通用电气绿带DOE读本

1、Fractional Factorial Designs2022/8/251The factorial strategy is an efficient approach to experimentation as compared to “one at a time.”This can result in a large number of runs, even with a relatively small number of factors.For a 2 level factorial with k factors.Number of run=2k.Number of Runs Req

2、uired.NumberNumber Of Factorsof Runs 1 2 2 4 3 8 4 16 5 32 6 64 . . 10 1024 . . 20 1,048,5762022/8/252Information From 2-level Factorial DesignsNumber ofFactors 1 2 3 4 5 6 7 8 910 15 20 1 2 3 4 5 6 7 8 910 15 20MainEffects2-wayInteractions136101521283645105190Higher OrderInteractions151642992194669

3、6832,6471,048,3652022/8/253Example of Information Available.from a Full Factorial (4 Factors)Overall AverageMain effects: A B C D2-way interactions: AB AC AD BC BD CD3-way interactions: ABC ABD ACD BCD4-way interactions: ABCDNumber146412022/8/254Reducing the Size of a Factorial ExperimentEquipment d

4、elays allow you time to run only 4 trials in the allotted time.Which 4 trials will you choose?+A12345678Std.Order+B+CABC783456122022/8/255Choosing the Half Fraction+AStd.Order+B+C12345678We can select:ABC78345612orThe 4 shaded trials,+-+A+B+CStd.Order2358 The 4 unshaded trials,+A+B+CStd.Order1467Eit

5、her set provides two occurrences of + and two occurrences of - for each Factor2022/8/256Choosing the Half Fraction+AStd.Order+B+C12345678ABC78345612Half Fraction designs use Half the runs of Full Factorial designs.DesignNumber of runs Full Factorial2k = 23 = 8Half Fraction 2k-1 = 23-1 = 22 = 42022/8

6、/257Properties of a Properly Selected Half FractionThe design is nicely balanced, that is, each factor is studied the same number of times at each level. (Equal number of + and -).The design collapses into a full factorial. Should any factor turn out not to matter, the result is a full factorial in

7、the other two factors.The design covers much of the region of interest.2022/8/258Constructing a Half Fraction for Four FactorsList the full factorial for three factors. This is called the base design.The fourth factor is assigned the values of the 3-factor interaction of the other three factors.Reca

8、ll: a 3-factor interaction column is obtained by multiplying the three main effect columns. So D = ABC.AB+C+D = ABCFull factorialfor 3 factors2022/8/259General Rule for Constructing a Half FractionDefine the base design as a full factorial for the first k-1 factors.Assign the kth factor to the inter

9、action of the first k-1 factors from the base design.This interaction is found by multiplying the factor level settings together for the first k-1 factors from the base design.Number of runs = N = 2k-1.2022/8/2510Exercise Half fraction of a 25 factorial.Generate a half fraction experiment fork = 5 f

10、actors, A, B, C, D, and E.ABCDEFactors2022/8/2511Answers Half fraction of a 25 factorial+A+B+C+D+E2022/8/2512Trade-offsBetween full factorial and half fraction designsNumber of Effects Computed EffectsMeanMain Factors2 Factor Int.3 Factor Int.4 Factor Int.5 Factor Int.Total EffectsComputedFullFactor

11、ialHalfFraction1510105132151016This is also the number of runs Are the additional runs worth it? What happens to the higher order interactions?2022/8/2513A 2-factor Experiment With Confounded Effects+Factor A12345678Run+Factor BThe effects of Factors A and B are confounded. There is no way to differ

12、entiate between the effect of Factor A and the effect of Factor B.13012513313050857993ResponseConfounding is the combining of the effects of two or more factors into one resulting number such that the magnitude of the effects of the individual factors cannot be separated.2022/8/2514Confounding in th

13、e Half FractionABCD AB AC ADmean=BCDACDABDABCCDBDBCABCDFor the half fraction, each letter must be present on one side of the equal sign or the other.+A+B+C+D+AB+CD2022/8/2515ResolutionUnderstanding the degree of confounding in a fractional factorial.The “cost” of running fractional factorials is tha

14、t effects and interactions will be confounded.We use a single roman numeral, called resolution, to describe the degree of confounding.Resolution tells us the size of the effects we expect to see confounded.Resolution III designs have main effects confounded with 2-factor and higher order interaction

15、s, but not with other main effects.Resolution IV designs have main effects confounded with 3-factor and higher order interactions, but not with other main effects or 2-factor interactions. In other words, main effects are clear of 2-factor interaction.Resolution V designs have main effects confounde

16、d with 4-factor and higher order interactions, but not other main effects, 2- or 3-factor interactions. Two-factor interactions are confounded not with each other, but with 3-factor or higher order interactions. In other words, 2-factor interactions are clear of each other.2022/8/2516Resolution “Han

17、d” MethodHold up number of fingers equal to design resolutionfor resolution V = 5 fingers.Use other hand to grab number of fingers equal to the main/interaction effects you wish to investigate for confoundingi.E., To determine what main effects are confounded with, grab one finger.The remaining numb

18、er of fingers is the lowest level of interaction effects which are confounded.In the example above, the result is “4 - factor interactions and higher” are confounded with main effectssince there are four (4) fingers remaining.Review: given a resolution IV design, what are the second order interactio

19、ns confounded with?2022/8/2517Exercise ResolutionIf you expect several 2-factor interactions to be important, what resolution design should you select?What is the advantage of a resolution V design over a resolution IV?What is the resolution of screening designs?2022/8/2518Answer ResolutionSelect a

20、design of either resolution IV or V.A resolution V design has all 2-factor interactions clear of each other. Thus, you will always be able to tell which 2-factor interactions are important. In a resolution IV design, 2-factor interactions are confounded together. If you get a large interaction effec

21、t, it will not always be clear which particular 2-factor interaction is the one that is important. Sometimes you can figure it out by what you know about the experiment, but sometimes additional runs are required to resolve the ambiguity.Screening designs are resolution III.2022/8/2519Selecting the

22、Appropriate Design2022/8/2520Exercise: Using the Look-up TablesAn engineer has 7 factors that she wants to study attwo levels. She believes that all of these factors will affect the responses of interest and that some two-way interactions are likely.A) which design should she select?B) how many runs

23、 does this design have?C) what size fraction is this?D) what are 2-factor interactions confounded ith in this design?Bonus question: what design would you use to study 5 factors when you can easily afford to make 32 runs?2022/8/2521Using the Look-up Tables (Cont.)1. a. b. c. This is a 2-3 = 1/23 = o

24、ne-eighth fraction. d. Since this design is resolution IV, 2-factorinteractions are confounded with each other.2IV73273 = 24 = 162022/8/2522Using the Look-up Tables (Cont.)Bonus question:Consider running the half fraction of the 5-factordesign. With the additional runs that you can easily make, repl

25、icate this half fraction.The half fraction is of resolution V. Resolution V designs are generally adequate (unless you expect 3-factor interactions), and with replication, you can additionally investigate if the factors affect the variability of the response.2022/8/2523Summary of the Half FractionThe half fraction of a full factorial can often provide the sa