II.2 Four Factors in Eight Runs II.2 Four Factors in Eight Runs Some Fiber Optics Examples and Some Fiber Optics Examples and Exercises* Exercises* Some 2 Some 2 4-1 4-1 DOE’s DOE’s Responses: Responses: – Shrinkage (S) Shrinkage (S) – Excess Length (L) Excess Length (L) Factors: some are trade secrets and others include Factors: some are trade secrets and others include – Time Time – Temperature Temperature – Fiber Tension Fiber Tension – Draw Ratio Draw Ratio – Line Speed Line Speed – Fiber Conditioning Fiber Conditioning * Examples are based on some 2 * Examples are based on some 2 4-1 4-1 DOE studies done at the Pirelli’s Lexington, SC DOE studies done at the Pirelli’s Lexington, SC plant plant
II.2 Four Factors in Eight Runs Some Fiber Optics Examples and Exercises* Some 2 4-1 DOE’s Some 2 4-1 DOE’s Responses: Responses: –Shrinkage (S) –Excess.
Dec 19, 2015
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II.2 Four Factors in Eight RunsII.2 Four Factors in Eight RunsSome Fiber Optics Examples and Exercises*Some Fiber Optics Examples and Exercises*
Some 2Some 24-14-1 DOE’s DOE’s Responses:Responses:
– Shrinkage (S)Shrinkage (S)– Excess Length (L)Excess Length (L)
Factors: some are trade secrets and others includeFactors: some are trade secrets and others include– TimeTime– TemperatureTemperature– Fiber TensionFiber Tension– Draw RatioDraw Ratio– Line SpeedLine Speed– Fiber ConditioningFiber Conditioning
* Examples are based on some 2* Examples are based on some 24-14-1 DOE studies done at the Pirelli’s Lexington, SC plant DOE studies done at the Pirelli’s Lexington, SC plant
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesShrinkageShrinkage
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects: -.39, -.07, -.07, -.05, -.03, -.01, .36
0.40.0-0.4
B+ACD
C+ABD
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects: -.39, -.07, -.07, -.05, -.03, -.01, .36
0.40.0-0.4
B+ACD
C+ABD
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.582 -1 -1 -1 1 1 1 -1
.509 1 -1 -1 -1 -1 1 1
.954 -1 1 -1 -1 1 -1 1
.910 1 1 -1 1 -1 -1 -1
.163 -1 -1 1 1 -1 -1 1
.210 1 -1 1 -1 1 -1 -1
.614 -1 1 1 -1 -1 1 -1
.423 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.55 -.07 .36 -.39 -.05 -.01 -.03 -.07
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.582 -1 -1 -1 1 1 1 -1
.509 1 -1 -1 -1 -1 1 1
.954 -1 1 -1 -1 1 -1 1
.910 1 1 -1 1 -1 -1 -1
.163 -1 -1 1 1 -1 -1 1
.210 1 -1 1 -1 1 -1 -1
.614 -1 1 1 -1 -1 1 -1
.423 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.55 -.07 .36 -.39 -.05 -.01 -.03 -.07
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesShrinkage - Discussion and InterpretationShrinkage - Discussion and Interpretation
The 2The 24-14-1 Fractional Factorial Used Here (D=ABC or ABCD=I) Fractional Factorial Used Here (D=ABC or ABCD=I) is called a Resolution IV Design. This Means That, is called a Resolution IV Design. This Means That, Assuming that 3-way and 4-way Interactions are Assuming that 3-way and 4-way Interactions are NEGLIGIBLE , Factors B and C Affect Shrinkage Without a NEGLIGIBLE , Factors B and C Affect Shrinkage Without a Significant Interaction. (More About Resolution Later)Significant Interaction. (More About Resolution Later)
We Estimate ThatWe Estimate That– Setting B Hi Increases Shrinkage .36 over Setting it LoSetting B Hi Increases Shrinkage .36 over Setting it Lo– Setting C Lo Increases the Shrinkage .39 over Setting it HiSetting C Lo Increases the Shrinkage .39 over Setting it Hi
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesShrinkage - Estimated Mean Response (EMR)Shrinkage - Estimated Mean Response (EMR)
