Q20 of 20 Page 12

A farmer mixes two brands P and Q of cattle feed. Brand P, costing ₹ 250 per bag, contains 3 units of nutritional element A, 2.5 units of element B and 2 units of element C. Brand Q costing ₹ 200 per bag contains 1.5 units of nutritional element A, 11.25 units of element B, and 3 units of element C. The minimum requirements of nutrients A, B and C are 18 units, 45 units and 24 units respectively. Determine the number of bags of each brand which should be mixed in order to produce a mixture having a minimum cost per bag? What is the minimum cost of the mixture per bag?

The above information can be expressed with the help of the following table:



Let ‘x’ bags of P and ‘y’ bags of Q be bought.


Cost of P = 250x


Cost of Q = 200y


Cost of mixture = 250x + 200y


Now,


⟹ 3x + 1.5y ≥18


i.e. the minimum requirement of element A from both P and Q combined is 18units, each of which has 3units and 1.5units of element A.


⟹ 2.5x + 11.25y ≥ 45


i.e. the minimum value of element B from both P and Q combined is 45units, each of which contains 2.5units and 11,25units of element B.


⟹ 2x + 3y ≥ 24


i.e. the minimum value of element C from both P and Q combined is 24units, each of which contains 2units and 3 units of element C.


Hence, mathematical formulation of the above LPP is as follows :


Find ‘x’ and ‘y’ that:


Minimises Z = 250x + 200y


Subject to the following constraints:


(i) 3x + 1.5y ≥18


(ii) 2.5x + 11.25y ≥ 45


(iii) 2x + 3y ≥ 24


(iv) x,y ≥0 (∵ quantity cant be negative)



The feasible region is unbounded


The corner points of the feasible region are as follows:



Z is minimised at B(3,6) i.e. 3 bags of P and 6 bags of Q should be purchased to achieve the minimum cost of the mixture per bag.


The minimum cost of the mixture is ₹1950.

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