ESD.77 – Multidisciplinary System Design OptimizationRobust DesignDan FreyAssociate Professor of Mechanical Engineering and Engineering Systemsmaineffectstwo-factor interactionskn-kk2kn21kABCAgain, run a resolutionIII on noise factors. If there is an improvement, in transmitted variance, retain the changeIf the response gets worse, go back to the previous state Run a resolution IIIon noise factors Stop after you’ve changed every factor onceChange one factor abcabcabcabcabcabcabcabcResearch OverviewComplex SystemsMethodology ValidationConcept DesignOutreach to K-12Adaptive Experimentation and Robust DesignBDABADABCABDABCDAACmain effectstwo-factor interactionsBCBDCDCthree-factor interactionsACDBCDfour-factor interactionsBDABADABCABDABCDABCDAACmain effectstwo-factor interactionsBCBDCDCthree-factor interactionsACDBCDfour-factor interactions01222221)2(2212112122222222212121)2(21210Pr dxdxnexerfnxxxINTMEINTnxxnINTijINTMEINTOutline• Introduction– History– Motivation• Recent research– Adaptive experimentation – Robust design“An experiment is simply a question put to nature … The chief requirement is simplicity: only one questionshould be asked at a time.”Russell, E. J., 1926, “Field experiments: How they are made and what they are,” Journal of the Ministry of Agriculture 32:989-1001.“To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of.”- Fisher, R. A., Indian Statistical Congress, Sankhya, 1938.Estimation of Factor Effects(abc)(bc)Say the independent il f+(b)(ab)experimental error of observations(a)(ab) et cetera isσB+(c)(ac)(a), (ab), et cetera is σε.We define the main effect []1AC-+--(1)(a)estimate Α to be[])1()()()()()()()(41−−−−+++≡ bccbaacababcAThe standard deviation of the estimate isεεσσσ221841==AThe standard deviation of the estimate isA factor of two improvement in efficiency as compared toεε24Aefficiency as compared to “single question methods”Fisher, R. A., 1926, “The Arrangement of Field Experiments,” Journal of the Ministry of Agriculture of Great Britain, 33: 503-513.“It will sometimes be advantageous deliberately to sacrifice all possibility of obtaining information on some points, these being confidently believed to be unimportant … These comparisons to be sacrificed will be deliberately confounded with certain elements of the soil heterogeneity… Some additional care should, however, be taken…”Fractional Factorial ExperimentsFractional Factorial ExperimentsABC+-++--132IIITrial A B C D E F G1 -1 -1 -1 -1 -1 -1 -12 -1 -1 -1 +1 +1 +1 +13 -1 +1 +1 -1 -1 +1 +14 -1 +1 +1 +1 +1 -1 -15 +1 -1 +1 -1 +1 -1 +16 +1 -1 +1 +1 -1 +1 -17 +1 +1 -1 -1 +1 +1 -18 +1 +1 -1 +1 -1 -1 +127-4Design (aka “orthogonal array”)Every factor is at each level an equal number of times (balance).High replication numbers provide precision in effect estimation.Resolution III.FG=-A+1+1+1+1-1-1-1-1Fractional Factorial ExperimentsRobust Parameter Design … is a statistical / engineering methodology that aims at reducing the performance variation of a system (i.e. a product or process) by choosing the setting of its control factors to make it less sensitive to noise variation.Robust Parameter DesignWu, C. F. J. and M. Hamada, 2000, Experiments: Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons, NY.Cross (or Product) ArraysA B C D E F G1-1 -1 -1 -1 -1 -1 -12-1 -1 -1 +1 +1 +1 +13-1 +1 +1 -1 -1 +1 +14-1 +1 +1 +1 +1 -1 -15+1 -1 +1 -1 +1 -1 +16+1 -1 +1 +1 -1 +1 -17+1 +1 -1 -1 +1 +1 -18+1 +1 -1 +1 -1 -1 +1Control Factorsa-1 -1b-1c-1 +1 +1 -1+1+1-1+1+1472III134722IIIIII132IIINoise FactorsTaguchi, G., 1976, System of Experimental Design.Identify Projectand TeamStep 1FormulateEngineeredSystem: IdealFunction / QualityCharacteristic(s)Step 2FormulateEngineeredSystem:ParametersStep 3Assign ControlFactors to InnerArrayStep 4Step 1 Summary:• Form cross function team of experts.• Clearly define project objective.• Define roles and responsibilities to team members.• Translate customer intent non-technical terms into technical terms.• Identify product quality issues.• Isolate the boundary conditions and describe the system in terms of its inputs and outputs.Step 2 Summary:• Select a response function(s).• Select a signal parameter(s).• Determine if problem is static or dynamic parameter. Note: Static has a single inputand one or more responses. Dynamic has multiple input signals and multiple outputsignals.• Determine the S/N function. See section Step2 e.g. Nominal-the-best, etc.Step 3 Summary:• Select control factor(s).• Rank control factors.• Select noise factors(s).Step 4 Summary:• Determine control factor levels.• Calculate the DOF.• Determine if there are any interactions between control factors.• Select the appropriate Orthogonal Array.Step 4 Summary:• Determine control factor levels• Calculate the DOF• Determine if there are any interactions• Select the appropriate orthogonal arrayAssign NoiseFactors to OuterArrayStep 5ConductExperiment andCollect DataStep 6Analyze Data andSelect OptimalDesignStep 7Predict andConfirmStep 8Step 5 Summary:• Determine noise factors and levels.• Determine noise strategy - Surrogate Noise Strategy - Compound Noise - Treat Noise Individually• Establish outer noise array matrix.Step 6 Summary:• Cross functional team will need to develop a step-by-step plan to carry out thelogistical activities necessary for successful completion of the data collectionphase of the optimization experiment.• Identify Facility Constraints.• Determine logistical/run order.Step 7 Summary:• Calculate the Following Values:• Mean• Variance• Signal to Noise Ratio• ANOVA• Factor Effects• Interpret the Results• Select Factor Levels providing the largest improvement in S/N should be• Use the mean effect values to predict the new mean of the improved system• If needed, select a value for the scaling factorStep 8 Summary:• TBD - CaniceOne way of thinking of the great advances of the science of experimentation in this century is as the final demise of the “one factor at a time” method, although it should be said that there are still organizations which have never heard of factorial experimentation and use up many man hours wandering a crooked path.Logothetis, N., and Wynn,