For B = + and C = -, EMR = .55For B = + and C = -, EMR = .55+ (+1)(.36)/2 + + (+1)(.36)/2 + (-1)(-.39)/2 =.55 +.375 = .925(-1)(-.39)/2 =.55 +.375 = .925
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.582 -1 -1 -1 1 1 1 -1
.509 1 -1 -1 -1 -1 1 1
.954 -1 1 -1 -1 1 -1 1
.910 1 1 -1 1 -1 -1 -1
.163 -1 -1 1 1 -1 -1 1
.210 1 -1 1 -1 1 -1 -1
.614 -1 1 1 -1 -1 1 -1
.423 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.55 -.07 .36 -.39 -.05 -.01 -.03 -.07
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.582 -1 -1 -1 1 1 1 -1
.509 1 -1 -1 -1 -1 1 1
.954 -1 1 -1 -1 1 -1 1
.910 1 1 -1 1 -1 -1 -1
.163 -1 -1 1 1 -1 -1 1
.210 1 -1 1 -1 1 -1 -1
.614 -1 1 1 -1 -1 1 -1
.423 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.55 -.07 .36 -.39 -.05 -.01 -.03 -.07
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesExcess LengthExcess Length
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.090 -1 -1 -1 1 1 1 -1
.190 1 -1 -1 -1 -1 1 1-.113 -1 1 -1 -1 1 -1 1-.207 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1-.100 1 -1 1 -1 1 -1 -1-.147 -1 1 1 -1 -1 1 -1-.125 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4-.057 -.01 -.18 -.09 .-.03 -.01 .12 .07
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.090 -1 -1 -1 1 1 1 -1
.190 1 -1 -1 -1 -1 1 1-.113 -1 1 -1 -1 1 -1 1-.207 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1-.100 1 -1 1 -1 1 -1 -1-.147 -1 1 1 -1 -1 1 -1-.125 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4-.057 -.01 -.18 -.09 .-.03 -.01 .12 .07
Effects
7 Effects Plot
1
2
3
4
5
6
7
0.10.0-0.1-0.2
Ordered Effects: -0.18, -0.09, -0.03, -0.01, -0.01, 0.07, 0.12
B+ACD
AD+BC
C+ABD
D+ABC
Effects
7 Effects Plot
1
2
3
4
5
6
7
0.10.0-0.1-0.2
Ordered Effects: -0.18, -0.09, -0.03, -0.01, -0.01, 0.07, 0.12
B+ACD
AD+BC
C+ABD
D+ABC
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesExcess LengthExcess Length - - Discussion and InterpretationDiscussion and Interpretation
The 2The 24-14-1 Fractional Factorial Used Here is Again the Resolution Fractional Factorial Used Here is Again the Resolution IV Design, D=ABC (or ABCD=I). Thus, Assuming that 3-way IV Design, D=ABC (or ABCD=I). Thus, Assuming that 3-way and 4-way Interactions are NEGLIGIBLE, Factors B, C and D and 4-way Interactions are NEGLIGIBLE, Factors B, C and D Affect Excess Length But There Appears To Be A Significant Affect Excess Length But There Appears To Be A Significant Interaction, (AD or BC, or Both). Interaction, (AD or BC, or Both).
Since B and C signaled but A did not, We Shall ASSUME that Since B and C signaled but A did not, We Shall ASSUME that the BC interaction is Significant for Interpretation Purposes.the BC interaction is Significant for Interpretation Purposes.
II.2 Four Factors in Eight Runs: ExampleII.2 Four Factors in Eight Runs: ExampleExcess LengthExcess Length - - Discussion and InterpretationDiscussion and Interpretation
Noise Factors/Control FactorsNoise Factors/Control Factors– If Factor C was Hard to If Factor C was Hard to
Control (a Noise Factor), But B Control (a Noise Factor), But B was not (a Control Factor), was not (a Control Factor), Setting B Hi mitigates the Setting B Hi mitigates the effect of Ceffect of C
– Similarly, Setting C Hi Similarly, Setting C Hi mitigates the effect of Bmitigates the effect of B
Ideally, The Excess Length Ideally, The Excess Length should be 0.should be 0.– Setting B Lo and C Hi Give the Setting B Lo and C Hi Give the
Smallest Magnitude of Excess Smallest Magnitude of Excess Length with Excess Length at Length with Excess Length at about -.07.about -.07.
– In addition, changing D from In addition, changing D from Lo to Hi increases Excess Lo to Hi increases Excess LengthLength
– Thus, Set B Lo, and C and D Thus, Set B Lo, and C and D both Hi. both Hi.
