Question
A manufacturer employs three inputs: man-hours, machine-hours and cloth material to manufacture two types of dresses. Type A dress fetches him a profit of Rs. 160/- per piece, while type B, that of Rs. 180/- per piece. The manufacturer has enough man-hours to manufacture 50 pieces of type A or 20 pieces of type B dresses per day, while the machine-hours he possesses suffice only for 36 pieces of type A or for 24 pieces of type B dresses. Formulate the linear programming model and solve it graphically.
Answer :
Word Count : 878
Let $x$ denote the number of type A dresses produced per day and $y$ the number of type B dresses produced per day. The manufacturer’s profit is Rs. 160 per A and Rs. 180 per B, so the objective is to maximize the total profit $$ Z \;=\; 160x \;+\; 180y. $$ The problem statement gives capacity information in terms of equivalent production: the available man-hours are enough to make either 50 pieces of A or 20 pieces of B in a day; similarly, available machine-hours suffice for either 36 pieces of A or 24 pieces of B. Translating these into linear constraints is easiest by observing that using up the entire available man-hours by producing only A means $x=50$, and producing only B means $y=20$. If both types are produced, the fraction of the man-hour resource used equals $x/50 + y/20$, which must not exceed ____ _________ ______ _________ ____ ________.
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