大米分级下料装置及整体结构的设计（含19张CAD图纸）
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由一个单一的存储/检索机服务的多巷道自动化立体仓库存在的拣选分拣问题Yaghoub Khojasteh-GhamariYaghoub Khojasteh Jae-Dong Son加马里筑波大学 崇实大学日本 韩国Abstract摘要随着Recent technological developments have revolutionized the design and operation of ware-现代化科技的发展，仓库式存储系统在设计与运行方面出现了巨大的改革。自动化立体仓库（AS / RS）嵌入计算机驱动正变得越来越普遍。由于AS / RS使用的增加对计算机控制的需要与支持也在提高。这项研究解决了在多巷道立体仓库的拣选问题，在这种存储/检索（S / R）操作中，每种货物可以在多个存储位置被寻址到。提出运算方法的目标是，通过S/R系统拣选货物来最大限度的减少行程时间。我们开发的遗传式和启发式算法，以及通过比较从大量的问题中得到一个最佳的解决方案。Keywords: Automated Warehouse, AS/RS, Order Picking, Genetic Algorithms. 关键词：自动化立体仓库，AS / RS系统，拣选，遗传算法。 1.Introduction导言 在现今的生产环境中，库存等级保持低于过去。那是因为这种较小的存储系统不仅降低库存量还增加了拣选货物的速度。自动化立体仓库（AS / RS），一方面通过提供快速响应，来达到高操作效率；另一方面它还有助于运作方面的系统响应时间，减少的拣选完成的总行程时间。因此，它常被用于制造业、储存仓库和分配设备等行业中。 拣选是仓库检索功能的基本组成部分。它的主要目的是，在预先指定的地点中选择适当数量的货物以满足客户拣选要求。虽然拣选操作仅仅是物体在仓储中装卸操作之一，但它却是“最耗时间和花费最大的仓储功能。许多情形下，仓储盈利的高低就在于是否能将拣选操作运行处理好”。 （Bozer和White）Ratliff和Rosenthal，他们关于自动化立体仓库系统（AS/RS）的拣选问题进行的研究，发明了基图算法，在阶梯式布局中选取最短的访问路径。Roodbergen 和 de Koster 拓展了Ratliff 和Rosenthal算法。他们认为，在平行巷道拣选问题上，应该穿越巷道末端和中间端进行拣选，就此他们发明了一种动态的规划算法解决这问题。就此Van den Berg 和 Gademann发明了一种运输模型(TP)，它是对于指定的存储和卸载进行测算的仪器。他们表示，最好的解决运输问题的方法是以机械的最佳布局来尽量减少运行时间。Elsayed对阶梯结构的立体仓库问题的研究表明，要在多巷道中拣选货物并拟定最佳方案，是非常困难和并且耗时的。 Elsayed 和 Stern提出了启发式算法，但据说，他们并没有在实际生产过程中得到满意的结果。黄禹锡等人，研究了立体仓库系统中的单巷道选道的问题，并提出决定了每个S /R系统拣选效率的启发式算法。Thealgorithms在聚集前人分析的基础上，采取了一些相似的措施。在1983年，通过仿真，把计算得到的参数与Elsayed和Sterns的结论进行了比较。Bozer、White、Han、Lee和Schaefer等人提出了一个程序，在检索测序的基础上进行优化，解决了线性分配的问题。Lee 和 Schaefer介绍了一些优化和启发式的测序方法，其中包括存储指令如何被分配到预先确定的存储位置。Mahajan通过对小件货物的贮存系统进行了改善，得到了一种新的检索测序方案，提出最近检索原则并开发了一个验证模型来预测效果。黄禹锡制作了非线性数学模型，开发出以一种启发式程序设计的自动化立体仓，与此同时还可以确定单位负载的大小。Van den Berg 和 Rouwenhorst 调查了仓库规划和控制的文献，规划文件包括存储位置的分配问题，仓库储存系统的控制问题包括路由、排序、调度、停留点的选择和秩序配料。 Goetschalckx 和 Wei提交1985年至1992年拣选系统的参考文献。Koh提出了一些关于在存储仓库中，带有塔式起重机的自动化立体仓库的模式。他们推论出的这个模式是建立在随机存储分配规则的基础上的一个单、双指令周期。他们还根据营业额的存储分配规则计算出相应行程时间。Koh提出了优化模式，在拣选系统的巷道最末端寻找到了一个最佳缓冲的区域，在那里S/R系统可提供多若干个通行巷道。Amato以colored timed Petri nets网站的资料为基础提出了对顺序检索的拣选优化算法。他们还提出了两项对于起重机和航天飞机的运作的优化控制算法。Hsu审议多巷道的仓库的顺序配料问题，提出了遗传算法来减少总旅行距离。Hwang 和 Cho提出了采摘的供应中心仓库秩序的绩效评估模式。他们研究的目的是通过减少运输数量、计算性能和设备利用率来减少尽量减少成本。在近期的研究中，De Koster 对设计与控制手册中拣选工程的典型决定问题进行了文献回顾。他们主要关注于存储分配方法、路径的选择、配料和分区。然而，我们没有这么多的文献上的知识，在处理自动化立体仓库的拣选问题上，每个物品都能够被储存在多个储存点里。事实上，许多厂家的产品有许多类型、种类和形状，这也是他们成品仓库面临的问题。例如一个瓷砖制造商，他的产品有两个类型（墙砖和地砖）,分别有7中不同的尺寸，4种不同耐久性（磨损差饷)和100多种不同的颜色、图案、颜色和形状，总共有5600多种不同的产品类型。作为存储策略，要一件刚进来的货物存放在最近的空仓位位置上。当一个来自仓库中物品，由于产品种类繁多，有很大的可能性从一个地方存入到另一个地方。因此，一件物品需要有几个在仓库中存储位置。换句话说，由于分类和分区，每个单独类型的产品在仓库中需要一个更大的空间，一个物品在几个地方存储时不可避免的。2.问题描述在本研究中，我们考虑到了小件物品的自动存储和检索系统，那有一个或多个巷道。每个巷道包含了关于巷道两旁仓储货架。每个巷道结束的地方都有一个输入/输出口（I/O）。在那里还有一个单独的存储/检索（S / R）的仪器来为所有巷道的系统服务，它可以同时在垂直和水平方向移动。因此，在两点之间的行程等于最小的水平和垂直行程。在收到命令之前S/R仪器已经定位了输入/输出口中的位置。仪器的起始位置取决于最后一件货物的最后一个命令的存储位置。S/R计算行程中以恒定的速度水平和垂直移动。一个命令可以由多个货物请求组成的。同样每个货物也可以在仓库中多个位置存储。当检索请求包括多个货物，并且这些货物在多个不同的仓库位置时，S/R仪器必须到多个不同的存储地点完成各个命令。本次研究的目的就是提出计算方法来减少S/R走过的总时间来完成命令程序。3.运算方法我们现在有两种运算方法来解决这个问题：一种是探索式算法，还有一种是遗传式算法。为了显示所提出算法的优越性，我们把它与其他方法进行了比较。由于我们的解决问题方法是新提出的，没有前人在这个领域进行过研究，那么我们最先提出的一种运算法，用它来获取的最佳的解决方案，这种方法我们称它为例证算法。其结果作为对于两种拟议算法比较的基准解决方案。在例证法中，我们确定所有可行的解决方法并将他们互相比较找出最好的解决方法来。为此，这个方案首先要找所有可行的方法来选择一个命令。然后，S/R系统的计算获得每个方法行程的总时间，最后，选取的解决方案要求在最短时间内完成要求。这个解决方案被认为是该问题的最佳解决方案。考虑到一个命令的由k种不同类型的货物组成，其中在ni(i = 1, 2, . . . , k)项货物中第i项货物被提出请求。