时间窗约束下的车辆路径问题遗传算法外文翻译(可编辑)

时间窗约束下的车辆路径问题遗传算法外文翻译
外文翻译
原文
Genetic Algorithms for the Vehicle Routing Problem with Time Windows Material Source: Special issue on Bioinformatics and Genetic AlgorithmsAuthor: Olli Br?ysy
1 Introduction
Vehicle Routing Problems VRP are all around us in the sense that many consumer products such as soft drinks, beer, bread, snack foods, gasoline and pharmaceuticals are delivered to retail outlets by a fleet of trucks whose operation fits the vehicle routing model. In practice, the VRP has been recognized as one of the great success stories of operations research and it has been studied widely since the late fifties. Public services can also take advantage of these systems in order to improve their logistics chain. Garbage collection, or town cleaning, takes an ever increasing part of the budget of local authorities A typical vehicle routing problem can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points cities, stores, warehouses, schools, customers etc. The routes must be designed in such a way that each point is visited only once
by exactly one vehicle, all routes start and end at the depot, and the total demands of all points on one route must not exceed the capacity of the vehicle The Vehicle Routing Problem with Time WindowsVRPTW is a generalization of the VRP involving the added complexity that every customer should be served within a given time window. Additional complexities encountered in the VRPTW are length of route constraint arising from depot time windows and cost of waiting time, which is incurred when a vehicle arrives too early at a customer location. Specific examples of problems with time windows include bank deliveries, postal deliveries, industrial refuse collection, school-bus routing and situations where the customer must provide access, verification, or payment upon delivery of the product or service[Solomon and Desrosiers, 1988].
Besides being one of the most important problems of operations research in practical terms, the vehicle routing problem is also one of the most difficult problems to solve. It is quite close to one of the most famous combinatorial optimization problems, the Traveling Salesperson ProblemTSP, where only one person has to visit all the customers. The TSP is an NP-hard problem. It is believed that one may never find a computational technique that will guarantee optimal solutions to larger instances for such problems. The vehicle routing problem is even more complicated. Even for small fleet sizes and a moderate number of transportation requests, the planning task is highly complex. Hence, it
is not surprising that human planners soon get overwhelmed, and must turn to simple, local rules for vehicle routing. Next we will describe basic principles of genetic algorithms and some applications for vehicle routing problem with time windows.
2 General principles of genetic algorithms
The Genetic Algorithm GA is an adaptive heuristic search method based on population genetics. The basic concepts are developed by [Holland, 1975], while the practicality of using the GA to solve complex problems is demonstrated in [De Jong, 1975] and [Goldberg, 1989]. References and details about genetic algorithms can also be found for example in [Alander, 2000] and [Mühlenbein, 1997] respectively. The creation of a new generation of individuals involves primarily four major steps or phases: representation, selection, recombination and mutation. The representation of the solution space consists of encoding significant features of a solution as a chromosome, defining an individual member of a population. Typically pictured by a bit string, a chromosome is made up of a sequence of genes, which capture the basic characteristics of a solution.
The recombination or reproduction process makes use of genes of selected parents to produce offspring that will form the next generation. It combines characteristics of chromosomes to potentially create offspring with better fitness. As for mutation, it consists of randomly
modifying genes of a single individual at a time to further explore the solution space and ensure, or preserve, genetic diversity. The occurrence of mutation is generally associated with low probability. A new generation is created by repeating the selection, reproduction and mutation processes until all chromosomes in the new population replace those from the old one. A proper balance between genetic quality and diversity is therefore required within the population in order to support efficient search.
