质量管理工具screening doe 3

1、DOEDesignofExperiment-Optimization DOEBy - Boron Chan11DOEIntroductionThe vital few important xs were identified and characterized in screening DOE. And a prediction equation being identified in screen DOE.The tools in this part will follow a methodology to find settings for the xs with lead to cent

2、ering process and minimize process variability. And an optimized transfer function will be identified in this part22DOEObjectivesAt the conclusion of this part, the participant should be able to: State the levels of the vital xs which optimize y. Design, perform and analyze optimization experiments.

3、33DOEFlowchart of Optimization DOEObtain Transfer Function y = f(x) from screen DOERefine Transfer Function with Additional Levels of the Vital xsChoose Level of Each Vital x to Optimize yRun Confirmation of Expected y for Optimal Levels of xsEstimate Error in Process (serror) via Trivial xsDevelop

4、Enhanced Transfer Function with Additional xsInclude Next Significant Group of Trivial xs(as per Screen DOE)Screen DOE-Optimization DOE-44Met goal? NoYesDOE endingDOEKey Concepts The transfer function from Screening DOE could have been either known, estimated or modeled. In the case of an estimated

5、transfer function, it may be necessary to explore additional levels of the vital xs, particularly if the x is continuous; in order to gain a better estimate of f(x). Either a numerical or graphical depiction of f(x) should provide sufficient information to estimate which x levels optimize y. Once th

6、e level of each x is chosen, it is always a good idea to verify the function. Confirmation runs should be made to verify that the predicted value of y occurs. The error in the transfer function should be estimated to affirm that the error is sufficiently small to achieve the goal set by customer (in

7、ternal or external). If it is not, then additional xs may be necessary to create an enhanced transfer function.55DOEObjectivesIn optimization designs, it is important to: Know how to use experimental designs to determine the factorsettings which will produce optimal results. Know common types of exp

8、erimental designs which use more than two factor levels. Understand the advantages/disadvantages of design types. Be able to analyze the results of optimization experiments and determine if further improvement is possible and/or desirable.66DOETransfer Function Refinement Cycle77Base linevSTOPUnders

9、tand current Capability& Performance2k-pCurrent Transfer Function (Screening DOE)2-Level Optimization DOE to explore operation areaRefined Transfer Function3- Level Optimization DOEYou Are HereOptimization Designs are used on continuous data for: Determining if “better levels” of performance exist-

10、Ascent/Descent Method- Simplex Method Understanding the area around the optimum Increased Local Information- 3k, CCD, Box-Behnken, D-Optimal designsDOERegion of Experimentation88Factor BScreen. DOE regionfeasibleregionPotential high nonfeasibleregionNon feasible SettingActual highActual low Potentia

11、l lownonfeasible regionregion of actual experimentpotential lowactual lowactual highpotential highFactor ADOE99Choosing factor levelsKey ConceptsThe rule for quantitative factors, in which a linear or monotone (i.e., non-increasing or non- decreasing) effect is expected, is to choose the two factor

12、levels as far apart as physically permissible. This choice of levels produces the strongest signal.The exceptions to this rule are: Some higher and lower potential levels are only feasible in combination with the proper choice of levels of other factors. See graphical example of a feasible and non-f

13、easible region. Prior experience suggests a particular reduced region of interest which is smaller than the feasible region. The experiment calls for future experimental runs outside the present region. This is particularly important in running central composite designs, which in a second block requ

14、ire star points outside the region of the first block.The actual region depends on the type of experiment. On-line experiments tolerate only small perturbations of the process due to limitations on the equipment and the risk of catastrophic results. Laboratory or pilot experiments often allow artifi

15、cially wide ranges of factor levels which also presents a danger, because they may lead to meaningless results with respect to a future application to a real process.An important consideration in choosing a level other than the present setting is the direction of change that appears to lead to the m

16、ost promising result for on-line processes.DOERS DesignResponse Surface Design1100DOEResponse Surface MethodologyResponse Surface methodology is a methodology which can examine the rough shape of the surface (within limitations that are often quite small in practical applications). Being able to loc

17、ate the optimum factor combination is a direct consequence of this feature. Within the sequential process of experimentation, here are some steps that could be followed:1. Conduct a screening experiment (2-level fractional factorial) in a region believed to contain the optimum.2. Fit a (linear) mode

18、l to the data collected in step 1.3. Use the fitted model to find the direction of the steepest ascent or descent (perpendicular(正交) to contour lines) in the response.4. Conduct a series of test runs along the path determined by the steepest ascent or descent direction until the changes in the respo

19、nse become small or reverse.5. Conduct a small scale experiment (2-level fractional factorial) and repeat steps 3-4 in the new region of the factors until the region of the optimum response is located or considerably narrowed down.6. Conduct a more extensive experiment that will permit the fitting o

