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Sometimes we have a problem where our choices are restricted by some linear inequalities, and we want to maximize or minimize some other linear expression within those restrictions. The process of setting up and solving this kind of problem is called linear optimization.
A baker wants to make brownies and chocolate chip cookies, but he’s running low on sugar and butter. He wants to know how many cookies and brownies he should bake in order to make the most profit. To solve his problem, he uses linear optimization.
He decides to let $b$ be the number of batches of brownies he bakes, and $c$ be the number of batches of chocolate chip cookies. Each batch of brownies uses 4 cups of sugar. Each batch of cookies uses 2 cups of sugar. So, the amount of sugar he’ll use is $4b+2c$. The baker only has 520 cups of sugar, so he knows that $4b+2c≤520$.
Click . The black line is a graph of a fixed profit level, $P=2b+1.8c$. $P$ is set to 40, so if the baker makes brownies and cookies corresponding to any point on the black line, he will make \$40 in profit. Use the slider to make $P$, the profit, larger and notice how the line moves.
A coffee distributor wants to make a blend of coffee using a mix of premium and regular 10-pound bags of coffee beans. She has high standards for the blend she wants to make. She also wants to minimize her cost while keeping her high standards, so she uses linear optimization. She lets $p$ be the number of bags of premium beans and $r$ be the number of bags of regular beans.
Each 10-pound bag of premium coffee contains 6 lbs. of Colombian coffee beans. Each 10-pound bag of regular coffee contains 3 lbs. of Colombian coffee beans. So, the amount of Colombian beans in her blend is $6p+3r$. The distributor wants her blend to contain at least 360 pounds of Colombian beans, so $6p+3r≥360$.
A company that makes bookcases and desks out of oak and maple has a limited amount of each kind of wood. Also, the company’s employees are available to work a set number of hours every week. This company wants to know how many bookcases and desks it should make to maximize its profits. They decide to use linear optimization to solve this problem.
First, they define their variables. They let $b$ be the number of bookcases that are made in a week and $d$ be the number of desks that are made. Each bookcase uses 2 board-feet of oak and each desk uses 1 board-foot of oak. So, the amount of oak that will be used is $2b+d$. Every week they get a supply of 1000 board-feet of oak, so $2b+d≤1000$.