在可行的解决办法总数挑选顺序可以给出：其中，mi是在第i项货物在仓库中的总库存，得出：通过例证法已经解决了各种类型的问题，并且确定了这种低金额低行程的最佳方案。我们发现，在当前巷道上存在货物（如：该巷道的S/R系统是在检索过程的起始端）是解决这个问题的关键技术。我们基于先前提到的运算结果发现了一种计算方法，称它为现有巷道探索式（CAH)算法。在现有巷道探索式算法中，在当前巷道中现存的货物是首先被检索的对象。其后，对该命令的其余部分（如果有的话）选中并运用各种检索方式进行研究计算。我们可以简单的对其进行表达，如果设r表示在现有巷道中指令货物的数目，那么如果r=0时，该运算方法就类似于原来的例证法。如果r=1时，该运算方法首先要通过S/R系统对行程时间进行计算，设t1表示在当前巷道中，现存货物为了避免与拣选中的货物冲突，对于其余的货物(如果有的话)进行同等于例证法的计算，以此来得到最小的计算行程时间。设t2表示在S/R系统中总的行程时间。最后将t1和t2之和作为最终的解决方案。如果r1时，则该方法首先分配拣选顺序，拣选所有的r货物，既巷道中的现存货物。在计算好行程时间之后，进入t1阶段开始移除列表中指令的货物。在这之后，其余货物(如果有的话)进行类似于例证法的运算,就如同，通过对每一个可行的方法计算出行程时间，最终选取其中最小的那个值，即t2阶段。最后，在S/R系统中将t1和t2的和设为最终的解决方案。Khojasteh-Ghamari详细的对在现有巷道中的货物的拣选顺序的分配方法进行了讨论。如果任何待命的货物存在于现有巷道中，那么就将仓库中现存货物的数目除以解决方案的数目。因此，这项任务目的就是降低总方案的数目，以此来减少CPU时间（程序的处理时间）。3.1.遗传算法遗传算法是一种优化过程，它将问题域比作基因类（个体或染色体），基因类是有多个基因体组成，其中基因体成符号形式串行。每一个基因类都有一个可能的解，通过对问题域中的染色体进行评估来寻求可能的解决方案。在每一代中，我们对每个染色体进行评估，选择一个分布优秀的区域，在其中对染色体进行变异和交叉操作，重新组合，得到新的染色体。这样几代之后，在进一步观察后没有得到新进展的情况下，那么就将所得到最具适应度的染色体视为（所有可能的）最佳解决方案。运算常常会在出现大量的迭代速度和资料后终止（Michalewicz）。表示法每一个染色体表示待求解问题的一个可能解，将其中每一个等位基因被归为一个货物序列中。如此类推，在染色体中的每个基因序列表示货物的种类和相对等位基因的存储位置。因此，每个解决方案包括一个染色体，其中基因的数量等于所收到命令的货物数目。如给出一个例子，图 1如图1可见，一个可行方案中的货物设为A,B,C和D代码，他们被检索位置为：货物C在5号位置，货物B在7号位置，货物A在4号位置，货物D在3好位置。图1.代表一个可行的解决方案其表格表示为，货物被拣选的顺寻也显示在其中。在这个例子中，在5号位置中货物C将被首先检索，其次是货物B，再是货物A，最后是货物D。初始化初始域是随机产生的。拥有随机序列的指令货物组成了染色体。在染色体中，每个货物被赋予一个随机代号。由此可见，每个可行方案所给予的条件是相同的。然而，在每一次重新运算过程中，都会有一套适合的程序来解决方案。因此，染色体中的指令货物将会无重复的随机分布，货物的地址代码也会随机选取，所分配的代号范围会在1到该货物的总仓库库存数之间。假设在仓库内现有总共A、B、C和D4件货物，它们分别对应代码是6、9、7和4。为了形成如图1所示的解决方案，首先，指令货物死随机选取的（C,B,A和D），然后，货物C选取1,7的随机整数，货物B在1，9中选取，A在1,6之间选取，最后D在1，4中间随机选取一个。交叉操作在置换问题的操作描述里，部分匹配交叉（简称PMX）常被用于拣选问题上，部分匹配交叉被视为一种交叉的排列，它确保所有的货物能迅速的被后裔所发现。也就是说，两个后裔全面的接受了父辈基因，接着再填充到其父辈的等位基因上。在图2中，两个父辈用p1和p2来表示，交叉点是1和3。根据在相应的M,R和E,A之间，重复做货物的取代，这就是说，在第一个父辈中的A和E由R和M所取代，而在第二个父辈中的R和M就由A和E来取代。生成的后代是O1和O2(图2)。同时，根据PMX中的拣选问题得知，交叉操作的关键是只交换在染色体中的货物区域并且不交换相关的等位基因。图2.PMX操作变异操作我们现在用二进制位(0和1)来表示基因。在拣选的问题上，相关联的等位基因通过变异操作，将库存中一个基因替代另外一个等位基因。换而言之，这个操作并没有对货物的序列起到任何作用，仅仅只是货物选择了另外一个序列代码。假设在O1中，第三个基因被选为变异基因。由于货物A在各储存位置上的总数有6个，通过变异操作在1,6范围里产生随机整数来代替原来的第三个基因(图 3)，当然，产生的代码等于现有代码时(如2)，则操作重复进行，直到取得一个新代码(除了2)。在这个范例中，4就是最后产生的代码。评估与选择在每代中，对于染色体的评估使用了一些有效的方法。图3.变异操作在大量的优化应用中，适应度是对目标客观本质的计算。在拣选问题中，目标函数的作用是将S/R系统的行程时间降低到最小。通过S/R系统对总行程时间做标准化的计算来得到下一代。Khojasteh-Ghamari对S/R系统计算的行程时间进行做了一下说明。由于这个问题是最小化的问题，所以我们可以将每个染色体的目标函数值改变成适应值，适应值大的染色体就更具适应能力，这样就能更清晰的表达他们的价值程度(cheng等人提出)：其中，eval(vk)是第K个染色体的适应函数，f(vk)是第k个染色体在S/R系统下总行程时间。问题域的大小(简称pop size)决定了每个染色体应被给的时间。现在来做个比喻，我们对下一代染色体的选择比作为(赌台上的)轮盘，适应度大的染色体在下一代遗传中被选的概率更高。在此方案中，行程时间短的更容易被选中作下一代的遗传。赌盘的执行如下：1.计算对于每个染色体的vk(k=1,2,.,最大范围值)在S/R系统的总行程时间。2.计算每个染色体的适应度eval(vk)(k=1,2,.,最大范围值)。3.求得所有适应的总数量4. 计算对于每个染色体Vk的选择概率pk(k=1,2,.,范围最大值)。5. 计算每个染色体vk的累积概率qk(k=1,2,.,范围最大值)。每次选择是在旋转的赌盘中进行的，其结果是动态的，被选中的染色体作为下一代的范围域。-生成的一个随机实数r在0，1范围内；-如果r1，那么选择的第一个染色体v1，否则选择第k个染色体vk(2 k pop size)，这样就有qk1 1, then the method first assigns the picking sequence to pick up all r items which exist in the current aisle. After calculating the travel time, t1, it removes the items from the list of ordered items. Then, for the remaining items (if any), it proceeds like the enumeration algorithm, i.e. finding all feasible ways followed by calculating the travel time for each one and finally selecting the minimum value among them, t2. The total traveled time of the S/R machine will be sum of t1 and t2 as the final solution.Khojasteh-Ghamari