desirability function

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A Sopadang Desirability Function 1

Desirability Function

Dr. Apichat Sopadang

Faculty of Engineering

Chiang Mai University

Principle of Management Science


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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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


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


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


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


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


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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desirability function

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