SIGMA

Six Sigma Black Belt Incremental Training Improve,2,Six Sigma: The Five Project Phases,Process Input Variables,Understand the real,process by gathering,relevant data,Analyze data to highlight significant factors and root causes,Develop and test improvements that,eliminate or control the root cause,Modify process to include improvements and controls that make a real impact,Narrow the focus,of the project and,clearly identify it,Measure,Analyze,Improve,Control,Define,Number of Variables,All phases are supported and driven by dataGut feel is not enough; critical assertions are validated,3,Six Sigma Tools and Techniques,Six Sigma OverviewProject DefinitionProject Scoping ToolsIntroduction to Minitab Statistics for Understanding Six SigmaEffective Teams,Process MappingTransactional Kaizen TopicsRoot Cause Analysis TechniquesFMEAMeasurement Systems AnalysisCapability AnalysisRolled Throughput Yield,Define,Measure,Analyze,Improve,Control,Graphical Data AnalysisCorrelation / Simple Linear RegressionConfidence IntervalsHypothesis TestingMeansVarianceProportionsContingencySample Size SelectionANOVABinary Logistic Regression,Solution Generation TechniquesDesign Of Experiments IntroductionFull Factorials2K Factorials 2K Center Points & BlockingFractional FactorialsResponse Surface MethodsMultiple Regression,Control MethodsControl ChartsVariable SPCAttribute SPCProject Closure,Six Sigma: Design of Experiment,Six Sigma : Design of ExperimentsDoE with ANOVA,5,DoE with ANOVA Application,ANOVA - study the effect of one or more factors on a response, each factor having two or more levels. Can be used todetermine the statistical significance of effectscalculate the components of varianceestimate the contribution to variation by each identified sourceestimate the underlying noise within the process,General use for Full Factorial Experiments (DoE):Used whenYou need to establish a prediction model or transfer function.You need understand the effects of inputs and their interactions on the output(s)You want to separate significant inputs from insignificant inputs.,6,2-Way ANOVA Application,The simplest experiment that is useful involves 2 factorsOne example of a useful 2-factor experiment is a Gage R&R study:Multiple parts measured by multiple operators on a single measurement systemMultiple parts measured by a single operator on multiple measurement systemsA single part measured by multiple operators on multiple measurement systems,for i=1,2,., a j=1,2,., b k=1,2, n,7,To determine whether or not there is enough statistical evidence to conclude that each factors effect on the output is different from zero.yij = + i + j +()ij+ ijHo: is = 0Ha: at least one i 0Ho: js = 0Ha: at least one j 0Ho: tb ijs = 0Ha: at least one t ij 0,Wherey = observed output = average response = effect due to factor 1 = effect due to factor 2 = random error,The Statistical Objective,8,Select the Experiment Design,Simple Comparative ExperimentsTest of two means1- Paired t-tests1- & 2-variance tests1- & 2-proportion testsExperiments with a Single Factor:1-Way ANOVAExperiments with multiple factors & levelsStatistically Designed Experiments DOEs (Factorial)Analysis with N-ways ANOVA, Regression,9,DOE Introduction: Experiments,Definition:An experiment is a test or series of tests in which purposeful changes are made to input variables of a process or system so that changes in the output responses can be observed and identifiedDouglas C. MontgomeryTwo Different Approaches:One-Factor-At-a-Time (OFAT)One factor is changed throughout the range of interest for that factor, while all other factors are held constant.Statistically Designed Experiments (a.k.a. DOE Design of Experiments)Full FactorialFractional Factorial,10,Most traditional experiments involve the investigation of a single factor (One-Factor-At-a-Time), even though more than one factor may be influencing the output of interestOFAT fails to consider interactions (the failure of one factor to produce the same effect at different levels of another factor)OFAT, though used extensively, is inefficient and usually inadequate to discover “breakthroughs”,Single Factor Vs Factorial Experimentation,Factorial Experiment - Factors are varied together instead of one at a time.Experiments are a complete set of experimental runs and are analyzed when all of the runs are complete,11,The GOAL is to obtain a mathematical relationship which characterizes: Y = F (X1, X2, X3, .).Mathematical relationships allow us to identify the most important or critical factors in any experiment by calculating the effect of each.Factorial Experiments allow investigation of multiple factors at multiple levels. This is much quicker than traditional hypothesis testing techniques known as OFAT (One Factor At a Time). Factorial experiments are particularly valuable in the early stages of a study to “screen” a large number of factors (fractional factorial).Factorial Experiments provide insight into potential “interactions” between factors. OFAT experiments do not. This is referred to as factorial efficiency.,Why Use Factorial Experiments?,12,The purpose of anexperiment is to betterunderstand the real world,not to understand theexperimental dataWilliam DiamondIBM - Retired Statistician,Experimentation A General Method,Step (1)Define the ProblemEstablish the ObjectiveSelect the Output (Response)Step (2)Select the Input Factors (Xs)Step (3)Choose the Factor LevelsStep (4)Select the Experiment Design & Sample SizeStep (5)Run the Experiments & Collect the DataAnalyze the DataDraw Statistical & Practical ConclusionsTranslate the Conclusions into Practical Terms,13,Step 1: Establish the Objective,What do you want to discover by conducting the experiment?Are you trying to establish