Although theoretical results that characterize the behavior of the GA have been obtained for bit-string chromosomes, not all problems lend themselves easily to this representation. This is the case, in particular, for sequencing problems, like vehicle routing problem, where an integer representation is more often appropriate. We are aware of only one approach by [Thangiah, 1995] that uses bit string representation in vehicle routing context. In all other approaches for vehicle routing problem with time windows the encoding issue is disregarded.
3 Applications for vehicle routing problem with time windows
[Thangiah, 1995] describes a method called GIDEON that assigns customers to vehicles by partitioning the customers into sectors by genetic algorithm and customers within each formed sector are routed using the cheapest insertion method of [Golden and Stewart, 1985]. In the next step the routes are improved using λ-exchanges introduced by [Osman,
1993]. The two processes are run iteratively a finite number of times to improve the solution quality. The search begins by clustering customers either according to their polar coordinate angle or randomly. The proposed search strategy accepts also infeasibilities during the search against certain penalty factors. In the GIDEON system each chromosome represents a set of possible clustering schemes and the fitness values are based on corresponding routing costs. The crossover operator exchanges a randomly selected portion of the bit string between the chromosomes and mutation is used with a very low probability to randomly change the bit values.
[Potvin and Bengio, 1996] propose a genetic algorithm GENEROUS that directly applies genetic operators to solutions, thus avoiding the coding issues. The initial population is created with cheapest insertion heuristic of [Solomon, 1987] and the fitness values of the proposed approach are based on the number of vehicles and total route time. The selection process is stochastic and biased toward the best solutions. For this purpose a linear ranking scheme is used. During the recombination phase, two parent solutions are merged into a single one, so as to guarantee the feasibility of the new solution. Two types of crossover operators are used to modify a randomly selected route or to insert a route into the other parent solution. A special repair operator is then applied to the offspring to generate a new feasible solution. Mutation operators are aimed at reducing the number of routes. Finally, in order to locally
optimize the solution, mutation operator based on Or-opt exchanges [Or, 1976] is included.
[Berger et al., 1998] propose a method based on the hybridization of a genetic algorithm with well-known construction heuristics. The authors omit the coding issues and represent a solution by a set of feasible routes. The initial population is created with nearest neighbor heuristic inspired from [Solomon, 1987]. The fitness values of the individuals are based on the number of routes and total distance of the corresponding solution and for selection purposes the authors use the so called roulette-wheel scheme. In this scheme the probability of selecting an individual is proportional to its fitness; for details see [Goldberg, 1989]. The proposed crossover operator combines iteratively various routes r1 of parent solution P1 with a subset of customers, formed by r2 nearest-neighbor routes from parent solution P2. A removal procedure is first carried out to remove some key customer nodes from r1. Then an insertion heuristic inspired from [Solomon, 1987] coupled to a random customer acceptance procedure is locally applied to build a feasible route, considering the partial route r1 as an initial solution. The mutation operators are aimed at reducing the number of routes of solutions having only a few customers and locally reordering the routes.
[Br?ysy, 1999a] and [Br?ysy, 1999b] extended the work of [Berger et al., 1998] by proposing several new crossover and mutation operators,