20、f higher order polynomials(多项式) in order to evaluate the shape of the response surface.1111DOEFinding Ascent (Descent) Directions from Screening DOE Consider a process that has two controllable factors: Reaction Time & Reaction Temperature The response that we are trying to maximize is yield (%). Th

21、e current conditions are as follows: Reaction Time 35 minutes Reaction Temperature 155 degreesResultant Yield approximately 40 percentThe Design Matrix for Reaction Example:Temperature112216015040.039.340.340.540.740.240.641.540.9Time3040Why run trials at center point?The 2x2 Design with 5 Center Po

22、intsDOEKey Points A single trial at the center point provides insight into the non-linear response. Multiple trials at the center point allows for an estimate of error. Are the current operating conditions optimum? If not, which direction provides improvement?1133DOEDetermination of Direction Least

23、squares regression can be used to determine estimates for the coefficients of the prediction equation. The regression fit is significant. The best fit linear model is:y = 24.94 + 0.155Time + 0.065Temp This line indicates the best linear direction for optimum performance.The Design Matrix: Best Fit L

24、ineTemperature114416015040.039.340.340.540.740.240.641.540.9Direction for ImprovementTime3040The 2x2 Design with 5 Center PointsDOEKey ConceptsOutput from STAT Regression RegressionRegression AnalysisThe regression equation isYield = 24.9 + 0.0650 Temp + 0.155 TimePredictorCoefStdevt-ratiop Constant

25、24.9442.7329.130.000Temp0.065000.017193.780.009Time0.155000.017199.020.000s = 0.1719R-sq = 94.1%R-sq(adj) = 92.1%Analysis of VarianceSOURCEDFSSMSFp Regression22.82501.412547.820.000Error60.17720.0295Total83.0022SOURCEDFSEQ SSTemp10.4225Time12.40251155DOEAnalysis from DOEThe analysis from Defining an

26、d Analyzing an RS Design for a Linear Fit yields:Response Surface RegressionThe analysis was done using uncoded units. Estimated Regression Coefficients for YieldTermCoefStDevTPConstant24.94442.731559.1320.000Temp0.06500.017193.7820.009Time0.15500.017199.0190.000(T=COEF/STDEV)S = 0.1719R-Sq= 94.1%R-

27、Sq(adj) =92.1%Analysis of Variance for YieldSourceDFSeq SSAdjSSAdj MSFPRegression22.825002.825001.4125047.820.000Linear22.825002.825001.4125047.820.000Residual Error60.177220.177220.02954Lack-of-Fit20.005220.005220.002610.060.942Pure Error40.172000.172000.04300Total83.00222(MS=SS/DF)1166DOEQuestions

28、The earlier page provided output from the regression model whereas the attached output is from the DOE Analysis. Is there a difference between the two outputs? Can one be used in place of the other? Which output is more useful to you?1177DOEProportional Ascent (Descent) We usually take the center of

29、 the design region as the base or origin point for the path of steepest ascent. Then steps along the path are chosen proportionally to the signs and magnitudes of the regression coefficients. Usually, the regressor coefficient with the largest magnitude is chosen as a unit step and all other coeffic

30、ients are then a fraction of this step.Temperature118816015040.039.340.340.540.740.240.641.540.9Time304050The 2x2 Design with 5 Center PointsWhen stop explore?DOEKey PointAdditional Test Points for Reaction ExampleSince x1 has the larger regression coefficient, it is chosen as the base step size; in

31、 this case, the step is chosen as the distance from the origin to the face (5 minutes).The step size for x2 is chosen in proportion as 5(B2/B1) = 5(0.065/0.155) 2.1 The following subsequent set of trials could now be run:x1x23515540157.145159.250161.355163.460165.565167.6and so on.Questions:How far

32、would you take these trials, i.e. when do you stop? How many replicates should be run at each point?What do you do if the cycle time cannot be increased beyond 50?1199DOESimplex StepsThere are other methods of running subsequent experiments. One other method is adjoining simplices.A second 22 experi

33、ment could be run with one of the previous corner points as a base point. The original design is shown as solid lines and this second simplex is shown as dashed.Temperature 170220016040.041.515039.33040.340.540.740.240.640.94050Time22 Simplex DesignDOEKey Concepts What are the advantages of this met

34、hod over the previous step search? Should additional center points be run (in this simplex)?2211DOETransfer Function Refinement Cycle2222Base LineUnderstand CapabilityvSTOP2k-pCurrent Transfer FunctionRefined Transfer Function2-Level3- Level& Performance(ScreeningDOE)Optimization DOEOptimization DOE

