Engineering Optimization: Theory and Practice, Fourth Edition

12 Introduction to Optimization

Thus the behavior constraints can be restated as

g 1 (X)=

2500

πx 1 x 2

− 005 ≤ 0 (E 6 )

g 2 (X)=

2500

πx 1 x 2

−

π^2 ( 50. 8 × 106 )(x^21 +x 22 )

8 ( 250 )^2

≤ 0 (E 7 )

The side constraints are given by

2 ≤d≤ 14

0. 2 ≤t≤ 0. 8

which can be expressed in standard form as

g 3 ( X)=−x 1 + 2. 0 ≤ 0 (E 8 )

g 4 (X)=x 1 − 41. 0 ≤ 0 (E 9 )

g 5 ( X)=−x 2 + 0. 2 ≤ 0 (E 10 )

g 6 (X)=x 2 − 0. 8 ≤ 0 (E 11 )

Since there are only two design variables, the problem can be solved graphically as

shown below.

First, the constraint surfaces are to be plotted in a two-dimensional design space

where the two axes represent the two design variablesx 1 andx 2. To plot the first

constraint surface, we have

g 1 (X)=

2500

πx 1 x 2

− 005 ≤ 0

that is,

x 1 x 2 ≥ 1. 593

Thus the curvex 1 x 2 = 1. 5 93 represents the constraint surfaceg 1 ( X)= 0. This curve

can be plotted by finding several points on the curve. The points on the curve can be

found by giving a series of values tox 1 and finding the corresponding values ofx 2

that satisfy the relationx 1 x 2 = 1. 5 93:

x 1 2.0 4.0 6.0 8.0 10.0 12.0 14.0

x 2 0.7965 0.3983 0.2655 0.1990 0.1593 0.1328 0.1140

These points are plotted and a curveP 1 Q 1 passing through all these points is drawn as

shown in Fig. 1.7, and the infeasible region, represented byg 1 ( X)> 0 orx 1 x 2 < 1. 5 93,

is shown by hatched lines.†Similarly, the second constraintg 2 ( X)≤ 0 can be expressed

asx 1 x 2 (x 12 +x^22 ) ≥ 47 .3 and the points lying on the constraint surfaceg 2 ( X)= 0 can

be obtained as follows forx 1 x 2 (x 12 +x 22 ) = 47 .3:

†The infeasible region can be identified by testing whether theorigin lies in the feasible or infeasible

region.