Q7 of 118 Page 31

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 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.


7.jpg


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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