Math484.2 September 29,2011 Name:__________Dr.V._______________________________
Midterm1, 5 problems, 15 points each. Return this page with your name on both sides.
1. Solve for x, y where a is a given number:
a2x - y= a2,
ax +ay = 1.
Solution. A row addition operation gives
a2x - y= a2,
(a + a3 )x = 1 +a3.
If a ≠ 0 ,then x = (1 +a3 )/(a +a3 ) and
y = (x - 1)a2 = a(1 - a)/(1+a2).
If a = 0 , then there are no solutions.
2. x + y 2 -> max,
x2 + y 2 = 10; x and y integers.
Solution There are 8 feasible solutions: (x, y) = (±1, ±3), (±3, ±1).
max= 10 at x = 1 y = ±3 (two optimal solutions).
3, 4. Solve the linear programs given by the following tableaux with all decision variables xi > 0:
|
x1 |
x2 |
x3 |
1 |
Problem 3 |
|
1 |
0 |
-1 |
-2 |
=- x4 |
|
1 |
0 |
1 |
-1 |
-> min |
Solution. The standard tableau is
|
x1 |
x2 |
x3 |
1 |
Problem 3 |
|
-1 |
0 |
1 |
2 |
= x4 |
|
1 |
0 |
1 |
-1 |
-> min |
It is optimal so min = -1 at x1 =x2 = x3 = 0, x4 = 2.
|
x1 |
x2 |
-x3 |
1 |
Problem 4 |
|
1 |
0 |
-1 |
2 |
= x4 |
|
1 |
0 |
1 |
-1 |
-> min |
Solution. The standard tableau is
|
x1 |
x2 |
x3 |
1 |
Problem 4 |
|
1 |
0 |
1 |
2 |
= x4 |
|
1 |
0 |
-1 |
1 |
-> min |
It is feasible, and the x3-column is bad, so LP is unbounded.
5.Find all logical implications between the following 5 constraints on x, y:
(a) x4= y4, (b) 0 > -2, (c) 0= 0, (d) x = -y, (e) x=y=0.
Solution.
(e) ⇒ (d) ⇒ (a) ⇒ (b) ⇔ (c) .