Question

In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in mg/tablet) are given as below

$\begin{array}{cccc}Tablets& Iron& Calcium& Vitamin\\ X& 6& 3& 2\\ Y& 2& 3& 4\end{array}$

The person needs atleast 18 mg of iron, 21 mg of calcium and 16 mg of vitamins. The price of each tablet of X and Y is Rs 2 and R1, respectively. How many tablets of each should the person take in order to stisfy the above requirement at the minimum cost?

Answer 1

Let the person takes x units of tablet X and y units of tablet Y.
So, from the given information,we have
$6x+2y\ge 18\Rightarrow 3x+y\ge \mathrm{9.....}\left(i\right)3x+3y\ge 21\Rightarrow x+y\ge \mathrm{7.....}\left(ii\right)$
and $2x+4y\ge 16\Rightarrow x+2y\ge 8.....\left(iii\right)$
Also, we know that here, $x\ge 0,y\ge \mathrm{0....}\left(iv\right)$
The price of each tablet of X and Y is Rs 2 and Rs 1. respectively.
So, the corresponding LPP is minimise Z=2x+y, subject to $3x+y\ge 9,x+y\ge 7,x+2y\ge 8,x\ge 0,y\ge 0.$
From the shaded graph, we see that for the shown unbounded rgion, we have coordinates of corner points A,B,C and D as (8,0),(6,1),(1,6), and (0,9), respectively.
[on solving x+2y =8 and x+y =7, we get x =6, y =1 and on solving 3x+y =9 and x+y =7, we get x=1, y =6]

$\begin{array}{cc}\text{Corner points}& \text{Value of Z =2x+y}\\ (8,0)& 16\\ (6,1)& 13\\ (1,6)& 8\leftarrow Minimum\\ (0,9)& 9\end{array}$
Thus, we see that 8 is the minimum value of Z at the corner point (1,6). Here, we see that the feasible region is unbounded. Therefore, 8 may or may not be the minimum value of Z.
To decide this issue, we graph the inequality
2x+y < 8...(v)
and check whether the resulting open half has points in common with feasible region or not. If it has common point. then 8 will not be the minimum value of Z, otherwise 8 will be the minimum value of Z.
Thus, from the graph it is clear that, it has no common point.
Therefore, Z=2x+y has 8 as minimum value subject to the given constraints.
Hence, the person should take 1 unit of X table and 6 units of Y tablets to satisfy the given requirements and at the minimum cost of Rs 8.