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.
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Raw materials
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I
|
II
|
III
|
|
A
|
2
|
3
|
5
|
|
B
|
4
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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
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Raw materials
|
I
|
II
|
III
|
Availability
|
|
A
|
2
|
3
|
5
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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