# Use the Karush-Kuhn- tucker conditions to solve

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1. Consider the LP Max z=c1x1+2x2+c3x3

Subject to x1 + 5x2+a1x3 ≤ b1

X1-5x2+a2x3≤ b2

X1, x2, x3 ≥ 0

The optimal tableau for this LP is

1 d1 2 1 0 30

0 d2 -8 -1 1 10

------------------------------------

0 d3 -7 d4 0 z - 150

Without using the simplex method, determine all unknown constants in this problem

(i.e., a1,a2,b1,b2, c1, c3, d1,d2, d3, d4)

2. Consider the following problem:

Maximize z= 6x1 + 8x2

Subject to 5x1 + 2x2 less than or equal to 20

x1 + 2x2 less than or equal to 12

x1, x2 greater than or equal to 0

(a) Sketch the feasible set and solve the program geometrically

(b) Determine all the basic solutions for this program and indicate them on the sketch in (a). Which ones are feasible? (there are 6 basic solutions, not all feasible).

(c) Solve the program by the simplex method. For each tableau, indicate which feasible solution corresponds to it

(d) Write the dual of this program. For each primal basic solution, determine the corresponding dual basic solution that satisfies the complementary slackness principle. Which ones are dual feasible?

3. Consider the system of equations -x1+x2-x3=3, -x1+2x2-x4=2, and x1+x2+x5=2

(a) by converting the equations into inequalities in two variables, show geometrically that there are no nonnegative solutions to this system.

(b) use the simplex method to show the same thing

4. Use the Karush-Kuhn conditions to solve:

Maximize z=2x1+x2+3x3

Subject to x1+2x2+x3 less than or equal to 12

X1,x2,x3 greater than or equal to 0

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