南京理工大学紫金学院-科幻作文500字

物流分拣中英文对照外文翻译文献


物流分拣中英文对照外文翻译文献

(文档含英文原文和中文翻译)





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
摘要
随 着现代化科技的发展，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SR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。
关键词：自动化立体仓库，AS RS系统，拣选，遗传算法。

物流分拣中英文对照外文翻译文献
1.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（ASRS）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。Roo dbergen
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。Thealgorit hms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、H an、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

物流分拣中英文对照外文翻译文献
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SR系
统可提供多若干个通行巷道。Amat o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10 0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< br>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（IO） 。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SR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。SR
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，SR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SR走过的总时间来完成命
令程序。
3.运算方法

物流分拣中英文对照外文翻译文献
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，SR
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n
i
(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SR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< br>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SR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SR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r>1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< br>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SR系统中将 t1和t2的和设为最终的解决方案。

物流分拣中英文对照外文翻译文献
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

物流分拣中英文对照外文翻译文献
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

物流分拣中英文对照外文翻译文献
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< br>标函数的作用是将SR系统的行程时间降低到最小。通过SR系统对总行程时间
做标准化的计算来 得到下一代。Khojasteh-Ghamari对SR系统计算的行程时间进
行做了一下说明。 < br>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SR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SR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

物流分拣中英文对照外文翻译文献

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r≤1，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qk−1 < r ≤qk。
在上一代中的染色体被新一代的染色体所替代。
4.仿真结果
我们制作了一个拥有 36种不同货物的立体仓库，在其中还有5种不同类型的指令，
对此比较3种运算法的性能。每个货物首 先先用例证法来解决。以获取最佳的行程时间
和CPU占用率。接着用另外两种解法来解决。研究结果如 下2表。
4.1.仿真模型
我们创建了一个在36种不同物理规格情况下的仓库，通过对于 每一个仓库施加5
种不同的指令来对这3种算法的性能进行比较。每种情况首先按例证法来得到最佳的行
程时间和CPU占用率，然后再通过另外两种计算方法来解决问题。研究结果显示在下
面两个表 格中。
利用仓库的主要3个参数（仓储容量、密度和形状）来设计36种不同存储的情况。
由 于仓储容量与仓库中的巷道成比例关系，我们将仓储容量划分为4种情况，分别是1、
2、3和4种巷道 的形式。每个仓储货架有780个存储位置。因为每个巷道有两个货架，
则一个巷道就拥有1560个存 储位置。由于一个系统对仓库中大量巷道进行服务的话，
将会大大降低其系统实际效率。所以在不考虑5 个或更多巷道的情况下，就由一个SR
系统对所有巷道进行服务。对于仓储密度，我们假定仓库中的使用 率为60%、75%和95%。
Bozer 和 White对仓储形状的配置进行了相关描述为，仓储 形状，它是一种对于货架高
度与长度的空间比例，假设仓储容量与SR系统的水平和垂直速度都是常数。 那么我们
将这3个值设定为（0.6,0.73和1）。

物流分拣中英文对照外文翻译文献
此外还要补充的是，对上述每种情况的描述 中，5种不同的指令为别是1，2，3，4
和5，5种所要求的货物编码分别是一，二，三，四和五。
4.2.结果
在个人电脑配置是：“奔腾III,1000MHz的处理器,512 MB内存和2 GB虚拟内存”
的情况下进行了试验。结果列于表1和表2中。表1表示在3种运算法下 ，4种类型“SR
系统平均行程时间”和“SR系统平均CPU占用率”。两种仓储参数（仓储密度和形 状）
的组合形成了每个仓库(仓库分别有1、2、3和4个巷道)的9种情况，每种情况下的值
表示了5种命令下的平均值。表2表示在仓储形状为0.6和1，4种巷道情况下的平均行
程时间和平均 CPU占用率。
在表格中，例证法、现有巷道探索式算法和遗传算法分别用“Enumeration ”，“CAH”，
“GA”所表示。
5.分析结果
通过对表1分析可知，在所有情 况下的各类仓库(1，2，3和4个巷道)，CAH算法
是能获得最大行程时间和最小CPU占用率的解 决方案。换而言之，它是占用较小CPU
使用率的方法。然而，它对SR系统的行程时间超过了其他两个 。
在仓库中只有一个巷道的情况下，通过遗传算法解决获得的方案中89%为最佳的方
案。其 余的方案里次优和最优的解决方案平均只相差0.09％（但需要更大的CPU时间）。
在拥有2个3个 和4个巷道的仓库中，遗传法提供的11%的解决方案为最佳方案，其余
方案里，获得方案与最佳方案差 别不大，分别是2巷道相差3.86%，3巷道相差4.83%
和4巷道相差4.69%。
仓 库中巷道的层架数目会影响到运算效率。由于增加的总数是实际问题中出现的，
例证法中要增加较大的C PU占用率才能获得最佳解决方案。然而在大多数情况下，遗
传法则需要相比于例证法较少的CPU占用 率就能完成SR系统的最佳方案。

