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DOE Practioner’s Guide to Achieve Robust Designs

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稳健参数设计的产品，在正确使用方法和环境下，在产品寿命周期里，能够

始终如一地满足并超过客户的期望。主要有两种方法可以来实现这个目标，第一种方法是在设计产品时考虑额外的极限条件或冗余，以避免产生缺陷产品性能的任何可能性，这种方法虽然有效的实现了稳健参数设计的效果，但结果通常是一种过于保守的设计方法，往往代价太高。第二种方法是在设计过程和运用统计方式对产品表现不佳的风险管理，对产品不确定性和可变的因素进行明确定义。这种方法被许多世界500强公司所倡导，如摩托罗拉，通用电气，联合信号，以及福特汽车公司。UTC积极提倡采用这种设计工具，其中“Process Certification ”和“Design Process Certification”作为企业获得竞争优势工具的重要组成部分。

. 实验室设计（DOE）是设计过程认证（Design ProCert）中的基础工具之一，在这里我将主要介绍下，试验设计是一项系统测试方法，为了是能够提高测试的效率和有效性，为了揭示输入（因素）和输出（响应）之间的关系。这种关系通常被称为系统传递函数，要求设计者顺着输出端的需求来限制输入端的因素，达到稳健参数设计的关键是传递客户的期望价值。DOE是一个技术手段发现关键系统的灵敏度， 因而使设计师可以有效的控制输入端的可变性来实现输出的一致性，以及客户的满意度；

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DOE是奥的斯整体稳健性设计策略中的一种工具就如图1.1说明的 。在产品开发初期，不符合客户需求的风险将被定义。产品设计时应进行设计失效模式及影响分析（FEMA ）以便能找出潜在的问题。当为了有效的完善产品性能应使用 Modeing ，分析，模拟和计算（MASC）等工具。DOE工具被用于去获得更深入地发现能确定产品关键质量特性（ CTQ ）的因素的设计性能。“UTC ACE DIVE”工具，如市场反馈分析（MFA） ，错误预防（ MP ） ，严格根源分析（RRCA ） ，质量诊断流程（ QCPC ） ，被用来跟踪和落实纠正措施作为必需的要求。在产品开发初期获得经验，然后通过细化的奥的斯标准，被认可的标准工作和工具进行制度化。为在下一产品开发周期满足客户需要减少初始风险影响。

奥的斯公司在实施稳健的设计上的根本策略是在产品开发过程中利用六西格玛（如DOE方法） 原则来识别、确定优先次序，关键的对客户有价值的表现

DOE技术在文件中被详细的描述成一种为构建高效的测试计划以处理多重输入可变因素的方法，，同样也是处理数据，通过测试揭示产品或过程的影响的分析方法，通过统计学的原理使风险可以满足客户要求。举一个例子，本文件（第4.1条）的一个案例研究 ，涉及通过DOE研究的一部电梯轿厢隔震器系统。关键系统的输出主要体现在轿厢噪声和支架的振动传递率（振动的比率体现在任何一侧的支架上） 。关键输入是轿厢隔震垫的刚度和位置，支架的刚度，轿厢的平衡、支架的规格。共有27种测试条件，有选择的定义输入变量，确定应用于电梯状态的DOE测试矩阵。测试结果被用于进行分析，然后在所有输入中选择最佳的设置，达到减小轿厢噪声水平TBD ；

这个文件的目的是帮助奥的斯公司的工程师成功地运用DOE方法，以达到稳健的电梯和自动扶梯的设计。作为对奥的斯公司的工程师的辅助，这份文件将呈现一个如何应用DOE进行电梯和自动扶梯设计的标准工作版本。参考文献被列在TBD章节为有兴趣的读者提供更多的DOE过程步骤详细资料；

商业软件包minitab已被确立为UTC 进行DOE的测试和分析的基本工具。它提供了一个自动化的方法进行统计分析，设计DOE测试矩阵，处理试验数据。在整个文件中提到了如何使用minitab来执行一些基本DOE步骤。在这个文件里就不再描述MiniTAB用户手册的内容。

在这个文件中有8个章节（被列举在图片1.2） 。读者可以参考任何一个章节来获取需求。例如，经理可以读章1和3来了解DOE的概念以及如何进行稳健设计。第2章包含这一文件的主要内容：DOE测试操作步骤到产品开发。第五章通过工作的列子来描述DOE的各个时段内容所需要进行的标准工作，包括使用MiniTab 。第4 ，第6 ，第7和第8条章主要是一些得心应手的参考资料。

DOE方法提供了一种了解关键的输入参数（通常称为因子）和一个输出参数（响应） 之间关系的方法。这方面的知识，通过有效的测试，被设计工程师用于优化系统，以确保有稳健的性能。举例来说，为确保电梯的横向振动不超过某一指定水平的，必须进行导轨，轿厢，构架，悬吊设计，并尽量减少导轨的不规则。DOE测试提供了一种方法去识别潜在的因素物理联系，这些因素（例如，滚轮跳动，导轨之间的距离，轿厢的平衡性） 都会影响到震动水平（即响应） 。设计者可以设置规格界限，以确定控制哪些对顾客价值反应影响最大的因素（构成关键质量特性因素）。这一数据成为关键参数管理（ CPM）资料库的一部分，是用来保证产品的稳健性，见图1.3 。

鲁棒性设计可以被认为是DOE的衍生模式通过控制降低可变因子来提高队客户有价值的产品关键质量特性。不能满足对客户有价值的风险需要通过统计学来确定规格限制，通过样品CPM 记分卡的数据来说明如图1.4 。CPM成为跟踪产品在测试过程中所制定的关键要求等级的方法。

在这个例子中最高层要求是电梯横向振动水平须低于13 mg。通过DOE测

试，下层次三个控制因素（滚轮跳动，导轨间隙 ，轿厢平衡性）的确定和实证模型，导出他们之间与横向振动的关系。DOE测试主要成果是识别这些因素

（40.4.20）的上限，以满足13mg振动规格。橙色的数据表示了对客户需求有价值的要求，黄色的数据表示产品能力在统计学的研究下的常规体现（对它们的定义将在第7章中详细讨论） 。稳健设计必须证明他的产品设计符合要求。DOE工具提供数据和知识严格界定其要求。统计过程控制（ SPC ）工具提供数据产品的能力。

在已出版的文献中有三种不同类别的DOE测试，他们与稳健设计都有着相关的联系就像图1.5所展示的；

筛选试验设计是一种从一个较大的范围内的潜在因素中判断那个因子对于系

统响应影响最大的方法。一组特殊的测试程序已经成熟的被正在进行产品开发的工程师所应用，并帮助他们完成典型分析。建模设计被用于开发因子以及系统响应两种功能模块之间的关系。这方面的资料从经验模型中获取，然后形成优化执行和分派要求的主要原则。最后一个主要的DOE种类被称为鲁棒性，用于减少敏感的关键系统输出噪声的因素。这一技术可用于直接的稳健性设计到产品。所有这三个技术DOE技术将在接下去的第二章内容中进行讨论；

