You and your roommate have timed the necessary physical operations. The ﬁrst physical production step is to wash out the mixing bowl from the previous batch, add all of the ingredients, and mix them in your food processor. The mixing bowls hold ingredients for up to three dozen cookies. You then dish up the cookies, one dozen at a time, onto a cookie tray. These activities take six minutes for the washing and mixing steps, regardless of how many cookies are being made in the batch. That is, to mix enough dough and ingredients for three dozen cookies takes the same six minutes as one dozen cookies. However, dishing up the cookies onto the tray takes two minutes per tray. The next step, performed by your roommate, is to put the cookies in the oven and set the thermostat and timer, which takes about one minute. The cookies bake for the next 9 minutes. So total baking time is 10 minutes, during the ﬁrst minute of which your roommate is busy setting the oven. Because the oven only holds one tray, a second dozen takes an additional 10 minutes to bake.

Question

You and your roommate have timed the necessary physical operations. The ﬁrst physical production step is to wash out the mixing bowl from the previous batch, add all of the ingredients, and mix them in your food processor. The mixing bowls hold ingredients for up to three dozen cookies. You then dish up the cookies, one dozen at a time, onto a cookie tray. These activities take six minutes for the washing and mixing steps, regardless of how many cookies are being made in the batch. That is, to mix enough dough and ingredients for three dozen cookies takes the same six minutes as one dozen cookies. However, dishing up the cookies onto the tray takes two minutes per tray. The next step, performed by your roommate, is to put the cookies in the oven and set the thermostat and timer, which takes about one minute. The cookies bake for the next 9 minutes. So total baking time is 10 minutes, during the ﬁrst minute of which your roommate is busy setting the oven. Because the oven only holds one tray, a second dozen takes an additional 10 minutes to bake.

Your roommate also performs the last steps of the process by ﬁrst removing the cookies from the oven and putting them aside to cool for ﬁve minutes, then carefully packing them in a box and accepting payment. Removing the cookies from the oven takes only a negligible amount of time, but it must be done promptly. It takes two minutes to pack each dozen and about one minute to accept payment for the order.
Baking trays hold exactly one dozen cookies, you will produce and sell cookies by the dozen. Should you give any discount for people who order two dozen cookies, three dozen cookies, or more? If so, how much? Will it take you any longer to fill a two-dozen order than a one-dozen order?

Note: Here are my answer.
Since the packaging and ingredient cost are the same it doesn’t matter if one buys 1,2 or 3 dozen what differs is the time.

I don’t know how to calculate the discount for the two dozen cookies, three dozen cookies, or more.

Answer 1

Also, would the number of trays and mixers needed be just one mixer since it is used to make 3 dozen cookies and afterward it is idle; 3 trays for the 3 dozen cookies the one mixer able to provide the dough for?

Answer 2

In addition, how would one calculate the the amount I would be willing to pay to rent an additional oven?

Answer 3

To determine whether a discount should be given for larger orders, you need to consider the time it takes to produce each order. Let's break down the time it takes to produce one dozen and two dozen cookies.

For one dozen cookies:
- Washing and mixing: 6 minutes
- Dishing up onto a tray: 2 minutes
- Roommate's tasks (setting oven, removing cookies, packing, accepting payment): 1 minute + 5 minutes + 2 minutes + 1 minute = 9 minutes
- Total time for one dozen: 6 + 2 + 9 = 17 minutes

For two dozen cookies:
- Washing and mixing: 6 minutes (same as one dozen)
- Dishing up onto a tray: 4 minutes (two trays, 2 minutes per tray)
- Roommate's tasks: 9 minutes (same as one dozen)
- Total time for two dozen: 6 + 4 + 9 = 19 minutes

Comparing the two, it takes 17 minutes to produce one dozen cookies and 19 minutes to produce two dozen cookies. Therefore, it takes slightly longer to fill a two-dozen order than a one-dozen order.

Now, let's consider whether a discount should be given for larger orders. Since the packaging and ingredient costs are the same for each dozen, the main factor is the labor and time involved.

If you decide to offer a discount, it should reflect the additional time and effort required to produce the larger order. For example, you could calculate the average production time per dozen and then add a certain percentage or fixed amount as the discount for each additional dozen. This discount would compensate for the increased time and effort.

However, without specific information on your production costs and target profit margins, I cannot provide an exact discount amount. It would be best to analyze your production costs and desired profit margin and consider market factors to determine an appropriate discount for larger orders.