# A manufacturing company makes two types of television sets, one is black and white and other is coloured. The company has resources to make at most 300 sets a weak. It takes Rs. 1800 to make a black and white set and Rs. 2700 to make a coloured set. The company can spend not more than Rs. 648000 a weak to make television sets. It makes a profit of Rs. 510 per black and white set and Rs. 675 per coloured set, how many sets of each type should be produced so that the company has maximum profit? Formulate this as LPP given that the objective is to maximise the profit.

Toolbox:
• Let $R$ be the feasible region for a linear programming problem and let $z=ax+by$ be the objective function.When $z$ has an optimum value (maximum or minimum),where variables $x$ and $y$ are subject to constraints described by linear inequalities,this optimum value must occur at a corner point of the feasible region
• If R is bounded then the objective function Z has both a maximum and minimum value on R and each of these occur at corner points of R
Step 1:
The information given can be tabulated as follows :
Now we can mathematically formulate the above information as follows :
The objective function $z=510x+675$ subjected to constraints $x+y\leq 300$
$1800x+2700y\leq 6,48,000$
$x,y \geq 0$
Step 2:
Let us plot the lines in the graph to obtain the feasible region and the corner points.
Clearly the shaded portion is the feasible region .The corner points are O(0,0),A(300,0),B(180,120),C(0,240)
Step 3:
Now let us find the value of the objective function $z=510x+675y$
At the points $(x,y)$ the value of the objective function subjected to $z=510x+675y$
At $O(0,0)$ the value of the objective function $z=0$
At $A(300,0)$ the value of the objective function $Z=510\times 300+675\times 0=153000$
At $B(180,120)$ the value of the objective function $Z=510\times 180+675\times 12 0=172800$
At $C(0,240)$ the value of the objective function $Z=510\times 0+675\times 240=162000$
Clealy $z$ has maximum value at $B(180,120)$
Hence 180 sets of black and white TV sets and 120 sets of color TV sets should be manufactured to obtain a maximum profit.