3. Factorial Experiments (Ch.5. Factorial Experiments)

Transcription

1 3. Factorial Experiments (Ch.5. Factorial Experiments) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University DOE and Optimization 1

2 Introduction to Factorials Most experiments for process and quality improvement involve several variables. Factorial experimental designs are used in such situations. Specially, by a factorial experiment we mean that in each complete trial or replicate of the experiment all possible combinations of the levels of the factors are investigated. Thus, if there are two factors A and B with a levels of factor A and b levels of factor B, then each replicate contains all ab possible combinations. 2

3 Some Basic Definitions Definition of a factor effect: The change in the mean response when the factor is changed from low to high Main effect of A Main effect of B Interaction effect between A and B A y y 21 A A B y y B B AB

4 The Case of Interaction: Main effect of A Main effect of B Interaction effect between A and B A y y A A B y y 9 B B AB

5 The Battery Life Experiment An engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select at this point is the plate material for the battery, and he has three possible choices. When the device is manufactured and is shipped to the field, the engineer has no control over the temperature extremes that the device will encounter. It is known from experience that temperature will probably affect the effective battery life. However, temperature can be controlled in the product development laboratory for the purposes of a test The engineer decides to test all three plate materials at three temperature levels,15, 70, 125 o F, because these temperature levels are consistent with the product end-use environment. DOE and Optimization 5

6 The Battery Life Experiment A = Material type; B = Temperature 1. What effects do material type & temperature have on life? 2. Is there a choice of material that would give long life regardless of temperature (a robust product)? DOE and Optimization 6

7 General Two-Factor Factorial Experiment a levels of factor A; b levels of factor B; n replicates This is a completely randomized design DOE and Optimization 7

8 Statistical Model of Two-factor Factorial Design The observations may be described by i 1,2,..., a yijk i j ( ) ij ijk j 1,2,..., b k 1, 2,..., n where is the overall mean effect, i is the effect of the ith level of factor A, j is the effect of the jth level of factor B, ( ) ij is the effect of the interaction between A and B. 2 ijk is a NID (0, ) random error component. DOE and Optimization 8

9 Hypotheses for Two-factor Analysis Hypotheses of no significant factor A effect, no significant factor B effect, and no significant AB interaction. That is, H H H H H o 1 o 1 o :... : : 1 2 at least one 1 2 at least one :( ) 0 for all i, j H : at least one ( ) 1 ij i 0 :... 0 ij b j a DOE and Optimization 9

10 Extension of the ANOVA to Factorials a b n a b ( yijk y... ) bn ( yi.. y... ) an ( y. j. y... ) i1 j1 k1 i1 j1 a b a b n 2 2 ( ij. i... j....) ( ijk ij. ) i1 j1 i1 j1 k1 n y y y y y y SS SS SS SS SS df T A B AB E breakdown: abn1 a1b1 ( a1)( b1) ab( n1) DOE and Optimization 10

11 DOE and Optimization 11

12 ANOVA Table Fixed Effects Case DOE and Optimization 12

13 The Battery Life Experiment SS a b n i1j1k 1 y 2 ijk 2 y... abn SS A a yi bn i y abn 2 b y 2 j y SSB..... j1 an abn DOE and Optimization 13

14 The Battery Life Experiment SS subtotals a b i1j1 y 2 2 ij.... n y abn SSAB SSsubtotals SSA SSB SS SS SS SS SS E AB A B DOE and Optimization 14

15 The Battery Life Experiment ANOVA Table DOE and Optimization 15

16 Residual Analysis DOE and Optimization 16

17 Residual Analysis DOE and Optimization 17

18 Interaction Plot DESIGN-EXPERT Plot Life X = B: Tem perature Y = A : M a te ria l 188 Interaction Graph A: M a te ria l A1 A1 A2 A2 A3 A3 146 Life B: Tem perature DOE and Optimization 18

19 MINITAB Practice An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment and obtains the following data: DOE and Optimization 19

20 MINITAB Practice Data input DOE and Optimization 20

21 MINITAB Practice Two way ANOVA Stat -> ANOVA -> Two way Select response, row factor (control factor), and column factor (uncontrollable factor) DOE and Optimization 21

22 MINITAB Practice Select Graph Select residual plot (Four in one) Select feedrate and depth of cut for residual DOE and Optimization 22

23 MINITAB Practice AVOVA result DOE and Optimization 23

24 MINITAB Practice Residual Analysis Result DOE and Optimization 24

25 MINITAB Practice Residual Analysis results (vs factor level) DOE and Optimization 25

26 MINITAB Practice Interaction plot Stat -> ANOVA -> Interactions plot.. Select response and factors DOE and Optimization 26