312 Introduction to Linear Optimization and Extensions with MATLAB
R
minimize 2x
1
+ 3x
2
subject to a
11
x
1
+ a
12
x
2
≥ b
1
a
21
x
1
+ a
22
x
2
≥ b
2
.
If the constraint and the right-hand side coefficients are uncertain, then
the robust counterpart RC is
minimize 2x
1
+ 3x
2
subject to a
11
x
1
+ a
12
x
2
≥ b
1
a
21
x
1
+ a
22
x
2
≥ b
2
∀
a
11
a
12
a
21
a
22
b
1
b
2
∈ U ⊂ R
2×2
× R
2×1
.
Note: An element of U is the direct product of a 2 × 2 matrix and a 2 ×1
vector, and so two sets of matrix brackets were used to highlight the dimension
of the direct product. From here on such an element will be represented using
only one bracket for ease of exposition, i.e.,
a
11
a
12
a
21
a
22
b
1
b
2
=
a
11
a
12
b
1
a
21
a
22
b
2
and
(a
i
, b
i
) = (a
i1
, a
i2
, b
i
).
If U =
0.95 1.95 0.95
2.95 1.95 1.95
,
1 2 1
3 2 2
,
1.05 2.05 1.05
3.05 2.05