Noise Factors/Control FactorsNoise Factors/Control Factors– If Factor C was Hard to If Factor C was Hard to
Control (a Noise Factor), But B Control (a Noise Factor), But B was not (a Control Factor), was not (a Control Factor), Setting B Hi mitigates the Setting B Hi mitigates the effect of Ceffect of C
– Similarly, Setting C Hi Similarly, Setting C Hi mitigates the effect of Bmitigates the effect of B
Ideally, The Excess Length Ideally, The Excess Length should be 0.should be 0.– Setting B Lo and C Hi Give the Setting B Lo and C Hi Give the
Smallest Magnitude of Excess Smallest Magnitude of Excess Length with Excess Length at Length with Excess Length at about -.07.about -.07.
– In addition, changing D from In addition, changing D from Lo to Hi increases Excess Lo to Hi increases Excess LengthLength
– Thus, Set B Lo, and C and D Thus, Set B Lo, and C and D both Hi. both Hi.
1-1
0.15
0.05
-0.05
-0.15
B
C-
C+
1-1
0.15
0.05
-0.05
-0.15
B
C-
C+
II.2 Four Factors in Eight Runs: ExamplesII.2 Four Factors in Eight Runs: ExamplesExcess LengthExcess Length - - Estimated Mean Response (EMR)Estimated Mean Response (EMR)
For B = - , C = +, D=+:For B = - , C = +, D=+: EMR = -.057EMR = -.057+ (-1)(-.18)/2 + (+1)(-.09)/2 + (+1)(.07)/2+ (-1)(-.18)/2 + (+1)(-.09)/2 + (+1)(.07)/2
+(-1)(.12)/2 = -.057 +(-1)(.12)/2 = -.057+ .02 = -.037+ .02 = -.037
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.090 -1 -1 -1 1 1 1 -1
.190 1 -1 -1 -1 -1 1 1-.113 -1 1 -1 -1 1 -1 1-.207 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1-.100 1 -1 1 -1 1 -1 -1-.147 -1 1 1 -1 -1 1 -1-.125 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4-.057 -.01 -.18 -.09 .-.03 -.01 .12 .07
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.090 -1 -1 -1 1 1 1 -1
.190 1 -1 -1 -1 -1 1 1-.113 -1 1 -1 -1 1 -1 1-.207 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1-.100 1 -1 1 -1 1 -1 -1-.147 -1 1 1 -1 -1 1 -1-.125 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4-.057 -.01 -.18 -.09 .-.03 -.01 .12 .07
II.2 Four Factors in Eight Runs: U-Do-It II.2 Four Factors in Eight Runs: U-Do-It ExerciseExercise
Completed Signs Tables and Normal Completed Signs Tables and Normal Probability Plots are Provided for Two Probability Plots are Provided for Two Other 2Other 24-14-1 Experiments (D=ABC or Experiments (D=ABC or ABCD=I) Involving Shrinkage and ABCD=I) Involving Shrinkage and Excess LengthExcess Length
Analyze These ExperimentsAnalyze These Experiments Calculate the EMR for Shrinkage and Calculate the EMR for Shrinkage and
Excess Length when A and B are both Excess Length when A and B are both Lo and C and D are both HiLo and C and D are both Hi
II.2 Four Factors in Eight Runs: U-Do-It II.2 Four Factors in Eight Runs: U-Do-It ExerciseExerciseShrinkageShrinkage
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
0.559 -1 -1 -1 1 1 1 -10.635 1 -1 -1 -1 -1 1 10.575 -1 1 -1 -1 1 -1 10.587 1 1 -1 1 -1 -1 -10.826 -1 -1 1 1 -1 -1 10.788 1 -1 1 -1 1 -1 -10.788 -1 1 1 -1 -1 1 -10.779 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.692 .01 -.02 .206 -.009 -.034 .023 -.004
Sy
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
0.559 -1 -1 -1 1 1 1 -10.635 1 -1 -1 -1 -1 1 10.575 -1 1 -1 -1 1 -1 10.587 1 1 -1 1 -1 -1 -10.826 -1 -1 1 1 -1 -1 10.788 1 -1 1 -1 1 -1 -10.788 -1 1 1 -1 -1 1 -10.779 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.692 .01 -.02 .206 -.009 -.034 .023 -.004
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects:
0.20.10.0
-.034, -.020, -.009, -.004, .010, .023, .206
C+ABD
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects:
0.20.10.0
-.034, -.020, -.009, -.004, .010, .023, .206
C+ABD