testing different forms of genetic algorithms, selection schemes, scaling schemes and the significance of the initial solutions. When it comes to recombination, an approach where customers within randomly generated segments in parent solution P1 are replaced with some other customers on the near routes of parent solution P2 is found to perform best. The best-performing mutation operator selects randomly one of the shortest routes and tries to eliminate it by inserting the customers into other longer routes. Regarding different forms of genetic algorithms it is concluded that it is important to create many new offspring each generation and it is enough to maintain only one population. For selection purposes so-called tournament selection is found to perform best. In the first phase two individuals are selected with a random procedure that is biased towards better fitness scores. In the second phase, the individual with better fitness is selected. However the differences between different schemes were minor. A new scaling scheme based on a weighted combination of number of routes, total distance and waiting time is found to perform particularly well. Finally to create the initial population, several strategies, such us heuristics of [Solomon, 1987] and randomly created routes were tried and it was concluded that the best strategy is to create a diverse initial population that also contains some individuals with better fitness scores.
[Homberger and Gehring, 1999] propose two evolutionary
metaheuristics for VRPTW. The proposed algorithms are based on the class of evolutionary algorithms called Evolution Strategies. Differences to GA exist with regard to the superior role of mutation compared to the recombination operators. Here the individual representation also includes a vector of so called strategy parameters in addition to the solution vector and both components are evolved by means of recombination and mutation operators. In the proposed application for VRPTW these strategy parameters refer to how often a randomly selected local search operator is applied and to binary parameter used to alternate the search between minimizing the number of vehicles and total distance.
Selection of the parents is done randomly and only one offspring is created through the recombination of parents. This way a number λ?μoffspring is created, where μis the population size. At the end fitness values are used to select μ offspring to the next population. Because the parents are not involved in the selection process, deteriorations during the search are permitted. The first out of the two proposed metaheuristics, evolution strategy ES1, skips the recombination phase. The second evolution strategy ES2 uses uniform order-based crossover to modify the initially randomly created mutation codes of the two parents and tries to improve the solution vector of a third randomly selected parent using the modified code. The mutation code is used to control a set of removal and insertion operators performed by Or-opt operator [Or,
1976]. The fitness values are based on number of routes, total travel distance and on a criterion that determines how easily the shortest route of the solution in terms of the number of customers on the route can be eliminated. The individuals of a starting population are generated by means of a stochastic approach, which is based on the savings algorithm of [Clarke and Wright, 1964].
[Br?ysy et al., 2000] describe a two-phase hybrid evolutionary algorithm based on the hybridization of a genetic algorithm and an evolutionary algorithm consisting of several local search and route construction heuristics inspired from the studies of [Solomon, 1987] and [Taillard et al., 1997]. In the first phase a genetic algorithm based on the studies [Berger et al., 1998] and [Br?ysy, 1999a] is used to obtain a feasible solution. The evolutionary algorithm used in the second phase picks every pair of routes in random order and applies randomly one out of the four local search operators or route construction heuristics. Finally, offspring routes generated by these crossover operators are mutated according to a user-defined probability by selecting randomly one out of two operators. Selecting each possible pair of routes, mating and mutation operators are repeatedly applied for a certain number of generations and finally a feasible solution is returned. To escape from local minimum, arcs longer than average are penalized, if they appear frequently during the search.
译文
时间窗约束下的车辆路径问题遗传算法
资料来源:生物信息和遗传算法特刊作者:奥利布瑞赛
1 简介
车辆路径问题(VRP)对我们身边许多消费品都很有意义,例如软性饮料,啤酒,面包,小吃,汽油和药品通过一个车队走合适的运输路线,运送至零售网点。

