Engineering Optimization: Theory and Practice, Fourth Edition

28 Introduction to Optimization

TD≤W 2 (E 5 )

TE≤W 3 (E 6 )

TF≤W 3 (E 7 )

Finally, the nonnegativity requirement of the design variables can be expressed as

x 1 ≥ 0

x 2 ≥ 0

x 3 ≥ 0 (E 8 )

Since all the equations of the problem (E 1 ) to (E 8 ), are linear functions ofx 1 , x 2 , and

x 3 , the problem is a linear programming problem.

1.5.5 Classification Based on the Permissible Values of the Design Variables

Depending on the values permitted for the design variables, optimization problems can

be classified as integer and real-valued programming problems.

Integer Programming Problem. If some or all of the design variablesx 1 , x 2 ,... , xn

of an optimization problem are restricted to take on only integer (or discrete) values,

the problem is called aninteger programming problem. On the other hand, if all the

design variables are permitted to take any real value, the optimization problem is

called areal-valued programming problem. According to this definition, the problems

considered in Examples 1.1 to 1.6 are real-valued programming problems.

Example 1.7 A cargo load is to be prepared from five types of articles. The weight

wi, volumevi, and monetary valueciof different articles are given below.

Article type wi vi ci

1 4 9 5

2 8 7 6

3 2 4 3

4 5 3 2

5 3 8 8

Find the number of articlesxiselected from theith type (i= 1 , 2 , 3 , 4 ,5), so that the

total monetary value of the cargo load is a maximum. The total weight and volume of

the cargo cannot exceed the limits of 2000 and 2500 units, respectively.

SOLUTION Letxi be the number of articles of typei(i=1 to 5)selected. Since

it is not possible to load a fraction of an article, the variablesxican take only integer

values.

The objective function to be maximized is given by

f (X)= 5 x 1 + 6 x 2 + 3 x 3 + 2 x 4 + 8 x 5 (E 1 )