有几个问题确实（见图1.6 ）是值得进行讨论的，在此概述性介绍。

一般来说，DOE测试应尽早出现在PDP的过程中。当设置的物理原型是有

效的，我们仍需要选择有代表性的例子，虽然DOE工具，可以有效地利用随着MASC工具，以驱动概念的制定和选择。此外，DOE测试物理原型，可用于完善好，发展好，并验证MASC模式，可以用在产品开发周期的疑问，以及任何后继产品设计。

DOE技术提供了一种有效方法去研究多个因子实验测试结果。这里有一个

问题将会在下面详细说明，就是在任何测试中都有可能出现大量的是需要考虑的因素。起初，它可能会出现超过十二个因素被确定为潜在的关键因子。传统的做法\一次验证一个因子\（ OFAT）将采取比DOE设计更多的测试（如采用Plackett-Burman DOE设计） ，以获取敏感因子之间的不同评估。此外，通过DOE技术分析收集到的数据，可以在一个非常快速的周期对鉴定结论的关键因素之间见的影响和模型。频繁的DOE试验证明在输入因子之间是有相互影响的，意思是\一个因子的变化怎么样导致另一个因子在最终结果上的变化？ \因子之间的相互影响意味着一个因子是其他因子的催化剂。一个因素交互作用的例子，轿厢的不平衡和导轨拼接直接导致轿厢振动。当这两个因子都影响大的，振动严重，但是，当只有一个是较大因子时，振动水平是适合的。因子交互直接影响的讨论和评价，在本文的DOE的测试标准工作概述。

许多书籍，文章和论文已以DOE设计和测试为主题。那些DOE的发表文

献都认可DOE的步骤两个共同点; DMAIC（定义，测量，分析，识别和控制） 用于精益制造和DMADC（定义，测量，分析，设计，核实）应用于典型设计中。在本文中对DOE大纲描述的标准工作步骤包括，具体提及相关的任务DOE设计和整合的DOE成果转变为CPM的数据库。因此，正如显示在图1.7 ，有7个DOE的基本步骤DOE将被详细描述在下一个的章节中。 （注：每一个DOE步骤在下一个章节中的具体描述可以参照图1.7））

作为任何一个实验性测试项目，定义去做什么测试，以及怎么做测试都需要

花费较多的时间和精力。但是使用了DOE工具后这就会有较大的改观；主要的操作步骤清单将会被自动列出，并有商业软件包进行支持，如minitab 。步骤2 ，第4 ，第5和第7 （即DOE设计，数据分析，系统优化，CPM规格）详细描述了一个多层面的团队，以高效率和有效地实施DOE测试。所有团队中的成员必须积极从事DOE过程中的所有步骤。 \技术专家\，将指导定义过程，完成测试，并验证设计。 \的专家\，将指导DOE设计，分析数据，优化系统，CPM规格。

关键点：成功的DOE测试需要技术与DOE领域的专家积极参与。这份文件将强调操作的主要步骤，以协助所有类别的DOE团队成员。

本章提概述了贯穿DOE进程的七个相关联的主要步骤。在每一个步骤的关键点，包括关键的成果，均在以下章节里给与概述。操作者应以适当的标准工作的流程图为方针。

第一，可以说是最重要的一步，在DOE

的测试中最重要的一步是要哪些参数对被测产品的输入和输出条件产生影响。这是一个成熟的过程示意图，且在P图中定义了三组因素（或称为因子）。

产品的关键特性的输出被称为\系统响应（简写为s ） \。一些预先定义的

规格界限须满足顾客期望值。例如已被用来在电梯相关系统中的案例，可能会有横向振动等级，在轿厢内噪声水平，或平层精度。而不止一个反应可以指定，最后一个单一的值被用来描述最佳等级（DOE步骤5 ） 。这可以通过建立一个数学函数把不同的输出值联系在一起。但是，大部分的时候一个单一的系统响应是已常见的，还应该确保该系统的反应是一个连续变量。举例来说，如果正在用DOE测试来验证设计是正确的，则我们希望输出该设计是“失败的”这个结论。例如测量应力水平的，应作为衡量输出状态。通过/失败这样的输出条件在DOE的测试里不是

有用的系统响应。一般的观念是，这个变数应该是连续的，可衡量的，并且有合适的产品或生产过程特性指标。

下一步是要确定影响系统的

回应的产品或过程的因子。这些因素一般都可以分解成两类；可控因子（所谓的控制因子） ，以及不可控因子（称为噪声因子） 。可控因素成为设计团队确保输出参数满足预定的可接受范围的方法。DOE测试的建模可以查看帮助文件中关于可控因子的详细说明。然后驱车规格放在系统响应规格上的

控制因素。例子电梯有关的可控制因素包括滚动轴承跳动（例如，在导向轮和缓冲器） ，弹簧弹性系数和及预紧力。 。

噪音因素所代表的不可控的输入，这些因素会是系统的输出不够稳定，应该

尽力避除，以确保有稳健的表现。例如一些噪音因素可能来自乘客的影响（如轿厢地板上载荷分布不均） ，供应商或装配的性能（例如，导轨之间的距离，导向变异性） ，寿命的变化（如轴承磨损） ，和环境因素（如温度和湿度的变化） 。鲁棒DOE测试可以用来选择控制因素，最大限度地减少噪音的对选定的系统响应的影响。

为在DOE的过程中选择适当的因子，必须遵循现有的产品或生产过程的专

家。在这一阶段在DOE的过程中对于产品的经验和产品知识是不可替代的。最后你会发现选择合适的P图来表述系统问题是整个DOE测试的关键。

很多UTC的ACE工具可以帮助这一进程。例如，卢索图，大象图，并marciano图表（以分析客户的问题，与现有的外观设计） ， 5Whys和鱼骨图。据此建议使用头脑风暴以尽快产生候选的 P -图。

关键点：要成功实施DOE首先要在权威专家的带领下建立P-图，以建立主要的输入输出因子。

建立好P图以后，下一步的工作是开发一个试验矩阵或通常被称为DOE设

计。右侧是一个通用的代表性测试矩阵说明以供参考。测试的数量被列在测试表格中第一列。在这个例子中有8个测试。已经确定的因子被列在下面的几列（这里我们有3个需研究的因素） 。为完成DOE设计，每一项因素的水平写在对应的单元格里。通常每个因子会两个水平，而视情况不同也可以选择3到5个水平。在所举的事例里，DOE设计每个因子有两个水平，在后面的学