在实践中,VRP被认为是一个非常成功的运行研究故事,自五十年代后期以来,它已被的广泛研究。

公共服务也利用这些系统的优点,来提高物流链。

垃圾收集,城镇的清洁,也需要地方当局需要增加预算的部分。

一个典型的车辆运输路径问题可以被描述为,设计一个最少成本的路线,即从一个仓库,到一系列地理上分散的点(城市,商店,仓库,学校,客户等)。

这些路线的设计必须遵循每一个点被一辆运输车访问一次,所有路线起点和终点在一个仓库,所有点的总需求量,不得超过车辆的运输能力。

在时间窗约束下的路径问题(VRPTW)是VRP的概括,规定应在一个客户给定的时间窗内送达,增加了复杂性。

在VRPTW中遇到的额外复杂过程是,从一个仓库时间窗的路线长度限制提高,以及等待时间的成本,这是发生在车辆到达客户点太早所造成的。

时间窗的具体问题结合实例,包括银行交付,邮递,工业垃圾收集,校巴路线和路况,这些客户必须在运输物品或提供服务后,提供通道,证明,或者付款[所罗门和戴斯洛赛斯,1988]。

在实际操作中,除了是一个最重要的操作研究问题外,车辆路径问题也是最难解决的问题之一。

它非常接近于最著名的组合优化问题??旅行商问题,即
一个商人必须访问所有客户,则拜访路线要怎么选择。

旅行商问题是一个非常难的问题。

它被认为是一个永远找不到计算结束去保证这是最佳解决方案的问题,而这些运输路径问题则更为复杂。

即使是小型车型和中型数量的运输要求,需规划的任务也是非常复杂。

因此,若策划者很快就会不堪重负,也是并不奇怪的,必须将其转化为简单的,适应本地规则的运输路径。

下一步,我们将介绍遗传算法的基本原理和一些时间窗约束下的车辆路径问题。

2 遗传算法的一般原则
遗传算法(GA)是一种在人口遗传基础上,适应启发式的探访方法。

基本概念是发展于[荷兰,1975],而实际上将遗传算法用于解决复杂的问题,被证明是在[德容,1975]和[哥德堡,1989]中。

文献和遗传算法的细节也在[阿兰德,2000]和[穆棱贝,1997]中找到。

新的一代独立创作,主要涉及四个步骤或阶段:代表,选择,重组和突变。

解决方案的代表由显著特点的染色体编码组成,确定为一个独立的个体。

通常染色体有一系列的基因组成,这就抓住了一个解决方案的最基本特征。

重组或者再生产过程可以使父母选择使用的基因,并形成下一代。

它结合染色体特征,能使创造出的后代更具有适应性。

至于突变,它由一个独立的随机更改的基因组成,能进一步探索解决方案和确保,或保存遗传多样性。

这个突变的发生通常概率很低。

新一代的创建,通过复制选择,再生产和突变过程,直到所有新的染色体在人类中产生,取代旧的。

在遗传质量和多样性之间的适当的平衡,是需要在人群中,用来支持高效的搜索。

虽然理论结果表明,遗传算法的独特已经获得染色特的位串,但并不能简单地代表所有问题。

这样的话,特别是,对于时常发生的问题,如车辆路径问题,
一个整数就代表更正确。

我们知道只有一个方式[衫葛,1995],在车辆路径文件中运用位串代表。

在其他时间窗约束下的路径问题中,可以忽视编码问题。

3 时间窗约束下的路径问题的应用
[衫葛,1995]描述了一个叫做基甸分配的方法,即运送指定客户,通过遗传算法将客户分割成几个部分,客户之间路线的形成要用最低级的插入方式[戈登和斯图沃特,1985]。

