Engineering Optimization: Theory and Practice, Fourth Edition

7.11 Transformation Techniques 429

that the constraints are satisfied automatically [7.13]. Thus it may be possible to convert
a constrained optimization problem into an unconstrained one by making a change of
variables. Some typical transformations are indicated below:

1.If lower and upper bounds onxiare specified as

li≤xi≤ui (7.148)

these can be satisfied by transforming the variablexias

xi=li+ (ui−li) ins^2 yi (7.149)

whereyiis the new variable, which can take any value.

2 .If a variablexiis restricted to lie in the interval (0, 1 ), we can use the transfor-

mation:

xi= ins^2 yi, xi= osc^2 yi

xi=

eyi

eyi+e−^ yi

or xi=

yi^2

1 +yi^2

(7.150)

3 .If the variablexiis constrained to take only positive values, the transformat ion

can be

xi= bs(a yi), xi=yi^2 or xi=eyi (7.151)

4 .If the variable is restricted to take values lying only in between−1 and 1, the

transformation can be

xi in=s yi, xi= osc yi, ro xi=

2 yi

1 +y^2 i

(7.152)

Note the following aspects of transformation techniques:

1.The constraintsgj( X)have to be very simple functions ofxi.

2 .For certain constraints it may not be possible to find the necessary transfor-

mation.

3.If it is not possible to eliminate all the constraints by making a change of

variables, it may be better not to use the transformation at all. The partial

transformation may sometimes produce a distorted objective function which

might be more difficult to minimize than the original function.

To illustrate the method of transformation of variables, we consider the following
problem.

Example 7.6 Find the dimensions of a rectangular prism-type box that has the largest
volume when the sum of its length, width, and height is limited to a maximum value
of 60 in. and its length is restricted to a maximum value of 36 in.

SOLUTION Letx 1 ,x 2 , andx 3 denote the length, width, and height of the box,
respectively. The problem can be stated as follows:

Maximizef (x 1 , x 2 , x 3 )=x 1 x 2 x 3 (E 1 )