Answer to Question #283387 in Operations Research for Andinet

Question

4. Bole cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Birr 5 and that from a shade is Birr 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximize his profit? (Solve by simplex model).

Answer 1

profit:

$Z=5x_1+3x_2$

for sprayer time:

$3x_1+2x_2\le 20$

for grinding/cutting time:

$2x_1+x_2\le 12$

where x₁ is number of lamps,

x₂ is number of wooden shades

After introducing slack variables:

Max $Z=5x_1+3x_2+0S_1+0S_2$

subject to

$3x_1+2x_2+S_1=20$

$2x_1+x_2+S_2=12$

$x_1,x_2,S_1,S_2≥0$

Negative minimum Z_j-C_j is -5 and its column index is 1. So, the entering variable is x₁.

Minimum ratio is 6 and its row index is 2. So, the leaving basis variable is S₂.

∴ The pivot element is 2.

Entering =x₁, Departing =S₂, Key Element =2

Negative minimum Z_j-C_j is -0.5 and its column index is 2. So, the entering variable is x₂.

Minimum ratio is 4 and its row index is 1. So, the leaving basis variable is S₁.

 ∴ The pivot element is 0.5.

Entering =x₂, Departing =S₁, Key Element =0.5

Since all Z_j-C_j≥0

Hence, optimal solution is arrived with value of variables as :

x₁=4, x₂=4

Max Z=32