Exercise 4. Consider the polyhedral set defined by the system of inequalities: a

ID: 3108726 • Letter: E

Question

Exercise 4. Consider the polyhedral set defined by the system of inequalities:

ax1 + (b + 1)x2 120

x1 + (a + b)x2 160

(a b)x1 + x2 30

x1 0

x2 0

Find the necessary conditions for the numbers a and b to obtain (If it is possible):

1. Edges

2. Extreme points No degenerate

3. Extreme points degenerate

4. Family of directions

5. Extreme directions.

6. No extreme directions.

7. No extreme points

8. Is it possible to find the sufficient conditions in order to characterize the extreme points (degenerate).

9. Find the necessary conditions for the numbers a, b to make the L.P problem have an optimal solution. (If it is possible)

consider x1+(a+b)x2<=160 and (a-b)x1+x2<=30. Solving both the equations we get a=95 and b=65.

Now substituting the values of a and b in ax1+(b+1)x2<=120 we get the following inequality 95x1+66x2<=120

Consider 95x1+66x2<=120 putting x1 = 0 , x2=0 we get two points (0,1.82) and (1.26,0)

next consider the second inequality x1+160x2<=160. Putting x1=0 and x2=0 we get two points (0,1) and (160,0).

Consider the third inequality 30x1+x2<=30. putting x1=0 and x2=0 we get two points (0,30) and (1,0).

With the above details we can find the optimal solution.

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