Question

A company manufactures two types of products $A$ and $B$. Each unit of $A$ requires $3$ grams of nickel and $1$ gram of chromium, while each unit of $B$ requires $1$ gram of nickel and $2$ grams of chromium. The firm can produce $9$ grams of nickel and $8$ grams of chromium. The profit is Rs. $40$ on each unit of product of type $A$ and Rs. $50$ on each unit of type $B$. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution

Answer 1

Type of product	$9$ gms Nickel (gms)	$8$ gms Chromium (gms)	Profit per unit
$A$	$3$	$1$	Rs. $40$
$B$	$1$	$2$	Rs. $50$

Let $x=$ number of units of type $A$
$y=$ Number of units of type $B$
Maximize $Z=40x+50y$
Subject to the constraints
$3x+y\le 9$
$x+2y\le 8$
and $x,y\ge 0$
Consider the equation,
$3x+y=9$

$x$	$=$	$3$
$y$	$=$	$0$

and $x+2y=8$

$x$	$=$	$8$
$y$	$=$	$0$

The solution set of this system is the shaded region in the diagram
Now, we can determine the maximum value of $Z$ by evaluating the value of $Z$ at the four points (vertices) is shown below

Vertices	$Z=40x+50y$
$(0,0)$	$Z=40\times 0+50\times 0=$ Rs. $0$
$(3,0)$	$Z=40\times 3+50\times 0=$ Rs. $120$
$(0,4)$	$Z=40\times 0+50\times 4=Rs.200$
$(2,3)$	$Z=40\times 2+50\times 3=Rs.230$

From graph,
Maximum profit, $Z=$ Rs. $230$
$\therefore$ Number of units of type $A$ is $2$ and number of units of type $B$ is $3$.