Question

A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5760 to invest and has space for at most 20 items. A fan costs him Rs 360 and a sewing machine Rs 240. His expectation is that he can sell a fan at a profit of Rs 22 and a sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy. How should he invest his money in order to maximize his profit ? Formulate this problem mathematically and then solve it.

Answer 1

Let dealer should purchase x fans and y sewing machines.

∴ the cost of x fans = Rs  360x

and the cost of y sewing machines = Rs  240y

∴ According to question,

360x + 240 ≤ 5760

According to place available to have x fans and y sewing machines x + y ≤ 20.

The profit earned by dealer on x fans = Rs 22x

and profit earned on y sewing machines = Rs 18y

∴ To earn maximum profits objective function

Z = 22 x + 18y

Mathematically formulation of LPP is as following :

Maximum Z = 22x + 18y

Subject to the constraints

360x + 240y ≤ 5760

x + y ≤ 20

x ≥ 0, y ≥ 0

Converting the given inequations into equations 360x + 240y = 5760

⇒ 3x + 2y = 48 …..(1)

and x + y = 20 ……(2)

Region represented by 3x + 2y ≤ 48 : 

The line 3x + 2y = 48, meets the coordinate axis at A(16, 0) and B(0, 24).

3x + 2y = 48

x	16	0
y	0	24

A(16,0);B(0,24)

Join the points A and B to obtain the line. 

Clearly (0, 0) stisfies the in equation 3 × 0 + 2 × o = 0 ≤ 48. 

So, the region containing the origin represents the solution set of the in equation.

Region represented by x + y ≤, 20 : 

The line x + y = 20 meets at the point E(20, 0) and F(0, 20).
x + y = 20

x	20	0
y	0	20

C(20, 0); D(0, 20)

Join the points E and F to obtain the line.

Clearly (0,0), satisfies the in equation 0 + 0 ≤ 20. 

So, the region containing the origin represents the solution set of the in equation.

Region represented by x ≥ 0, y ≥ 0 : 

Since every point in the first quadrant satisfies these in equation. 

So, the first quadrant is the region represented by the in equations x ≥ 0 and y ≥ 0.

The shaded region OAED represents the common region of the above in equations. This region is the feasible region of the given LPP.

The coordinates of the comer points of this feasible region are O(0,0), A(16,0), E(8, 12) and D( 0,20) where E( 8,12) is the point of intersection of lines 3x + 2y = 48 and x + y = 20.

The value of objective function on these points are given in the following table:

Point	x-coordinate	y-coordinate	Objective function Z = 22x + 18y
O	0	0	ZO = 22 x 0 + 18 x 0 = 0
A	16	0	ZA = 22 x 16 + 18 x 0 = 352
E	8	12	ZE = 22 x 8 + 18 x 12 = 392
D	0	20	ZD = 22 x 0 + 18 x 20 = 360

From the table it is clear that the objective function has maximum value Rs 392 at point E(8,12) where x = 8 and y = 12.

Hence dealer should purchase 8 fans and 12 sewing machines to get maximum profit Rs 392.