Question

The doctor advises a patient visited him that the patient is weak in his health due to shortage of two vitamins, i.e., vitamin A and vitamin D. He advises him to take at least 40 units of vitamin A and 50 units of Vitamin D every day. He also advises that these vitamins are available in two tonics X and Y. Each unit of tonic X consists of two units of vitamin A and three units of vitamin D. Each unit of tonic Y consists of four units of vitamin A and two units of vitamin D. Tonic X and Y are available in the medical shop at a cost of Birr three per unit of X and Birr 2.50 per unit of Y. The patient has to fulfill the need of vitamin by consuming X and Y at a minimum cost.

a. Formulate the primal linear programming model for the patient’s problem and solve the problem using the simplex method

b. Indicate the range over which objective function coefficient of basic decision variables can change without changing their optimal values.

Answer 1

Answer

Let us first add the table

Let the variables $x_{1}$ and $x_{2}$ represent units of tonic $x$ and $y$ respectively.

The total cost of diet consisting of $x_{1}$ units of $x$ and $x_{2}$ units of $y$ is given by, $z=5 x_{1}+3 x_{2}$

Since, the tonics $x$ and $y$ cost Rs 5 per unit and Rs. 3 per unit respectively.

Now, total amount of vitamin A required for the tonic $x$ and $y$ is $2 x_{1}+4 x_{2}$ .

Total amount of vitamin D required for the tonic $x$ and $y$ is $3 x_{1}+2 x_{2}$ .

Again, Since, the minimum daily requirement of vitamin A is 40 units and that of vitamin B is 50 units, therefore must have, $\quad 2 x_{1}+4 x_{2} \geqslant 40$

and $3 x_{1}+2 x_{2} \geqslant 50$

Also since, $x_{1}$ and $x_{2}$ are either positive or zero, we should have $x_{4} \geqslant 0, x_{2} \geqslant 0$

Hence, the formulation of the L.P.P. is

minimize $z=5 x+3 x_{2}$

Subjected to

$2 x+4 x_{2} \geqslant 40 \\ 3 x+2 x_{2} \geqslant 50 \\ x_{1}, x_{2} \geqslant 0$