Q1 of 118 Page 31

A diet of two foods F1 and F2 contains nutrients thiamine, phosphorous and iron. The amount of each nutrient in each of the food (in milligrams per 25 gms) is given in the following table :


The minimum requirement of the nutrients in the diet is 1.00 mg of thiamine, 7.50 mg of phosphorous and 10.00 mg of iron. The cost of F1 is 20 paise per 25 gms while the cost of F2 is 15 paise per 25 gms. Find the minimum cost of diet.

The information above can be expressed in the following table:



Let the amount of food F1 and F2 required to be ‘x’ and ‘y’ units.


Cost of F1 = 0.20x


Cost of F2 = 0.15y


So, Cost of diet = 0.20x + 0.15y


Now,


⟹ 0.25x + 0.10y ≥ 1.00


i.e. the minimum requirement of Thiamine should be 1.00mg, from both the foods F1 and F2, each of which have 0.25mg and 0.10mg of Thiamine respectively. So, this is the first constraint.


⟹ 0.75x +1.50y ≥ 7.50


i.e. the minimum requirement of Phosphorous should be 7.50mg, from both the foods F1 and F2, each of which have 0.75mg and 1.50mg of Phosphorous respectively. This is the second constraint.


⟹ 1.60x + 0.80y ≥ 10.00


i.e. the minimum requirement of Iron should be 10.00mg, from both the foods F1 and F2, each of which have 1.60mg and 0.8mg of Iron respectively.


Hence, mathematical formulation of LPP is as follows:


Find ‘x’ and ‘y’ which


Minimise Z = 0.20x + 0.15y


Subject to the following constraints:


(i) 0.25x + 0.10y ≥ 1.00


(ii) 0.75x +1.50y ≥ 7.50


(iii) 1.60x + 0.80y ≥ 10.00


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



The feasible region is unbounded


The end points of the feasible region are as follows:



So, Z is smallest at B(5,2.5)


Let us consider the inequation 0.20x + 0.15y ≤ 1.375


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


So, the minimum cost of diet is ₹1.375


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