Supposedly, gun-offender registry have had “an impact.” I am not surprised. But they surely come with a cost. The cost in this case is a decreased manpower of already-understaffed police force. No one wants to appear soft on crime, but no one wants to pay for it either. I was surprised that the NRA representative quoted in the article appeared to be the most sensible in his assessment of the tradeoffs:

Andrew Arulanandam, the National Rifle Association’s director of public affairs, said his group advocates vigorous enforcement of existing gun laws rather than the creation of new databases. Running a gun-offender registry, he said, requires taxpayers to foot the bill for computer equipment and IT professionals, and it could take officers off the streets and place them in administrative jobs.

Gun registries could be effective despite the costs. If those on the gun registries are “dangerous,” then monitoring those individuals more closely could be an effective use of police manpower, since it focuses police attention on those most likely to commit dangerous crimes. That could more than make up for the policing that is foregone due to the new responsibilities associated with the registry. But I need to see some research that honestly shows the level of impact that registries could have. The police chief in Prince George does indeed intend to use some of their existing 1400 police officers to monitor the gun registry.

[Deputy Police Chief Craig] Howard said police would initially assign a sergeant and four or five detectives to handle the gun registry, pulling them from other jobs in the department. He said the creation of the computer database could be handled by those who run the department’s sex-offender registry and do other IT work.

The 5-6 officers needed for the registry are less than 1% of the police force. But the current registry level is zero and the system is not in steady-state. The registry will likely grow in time as gun offenders are added, necessitating an increasing number of police officers to monitor the registry. I’m not sure that monitoring the registry in the future will be sustainable.

I am agnostic on gun registries. I am not agnostic about honesty regarding the true cost of gun registries and their impacts. Certainly, different municipalities and counties need to weigh the costs with the rewards and make a decision that they can live with. Previous attempts to curb crime—such as the Three Strikes Laws and minimum sentencing—have lowered crime at an enormous cost. The costs here are associated with an exploding prison population. I suspect that the same might be true for gun registries.

The project is a great way to teach about the MODA process. The process not only includes the modeling, but also the craft of working with decision makers and iteratively improving the model. It’s useful for students to be exposed to the entire analysis process. I don’t do this in my other classes.

On the first day of class, we came up with our objectives hierarchy. I did this by passing out about five Post It notes to each student. They each wrote one criteria for selecting a restaurant on each Post It note. They stuck their Post It notes to the wall. Together, we regrouped and organized our criteria into an objectives hierarchy.  Some of the objectives because “weed out criteria,” such as making sure that the restaurant could accommodate all of us and comply with dietary restrictions.

Our initial criteria were:

1. Distance
2. Quality of food
3. Variety of food
4. Service: Fast service
5. Service: Waiting time for a table
6. Service: Friendly service
7. Atmosphere: Noise level
8. Atmosphere: Cleanliness
9. Cost

Our final criteria were as follows (from most to least important):

1. Quality of food
2. Cost (tie with #3)
3. Distance
4. Fast service (tie with #5)
5. Noise level
6. Cleanliness

We removed variety of food, waiting time, and friendly service because classroom discussions indicated that they weren’t important compared to the other criteria. Variety, for example, was less important if we were eating delicious food at an ethnic restaurant that had less “variety” (variety in quotes here, because it depends on you you measure it).

In the next few weeks, we worked on identifying how we would actually measure our criteria. Then, we came up with a list of our favorite restaurants. During this process, we removed objectives that no longer made sense.

We collaboratively scored each of the restaurants in each of the six categories by using a google docs spreadsheet.

1. Quality of food = average score (1-5 scale)
2. Cost (tie with #3) = cost of an entree, drink, tax, and tip
3. Distance = distance from the class (in minutes walk/drive)
4. Fast service (tie with #5) = three point scale based on fast service, OK service, or very slow service
5. Noise level = four point scale based on yelp.com ratings
6. Cleanliness: based on the last inspection. Score = # minor violations + 4*# major violations.

A real challenge was to come up with:

• the single dimensional value functions that translated each restaurant score for an objective into a value between 0 and 1.
• the weights that balanced our preferences across objectives using swing weight thinking. FYI, we used an additive model.

I won’t elaborate on these parts of the process further. Ask me about these if you are interested.

