Question

Model Problem

Coffee vs. Tea
A café by Lindsay's house sells coffee and tea. They are trying to make the most amount of money
possible. In other words they are trying to maximize profit. They asked me to look at their sales
receipts and figure out what they should do. But they had a few limitations on what could be done.
●
Constraint #1: The Baristas (the people who makes the coffee) can only reasonably serve 90 cups
per shift. It can be 90 cups of coffee, 90 cups of tea or a mixture of the two.
Constraint #2: Coffee costs $3 per cup and Tea costs $2. In order to stay on track to pay their
bills, the owners would like to sell at least $120 worth of beverages every shift.
Constraint #3: They have a limited supply of coffee and tea that gets delivered every day. If they
don't want to sell out (And annoy customers) they can sell no more than 70 cups of tea per shift.
Constraint #4: Similar to the last constraint, they can only sell 60 cups of coffee per shift before
they sell out.

Answer 1

To help the café maximize its profit while adhering to the given constraints, we can set up a linear programming problem. Let's define some variables:

Let:

- \(C\) be the number of cups of coffee sold per shift.

- \(T\) be the number of cups of tea sold per shift.

Now, we can establish the objective function and constraints:

Objective Function (to maximize profit):

\[P = 3C + 2T\]

This objective function represents the profit generated from selling \(C\) cups of coffee at $3 each and \(T\) cups of tea at $2 each.

Constraints:

1. Barista constraint: \(C + T \leq 90\)

- The total number of cups of coffee and tea sold should not exceed 90.

2. Revenue constraint: \(3C + 2T \geq 120\)

- The total revenue (profit) should be at least $120.

3. Tea supply constraint: \(T \leq 70\)

- The café can sell at most 70 cups of tea per shift.

4. Coffee supply constraint: \(C \leq 60\)

- The café can sell at most 60 cups of coffee per shift.

5. Non-negativity constraint: \(C, T \geq 0\)

- The number of cups sold cannot be negative.

Now, you can use linear programming techniques to solve this problem and find the values of \(C\) and \(T\) that maximize the profit while satisfying all the constraints. The solution will provide the optimal combination of coffee and tea sales to achieve the highest profit.