Tag Archives: cheese

on cheese and industrial engineering

At the end of last semester, I took my graduate level course on stochastic processes for a tour of the Babcock Dairy Hall, a campus dairy facility that makes numerous dairy products and has an observation deck for viewing.  Cheese inspired my midterm exam, and we wanted to learn more about cheese and industrial engineering.

I cheese Wisconsin. Um, I might have bought this shirt for members of my family.

It turns out that making cheese draws upon industrial engineering methodologies in a big way.

Main challenge: variable prices and variable demand

The price of milk is variable. Cheese makers buy milk now, but the selling price after the cheese has aged is based on future milk prices. So cheese makers must forecast selling prices and work their way backward to schedule production now (I have more on cheese forecasting later, too). Cheese ages from months to years, so they look fairly far into the future. To complicate matters, they do not know the price of milk now. They are billed a month after they purchase and then must correct their forecast error.

It was somewhat surprising that the decisions are made quickly. The milk must be processed within 24 hours of when it is purchased. This rules out the possibility to stockpile milk when it is cheap to address rising forecasts, spikes in demand, etc.

The cost of milk isn’t the only cost. The other (and only other) major cost is labor cost. At a small scale cheese plant like that at UW-Madison, labor represents a larger share of cheese cost than what is typical. This is because the UW’s cheese plant is flexible and non-specialized, both of which hinder the use of automation to keep costs down.

Matching supply and demand

Another interesting challenge is that there is a craft to matching supply and demand. Demand for cheese is very cyclic and somewhat predictable from what I’ve been told. Still, the forecasting process is not perfect. There is a window for selling cheese that is approximately 3-6 months. After this point, the cheese is not as fresh. The Babcock Dairy Hall sometimes sells cheese at 20% off to increase demand and reduce their inventory of cheese that is getting close to the end of its freshness window. I got the impression that other pricing strategies was an area of interest.

Finding the right mix of cheese

There are two main types of constraints for scheduling cheese production. The first is scheduling how cheese is made. The cheese makers at the Babcock Dairy Hall typically make one type of cheese per day. Some types of cheese like cheddar are so popular that they might be made every week. The second constraint is scheduling how cheese is stored. The huge refrigerator for aging cheese is almost always near its capacity. Different types of cheese are aged at different times. Take cheddar for example. Mild cheddar is aged three months whereas extra sharp is aged for two years. The recipes are different, so the cheese makers do not use the same batch of cheddar to make mild and extra sharp simply by aging it longer.  Storing too much extra sharp cheddar would not leave enough room for havarti or Colby. This reminded me of a linear programming problem I once saw about wine production, where a wine maker had to use a limited number of barrels to make the right mix of red and white wine. And yes, I will certainly ask a question about finding the optimal cheese mix when I teach introductory linear programming.

Everything works perfectly until you experiment with new cheese

If a cheese is popular and sells out quickly, it is not possible to ramp up production and instantly meet demand, as can be done for ice cream, which is made one day and eaten the next. The Dutch Kase cheese ended up extremely popular and selling out quickly. The problem was that it takes 18 months to age, and therefore, customers were irate as they waited for cheese. Forecasting the demand for new cheese is close to impossible. I’m not sure what can be done here. Everyone loving a new type of cheese is a good problem to have. The Babcock Dairy Plant experiments with new products all the time. Lately, they have been pushing lactose free ice cream (yeah!!), which I hope they continue for people like me who are lactose challenged.

Other issues are scheduling cleaning (a minimum of 2.5 hours per day), quality control tests, and taste tests (expert tasters cannot taste too much cheese in one sitting or it all tastes the same).

From linear programming to forecasting and pricing and manufacturing, industrial engineering is important for making cheese.

On Wisconsin! Three types of cheese all invented in the state of Wisconsin.

