Question

To state if the statements are true or false with a short proof or a counter example in support of the answer.

1.) The optimal solution for the following LLP is Z* = 30 :

Maximise Z = x₁ - x₂ + 3x₃

Subject to : x₁ + x₂ + x₃ ≤ 10

x₁ , x₂ , x₃ ≥ 0

2.) For the mixed generator r₍ₙ₊₁₎=(5r+7)(mod 8), if r₀=4, then rₙ=0

3.) If the availabilities and requirements of a transportation problem are integers, the optimal solution to the problem will have integer values.

4.) The optimal solution of ILLP can be obtained by rounding off the optimal solution of its LP relaxation.

Answer 1

1) True

$Z_{max}=x_1-x_2+3x_3$

Solve by simplex method using online calculator:

$Z_{max}=30,x_1=x_2=0,x_3=10$

2) false

$r_{n+1}=(5r_n+7)mod8$

$r_1=27mod8=3$

$r_2=22mod8=6$

$r_3=37mod8=5$

3) True

For balanced problem: total supply=total demand.

In process we perform only addition and subtraction with integers.

4) false

The rounded LP relaxation solution is almost certainly non-optimal and may be very non-optimal (in the sense that the objective function value of the rounded LP relaxation solution is very far away from the objective function value of the optimal integer solution).