discussed in detail the method of assigning a picking sequence of the ordered items existing in the current aisle. If any ordered items exist in the current aisle, then the number of ways studied will be divided by the number of the items existing in the warehouse. Therefore, this task causes the total number of potential solutions to decrease dramatically, and hence, the CPU time (process time of the program) would be decreased as well.3.1. Genetic algorithmA genetic algorithm is an optimization process that employs genotypes (individuals or chromosomes) in a population, and the genotypes are made of units called genes arranged in linear succession. Each genotype would represent a potential solution to a problem; an evaluation process run on a population of chromosomes corresponds to a search through a space of potential solutions.In each generation, we evaluate each chromosome, select a new population with respect to the probability distribution based on fitness values, and recombine the chromosomes in the new population by mutation and crossover operators. After a number of generations, when no further improvement is observed, the best chromosome represents an optimal (possibly the global) solution. The algorithm is often terminated after a fixed number of iterations depending on speed and resource criteria (Michalewicz).RepresentationA chromosome represents a potential solution, where each one is viewed as a sequence of items each with its own associated allele. By analogy, each gene in a chromosome represents the item type, and its associated allele represents the storage location. Therefore, each potential solution consists of a chromosome, in which the number of genes is equal to the number of items in the received order. An example is given in Figure 1.Figure 1 shows a potential solution in which the items with codes A, B, C and D have been ordered. Item C with location number 5, item B with location number 7, item A with location number 4, and item D with location number 3 have been selected for retrieval.Figure 1. Representation of a potential solution.It should be noted that the sequence of picking items has also been considered in the representation. In this example, item C with location number 5 will be retrieved first, followed by items B, A, and, finally, D.InitializationThe initial population is randomly generated. Each chromosome consists of a randomly generated sequence of the ordered items. In each chromosome, a number is assigned to each item. It should be noted that the condition of feasibility for each solution is necessary. Therefore, in the population-generating process, a suitable procedure is required to make each solution feasible. Thus, in each chromosome, the ordered items are randomly distributed without repetition, and the location numbers for each item are randomly selected. The assigned numbers range from 1 to the total warehouse inventory of that item.Suppose that the existing total number of items A, B, C and D within the warehouse is 6, 9, 7 and 4, respectively. In order to form the solution depicted in Figure 1, first the ordered items are randomly selected (C, B, A and D), then for item C, the selected integer number has been generated randomly between 1, 7, also for item B a number between 1, 