Numericals on Dual Problem of Linear Programming

 (Dual Problem)

1.        Write the dual of

Maximize     Z = 5x₁ + 3x₂

Subject to    x₁ + x₂ \< 2

5x₁ + 2x₂ \< 10

3x₁ + 8x₂ \< 12

x₁, x₂  \<0                                     (Ans :  Minimize Z*= 2y₁ + 10y₂ + 12y₃

Subject to          y₁ + 5y₂ + 8y₃

5x₁ + 2x₂  10

y₁ y₂ y₃  >/0)

2.        Write the dual of

Maximize     Z = 4x₁ + 3x₂

Subject to    3x₁ + 2x₂ \< 15

2x₁ + 5x₂  \<20

x₁, x₂  >/0                                        (Ans :  Minimize Z* = 15y₁ – 20y₂

Subject to          3y₁ + 2y₂ >/ 4

5x₁ + 2x₂  >/10

2y₁ + 5y₂ >/3

y₁ y₂  >/0)

3.        Solve the following LPP using dual and interpret the optimal solution.

Minimize      Z = x₁ + x₂

Subject to    0.12x₁ + 0.04x₂  >/600

0.10x₁ + 0.40x₂ >/ 1000

x₁, x₂ >/ 0                                      (Ans :  x₁ = 50000/11, x₂ = 15000/11)

4.        Minimize Z = 120x₁ + 180x₂

Subject to    6x₁ + 3x₂  >/45

4x₁ + 10x₂ >/ 55

x₁, x₂  >/0

(a)      Write the dual of the given lpp.

(b)      Find dual optimal solution.

(c)      Mention unutilized capacity of slack variables if any.

(d)      State with reasons whether degeneracy occurs in the sum.