if all constraints of VRP are satisfied. Furthermore a vehicle may not return to the depot in the same day it departs. Over the M-day period, each customer must be visited at least once.
Capacitated VRP (CPRV)
• CVRP is a VRP in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. That is, CVRP is like VRP with the additional constraint that every vehicles must have uniform capacity of a single commodity. We can find below a formal description for the CVRP: • Objective: The objective is to minimize the vehicle fleet and the sum of travel time, and the total demand of commodities for each route may not exceed the capacity of the vehicle which serves that route. • Feasibility: A solution is feasible if the total quantity assigned to each route does not exceed the capacity of the vehicle which services the route.

物流配送路径优化研究论文

物流配送路径优化研究论文摘要高效率合理的配送是物流系统顺利运行的保证，配送线路安排的合理与否对配送速度、成本、效益影响很大。

所以正确合理地安排车辆的配送线路，实现合理的线路运输可以使企业达到科学化的物流管理, 这也是企业提高自身竞争力的有效途径之一。

本文以帝峰模具有限公司的配送方案为例，对其配送现状中存在的问题进行分析，并运用节约算法、扫描算法以及改进后的最近插入法对配送线路进行优化，提出物流配送线路优化的方案，并且得到了相对满意的结果。

优化后的配送线路有效提高了帝峰模具有限公司的作业效率，降低物流成本，从而提升企业的经济效益，并让公司能够在激烈的竞争市场立足，同时，也可以给同类企业提供参考。

[关键词]：帝峰模具配送路径优化最近插入法扫描法节约算法[Abstract]Reasonable and efficient distribution is theinsurance of a smooth running logistics system,distribution line arrangement is reasonable or not has a great influence on the speed of delivery, costs and benefits.Therefore,to arrange a reasonable and correct delivery line for vehicle and achieve a reasonable transport line can enable enterprises to achieve scientific logistics management, which is one of the effective way for an enterprise to improve its competitiveness.This paper take Difeng Mold Co,Ltd. distribution as an example to analysis of the problems existing in the status of its distribution,through the saving algorithm, the improved insert method and scanning method of these three methods are optimized for distribution lines, logistics distribution route optimization scheme is put forward, and a relatively satisfactory results are obtained.Optimized distribution lines effectively improve the gravelslogistics company's efficiency, reduce logistics costs, thereby improve enterprise economic benefits, and gravels can in the fierce market competition, at the same time, also can provide a reference for similar enterprises.[Key words]Difeng Mold Co,Ltd Distribution route optimization scanning method the improved insert method the saving algorithm目录TOC \o "1-3" \h \z \u 第1章绪论 1.1 研究背景 61.2国内外研究现状7 1.2.1 国外研究现状7 1.2.2 国内研究现状9 1.3研究目的、意义和方法11 1.3.1 研究目的11 1.3.2 研究意义11 1.3.3 研究方法12 1.4本文研究内容12第2章相关理论概述2.1 物流配送14 2.1.1 物流配送的概念14 2.1.2 物流配送的功能14 2.1.3 物流配送的要素15 2.2 配送路径优化问题16 2.2.1 配送路径优化的目标16 2.2.2 配送路径优化问题的分类18 2.2.3 配送路径优化问题的解法分类192.3 本文配送路径优化方法20 2.3.1建立VRP模型20 2.3.2最近插入法21 2.3.3 扫描法22 2.3.4节约算法23节约里程算法主要步骤：24 第3章帝峰模具公司物流配送路径现状分析 3.1公司简介25 3.2 公司物流配送路径现状25 3.3 公司物流配送路径存在的问题路径分析28 3.3.1 路径迂回28 3.3.2对流运输29 3.3.3经验化操作过多30第4章帝峰模具公司物流配送路径优化策略4.1建立VRP模型优化配送路径31 4.2公司物流配送路径的优化31 4.3.1运用最近插入法优化314.3.2运用扫描法法优化35 4.2.3运用节约算法优化39 4.4三种优化方案比较分析44结论致谢参考文献第1章绪论 1.1 研究背景物流是为了满足消费者需要而进行的从供应地到接收地的原材料、中间产品、最终产品及相关信息的有效流动和储存计划、实施和控制的管理过程。

VRP

一、VRP (Vehicle Routing Problem)车辆路径问题车辆路线问题（VRP）是现代物流配送中心末端送货线路研究的一项重要内容，由Dantzig和Ramser于1959年首次提出。

它是指在一定的约束下，根据已知的待服务客户的网点布局、物流配送中心的位置、车辆的最大负荷等信息，为车队组织出适当的行车路线分送货物。

使得在满足客户的需求的同时，实现诸如路程最短、成本最小、耗费时间最少等目标。

最基本的车辆路径问题是从一个服务中心向离散分布在某一区域的n个客户派遣m辆车辆来提供货物，要求确定各车辆的行走路线使总的运输成本最小，并保证每个服务需求点只被其中的一辆车辆访问过一次。

TSP（旅行商问题）是由一辆车来串联多个派货点，以完成派送任务，而VRP是由一个车队来完成。

所以TSP只是VRP的一个特例。

而Gaery已经证明了TSP是NP问题（全名：NP完全问题），所以VRP问题自然也是NP问题，而且还是比TSP更加复杂的NP问题。

Savelbergh和Solomon也指出带时间窗的车辆路优化问题（VRPTW：Vehicle Routing Problem with Time Window）是NP问题，并且比一般的VRP更加复杂。

但是，人们非但没有因为这个问题的复杂性而放弃对他的研究，更是由于其使用范围的广泛性和问题的复杂性，更多的人将目光投注在他的身上。

并且，VRP被进一步的实例化，更多的算法也被提了出来。

例如带能力约束的车辆路径问题、带时间窗的车辆路径问题、追求最佳服务时间的车辆路径问题、多车种车辆路径问题、车辆多次使用的车辆路径问题等。

被提出的算法大致可以分为两类：精确算法和启发式算法。

精确算法顾名思义就是可以求出精确的最优解的算法。

然而，对于比TSP还要复杂的VRP来说，目前为止，最有效的精确算法最多也只能包含50个派送点。

因此，人们把主要的精力还是集中在了启发式算法上。

启发式算法是基于直观或者经验构造出来的算法，且一般不要求将问题描述成标准的数学模型，在可以接受的计算量之内，他得出结果的具有很强的不可预知性，不能保证得到的解就是最优解。

同尺寸长方体物品装箱问题的一种求解算法

同尺寸长方体物品装箱问题的一种求解算法丁莎;谢海江;潘立武【摘要】研究同尺寸长方体物品的装箱问题,即在一个给定的箱子中装入尽可能多的同尺寸长方体物品.采用分层装载方案简化装载操作,首先运用递归算法确定层中长方体物品的布局方式;然后求解整数规划模型确定箱中层的最优组合,得到最优装载方案.采用随机测题,将文中算法与文献中装箱算法进行对比.实验结果表明文中算法生成的装载方案箱体空间利用率由文献中装箱算法的99.35％提高到了99.77％.文中算法可以在合理的时间内得到装载操作简单,箱体空间利用率较高的装载方案.【期刊名称】《机械设计与制造》【年(卷),期】2016(000)007【总页数】4页(P12-14,19)【关键词】装箱;层装载;递归算法;整数规划【作者】丁莎;谢海江;潘立武【作者单位】四川大学锦江学院,四川眉山620860;郑州职业技术学院汽车工程系,河南郑州450121;河南牧业经济学院自动化与控制系,河南郑州450011【正文语种】中文【中图分类】TH16;TP391装箱问题又称三维布局问题[1-5]广泛的出现在机械设计与制造领域的产品包装环节。

在实际生产中，很多装箱问题是由工人凭借经验进行手工装箱完成的。

手工装箱不能充分利用箱体空间，运用计算机辅助设计技术（CAD）进行装箱可以明显的提高箱体空间利用率。

由于装箱问题从计算复杂性理论上看是属于一类很难解决的NP难度组合优化问题[6-8]，故精确算法只能够解决规模很小的装箱问题，实际生产中的装箱问题由于规模较大往往采用启发式算法使得在合理的时间内得到最优解或近似最优解[9]。

按照箱体中可以装入的物品种类数，装箱可以分为同构装箱和异构装箱[4]。

箱体中只装入一种物品称为同构装箱。

箱体中可装入多种物品称为异构装箱。

在实际应用中，同构装箱由于装载操作相对简单，往往更容易被用户所接受。

讨论同构装箱（Isomorphism bin packing，IBP）问题：在长宽高分别为L、W、H的方型箱子中装入长宽高分别l、w、h的方型物品，优化目标是使得装入的物品个数最多。

