Question

A firm manufactures headache pills in two sizes A and B. Size A contains 2 grams of aspirin 5 grams of bicarbonate and 1 gram of cofeine; size B contains 1 grams of a sprin of 8 grms of bicarbonate and 6 grains of cofeine. It has been found by users that it requires at least 12 grams of aspirin 74 grams of bicarbonate and 24 grams of cofeine for providing immediate effects. Deternfine graphically the least number of pills a patient should have to get immediate relief.

Answer 1

Let the patient should have x pills of size A and y pills of size B.

∴ Maximum number of pills = x + y

According to question,

A type pills have quantity of Aspirin = 2x gramn and B type pills have quantity of Aspirin = 1 y grams

∴ Constraints for quantity of Aspirin

2x + y ≥ 12 grams

Similarly constraints for Bicarbonate

5x + 8 ≥ 74 grams

and constraints for quantity of cofeine

x + 6.6y ≥ 24 grams

∵ x and y are the number of pills.

∴ x ≥ 0 and y ≥ 0

So mathematical formulation of linear programming problem is the following :

Minimize Z = x + y

Subject to the constraints

2x + y ≥ 12

5x + 8 ≥ 74

x + 6y ≥ 24

x ≥ 0, y ≥ 0

Converting the constraints into the equations

2x + y = 12 …..(1)

5x + 8y = 74 …..(2)

x + 6y = 24 …..(3)

Region represented by 2x + y ≥ 12 : 

The line 2x + y = 12 meets the coordinate axis at the points A(6,0) and B(0, 12).

Table for 2x + y = 12

x	6	0
y	0	12

A(6,0); B(0, 12)

Join the point A to 5 to obtain the line. 

Since the origin (0, 0) does not satisfy the in equation 2(0) + (0) = 0 ≥ 12. 

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

Region represented by 5x + 8y ≥ 74 : 

The line 5x + 8y = 74 meets the coordinate axis at the point C(74/5, 0) and D(0, 74/8).

Table for 5x + 8y = 74

x	74/5	0
y	0	74/8

Join the points C and D to obtain the line. 

Since the origin (0,0) does not satisfy the in equation 5(0) + 8(0) ≥ 74. 

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

Region represented by x + 6y ≥ 24 : 

The line x + 6.6y = 24 meets the coordinate axis at the point E(24,0) and F(0,4).

The table for x + 6y = 24

x	24	0
y	0	4

A(24, 0); B(0, 4)

Join the points E and F to obtain the line. 

Since the origin (0, 0) does not satisfy the in equation. 

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

Region represented by x ≥ 0, y ≥ 0 : 

Since every point in first quadrant satisfies the in equations so the first quadrant represents the solution set of these in equations.

The point of intersection of lines 2x +y – 12 and 5x + 8y = 74 is G(2, 8).

The point of intersection of lines 5x + 8y = 74 and x + 6y = 24 is H (126/11, 23/11).

The shaded region BGHE is the common region of the in equations. This is an open feasible region of solution.
The corner points of this open feasible solution region
are B(0, 12), G(2, 8), H(126/11, 23/11) and E(24, 0).

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

From the above table the value of objective function is minimum at the point G(2, 8) where x = 2 and y = 8 and z = 2 + 8 = 10.

Since feasible region of solution set is an open region. 

So the graph of x + y ≤ 10 passes through point G(2,8) which satisfies the in equation z + 8 = 10 so the solution of this LPP is G(2, 8) where x = 2 and y = 8.

Hence the patient should have minimum 2 pills of type A and 8 pills of type B to get immediate relax.