the relationship between the input factors (Xs) and the output (response-Y)?Are you trying to establish the vital few Xs from the trivial many (possible factors)?Are you interested in knowing if several input factors act together to influence the output (Y)?Are you trying to determine the optimal settings of the input factors?,14,Usually state in terms of the effects of inputs on outputs,Possible Experimental Objectives,To determine the effects of material variation on product reliabilityTo determine sources of variation in a critical processTo determine the effects of less expensive materials on product performanceTo determine the impact of operator variation on the productTo determine cause-effect relationships between process inputs and product characteristicsDetermine the equation which models your process,15,Different Operators Different Machines Different Shifts Suppliers/Parts,Noise Inputs(Discrete),Noise Inputs(Continuous),Key ProcessOutputs,ControllableInputs,Temperature Pressure,Toolsto ID Inputs/Output C&E Matrix/FMEA Fishbone Diagram Short-term Capability,Process,Room Temperature Barometric Pressure Relative Humidity Raw Material Characteristics,A factor is one of the controlled or uncontrolled inputs into a process whose influence upon a response is being studied in the experiment.,A factor may be quantitative (variables data), e.g., temperature in degrees, time in seconds.A factor may also be qualitative (attributes data), e.g., different machines, different operators, clean or not clean,Step 2: Selecting the Input Factors (Xs),16,Possible Sources to Narrow Xs,FMEAMulti-Vari & Hypothesis TestingProcess MappingBrainstormingLiterature ReviewEngineering KnowledgeOperator ExperienceScientific TheoryCustomer/Supplier Input,17,Step 3: Choosing Input Levels Based on Objective,Objective: Determine vital few inputs from a large number of variables (Screening)Set “Bold” levels at extremes of current capabilitiesGoal: if we vary the Input to extremes we will be assured of seeing an effect on the output if there is one Will exaggerate variationObjective: To better understand factor interactions (Mathematical Relationship)Once critical inputs are identified, reduced spacing of the levels is used to identify interactions among InputsThis approach usually leads to a series of sequential experimentsObjective: To identify the operating window of a set of input variables (Process Optimization)Close settings are again usedSequential experimentation is also used,18,The Higher You Get, The More You Will Learn Y= f(X)!,Step 4: Selecting the Type of Experiment Design & Sample Size,DOEsResponse Surface MethodsFull Factorials with ReplicationFull Factorials with RepetitionFull Factorials without Replication or RepetitionScreening or Fractional DesignsSingle Factor ExperimentsAnalysis of VarianceSimple Comparative ExperimentsTest of two means, 1- & 2-variance tests, 1- & 2-proportion tests,19,Recall that the sample size is dependent upon:Appropriate risk (a and b) levelsCritical difference to be detectedRather than using sample size to indicate the number of observations you need, factorial designs are expressed in terms of the number of replicates. A replicate is a repeat of each of the design points (experimental conditions) in the base design. In most practical cases, combination of repeats & replicates are utilized.,Choosing Sample Size & Number of Replicates,20,Step 5: Performing the Experiment,Document initial informationVerify measurement systemsEnsure baseline conditions are included in the experimentMake sure clear responsibilities are assigned for proper data collectionAs far as possible, perform a pilot run to verify and improve data collection proceduresWatch for and record any extraneous sources of variationAnalyze data promptly and thoroughlyGraphicalDescriptiveInferentialAlways run one or more verification runs to confirm your results (go from Narrow to Broad Inference),21,A Form to Help With Planning,DOEPlan.doc,22,Report should include the following sections:Executive Summary or AbstractProblem Statement and BackgroundObjectivesOutput VariablesInput VariablesStudy DesignProceduresResults and Data AnalysisConclusionsAppendicesDetailed data analysisOriginal data if practicalDetails on instrumentation or procedures,Y= f(X),The Final Report,23,Objective: It is important that the purpose of running the experiment is clear and that the defined experiment design is followed. When planning the experiment it should be made clear which factors are under investigation, what are the levels, how could other influences affect my experiment, what noise should be minimized, how should the data be collected, etc. Measurement SystemEffective measurement systems are vital for correct data analysis. Gage systems must be capable.Noise variables:Uncontrollable factors that may influence your experimental results - all attempts should be made to minimize the effect of such factors when running the experiment.,Considerations when running a DOE,24,Randomization: Randomizing the runs in an experiment can be used as a method to minimize the effects of noise variables such as time and machine or material degradation.Repeats: Immediately repeating an experiment run without changing the factor levels/settings - these give a measure of the sample and analysis variance Replicates: Performing the whole or part of an experiment more than once, at different times and in different orders - more valuable than repeats, they give an estimate of the total variability affecting the experiment.,Considerations when running a DOE,25,DOE - Other Considerations,Inference SpaceExternal & Internal Validity,26,Area within which you can draw your conclusionsTwo classifications: Broad and