Answer to Question #291483 in Operations Research for Danny

Question

A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Birr 25000 and Birr 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Birr 70 lakhs and if his profit on the desktop model is Birr 4500 and on portable model is Birr 5000. (Find optimal Solution by simplex Model)

Answer 1

Solution:

Let number of desktop model be x and number of portable model be y

According to question,

Since, monthly demand doesn't exceed 250 units.

∴x₁+x₂≤250   ...(1)

Since, maximum invest is 70 lakhs.

∴25000x₁+40000x₂≤7000000⇒5x₁+8x₂≤1400   ...(2)

Also, quantity can't be negative.

∴x₁≥0,x₂≥0   ...(3)

We have to maximize profit Z

where Z=4500x₁+5000x₂

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '≤' we should add slack variable S₁

2. As the constraint-2 is of type '≤' we should add slack variable S₂

After introducing slack variables

Max Z=x₁+5000x₂+0S₁+0S₂

subject to

x₂+S₁=250

8x₂+S₂=1400 and

x₁,x₂,S₁,S₂≥0

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropiate

1. As the constraint-1 is of type '≤' we should add slack variable S1

2. As the constraint-2 is of type '≤' we should add slack variable S2

After introducing slack variables

Max Z=4500x1+5000x2+0S1+0S2

subject to

x1+x2+S1=250

5x1+8x2+S2=1400 and

x1,x2,S1,S2≥0

Since all Z_j-C_j≥0

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

$x_1=200,x_2=50$

Max Z=1150000