Q10 of 118 Page 31

A manufacturer of Furniture makes two products : chairs and tables. Processing of these products is done on two machines A and B. A chair requires 2 hrs on machine A and 6 hrs on machine B. A table requires 4 hrs on machine A and 2 hrs on machine B . There are 16 hrs of time per day available on machine A and 30 hrs on machine B. profit gained by the manufacturer from a chair and a table is ₹ 3 and ₹ 5 respectively. Find with the help of graph what should be the daily production of each of the two products so as to maximize his profit.

Let daily production of chairs and tables be x and y respectively.


Since, profits of each chair and table is Rs. 3 and Rs. 5 respectively. So, profits on x number of type A and y number of type B are 3x and 5y respectively.


Let Z denotes total output daily, so,


Z = 3x + 5y


Since, each chair and table requires 2 hrs and 3 hrs on machine A respectively. So, x number of chair and y number of table require 2x and 4y hrs on machine A respectively. But,


Total time available on Machine A is 16 hours. So,


2x + 3y 16


x + 2y 8 {First Constraint}


Since, each chair and table requires 6 hrs and 2 hrs on machine B respectively. So, x number of chair and y number of table require 6x and 2y hrs on machine B respectively. But,


Total time available on Machine B is 30 hours. So,


6x + 2y 30


3x + y 15 {Second Constraint}


Hence mathematical formulation of the given LPP is,


Max Z = 3x + 5y


Subject to constraints,


x + 2y 8


3x + y 15


x,y 0 [Since production of chairs and tables can not be less than zero]


Region x + 2y 8: line x + 2y = 8 meets the axes at A(8,0), B(0,4) respectively.


Region containing the origin represents x + 2y 8


as origin satisfies x + 2y 8.


Region 3x + y 15: line 3x + y = 15 meets the axes at C(5,0), D(0,15) respectively.


Region containing the origin represents 3x + y 15 as origin satisfies 3x + y 15


Region x,y 0: it represents the first quadrant.


10.jpg


The corner points are O(0,0), B(0,4), E(), and C(5,0).


The values of Z at these corner points are as follows,



The maximum value of Z is 22.2 which is attained at E().


Thus the maximum profit of Rs 22.2 when units of chair and units of table are produced.


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