习中你讲进一步了解，这是一个3个因子，两个水平的设计。完成后，这个测试矩阵，将用于指导整套测试程序的运行。设立DOE文本矩阵，可被分解成一系列的子步骤，其中详述如下

如在图2.1是一个流程图，它可以用来选择适当的DOE设计。若P -图包括五个以上控制因素的话，则建议重新考虑每个因子，使因子数量减少到5个以下。该DOE设计被称Plackett-Burman DOE，他是DOE的首选，如右所展示的。它通常涉及12个试运行，它可以用于11个潜在的控制因素。在这里使用者必须给每个因子指定高和低的值。可以利用一些商业DOE软件包或者UTC推荐的MiniTAB来生成测试

矩阵。 （见TBD部分对Plackett-Burman DOE设计的叙述） 。建立DOE流程图后，就可以用一些标准数据的处理技术将DOE的控制因子数降低到5个以内，以此服务于后面的DOE测试。（对于标准数据的处理技术将会在TBD的部分给与描述）。

主要有两个DOE设计，将被视为指南去进行稳健产品设计。建模设计是用于开发一种功能性的关系，控制因素和系统响应。这方面的资料，获取一个实证模型，然后形式的基础上，为优化系统性能和配置的要求，以控制因素。鲁棒性设计，设法减少敏感性系统响应的噪音因素，选择最好的搭配得当控制因素。这两种DOE将在余下详细讨论

见图2.1 ，典型的DOE设计被要求建构一个经验模型来针对产品或生产过

程中因子的响应，因子交互互作用和非线性的的问题。交互作用即其中一个因素改变后，对系统内另一个因子对系统的响应有影响。讲一个关于电梯导轨拼接和轿厢不平衡两个因子之间的交互作用。根据近平衡的条件下，悬架弹簧比较软，导轨对接水平不会对水平振动产生较大影响。不过，根据不平衡的条件下，弹簧变的僵硬，导轨对接水平变的十分重要。

设计者在构造模型设计同时必须对存在潜在影响的因子间的交互影响做出

评估。如果显着，控制因子交互将需要更多的测试，以方面通过统计学来分析交互作用的条件。如果专家没有洞察到因素之间的相互作用，那么，最安全的做法是假设这些因子都存在交互作用，然后进行DOE测试，以证实这个假设。

第二个响应的性质是对DOE设计选型具有重要的影响力是评估非线性系统传递函数。非线性是因子在指当一个因子的水平改变，因子将影响到系统响应；以一部电梯为例，非线性因子可能是隔音垫的刚度，其影响力就在轿厢噪声水平。如果是一个非常柔软的隔音垫，它可能是一个非常好的噪音传输隔离器，但由于其刚度增加，它变得不那么有效，整个结构的能量传输将会使的轿厢内的噪音以一个较大的速率不断增加。

控制因子之间如果有显着的非线性效应将需要进行两个水平（高，低）以上的测试。在多数情况下，第三个水平的出现意味着讲需要进行更多的测试来证明之间的影响。

因此，四个不同类型的DOE设计，有可以进行建模设计测试。这些设计宗旨是以最少的测试次数，从整个测试矩阵中获取期望得到的经验值；

第二个主要分支在图2.1涉及到被Dr. Genichi Taguchi 普及的鲁棒性DOE设计。你会记得，他的做法是定义噪音因子再去控制因子。DOE测试的目的，在这种情况下，是要去控制那些对于系统响应灵敏度最小的因子。

鲁棒性设计的第一步是确定那些因子的组合将会对系统响应造成的最大的典型变化。有时，这可以充分利用工程判断。其他时候并不明显，因此需要进行DOE筛选实验。在筛选DOE中， Plackett - Burman设计，可用于如大量的噪音因素，全析因或者Taguchi 2-level设计，可用于如只有少数的噪音因

素。在这些初步筛选试验的目标是确定所有测试噪音的因素的方向性。对于这些因子进行分析，然后判断出这些因子进行系统响应的最大价值以及最小价值。统称对这些因子矢量设置low and high。

接下去是通过完成每个因子的2个水平运行（所谓外阵）来完成对于所有

所有控制因素（所谓内阵列）的Taguchi DOE设计。测试矩阵被例在右边的图表中

NN

Exect

在完成了DOE设计的第2步，下一步的工作是执行。在测试矩阵中的每一

行，代表为各种因素的一种结合，它必须经过文章中所涉及问题的针对性测试。不过，在进行这些试验前需要制备一套相应的补步行动方案；

首要的行动是进行涵盖整个测试计划

的工作危害分析（JHA）。该JHA的功能是要找出在试验中的潜在危险，并制定适当的计划（例如，使用人员保护装置，挂牌上锁等） 。当然， hazscan技术应该用来在整个测试，以确定在JHA中没有被考虑到的新的风险。

其次，评估有可能反复

的，需要对每一个测试案例（即每排的DOE测试矩阵） ，需要加以估计。这里需要两个输入去做出决定；一个是对于不同过程响应的期望值的估计，另一个是我们希望去发现的系统相应的最小差异性，我们希望的得到最小差异的测试总数是32次。这是

一个准则，即应当遵循如果可能的话，但很多时候是重复试验室不切合实际的。

可以非常有趣的发现，DOE测试是一个数值模拟（即建模，分析，仿真和

计算（ MASC ）工具） 。强大的分析工具来优化系统响应，判断自己的分析模型。在大多数情况下， MASC工具，不会有任何的过程中的变异性，所以对任何一个DOE测试条件都不存在需要进行重复性试验 DOE运行测试的顺序，应尽可能的随机，以此来减少外界环境对于最终结果的潜在影响。另一个因素是我们在测试过程中不知道会发生什么事，而不得不去停止测试矩阵；运行随即程序的话，我们不会遗漏任何东西；如果，从另一方面，例如采用标准表上的全因子分析进行第一个因子的分析，或者只是前一半，那么，如果测

试是停止在这一部分，就不会从第一个因子里得出任何结论。但是，这个随机增加了相当多的时间和费对于DOE测试项目，而且可能不值得。

在进行实际测试之前所需要讲的最

后一个重要环节是进行测量系统的评估。事实上，这项评估工作应尽可能早的进行，以确保获取高质量的数据。标准DOE术语称这部分工作为可重复性和可再现性（Gage R&R） 。重复性定义由一个人用同一个测量系统多次重复测量同

一个系统。再现线定义不同的人用同一个测量系统测量同一个系统。商业DOE的软件包，如minitab ，提供最先进的方法来分析Gage R&R，一般准则，是测量差异小于整体期望差异的20%。如果不是，那么在DOE测试即可开始前必须找到一种改进或替代手段；