9, for item A between 1, 6 and finally for item D, a number between 1, 4 has been randomly selected.CrossoverAmong the described operators for permutation problems, the Partially Matched Crossover (PMX) is used for the order picking problem. Partially matched crossover is viewed as a crossover of permutations, which guarantees that all items are found exactly once in each offspring. That is, both offspring receive a full complement of genes, followed by the corresponding filling in of alleles from their parents. In Figure 2, there are two parents denoted by p1 and p2, and the crossover points are 1 and 3. According to the corresponding between M,R and E,A, the repeated items are replaced. That is, A and E in the first parent are replaced by R and M, respectively; while in the second parent, R and M are replaced by A and E, respectively. The generated offspring are o1 and o2 (Figure 2).Meanwhile, according to PMX for the order picking problem, the role of crossover operator in this problem is to change the sequence of the items in a chromosome, without changing the associated alleles.Figure 2. PMX operator.MutationContrary to binary implementation that each gene is replaced with a complementary amount (0 with 1 and vice versa), in the order picking problem, the associated allele of each gene that has been selected by a mutation operator can be replaced with another allele in the range of total inventory of the item. This operator, on the other hand, does not have any role in changing the sequence of items, but can only select another number (storage location) for an item.Suppose that in the o1, the third gene has been selected by mutation. Since the total number of item A in various storage locations within the warehouse is six, the mutation operator generates an integer random number between 1, 6 to replace the third gene (Figure 3). Of course, when the generated number is equal to the current number (namely 2), the operator repeats generating the random numbers until obtaining a number except 2. In this example, the number 4 has been generated.Evaluation and selectionDuring each generation, chromosomes are evaluated using some measure of fitness.Figure 3. The mutation operator.In most optimization applications, fitness is calculated based on the original objective function. In the order picking problem, the objective function is to minimize the travel time of the S/R machine. The total time traveled by the S/R machine is the criteria to select the chromosome for the next generation. Khojasteh-Ghamari explained the procedure for the calculation of travel time of the S/R machine.Because this is treated as a minimization problem, we must convert the objective function value for each chromosome into a fitness value, so that a fitter chromosome has a larger fitness value. This can simply be done by the inverse of its value as follows (Cheng et al.):where, eval(vk) is the fitness function for the k th chromosome and f(vk) is the total time traveled by the S/R machine for the k th chromosome. Population size (pop size) determines how many chromosomes should be in the population at any given time.We use a roulette wheel as the basic selection method to reproduce the next generation based on the current enlarged population, in which a fitter chromosome