第十课：2k Factorial Experiment

SIX SIGMA OPERATIONAL2K Factorial Experiments2K 因子实验,May 2015,Kenny,Operational ExcellencePanyu / China,Improve Phase 改进阶段,Project Definition: 项目定义Problem Description 问题描述Project Metrics 项目指标Remaining Session 1 Deliverables Remaining Session 2 DeliverablesDOE Planning DOE 计划Inputs List 输入清单DOE Planning Sheet DOE计划表Designed Experiments 实验设计Analysis of Experiment(s) 实验分析Y = F ( X1, X2, X3,.)Project Summary 项目概述Conclusion(s) 结论Issues and barriers 问题点和障碍Next steps 下一步Completed “Local Project Review”,2,2k Factorial Experiments Agenda: 2k 因子实验议程,Why do we need to use 2k factorial experimentation? 目的2k Vocabulary 术语Steps for DOE Analysis DOE 分析步骤2k Standard Order Designs 2k标准次序设计Exercise: 练习Calculating main effects 计算主要Calculating interactions 计算交互影响2k Example (using Minitab) 2k示例(使用Minitab)2k Exercise (using Minitab) 2k练习(使用Minitab)Adding Center Points 添加中心点2k Example with center points (using Minitab) 中心点2k示例(使用Minitab)Adding Blocking 添加模块2k Example with Blocking 模块2k示例Catapult Exercise 弹弓练习,3,Why use 2k Factorial Experiments?为什么使用2k因子实验?,The GOAL is to obtain a mathematical relationship which characterizes:Y = f (X1, X2, X3,.).实验目标是通过实验得到一个函数关系式Y = f (X1, X2, X3,.).The mathematical relationships allow us to identify the most important or critical factors in the experiment.这个函数关系可以帮助我们判定实验中最重要或关键的因素.2k Factorial Experiments allow investigation of multiple factors each at only two levels. These designs require relatively few runs per factor studied as compared to the full factorial approach. 2k因子实验仅用于多因素两水平的研究, 和全因子实验相比, 这些设计仅要求每个因子重复少数几次实验.2k Factorial Experiments also allow us to investigate a large number of factors simultaneously in relatively few runs. 2k因子实验也可用于研究大量因子的同时进行少数的几次实验.Finally, 2k designs are used most frequently in industrial DOE applications because they are very easy to analyze and lend themselveswell to sequential studies. 最后, 2k因子设计最频繁的用于工业DOE, 因为它很容易分析, 并可借助它做进一步的实验.,4,DOE Vocabulary:DOE 术语,Center Points: n repeats or replicates run at the center or midpoint of all quantitative factor levels. Center points can often represent the process parameter levels at their current operating conditions. These points allow us to check the accuracy of a linear fit.中心点: 对所有量化的因子水平设置中心或中点进行n次重复或复制实验. 中心点经常反映当前工作条件下的过程参数水平, 这些点也可以用于检查线性度的准确性.Curvature: The situation when the output of the process does not act linear at center levels (center points) of all factors.弯曲: 过程的输出所有因子的中心水平不是线性的情形.Experimental Error: The natural variation that occurs when collecting the data实验误差: 收集数据时存在的内在变异Pure Error: The variation in the data due to repeatability. It can only be estimated via true repeat or replicate measurements.纯误差:由于重复性产生的变异. 它只能通过重复或复制测量来评估.,5,DOE Vocabulary:DOE 术语,Blocking Variable: A factor that has an undesired influence as a source of variability is called a “blocking variable .” A “block” is a set of conditions likely to produce experimental runs that are more homogenous within the block than between the blocks. For example, parts from a single batch of material are likely to be more uniform than parts from a different batch. The batches would be regarded as the blocking variable.模块变量:如果实验中的一个因子存在负面影响, 并可以作为变异的一种来源, 这个因子被称为”模块变量”. 一个模块可以是一批材料或一整套条件, 这种条件常指把一个模块(或同批)中同质的因素进行实验设计. 比如不同批的材料, 不同的员工或班次. Confounding: One or more effects that cannot unambiguously be attributed to a single factor or interaction.