Vehicle routing problem with time windows

Discrete OptimizationVehicle routing problem with time windows and alimited number of vehiclesHoong Chuin Laua,*,Melvyn Sim b ,Kwong Meng TeocaDepartment of Computer Science,School of Computing,National University of Singapore,3Science Drive 2,117543SingaporebOperations Research Center,Massachusetts Institute of Technology,Cambridge,MA 02139,USAcSavy Technology Asia Pte Ltd.,Technology Park@Chai Chee,469001SingaporeAbstractThis paper introduces a variant of the vehicle routing problem with time windows where a limited number of vehicles is given (m -VRPTW).Under this scenario,a feasible solution is one that may contain either unserved customers and/or relaxed time windows.We provide a computable upper bound to the problem.To solve the problem,we propose a tabu search approach characterized by a holding list and a mechanism to force dense packing within a route.We also allow time windows to be relaxed by introducing the notion of penalty for lateness.In our approach,customer jobs are inserted based on a hierarchical objective function that captures multiple objectives.Computational results on benchmark problems show that our approach yields solutions that are competitive to best-published results on VRPTW.On m -VRPTW instances,experiments show that our approach produces solutions that are very close to computed upper bounds.Moreover,as the number of vehicles decreases,the routes become more densely packed monotically.This shows that our approach is good from both the optimality as well as stability point of view.Ó2002Elsevier Science B.V.All rights reserved.Keywords:Tabu search;Heuristics;Routing;Combinatorial optimization;Vehicle routing problem with time windows1.IntroductionMany practical transport logistics and distri-bution problems can be formulated as a vehicle routing problem whose objective is to obtain a minimum-cost route plan serving a set of customers with known demands.Each customer is assigned to exactly one vehicle route and the total demand of any route must not exceed the vehicle capacity.To date,most of the proposed algorithms as-sume that the number of vehicles is unlimited,and the objective is to obtain a solution that either minimizes the number of vehicles and/or total travel cost.However,transport operators in the real world face resource constraints such as a ﬁxed ﬂeet.The question we like to ask is,if the given problem is over-constrained in the sense of insuf-ﬁcient vehicles,what constitutes a good solution and how may we ﬁnd one?In this paper,we provide some insights to this question.In our view point,it is desirable to have an algorithm that not only performs well given a*Corresponding author.E-mail addresses:lauhc@.sg (u),melvyn@ (M.Sim),kmteo@ (K.M.Teo).0377-2217/03/$-see front matter Ó2002Elsevier Science B.V.All rights reserved.doi:10.1016/S0377-2217(02)00363-6European Journal of Operational Research 148(2003)559–569standard VRPTW problem,but also handles over-constrained problems well in the following sense: 1.Optimality:It returns solutions which serve(orpack)as many customers as possible as the pri-mary objective,while optimizing standard crite-ria such as the number of vehicles and distance travelled.2.Stability:It degrades gracefully under con-strainedness,i.e.when the number of vehicles is reduced,the customer packing density,deﬁned as the average number of customers per vehicle in service,must be monotically increasing,al-though the total number of customers served will become less.This paper proceeds as follows.Weﬁrst introduce the problem(m-VRPTW)and a computable upper bound to the problem.We then present a tabu search approach with the following characteristics: (a)a holding list to accommodate unserved cus-tomers;(b)a mechanism that introduces new vehicles in stages so as to force denser customer packing within a route.We then extend the algo-rithm to a generalization of the problem with re-laxed time windows.In terms of computational results,experiments on VRPTW benchmark problems show that our approach can produce solutions that are very close to previous best-published results.What is more interesting perhaps is the performance on m-VRPTW instances.Results show that our approach produces solutions that are very close to computed upper bounds.Moreover,as the number of vehicles is reduced,the average number of customers per route is monotically increasing.This shows that our approach is good from both the optimality as well as stability point of view.2.Literature reviewThe primary objective of m-VRPTW is to maximize the number of customers served,which is NP-hard,sinceﬁnding it is a generalization of the multiple constrained knapsack problem.Al-though the classical VRPTW has been the subject of intensive research since the80s,to our knowl-edge,there has been little research work on m-VRPTW.We give some research developments in VRPTW.SolomonÕs insertion heuristics[18]is the seminal work behind heuristic construction algo-rithms.Many eﬃcient heuristic and meta-heuristic approaches have been proposed recently,including the works of Chiang and Russell[5],Potvin and Rousseau[14],Rochat and Taillard[15],Taillard et al.[20],and Thangiah et al.[21].More recently, Schulze and Fahle[19],Gehring and Homberger [9]proposed new parallel tabu search heuristics that enable large-scale VRPTW instances to be solved.Several works have been carried out advocat-ing the hybrid use of constraint programming and local search.For example,Pesant and Gendreau [13]applied constraint programming to evaluate the local neighborhood toﬁnd the best local moves.There is also constrained-directed local search proposed in[1,11,17].In[17],for example, the author presented a method called large neigh-borhood search(LNS)for VRP in which a part of a given solution is extracted and then reinserted into the partial solution using a quasi-complete search process.If the reinsertion procedure gen-erated a better solution,then the solution is kept. This process is repeated until certain stopping criterion is met.The result produced with this technique is competitive with other meta-heuristic approaches.In terms of exact algorithms,Desrochers et al.[6]has applied column generation that was able to solve some100-customer problems optimally. Based on this,Kohl et al.[12]developed a more eﬃcient optimization algorithm by introducing a new valid inequality within a branch-and-cut algorithm,called k-path cuts,which solves70of the87Solomon benchmark problems to optimal-ity.However,due to the exponential size of the solution space,it is unlikely that these optimization procedures can be used for larger-scale problems.3.Problem deﬁnition and notationThe standard VRPTW problem is deﬁned for-mally as:Given an undirected graph GðV;AÞu et al./European Journal of Operational Research148(2003)559–569where V ¼f v 0;v 1;...;v n g ,v 0is the depot,v i ,i ¼0is a customer with demand d i ,time windows (e i ;l i )and service duration s i ;A ¼fðv i ;v j Þ:i ¼j ,v i ;v j 2V g ,each arc (v i ;v j )having a travel distance (time)t ij ;and vehicle capacity Q ,ﬁnd a minimum set of vertex-disjoint routes starting and ending at depot v 0such that each customer v i is served by exactly one vehicle within its time windows,P d i for all customers v i served by each vehicle is less than Q ,and the total distance travelled is mini-mized (as the secondary objective function).m -VRPTW is deﬁned formally as:Given m (number of vehicles)and a VRPTW instance,ﬁnd m or less routes with the primary objective func-tion of maximizing total number of customers served,and the secondary objective function of minimizing the total distance travelled.4.Upper bound for m -VRPTWIn this section,we determine an upper boundfor the total number of customers that can be served by a given ﬁxed number of vehicles.We propose an integer programming (IP)formulation.The IP formulation should be able to solve large-scale problems,yet not be overly simpliﬁed such that the gap of the bound from the optimum is too wide.We have adopted a formulation that ac-counts for the capacity constraints of the vehicles as well as the time constraints imposed by the latest return times of every vehicle to the depot.The upper bound is derived by solving a relaxation of m -VRPTW,formulated as follows.Deﬁne U ¼f 1;2;...;m g to be the indices of the set of m serving vehicles and V c to be the set of customer nodes (excluding the node at the depot).Deﬁne r i ¼min j ;j ¼i t ij ,i ;j 2V which is the travel-ing time from node i to its nearest neighbor.This quantity is used to lower bound the travel time from node i to any other node.Let w i ¼l i þs i þt i 0,i 2V c denote the time of return to the depot after serving node i as its last customer at its latest start time.Without loss of generality,assume that all w i s do not exceed the depot close time.Let G ¼½g 1;g 2;...;g m be a list of m unique customers in V such that w i P w j for all i 2G and j 2G .Since vehicles are identical,the melements of G represent the latest possible times of return to the depot for each of the m vehicles,for a solution to be feasible.The decision variables are denoted by x ij 2f 0;1g ,i 2V c ,j 2U where x ij ¼1if and only if the customer at node i is served by vehicle j .The following IP returns the upper bound of the total number of customers served by all the vehicles:max X i 2V c Xj 2Ux ij s :t :Xj 2Ux ij 61;8i 2V cð1ÞXi 2V cx ij d i 6Q ;8j 2Uð2ÞXi 2V cx ij ðs i þr i Þþr 06w g j ;8j 2Uð3Þx ij 2f 0;1g ;8i 2V c ;j 2UConstraints (1)state that all customers must be assigned to at most 1vehicle.Constraints (2)ensure that the vehicle capacity constraint is not violated.Constraints (3)impose some linear tim-ing constraints:it says that for each vehicle,the earliest possible time of returning to the depot (induced by the assignment x )cannot exceed the latest possible return time (imposed by G ).Note that in this formulation,we ignore the full con-siderations of time window constraints and the actual travel time between two nodes.The bound is thus expected to be less eﬀective on test cases with tight time window constraints.Note that the above formulation gives us a constrained knapsack problem ,which is NP-hard.Fortunately,many variants and generalizations of the knapsack problem have been well-studied there exist exact algorithms which are computationally eﬃcient,such as [22].5.Standard two-phase methodMost of recently published VRPTW heuris-tics are two-phase approaches.First,a construc-tion heuristic is used to generate a feasible and as good as possible initial solution.Then,an iterative improvement heuristic is applied to this solution.u et al./European Journal of Operational Research 148(2003)559–569561It generates successive solutions by searching the neighborhood of the current solution.In the sec-ond phase,various methods are then used to pre-vent the algorithms from being trapped at local optimal and to explore a larger search space.The construction phase involves insertion of all the customers into a set of feasible vehicles routes. The purpose of the construction phase is to pro-vide an initial feasible ually,each customer will be inserted in turn to the route that gives the minimum additional cost or distance at that instance.The order of customer selection then deﬁnes the heuristics,some of which are listed follows:•Nearest insertion rule:Next nearest unserved customer will be selected.•Earliest ready time rule:Next unserved cus-tomer with the earliest ready time will be se-lected.•Window tightness rule:Next unserved customer with the tightest time window will be selected. Observe that these heuristics assume that there are enough vehicles to serve all the customers.As such,for over-constrained problems,these heu-ristics may fail to deliver satisfactory solutions.Given the initial feasible solution from phase I, the phase II route improvement phase involves an iteration of moving from a feasible solution to its feasible neighborhood until certain termi-nating condition is met.In this phase,the heuris-tics are deﬁned by the neighborhood structure,the choice of the next move,and the terminating condition.The simplest approach of the steepest descent algorithm chooses the best and improved solution among all the neighboring solutions at every iter-ation.However,the algorithm would very quickly be trapped within a local minimum.The neighborhood structure used is usually a k-opt local search procedure,where k refers to the number of customers/arcs that can be inter-changed from the initial solution to its neighbor-hood solutions.Some of these interchanges are •Relocate:Customer from one route transfers to another route.•Exchange:Customer from one route exchanges position with another customer in another route.The quality of a two-phase method depends on whether the choice of construction and improve-ment heuristics is a goodﬁt to the nature of the search space.The construction heuristics should produce a good enough initial location such that the improvement algorithm starts in a region where good solutions can be achieved.Subse-quently,the improvement heuristics would need to be able to bypass suﬃciently many local minima to terminate at a good solution.6.Proposed algorithmThe above-mentioned two-phase method would normally have to work diﬀerently for an over-constrained problem.One way is to use the inser-tion heuristic to determine whether the problem instance is feasible.Following which,we have two sets of heuristics to handle separately the infeasi-ble case and the feasible case.Another approach is to increase enough vehicles so as to serve all customers,and then,through subsequent heuris-tics,try to obtain a subset of the solution that maximizes the number of customers that can be served.However,there are pitfalls in both of these approaches.Though theﬁrst seems credible,it demands extensive use of heuristics and does not value-add in terms of algorithm development.The second approach is not addressing the issue of infeasibility directly,and is therefore unlikely to give a good solution consistently.We therefore seek to have a generalized method to handle both feasible and infeasible problem in-stances.The approach to such an algorithm lies in the introduction of the holding list,the data structure that contains unserved customers.Al-though the idea of holding list is not new atﬁrst sight(for example,the ILOG Dispatcher product uses the same idea),our overall algorithmic strat-egy of transferring customers back and forth the holding list under our tabu search strategy(see below)is,to our knowledge,a novel idea.u et al./European Journal of Operational Research148(2003)559–5696.1.Holding listThe holding list contains the list of the cus-tomers that are not served in the current solution. The idea of introducing a holding list is triggered by the role of artiﬁcial variables in the phase I of the simplex algorithm.A feasible solution of the VRPTW is found when all the customers are dri-ven out of the holding list,which is analogous to driving out all the artiﬁcial variables from the simplex tableau.The holding list will induce an extended neigh-borhood search space,which includes the follow-ing moves,in addition to the basic relocate and exchange moves discussed in Section5:•Relocate from holding list:Transferring a cus-tomer from holding list to an existing route.