has a large chance to be reproduced into the next generation. In this selection method, solutions with short travel times have higher probabilities of being chosen for the next generation.The roulette wheel is performed as follow:1. Calculate the total time traveled by the S/R machine f(vk) for each chromosome vk(k = 1, 2, . . . , pop size)2. Calculate the fitness value eval(vk) for each chromosome vk (k = 1, 2, . . . , pop size)3. Find the total fitness of the population4. Calculate the probability of a selection pk for each chromosome vk (k = 1, 2, . . .,pop size):5. Calculate the cumulative probability qk for each chromosome vk (k = 1, 2, . . .,pop size):The selection process is based on spinning the roulette wheel for pop size times; each time a single chromosome is selected for the new population in the following way.- Generate a real random number r between 0, 1;- If r q1, then select the first chromosome v1; otherwise select the k th chromosome vk (2 k pop size) such that qk1 r qk.All chromosomes of the last population are then replaced by the newly generatedchromosomes.4. Simulation ResultsWe constructed a set of 36 different cases of physical specifications of a warehouse, each of which with 5 different kinds of orders to compare the performance of the three algorithms. Each case was first solved by the enumeration algorithm to obtain the optimal travel time and the required CPU time, and then solved by the other two algorithms. A summary of results are presented in the following two tables.4.1. Simulation modelsThe three main warehouse parameters used to design the 36 different cases are warehouse capacity, warehouse density and shape factor. Warehouse capacity is proportional to the number of aisles in the warehouse. We consider four cases for warehouse capacity;a warehouse with one, two, three and four aisles. Each storage rack contains 780 storage locations. Since each aisle contains two racks, the capacity of each aisle is 1560 storage locations. The main reason not to consider a warehouse with five or more aisles is the assumption that there is only one S/R machine serving all aisles. Assigning a machine to a warehouse with a high number of aisles will decrease the practical efficiency of the system. As warehouse density, we assume that either 60%, 75% or 90% of total capacity of the warehouse is used. The configuration of rack or shape factor, b, as described by Bozer and White is the time ratio of length and height of the rack, supposing that rack capacity and both horizontal and vertical velocity of S/R machine are constant. Similar to Han et al. we use three values (0.6, 0.73 and 1) for the shape factor.In addition, for each aforementioned case, we consider five kinds of orders so that, for orders 1, 2, 3, 4 and 5, the number of requested items to retrieve is one, two, three, four and five items, respectively.4.2. ResultsA personal computer with following specification was used to run the experiments: Pentium III, CPU equal to 1000MHZ, 512 MB RAM and 2 GB of assigned virtual memory. The results are given in Tables 1 and 2. Table 1 shows the comparison between the average travel time of the S/R machine and the average CPU time of the three algorithms for four kinds of warehouses. For each kind of warehouse (a warehouse