Question

A diet of a sick person must contain at least 4000 units of vitamins, 50 units of minerals and 1400 of calories. Two foods A and B are available at a costs Rs 4 and Rs 3 per unit respectively. If one unit of A contains 200 unit of vitamin, 1 unit of mineral and 40 calories and one unit of B contains 100 unit of vitamin, 2 unit of minerals and 40 calories, find what combination of foods should be used to have the least cost ?

Answer 1

Let x unit of food A and y unit of food B are taken in combination of food.

∴ According to question,

Objective function to get minimum cost of combination is

Z = 4x + 3y

Constraints in problem :

For vitamin 200x + 100y ≥ 4000

For mineral x + 2y ≥ 50

and for Calorier 40x + 40y ≥ 1400

∵ x and y are quantity.

∴ x ≥ 0 andy ≥ 0

Converting the given in equations into the equations

200x + 100y = 4000

⇒ 2x + y = 40 …..(1)

x + 2y = 50 …..(2)

40x + 40y = 1400

⇒ x + y = 35 …..(3)

x= 0 …..(4)

y = 0 …..(5)

Region represented by 200x + 100y ≥ 4000 : 

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

2x + y = 40

x	20	0
y	0	40

A(20, 0); B(0, 40)

Join points A and B to obtain the line. 

Since the origin (0, 0) does not satisfy the given in equation 2(0) + 0 = 0 ≥ 40. 

So the region opposite to the origin represents the solution set of the in equations.

Region represented by x + 2y ≥ 50 : 

The line x + 2y = 50 meets the coordinate axis on the points C(50, 0) and D(0, 25).
x + 2y = 50

x	50	0
y	0	25

C(50, 0);D(0, 25)

Join points C(50,0) and D(0,25) to obtain the line.

Since the origin (0, 0) does not satisfy the given in equation 0 + 2(0) = 0 ≥ 50. 

So the solution set of the in equation in opposite to the origin.

Region represented by x + y ≥ 35 : 

The line x + y = 35 meets the coordinate axis are C(35, 0) and D(0, 35).

x + y = 35

x	35	0
y	0	35

E(35, 0); F(0, 35)

Join the points E and F to obtain the line. 

Clearly (0, 0) does not satisfy the in equation x + y ≥ 35.

So, the region opposite to the origin represents the solution set of this in equation.

Region represented by x ≥ 0 and y ≥ 0 : 

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

So the first quadrant represents the solution set of the in equations x ≥ 0 and y ≥ 0.

The coordinates of the point of intersection of lines 2x + y = 40and x + 2y = 50 are x = 10 and y = 20.

The coordinates of the point of intersection of lines x + 2y = 50 and x + y = 35 are x = 20 and y = 15 and the coordinates of the point of intersection of lines 2x + y = 40 are x + y = 35 and x = 5 and y = 30.

The shaded region CHJB represents the common region of the in equations. This is an open feasible solution region whose coordinates of the corner points are C(50, 0), H(20, 15), J(5, 30) and 5(0, 40).

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

Point	x-coordinate	y-coordinate	Objective function Z = 4x + 3y
C	50	0	ZC = 4(50) + 3(0) = 200
H	15	15	ZH = 4(20)+ 3(15) = 125
J	5	30	ZJ = 4(5) + 3(30) = 110
B	0	40	ZB = 4(0) + 3(40) = 120

From table the value of Z is minimum at point J(5, 30) which is 110.

∵ Feasible region is open, so on plotting the graph of line 4x + 3y = 110 which is an open feasible semi region which have a common point J(5,30).

Hence the proper solution of LPP is point J(5, 30) to get the minimum price of combination.

The minimum price of combination is Rs 110 and combination has food A = 5 unit and food B = 30 unit.