Question

Vitamins A, B and C are found in Foods F1 and F2. One unit of F1 contains 1 mg of A, 100 mg of B and 10 mg of C. One unit of F2 contains 1 mg of A, 10 mg of B and 100 mg of C. The minimum daily requirement of A, B and C are 1 mg, 50 mg, and 10 mg respectively. The cost per unit of F1 and F2 are Rs. 1 and Rs. 1.5 respectively. You are required to (i) formulate the above as a linear programming problem minimizing the cost per day, (ii) write the dual of the problem and (iii) solve the dual by using simplex method and read from there the answer to the primal.

Answer 1

Answer:

Given :  

Vitamins A, B and C are found in Foods F1 and F2.  

F1 contains:   A = 1 mg, B = 100 mg and   C = 10 mg. (One unit)  

F2 contains:  A = 1 mg,  B = 10 mg  and  C = 100 mg . (One unit)

The minimum daily requirement of A, B and C are 1 mg, 50 mg, and 10 mg respectively.  

The price per unit of F1 and F2 are Rs. 1 and Rs. 1.5 respectively.

To find: minimizing the cost per day,

Solution:

Let say F1 =  X unit

F2  =  Y  unit

X +  Y  ≥ 1

100X + 10Y    ≥  50   =>  10X + Y   ≥  5

10X +  100Y  ≥  10     => X  + 10Y  ≥  1

Cost = X  + 1.5Y  

X =  0     Y  = 5      Cost  =  7.5

X = 4/9   Y = 5/9   Cost   = 11.5/9  = 1.278 Rs

X = 1       Y = 0       Cost =  1   Rs

Minimum Cost by Having one unit of F1