物流分拣中英文对照外文翻译文献

表格1. 3种算法的性能

表格2.3中算法在仓储形状上的比较
此外 ，运算方法的性能是受货架配置所影响的。表2显示了通过对SR系统的平均
行程时间和平均CPU占用 率在多巷道中的两种仓储形状（0.6和1）的比较。在此表中

物流分拣中英文对照外文 翻译文献
显示了当仓储容量增加时，两个货架配置的算法比较。在一个仓库只有一个巷道时，例
证法提供了最佳的方案，并且它的CPU占用率低于遗传法。然而，如果仓库有多个巷
道时，遗传算法 需要的CPU占用率低于例证法。由于各种仓储形状B的结果相似，我
们将仓储形状B设为0.73。因 为对B的3种算法性能大致相同，所以在仓库里的货架
配置对算法性能没有影响。
6. 总结
在本次研究中，我们讨论了多巷道自动化立体仓库系统，并得到了结果。就同类货
物在不同存储 位置被寻找的情况下，我们发明了两种算法来解决这个问题，我们将第一
种探索式算法命名为现有巷道探 索式算法（简称CAH），第二种命名为可接受遗传算
法。为显示每种算法的实际效率，我们将他们与例 证法做了对比，例证法在获得最佳方
案的同时需要大量的CPU占用率，因此它并不是最理想的解决方案 。CAH算法需要较
小的CPU占用率，但获得的方案大多数是需要较长的SR系统行程时间的次佳的方 案。
而遗传算法提供的方案大多是最佳和准佳(平均占3.37%)的方案。因此，模拟的遗传算
法显示，它的效率高于其他两种算法。
不久的将来，在功效和双命令（DC）的自动化仓库系统领域 中，将对元启发式方
法和分支定界算法进行评估，以便能在自动化仓库拣选问题上创造最佳的解决方案。
7. 鸣谢
我们感谢来自Tarbiat Modarres 大学M.M. Sepehri教授的宝贵建议。我们也同样的感谢
为我们提出建议的匿名审稿人。
参考文献
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locations. .
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system with a tower crane SR machine, Computers & Industrial Engineering, Vol. 43,
pp.495-507, 2002.
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picking system: Multi-aisle served by a single order picker, International Journal of
Production Economics, Vol. 98, pp.162-171, 2005.
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storage and retrieval systems with multiple openings, International Journal of
Production Research, Vol. 34, pp.2943-2962, 1996.
[15] Lee, H. F. and Schaefer, S. K., Sequencing methods for automated storage and
retrieval systems with dedicated storage, Computers & Industrial Engineering, Vol. 32,
pp.351-362, 1997.
[16] Mahajan, S., Rao, B. V. and Peters, B. A., A retrieval sequencing heuristic for
miniload end-ofaisle automated storageretrieval systems, International Journal of
Production Research, Vol. 36, pp.1715-1731, 1998.
[17] Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs,
1992 (SpringerVerlag: Berlin)
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solvable case of the traveling salesman problem, Operations Research, Vol. 31,
pp.507-521, 1983.
[19] Roodbergen, K. J. and de Koster, R., Routing order pickers in a warehouse with a
middle aisle, European Journal of Operational Research, Vol. 133, pp.32-43, 2001.
[20] Rouwenhorst, B., Reuter, B., Stockeahm, V., van Houtum, G. J., Mantel, R. J.
and Zijm, W. H. M.,Warehouse design and control: framework and literature review,
European Journal of Operational Research, Vol. 122, pp.515-533, 2000.
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1999.
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minimizing travel distance with genetic algorithms, Computers in Industry, Vol. 56,
pp.169-178, 2005.

