MIE364H1S Methods of Quality Control and Improvement

Course Instructor: Prof. V. Makis Handout #6

Factorial Design: 4 Factors, Each at 2 Levels, n = 1. A milling operation was evaluated for finish milling of a particular part. The critical controllable factors affecting surface finish were identified as: A = cutting speed (80, 100) B = depth of cut (1, 8) C = cutter diameter (100, 200) D = feed per tooth (0.25, 0.65) For this experiment, smaller values are better (smoother). Factorial Design Full Factorial Design Factors: 4 Base Design: 4, 16 Runs: 16 Replicates: 1 1 Blocks: none Center pts (total): 0 All terms are free from aliasing

Fractional Factorial Fit Estimated Effects and Coefficients for Y (coded units) Term Effect Coef Constant 69.75 A -14.5 -7.25 B 9.5 4.75 C 3.5 1.75 D 2.25 1.125 A*B 1.5 0.75 A*C 1 0.5 A*D -1.75 -0.875 B*C 2 1 B*C -0.75 -0.375 C*D -17.75 -8.875 A*B*C 2.5 1.25 A*B*D -3.75 -1.875 A*C*D 1.75 0.875 B*C*D 21.75 10.875 A*B*C*D -16.75 -8.375

Analysis of Variance for Y (coded units)

Source DF Seq SS

Adj SS Adj MS F P

Main Effects 4 1271 1271 317.8 * * 2-Way Interaction 6 1304 1304 217.3 * * 3-Way Interaction 4 1986 1986 496.4 * * 4-Way Interaction 1 1122 1122 1122.3 * * Residual Error 0 0 0 0 Total 15 5683

Data Display

Row A B C D Y 1 -1 -1 -1 -1 42 2 1 -1 -1 -1 44 3 -1 1 -1 -1 86 4 1 1 -1 -1 60 5 -1 -1 1 -1 103 6 1 -1 1 -1 65 7 -1 1 1 -1 69 8 1 1 1 -1 80 9 -1 -1 -1 1 101 10 1 -1 -1 1 70 11 -1 1 -1 1 74 12 1 1 -1 1 67 13 -1 -1 1 1 46 14 1 -1 1 1 49 15 -1 1 1 1 95 16 1 1 1 1 65

Yates’ Algorithm 24 design, n = 1, N = 16

A B C D Y (i) (ii) (iii) (iv) effect SS F=SS/MSE (1) - - - - 42 86 232 549 1116 µ = 69.75 77841

a + - - - 44 146 317 567 -116 A = -14.5 841 AF =26.98 *

b - + - - 86 168 312 -51 76 B = 9.5 361 BF =11.58 *

ab + + - - 60 149 255 -65 12 ˆAB = 1.5 9 c - - + - 103 171 -24 41 28 3.5 49 ac + - + - 65 141 -27 35 8 1 4 bc - + + - 69 95 -38 21 16 2 16 abc + + + - 80 160 -27 -9 20 2.5 25 d - - - + 101 2 60 85 18 2.25 20.25 ad + - - + 70 -26 -19 -57 -14 -0.175 12.25 bd - + - + 74 -38 -30 -3 -6 -0.75 2.25 abd + + - + 67 11 65 11 -30 -3.75 56.25

cd - - + + 46 -31 -28 -79 -142 -17.75 1260.25 CDF =40.43 * acd + - + + 49 -7 49 95 14 1.75 12.25

bcd - + + + 95 3 24 77 174 21.75 1892.25 BCDF =60.7 *

abcd + + + + 65 -30 -33 -57 -134 -16.75 1122.25 ABCDF =36 *

Effect

Perc

ent

20100-10-20

99

95

90

80

7060504030

20

10

5

1

Factor

D

NameA AB BC CD

Effect TypeNot SignificantSignificant

ABCD

BCD

CD

B

A

Normal Probability Plot of the Effects(response is Y, Alpha = .05)

Lenth's PSE = 2.8125

TSS = 83524 TCSS = 5683 = TSS – SS(µ ) SSE = SS(ABC) + SS(ABD) + SS(ACD) = 93.5 MSE = SSE/3 = 31.17

0.05,1,3 10.13.F =