7.22 Test Problems 467
7.22 Test Problems
As discussed in previous sections, a number of algorithms are available for solving
a constrained nonlinear programming problem. In recent years, a variety of computer
programs have been developed to solve engineering optimization problems. Many of
these are complex and versatile and the user needs a good understanding of the algo-
rithms/computer programs to be able to use them effectively. Before solving a new
engineering design optimization problem, we usually test the behavior and conver-
gence of the algorithm/computer program on simple test problems. Five test problems
are given in this section. All these problems have appeared in the optimization literature
and most of them have been solved using different techniques.
7.22.1 Design of a Three-Bar Truss
The optimal design of the three-bar truss shown in Fig. 7.21 is considered using two
different objectives with the cross-sectional areas of members 1 (and 3) and 2 as design
variables [7.38].
Design vector:
X=
{
x 1
x 2
}
=
{
A 1
A 2
}
Objective functions:
f 1 ( X)=weight= 2
√
2 x 1 +x 2
f 2 ( X)=vertical deflection of loaded joint=
P H
E
1
x 1 +
√
2 x 2
Constraints:
σ 1 (X)−σ(u)≤ 0
σ 2 (X)−σ(u)≤ 0
σ 3 (X)−σ(l)≤ 0
xi(l)≤xi≤xi(u), i= 1 , 2
Figure 7.21 Three-bar truss [7.38].