•Relocate to holding list:Transferring a cus-tomer from an existing route to the holding list.•Exchange with holding list:Exchanging a cus-tomer from an existing route with another cus-tomer in the holding list.The holding list is similar to a‘‘phantom’’route which participates in the regular local search,with a variant that insertion of a customer to the hold-ing list is always feasible and does not incur any cost.The customers of a selected route will be searched completely for possible of transfer to/ from or exchange with customers in the holding list.The next accepted move is determined by using a best improvement strategy depicted in the hierarchical cost structure.6.2.Hierarchical cost structurem-VRPTW introduces the additional objective of maximizing the number of served customers and minimizing lateness(if time windows can be relaxed).This implies that the objective function becomes a composite function:•maximize total number of customers served,•minimize total number of customer served late (if allowed),•minimize total lateness duration(if allowed),•minimize total number of vehicle used,•minimize total distance traveled.One possible way of dealing with multi-criteria objective is to deﬁne a composite cost function with diﬀerent weights for the diﬀerent cost pa-rameters.However,setting the proper weights becomes a tricky(or almost impossible)mission, and the resulting function becomes meaningless to interpret.Our approach is to deﬁne a hierarchical cost structure.For example,serving more customers is always better regardless of the number of vehicles used.Although one can argue that theÔbig MÕapproach on the composite cost function can be used to enforce the hierarchy,we believe our ap-proach is a cleaner way.We propose a hierarchical cost structure in decreasing order of priority of the above list of objectives.In hierarchical cost comparison between two states(during local search),the state with greater preference down the hierarchy of importance as-sumes aÔlower costÕ.As opposed to the composite cost function,where the total composite cost is computed,hierarchical cost is never computed. This is because in local search techniques,com-putation of the absolute cost is not needed. Rather,the hierarchical cost is used for comparing the current solution state with the previous best solution state.6.3.Tabu search strategyAlthough many construction and local im-provement techniques have been reported to solve VRPTW problems,it is unclear how these heuris-tics will behave when constrained by a limited vehicleﬂeet.Particularly,it is unclear how the two-phase approach can work co-operatively in ensur-ing the ultimate solution to have good customer packing while not exceeding the prescribed vehicle limit.For instance,if we use too few vehicles in the construction phase,it may limit the search space of the local improvement phase;on the other hand,if we use up to the maximum allowable number of vehicles,then it leaves little room for the local search phase to add more unserved customers, since all routes have been used up.The outcomeu et al./European Journal of Operational Research148(2003)559–569563may be a solution where some of the routes are relatively loosely packed,but no more customers can be added to any one of them unless drastic changes are made to the solution,which local search,by its nature,is incapable of realizing.To deal with limited vehicleﬂeet,our strategy is to meld the two-phase approach into a nested approach.The idea is to increase the number of vehicles in stages and at each stage,apply standard tabu search to maximize the number of customers to be inserted onto those vehicles.Within each stage,the number of vehicles isﬁxed and hence the search will not consider adding a vehicle to serve unassigned customers.In other words,we steer the tabu search to favor packing of customers within the existing routes.With certain abuse of termi-nology to draw parallelism with duality,the dual of the vehicle minimization problem is the maxi-mization of customer packing problem.Empirical results have shown that this strategy has tremen-dous improvement of packing density on Solomon test cases,without having to rely on a set of good construction heuristics.With the incorporation of the holding list,it is easy to implement the tabu search strategy dis-cussed above.The hierarchical cost structure fa-vors transfer of customers from the holding list to the routes.Unlike many tabu search algorithms, where the search space is always feasible,the holding list is a neat way of incorporating local search towards a path of feasibility that favors customer packing.Let TS denote one iteration of a standard tabu search procedure with the search neighborhood and hierarchical cost structure discussed in Sec-tions6.1and6.2respectively.The tabu list stores customers that have been moved within the preced-ing number of iterations deﬁned by the tabu length.A move is tabu if and if only the customers are in the tabu list and it is not aspired by being better than the best solution so far,in the sense of the hierarchical cost function deﬁned in Section6.2.Let StepSize denote the additional number of vehicles introduced in each stage.Let numVeh denote the current number of vehicles used,ini-tialized to StepSize.Let CountLimit denote the maximum number of non-improving moves using numVeh vehicles.The algorithm proceeds as follows(refer to Algorithm A).The holding list initially contains all customers of the given instance.We introduce StepSize additional number of vehicles(i.e.empty routes)in each stage.Each stage is implemented by the while loop(i.e.steps(3)to(6)).When TS is called in step(4),it will return a solution which diﬀers from the previous solution by a local move made with respect to the neighborhood and the hierarchical cost structure.In step(5),a better solution means that the hierarchical cost objective value is better than that of the best solution found so far.If a better solution has not been found after CountLimit consecutive tries,then the stage ends, and the algorithm proceeds to the next stage by adding more vehicles.Algorithm A(1)until holding list is empty or numVeh¼m(2)set Count¼0(3)while Count6CountLimit(4)call TS based on numVeh vehicles(5)if better solution found then set Count¼0else set Count¼Countþ1(6)endwhile(7)set numVeh¼minðnumVehþStepSize;mÞ(8)end until6.4.m-VRPTW with relaxed TWsThe logical extension to solving m-VRPTW problems is to relax the time window constraints. In other words,we allow late arrivals after the intended due time to increase the number of cus-tomers that could be served.Although one could perceive this as a separate objective that may entail a separate algorithm,our challenge is to incorpo-rate this feature into one seamless generalized al-gorithm.We deﬁne late period to be the amount of late-ness between the time windows upper bound and the actual arrival time(and0if the arrival time lies within the time window).We impose the late pe-riod as a soft time constraint which can be vio-lated.This is done by incorporating the number of late arrivals and the total late period into the hi-erarchical cost function.u et al./European Journal of Operational Research148(2003)559–569Intuitively,by relaxing the time window con-straints,we would expect to have improved solu-tions(in terms of its objective value).However, from experimentation with our proposed local search technique,we discovered an anomaly that the solution for some cases become worse if we relax the constraints as it is.This is illustrated in the sample run in solving an over-constrained Solo-mon benchmark problem R103with13vehicles. Without relaxing the time windows,the total late period is0,while with relaxed time windows,the late period became77.8units!This contradicts with the intuition that with relaxed time windows,the solution obtained from local search will always be an improvement.As local search techniques do not guarantee global optimality,it is likely that even with relaxed constraints the reported solution may be worse oﬀ.To deal with this anomaly,we propose Algo-rithm B.Here,weﬁrst solve the problem without relaxing the time windows.The solution then be-comes the initial feasible solution for the problem with relaxed time windows.In the latter case,more customers may be inserted from the holding list, albeit at the expense of relaxed time windows.In this manner,we can always guarantee improve-ment,if any,on the relaxed problem.Table1pre-sents the new result obtained for R103under this new scheme.Algorithm B(1)call Algorithm A(2)if holding list not empty(3)relax time windows constraints(3)set Count¼0(4)repeat steps(2)to(7)of Algorithm A 7.Results and analysisIn this section,we present experimental results of applying our algorithm to solve both the stan-dard VRPTW as well as m-VRPTW problems. In our experiments,we set the tabu length to be 100.We set the values of CountLimit and Step-Size in Algorithm A to be500and1respec-tively.We refer to our implementation as the OV method.7.1.Performance on VRPTW problemsHere,we test the performance of Algorithm A on the set of56SolomonÕs test cases with100 customers.The run time of the algorithm tested on a Pentium II433machine is about1min on average.Weﬁrst compare our results with the overall best-published heuristics results.1The comparison of solutions is presented in Table2,where the Best and OV columns contain the best published and our results respectively.Next,we also compare our results with the speciﬁc results of(a)Rochat and Taillard[15] (RT),(b)Chiang and Russell[3](CR),(c)Taillard et al.[20](TBGGP),(d)Homberger and Gehring [10](HG),and(e)Cordeau et al.[4](CLM).A summary of the comparison is given in Table3. We observe that although our results are in general inferior compared to these results,they are within only several percent worse on average.We believe this is justiﬁable,in two sense.First,our goal is in obtaining results within1min on average and hence set the maximum number of iterations to 500.This is in contrast to the other methods which typically require hours of CPU time on compatibleTable1Consistency of solution with relax TWsWithout relaxed time windows With relaxed time windows#Vehicles1313#Customersinserted99100#Customersserved late02 Total late period015.7921Contributions to the best-published results are taken from: Cordeau et al.[4]––R107,R108,RC104,RC106,R204,RC201, RC207,Chiang and Russell[3]––R207,Homberger and Geh-ring[10]––R103,R109,R112,R201,R203,R208,R210,R211, C202,RC203,RC204,RC205,R110,Rousseau et al.[16]––R111,RC105,R202,R205,R209,RC206,RC208,Rochat and Taillard[15]––R101,R104,R105,R106,C101,C102,C103, C104,C105,C106,C107,C108,C109,RC103,R206,C201, C202,C203,C204,C205,C206,C207,C208,R102,Taillard et al.[20]––RC101,RC102,RC107,RC108.u et al./European Journal of Operational Research148(2003)559–569565machines.Second,this work is not aimed at beating the best VRPTW results,but rather on proposing an algorithm that works well when limited to a number of vehicles.7.2.Performance on m-VRPTWTo assess the performance of our algorithm on m -VRPTW,we will measure the total number ofTable 3Summary of comparisonRTCR TBGGP HG CLM OV C1mean no of vehicles 10.0010.0010.0010.0010.0010.00C1mean distance832.59828.38828.38828.38828.38832.13C2mean no of vehicles 3.00 3.00 3.00 3.00 3.00 3.00C2mean distance595.38591.42589.86589.86589.86589.86R1mean no of vehicles 12.8312.1712.1711.9212.0812.16R1mean distance1208.431204.191209.351220.971210.141211.55R2mean no of vehicles 3.18 2.73 2.82 2.73 2.73 3.00R2mean distance999.63986.32980.27968.55969.581001.12RC1mean no of vehicles 12.7511.8811.5011.5011.5012.25RC1mean distance1381.331397.441389.221388.241389.781418.77RC2mean no of vehicles 3.62 3.25 3.38 3.25 3.25 3.37RC2mean distance1207.371229.541117.441140.431134.521170.93Table 2OV solutions against best-published resultsBest OV Best OV VehiclesDistance Vehicles Distance Vehicles Distance Vehicles Distance C10110828.9410828.94R11291003.73101021.95C10210828.9410834.64R20141252.3741292.53C10310828.0610834.56R20231191.7031158.98C10410824.7810846.32R2033942.643980.70C10510828.9410828.94R2042849.623847.74C10610828.9410828.94R2053994.4231146.80C10710828.9410828.94R2063912.9731007.00C10810828.9410828.94R2072914.393869.94C10910828.9410828.94R2082731.232790.46C2013591.563591.56R2093909.8631020.06C2023591.563619.36R2103955.3931032.65C2033591.173604.01R2112910.093866.10C2043590.63644.23RC101141696.94151657.46C2053588.883601.43RC102121554.75131535.79C2063588.493588.88RC103111262.02121386.03C2073588.293608.94RC104101135.48101213.25C2083588.323591.83RC105131633.72151625.13R101191650.8201765.00RC106111427.13121426.07R102171486.12181548.61RC107111230.54111330.59R103131292.85141258.34RC108101139.82101175.88R10410982.01101018.48RC20141406.9441468.46R105141377.11151462.69RC20231389.5741222.69R106121252.03121328.66RC20331060.4531171.88R107101113.69121160.08RC2043799.123839.32R1089964.38101045.83RC20541302.4241338.70R109111194.73131259.09RC20631153.9331201.27R110101124.4111127.70RC20731062.0531139.48R111101096.72111097.10RC2083829.693985.60u et al./European Journal of Operational Research 148(2003)559–569。

