4.4 Decomposition Principle 205subject to
A 1 X 1 +A 2 X 2 ≤b 0
B 1 X 1 ≤b 1
B 2 X 2 ≤b 2
X 1 ≥ 0 , X 2 ≥ 0(E 5 )
whereX 1 ={
x 1
x 2}
, X 2 =
{
y 1
y 2}
, c 1 ={
1
2
}
, c 2 ={
2
3
}
,
A 1 =
[
1 1
1 0
]
, [A 2 ]=
[
1 1
1 0
]
, b 0 ={
1000
500
}
,
B 1 =
[
1 1
1 − 2
]
, b 1 ={
600
0
}
, B 2 =
{
− 2 1
}
,b 2 ={ 0 },X=
{
X 1
X 2
}
Step 1We first consider the subsidiary constraint sets
B 1 X 1 ≤b 1 , X 1 ≥ 0 (E 6 )
B 2 X 2 ≤b 2 , X 2 ≥ 0 (E 7 )The convex feasible regions represented by (E 6 ) nd (Ea 7 ) re shown in Fig. 4.1a a
andb, respectively. The vertices of the two feasible regions are given byX( 11 )= ointp P={
0
0
}
X( 21 )= ointp Q={
0
600
}
X( 31 )= ointp R={
400
200
}
Figure 4.1 Vertices of feasible regions. To make the feasible region bounded, the constraint
y 1 ≤1000 is added in view of Eq. (E 2 ).