Q. 145.0( 1 Vote )

A factory uses th

Answer :

Let number of product I and product II are x and y respectively.


Since, profits on each product I and II are 2 and 3 monetary unit. So, profits on x number of Product I and y number of Product II are 2x and 3y respectively.


Let Z denotes total output daily, so,


Z = 2x + 3y


Since, each I and II requires 2 and 4 units of resources A. So, x units of product I and y units of product II requires 2x and 4y minutes respectively. But, maximum available quantity of resources A is 20 units.


So,


2x + 4y 20


x + 2y 10 {First Constraint}


Since, each I and II requires 2 and 2 units of resources B. So, x units of product I and y units of product II requires 2x and 2y minutes respectively. But, maximum available quantity of resources A is 12 units.


So,


2x + 2y 12


x + y 6 {Second Constraint}


Since, each units of product I requires 4 units of resources C. It is not required by product II. So, x units of product I require 4x units of resource C. But, maximum available quantity of resources C is 16 units.


So,


4x 16


x 4 {Third Constraint}


Hence mathematical formulation of LPP is,


Max Z = 2x + 3y


Subject to constraints,


x + 2y 10


x + y 6


x 4


x, y 0 [ Since production for I and II can not be less than zero]


Region represented by x + 2y 10: The line x + 2y = 10 meets the axes at A(10,0), B(0,5) respectively.


Region containing the origin represents x + 2y 10 as origin satisfies x + 2y 10.


Region represented by x + y 6: The line x + y = 6 meets the axes at C(6,0), D(0,6) respectively. Region containing the origin represents x + y 6 as origin satisfies x + y 6


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


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The corner points are O(0,0), B(0,5), G(2,4), F(4,2), and E(4,0).


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



The maximum value of Z is 16 which is attained at G (12,4).


Thus, the maximum profit is 16 monetary units obtained when 2 units of first product and 4 units of second product were manufactured.


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