A manufacturer makes two types A and B of tea - cups. Three machines are needed for the manufacture and the time in minutes required for each cup on the machines is given below :
Each machine is available for a maximum of 6 hours per day. If the profit on each cup A is 75 paise and that on each cup B is 50 paise, show that 15 tea - cups of type A and 30 of type B should be manufactured in a day to get the maximum profit.
Let the required number of tea cups of Type A and B are x and y respectively.
Since, the profit on each cup A is 75 paise and that on each cup B is 50 paise. So, the profit on x tea cup of type A and y tea cup of type B are 75x and 50y respectively.
Let total profit on tea cups be Z, so
Z = 75x + 50y
Since, each tea cup of type A and B require to work machine I for 12 and 6 minutes respectively so, x tea cups of Type A and y tea cups of Type B require to work on machine I for 12x and 6y minutes respectively.
Total time available on machine I is 6
60 = 360 minutes. So,
12x + 6y 
360 {First Constraint}
Since, each tea cup of type A and B require to work machine II for 18 and 0 minutes respectively so, x tea cups of Type A and y tea cups of Type B require to work on machine IIII for 18x and 0y minutes respectively.
Total time available on machine I is 6
60 = 360 minutes. So,
18x + 0y 
360
x 
20 {Second Constraint}
Since, each tea cup of type A and B require to work machine III for 6 and 9 minutes respectively so, x tea cups of Type A and y tea cups of Type B require to work on machine I for 6x and 9y minutes respectively.
Total time available on machine I is 6
60 = 360 minutes. So,
6x + 9y 
360 {Third Constraint}
Hence mathematical formulation of LPP is,
Max Z = 75x + 50y
subject to constraints,
12x + 6y 
360
x 
20
6x + 9y 
360
x,y 
0 [Since production of tea cups can not be less than zero]
Region 12x + 6y 
360: line 12x + 6y = 360 meets axes at A(30,0), B(0,60) respectively. Region containing origin represents 12x + 6y 
360 as (0,0) satisfies 12x + 6y 
360
Region x 
20: line x = 20 is parallel to y axis and meets x - axes at C(20,0). Region containing origin represents x 
20
as (0,0) satisfies x 
20.
Region 6x + 9y 
360: line 6x + 9y = 360 meets axes at E(60,0), F(0,40) respectively. Region containing origin represents 6x + 9y 
360 as (0,0) satisfies 6x + 9y 
360.
Region x,y 
0: it represents the first quadrant.
The shaded region is the feasible region determined by the constraints,
12x + 6y 
360
x 
20
6x + 9y 
360
x,y 
0
The corner points are F(0,40), G(15,30), H(20,20), C(20,0).
The values of Z at these corner points are as follows
Here Z is maximum at G(15,30).
Therefore, 15 teacups of Type A and 30 tea cups of Type B are needed to maximize the profit.