35、You Are Here Region of optimum performance has been defined. Conduct more extensive experiment(s) to determine the shape of the response curve, to identify optimized response.DOEKey Points Sets of sequential screening designs are used to reach the vicinity of the optimum factor settings. Since the s

36、creening designs only model linear surfaces, an augmented design is needed to quantify the curvature of the surface.2233DOEThree-Level Optimization Designs2244Two Levels Not Sufficientto Uniquely Define CurvesThree or More LevelsUniquely Defines CurvePrecision Required in OptimizationDOEKey Points T

37、he purpose of many experiments is to determine the factor settings which will produce the best, or optimal, result. Two-level experiments will help to identify trends in the process data but are not particularly useful for identifying the factor levels producing desired maximums or minimums. Experim

38、ents with factors at three or more levels are required to identify these optimums.2255DOENonlinearity Detection using a Center PointDoes Factor A or Factor B nonlinearly affect the response?Response50502266Factor BHigh705050Low TimeLowFactor AHighFactor BFactor AFactor AFactor BFactor AFactor BDOEKe

39、y Points Use of a center point helped in forming a direction of improvement, however, it does not clarify which factor, if not all, contribute to the nonlinearity. Other designs are needed to quantify the nonlinear terms. A number of experimental designs exist for determining optimal factor settings

40、, such as 3k, Central Composite Designs (CCD) and D-Optimal Designs.2277DOEDesign Matrix Types22882 factors9 trials2 factors9 trialsCCF/3KCCD3 factors15 trials3 factors13 trialsBox-BehnkenDOEKey PointDesign Matrix TypesTypeCharacteristics Excellent spatial coverage2299Central Composite(CCD)kCCF/3 Or

41、thogonal and rotatable Good size/cost tradeoff On Minitab(Central Composite Face) Special cases of the CCDBox-Behnken (Center Edge Design) Good spatial coverage Orthogonal and (nearly) rotatable On MinitabD-Optimal Accommodates practical constraints Useful for non-symmetric space Not on MinitabDOESu

42、mmary of Design Attributes Desirable aspects of a design matrix:- Adequate Spatial Coverage- Cost-effective yet Informative Experimental Size- Separability of Factor Effects There are a number of useful properties that may help choose a particular design matrix:- Orthogonality- Rotatability- Uniform

43、 Precision3300DOEKey PointsDesign AttributesThe relationship between the size of the experiment and the information content plays an important role in the choice of the design matrix. Available resources may allow for a larger experiment, and thus more information. Other scenarios may result in frug

44、al (简朴的) designs.Regardless of the number of trials in the experiment, the investigator should be able to construct a design matrix that will spatially cover the region of interest. Hopefully, the region can be covered with a matrix that also has desirable properties such as orthogonality and rotata

45、bility. While the presence of these properties is not necessary to forming a meaningful analysis of the process, such designs are stronger in terms of their information content.About the Design MatrixThe class of orthogonal designs allows for the separability of effects among the factors of interest

46、. This class of designs minimizes the variance of the regression coefficients (and thus translates to a higher degree of confidence in the estimated function). A first order design that is orthogonal is the 2k design, where there are no cross-product terms and no aliasing.In moving to second order m

47、odels, the variance is typically smallest at the (geometric) center of the design. In moving away from the center however, if the variance does not increase uniformly, then the design will not be rotatable. In other words, circular or spherical variance contours yield rotatable designs, and are repr

48、esented by central composite designs.Designs whose variance at the center and at unit distance are the same are classified as uniform-precision designs. Central composite designs can also be made to be uniform-precision.3311DOEMisleading DesignsExperimenters must be aware of designs that can provide

49、 misleading, or even incorrect, results. Dont investigate interaction between factors Strong correlation between factors Insufficient spatial coverageSpatial Coverage3322DOEKey ConceptsExperimenters must be aware of designs that can provide misleading, or even incorrect, results.There are three main

50、 areas to be discussed in terms of misleading designs. One trap involves the use of designs that do not investigate interaction effects among the factors of interest. A second issue occurs when high correlations exist among the factors of interest. The dependency of supposedly independent factors, c

51、lassified as multicollinearity, causes high standard errors and very misleading results. This relationship occurs naturally in mixture experiments and should be recognized. If sufficient spatial coverage is not represented in the experimental trials, there is a danger of attempting to draw conclusio

52、ns for areas (that is, combinations of the factor settings) that are not well-represented by the experimental trials. A design which is taken along a very narrow spatial band will only provide information on the process in that region. Thus, the area of high confidence of results will be within the cylindrical region (sometimes represented as an ellipsoid). Projected results can be determined for outside the region, however with very low confidence. As the number of factors increases, the spatial area of coverage will not