Stochastic Programming - Index of

BASIC CONCEPTS 47
Hence the feasible set
B(α) ={x | P ({ξ | g(x, ξ) ≤ 0}) ≥ α}
is the union of all those vectors x feasible according to (6.2), and consequently
may be rewritten as
B(α) = ⋃ ⋂
{x | g(x, ξ) ≤ 0}. (6.3)
G∈G
ξ∈G
Since a union of convex sets need not be convex, this presentation
demonstrates that in general we may not expect B(α) to be convex, even
if {x | g(x, ξ) ≤ 0} are convex ∀ξ ∈ Ξ. Indeed, there are simple examples for
nonconvex feasible sets.
Example 1.4 Assume that in our refinery problem (3.1) the demands are
random with the following discrete joint distribution:
( )
h1 (ξ
P
1 ) = 160
h 2 (ξ 1 =0.85,
) = 135
( )
h1 (ξ
P
2 ) = 150
h 2 (ξ 2 =0.08,
) = 195
( )
h1 (ξ
P
3 ) = 200
h 2 (ξ 3 =0.07.
) = 120
Then the constraints
x raw1 + x raw2 ≤ 100
x raw1 ≥ 0
x raw2 ≥ 0
( )
2xraw1 +6x
P
raw2 ≥ h 1 (˜ξ)
≥ α
3x raw1 +3x raw2 ≥ h 2 (˜ξ)
for any α ∈ (0.85, 0.92] require that we
• either satisfy the demands h i (ξ 1 )andh i (ξ 2 ),i=1, 2 (enforcing a reliability
of 93%) ( and ) hence choose a production program to cover a demand
160
h A =
195
• or satisfy the demands h i (ξ 1 )andh i (ξ 3 ),i=1, 2 (enforcing a reliability of
92%) such ( that ) our production plan is designed to cope with the demand
200
h B = .
135