Math484.2  September 29,2011  Name:__________Dr.V._______________________________

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) .