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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

 

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