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my stochastic processes midterm: Wisconsin edition

Based on the curiosity over my cheese-related exam question on twitter, I have decided to post the midterm for my graduate level course on stochastic processes. My favorite questions are #1 and #2. I should note that #1 was inspired by actual leftover cheese that is packaged and sold at a discount at the Babock Dairy Store on campus (picture is below). If there is enough extra leftover cheese, it is poured into a bag, leading to cheese that has layers like an onion. It is apparently not cost-effective to repress the leftover cheese into a smooth brick of cheese. As someone who didn’t grow up wearing a foam cheese hat, I find that cheese production, quality control, and inventory is the right avenue for me to learn about cheese.

The Midterm

1. The dairy plant in Babcock Hall makes one batch of cheese six days per week. The amount of cheese (in pounds) left over after each batch is distributed according to an exponential distribution with parameter 1. Cheese production on each day is independent. The Babcock Dairy store will package and sell any leftover cheese in a batch (i.e., a day) that is more than 1 pound—a “cheese factory second.” Let Ei be the event that there is more than 1 pound of cheese available on day i, with i = 1, 2, 3, 4, 5, 6.

Therefore, Ei is a random variable:
Ei = 1 if there are cheese factory seconds for the type of cheese produced on day i and 0 otherwise (i.e., ignore how much cheese is leftover – we are instead interested in the binary outcome of whether cheese is leftover).

E = the total number of types of cheese factory seconds across the 6 day week.

a) Define E in terms of Ei.
b) What is the sample space for E?
c) What is the probability that there is at least a pound of leftover cheese on day 1?
d) What is the probability that four or more days in the week produce cheese factory seconds?

2. The Lightsaber Manufacturing Company (LMC) operates their manufacturing plant in a galaxy far, far away. They need to decide to how many lightsabers to stock for the next Jedi Convention, where they will sell lightsabers to Jedi apprentices. Due to the expense associated with interstellar travel, LMC will discard the unsold lightsabers after the convention. Lightsabers cost 413 galactic credits to produce and are sold for 795 galactic credits (the unit of currency in the Empire). If the demand for lightsabers follows an exponential distribution with an average of 75 lightsabers, how many lightsabers should the LMC bring to the Jedi Convention to maximize its profit?

3. A student has a hard time figuring out how to get started on homework for Stochastic Modeling Techniques. The student randomly selects one of 3 potential places to start a homework problem with equal probability. The first approach is not fruitful; the student will return to the starting point after 1 hour of work. The second approach will return the student to the starting point after 3 hours. The third will lead to the solution in 15 minutes (1/4 hour). The student is confused, so he/she always chooses from all 3 available approaches each time. What is the expected amount of time it takes this diligent student to solve the homework problem?

4. A mysterious illness called “badgerpox” has affected the local badger population near Madison. The exposure level (X) largely determines whether a badger contracts the disease (D). The probability distribution for the exposure level and the conditional probability of disease given the exposure level are given in the table below.

Exposure level (Xi) P(Xi) P(D | Xi)
10 0.7 0.001
100 0.15 0.005
1000 0.12 0.12
10000 0.03 0.78

Find:

(a) The conditional distribution P(Xi | D) for each value of Xi
(b) The probability that a badger contracts the disease P(D)
(c) The expected exposure level for badgers that have contracted the disease.

5. A student likes to come to ISyE 624 on time, which is possible as long as the student can travel from left to right in the diagram below. There are two paths to class; the student can pass through if and only if all components along its path are open. Due to construction, the probability that component i is open has probability pi, i = 1,2,3,4. Assume components are independent. If there is not a path to class, the student will arrive to class late, and the professor will be sad. What is the probability the student gets to class on time?

midterm1_prob1

6. The Wisconsin-Minnesota football rivalry dates back to 1890. The teams play once per year for the trophy of “Paul Bunyan’s Axe,” which replaced the first trophy (the “Slab of Bacon,” 1930-1943)*. The teams are unevenly matched, with Wisconsin winning 16 of the last 20 games. Let’s say that Wisconsin wins each game independently with probability 0.8. The teams play next on 11/23/13.

(a) What is the expected number of games/years until Wisconsin loses next?
(b) What is the expected number of games/years until Wisconsin loses 2 games in a row?
(c) What is the probability that it Wisconsin loses for the 3rd time in the 5th game in the series?

* I didn’t make that up!

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