动态车辆路径问题模型与优化算法的开题报告

动态车辆路径问题模型与优化算法的开题报告一、研究背景随着交通网络不断扩展和城市化程度的加深，交通流量的快速增长，交通拥堵已成为城市生活中的一个普遍问题。

此时，动态车辆路径问题(Dynamic Vehicle Routing Problem, DVRP)作为最基本、最核心的运输问题之一，越来越得到交通规划和管理领域的关注。

DVRP是指在一个动态环境中，为一批客户安排最优的配送路径和调度方案，以使得运输成本达到最小化。

DVRP的求解对许多商业和公共部门都有着重要的意义，如生产调度、快递配送、军事物流等。

二、研究目标本文旨在研究动态车辆路径问题的优化算法，主要包括以下研究目标：1.设计一个DVRP模型，考虑多个时间窗口、多个车辆和多个目标地点。

2.针对所设计的DVRP模型，提出多种求解DVRP问题的优化算法。

3.通过实验研究，比较不同的优化算法的效果，找出最优解。

三、研究内容1.综述DVRP问题及其主要研究方法，分析相关文献，探讨其优化难点。

2.设计基于遗传算法和模拟退火算法的DVRP优化模型，分析模型求解的时间复杂度和准确性，并进行实验验证。

3.设计基于分支定界法和粒子群优化算法的DVRP优化模型，比较各种算法的效果，并进行实验验证。

4.从结果上加以比较，并对最佳算法进行改进，以获得更好的性能。

四、研究方法1.文献研究法。

对DVRP问题的背景、历史、研究现状等进行深入了解。

2.算法设计法。

提出基于遗传算法、模拟退火算法、分支定界法和粒子群优化算法的DVRP优化模型，实现代码开发。

3.实验研究法。

比较不同算法的效果，在多个数据集上进行计算实验并分析结果。

五、论文结构安排本文预计分为引言、研究背景和意义、DVRP模型设计、优化算法设计、实验验证、结果分析与讨论、结论等七个部分。

其中：1.引言：介绍研究原因、研究现状、本文的研究目的和研究方法。

2.研究背景和意义：对DVRP问题的相关知识，及其在实际应用中的重要性进行介绍。

车辆路径问题概念、模型与算法(五星推荐)

车辆路径问题概念、模型及算法
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1、定义
车辆路径问题(VRP)一般定义为：对一系列装货点和卸货点，组织适当的行车线路，使车辆有序地通过它们，在满足一定的约束条件(如货物需求量、发送量、交发货时间、车辆容量限制、行驶里程限制、时间限制等) 下，达到一定问题的目标(如路程最短、费用最少、时间尽量少、使用车辆数尽量少等)。
约束条件，求松弛问题的最优解，如果获得整数最优解，
即为所求，运算停止。如果所得到最优解不满足整数约
束条件，则在此非整数解的基础上增加新的约束条件重
分支过程得到的整数解中，目标函数值最优的一个叫做整数规划目标函数值的“界”。分支过程中非整数的线性规划的最优解，如果目标函数值劣于或等于这个 “界”，就停止继续分支。这个过程，叫做“定界”。
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割平面法(Cutting Planes Approach)
用割平面法求解整数规划的基本思路是：先不考虑整数
(5) 相容性约束：引出相容性约束车辆路径问题 (VehicleRouting Problem with Compatibility Constraints， VRPCC)。
(6) 随机需求：引出随机需求车辆路径问题 (VehicleRouting Problem with Stochastic Demand， VRPSD)。
(3) 车型约束：引出多车型车辆路径问题 (Mixed/Heterogeneous Fleet Vehicle Routing Problem， MFVRP/ HFVRP)。
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(4) 时间窗约束：包括硬时间窗(Hard Time windows)和软时间窗(Soft Time windows) 约束。引出带时间窗(包括硬时间窗和软时间窗)的车辆路径问题(Vehicle Routing Problem withTime windows，VRPTW)。