NarrowNarrow InferenceExperiment focused on specific subset of overall operationExamples: single shift, one operator, one machine, one batch, etc.Narrow inference studies are not as affected by Noise variablesBroad InferenceUsually addresses entire process (all machines, all shifts, all operators, etc.)Generally, more data must be taken over a longer period of timeBroad inference studies are affected by Noise variables,Generally, Narrow inference studies are done first to controlNoise variables. Broad Inference Studies are used to verifyresults of the Narrow Inference studies,Inference Space,27,Statistical Validity,Ensuring Internal ValidityRandomization of experimental runs “spreads” the noise across the experimentBlocking ensures Noise is part of the experiment and can be directly studiedHolding Noise Variables constant eliminates the effect of that variable but limits Broad InferenceEnsuring External ValidityInclude representative samples from possible Noise VariablesExamples:Ensure experimental units represent supplier variabilityDo experiment across shifts and daysInclude different product families,28,Threats to Statistical Validity,Low statistical power: sample size too smallInadequate measurement systems - inflates variability of measurements (GR&R)Noise factors in the experimental setting - inflates variability of measurement,Randomization and proper sample size minimize threats,29,A few Tips for Experimentation,Use your process knowledge of the problemValuable for choosing factors & levelsKeep the design and analysis as simple as possibleComplex experiments and analyses are often fraught with errorsUse two-level designs in early stages of experimentationKnow the difference between statistical & practical importanceA statistically significant difference in a process change does not mean it is importantExperiments are usually iterativeOur knowledge increases with time. Expect to use several experiments to arrive at the optimum process.Use not more than 25% of resources/budget in 1st experiment.,*,Six Sigma : Design of ExperimentFull Factorial,31,Learning Objectives,Introduce the full factorial method of DOE.Explain the concepts of main effects and interactions.Demonstrate and practice the full factorial method using Minitab.Explain steps in conducting successful DOE.Introduce DOE terms and definitions.,32,Full Factorials,Evaluates all possible combination of factors in a single experiment.Every possible data point is collected. Draw backs: ExpensiveTime consuming,33,Definition & Notation,In a full factorial experiment all levels of each factor occur with all possible combinations of all other factors,We can describe a 2k (“two to the k”) experiment with 3 factors, each at 2 levels by this notation:,To calculate the number of runs needed just multiply out the levels for each factor:23 = 2 x 2 x 2 = 8 runs2 x 3 = 6 runs, 2 x 2 x 3 = 12 runs,(Levels),Factors,(2),3,=,= 8 runs,34,The Golf Example a Simple 2x2 Factorial,A golfer experimented with two different club manufacturers and two different balls. He played each set of clubs with each type of ball and recorded his score.This is called a full factorial design where each level of all factors is run with every level of all other factors.In this experiment, what are the factors, factor levels and response?,Golf Clubs,Golf Balls,35,Calculation of Main Effects,Main Effect the average change in the response variable produced by a change in the level of the factor.,Main EffectBalls = ResponseTitleist- ResponseTopFlite,Main EffectClubs = ResponseTitleist- ResponsePing,Golf Clubs,Golf Balls,36,What is a Main Effect?,Main Effect of Golf Balls,Average Score,8685.5 8584.5 8483.5 83,1.5 Strokes,TopFlite,Titleist,Type of Golf Balls,The Main Effect for golf balls is the difference in the average score when playing with TopFlite golf balls and Titleist golf balls.,Golf Balls,Golf Clubs,37,Other Main Effects,Consider the experiment with walking/riding and drinking beers.In this experiment, what are the factors, factor levels and response?What are the main effects?,Main EffectTrans = ResponseRide- ResponseWalk,Main EffectBeers = Response0 - Response4,Transport,Beers,38,Main Effects Plots,For the two examples, plot the data using Minitabs Main Effects plots. (Stat ANOVA Main Effects Plots) Golf.mtw.,39,Interactions,Interaction when the difference in the response between levels of one factor is not the same at all levels of the other factors.At the low level of Beers, the effect of Transport is: 84-85= -1At the high level of Beers, the effect of Transport is: 85-92= -7The magnitude of the Beers/Transport interaction is the average difference in these two effects or: (-7-(-1)/2= -3,Transport,Beers,Interaction Effect = ResponseHigh - ResponseLow,40,Interaction Plots,For the two examples, plot the data using Minitabs Interaction plots (Golf.mtw)Stat ANOVA Interactions Plot,The larger the deviation from parallelism, the greater the likelihood of a significant interaction.,41,Another Look at Interactions,Here we see the additive model for the Two-way ANOVA analysisRemember this model did not fit reality as shown by the Residuals vs. the Fitted Values plotThis poor model fit was corrected by adding the interaction term.The final model:,yij = m + ti + bj + eij,yij = m + ti + bj + (t b)ij +eij,Main Effects,Interaction Effect,Error Term,42,Analysis with Minitab,Use Minitabs Balanced ANOVA analysisStat ANOVA Balanced ANOVA,* ERROR * Not enough data for this model.,The data set contains only enough data points to estimate the Ball and Club effects. There is not enough data to calculate the effect of interaction and still have a degree of freedom left for the error term. This term is required to calculate an F-statistic