A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 h of machine time and 3 h of machine time in its making while a cricket bat takes 3 hours of machine time and 1 h of craftsman's time. In a day, the factory has the availability of not more than 42 h of machine time and 24 h of craftsman's time.

(a) What number of rackets and bats must be made, if the factory is to work at full capacity?

(b) If the profits on rackets and on bats are Rs. 20 and Rs. 10 respectively, find the maximum profit of the factory when it works at full capacity.

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Solution

Let the number of rackets and the number of cricket bats to be made in a day be x and y respectively.

Construct the following table :

$\begin{matrix} I t e m & N u m b e r & M a c h i n e t i m e & C r a i t s m a n^{'} s & P r o f i t (i n R s .) (i n h) & t i m e (i n h) T e n n i s r a c k e t s & x & 1.5 x & 3 x & 20 x C r i c k e t b a t s & y & 3 y & 1 y & 10 y T o t a l & x + y & 1.5 x + 3 y & 3 x + y & 20 x + 10 y A v a i l a b i l i t y & 42 & 24 \end{matrix}$

The machine time is not available for more than 42 h. $∴ 15 x + 3 y \leq 42$

The craftman's time is not available for more than 24 h. $∴ 3 x + y \leq 24$

The profit on rackets is Rs. 20 and on bats is Rs. 10.

$∴$ Maximum Z = 20x + 10y .........(i)

Subject to constraints $1.5 x + 3 y \leq 42$ ......(ii)

$3 x + y \leq 24$ ...............(iii)

$x \geq 0, y \geq 0$ .........(iv)

Firstly, draw the graph of the line 10.5x + 3y = 42

$\begin{matrix} x & 0 & 28 y & 14 & 0 \end{matrix}$

Putting (0, 0) in the inequality 1.5x + 3y $\leq$ 42, we have

$1.5 \times 0 + 3 \times 0 \leq 42 \Rightarrow 0 \leq 42$ (which is true)

So, the half plane is towards the origin. Since, x, y $\geq$ 0

So, the feasible region lies in the first quadrant.

Secondly, draw the graph of the line 3x + y = 24

$\begin{matrix} x & 0 & 8 y & 24 & 0 \end{matrix}$

PUtting (0, 0) in the inequality 3x + y $\leq$ 24,

we have $3 \times 0 + 0 \leq 24 \Rightarrow 0 \leq 24$ (which is true)

So, the half plane is towards the origin.

On solving equations 1.5x + 3y = 42 and 3x + y = 24, we get B (4, 12).

$∴$ Feasible region is OABCO.

The corner points of the feasible region are O(0, 0), A(8, 0), B(4, 12) and C(0, 14). The values of Z at these points are as follows:

$\begin{matrix} C o r n e r p o i n t & Z = 20 x + 10 y O (0, 0) & 0 A (8, 0) & 160 B (4, 12) & 200 \to M a x i m u m C (0, 14) & 140 \end{matrix}$

Thus, the maximum profit of the factory when it works to its full capacity is Rs. 200.

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A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

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