Analyze Data

完成DOE试验并填完矩阵中的响应后，下一步就是对数据的分析，有一大堆的分析工具可以用来对DOE的数据进行分析。这章将着重讲述这些工具，以及如何在主要的3类DOE试验中使用它们（筛选型、建模型、鲁棒型DOE设计）。 让我们从筛选型DOE试验的数据开始。一个典型的12次筛选型试验设计，可用于筛选6-11个潜在因子。筛选型DOE的目的是找出对结果影响最大的因子，来减少试验中的因子数目。两类图对该评价效果显著，主效应图和帕累托图。 一个简单的9因素案例如右图所示。每个因素都有2水平（在这里显示为+1和-1）首先，我们可以用主效应图找出9个因素中对结果影响最大的因子，如图2.2所示.这幅图

里，每个因素都有其对结果影响的一幅小图，显示两个水平对结果音响的高低。筛选型DOE设计通过使因子有相同多的相应数据，来保持其评价的公平性。 主效应图因此被用来确定哪个因素影响最大。因此，这些有着最大斜率的因素也对结果影响较大，他们将被延用至下依次的试验。在这个案例中，因素A和G是最显著的

帕累托图示另一类典型的DOE分析图，他也可被用在筛选型DOE试验中，来寻找关键因素。他把主效应的斜率直接反映在图上来进行对比。图2.3是一个简单的帕累托图，通过它也可看出A和G是显著因素。

接下来，建模型的DOE设计将被讨论。这类DOE的目的是通过观测试验反应来建立数学模型。该模型也有对结果影响明显的2水平，并得出经验性，线性和非线性的模型。两类模型都可以用简单的图形工具看出来。 例如，一个2因素，3水平，2仿行的全析因的测试数据如右图所示。在这个案例中，潜在的线性和非线性被考虑在内。

左边的主效应图显示当X变化时结果不在同一体条线上，表示其存在明显的非线性。

同理，左边的图交互作用图，显示XY的共同变化存在着明显的线性。

这些图表提供视觉洞察到系统响应。实际估计的参数值为经验模型，可利用方差分析（ANOVA）技术，这是一个支持DOE的商业包的线性回归法软件

一个例子，右边图标的样本建模设计是一个典型的方差分析。评估可以为六个系数（ 1个斜线，2个主要的影响项，2非线性项，和1个交互作用项） 。首先， 我们应该通过看R-Sq来评估整个经验模型。这是一个通过实际标准数据来评估整个经验模型相互关系是否恰当。当R-Sq > 80 ％ ，是理想的建模DOE测试。在这种情况下，模型的拟合优度非常好（即99.9 ％ ） 。

估算的六个潜在经验模型系数在表中的“Coef”栏中体现。所有DOE软件包用\值提供一个统计显著性。一般规则是，如果P值小于0.1 ，这一项是有巨大意义的，应该被保留在DOE经验模型中。

最终的DOE数据分析程序，以涵盖所有与之相关的鲁棒性。你会回想起，鲁棒性设计的目的是否是要选出最佳的组合控制因子值（所谓内排列） ，使得因子对于系统响应的敏感性达到最小（称为外阵） 。第一步就是要转换成测量系统响应值在两个噪音水平，设定一个适当的S / N比。三种常用的S / N值给在右边的表上给出指示。\较大的是更好地\， \正常是最好的\， \较小就是更好的\，指的响应值。举例来说，如果我们正在处理

的系统响应时噪音水平，我们自然会选择\较小，是较好的\比率，因为我们想尽量减少噪音水平，同时也最大限度地减少其差额。

为了说明这个过程中，假设我们可以得到鲁棒性DOE数据集如下图右。Taguchi L4 2-factor DOE被用来提出系统响应中高的或者低的期望值。我们将用一个\较小是较好的\值。最适宜的控制因子值

可以通过有S/N值输出变量的主因素图来定义就像图2.5中所表示的。目的是为了S / N的比率最大化，而不论是采用了那种S / N比率。对于这个例子最适宜鲁棒性设计是x1 = 1和X2 = 1 ，虽然对于X2因子来说，它的斜率特别小，如果它有潜在的成本降低它也是被认同的。

因此，鲁棒性设计中的输出有两个种类。第一个是确定关键因子的最佳值。其次，是通过公式确定的噪音因子水平的范围。

第5步，基于以上的测试和分析对系统进行优化。对于鲁棒性DOE测试，这一步就是通过主效应图找出哪组可控因素的信噪比最大。对于建模型DOE测试，这一步就是对通过数据得到的模型进行进一步分析。 对于单个响应的模型DOE，可以多次使用简单图解来优化系统。以以前的一个例子为例。是一个，2因素，3水平，2仿行的模型。它既有分线性变量，又有X1/X2的线性关系。如右图所示，通过等高线找出两因子共同作用的效果。图

示中的Y是与X1、X2相关的。给定输出数据（例如80）可以在曲线上找出X1和X2的对应值。例如，确定X2更加显著，可以通过调整起来获得跟好的结果。 对于多个响应的系统这里做个简单介绍。任何优化方法最终需要一个单一的标量最优标准来加以界定。很多关于DOE的软件都带有“趋势分析功能”比如

MiniTAB的GUI-driven（添加一些参数）能够实现可能性分析。通过选择目标值（最大化、最小化、趋向某值）系统将反响计算其他目标值将受到的影响

可能性分析，加上模型验证功能和数值优化方法，这使得DOE分析师有能力在条件不完全清晰的情况下，是选对系统的优化。

Verify Design

一般的，通过稳定性设计和DOE试验分析得到的对于可控因素的最优化设置，不一定会是原测试矩阵中的某一组设定的测试条件。因此，进行一系列后续的试验来对数据的预判断进行验证就显得非常重要。一般建议的测试次数是4到20次（如果测试费用较高就选择4次， 如果测试费用低就选择20次）。如果使用MASC模型来进行DOE测试，那么只需要在硬件达到测试要求的情况下进行一次优化条件下的模拟运行即可。

通过统计分析，可以用来确认我们的DOE过程模型及其相关的优化设想。

通过统计分析，可以用来确认我们的DOE过程模型及其相关的优化设想。

如果这些验证试验没有达到预计的结果，那么开发团队就需要重新考虑以下潜在的原因：

? 在原先的DOE测试方案中，还有其他影响测试结果的因素没有被约束或

者监控。

? 在数据收集和测量时可能发生错误（例如抄写错误）。 ? 在最初的DOE设计中，可能有被疏忽的显著交互作用。

? 在最初的DOE设计中，可能有未被考虑的显著非线性影响。 ? 在实际的试验过程中可能存在与原DOE设计不同的情况。 ? 测试的重复性较差，可能会造成对试验结果有较大的变化。

