how do i use linear programming to solve this problem and find the constraints and variables.

Question

You make custom T-shirts with inspirational sayings on them. You just found out about a community flea market sale that is starting tomorrow. You have just 8 hours to prepare product for this sale. You start with a plain white T-shirt. This is your most popular color. But you can dye the white T-shirt blue, yellow, or red—but only one shirt at a time. Your current inventory is 50 white T-shirts and you have enough dye to make 12 red shirts, 10 yellow shirts, and 15 blue shirts. You take the various color shirts to the sale and then stencil on an inspirational saying—whatever the customer wants up to 35 characters.

Based on experience, you know that at a sale like this, you will be able to sell all 50 shirts. Use the data in the table below to determine how many shirts of each color you should bring to the sale to maximize your profits.

ColorTime Required to Dye (Minutes)Your Cost of Materials Including Dye and StencilYou Have Enough Dye to Make This Many ShirtsSelling PriceProfit
White	0 min	$5 cost of mat	50 enough dye to make	$12 selling price	$7 profit
Blue	20 min	$7 cost of mat	15 enough dye to make	$15 selling price	$8 profit
Yellow	20 min	$7 cost of mat	10 enough dye to make	$15 selling price	$8 profit
Red	40 min	$10 cost of mat	12 enough dye to make	$16 selling price	$6 profit

Answer 1

You don't need linear programming to solve this problem, although you can set it up for that purpose. The object is to maximize profits. Blue and yellow shirts yield $8 profit per shirt.

So make as many of them as you have time for. They both take 20 minutes per shirt. 8 hours / 20 minutes per shirt = 8 hours * 60 minutes per hour / 20 minutes per shirt = 8 * 3 = 24 shirts. Any combination of 24 yellow and blue shirts will work as long as there are no more than 15 blue shirts and no more than 10 yellow shirts.

15 blue and 9 yellow shirts is one such combination. The only other combination possible is 14 blue shirts and 10 yellow shirts; this leads to the same amount of profit.

Red shirts take 40 minutes per shirt and provide only $6 profit per shirt. White shirts take no time but provide $7 profit per shirt, so there is no point in making any red shirts. 50 - 24 = 26 shirts can be made, so make them all from white material.

24 shirts * $8 profit per shirt = $192 profit.

26 shirts * $7 profit per shirt = $182 profit.

This is a total profit of $374 for the 50 shirts detailed above.

The same conclusion can be developed by using linear programming, but it is not necessary.