# A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftmans time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftmans time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsmans time. (i) What number of rackets and bats must be made if the factory is to work at full capacity? (ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

$\begin{array}{1 1} 4 Rackets, 12 Bats, Profit = 200 \\ 8 Rackets, 0 Bats, Profit = 160 \\ 12 Rackets, 4 Bats, Profit = 160 \\10 Rackets, 10 Bats, Profit = 300 \end{array}$

Toolbox:
• To solve a Linear Programming problem graphically, first plot the constraints for the problem. This is done by plotting the boundary lines of the constraints and identifying the points that will satisfy all the constraints. One we graphically plot the area bounded by the constraints, it’s easy to see which points satisfy all constraints. This common region determined by all the constraints including non-negative constraints of a linear programming problem is called the $\textbf{Feasible Region (or solution region).}$
• Now, any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an $\textbf{Optimal Solution}$. We see that every point in the feasible region satisfies all the constraints, and there are infinitely many points.
• Since we know from theory that the optimal value must occur at a corner point (vertex) of the feasible region, calculate the objective function values associated with the coordinates of all the extreme points. This method is called the $\textbf{Corner Point Method}$.
• If the feasible region is bounded (if it can be enclosed), the point with the best objective function value is the best optimal solution. If the feasible region is unbounded (means that the feasible region does extend indefinitely in any direction), the then a maximum or a minimum value of the objective function may not exist. However, if it exists, it must occur at a corner point of the feasible region, which can be calculated.
We are given 42h or machine time and 24h of craftsmen time. Let x be the number of tennis rackets and y be the number of cricket bats that we can make. Our problem is to find out the maximum profit at full capacity.
Clearly, x, y ≥ 0. Let us construct the following table from the given data.
 Tennis Rackets (x) Cricket Bats (y) Requirements Machine Time (h) 1.5 3 42 Craftsmen Time (h) 3 1 24 Profit (Rs) 20 10
Since we have only 42h of machine time and 24h of craftsmen time, we have to maximize x+y, given the following constraints:
$(1): 1.5x + 3y \leq 42$
$(2): 3x + y \leq 24$
$(3): x \leq 0, \; (4): y \leq 0$
$\textbf{Plotting the constraints}:$
Plot the straight lines 1.5x + 3y = 42 and 3x + y = 24
First draw the graph of the line 1.5x + 3y = 42. If x = 0, y = 14 and if y = 0, x = 28. So, this is a straight line between (0,14) and (28,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is $\leq$ 0. So the area associated with this inequality is bounded towards the origin.
Similarly, draw the graph of the line 3x + y = 24. If x = 0, y = 24 and if y =0, x = 8. So, this is a straight line between (0,24) and (8,0).
At (0,0), in the inequality, we have 0 + 0 = 0 which is ≤ 0. So the area associated with this inequality is bounded towards the origin.
$\textbf{Finding the feasible region}$:
We can see that the feasible region is bounded and in the first quadrant.
On solving the equations 1.5x + 3y = 42 and 3x + y = 24, we get,
1.5x + 3 (24-3x) = 42 $\to$1.5x + 72 – 9x = 42 $\to$-7.5x = -30 $\to$x =4 .
If x = 4, y = 24 – 12 = 12.
$\Rightarrow x = 4, y = 12$
Therefore the feasible region has the corner points (0,0), (0,14), (4,12), (8,0) as shown in the figure.
$\textbf{Solving the objective function using the corner point method}$:
The values of Z at the corner points are calculated as follows:

Corner Point

Z = 20x + 10y

O (0,0)

0

A (0,14)

160

E (4,12)

200 (Maximum value)

B (8,0)

140

$\textbf{A) The number of rackets and bats we can make at full capacity are 4 and 12. }$
$\textbf{B) The maximum profit we can make that meet the constraints is 200. }$

edited Apr 17, 2013