656 Stochastic Programming
the objective function can be expressed asf (Y)= 5 W+ 2 d= 5 ρlπdt+ 2 d. SinceY=
P
E
ρ
fy
l
d
=
2500
0. 85 × 106
0 025. 0
500
250
d
f (Y)= 5 ρlπdt+ 2 d= 9. 8175 dt+ 2 d
∂f
∂y 1∣
∣
∣
∣Y=
∂f
∂y 2∣
∣
∣
∣Y=
∂f
∂y 4∣
∣
∣
∣Y=^0
∂f
∂y 3∣
∣
∣
∣Y=^5 πldt=^3927.^0 dt∂f
∂y 5∣
∣
∣
∣Y=^5 πρdt=^0.^03927 dt∂f
∂y 6∣
∣
∣
∣Y=^5 πρlt+^2 =^9.^8175 t+^2.^0Equations (11.87) and (11.88) give
ψ(Y)= 9. 8175 dt+ 2 d (E 1 )σψ^2 927 =( 3. 0 dt)^2 σρ^2 + ( 0. 03927 dt)^2 σl^2 + ( 9. 8175 t+ 2. 0 )^2 σd^2= 0. 9835 d2
t^2 + 0. 0004 d2
+ 0. 003927 d2
t (E 2 )Thusthe new objective function for minimization can be expressed as
F (d, t)=k 1 ψ+k 2 σψ=k 1 ( 1759. 8 dt+ 2 d)+k 2 ( 8350. 9 d
2
t^2 + 0. 0004 d
2
+ 0. 003927 d
2
t)^1 /^2 (E 3 )wherek 1 ≥ and 0 k 2 ≥ indicate the relative importances of 0 ψandσψfor minimiza-
tion. By using the expressions derived in Example 1.1, the constraints can be expressed
as
P[g 1 ( Y)≤ 0 ]=P(
P
π dt−fy≤ 0)
≥ 0. 95 (E 4 )
P[g 2 ( Y)≤ 0 ]=P[
P
π dt−
π^2 E
8 l^2(d^2 +t^2 )≤ 0]
≥ 0. 95 (E 5 )
P[g 3 (Y)≤ 0 ]=P[−d+ 2. 0 ≤0]≥ 0. 95 (E 6 )
P[g 4 (Y)≤ 0 ]=P[d− 14. 0 ≤0]≥ 0. 95 (E 7 )
P[g 5 (Y)≤ 0 ]=P[−t+ 0. 2 ≤0]≥ 0. 95 (E 8 )P[g 6 (Y)≤ 0 ]=P[t− 0. 8 ≤0]≥ 0. 95 (E 9 )