混淆: 一个或多个影响不能用单一因子或交互作用来清楚的表达.,6,Method for 2k Analysis (1-4): 2k分析方法(1-4),Step 1:State the practical Problem步骤1: 阐述实际问题Step 2:State the factors and levels of interest, create a Minitab experimental datasheet such that all values for each response variable are in one column. Each inputvariable, or factor, is assigned to another column, which designates the two levels of that factor.Stat DOE Create Factorial Design步骤2: 选择因子和因子水平, 在Minitab里建立数据表, 所有的结果变量放在一列. 每个输入或因子, 放在其他列.Stat DOE Create Factorial DesignStep 3:Select appropriate sample size. Randomize experimental runs in the datasheet. Run the experiment.步骤3: 选择恰当的样本数, 随机安排实验的运行次序并进行实验Step 4:Construct the ANOVA Table for the FULL model:Stat DOE Analyze Factorial (or Custom) DesignGraphs Effets Plots (Select Normal or Pareto, and Set Alpha)步骤4: 为运行模式建立ANOVA数据表Stat DOE Analyze Factorial (or Custom) DesignGraphs Effets Plots (Select Normal or Pareto, and Set Alpha),7,Method for 2k Analysis (5-6): 2k分析方法(5-6),Step 5:Run a REDUCED model by eliminating:effects with non-significant P-Values, oreffects plotted low on the “effects Pareto” chart from Step 4Stat DOE Analyze Factorial DesignStorage Residuals & Fits步骤5: 通过消除高次交互影响运行减少模式effects with non-significant P-Values, oreffects plotted low on the “effects Pareto” chart from Step 4Stat DOE Analyze Factorial DesignStorage Residuals & FitsStep 6:Investigate the residuals plots to ensure model fitStat Regression Residual Plots orStat DOE Analyze Factorial Design - Graphs Residual Plots步骤6: 研究余差图以确认模式适合要求Stat Regression Residual Plots orStat DOE Analyze Factorial Design - Graphs Residual Plots,8,Method for 2k Analysis (7 - 8): 2k分析方法(7-8),Step 7:Assess the significance of the interaction termsUse Stat DOE Factorial Plots Interaction Plot (for 2-way interactions)Unstack data for 3-way interactionsOnce the highest order interaction is interpreted, analyze the next set of lower order interactions.步骤7: 研究交互作用的影响Use Stat DOE Factorial Plots Interaction Plot (for 2-way interactions)Unstack data for 3-way interactionsOnce the highest order interaction is interpreted, analyze the next set of lower order interactions.Step 8:Assess the significance of the main effectsUse Stat DOE Factorial Plots Main Effects Plot (for a graphical interpretation)Use Stat DOE Factorial Plots Cube Plot步骤8: 研究主要影响Use Stat DOE Factorial Plots Main Effects Plot (for a graphical interpretation)Use Stat DOE Factorial Plots Cube Plot,9,Method for 2k Analysis (9 -11): 2k分析方法(9-11),Step 9:State the mathematical model obtained. If possible calculate epsilonsquared and interpret for relative significance.步骤9: 阐述得到的函数关系式, 可能的话计算平方和, 并解释相关的因子的影响Step 10: Translate the mathematical model into process terms. Formulate conclusions and recommendations.步骤10: 把函数关系式应用到实际的过程中, 用公式陈述结论和改进建议.Step 11: Replicate optimum conditions. Plan the next experiment or institutionalizethe change.步骤11: 将最佳设置应用到实际过程中, 计划进行下一个实验或制度化变更.,10,Standard Order of 2k Designs: 2k Designs设计的标准次序,2K factorials refer to k factors, each with 2 levels. A 22 factorial is a 2x2 factorial.This design has two factors with two levels and can be executed in only 2x2 or 4runs. Likewise a 23 factorial has 3 factors, each with two levels. This experimentcan be done in 2x2x2 or 8 runs.2K 因子是指两水平, K个因子. 一个22 因子指的是2x2 的阶乘.