开发团队需要逐一研究这些潜在的问题。在有些情况下，采取纠正行动，是简单的（例如纠正抄写错误，增加一些中间值的测量结果来识别和模拟非线性，增加更多的测试点来改善交互作用），在某些情况下，就会比较复杂（新的因素及相关的控制和测量方法需要被综合定义到测试平台里）。

一个长时间的DOE建议应该包括被讨论的人口设计统计取样。市场反馈分析必须成为开发质量计划的一部分，其中涉及到统计质量控制顺着整个价值流，通过适当的监督和检查，以确定的产品或生产过程的控制因素。

DOE过程中的最后一步，

是对已确定的关键因素详细定义规格。这个在建模设计中使用检查验证经验模型以及蒙特卡洛模拟（MCS）来完成是十分简单的。MCS过程包括重复的数值计算，再加上一个随机数运行 ，以假定一个对于系统可变性响应的关键因子的可变性。

使用MS-Excel将可以十分简单

的进行MCS分析过程。一个简单的电子数据表可以象图2.6一样被创造，第一栏代表在“M”中的因子的预期可变性。系统响应通过在数据分析的经验模型来计算。最后一栏表示为系统响应一个预测的潜在价值，。

DOE过程的最后一步是制定一个计划去监控那些关键因子，以确保系统响应是符合客户期望的。两种方法都可以被使用；

第二个常用方法（如下所示） ，是使用蒙特卡罗模拟设计（MCS）去观察随机变化的关键因子对于系统输出的影响。在这种情况下，各关键因子被随机分布，同时通过数据发生器合成因子的概率分布。通过限制输出参数变化，以确定每个的关键控制因素。

给定的上限（例如显示在蓝色格子）是用在conjection与DOE经验模型（在系统响应栏） 以产生预期的系统响应。

这里有几个统计标准（平均值，标准差，过程能力，过程能力指数） ，这些是用来量化可变性和在范围内归规范风险。其定义体现在右边的表中。我们给出的建议是如果工序能力指数是1.33或更高，那么过程是在控制下的。 [这可以被转换成一种估计缺陷率的每百万中存在66个缺陷。]备注，6Sigma的设计目标是CPK 达到2；

当所有的关键因子都在上下限内，而且CPK通过控制因子的可变性分布的有效评估，我们认为产品的规范是完成的。

学习一样东西的最好途径是看他人如何使用它，接下来三个例子将展示DOE方法在otis产品开发中的运用。项目小组很慷慨的在otis内部与我们分享这些具有代表性的案例。

很多工地出现了传动带噪声过大，和磨损过快的问题。为此，在工地进行了一些研究其潜在原因的实验。开始时项目组进行头脑风暴、鱼骨图、5W分析和一些前期探索性的DOE试验，以帮助项目组最终确定了5个因素来进行该项DOE建模。该DOE设计共21次试验，3水平，响应曲面设计。该试验的目的在于确定关键因素，该关键因素影响到其位置水平度，定定义确定橡胶传动带工地使用小朵的评价表准。也就是说，传动带的负载将小于5千克力。 5个可控因素分别是：X1： 固定点X2 ：垂直高度差X3：线性扭转X4：Y方向偏转X5：X方向偏转，参考在NSE试验中取得的显著效果，一种特殊的夹具被制作已进行该项测试。每天做一次试验，该试验耗时1个月。

Next Step Escalator Belt Tracking Control Factors

虽然产生噪声的因素还未被正式确定，但项目组已确定传动带的新旧程度是一个可变因素。当扶梯上下运行时所产生的左右不同的负载（在安装单元测试）也将作为一项输出。

小组对DOE测试进行了一些先期准备，使之更能体验可控因素的影响。实施试验的一个关键是试验的可重复性。对此项目组进行了重复性和再现性研究。

完成了21次试验后，进行了一个两因素方差分析。可控因素结合经验可建立如下系统模型。

该模型被用来确定以减少传动带问题的出现的参数。基于模型一个关于因素效果显著程度的表格如下，最佳参数是：

? Unequal position +/- 0.7mm ? Line-up +/- 1.0mm

? Y-Axis misalignment +/- 0.3mm, and ? X-Axis misalignment +/- 0.3mm.

该DOE试验用于验证一格新型竖直放置的位置传感器是否能满足公差需要。该DOE的特别之处在于，它是用于产品开发过程中，而不是基于已有的工地反馈。所以该案例用于体现DOE试验如何把客户需求如入产品。该产品的简单CAD模型如下：

项目组使用鱼骨图和专家分析法确定了2个关键因素（容器的直径和长度）3个干扰因素也被考虑在内（霍尔传感器的质量变异，磁体质量变异，空气间隙变化），通过一些前期测试证实了干扰因素对试验结果的影响（如2.4中所述）然后项目组确定了2因素，3水平，全析因的DOE试验方案。使用田口试验法，2个干扰因素在实施时被考虑。

主效应和信噪比如下图被进行对比，这一结果显示容器的直径和长度越大，传感器的鲁棒性越好，根据最大化鲁棒性并参考信噪比，决定容器直径为B。长度为Y。

这一项目研究的是客户反馈问题，GeN2系统在韩国的使用中收到大量的，关于异常噪声的客户抱怨，于是项目组从2004年开始对76个工地的113台扶梯及行了调查。从而总结了一批纠正措施（调整稳压电源、调整CSB涨紧度、优化主机软件、电流编成设置等）来减少整体噪音，使之达到48分贝的要求。

Pareto分析显示对钢丝绳稳定器的调整可以作为减少噪音的关键因素。为进一步探讨这一问题，一个关于鲁棒性设计的DOE试验将被进行，具体步骤将在下面被讨论。

“定义”阶段，也就是定义一个过程图，这花去了该项目开始的一年。

通过大量在工地和试验塔的试验和根源分析、鱼骨图、5W分析法、头脑风暴，项目组最终得出了一张P-图和含有4个可控因素，1个干扰因素，3水平，田口式的DOE方案。如下表所示。

3个不同的稳定器构造（现有形状、C型、L型）将被进行如下的：27次测试，4因素，3水平的DOE试验。将使用两项指标对其结果进行评价：稳定器的传送率和噪声水平。传送率通过给钢丝绳和支架24的加速度进行测试。在每次可控因素测试中三水平的3个干扰因素也被设置在内（通过移动位置负荷在轿厢内的位置进行） 该测试的实施是分耗时，安装和校准的加速度计和麦克风需要几周时间。制备符合测试要求得样品需要数周时间，每次测试都需要重新安装和调整位置。

经过一个月的测试，得出了以下图形。可控因素的影响被通过鲁棒性和信噪比惊醒评价，通过主效应图和信噪比可以看出（如下图）。鲁棒性最好的设计参数是无论C型或L型，稳定器在其最小硬度（50%正常硬度）中心矩156mm压强250千克力/厘米）

As an aid to the DOE practitioner, this chapter contains FAQs about DOE testing. These questions were formulated to highlight some of the potentially confounding aspects of DOE testing and were based on discussions with

participants in UTC’s Design Process Certification 201 two week training classes.