with one, two, three, and four number of aisles), a set of 9 cases are considered, which are combination of the two assumed warehouse parameters, warehouse density, and shape factor. The values in each case represent the average values of five kinds of order types. Table 2 gives the comparison of the average travel time and the average CPU time with respect to the number of aisles, and shape factors of 0.6 and 1.In the tables, the enumeration, the current-aisle heuristic, and the genetic algorithm are denoted by Enumeration, CAH and GA respectively.5. Analysis of the ResultsAs it can be inferred from the Table 1, for all kinds of warehouses (with one, two,three and four aisles) across all cases, the CAH algorithm attains solutions with the largest travel time calculated for the S/R machine, and a less CPU time than other two algorithms. In other words, the CAH algorithm requires a less CPU time to provide a solution to the problem. However, the travel time of the S/R machine is much longer than those of the other two algorithms.In the case of a warehouse with one aisle, 89% of the solutions obtained by the genetic algorithm are optimal. In the remaining, the average difference between the suboptimal and optimal solutions is only 0.09% (but requires larger CPU times). Where the warehouse consists of two, three and four aisles, 11% of the solutions generated by the genetic algorithm are optimal. In the remaining cases, however, the average difference between the obtained solution and the optimal one is trivial and equivalent to 3.86%,4.83% and 4.69% in the warehouse with two, three and four aisles, respectively.Increasing the number of aisles in a warehouse affects the performance of these algorithms. Due to the increase of the total number of feasible solutions, the CPU time of the enumeration algorithm increases dramatically in order to obtain the optimal solution. The genetic algorithm, however, requires a less CPU time than that of the enumeration algorithm to achieve a solution with a reasonable travel time for the S/R machine, which is optimal in most cases or quasi-optimal.Table 1. Performance of the three algorithms.Table 2. Performance of the three algorithms with respect to the shape factor, b.In addition, the performance of the algorithms is influenced by the rack configuration,b, as well. Table 2 shows a comparison of the average travel time of the S/R machine and the average CPU time with respect to the number of aisles, in two shape factors 0.6 and 1. This table provides comparisons between performances of the algorithms in two rack configurations as warehouse capacity increases. In the case of a warehouse with only one aisle, the enumeration algorithm provides the best solution in the both levels of b, with a CPU time less than that of the genetic algorithm. However, if the warehouse has more aisles, the genetic algorithm requires less CPU time than the enumeration algorithm. Since the results for various b levels were similar, we omitted the results corresponding to b = 0.73. As the performance of the three algorithms for various b are approximately the same, the rack configuration in the warehouse shows essentially no effect on the performance of the algorithms.6. ConclusionsIn this research, we addressed an order picking