midterm1_Qiong_prep - Problem 1: consider the following LP...

Problem 1: consider the following LP
max
x
1
;x
2
;x
3
f cx
1
° 3 x
2
g
s.t. x
1
° 2 x
2
± 6
x
2
° x
3
± 1
x
1
² 0 ; x
2
² 0 ; x
3
² 0
³ can you °nd a value of c to make the LP unbounded?
³ suppose that the LP has an additional constraint
x
3
± b
can you give a value of b to make the LP infeasible?
Problem 2 : Draw a feasible region of the following two-variable LP
max
x
1
;x
2
f 2 x
1
° x
2
g
s.t. x
1
+ x
2
² 1
x
1
° x
2
± 1
3 x
1
+ x
2
± 6
x
1
² 0 ; x
2
² 0
Determine the optimal solution to this LP, using whichever method you °nd the
easiest.
Problem 3 Formulate the augmented LP for the following
max
x
1
;x
2
;x
3
f 4 x
1
+ x
2
° x
3
g
s.t. x
1
+ 3 x
2
± 6
3 x
1
+ x
2
° x
3
± 9
x
1
² 0 ; x
2
² 0 ; x
3
² 0
how many basic variables does this LP has? How many constraints (excluding
non-negativity constraints) does it dual LP has? How many basic variables does
the dual LP has? If x
1
and x
2
are basic variables in the above, what is matrix
B , and verify that
B
° 1
=
°
° 1 = 8 3 = 8
3 = 8 ° 1 = 8
±
; B
° 1
b =
°
21 = 8
9 = 8
±
1