V.            Six Sigma – Improve & Control (15 Questions)

                              A.            Design of experiments (DOE)

                                                       1.            Basic terms
Define and describe basic DOE terms such as independent and dependent variables, factors and levels, response, treatment, error, repetition, and replication. (Understand)

                                                       2.            Main effects
Interpret main effects and interaction plots. (Apply)

DOE (Design of experiments) – a methodology of varying a number of input factors simultaneously, in a careful planned manner, such that their individual and combines effects on the output can be identified. The term SDE (statistical design of experiment) is also widely used.

·          Experimental objectives

o         Comparative – 1-factor completely randomized design; 2 or more factors randomized block design.

o         Screening – the primary purpose is to select or screen-out the few important main effects from the many lesser important ones. It identifies the key input factors. Design choices: full or fractional factorial; Plackett-Burman

o         Response surface – designed to let an experimenter estimate interaction effects, and, therefore, give an idea of the shape of the response surface. After screening, response surfaces will then help optimize the model provided that all variables are quantitative. Design choices: central composite; Box-Behnken

·          Iterative – the recognition that sequential experimentation will often yield more satisfactory results than “one big experiment.” There may be times in which an experiment is constructed with one factor at a time testing. However, the main disadvantage to “one factor” testing is the failure to account for interactions.

·          Experimental assumptions

o         Measurement system capability

o         Process stability

o         Residuals well behaved – one expects them to be normally and independently distributed with a mean of 0 and some constant variance.

§          Graphical methods suitable for judging the normality of the distribution are used to examine residuals. The three most common types are: histograms, normal probability plots, and dot plots.

Basic Terms

·          Alias – occurs when two factor effects are confused or confounded with each other.

·          Block – a subdivision of the experiment into relatively homogenous units. The term is from agriculture where a single field would be divided into blocks for different treatments.

·          Blocking – is used to account for variables that the experimenter wished to avoid. A block may be a dummy factor which doesn’t interact with the real factors.

·          Classical – 1FAT (one factor at a time) at two or three levels and attempt to hold everything else constant which is impossible to do in a complicated process. This traditional approach can yield invalid or inconclusive results as compared to modern design methods that squeeze a large amount of valid information from a few trials.

·          Confounded – when the effects of two factors are not separable. Normally, no experiment would confound one main effect with another main effect.

·          Collinear – this condition occurs when two variable are highly correlated. This condition would make it difficult or impossible to detect which factor really affects the response, so one variable must be eliminated from the analysis for valid results.

·          Covariate – factors that change during an experiment but were not planned to change.

·          Independent variable – an input or process variable that can be set directly to achieve a desired output.

·          Dependent variable – a variable that can change a desired output.

·          Error – experimental error, also called residual error, refers to variation in observations made under identical test conditions, or the amount of variation that cannot be attributed to the variables included in the experiment. Every experiment has inherent variability.

·          EVOP – stands for evolutionary operation, a term that describes the way sequential experimental designs can be made to adapt to system behavior by learning from present results and predicting future treatments for better response. EVOP trials are conducted in the near vicinity of an already satisfactory process, and as such, are normally used at the end of experimentation when the process essentially has statistical control. Often, small response improvements may be made via large sample sizes.

·          Factors – an independent variable which may affect a (dependent) response variable and is included at different levels in the experiment.

·          Graeco-Latin Design – is an extension of the Latin square design, but one extra blocking variable is added for a total of three blocking variables.

·          Inner Array, Outer Array – in Taguchi style fractional factorial experiment, Inner Array are the factors that can be controlled in a process; and Outer Array are the factors that cannot be controlled.

·          Interaction – occurs when the effect of one input factor on the output depends upon the level of another input factor.

·          Latin Square Design – are highly fractional factorial designs which only permit analysis of main effects only. Interaction effects cannot be determined, they are confounded with main effect results. A single factor experiment containing 2 specific nuisance (blocking) factors.

·          Levels – a given factor or a specific setting of an input factor.

·          Mixture Design – experiments in which the variables are expressed as proportions of the whole and sum to 1.0

·          Precision – the closeness of agreement between test results.

·          Randomized – frees the experiment from the environment and eliminates biases.

·          Repetition – is the variation in measurements obtained when one person takes multiple measurements using the same instrument and techniques on the same parts or items.

·          Replication - Replication occurs when an experimental treatment is set up and conducted more than once. Replicates are equal experiments run in exactly the same combination of factors. If you collect two data points at each treatment, you have two replications. Replication is done to reduce the impact of the inherent variation in the process, whereas repetition reflects the uncontrolled variability in the measurements.
In other words Repetition is equivalent to Repeatability, whereas Replication is equivalent to Reproducibility.

·          Residuals – (see error) are estimates of experimental errors obtained by subtracting the observed response from the predicted response. Residuals can be thought of as elements of variation unexplained by the fitted model. The differences between the response data and the model data.

·          Response (variable) – the variable that shows the observed results of an experimental treatment. Also know as the output or dependent variable.

·          Treatment – in an experiment the various factor levels that describe how an experiment is to be carried out.

·          Residuals – (see error) are estimates of experimental errors obtained by subtracting the observed response from the predicted response. Residuals can be thought of as elements of variation unexplained by the fitted model. The differences between the response data and the model data.

·          Screening experiment – a technique to discover the most (probable) important factors in an experiment. In screening experiments, highly fractional factorial designs are used to look for factor main effects only. They are called screening because they attempt to eliminate seemingly unimportant factors.

Main effects - A main effect is a measurement of the average change in the output when a factor is changed from its low level to its high level. An estimate of the effect of a factor independent of any other factors.

·          Example, the smallest run number possible to examine the main effects of 22 factors at 2 levels, using the Plackett-Burman design = 24. 22 factors require 24 tests. There must be at least one more run than there are factors, and the number of runs is divisible by 4.

Full factorial DOE - A full factorial design of experiment (DOE) measures the response of every possible combination of factors and factor levels. These responses are analyzed to provide information about every main effect and every interaction effect. A full factorial DOE is practical when fewer than five factors are being investigated. Testing all combinations of factor levels becomes too expensive and time-consuming with five or more factors.

·          The number of trials required for a full factorial is the number of levels raised to the number of factors. For example, 4 factors at 3 levels = 3 raised to the 4^th power = 81.

·          Full factorial and fractional factorial designs allow any number of quantitative as well as qualitative variables. The arbitrary “low” and “high” levels need only an ordinal feature to differentiate between them. All calculations are based upon the responses that result from the combination of factors and not on the factors themselves.

Fractional factorial DOE - A fractional factorial design of experiment (DOE) includes selected combinations of factors and levels. It is a carefully prescribed and representative subset of a full factorial design. A fractional factorial DOE is useful when the number of potential factors is relatively large because they reduce the total number of runs required. By reducing the number of runs, a fractional factorial DOE will not be able to evaluate the impact of some of the factors independently. In general, higher-order interactions are confounded with main effects or lower-order interactions. Because higher order interactions are rare, usually you can assume that their effect is minimal and that the observed effect is caused by the main effect or lower-level interaction. A fractional factorial studies all factors involved.

Post improvement considerations

·          Brainstorming

·          FMEA

·          Multi-vari-re-analysis

·          Post improvement capability analysis

·          DOE improvement analysis

·          Measurement system re-analysis