Engineering Optimization: Theory and Practice, Fourth Edition

1.5 Classification of Optimization Problems 33

Multiobjective Programming Problem. A multiobjective programming problem can
be stated as follows:

FindXwhich minimizesf 1 ( X), f 2 ( X),... , fk(X)

subject to

gj( X)≤ 0 , j= 1 , 2 ,... , m

(1.13)

wheref 1 , f 2 ,... , fkdenote the objective functions to be minimized simultaneously.

Example 1.10 A uniform column of rectangular cross section is to be constructed
for supporting a water tank of massM(Fig. 1.14). It is required (1) to minimize the
mass of the column for economy, and (2) to maximize the natural frequency of trans-
verse vibration of the system for avoiding possible resonance due to wind. Formulate
the problem of designing the column to avoid failure due to direct compression and
buckling. Assume the permissible compressive stress to beσmax.

SOLUTION Letx 1 = bandx 2 = d denote the cross-sectional dimensions of the
column. The mass of the column(m)is given by

m=ρbdl=ρlx 1 x 2 (E 1 )

whereρis the density andlis the height of the column. The natural frequency of
transverse vibration of the water tank (ω), by treating it as a cantilever beam with a
tip massM, can be obtained as [1.118]

ω=

[

3 EI

(M+ 14033 m)l^3

] 1 / 2

(E 2 )

Figure 1.14 Water tank on a column.