Robust vehicle routing problem with deadlines and traveltime_demand uncertainty

Robust vehicle routing problem with deadlines and travel time/demand uncertaintyC Lee1,K Lee2and S Park3Ã1ETRI,Daejeon,Republic of Korea;2Hankuk University of Foreign Studies,Gyeonggi-do,Republicof Korea;and3KAIST,Daejeon,Republic of KoreaIn this article,we investigate the vehicle routing problem with deadlines,whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity.It is assumed that the travel time between any two customers and the demands of the customer are uncertain.Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand.The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand deﬁned in the uncertainty sets.We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem.A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty.The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.Journal of the Operational Research Society(2012)63,1294–1306.doi:10.1057/jors.2011.136Published online7December2011;corrected online11January2012Keywords:vehicle routing;robust optimization;column generationIntroductionThe vehicle routing problem(VRP)calls for the determi-nation of the optimal set of routes to be performed by aﬂeet of vehicles to serve a given set of customers(Toth and Vigo,2002).It can be considered to be one of the more important problems—both in theory and practice—in theﬁelds of transportation,distribution,and logistics. Generally speaking,the goal of VRP is toﬁnd the minimum cost routes visiting(or serving)a given number of customers while respecting various resource constraints, including deadlines,vehicle capacities,and number of available vehicles.Several variations of the VRP exist. In the VRP with deadlines(VRPD),a deadline is imposed on each customer,with the requirement that the service to the customers must be provided before the deadline. Each customer has a given amount of demand and is visited by a vehicle exactly once.Each vehicle should have sufﬁcient capacity to serve the demands of all of the customers it visits/serves.Since the VRPD is a generalization of the VRP,which is a generalization of the TSP(travelling salesman problem),VRPD is N P-hard.The VRPD is a special case of the VRP with time windows(VRPTW).Therefore, any solution algorithm for the VRPTW can be used to solve the VRPD.Kolen et al(1987)presented an optimization method for the VRPTW using a dynamic programming approach.The problem size was,however, limited to15customers.Fisher et al(1997)and Kohl and Madsen(1997)proposed Lagrangian relaxation approaches which were efﬁcient for solving problems involving up to100customers.Ioachim et al(1998) developed a dynamic programming algorithm for the shortest path problem with time windows and linear node costs.At the present time,the most dominant and widely studied optimization approach for the VRPTW is the column generation-based approach.This approach wasﬁrst presented by Desrochers(1986),who applied it to problems with as many as100customers.Fukasawa et al(2006)developed a branch-and-price-cut algorithm for the capacitated VRP(CVRP)by incorporating the cutting plain method into the branch-and-price algorithm. Many heuristic methods have also been developed in the context of the VRP,including the local search method (Lin and Kernighan,1973;Dethloff,2002),the tabu-search algorithm(Gendreau et al,1994,1996;Rego,1998; Cordeau et al,2002),the genetic algorithm(Berger and Barkaoui,2003),and the ant colony algorithm(Dorigo et al,1996;Yu et al,2010).Journal of the Operational Research Society(2012)63,1294–1306©2012Operational Research Society Ltd.All rights reserved.0160-5682/12/jors/ÃCorrespondence:S Park,Department of Industrial Engineering,KAIST,291Daehak-ro,Yuseong-gu,Daejeon305-701,Republic of Korea.E-mail:sspark@kaist.ac.krAlthough there have been many advances in the optimization method for the VRP,most studies have not considered the uncertainty of the data.In practice, however,uncertainty in customer demand and/or travel time is inevitable.The feasibility of the solution obtained may not be guaranteed unless the uncertainty is incorpo-rated directly in the optimization methodology.The robust optimization methodology deals directly with the robustness of the solution byﬁnding a solution which is immune to variations in the data.The robust optimization approach differs from that of the stochastic optimization in that with the former it is not required to know the probability distribution of uncertain data a priori.In many cases,it may be very difﬁcult—or even impossible—to estimate fairly accurate probability distributions of the data.In the robust optimization,instead of estimating the prob-ability distributions,an uncertainty set is introduced to control the robustness of the solution;see Bertsimas and Sim(2004)for details.Here,we propose an optimization method for the VRPD under travel time and customer demand uncertainty.In terms of the VRP with data uncertainty,the most studied problem may be the VRP with stochastic demands (VRPSD)(Gendreau et al,1995,1996a,b;Christiansen and Lysgaard,2007;Tan et al,2007).Bertsimas and Simchi-Levi(1996)developed a set of analytical results for the VRP with random demands and proposed several heuristic algorithms.In the VRPSD,the customer demands are assumed to be uncertain and to have stochastic properties,characterized by certain probability distributions.A number of studies on the travel time uncertainty in the VRP have used the stochastic program-ming approach to handle the uncertainty of travel time (Jula et al,2006;Chang et al,2009).The basic assumption of such studies is that the stochastic properties of travel time are known in advance;the goal is to obtain a solution with the best expected cost(distance or travel time),while the solution guarantees certain service levels.The functions of the expected arrival times(and/or penalties)at the customers’locations are deﬁned and a number of methods for efﬁciently calculating the expected arrival time pro-posed.Since the exact calculation of the expected arrival times is often quite complicated,in many cases the calculation is done approximately(Jula et al,2006;Chang et al,2009)or heuristically(Cheung and Hang,2003). Russell and Urban(2008)derived closed-form expressions of a penalty function for the Erlang travel times and developed a tabu-search-based algorithm.A number of papers have appeared on handling the recent changes in data in which the problem was iteratively solved to reﬂect these recent changes.To hedge for possible future changes,certain expected costs were calculated at each stage,so that the routes would be iteratively updated based on the best expected costs.Hvattum et al(2006)assumed that the presence of each customer is uncertain—a customer might place orders,or not,at any time during the planning horizon.They developed a multi-stage heuristic method to construct the routes gradually by considering future customer demands deﬁned by pre-estimated probability distributions.Campbell and Thomas(2008,2009)considered a similar problem but with customer deadlines.They proposed several different models to measure the penalty function with violations of the deadlines.In their settings,however,the uncertain-ties originate from the stochastic presence of the customers, not from the travel times.The stochastic approach is limited to cases in which the stochastic properties of uncertainty can be measured precisely,which may be exceedingly problematic,especially when data are scarce. Moreover,in many cases,the service level is expressed by the nonlinear(and nonconvex)chance constrained model,which can make the problem hard to solve.Deﬁning the uncertainty set in the robust optimization approach has a number of practical advantages over estimating probability distributions in the stochastic optimization approach.Firstly,in many cases,it is easier to deﬁne the uncertainty set than to estimate the probability distributions.For example,we simply can take all past realizations of data as the uncertainty set. Secondly,under certain conditions,the robust approach does not signiﬁcantly escalate the complexity of the problem.For example,Bertsimas and Sim(2004)demon-strate that the robust counterpart problem of a linear programming problem is also a linear programming problem with a polynomially bounded problem size. Lastly,even though we already have some knowledge of the probability distributions,the uncertainty set can be easily constructed from these.For example,the conﬁdence intervals of random data can be used as the intervals of uncertain data in the uncertainty set.Never-theless,there have been only a few studies on the robust approach to the VRP.This scarcity of research on this topic may be due to(1)the VRP on its own being a hard problem and(2)the existing solution approaches for the deterministic problem being no longer valid for the robust version of the problem.The contributions of this paper to research on the VRP are as follows:(1)proposal of a robust optimization approach to the