II.2 Four Factors in Eight Runs: U-Do-It II.2 Four Factors in Eight Runs: U-Do-It ExerciseExercise
Excess LengthExcess Length
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.072 -1 -1 -1 1 1 1 -1
.146 1 -1 -1 -1 -1 1 1
.083 -1 1 -1 -1 1 -1 1
.134 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1.130 1 -1 1 -1 1 -1 -1
-.030 -1 1 1 -1 -1 1 -1.098 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.074 .107 -.005 -.070 -.018 .045 -.006 -.004
Ly
A+BCD
B+ACD
C+ABD
AB+CD
AC+BD
AD+BC
D+ABC
.072 -1 -1 -1 1 1 1 -1
.146 1 -1 -1 -1 -1 1 1
.083 -1 1 -1 -1 1 -1 1
.134 1 1 -1 1 -1 -1 -1-.045 -1 -1 1 1 -1 -1 1.130 1 -1 1 -1 1 -1 -1
-.030 -1 1 1 -1 -1 1 -1.098 1 1 1 1 1 1 1
8 4 4 4 4 4 4 4.074 .107 -.005 -.070 -.018 .045 -.006 -.004
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects: -.070, -.018, -.006, -.005, -.004, .045, .107
0.120.02-0.08
A+BCD
AC+BD
C+ABD
Effects
7 Effects Plot
1
2
3
4
5
6
7
Ordered Effects: -.070, -.018, -.006, -.005, -.004, .045, .107
0.120.02-0.08
A+BCD
AC+BD
C+ABD
II.2 Four Factors in Eight Runs: U-Do-It II.2 Four Factors in Eight Runs: U-Do-It Exercise Exercise
Shrinkage Solution - Discussion and InterpretationShrinkage Solution - Discussion and Interpretation C+ABD is Statistically Significant. Thus, Assuming that 3-C+ABD is Statistically Significant. Thus, Assuming that 3-
way and 4-way Interactions are NEGLIGIBLE , Factor C way and 4-way Interactions are NEGLIGIBLE , Factor C Affects Shrinkage. There are No Significant Two-way Affects Shrinkage. There are No Significant Two-way Interactions. Interactions.
We Estimate ThatWe Estimate That– Setting C Hi Increases the Shrinkage .206 over Setting it LoSetting C Hi Increases the Shrinkage .206 over Setting it Lo
When C = +,When C = +,
EMR = .692EMR = .692+ (+1)(.206)/2 = .692+ .103 = .795+ (+1)(.206)/2 = .692+ .103 = .795
II.2 Four Factors in Eight Runs: U-Do-It II.2 Four Factors in Eight Runs: U-Do-It Exercise Exercise
Excess Length Solution - Discussion and Excess Length Solution - Discussion and InterpretationInterpretation A+BCD, C+ABD and AC+BD are Statistically Significant. A+BCD, C+ABD and AC+BD are Statistically Significant.
Thus, Assuming that 3-way and 4-way Interactions are Thus, Assuming that 3-way and 4-way Interactions are NEGLIGIBLE , Factors A and C Affect Excess Length. To NEGLIGIBLE , Factors A and C Affect Excess Length. To interpret AC+BD, we will ASSUME that BD is Negligible interpret AC+BD, we will ASSUME that BD is Negligible Since A and C are Significant. Since A and C are Significant.
II.2 Four Factors in Eight Runs: ExercisesII.2 Four Factors in Eight Runs: Exercises
Excess Length SolutionExcess Length Solution - - Discussion and Discussion and InterpretationInterpretation
Ideally, Excess Length should be 0.Ideally, Excess Length should be 0.– Setting A Lo and C Hi Reduces the Magnitude of Excess Length with Setting A Lo and C Hi Reduces the Magnitude of Excess Length with
Excess Length at about -.04.Excess Length at about -.04.– For A = - and C = +,For A = - and C = +,
EMR = .074EMR = .074+ (-1)(.107)/2 + (+1)(-.07)/2 +(-1)(.045)/2 + (-1)(.107)/2 + (+1)(-.07)/2 +(-1)(.045)/2 = .074= .074 - .111 = -.037- .111 = -.037
Ideally, Excess Length should be 0.Ideally, Excess Length should be 0.– Setting A Lo and C Hi Reduces the Magnitude of Excess Length with Setting A Lo and C Hi Reduces the Magnitude of Excess Length with
Excess Length at about -.04.Excess Length at about -.04.– For A = - and C = +,For A = - and C = +,
EMR = .074EMR = .074+ (-1)(.107)/2 + (+1)(-.07)/2 +(-1)(.045)/2 + (-1)(.107)/2 + (+1)(-.07)/2 +(-1)(.045)/2 = .074= .074 - .111 = -.037- .111 = -.037
1-1
0.15
0.10
0.05
0.00
A
C-
C+
-0.05
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