在下一步中,路线用λ转变得到改善[奥斯曼,1993]。

这两个进程是有限次数的迭代运行,能提高解决方案的质量。

探索从聚类的客户开始,根据他们的极坐标角度或随机都可以。

这个探索策略建议的接受,对于探索中的反抗特定惩罚因素是不可行的。

在基甸系统中,根据相应的路径成本,每一个染色体代表了一系列聚类计划以及适应价值的可能。

交叉操作与随机选择部分在染色体和变异之间交换,被用于随机的价值的概率是非常低的。

[波文和班戈,1996]提出了一种遗传算法,即直接使用遗传操作去解决问题,从而避免了编码问题。

最初的人口计算是最低级的插入启发式创建的[所罗门,1987],适应价值的接受方法是基于路线总共的车辆数和总的所用时间。

选择过程是随机的并偏向最好的解决方案。

为此,使用线性排名计划。

在重组阶段,两个起始方法,合并为一个,因此能保证新方案的可行性。

两种交叉运行用来修饰随机选择路径,或在其他起始方案中插入一种路径。

一个特约维修商,是将其运用到产生下一代的新的可行方案。

变异算子的目的在于减少路径的数量。

最终,为了局部优化的解决方案,变异算子基于or-opt[奥尔,1976]。

[贝格等,1998]提出基于良好构架的启发式的杂交遗传算法。

作者省却了编码问题,并代表一系列可行路径的解决方案。

带有近邻的初始种群的创建灵感来自于[所罗门,1987]。

个人的适应价值基于路径数量,相应的总距离和选择目的,因此作者称其为赌轮计划。

在这项计划中,选择一个个
体的概率是与它的适当价值成正比;详情参见[哥德堡,1989]。

建议的解决方案是,交叉算子与迭代不同路径r1想结合,母体方案P1连带客户子系统,形成从母体方案P2中的近邻路径r2。

一个删除过程是首先从r1中消除一些关键客户节点。

然后插入一个启发式灵感[所罗门,1987]连接一个随机的客户接受程序,是建立一个可行路径,考虑部分路径r1作为一个初始解决方案。

该变异算子的目的是减少那些只有一点客户并且局部的重新安排路线的解决方案。

[布瑞赛,1999a]和[布瑞赛,1999b]将工作[伯格等,1998]通过提出一些新的交叉和变异算子,测试不同形式的遗传算法,选择计划,缩放计划和重要的初始方案。

当涉及到重组,用起始方案P1中的随机遗传环节的客户取代另外一些在起始方案P2上的靠近路线的客户效果是最好的。

最佳的变异算子随机选择是最短路径,并尝试通过插入客户到其他较长路径中,以消除它。

对于不同形式的遗传算法,它被断定对创造许多新后代非常重要,它也足以维持一族人群。

选择的目的,即所谓锦标赛选择是最好的形式。

在第一阶段,两个个体的随机选择,偏向更好的健康分数。

在第二阶段,更适合的个体被选择。

然而,不同计划之间的差异是次要的。

一个新的拉伸计划,是基于路线数目,总距离和等待时间结合的一个加重量,总被发现进行的特别好。

最后为了创建初始人群,一些策略,例如启发式[所罗门,1987]和尝试随机创建路线,它被认定是创建多样化初始人群的最好策略,并且包含了更好的健康分数。

[厚堡葛和格瑞,1999]提出两个VRPTW的进化启发式。

这个算法的提出是基于进化算法的一种叫做进化策略。

与重组运营相比,不同于遗传算法在异性方面存在突变的优越地位。

在这里,个人代表还包括一种向量叫做策略参数,将其加入到方案向量,这两个组成部分往往都是公司重组和突变的手段。

在为VRPTW这些策略参数所提出的应用中,提供随机选择局部研究的频率和二进制参数用于替
代最少车辆和最短距离的研究。

随机的母体选择,只有一个后代是通过母体重组得到的。

后代的创造是这样一个数据λμ,其中μ表示人口规模。

最终的健康价值被用于选择μ中后代至下一代人口。

由于母体不涉及选择过程,退化是被允许的。

首先得出的是两个启发式建议,进化策略ES1,跳过重组阶段。

第二个进化策略ES2用于统一以秩序为基础的交叉算子,修改初始创建的随机突变代码,这些代码是两个母体和试图提高解决第三次随机向量,并用随机选择母本修改代码。

突变代码是用来控制一系列的清楚和插入操作,这些是通过or-opt[奥尔,1976]实现的。

健康价值是基于路径数量,总距离和用什么标准来决定如何容易地选择最短路径,客户数量在路径方案上可以被除去。

人群中开始产生个体是通过随机方式形成的,这是基于储存算法[克拉克和赖特,1964]。

[布瑞赛等,2000]描述了一个两阶段混合进化算法,这是基于杂交遗传算法和进化遗传。

其中进化遗传包括一些局部研究和路线结构的启发式灵感[所罗门,1987]和[泰拉德等,1997]。

在第一阶段,基于[伯格等,1998]的遗传算法和[布瑞赛,1999a]用于获得一个可行的解决方案。

第二阶段的进化算法在随机顺序中选取每对路线,适用于随机的四个局部搜索营运商或启发式路径结构。

最后,这些后代路径所产生的交叉算子突变,是根据通过一两个营运商的用户定义概率来判定。

选择每一个可能配对的路线,交配和突变的运营商一再申请了确定的后代数量和最终返回的可行方案。

为了逃避局部最小,如果在搜索中平凡出现,弧线则长于平均处罚。