Q2 of 118 Page 31

A diet for a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 of calories. Two foods A and B, are available at the cost of ₹ 4 and ₹ 3 per unit respectively. If one unit of A contains 200 units of vitamin, 1 unit of mineral and 40 calories and one unit of food B contains 100 units of vitamin, 2 units of minerals and 40 calories, find what combination of foods should be used to have the least cost?

The above information can be expressed in the form of the following table:



Let the quantity of the foods be ‘x’ and ‘y’ respectively.


Cost of food A = 4x


Cost of food B = 3y


Total cost of the combination = 4x + 3y


Now,


⟹ 200x + 100y ≥ 4000


i.e. the minimum requirement of vitamins from the two foods should be 4000.


⟹ x + 2y ≥ 50


i.e. the minimum requirement of minerals from the two foods should be 50.


⟹ 40x + 40y ≥ 1400


i.e. the minimum requirement of calories from the two foods should be 1400


Hence, mathematical formulation of LPP is as follows:


Find ‘x’ and ‘y’ which


Minimize Z = 4x + 3y


Subject to the following constraints:


(i) 200x + 100y ≥ 4000


i.e. 2x + y ≥ 40


(ii) x + 2y ≥ 50


(iii) 40x + 40y ≥ 1400


i.e. x + y ≥ 35


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



The feasible region is unbounded.


The corner points of the feasible region is as follows:



Z is smallest at B(5,30)


Let us consider 4x + 3y ≤ 110


As it has no intersection with the feasible region, the smallest value is the minimum value.


The minimum cost of foods is ₹110


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