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  • Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg.

1.Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs.80/kg. Food P contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

Answers (1)

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Let mixture contain x kg of food P and y kg of food Q. Thus, x\geq 0,y\geq 0.

The given information can be represented in the table as :

  Vitamin A  Vitamin B  Cost 
Food P   3     5     60
Food Q    4      2     80
 requirement     8     11  

The mixture must contain 8 units of Vitamin A and 11 units of Vitamin B.

Therefore, we have 

 3x+4y\geq 8

 5x+2y\geq 11

Total cost is Z. Z=60x+80y

Subject to constraint,

  3x+4y\geq 8

  5x+2y\geq 11

  x\geq 0,y\geq 0

The feasible  region determined by constraints is as follows:

    Chapter 12 – Linear Programming

It can be seen that a feasible region is unbounded.

The corner points of the feasible region are A(\frac{8}{3},0),B(2,\frac{1}{2}),C(0,\frac{11}{2})

The value of Z at corner points is as shown :

 corner points  Z=60x+80y  
   A(\frac{8}{3},0)             160 MINIMUM

B(2,\frac{1}{2})

            160 minimum
C(0,\frac{11}{2})             440  

Feasible region is unbounded, therefore 160 may or may not be the minimum value of Z.

For this, we draw 60x+80y< 160\, \, or \, \, \, 3x+4y< 8 and check whether resulting half plane has a point in common with the feasible region or not.

We can see a feasible region has no common point with.\, \, 3x+4y< 8

Hence, Z has a minimum value 160  at line segment joining points  A(\frac{8}{3},0)     and   B(2,\frac{1}{2}).

Posted by

seema garhwal

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