Stochastic Programming - Index of

PROBABILISTIC CONSTRAINTS 251
For the lower bound we choose
min{v 1 + v 2 + ···+ v n }
s.t. v 1 +2v 2 + ···+ ( ) nv n = S 1,n ,
n
v 2 + ···+ v
2 n = S 2,n ,
v i ≥ 0, i=1, ···,n.
and correspondingly for the upper bound we formulate
max{v 1 + v 2 + ···+ v n }
s.t. v 1 +2v 2 + ···+ ( ) nv n = S 1,n ,
n
v 2 + ···+ v
2 n = S 2,n ,
v i ≥ 0, i=1, ···,n.
⎫
⎪⎬
⎪⎭
⎫
⎪⎬
⎪⎭
(3.3)
(3.4)
These linear programs are feasible and bounded, and therefore solvable. So,
there exist optimal feasible 2 × 2basesB.
Consider an arbitrary 2 × 2 matrix of the form
⎛
(
B = ⎝ i i+ r ⎞
) ( )
i i + r ⎠ ,
2 2
where 1 ≤ i 0
for all i and r such that 1 ≤ i