Get Solved LP Problems

Question

Solve Linear Programming Questions

A producer manufactures 3 models (I, II and III) of a particular product. He uses 2 raw materials A and B of which 4000 and 6000 units respectively are obtainable. The raw materials per unit of 3 models are listed below.

Raw materials	I	II	III
A	2	3	5
B	4	2	7

The labour time for each unit of model I is two times that of model II and thrice that of model III. The whole labour force of factory can manufacture an equivalent of 2500 units of model I. A model survey specifies that the minimum demand of 3 models is 500, 500 and 375 units correspondingly. However the ratio of number of units manufactured must be equal to 3:2:5. Suppose that gains per unit of model are 60, 40 and 100 correspondingly. Develop a LPP.

 

Answer

Assume

x₁ - number of units of model I

     x₂ - number of units of model II

     x₃ - number of units of model III

 

 

Raw materials	I	II	III	Availability
A	2	3	5	4000
B	4	2	7	6000
Profit	60	40	100

 

x₁ + 1/2x₂ + 1/3x₃ ≤ 2500                                                       Labour time

 

x₁ ≥ 500, x₂ ≥ 500, x₃ ≥ 375                                                    Minimum demand

 

The given ratio is x₁: x₂: x₃ = 3: 2: 5

x₁ / 3 = x₂ / 2 = x₃ / 5 = k

x₁ = 3k; x₂ = 2k; x₃ = 5k

x₂ = 2k → k = x₂ / 2

So x₁ = 3 x₂ / 2 → 2x₁ = 3x₂

Likewise 2x₃ = 5x₂

 

Maximize Z= 60x₁ + 40x₂ + 100x₃

Subject to 2x₁ + 3x₂ + 5x₃ ≤ 4000

                  4x₁ + 2x₂ + 7x₃ ≤ 6000

x₁ + 1/2x₂ + 1/3x₃ ≤ 2500

2 x₁ = 3x₂

2 x₃ = 5x₂

& x₁ ≥ 500, x₂ ≥ 500, x₃ ≥ 375