物流分拣中英文对照外文翻译文献
Order Picking Problem in a Multi-Aisle Automated
Warehouse Served by a Single StorageRetrieval
Machine
Yaghoub Khojasteh-Ghamari Jae- Dong Son
University of Tsukuba Soongsil University
Japan Korea
Abstract
Recent technological developments have revolutionized the design and operation of ware-housing systems.
Automated storage and retrieval systems (ASRS) driven by embedded computers are becoming increasingly more
prevalent. The increased use of ASRS is creating the need for computerized control algorithms to support the
scheduling and picking research addresses an order picking problem in a multi-aisle automated
warehouse, in which a single storageretrieval (SR) machine performs storage and retrieval operations, and each
item can be found in several storage locations. Our objective is to propose algorithms that minimize the total time
traveled by the SR machine to complete the retrieval process of orders. We develop a genetic algorithm and an
ordinary heuristic, and provide a performance comparison of them with optimal solution. Numerical results from a
large set of problems are reported.
Keywords: Automated Warehouse, ASRS, Order Picking, Genetic Algorithms.
1. Introduction
In today’s manufacturing environments, inventories are maintained at lower levels than in the
past. These reduced inventories have led to smaller storage systems, which in turn have
created the need for quick access to the material being held in , automated
storage and retrieval systems (ASRS) used in manufacturing, ware-housing, and distribution
applications must be designed to provide quick response times to service requests in order to
keep the system operating efficiently. One important operational aspect of the ASRS, which
contributes to the system response time, is to minimize the total time traveled by the SR
machine to complete the retrieval process of orders.
Order picking is a fundamental component of the retrieval function performed in warehouses.
The main purpose of an order picking system is to fill customer orders by selecting the
appropriate amount of material from a pre-designated storage medium known as the picking
or forward area. Order picking represents only a subset of the material handling operations
performed in warehousing. However, it is ‘one of the most costly and time-consuming
functions of warehousing. In many warehouses, the difference between profit and loss
depends on how well the order picking operation is run’ (Bozer and White ).
There are many studies on order picking problems in ASRS and automated ware- housing
systems. Ratliff and Rosenthal developed a graph-based algorithm to find the shortest path
to visit a set of pick locations in a ladder layout. Roodbergen and de Koster extended the
work of Ratliff and Rosenthal. They considered the order picking problem in a parallel aisle
warehouse in which order pickers can cross over the aisles at the ends of aisles as well as at a
middle cross aisle. They developed a dynamic programming algorithm to solve the problem.
Van den Berg and Gademann developed a transportation problem (TP) model for a block