VRPD that produces robust solutions under the uncertainty of travel time and demand;(2)demonstration that the robust version of the problem can be solved by the well-known branch-and-price algo-rithm,while consideration of the uncertainty is solely encapsulated in the column generation subproblem;(3)proposal of a new uncertainty-aware dominance rule for the labelling algorithm that enables the column generation subproblem to be solved efﬁciently;(4)report-ing of an extensive computational study,which shows that in many cases the gains in robustness are rather large with small penalties in the objective values.C Lee et al—Robust vehicle routing problem with deadlines and travel time/demand uncertainty1295Problem description and formulationIn this section,a mathematical formulation for the case with no uncertainty(deterministic case)is presented. We then extend the formulation to the case of uncertain data.The problem is deﬁned with the following para-meters:N{1,...,n},set of customersN0N,{0,nþl},where0and nþ1are depotsM{1,...,m},set of vehiclesQ capacity of a vehicler j demand of customer i A NA{(i,j)|i,j A N0and i a j},set of arcs.c ij travel distance from i to j,where(i,j)A At ij travel time from i to j,where(i,j)A Ab i deadline for delivery at customer i,where i A Nx ij k decision variable.1if vehicle k travels from i to j, and0otherwises i k decision variable.Arrival time of vehicle k at customer iWithout any loss of generality,it can be assumed that the service time for customer i is included in the travel time t ij.Any vehicle should depart from depot0and arrive at depot nþ1after visiting a subset of customers.The deadline of customer i can also be represented as time window[0,b i].We may assume that the deadlines for the depots0and nþ1are0and N,which are equivalent to time windows[0,0]and[0,N],respectively,and implying that a vehicle departs depot0at time0and can arrive at depot nþ1at any time.The arrival time at customer i cannot be greater than deadline b i.The travel cost c i0for any customer i is very large in order to prevent the vehicle from returning to depot0.Similarly,c nþ1,i for any customer i is very large,to prevent the vehicle from leaving from depot nþ1.The mathematical model has two types of decision variables.Theﬁrst type of variable determines the routes of the vehicles,that is, visiting sequences of the customers.Let x ij k be1if vehicle k travels from i to j,and0otherwise.The route of vehicle k is determined by the variables x ij k,8(i,j)A A.The second type of variable determines when a vehicle arrives at each customer.Let s i k be the arrival time of vehicle k at customer i.Obviously,s i k must be less than or equal to b i if vehicle k visits i;it is meaningless otherwise.Since all vehicles depart from depot0at time0,s0k is0for all k A M.In addition, s nþ1k is the arrival time of vehicle k at the depot nþ1. The(deterministic)VRPD can be stated as follows:ðVRPDÞminXk2M Xi2N0Xj2N0c ij x kijð1Þsubject toXk2M Xj2Nx kij¼18i2N;ð2ÞXj2N0x k0j¼18k2M;ð3ÞXj2N0x kijÀXj2N0x kji¼08i2N;k2M;ð4ÞXi2N0x ki;nþ1¼18k2M;ð5ÞXi2Nr iXj2N0x kijp Q8k2M;ð6Þs kiþt ijÀKð1Àx k ijÞp s k j8i;j2N0;k2M;ð7Þ0p s k i p b i8i2N0;k2M;ð8Þx kij2f0;1g8i;j2N0;k2M;ð9Þwhere K is a large number.The objective function is thesum of all routing distances of every vehicle.Constraints(2)ensure that each customer is served by exactly onevehicle.Constraints(6)are the capacity constraints for thevehicles;constraints(3),(4),and(5)are theﬂowconservation constraints,which ensure that each vehicle0sroute should start from depot0and end at depot nþ1;constraints(7)and(8)together guarantee that eachcustomer is served before the deadline.Robust VRPDWhen a real-world logistics company uses the VRPD fordesigning its operational planning system,it must compileVRPD data,such as customer demands,vehicle capacities,deadlines,and vehicle travel times and distances.It iscommon practice that data for deadlines and vehiclecapacities are speciﬁed or given,while those for travel timesand customer demands are estimated or forecasted.Ingeneral,requested or speciﬁed data can be considered to bemore accurate than estimated data.Moreover,in terms ofthe graph of all customers and depots,one should estimatethe travel times for all pairs of nodes,since the graph iscomplete.Even when the estimations are derived viastatistical methods,the estimated nominal values may bepoor representations of true values when the variances inthe data are considerably large.There are a number of published studies on the VRPwith travel time uncertainty.Jula et al(2006)considereda nonstationary stochastic travelling salesman problem.They call a route acceptable if the probability of visitingevery node on a route before its deadline time is greaterthan a given constant,which is called the service level at thenode.Given the probability distribution of travel times,they propose a simpliﬁed way to calculate the expectedarrival time at each node of a route by approximating theexpected arrival times.Although their approach does not1296Journal of the Operational Research Society Vol.63,No.9exploit the probability distribution directly,the approx-imating procedure depends on the probability distribution assumption,requiring that the probability distributions be precisely determined,which may be a hard task to accomplish.The aim of the robust optimization is to obtain the solution that is feasible for all realizations of uncertain data.In this approach,the probability distributions for uncertain data are assumed to be unknown,and only the nominal and maximum possible deviation values are speciﬁed.The uncertainty of data is represented by the uncertainty set,which contains all possible realizations of random data.To obtain a robust—but not too conservative—solution,it is necessary to introduce some parameters to control for the degree of robustness(the reader is referred to Bertsimas et al(2004)for details). Sungur et al(2008)considered robust capacitated VRP (CVRP)with uncertain demand.They modiﬁed the original CVRP formulation to incorporate the demand uncertainty and solved the problem directly by an off-the-shelf mixed integer programming(MIP)solver.Their results demonstrate that the robust optimization approach is attractive as it produces a much more robust solution with only a small penalty in the objective value.In terms of the travel time uncertainty,it is highly unlikely that every segment on a route is delayed;in fact,it is much more likely that some segments are delayed while others are not.This observation indicates that we may restrict the number of delayed segments on a route so that we can control how much of the route should be robust.In other words,we want to protect the routes of vehicles against the given number of delays in the travel time,which yields the following deﬁnition of the uncertainty set of travel time.Deﬁnition1.Model of Travel time Uncertainty set U t. For each arc(i,j)A A,the travel time takes values in [tˆij,tˆijþd ij],where d ij represents the maximum deviation from the nominal travel time tˆij.We introduce a nonnegative integer G as a parameter for controlling the degree of robustness for the travel time uncertainties.Then, the uncertainty set of travel time data is given asU t¼(~t2R j A j j~t ij¼^t ijþd ij v ij;Xði;jÞ2Av ij p G;0p v ij p18ði;jÞ2A):Similarly,we deﬁne the uncertainty set of demand as follows.Deﬁnition2.Model of Demand Uncertainty set U r.For each customer i A N,the demand takes values in[rˆi,rˆiþo i];where o i represents the maximum deviation from the nominal demand value rˆi.We introduce a nonnegative integer L as a parameter for controlling the degree of robustness for the demand uncertainties.Then,the uncertainty set of demand data is given asU r¼(~r2R j N j j~r i¼^r iþo i w i;Xi2Nw i p L;0p w i p1;8i2N):The robust version of VRPD can then be formulated as follows:ðRCVRPDÞminXk2MXi2N0Xj2N0c ij x kijð10Þsubject to(2),(3),(4),(5)Xi2Nr iXj2N0x kijþL z kþXi2Np ikp Q8k2M;ð11Þz kþp i k X o iXj2N0x kij8k2M;i2N;ð12Þðs k iÞBþ^t ijþd ij u B ijÀKð1Àx k ijÞpðs k jÞB8ði;jÞ2A;k2M;8B A;j B j p G;ð13Þ0pðs kiÞB p b i8i2N0;k2M;8B A;j B j p G;ð14Þp ikX08i2N;k2M;ð15Þz k X08k2M;ð16Þx kij2f0;1g8ði;jÞ2A;k2M;ð17Þwhere u ij B is an indicator function which is1if(i,j)A B, 0otherwise.Constraints(11)and(12)and variables(15)and(16)are obtained from the reformulation of Pi2Nr iPj2N0x kijþmax S N;j S j p LPi2So iPj2N0x kijp Q(see Bertsimas and Sim,2003).Note that a vehicle might arrive early or late at a certain customer location on the same vehicle route due to the uncertainty of the travel time. Therefore,we introduce additional variables,namely,(s i k)B, 8B D A,|B|o G,for different realizations of travel time. It should be noted that we assume the absence of uncertainty in the distance data c ij and that we still try to minimize the sum of travel distances.The underlying motivation of this formulation is to distribute the risks of late arrivals over all of the routes as evenly as possible.C Lee et al—Robust vehicle routing problem with deadlines