Engineering Optimization: Theory and Practice, Fourth Edition

11.4 Stochastic Nonlinear Programming 657

The mean values of the constraint functions are given by Eq. (11.92) as

g 1 =

P

πdt

−fy=

2500

πdt

− 500

g 2 =

P

πdt

−

π^2 E(d

2

+t^2 )

8 l

2 =

2500

πdt

−

π^2 ( 50. 8 × 106 )(d

2

+t^2 )

8 ( 250 )^2

g 3 = −d+ 2. 0

g 4 =d− 14. 0

g 5 = −t+ 0. 2

g 6 =t− 0. 8

The partial derivatives of the constraint functions can be computed as follows:

∂g 1

∂y 2

∣

∣

∣

∣Y=

∂g 1

∂y 3

∣

∣

∣

∣Y=

∂g 1

∂y 5

∣

∣

∣

∣Y=^0

∂g 1

∂y 1

∣

∣

∣

∣Y=

1

πdt

∂g 1

∂y 4

∣

∣

∣

∣Y= −^1

∂g 1

∂y 6

∣

∣

∣

∣Y= −

P

πd

2

t

=−

2500

πd

2

t

∂g 2

∂y 3

∣

∣

∣

∣Y=

∂g 2

∂y 4

∣

∣

∣

∣Y=^0

∂g 2

∂y 1

∣

∣

∣

∣Y=

1

πdt

∂g 2

∂y 2

∣

∣

∣

∣Y= −

π^2 (d

2

+t^2 )

8 l

2 = −

π^2 (d

2

+t^2 )

500 , 000

∂g 2

∂y 5

∣

∣

∣

∣Y=

π^2 E(d

2

+t^2 )

4 l

3 =^0.^0136 π

(^2) (d^2 +t (^2) )
∂g 2
∂y 6

∣

∣

∣

∣Y= −

P

πd

2

t

−

π^2 E( 2 d)

8 l

2 = −

2500

πd

2

t

−π^2 ( 3. 4 )d

∂g 3

∂yi

∣

∣

∣

∣Y=0 fori=1 to 5

∂g 3

∂y 6

∣

∣

∣

∣Y= −^1.^0

∂g 4

∂yi

∣

∣

∣

∣Y=0 fori=1 to 5