象23 因子有3个因子, 两水平, 可以运行2x2x2或8次.The design matrix for 2k factorials are usually shown in standard order. The lowlevel of a factor is designed with a “-” or -1 and the high level is designated with a“+” or 1. An example of a design matrix for 2k 因子的矩阵设计通常是按标准次序排列的, 因子的低水平一般用”-”或-1表示, 高水平用”+”或1表示, 范例如下:,11,TempConc -1 -1 1 -1 -1 1 1 1,TempConcCatalyst -1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 1 -1 1 1 1 1 1,Class Exercise: 课堂练习: Create a 24 Factorial Design Matrix 建立一个24 因子矩阵设计 What are the minimum number of runs needed? 最少应运行多少次? Verify your results using Minitab: Stat DOE Create Factorial Design 通过Minitab验证结果: Stat DOE Create Factorial Design,22 Factorial:,23 Factorial:,23 Factorial:,Exercise: Calculating Main Effects:练习:计算主要影响,12,This is an example of a Full Factorial Experiment with only one observation per Treatment Combination (Cell).,Step 1: Problem Statement. The head lab chemist would like to determine the effect of Temperature (quantitative), Concentration (quantitative), and Catalyst (qualitative) on the yield of a chemical process.步骤1: 问题描述. 化学家希望通过对温度, 浓度和催化剂进行研究, 发现它们对化学药品的影响Step 2: The factors and levels:Temp: 160C (-1), 180C (1)Concentration (%): 20 (-1), 40 (1)Catalyst: Brand A (-1), Brand B(1)步骤2: 因子和水平是:温度: 160C (-1), 180C (1)浓度(%): 20 (-1), 40 (1)催化剂: Brand A (-1), Brand B(1)Step 3: The Design Matrix with results looks like:步骤3: 矩阵设计是:,Exercise: Calculating Main Effects:练习: 计算主要影响,We will now calculate the effects of the experiment by hand. First well look at Temperature. 我们将手工计算实验影响, 先看温度. We simply add the yields associated with (-1)and the Yields associated with (1) and calculate the average (Sum/4). The “1s” and “-1s” in the Temperature column are called the “contrast” for the main effect of temperature.将全部(-1)的结果相加, 所有1的结果相加, 从而反映温度的影响,13,The yield increases, on average, by 23 points as temperature moves from Low to High (160 to 180)温度从低到高(160到180)使得实验结果平均增长23点.,= 75.75 - 52.75 = 23,Exercise: Calculating Main Effects:练习:计算主要影响,Now use the contrast forConcentration to calculatethe effect Concentrationhas on yield.计算浓度对结果的影响,14,As the Concentration moves from 20% to 40%, on average the yield drops by 5 points.浓度从20%增长到40%, 结果平均降低5点.,= 61.75 - 66.75 = -5,Exercise: Calculating Main Effects:练习: 计算主要影响,Next use the contrast forCatalyst to calculate theeffect that Catalyst type has on yield.计算不同的催化剂对结果的影响,15,By changing the Catalyst from Brand A (-1) to Brand B (1), on average the yield improves by 1.5 points.催化剂从A换成B, 结果平均增长1.5点.,= 65 - 63.5 = +1.5,Interpretation: _,Exercise: Calculating Interactions:练习: 计算交互作用,We have just finished calculating the Main Effects for this experiment. Weve only investigated the independent Effects of Temperature, Concentration and Catalyst. 我们刚刚计算了实验的主要影响, 仅仅是独立的考虑了温度,浓度和催化剂的影响.This is similar to conducting three 2-Sample comparisons.相似的要进行3次2因子的比较The benefit of factorial experiments is that they provide the ability to assess “interactions” between factors. “Is there a particular combination of input settings that improve yields over and above the singular (main) effects?”因子实验的好处在于实验能够提供因子间交互作用的研究. “除了单个因子的影响外, 有没有特定的输入设置可以改善结果?”The interaction contrast is derived by multiplying the columns to be represented.To simplify, lets designate the factors Temperature, Concentration and Catalyst as T, C and K. The interactions we can test will be T*C, T*K, C*K and T*C*K.简单的说, 我们把温度,浓度和催化剂用”T”, “C”, 和”K”来表示, 那么它们间的交互影响表示为:T*C, T*K, C*K and T*C*K.,16,Calculate the Interaction Contrasts below. 交互影响计算对照表如下:,Exercise: Calculating Interactions:练习: 计算交互作用,Verify the “Contrast” for each interaction. 