Midterm_ Practice Problems (1).pdf - IE 310: Operations...

IE 310: Operations Research
Practice Problems_ Midterm
1. Consider the following LP:
max Z = 10 x
1
+20 x
x
1
-2 x
2
≥ - 1 5
x
1
+ x
2
≤ 1
-5 x
1
-3 x
2
≥-45
x
1
, x
2
≥ 0
(a) Is the LP in standard form? If not, write the standard form of the LP.
(b) Draw the feasible region.
(c) Determine the optimal solution to this LP, using whichever method you find the easiest.
2 . Consider the following LP:
max Z = 4 x
1
+ x
2
- x
3
x
1
+ 3 x
2
≤ 6
3 x
1
+ x
2
- x
3
≤ 9
x
1
, x
2
, x
3
≥ 0
(a) Formulate the augmented LP.
(b) How many basic variables does this LP have?
(c) How many constraints (excluding non-negativity constraints) does the dual LP have?
(d) How many basic variables does the dual LP have?
(e) If x
1
, x
2
are basic variables in the above, then what are B , B
- 1
and B
- 1
b ?
3 Consider the following LP:
min Z = 2 x
1
+ 3 x
2
x
1
+ x
2
≥ 5
x
1
≥ 1
x
2
≥ 2
Show that x
1
= 3 , x
2
= 2 is an optimal solution by showing that the objective value of any
feasible solution is at least 12. (Hint: Duality)
2
2
1
Spring 2019