物流分拣中英文对照外文翻译文献
sequencing in an ASRS with dedicated storage and a single- load machine. They proved that
the optimal solution of the TP problem is the optimal sequence of the machine to minimize
the travelling time.
Elsayed made a chain of studies on the problem of optimally batching several orders in a
two-dimensional warehouse with ladder structure. Recognizing that the exact solutions of the
problem are very difficult and time consuming to obtain, Elsayed and Stern presented some
heuristic algorithms, but reported that none of them produces consistently superior results
through experimentations. Hwang et al. studied a similar order picking problem in a
single-aisle ASRS and presented heuristic algorithms,which determine an efficient batching
of orders for each tour of the SR machine. Thealgorithms were based on cluster analysis with
some similarity measures. Through simulation, they compared performances of the proposed
algorithms with Elsayed and Sterns’ results in 1983.
Bozer and White, Han et al., and Lee and Schaefer proposed a procedure to optimize the
sequencing of retrieval requests based on the solution of a linear assignment problem. Lee and
Schaefer also presented several optimum and heuristic sequencing methods, where a storage
request is assigned to a predetermined storage location. Mahajan et al. developed a retrieval
sequencing scheme aimed at improving the throughput of miniload ASRS. They proposed a
nearest-neighbor retrieval sequencing heuristic and developed an analytical model to predict
its performance. Hwang et al. formulated a nonlinear mathematical model and developed an
efficient heuristic solution procedure to design the ASRS and determine the unit load size of
the vehicle simultaneously. Van den Berg and Rouwenhorst et al. surveyed literature on
warehouse planning and control. Planning includes the storage location assignment problem,
and the control of a warehousing system includes routing, sequencing, scheduling,
dwell-point selection, and order batching. Goetschalckx and Wei presented a bibliography on
order picking systems for 1985 through to 1992.
Koh et al. proposed some models for travel times of the SR machine in a warehouse with a
tower crane. They derived the models for both single and dual command cycles based on the
randomized storage assignment rule. They also calculated the travel time under the
turnover- based storage assignment rule through a numerical approach. Koh et al. proposed an
optimization model to find an optimal buffer size in end-of- aisle order picking system, where
a single SR machine serves several et al. proposed an algorithm to optimally
sequence the retrieval orders based on colored timed Petri nets framework. They also
proposed two control algorithms to optimize the operations of the cranes and shuttle. Hsu et al.
considered the order batching problem in a multi-aisle warehouse and proposed a genetic
algorithm to minimize the total travel distance. Hwang and Cho presented a performance
evaluation model for the order picking warehouse in a supply center. The objective of their
study was to minimize the cost by minimizing the number of transporters and to calculate the
performance and facility utilization rate. In a recent study, De Koster et al. carried out a
literature review on typical decision problems in design and control of manual order picking
processes. They focused on optimal layout design, storage assignment methods,routing
methods, order batching and zoning.
However, we have no knowledge of papers in the literature that address the order picking
problem in automated storage and retrieval systems, where each item can be stocked at
several storage locations. In fact, some manufacturers whose products have a large variety of

物流分拣中英文对照外文翻译文献
types, shapes, and sizes are faced with this problem in their finished goods warehouses. A tile
manufacturer, for example, whose products are in two types (wall and floor), each of which in
7 different sizes, 4 groups of durability (P.E.I. Wear Rating), and 100 different colors, designs
and shades has totally 5600 different product types. As a storage policy, a coming pack of a
product into the warehouse is placed in the nearest empty storage location. When an item is
retrieved from the warehouse,because of the large variety of the products, there is a strong
probability that the place be filled with another type. Therefore, an item can be found in
several storage locations in the warehouse. In other words, since classifying and zoning each
individual type of product in the warehouse needs a warehouse with a large space, the storage
of an item in several places is unavoidable.
2. Problem Description
In this research, we consider a miniload automated storage and retrieval system,where there
are one or more aisles. Each aisle contains a storage rack on both sides of the aisle. There is
an inputoutput (IO) station at the end of each aisle. There is also a single storageretrieval
(SR) machine dedicated to all aisles of the system, which can simultaneously move in
vertical and horizontal directions. Hence, the travel time between two points is equal to the
maximum of the horizontal and vertical travels. The SR machine is positioned at one of the
IO stations before the receipt of an order. The starting place of the machine depends on the
storage location (aisle) of the last item of the last order. In calculating the travel time of the
SR machine, constant velocities are used for horizontal and vertical travels. An order can be
a request for more than one item. Also each item can be in several storage locations in the
warehouse. When retrieval requests consist of multiple items and the items are in multiple
stock locations,the SR machine must travel to numerous storage locations to complete each
order. The aim of this research is to propose algorithms to minimize the total time traveled by
the SR machine to complete the retrieval process of an order.
3. The Algorithms
We present two algorithms to solve the problem: an ordinary heuristic, and a suitable genetic
algorithm. To show the superiority of the presented algorithms, it is necessary to compare
them with other methods as well as the optimal solution. Since our problem is new, no
research has been conducted in this field, we first present an algorithm to obtain the best
solution for the problem, so called the enumeration algorithm. Its results are used as
benchmark solutions for performance comparison of the two proposed algorithms.
In the enumeration algorithm, we identify all feasible solutions and compare them with each
other to find the best solution. To do this, the method first finds all feasible ways to pick an
order. Then, it calculates the total time traveled by the SR machine for each one, and finally
selects the solution requiring the least amount of time to accomplish the order. This solution is
considered as the optimum solution for the problem. Consider an order consists of k distinct
types of items, in which n
i
(i = 1, 2, . . . , k) items of the ith type are requested. The total
number of feasible solutions to pick the order can be given by：

