Answer :

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 660 = 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 660 = 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 660 = 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.

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