验证每个交互影响的对比Enter these Contrasts into Minitab 输入MinitabStat DOE Create Factorial DesignNumber of Factors = 3Choose Designs, Choose Full Factorial (OK)Choose Factors, Type Temp, Conc, Cat (OK)Choose Option, do not fold design & do not check randomize runs Calc Calculator (create columns for interactions) Create and Enter the response (Yield) data,17,We now know where the main effects and interactions come from. The challenge is deciding which effects are important (significant).现在我们已经知道了主要影响和交互影响是怎么来的, 接下来的挑战是哪个影响是重要的(或有影响的),Example, Analysis with Minitab:示例, 用Minitab分析,18,This experiment only has one observation per treatment combination. Therefore we cant analyze the “full factorial” using ANOVA procedures until we learn a few analysis tricks.这个实验是只有一次实验结果的一次循环, 因此, 我们不能使用ANOVA分析”全因子”实验, 除非我们学会了一些分析的窍门. When there is only one observation per treatment combination, we can use both the normal probability plot and Pareto chart to interpret which effects are likely to be significant as shown in slide 17. 当每个循环单元只有一次实验结果时, 我们可以同时使用正态概率图和柏拉图来解释哪个影响是重要的. If a factor or its interaction is insignificant (the null hypothesis, effect = 0, is true), then we expect to see the effects normally distributed around a mean of zero. Any outlying effect is considered “important” or significant. 如果一个因子或它的交互作用是有影响的(零假设=0是真的), 那么我们可以认为影响是一个均值为0的正态分布. 偏离分布中心的影响认为是重要的或有影响的.,Example, Analysis with Minitab:示例, 用Minitab分析,Step 4: Construct the ANOVA Table for the FULL model:Stat DOE Analyze Factorial DesignGraphs Effects Plot (Normal or Pareto, set Alpha)步骤4: 建立ANOVA数据表,19,Example, Analysis with Minitab:示例, 用Minitab分析,Step 4: Construct the ANOVA Table for the FULL model:Stat DOE Analyze Factorial DesignGraphs Effects Plots (Normal or Pareto, set Alpha)步骤4: 建立ANOVA数据表,20,Effect,Example, Analysis with Minitab:示例, 用Minitab分析,Step 5: Run a REDUCED model by eliminating: 运行减少模式effects with non-significant P-Values, oreffects plotted low on the “Effects Pareto” chart from Step 4Stat DOE Analyze Factorial DesignStorage Residuals & Fits,21,Question: Why do we leave Catalyst in the model? 为什么在模式里保留催化剂?,Note: It is always prudent to remove a few factors of lowest significance first & check for the marginal ones upon re-calculating the p-values instead of removing all in one attempt. 注: 首先驱除一些最低影响的因素, 或,Example, Analysis with Minitab:示例, 用Minitab分析,Step 6: Investigate the residuals plots to ensure model fitStat Regression Residual Plots步骤6: 研究余差图看模式是否适合Stat Regression Residual Plots,22,Normality TestP-value=0.267正态测试,Example, Analysis with Minitab:示例, 用Minitab分析,Step 7-8: Using the ANOVA Table, investigate the P-Values of significant interactions first, then Main Effects.步骤7-8: 使用ANOVA表, 先看交互影响的P值, 其次看主要影响;If the P-Value is less than 0.05, then summarize the data as follows:Use Stat DOE Factoria Plots Interaction PlotUse Stat DOE Factorial Plots Main Effects PlotUse Stat DOE Factorial Plots Cube,23,Example, Analysis with Minitab:示例, 用Minitab分析,Step 9: Calculate epsilon squared for the main effects & interactions left inthe model. State the mathematical model obtained.步骤9: 计算模式中剩余的主要影响&交互影响的平方和, 解释所得到的数学模式,24,2,We can use the Coefficients from theanalysis to derive the following reducedmathematical model:可以用系数来构成下面的公式:,Yield = 64.250 + 11.500 (Temp) + 0.750 (Cat) + 5.00 (Temp*Cat),Question 1: What is the yield when everything is set to zero? 问题1: 当每一个项都等于零时, 结果是什么?Question 2: What does that value represent? 问题2: 那个结果代表什么意思?Question 3: Estimate Yield when all coefficients are at (+1) 问题3: 当所有系数都设置在(+1)上, 结果会怎么样?Question 4: What tool can we use to obtain a further break down on the individual SeqSS ? 