Engineering Optimization: Theory and Practice, Fourth Edition

658 Stochastic Programming

∂g 4

∂y 6

∣

∣

∣

∣Y=^1.^0

∂g 5

∂yi

∣

∣

∣

∣Y=

∂g 6

∂yi

∣

∣

∣

∣Y=0 fori=1 to 6

Since the value of the standard normal variateφj(pj) orresponding to the probabilityc

pj= 0. 9 5 is 1.645 (obtained from Table 11.1), the constraints in Eq. (11.98) can be

expressed as follows.

Forj= 1 †:

2500

πdt

− 500 − 1. 645

[

σP^2

π^2 d

2

t^2

+σf^2 y+

( 2500 )^2

π^2 d

4

t^2

σd^2

] 1 / 2

≤ 0

795

dt

− 500 − 1. 645

(

25 , 320

d

2

t^2

+ 5002 +

63. 3

d

2

t^2

) 1 / 2

≤ 0 (E 10 )

Forj=2:

2500

πdt

− 16. 78 (d

2

+t^2 ) − 1. 645

[

σP^2

π^2 d

2

t^2

+

π^4 (d

2

+t^2 )^2 σE^2

25 × 1010

+ ( 0. 0136 π^2 )^2 (d

2

+t^2 )^2 σl^2 +

(

2500

πd

2

t

+ 3. 4 π^2 d

) 2

σd^2

] 1 / 2

≤ 0

795

dt

− 16. 78 (d

2

+t^2 ) − 1. 645

[

25 , 320

d

2

t^2

+ 2. 82 (d

2

+t^2 )^2

+ 0. 113 (d

2

+t^2 )^2 +

63. 20

d

2

t^2

+ 0. 1126 d

4

+

5. 34 d

t

] 1 / 2

≤ 0 (E 11 )

Forj=3:

−d+ 2. 0 − 1 .645[( 10 −^4 )d

2

]^1 /^2 ≤ 0

− 1. 01645 d+ 2. 0 ≤ 0 (E 12 )

Forj=4:

d− 14. 0 − 1 .645[( 10 −^4 )d

2

]^1 /^2 ≤ 0

0. 98335 d− 14. 0 ≤ 0 (E 13 )

Forj=5:

−t+ 0. 2 ≤ 0 (E 14 )

Forj=6:

t− 0. 8 ≤ 0 (E 15 )

†The inequality sign is different from that of Eq. (11.98) due to the fact that the constraints are stated as

P[gj( Y)≤ 0 ]≥pj.