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A Sopadang Desirability Function 1
Desirability Function
Dr. Apichat Sopadang
Faculty of Engineering
Chiang Mai University
Principle of Management Science
A Sopadang Desirability Function 2
Introduction
The trade-off problem is particularly chronic in formulationrelated products. For example, the abrasion resistance of arubber shoe sole formulation might be improved by adding
certain tillers. But this change might also result in morecomplex production processes and higher production costs.
The problem becomes considerably more complex whenseveral components of the formulation are being varied.
Balance problems can be solved by using a modified formulaof E.C. Harrington's desirability function and combining it
with response surface methodology (RSM) to form amethodology called desirability optimization methodology
(DOM). Computer implementation of DOM further enhancesits power.
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A Sopadang Desirability Function 3
The desirability function can be used to combine
multiple responses into one response called thedesirability function by choice of value from 0 (one or
more product characteristics are unacceptable) to 1 (allproduct characteristics are on target).
The method is attractive because it is intuitive andsimple.
The inputs are mean response estimates, target value,and upper and lower acceptability bounds.
Nature of Desirability Function
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The individual desirability is combined using the geometric
mean. The desirability of a product characteristic value
depends on the lower and upper ranges of productspecification. Improper selection of ranges can result in avery different optimum.
The basic idea of the desirability function approach is to
transform a multiple response problem into a single
response problem by means of mathematicaltransformations.
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The desirability function involves transformation of
each estimated response variable Yjto a desirabilityvalue dj,where 0 di 1.The value ofdi increases as
the desirability of the corresponding response
increases.
D = (d1
d2
dk
)1/k
= jjww
j
wwdddD/1
21 ][21K
Transformation Methodology
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A function dj(Yj(x))with a range of value between 0.0 and 1.0
can be produced, where djis a desirability function ofYj
using the transformation given by One-Side Transformations
=
jj
jjj
r
jj
jj
jj
j
YxY
YxYYYY
YY
YxY
d
max
maxmin
minmax
min
min
)(if1
)(if
)(if0
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Two-Side Transformations
=
otherwise0
)(if
)(if
max
max
max
min
min
min
jjj
t
jj
jj
jjj
s
jj
jj
j YxYTYT
YY
TxYYYT
YY
d
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Graph of transformation for
various values of r, s, and t
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Example
In the development of a tire tread compound, the optimal
combination of three ingredient (independent)variables-hydrated silica levelX1, silane coupling agent levelX2, and
sulfur levelX3-was sought.The properties to be optimized
and constraint levels were as follows.
PICO Abrasion Index,Y1 120
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Regresstion Coefficients and Standard Error for Responses
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Solution
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Regression Analysis Using Minitab
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32321 00.1962.338.2367.180 xxxxY ++=
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3212 12.1402.2100.600.141 xxxY +++=
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Model generated (in term of coded factors)
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Hi
Lo0.60653D
Optimal
Cur
d = 0.91501
Targ: 190.0
Y4
d = 0.66846
Targ: 185.0
Y3
d = 0.50903
Targ: 185.0Y2
d = 1.0000
Targ: 190.0
Y1
y = 190.4250
y = 180.0269
y = 177.6354
y = 190.0
150.0
350.0
200.0
450.0
40.0
120.0Flow Tem Block TeFlow Rat
[102.3538] [420.9434] [350.0]
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160
170
180190
200
120110100908070605040
450
400
350
300
250
200
Flow Rate
FlowTemp
Contour Plot of Y1
Hold values: Block Te: 250.0
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140
150
160
170
Y1
Fl70 80 90 100 1w Rate
0
180
190
200
10200
120
40350
300250
4500
low Temp
Surface Plot of Y1
Hold values: Block Te: 250.0
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