problem in a multi-aisle automated warehouse served by a single storage/retrieval (S/R) machine, with a novel property: each item can be found in several storage locations. We developed two algorithms to solve the problem: an ordinary heuristic called the current-aisle heuristic (CAH) algorithm, and a suitable genetic algorithm. In order to show the efficiency of the proposed algorithms, we compared them with the enumeration algorithm, which attains the optimal solution, but does so with a long CPU time, hence making the method unsatisfactory. The CAH algorithm requires less CPU time, however, the achieved solutions are mostly sub-optimal with dramatically longer travel times for the S/R machine. With the genetic algorithm, most of the solutions are optimal or quasi-optimal solution (3.37% difference in average). Consequently, the proposed genetic algorithm showed more efficient than the other two algorithms.In the future, meta-heuristic methods, as well as branch and bound algorithms would be evaluated against various storage methods for their utility and a dual command (DC) S/R machine cycle in generating optimal solutions for order picking problems in automated warehouses.AcknowledgementsWe thank Professor M.M. Sepehri, Tarbiat Modarres University for his valuable comments. We are also grateful to anonymous referees for helpful suggestions that improved the presentation of the paper.References1 Amato, F., Basile, F., Carbone, C. and Chiacchio, P., An approach to control automated warehouse systems, Control Engineering Practice, Vol. 13, pp.1223-1241, 2005.2 Bozer, Y. A. and White, J. A., Travel-time models for automated storage/retrieval systems, IIE Transactions, Vol. 16, No. 4, pp.329-338, 1984.3 Bozer, Y. A. andWhite, J. A., Design and performance models for end-of-aisle order picking systems, Management Science, Vol. 36, No. 7, pp.852-866, 1990.4 Cheng, R., Gen, M. and Sasaki, M., Film-copy deliverer problem using genetic algorithms, Computers & Industrial Engineering, Vol. 29, pp.549-553, 1995.5 Elsayed, E. A., Algorithms for optimal material handling in automatic warehousing systems, International Journal of Production Research, Vol. 19, pp.525-535, 1981.6 Elsayed, E. A. and Stern, R. G., Computerized algorithms for order processing in automated warehousing systems, International Journal of Production Research, Vol. 21, pp.579-586, 1983.7 Goetschalckx, M. and Wei, R., Bibliography on order picking systems, Vol. 1, pp.1985-1992, 1994, available at /people/faculty/Marc Goetschalckx/research.html.8 Han, M.-H., McGinnis, L. F., Shieh, J. S. andWhite, J. A., On sequencing retrievals in an automated storage/retrieval system, IIE Transactions, Vol. 19, pp.56-66, 1987.9 Hwang, H., Baek, W. and Lee, M.-K., Clustering algorithms for order picking in an automated storage and retrieval system, International Journal of Production Research, Vol. 26, pp.189-201,1988.10 Hwang, H., Moon, S. and Gen, M., An integrated model for the design of end-of-aisle order picking system and the determination of unit load sizes of AGVs, Computers & Industrial Engineering, Vol. 42, pp.249-258, 2002.11 Khojasteh-Ghamari, Y., Order picking problem in an AS/RS with multiple stock locations. M.Sc.thesis, Tarbiat
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