物流分拣中英文对照外文翻译文献
where, m
i
is the total inventory of the ith type in the warehouse, and.

Having solved various problems by the enumeration algorithm and identifying the best
solution that had the minimum amount of travel time, we found that the existing items in the
current aisle (i.e. the aisle in which the SR machine is in at the beginning of the retrieving
process) are the key to the final solution. We develop an algorithm on the basis of the
mentioned results, and call it the current-aisle heuristic (CAH) algorithm.
In the CAH algorithm, the existing items in the current aisle are selected first for retrieval.
Afterwards, the remainder of the order (if any) is selected and all the various retrieval
methods are studied. We can simplify the previous statements by stating that if r denotes the
number of the ordered items existing in the current aisle, and if r = 0, the method then
becomes similar to the enumeration algorithm. If r = 1, then the method first calculates the
time traveled by the SR machine, t
1
, for only one existing item in the current aisle, and
removes that item from the list of ordered items, and then for the remaining items (if any), it
proceeds as same as the enumeration algorithm to obtain the minimum travel time, t
2
. The
total travel time of the SR machine will be sum of t
1
and t
2
as the final solution.
If r > 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, t
1
, 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, t
2
. The total traveled time of the SR
machine will be sum of t
1
and t
2
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 algorithm
A 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).
Representation
A chromosome represents a potential solution, where each one is viewed as a sequence of

物流分拣中英文对照外文翻译文献
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.
Initialization
The 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.
Crossover
Among 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 p
1
and p
2
, 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 o
1
and o
2
(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.

物流分拣中英文对照外文翻译文献

Figure 2. PMX operator.
Mutation
Contrary 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 o
1
, 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 selection
During each generation, chromosomes are evaluated using some measure of fitness.

物流分拣中英文对照外文翻译文献
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 SR machine. The total time traveled by the SR 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 SR 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(v
k
) is the fitness function for the k th chromosome and f(v
k
) is the total time
traveled by the SR 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 roulette wheel is
performed as follow:
1. Calculate the total time traveled by the SR machine f(v
k
) for each chromosome v
k
(k = 1,
2, . . . , pop size)

2. Calculate the fitness value eval(v
k
) for each chromosome v
k
(k = 1, 2, . . . , pop size)
3. Find the total fitness of the population

4. Calculate the probability of a selection p
k
for each chromosome v
k
(k = 1, 2, . . .,
pop size):

5. Calculate the cumulative probability q
k
for each chromosome v
k
(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 ≤ q
1
, then select the first chromosome v
1
; otherwise select the k th chromosome v
k
(2 ≤ k
≤ pop size) such that q
k−1
< r ≤q
k
.
All chromosomes of the last population are then replaced by the newly generated
chromosomes.
4. Simulation Results

物流分拣中英文对照外文翻译文献
We 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 models
The 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 SR 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 SR 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. Results
A 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 SR 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 Results
As 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 SR 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 SR 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

物流分拣中英文对照外文翻译文献
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 SR machine, which is optimal in most
cases or quasi-optimal.

Table 1. Performance of the three algorithms.

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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 SR 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. Conclusions
In this research, we addressed an order picking problem in a multi-aisle automated warehouse
served by a single storageretrieval (SR) 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 SR 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) SR
machine cycle in generating optimal solutions for order picking problems in automated
warehouses.
Acknowledgements
We 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.
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