Linear Optimization under Uncertainty 307
The optimal solution is x
3
=
x
1
x
2
=
66.8276
53.1724
and θ
3
= −7849.
Step 2: Not necessary.
Step 3: Solve recourse sub-problems given x
3
.
For ξ(1), the recourse problem is
minimize −12y
1
− 14y
2
subject to 3y
1
+ 5y
2
≤ 2004.8000
y
1
+ 0.625y
2
≤ 531.7328
y
1
≤ 500
y
2
≤ 100
y
1
≥ 0, y
2
≥ 0.
The optimal objective value is w
1
= −7030.7000 and solution y
1
=
469.2238
100
and π
T
1
=
0 −12 0 −6.5
.
For ξ(2), the recourse problem is
minimize −14y
1
− 16y
2
subject to 3y
1
+ 5y
2
≤ 2004.8000
y
1
+ 0.625y
2
≤ 531.7328
y
1
≤ 300
y
2
≤ 300
y
1
≥ 0, y
2
≥ 0.
The optimal objective value is w
2
= −7735.5 and solution y
2
=
300
220.9657
and π
T
2
=
−3.2 0 −4.4 0
.
Now e
3
= 0.4π
T
1
h
1
+ 0.6π
T
2
h
2
= 0.4(−650) + 0.6(−1320) = −1052 and
E
3
= 0.4π
T
1
T + 0.6π
T
2
T =
57.6 48
.
Then, w
2
= e
3
−E
3
x
3
= −1052