问题4:,Example, Analysis with Minitab:示例, 用Minitab分析,Step 10: Translate the mathematical model into process terms. Formulate conclusions and recommendations.步骤10: 把得到的数学模式应用到实际过程中去, 并阐述结论和改进建议.Conclusions: from the mathematical model we can see that if no change is made, then our process, on average, will run to 64.25% yield. We also see thatthe level of Temp is key in controlling the yield of the process.结论: 从数学模式中我们可以看到, 如果没有变化, 过程的平均结果是64.26%, 温度的设置是控制过程结果的关键.Recommendations: to improve yield, we recommend the process be run withboth Temperature and Catalyst set to their high values. Concentration shouldbe set to whatever is cost effective for the organization.Temp = 180CCatalyst = Brand BConcentration = whatever is cost effective改进建议: 如果要提高结果, 温度和催化剂应设置在高水平, 浓度的设置要考虑成本的高低;Temp = 180CCatalyst = Brand BConcentration = 成本是否接受Step 11: Replicate optimum conditions. Plan the next experiment orinstitutionalize the change.步骤11: 把最优化的结果应用到实际过程中, 计划进行下一步的实验或把变更规范化.,25,2k Factorial Class Exercise: 2k 因子课堂练习,1) Problem 1: Use the 11 step methodology to analyze the data for theexperiment in bhh325.mtw. Use: 问题1: 在bhh325.mtw中, 运用于11步方法分析实验数据, 步骤如下:Stat DOE Analyze Custom Design 2-Level FactorialHint: You will need to “Define Custom Factorial Design” firstInputs: Cat-Charge, Temp, Press, ConcOutput: % Converted2) Problem 2: Objective: To determine the effect of Moisture of Fiber, Process Temperature and Fiber Crimp Condition on Dyeability of theCarpet. Use the 11 step methodology outlined here to analyze the datain Carpet2.mtw using: 问题2: 目标: 测定光线湿度, 过程温度和光纤弯曲度对地毯可染性的影响. 在 Carpet2.mtw中, 运用11步方法分析实验数据:Stat DOE Analyze Custom Design 2-Level Factorial,26,Adding Center Points to 2k Factorials:增加中心点,27,Inputs: 输入 Reaction Temp: 反应温度 水平Levels: 150, 155, 160 Reaction Time:反应时间 水平Levels: 30, 35, 40,Product 1 Product 2产品1 产品2,Note: There are 5 center points added in this example. If the experimenter is familiar with the process, more efficient experiment & less center points may be used.注: 在这个实验里有5个中心点. 如果实验与过程相似, 可以使用更少的中心点&得到更有效的实验.,By only including two levels of the input variable, there is always a risk in 2-level designs of missing a curvilinear relationship; 如果输入变量只包含两个水平, 这就会存在忽略弯曲度的风险. The addition of “Center points” is an efficient way to test for curvature without adding a large number of experimental runs. 添加中心点来测试弯曲度的有效方法, 而不需要添加大量的实验次数. Example: A chemical engineer wants to improve yield for two different products. There are two inputs of interest: reaction time and reaction temperature. The engineer decides to conduct the experiment using a 2X2 design augmented with five center points to estimate experimental error and curvature. 例子: 一个化学工程师希望改进两个不同的产品. 输入变量有两个: 反应时间和温度. 工程师决定采用2X2的实验设计, 并添加5个中心点来测试实验偏差和弯曲度.,2k Example with Center Points (1-3): 中心点2k 示例(1-3),Step 1: State the problem: A chemical engineer wants to improve yieldfor two different products. There are two inputs of interest: reactiontime and reaction temperature.步骤1: 问题描述: 一个化学工程师想要改进两个不同产品的产量. 两个输入是: 反应时间和反应温度.Step 2: State the factors and levels of interest, create a Minitab experimental data sheet.步骤2: 建立因子和水平, 在Minitab里建立实验数据表Temperature: 150, 155, 160Time: 30, 35, 40Use Stats DOE Create Factorial DesignDesigns: 5 Center pointsOptions: No randomization of runsSpecify names and levels of factorsStep 3: Select appropriate sample size. Randomize experimental runs inthe data sheet. Run the experiment. We assume this was done.步骤3: 选择