In this project, you will use the techniques we have learned to maximize profit while minimizing your cost.

You are the new plant manager of The Great American Cookie Company's main production facility. You have a manufacturing plant that produces two varieties of cookies, and sadly that plant has been losing money for the company. It is your task to make some changes in their production schedule so they can start to turn a profit. Here is some information about your plant:

The plant that you manage produces two types of cookies, chocolate chip and oatmeal raisin. The chocolate chip cookies sell for $$70$ per case, and the oatmeal raisin cookies sell for $$65$ per case.

There are three machines in the plant that can make either type of cookie. To keep things simple, you have labeled each of these machines A$,$ B$,$ and C$.$ Each of these machines has a supervisor who is in charge of making sure that the production schedules are met. You have asked each of them to deliver you a report on the amount of time it takes to make a case of each kind of cookie, and the cost involved per hour.

The plant is open for $18$ hours every day. For the first hour, maintenance crews set up the machines for use. For the last hour, they clean the machines after use and prepare them for the next day, so the machines are only operational for $16$ hours a day.

Machine A$\text{'}$s supervisor says that it takes $$ of an hour to make a case of chocolate chip, and $$ of an hour to make a case of oatmeal raisin.

Machine B$\text{'}$s supervisor says that it takes $(1)/(3)$ hour to make a case of chocolate chip, and $1/6$ of an hour to make a case of oatmeal raisin.

Machine C$\text{'}$s supervisor says that it takes $1/7$ of an hour to make a case of chocolate chip, and $2/7$ of an hour to make a case of oatmeal raisin.

PART ONE:

Write and solve a linear programming problem which tells the number of cases of each type of cookie that maximizes your sales. Take the information above about your machines and organize a table, if necessary. Write your objective function and your constraints. $($HINT: Let x be the number of cases of chocolate chip, and y be the number of cases of oatmeal raisin; also keep in mind that you can't produce a negative number of cases$)$ Be sure to identify the corner points of the feasible region and compare each point with the objective function to determine the maximum sales.

Objective Function:

Constraints:

Picture of your graph $($Importing from Desmos or taking a screenshot or picture of your work on paper is OK:

Vertices $($take a picture of your work showing how you found the vertices and put it here$)$:

PART TWO:

The supervisor for each machine have come back to you with a report on the cost of running each machine per hour; Machine A costs $$70$ per hour to operate, Machine B costs $$90$ per hour to operate, and Machine C costs $$120$ per hour to operate. The setup and cleanup crews at the beginning and end of each day cost the company $$300.$

Answer the questions below by typing in your answers below the question.

Which machine is the most cost$-$effective machine to produce a case of chocolate chip cookies? Which machine is the most cost$-$effective to produce a case of oatmeal raisin cookies? Show all of your work that leads to your conclusion.

Make a table that shows the maximum number of cases of each kind of cookie that each machine can make in one day.

Take your answer to Part One and create a production schedule$-$ that is$,$ assign each line the task of producing enough cookies to meet your optimum sales amount. You can assign more than one type of cookie to the same machine $-$ but keep in mind you will have one hour of down time so the machine can be cleaned to produce the other type of cookie $($nobody wants raisins when they are expecting chocolate chips$).$ If you assign and hour of $$down time$$ to switch between recipes, you won't have to spend the hourly cost of running the machine, but it will cost $$200$ in labor and maintenance to clean the machine before making the other type of cookie. Additionally, you will no longer be able to use that machine during that hour!

Find the amount of PROFIT you will make under this production schedule. Remember, profit is sales $($your answer to Part One$)$ minus expenses $($the costs of running or$/$and cleaning your machines$)..$

The CEO has come to you with the news that a famous TV Chef will be doing commercials for your chocolate chip cookies. The CEO wants to know how many more cases of chocolate chip cookies you can produce in a day $-$ and how much more PROFIT the company will make.