Question

One kind of cake requires 300 kg of flour and 15 g of fat and another kind of cake requires 150 g of flour and 30 g of fat, find the maximum number of cakes which can be made from 7.5 kg of flour and 600 g of fat assuming that there is no shortage of the other ingredients used in making the cakes. Formulate the LPP solve the problem by graphical method.

Answer 1

Let x cakes are made of one kind and y cakes are made of another kind. 

Therefore objective function of maximum limit = x + y.

⇒ Z = x + y

Firstly according to question, there is 300x gm flour in one kind and 150y gm in another kind.

∴ 300x + 150y ≤ 7500 gm

Secondly according to question, there is 15x gm fats in one kind of cake and 30y gm fats is in another kind.

∴ 15x + 30y ≤ 600 gm

∵ The number of cakes can never be negative so x ≥ 0 and y ≥ 0.

Therefore mathematically formulation of Linear

Programming Problem is the following :

Maximum Z = x + y

Subject to the constraints

300x + 150y ≤ 7500

15x + 30y ≤ 600

x ≥ 0, y ≥ 0

Converting the given inequations into equations

300x + 150y ≤ 7500

⇒ 2x + y = 50 …..(1)

and 15x + 30y = 600

⇒ x + 2y = 40 …..(2)

Region represented by 2x + y ≤ 50 : 

The line 2x + y = 50, meets the coordinate axis at A(25, 0) and B(0, 50).
2x + y = 50

x	25	0
y	0	50

A(25, 0); B(0, 50)

Join the points A and B to obtain the line. 

Clearly, (0,0) satisfies the in equation 2 × 0 + 0 ≤ 50. 

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

Region represented by x + 2y ≤ 40 : 

The line x + 2y = 40 meets the coordinate axis at C(40,0) and D(0, 20).
x + 2y = 40

x	40	0
y	0	20

C(40, 0); D(0, 20)

Join the points C and D to obtain the line. 

Clearly (0, 0) satisfies the in equation (0 + 2 × 0) < 40. 

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

Region represented by x ≥ 0, y > ≥ : 

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

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 the shaded feasible region are O(0,0), C(25,0), E(20,10) and B(0, 20).
The point E has been obtained by solving the equations of the corresponding intersecting lines, simultaneously.

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

Point	x-coordinate	y-coordinate	Objective function Z = x + y
O	0	0	Z = x + y
A	25	0	ZA = 25 + 0 = 25
E	20	10	ZE = 20 + 10 = 30
D	0	20	ZD= 0 + 20 = 20

It is clear from the table that objective function has maximum value at point E(20, 10).

Hence there are 20 cakes of one kind and 10 cakes of another kind. Maximum number of cakes = 20 + 10 = 30.