Engineering Optimization: Theory and Practice, Fourth Edition

654 Stochastic Programming

By introducing the new variable

θ=

gj−gj

σgj

(11.94)

and noting that

∫∞

−∞

1

√

2 π

e−t

(^2) / 2
dt= 1 (11.95)
Eq. (11.90) can be expressed as
∫∞
−(gj/σgj)

1

√

2 π

e−θ

(^2) / 2
dθ≥

∫∞

−φj(pj)

1

√

2 π

e−t

(^2) / 2
dt (11.96)
whereφj(pj) s the value of the standard normal variate corresponding to the proba-i
bilitypj. Thus
−
gj
σgj
≤ −φj(pj)
or
−gj+σgjφj(pj)≤ 0 (11.97)
Equation (11.97) can be rewritten as
gj−φj(pj)

[N

∑

i= 1

(

∂gj

∂yi

∣

∣

∣

∣Y

) 2

σy^2 i

] 1 / 2

≥ 0 , j= 1 , 2 ,... , m (11.98)

Thus the optimization problem of Eqs. (11.83) and (11.84) can be stated in its equivalent

deterministic form as: minimizeF (Y)given by Eq. (11.89) subject to themconstraints

given by Eq. (11.98).

Example 11.6 Design a uniform column of tubular section shown in Fig. 11.5 to

carry a compressive loadP for minimum cost. The column is made up of a material

that has a modulus of elasticityEand densityρ. The length of the column isl. The

stress induced in the column should be less than the buckling stress as well as the yield

stress. The mean diameter is restricted to lie between 2.0 and 14.0 cm, and columns

with thickness outside the range 0.2 to 0.8 cm are not available in the market. The

cost of the column includes material costs and construction costs and can be taken as

5 W+ 2 d, whereWis the weight anddis the mean diameter of the column. The

constraints have to be satisfied with a probability of at least 0.95.

The following quantities are probabilistic and follow normal distribution with mean

and standard deviations as indicated:

Compressive load=(P , σP) =( 2500 , 500 )kg

Young’s modulus=(E, σE) =( 0. 85 × 106 , 850. 0 × 106 ) gkf/ mc^2

Density=(ρ, σρ) =( 0. 0025 , 0. 00025 )kgf/ mc^3

Yield stress=(fy, σfy)=( 500 , 50 )kgf/ mc^2