and travel time/demand uncertainty1297This can be done by distributing the timely tight visits to several different vehicles and making the vehicles visit those customers whose deadlines are tight at the early stages of the routes.The large number of constraints(13)and(14)makes the formulation intractable.Therefore,instead of using the formulation(RCVRPD),which is based on the edge variables,we consider the path-based formulation using the Dantzig-Wolfe style decomposition scheme(Barnhart et al,1998).We say that a route is robustly feasible if it remains feasible for all realizations of the uncertainty sets U t and U r.Formally speaking,a robustly feasible route is a path(0,i1,...,i l,nþ1)which meets the deadline and capacity constraints at each customer location,while most G(travel times)and L(demands)of uncertain data can be at their maximum deviations,where i1,...,i l A N.Let R denote the set of all robustly feasible routes.The path-based formulation for the robust VRPD can then be given as:ðPath-RCVRPDÞminXr2Rc r x rð18Þsubject toXr2Rd ir x r X18i2N;ð19ÞXr2Rx r p m;ð20Þx r2f0;1g8r2R;ð21Þwhere d ir is the number of visits to customer i in route r,m is the number of vehicles,and c r is the travel distance of route r A R,deﬁned as the sum of the distances of the arcs of the route.Since the number of robustly feasible routes of R can be exponentially large,we use a column generation method.In the column generation method,the linear relaxation of the above set covering model with a restricte d set of routes is solved,and the column generation subproblem is solved toﬁnd a column which has a negative reduced cost.When no column has a negative reduced cost, the column generation procedure is terminated.One advantage of(Path-RCVRPD)is that the same determi-nistic column generation method can be used as long as the subproblem correctly identiﬁes robustly feasible routes with negative reduced costs.In the following section,we present an algorithm forﬁnding those robustly feasible routes with negative reduced costs.Solution methodologyLet R0C R be the restricted set of robustly feasible routes, and(RM),which is the restricted master problem,denote the linear relaxation of this problem after R has been replaced with R0.ðRMÞminXr2R0c r x rð22Þsubject toXr2R0d ir x r X18i2N;ð23ÞÀXr2R0x r XÀm;ð24Þx r X08r2R0:ð25ÞNote that we do not need x r p1constraints,since at optimality,x r cannot be greater than one because we minimize the objective function.Column generation subproblemUsing the restricted set of routes R0C R,we now attempt toﬁnd the new routes—columns—entering(RM)by pricing their reduced costs.Let p i and p0denote the dual variables associated with constraint i of(23)and(24), respectively.Based on the linear programming theory, the reduced cost of route r is given as follows:rcðrÞ¼c rÀXi2Nd ir p iþp0:ð26ÞFinding routes with negative reduced cost is thereby reduced toﬁnding the shortest route with the following arc cost and respecting the deadline and vehicle capacity constraints.c0;i¼c0;iÀp i8i2N;ð27Þc ij¼c ijÀp j8i2N;j2N;ð28Þc i;nþ1¼c i;nþ1þp08i2N;ð29Þc0;nþ1¼0:ð30ÞThe travel time and vehicle capacity can be generalized as the resources consumed(or accumulated)in the route. When a vehicle visits customer j,the resource of vehicle capacity is consumed by the amount of the demand r j. Similarly,when a vehicle moves to the location of customer j from that of customer i,the resource of time is accumulated by the travel time t ij.The vehicle can move to j if the consumed(or accumulated)resources are not greater than the resource constraints at customer j. Generally speaking,more resources can be deﬁned,and1298Journal of the Operational Research Society Vol.63,No.9the shortest path problem respecting the resource con-straints is often referred to as the shortest path problem with resource constraints (SPPRC)(Irnich and Desaulniers,2004).Here we introduce our robust version of SPPRC (RSPPRC).In RSPPRC,the amount of resource to be used is uncertain,and possible resource usages are deﬁned by the uncertainty sets.A path is robustly feasible if—and only if—all of the resource constraints at every customer are satisﬁed at all realizations of the resource usages deﬁned by the uncertainty sets.We consider a graph G whose nodes are customers (N )and depots (0,n þ1),and (directed)arcs have costs of (27),(28),(29),and (30).In the standard algorithm for the SPPRC,each possible partial path is associated with a label,which represents the consumption (or accumula-tion)of the resources of the partial path.At the extending stage,all new partial paths are extended toward every possible successor node.At the elimination stage,a label is eliminated if it is dominated by some other label.A brief description of Desrochers’labelling algorithm and our adaptation to the RSPPRC are given in the following section.Desrochers’labelling algorithmFor a partial path p which ends at node i ,we associatea label E p ¼[c p ,R p 1,R p 2,...,R p L]with the path,where L is the number of resources.Let c p be the cost of path p .The resources can include vehicle capacity,travel time,and distance,among others.Here R p l represents the accumu-lated value of resource/at the last node of path p .Let v (p )denote the last node of path p ,that is v (p )¼i .A path q is a feasible extension of p if the path (p ,q ) {0,p 1,p 2,...,v (p ),q 1,q 2,...,v (q ))satisﬁes all resource constraints at every node in the path.Let E (p )be the set of all feasible extensions of p .The nonnegative value r ij l is deﬁned as the usage value of needed resource l when we travel from i to j ,that is R p l ¼P (i ,j )A A (p )r ij l,where A (p )is the set of arcs of path p .Let p and q be two distinct paths from 0to i with associated label E p and E q ,respectively.We say pdominates q if and only if c p p c q ,R p l p R q l,for all l ¼1,...,L .The domination is strict if c p o c q or thereexists l such that R p l o R q l.We denote these by E p o E q and E p o E q ,respectively.According to these deﬁnitions,the dominance rule is established as follows:Deﬁnition 3.Dominance Rule for SPPRC.Given two distinct paths p and q such that v (p )¼v (q ),the path q can be discarded if p strictly dominates q ,that is,E p o E q .Either p or q can be discarded if they dominate each other,that is,E p p E q and E p X E q .The validity of the dominance rule can be easily demon-strated.When path p dominates q ,clearly E (p )+E (q )and for any q þA E (q ),there is a p þA E (p )such that (p ,p þ)dominates (q ,q þ).The labelling algorithm is an iterative process of the path extending procedure (E XTENDING )and the label elimination procedure (E LIMINATE ).Note that the algorithm is quite general,in the sense that many types of resource constraints can be modelled within this frame-work.This algorithm allows multiple visits to a node,so the resulting path may contain cycles.Uncertain resources caseThe dominance rule of Deﬁnition 3is no longer valid when the resource usage value r ij l is uncertain.There are two types of uncertainty sets,namely,U t and U r ,in our setting.From now on,we only consider the travel-time uncertainty set U t ,since the following line of reasoning on U t can be readily extended to demand uncertainty set U r .For any resource l ,we apply the deﬁnition of travel-time uncertainty set U t (Deﬁnition 1).For a path p and givenG ,let ~R l p¼Pði ;j Þ2A ðp Þ^r l ijþmax f S &A ðp Þjj S j p G g P ði ;j Þ2Sd lijwhere r ˆijl is the nominal value of the consumption ofresource l when we move from node i to j ,and d ij lis the value of the maximum deviation.A path p ¼(0,p 1,p 2,...,p o ,n þ1)is robustly feasible if and only if,for alll A {1,...,L },~R l pk satisﬁes the resource constraints for all partial paths p k ¼(0,p 1,p 2,...,p k ),8k A {0,1,...,o ,n þl}.Without any loss of generality,we assume that the resource 1is subject to uncertainty and the degree of robustness G is given.For a partial path p ,such that v (p )¼i ,an associated label is deﬁned as follows:~E p ¼c p ;^R 1p ;D 1p ;...;D 2p ;...;D G p zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{G;R 2p ;...;R L p 264375;where D p k ¼max {S C A (p )||S |p k P (i ,j )A S d ij 1,k ¼1,...,G ,that is,the sum of k largest d ij 1s on path p ,R ˆp 1¼P (i ,j )A A (p )r ˆij 1and L the number of resources.Proposition 1.For two distinct paths p and q ,such thatv (p )¼v (q ),~Ep p ~E q if and only if p dominates q .Proof For the sufﬁcient condition (~Ep p ~E q )p domi-nates q ),we have to show that REðp Þ REðq Þwhen ~Ep p ~E q ,where REðp Þis the set of all robustly feasible extensions of p .Consider a robustly feasible path Q ¼(q ,q þ).By deﬁnition,q þ2REðq Þ.Let A þ¼A (Q )\A (q )and S k A ¼{S C A ||S |p k },then it is easily seen that the following holds:maxS 2S A ðQ ÞGXði ;j Þ2Sd 1ij¼maxk ¼0;...;G ;S 2S A ðq Þk [S A þGÀkXði ;j Þ2S d 1ij ¼maxk ¼0;...;G maxS 2S A ðq Þk [S A þG ÀkXði ;j Þ2Sd 1ij ¼max k ¼0;...;GðD k q þD G Àkq þÞ:C Lee et al —Robust vehicle routing problem with deadlines and travel time/demand uncertainty 1299。

