Tailoring Profits
In the month of January, a tailoring company has 1000 yards of fabric and 1800 man-hours to produce shirts and suits. It takes 1 yard of fabric with 2 hours to time to make a shirt, and 2 yards of fabric with 3 hours of time to make a suit.
Given the costs of inputs, they can make $12 selling a shirt and $20 selling a suit. What is the maximum profit of the tailoring company in January?
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1 solution
Let x be the total number of shirts & y be the number of suits produced. We now formulate the following linear program model:
M A X 1 2 x + 2 0 y
SUBJECT TO:
x + 2 y ≤ 1 0 0 0
2 x + 3 y ≤ 1 8 0 0
x , y ≥ 0
where the first and second constraints represent the necessary fabric and manhours respectively. If we plot the above constraints in the x y − plane, we obtain the feasible region containing the vertices ( x , y ) = ( 0 , 0 ) ; ( 0 , 5 0 0 ) ; ( 6 0 0 , 2 0 0 ) ; ( 9 0 0 , 0 ) . The profit is maximized at ( x , y ) = ( 6 0 0 , 2 0 0 ) , or ( $ 1 2 ) ( 6 0 0 ) + ( $ 2 0 ) ( 2 0 0 ) = $ 1 1 , 2 0 0 after checking each critical vertex.