Materi :

DOE Minggu XIII

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1. Introduction 2. Simple Comparative Experiments 3. Experiments with a Single Factor 4. The Randomized Complete Block Design 5. The Latin Square Design 6. Factorial Design 7. The 2 k Factorial Design 8. Two-Level Fractional Factorial Design 9. Nested or Hierarchial Design 10. Response Surface Methods



Introduction to Response Surface Methodology

Response Surface Methodology  or RSCM : A collection of mathematical are useful for modeling and statistical and analysis response of interest is influenced by several variables and objective is to optimize this response .

techniques that of problem in which a the  For example, suppose that a chemical engineer wishes to find the levels of temperature (x 1 ) and pressure (x 2 ) that maximize the yield (y) of a process. The process function of yield : y = f(x 1 ,x 2 ) +  where  represent the error.

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Introduction

to Response Surface Methodology … [ continued ]  If we denote the expected response by E (y) = f(x 1 ,x 2 ) =  then the surface represented by ,  = f(x 1 ,x 2 ) is called a response surface  If the response is well modeled by a linear function of the independent variables, the the approximating function is the first-order model y =  0 +  1 x 1 +  2 x 2 + … +  k x k +  4/27/2020

Introduction

to Response Surface Methodology … [ continued ]  If there is curvature in the system , then a polynomial of higher degree must be used, such as the second-order model   RSM is a sequential procedure The eventual objective of RSM is to determine the optimum operating conditions for the system of the factor space in which operating requirements are satisfied .

or to determine a region 4/27/2020

THE METHOD OF STEEPEST ASCENT [ DESCENT ]

The method of steepest ascent is  A procedure for moving sequentially along the path of steepest ascent, that is, in the direction of the maximum increase in the response .

If minimization is desired, then we call this technique the method of steepest descent.

Experiments are conducted along the path of steepest ascent until no further increase is response is observed continued. Eventually , . Then a new first-order model may be fit, a new path of steepest ascent determined, and the procedure the experimenter will arrive in the vicinity of the optimum . This is usually indicated by lack of fit of a first-order model estimate of the optimum.

. At that time additional experiments are conducted to obtain a more precise 4/27/2020

THE METHOD OF STEEPEST ASCENT

[ continued ] Path of steepest ascent is proportional to the signs and magni tudes of the regression coefficients in the fitted first-order model 1. Choose a step size in one of the process variables, say  x j . Usually, we would select the variable we know the most about, or we would select the variable that has the largest absolute regression coefficient |  j |.

2. The step size in the other variables is 3. Convert the  x i from coded variables to the natural variables.

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An Example : Process Data for Fitting the First  Order Model [ Ex. 11.1; page 431 ] Natural Variables  1 30 30 40 40 35 35 35 35 35  2 150 160 150 160 155 155 155 155 155 Coded variables x 1  1  1 1 1 0 0 0 0 0 x 2  1 1  1 1 0 0 0 0 0 Response y 39.3

40.0

40.9

41.5

40.3

40.5

40.7

40.2

40.6

Statement problems :  Which reaction time (  1 ) and reaction temperature (  2 ) do satisfy the operating conditions that maximize the yield of a process ? 4/27/2020

ANOVA for the First  Order Model : MINITAB output …  4/27/2020

ANOVA for the First  Order Model : MINITAB output …  4/27/2020

Steepest Ascent Experiment for Example 11.1

4/27/2020 Steps Origin  Origin + 1  Origin + 2  Origin + 3  Origin + 4  Origin + 5  Origin + 6  Origin + 7  Origin + 8  Origin + 9  Origin + 10  Origin + 11  Origin + 12  Coded variables x 1 1 2 0 1 3 4 5 6 7 8 9 10 11 12 x 2 0 0.42

0.42

0.84

1.26

1.68

2.10

2.52

2.94

3.36

3.78

4.20

4.62

5.04

Natural Variables  1 35 5 40 45 50 55 60 65 70 75 80 85 90 95  2 155 2 157 159 161 163 165 167 169 171 173 175 177 179 Response y 41.0

42.9

47.1

49.7

53.8

59.9

65.0

70.4

77.6

80.3

76.2

75.1

Figure

: Yield versus steps along the path of steepest Ascent for Example 11.1

(  1 = 85,  2 = 175) 4/27/2020

Analysis of A Second-Order Response Surface : Process Data for Fitting the Second  Order Model [ Ex. 11.2; page 440 ] Natural Variables  1  2 80 80 90 90 85 85 85 85 85 92.07

77.93

85 85 170 180 170 180 175 175 175 175 175 175 175 182.07

167.93

4/27/2020 Coded variables x 1 x 2  1  1 1 1 0 0 0 0 0 1.414

 1.414

0 0  1 1  1 1 0 0 0 0 0 0 0 1.414

 1.414

Response y 76.5

77.0

78.0

79.5

79.5

80.3

80.0

79.7

79.8

78.4

75.6

78.5

77.0

Graphical Analysis : Main Effect Plot …

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ANOVA for the Second  Order Model : MINITAB output … interaction  4/27/2020

ANOVA for the Second  Order Model : output [continued] …

Original scale (data)

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The Contour Plot of Response Surface … [MINITAB output] 4/27/2020

The three-dimensional Response Surface … [MINITAB output] 4/27/2020