Engineering Optimization: Theory and Practice, Fourth Edition

14 Introduction to Optimization

Next, the contours of the objective function are to be plotted before finding the

optimum point. For this, we plot the curves given by

f (X)= 9. 82 x 1 x 2 + 2 x 1 =c=constant

for a series of values ofc. By giving different values toc, the contours off can be

plotted with the help of the following points.

For 9. 82 x 1 x 2 + 2 x 1 = 05 .0:

x 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x 1 16.77 12.62 10.10 8.44 7.24 6.33 5.64 5.07

For 9. 82 x 1 x 2 + 2 x 1 = 04 .0:

x 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x 1 13.40 10.10 8.08 6.75 5.79 5.06 4.51 4.05

For 9. 82 x 1 x 2 + 2 x 1 = 13 .58 (passing through the corner pointC):

x 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x 1 10.57 7.96 6.38 5.33 4.57 4.00 3.56 3.20

For 9. 82 x 1 x 2 + 2 x 1 = 62 .53 (passing through the corner pointB):

x 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x 1 8.88 6.69 5.36 4.48 3.84 3.36 2.99 2.69

For 9. 82 x 1 x 2 + 2 x 1 = 02 .0:

x 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

x 1 6.70 5.05 4.04 3.38 2.90 2.53 2.26 2.02

These contours are shown in Fig. 1.7 and it can be seen that the objective function

cannot be reduced below a value of 26.53 (corresponding to pointB) without violating

some of the constraints. Thus the optimum solution is given by pointBwithd∗=

x∗ 1 = 5. 4 4 cm andt∗=x∗ 2 = 0. 2 93 cm withfmin= 62 .53.

1.5 Classification of Optimization Problems

Optimization problems can be classified in several ways, as described below.

1.5.1 Classification Based on the Existence of Constraints

As indicated earlier, any optimization problem can be classified as constrained or uncon-

strained, depending on whether constraints exist in the problem.