动态车辆路径问题-定义

动态车辆路径问题定义
动态车辆路径问题（Dynamic Vehicle Routing Problem）是指在车辆运输过程中，需要根据实时变化的需求和路况等因素，动态地调整车辆的行驶路径和任务分配的问题它与传统的静态车辆路径问题不同，静态车辆路径问题是在运输任务和路况等信息已知的前提下进行路径规划，而动态车辆路径问题则需要考虑到任务的实时变化和路况的不确定性。

在动态车辆路径问题中，车辆可能需要实时接收新的任务、处理任务取消或延迟等情况，同时还需要考虑到交通拥堵、道路施工等实时路况信息，以优化行驶路径，提高运输效率和客户满意度这个问题在物流配送、出租车调度、急救车辆派遣等领域都有广泛的应用。

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Keywords: Vehicle routing; Time dependent; Travel time; Genetic algorithm; Optimization; Network
1. Introduction Travel times in urban areas uctuate due to a variety of factors, such as accidents, tra c conditions and weather conditions. Ignoring these travel time variations when developing route plans for pick-up
Abstract In this paper we present a formulation for the dynamic vehicle routing problem with time-dependent travel times. We also present a genetic algorithm to solve the problem. The problem is a pick-up or delivery vehicle routing problem with soft time windows in which we consider multiple vehicles with di erent capacities, real-time service requests, and real-time variations in travel times between demand nodes. The performance of the genetic algorithm is evaluated by comparing its results with exact solutions and lower bounds for randomly generated test problems. For small size problems with up to 10 demands, the genetic algorithm provides almost the same results as the exact solutions, while its computation time is less than 10% of the time required to produce the exact solutions. For the problems with 30 demand nodes, the genetic algorithm results have less than 8% gap with lower bounds. This research also shows that as the uncertainty in the travel time information increases, a dynamic routing strategy that takes the real-time tra c information into account becomes increasingly superior to a static one. This is clear when we compare the static and dynamic routing strategies in problem scenarios that have di erent levels of uncertainty in travel time information. In additional tests on a simulated network, the proposed algorithm works well in dealing with situations in which accidents cause signiÿcant congestion in some part of the transportation network. ? 2004 Elsevier Ltd. All rights reserved.
a
Department of Civil and Environmental Engineering, University of Maryland, College Road, MD, USA b 600 Maryland Ave. SW, Suite 755, Washington, DC 20024, USA
Computers & Operations Research 32 (2005) 2959 – 2986
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A dynamic vehicle routing problem with time-dependent travel times
Ali Haghania ; ∗ , Soojung Jungb
0305-0548/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cor.2004.04.013
2960
A. Haghani, S. Jung / Computers & Operations Research 32 (2005) 2959 – 2986
Travel Time
ABa来自cbd
Time of a day
Fig. 1. Travel time variation as a step function.
and/or delivery vehicles can result in route plans that direct the vehicles into congested urban tra c conditions. Due to these travel time variations, in some cases the vehicles waste valuable time in tra c jams and customers have to wait unreasonably long without having any reliable information about the actual arrival times of vehicles. In these circumstances, it becomes di cult to satisfy the time windows during which the demand nodes must be visited. In addition, insertion of new demands for pick-up that arise after completion of route planning, in the planned vehicle routes in real time, may result in signiÿcant savings. Considering time-dependent travel times as well as information regarding demands that arise in real time in solving vehicle routing problems can reduce the costs of ignoring the changing environment. While considerable research has been devoted to the general traveling salesman problem and vehicle routing problem, research in time-dependent vehicle routing problem is sparse. The existing literature in dynamic vehicle routing problem is focused on dynamic traveling salesman problem [1–3], dynamic traveling repairman problem [2,4,5], dynamic dial-a-ride problem [1,6,7], and dynamic vehicle allocation problem [8–13], Bowman [14], Picard and Queyranne [15], Lucena [16], Malandraki and Daskin [17], Hill and Benton [18], Malandraki and Dial [19], and Ichoua et al. [20] examined the time-dependent vehicle routing problem. The most similar papers are Malandraki and Daskin [17], Hill and Benton [18], Malandraki and Dial [19], and Ichoua et al. [20]. The major weakness of these papers is that they cannot deal with “ÿrst-in-ÿrst-out” (FIFO) property. In these papers the planning period is divided into very few intervals and travel time is considered a step function as shown in Fig. 1. In this case, it is possible that a vehicle that departs at time d can arrive earlier than a vehicle that departs earlier at time c. Because any departure between a and b will arrive at the destination later than any departure at time b, all departures have to wait until time b. This is not a realistic situation and implies that real-world link travel times cannot be represented by a step function. Ichoua et al. [20] proposes a solution methodology to solve time-dependent vehicle routing problems with time windows using Tabu search, which satisÿes the FIFO assumption. This paper is very similar to ours in terms of methodology. Experimental results show that the time-dependent model provides signiÿcant improvements when results from the time-dependent speed model and the constant speed model are compared. Ichoua et al. divide the planning horizon into three time