Engineering Optimization: Theory and Practice, Fourth Edition

10.8 Generalized Penalty Function Method 623

Figure 10.11 Three-bar truss.

A general convergence proof of the penalty function method, including the integer
programming problems, was given by Fiacco [10.6]. Hence the present method is
guaranteed to converge at least to a local minimum if the recovery procedure is applied
the required number of times.

Example 10.6 [10.24]Find the minimum weight design of the three-bar truss shown
in Fig. 10.11 with constraints on the stresses induced in the members. Treat the areas
of cross section of the members as discrete variables with permissible values of the
parameterAiσmax/P given by 0.1, 0.2, 0.3, 0.5, 0.8, 1.0, and 1.2.

SOLUTION By defining the nondimensional quantitiesf andxias

f =

Wσmax

Pρl

, xi=

Aiσmax

P

, i= 1 , 2 , 3

whereW is the weight of the truss,σmaxthe permissible (absolute) value of stress,
P the load,ρthe density,lthe depth, andAi the area of cross section of member
i(i= 1 , 2 , 3 ), the discrete optimization problem can be stated as follows:

Minimizef= 2 x 1 +x 2 +

√

2 x 3

subject to

g 1 (X)= 1 −

√

3 x 2 + 1. 932 x 3

1. 5 x 1 x 2 +

√

2 x 2 x 3 + 1. 319 x 1 x 3

≥ 0

g 2 (X)= 1 −

0. 634 x 1 + 2. 828 x 3

1. 5 x 1 x 2 +

√

2 x 2 x 3 + 1. 319 x 1 x 3

≥ 0