How many replicates do I need to have?

Replicates are the number of times a specific row entry in the DOE test matrix is repeated. Section 2.4 deals with this question. There are two estimated parameters (variance in the process and the minimum practical difference) that can be used to determine the statistical requirements for the number of tests. This number, divided by the number of rows in the DOE test matrix is an estimate of the required number of replicates. See Section 2.3 for more details. Obviously for DOEs using MASC tools no replication is needed.

If I don’t know about nonlinear effects and factor interactions, how should I create a test matrix?

When in doubt one should assume both are significant in Modeling DOEs. That is, Taguchi testing or Fractional factorials design at multiple levels should be used if at all feasible. In general Screening and Robustness DOEs do not consider these higher order terms.

How do Gage R&R results influence my test plan?

Gage R&R is a method for establishing the measurement variance prior to conducting the DOE testing. A general rule is that the measurement system variance should be less than 20% of the estimated process variance.

Do I need to randomize the order of my test matrix?

Ideally one should randomize the test matrix runs, this helps to minimize the potential risk of external noise factors (such as operators, test setups, etc.). It also maximizes one’s ability to extract useful data from prematurely terminated DOE testing. However, if the costs in time and resources required to randomize the testing are significant this might not be practical.

I may have to stop in the middle of testing, what’s the best way to construct a test matrix so this isn’t a problem?

As mentioned in the last question, randomizing the test runs is the best hedge against a premature termination of the DOE testing.

What do I do if the verification runs differ from the expected outcome? There are many reasons this might happen. The first thing to check is the integrity of the data, make sure that no errors were made in filling out the data in the test matrix cells. Another common reason is that factor interactions and/or nonlinearities are not properly considered in the DOE testing. A common practice to check nonlinearities (if you did a 2-level DOE initially) is to add a middle point and check for the deviation of that data from the fixed slope.

Interactions can be checked by including some more datapoints in the DOE test matrix. Consult Section 2.6 for more discussion on this topic.

Should I put in the actual values of the factors into the test data or the intended values?

The standard post-processing plots (i.e., Main effects and Interaction plots) will not work properly if the DOE input factor settings are modified. However, the empirical model construction via linear regression analysis will work fine with altered factor settings.

How can I flow requirements to control factors based on the DOE testing? This, afterall, is the main output of the DOE testing process (i.e., DOE step 7, Specs for CPM). A standard method, as documented in Section 2.7, is to use Monte Carlo Simulation to propagate control factor variability into the system response variability. This can be readily done using MS-Excel and the developed empirical process model from Modeling DOEs.

Does a correlation coefficient greater than 0.95 always indicate a good model?

It is generally recommended that the correlation coefficient should be as a minimum greater than 0.80. Values as high as .95-.99 are routinely achieved in DOE testing. A word of caution though, this is only a necessary condition for a good model. A high correlation coefficient only means that the collected data can be well explained by the empirical model. Nonlinearities could be present but ignored by 2-level testing. Interactions could be present but not uncovered by certain DOE designs. A correlation coefficient is a necessary but not sufficient condition for effective modeling. Full confidence in the results is only possible with verification runs that confirm the model predictions.

How do I determine the value ranges of factors?

There is no standard criteria for this, although a few of the typical issues that can affect the factor ranges can be discussed. If the range is too small there is a danger that the estimates of the response sensitivities will be corrupted

significantly by process and measurement noise. If the range is too large than the response will likely be nonlinear and a third level should be added to account for this. The range should be larger than the potential range of control factor settings (that is, we don’t want to have to extrapolate when setting their optimal levels). The domain expert who is most familiar with the process will have the best feel for factor ranges.

What is a reasonable set of factors to consider in any one test matrix?

For Modeling or Robustness DOEs it is recommended that 5 or less factors are usee. For Screening DOEs many more factors are routinely considered.

What is the right DOE design for my interest?

Yes there is, consult section 2.2 of this document that gives a flowchart for selecting the best DOE design.

What is CPM and how is it related to DOE testing?

CPM, which stands for Critical Parameter Management, is one of the tools in the overall Design Process Certification strategy. It represents an approach to controlling and managing variability in the product development process.

What are D-optimal designs and how can they be used in DOE testing? D-optimal designs are a fairly recent addition (in the last 20 years) to DOE designs that are based on the notion of using computer optimization methods to create a test matrix that is tailored to a precise situation. That is, the user must define what interactions are important and which ones are not. Computer software packages then can customize a DOE design to meet these specific requirements. The advantages of D-optimal designs is that they represent the minimum set of tests required for any category of DOE testing (screening or modeling). The disadvantages are that they require software packages to

generate the tables and analysis must be done with regression which can compound the problem of results interpretation.

What does the p value mean in the ANOVA summary?

The “p value” is a statistical check of the relevance of the candidate term in the overall regression analysis. In effect, “1 – (p-value)” is the likelihood that the added term is real and significant and not the artifact of noise. A general rule is that the term of interest should be retained in the regression analysis if the p-value is less than 0.1.

What is the recommended means of archiving DOE results?

At present there is no standard in place for archiving DOE results. It is

recommended that an MS-powerpoint presentation be created that captures the results of the seven step DOE process. All data should be saved in the

appropriate format. For example, graphs and data can be stored as a project file in MiniTAB.

What software tool is recommended for DOE processing?

Many commercial software packages are available to the DOE practitioner, including MATLAB, JMP, DOE KISS, and DOE Wisdom. The recommended software from UTC is MiniTAB.

How do I handle discrete data?

The most common approach is handle discrete input factor data is to just assume it is continuous and then to round any optimization predictions to the nearest effective input setting. Discrete output data can present many issues in DOE testing. It is preferable to have continuous system response measurements. For example, many times DOE testing is used when reliability issues are detected in a product design. Where possible a continuous output response should be used, for example strain rates, instead of a pass-fail ranking.

How can I effectively handle multiple output variables?

Yes using Desirability Functions as illustrated in Section 2.5 of this document. The concept is that a single scalar variable is created by introducing a

mathematical equation (i.e, the Desirability Function) that combines the multiple output variables.

