Q4 of 118 Page 31

A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamine and at least 600 calories. Each unit of A contains 0.12 milligram of thiamine and 100 calories, while each unit of B contains 0.10 milligram of thiamine and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?

The above information can be expressed using the following table:



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


Cost of food A = 0.10x


Cost of food B = 0.10y


Cost of diet = 0.10x + 0.10y


Now,


⟹ 0.12x + 0.10y ≥ 0.5


i.e. the minimum requirement of thiamine in the foods is 0.5mg


⟹ 100x + 150y ≥ 600


i.e. the minimum requirement of calories in the foods is 600.


Hence, mathematical formulation of the LPP is as follows:


Find ‘x’ and ‘y’ that:


Minimises Z = 0.10x + 0.10y


Subject to the following constraints:


(i) 0.12x + 0.10y ≥ 0.5


(ii) 100x + 150y ≥ 600


i.e. 2x + 3y ≥ 12


(iii) 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 A(1.875,2.75)


Let us consider 0.1x+0.1y ≤ 0.4625


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


The minimum cost of the foods is ₹0.4625


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