What should I do if I can’t physically run every point in the DOE design space? This can occur when implementing DOE testing in a product development. There are two ways to handle this. The first is just to assume that the difference between the desired factor setting from the original DOE design and what can be implemented is small and thus the intended value can be used in the data matrix. In this case all the traditional DOE analysis tools will work. If the differences are

considered significant, then the practitioner can modify the value in the data matrix and use a standard linear regression method post-process the data to extract an empirical model.

What is a S/N ratio and how does it help me in DOE testing?

The S/N (signal-to-noise) ratio is a concept developed by Dr. Taguchi for Robustness DOE testing. It is a way of transforming two attributes of an output (it’s mean and standard deviation) into one metric that is used in Robustness DOE data analysis as discussed in detail in Section 2.5.

What do the terms “interaction”, “resolution”, and “aliasing” mean and how do they impact my DOE testing?

These are terms that are used to discuss how specific DOE test matrices can (or can not) account for the significance of various combinations and functional forms of control factors. An interaction means that the levels of one control factor influence the system response sensitivity of another. Mathematically this means that a term of the form K12x1x2 is present in the equation which links the control factors (x1,x2) to the system response Y. When choosing the DOE design (see Section 2.2 for more details) the practitioner must make a prediction about what interactions might be significant or not. If it is believed they are not, then a

reduced number of runs can be made. This fractional factorial design will not be able to distinguish between some factor interactions which will show up in the same way as other potential regression equation terms. This overlapping of potential terms is called “aliasing”. Lastly, resolution is a term applied to DOE designs describing the amount of aliasing present. For example a Resolution III design is one that does not alias main effects with other main effects, does alias main effects with 2-way interactions, and 2-way interactions are aliased with each other. The higher the Resolution, the more terms in the potential empirical model are not aliased.

Why is “one-factor-at-a-time” testing so bad?

One of the main problems with one-at-a-time (OFAT) testing is that a

suboptimal result is likely to be obtained. DOE testing allows the practitioner to uncover details about the process as captured by a response surface model via a Modeling DOE approach that can then be used to optimize the system behavior. OFAT testing in a sense is executing a localized gradient search optimization that will terminate at a local minima.

An additional issue with OFAT testing is that more test runs can be required due to process variability then for a balanced DOE design such as the Plackett-Burman screening design.

What is “Monte Carlo Simulation” and what does it have to do with DOE testing?

Monte Carlo Simulation (MCS) is a computational technique that can be used to predict the statistical variability of an output given variability in process inputs. An empirical equation, such as would be generated using a Modeling DOE

approach, that captures the functional relationship between input control factors and the desired system response is used in conjunction with a random number generator to make a large number of trial runs. The statistics of the output

variability can then be compiled. Section 2.6 illustrate how MCS can be used to generate specifications for control inputs as the final step in the DOE testing process.

I’ve heard that UTC is introducing a new tool into ACE called “Design Process Certification”. How does DOE fit in with this tool?

DOE is one of the key tools in Design Process Certification (DPC or Procert), UTC’s new approach for Robust Design. The intent of Procert is to apply

Process Certification techniques to the PDP to reduce variation and improve the robustness of our products and to improve the efficiency and effectiveness of the passport process.

What is “Six Sigma” and how does it relate to DOE testing?

“Six Sigma” has become a term associated with a process of ensuring quality in the presence of variability of subcomponents that was developed at Motorola in the 1980s. It was later deployed at GE and Allied Signal in the 1990s primarily in manufacturing using a five step process (Define, Measure, Analyze, Improve, and Control (DMAIC) . Design for Six Sigma (DFSS), with the steps (Define, Measure, Analyze, Design, and Verify (DMADV)) is a variant of SixSigma directed at improving the product development process. The goal of both methods is to manage and control variability to minimize defects to be smaller than 3.4 defects per million (the rate of escapes for a process with a capability index of +/- 6 sigma [Note: this would be a Cp value of 2.0]).

The DOE testing process described in this document is part of UTC’s Design Process Certification tools that will help reduce defects and thereby increase quality and customer satisfaction.

Can DOE be used to control time-varying transients?

Yes, by using Desirability Functions (see Section 2.5) coupled to a parametric characterization of the time transient waveform. For example, some typical parameters that can be used to desribe a time transient are overshoot, setting time, and delay. Each of these could be extracted from the DOE testing and then linked to a single scalar Desirability Function.

Can DOE methods be applied to numerical simulation predictions?

Yes, the Modeling DOE principles discussed in the document are directly applicable to numerical simulations. They represent the means to efficiently extract an empirical model response surface from computer simulations which can then be used to drive lower level control factor specifications.

What are some of the keys to successful DOE testing?

Some key points are: set clear objectives, if at all possible use quantitative measures, carefully consider process variations and use replications to minimize their impact, randomize the test run order if feasible to minimize external noise sources, consider potential aliasing-induced errors, and confirm results using verification runs.

What is “pooling” and how can I use it in DOE testing? What is the F-ratio?

When should I use a Robustness DOE vs. a Modeling DOE? What is the difference between Static and Dynamic SN ratios?

This chapter will present an illustration of the DOE process in a worked example that the reader can follow and execute themselves. All steps in the DOE process are executed. A numerical simulation tool is used as the test article that the reader can access if they want to reproduce the results and use this chapter as a Tutorial. The presented analysis is done in MiniTAB with the execution steps documented. Imagine yourself for a moment employed in the military over 1,000 years ago. Your job, as an

“Ingeniator” (engineer), is to determine the optimal setup for a newly designed siege weapon called a “Ingenium” or “trebuchet” (see illustration at the right), which harnesses the potential energy of a suspended weight to throw objects great distances. Advanced scouting has determined that in the next upcoming siege your trebuchet can be safely positioned in a wooded area outside the Evil Emperior’s castle of the SixSigma Dynasty. Your Commanding Officer and his System’s Engineering Lieutenant gives you the assignment of determining all the trebuchet’s critical factors and their associated settings to consistently hit a range of 390 to 410 feet, the anticipated requirement to smash the castle walls. The Quality Officer goes on to further define

“consistently” as missing less than 1 time in 100 attempts. Your mission, as you therefore understand it, is to flow an Upper Level Requirement (range within target of 390-410 feet 99% of the time) into Lower Level Specifications for the trebuchet design and settings. After consulting with

fellow Ingeniators, you discover that no one has an analytical model to predict the trebuchet’s distance as a function of settings, but everyone has their opinion of what are the most critical and useful adjustments. Problem is, there is no

agreement among the experts and no one has any data to defend their viewpoint. Based on careful considerations, you decide to perform DOE testing to answer your CO’s request.

You hold a meeting with your fellow Ingeniators and talk about what factors influence the range of the trebuchet’s toss. As you feared there a many variables to consider. In the end your

list includes nine potential control factors (see the

diagram at the right). Three weight values are considered (W1-W3), five dimensions (L1-L4 and H), and one

geometric variable (RA). Only one output response is considered, the horizontal distance the object travels. [At this point you hope that

there are no other walls which must be cleared to hit the castle wall.] So, you are done with DOE Step 1, the process diagram with 9 inputs (some of which might be considered noise factors at a later time) and 1 response is set.

The next step is to formulate the test matrix and plans. Clearly the first step in testing should be to narrow down the set of factors which must be considered (see Figure 2.1) so a

Screening DOE is recommended. A Placket-Burman design is chosen and can be created using MiniTAB using the pulldown menus as illustrated below.

You then select the Placket-Burman design with 9 factors using the popup menu, click OK and then accept the default parameters (0 corner points and only 1 replicate, this is sufficient for a Screening DOE). Next you must enter the factor names and variable ranges. The Screening DOE will use only 2 levels, so a High

and Low value must be used. The factor information entered by clicking on the “Factors” button. For this particular case it was decided that the Low levels would be at a baseline setting and High values would be a 10% increase in the factor levels. Finally, you can choose to randomize or not randomize the order that the test runs are stored. Here, by clicking on “Options” and unchecking the option “Randomize Runs”, and then clicking OK you will end up with the Screening DOE test

matrix as illustrated in Figure 5.1.

Now we are ready to execute the tests on the trebuchet. Before conducting the actual tests a Gage R&R analysis must be performed to ensure that the measurement system is adequate for the testing. MiniTAB provides tools to perform this analysis. A few test runs on made with the trebuchet at its nominal settings. Initially it was considered that fellow Ingeniators could stand out on the firing range and drop stones where they thought the 500 lb. projectile first hit. A calibrated string (a precursor to our modern

tape measures) was then used to measure the shot distance. Results indicated a sta

Rather than construct your own trebuchet, the user can use a trebuchet projectile prediction tool (Trebuchet for Windows V2.0 by Major Stephen J.

Ressler from West Point) that has been constructed using numerical simulation. The GUI for this simulation is the illustration used earlier to define the trebuchet factors. The measured distance for this Screening DOE test is listed in the test matrix in Figure 5.1.

A simple analysis can be performed in MiniTAB to identify the most significant factors. The Main Effects plot, as shown in Figure 5.2, shows three control factors are most significant – the release angle (RA), the length of the arm (L1), and the sling length (L4). This result was interesting, many of the Ingeniators were convinced their adjustment recommendations were right, but after reviewing this data and its systematic analysis they agreed with the conclusions. Surprisingly many were sure that

the projectile weight was critical to the range, but this testing refuted that claim.

Having reduced the number of control factors to 3, the next step is to construct a new Modeling DOE design. It is not known if interactions and nonlinearities are significant, so a Box-Behnken design is created (using the recommendations from Figure 2.1). Nominal values for the other control factors were set as follows: W1=11,000lbs, W2=6,200lbs, W3=500lbs, L2=29ft, L3=23ft, and H=25ft. The developed DOE design is shown in

Figure 5.3. In MiniTAB you can select the Box-Behnken design using the pull-down menu sequence STAT>DOE>Response Surface>Create Response

Surface Design. You can then select the Box-Behnken option, define the factors and levels and a test matrix will be automatically generated.

The test runs were made (see the D entry in Figure 5.3). Now the analysis can be made to determine the best empirical model fit to the data. In MiniTab you can select the option by the following menu picks, STAT>DOE>Response Surface>Analyze Response Surface Design. You then select the output response of interest (D in this case), and check to make sure all the terms

are included in this initial regression analysis. After the terms are selected, you can then click OK and MiniTAB will perform a regression analysis to estimate the best model fit to the collected Modeling DOE data.

For this particular case these initial results are shown in Figure 5.4. The first thing to notice is the correlation coefficient (R-Sq) is 99%, which is greater than the 80% minimum standard, very good results. Of the ten potential terms in the model, the p-values indicate that 6 are significant. [Remember p-values > 0.1 indicate the contribution of that term is not significant.]

The revised model fit should then be created by deselecting the four terms with large p-values. The revised model is then generated along with the coefficients of the six retained terms as illustrated in Figure 5.5.

The next step is to optimize the system. In this case we need to find a set of nomimal settings for the three control factors that meet the range goal of 400 feet. This can be

accomplished by using either a

graphical method using contour plots or by working directly with the surface response model.

MiniTAB allows you to

generator contour plots. On the right

is a plot of distance D versus L1 and RA for a fixed value of L4 (=32ft). The red line indicates all possible combinations of L1 and RA that result in a predicted range of 400 feet.

In this case we decide that the smaller values of L1 are desired, so if we set L1=60ft, and L4=32ft, we can solve for RA to be 169.3 degrees.

We are now ready to verify the predictions. In this case we set up the trebuchet at the optimized settings for all control factors and run a series of tests. In our case, because of the

importance of the impending siege, we are able to make 20 runs at this test condition. In this case the data fails to reject the hypothesis that the correct range of 400 feet was achieved, thus we can claim the system models were verified.

The final step in this process is to flow the system

requirement (that is, the Commanding Officer says we must hit the range of (390-410 feet) with only one or less misses per 100 attempts). The Monte Carlo Simulation method will be used to allocate the requirements to the identified control factors (L1, L4,

and RA). The first step is to convert the defect rate into a statistical measure. It can be shown that for normally distributed variables that a 99%two-sided confidence interval (can miss long or short) corresponds to +/- 2.57 standard deviations for centered values. This corresponds to a Cp value of (2.57/3 or 0.86).

Using the derived empirical model and the MCS template shown earlier in Section 2.7 (Figure 2.6) an allocation can be made. The adjustable parameters are the assumed distributions for the three random variable inputs (L1, L4, and R4). In this case all were assumed to be normally distributed (so in the cells the formula = norminv(rand(),M1,S1) was used, where M1 is the mean value, and S1 is the standard deviation). These values were adjusted until the Cp and Cpk values for the output response (estimated distance) were at the desired value of 0.87.

After finding an acceptable set of variables, one can then define the specification limits (LSL and USL) and their required Cp and Cpk values. The Ingeniators talked with the craftsmen who build the trebuchets and determined that these values could be controlled with proper monitoring and inspections.

Of course, the rest is history. We institute the above requirements as part of an overall CPM strategy which flows down to the craftsmen and their

Statistical Quality Control program. We defeat the SixSigma Dynasty and our trebuchet becomes the best selling product in the world. All because some

Ingeniator decided to use a systematic data-driven approach (which 1000 years later would come to be called “Design of Experiments”).

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