When we finished our model, the “best” decision was to forego a restaurant and do a potluck instead. No one was happy with this. We examined why this happened. This was great: ending up with a bad solution was a great opportunity for learning. We concluded that we didn’t account for the hidden costs associated with a potluck. Namely, it would entail either making a trip to the grocery store or cooking, approximately a 30 minute penalty. We decided that this was equivalent to driving to a distant restaurant, a 26 minute drive in our model.  It was also hard to evaluate cleanliness since the state do not inspect classrooms like they do restaurants. But since cleanliness didn’t account for much of our decision, we decided not to make adjustments there.

The final model is in a google docs spreadsheet.

We performed a sensitivity analysis on all of the weights. Regardless of what they were, most of the restaurants were dominated, meaning that they would not be optimal no matter what the weights were. The sensitivity was not in google docs, since we downloaded the document and performed sensitivity on our own. I show the sensitivity wrt to the weight for quality below. The base weight for quality is 0.3617. When the weight is zero and quality is not important, Chipotle would have been our most preferred restaurant. The Local would be preferred only across a tiny range.

We celebrated in Ipanema, a semi-vegetarian restaurant in Richmond. I think our model came up with a great restaurant. We all enjoyed a nice meal together. Interestingly, Mamma Zu scored almost identically to Ipanema (see the figure below).

I cannot claim credit for this fun class project. I shamelessly stole this idea from Dr. Don Buckshaw, who uses it in MODA short courses.  We use the Craig Kirkwood’s Strategic Decision Making as the textbook for the course. I also recommend Ralph Keeney’s Value Focused Thinking and John Hammond’s Smart Choices.

How do you choose a restaurant?

Sensitivity with respect to the weight for quality (0.3617 in the base case).

Let R = (Likelihood of being born in the “hot” 3 months) / (Likelihood of being born in the remaining 9 months).

The Birthday Problem assumes that R = 1. I consider 1 <= R <= 2. This post courtesy of Chris Rump indicates that R < 1.2, meaning that humans don’t have much of a mating season.

The simulations below show the average value of P(n), where P(n) = the probability that someone shares a birthday in a group of n people. The simulations are performed over 1M replications for each value of n. The probability of shared birthday goes up when people are more likely to be born in the birth months associated with “mating season.” But the effects are small, as can be seen by a fairly compressed y-scale. The simulations were performed in Matlab and the program is here.

The Birthday Problem probability for n=5.

The Birthday Problem probability for n=10.

The Birthday Problem probability for n=20.

The Birthday Problem probability for n=30.

The Birthday Problem probability for n=40.

The Birthday Problem probability for n=50.

Here, we are only concerned with birth day and month (not year). The solution assumes that a person is equally born on any of the 365 days in the year, thus ignoring leap years.

Let P(n) = the probability that someone shares a birthday in a group of n people and let Q(n) = the probability that everyone has unique birthdays. There are 365^n ways for n people to be born on any of the 365 days.Then

P(n) = 1 – Q(n) = 1 – (365*364*…*(365-n+1))/365^n.

P(n)

P(2) = 0.0028

P(5) = 0.0271

P(10) = 0.1169

P(20) = 0.4114

P(30) = 0.7063

P(40) = 0.8912

P(50) = 0.9704

P(60) = 0.9941 –> in a room with 60 people, you are almost certain to have at least two people that share a birthday!

The key assumption is that all birth dates are equally likely. This NPR article shows that humans have a “mating season” that makes July – September birthdays more likely. I posted the image below.

This will, of course, change our answer above. The probabilities depend on who is in the room. Have you simulated the Birthday Problem with an unequal birthday distribution? If so, please shed light on realistic numbers for P(n).

On a side note, the image below suggests that babies are induced on December 27-30 for a tax break. I’m not sure how I feel about that.

How likely are people to be born on different birth dates?

In response to fraud, two significant changes were made to the van pool contracts:

1. Van riders have to ride the van at least 50% of the working days each month (instead of each quarter).
2. The van needs to be more than 50% full (not including the driver) at least 80% of the time.

First requirement

The first requirement is a problem. Vacations, has off-site training, goes to a conference, etc, interfere with ridership. Add two of these events together, and you end up with a severely inflexible policy. Employees who have to travel or who want to take a vacation may not be able to abide with the new requirements. Employees that travel a lot may not be the best candidates for the van pool, but employees who have one long trip ever should be able to ride the van.

Second requirement

The second change is even more restrictive. My husband’s van holds 14 people. They need to have at least 8 people ride (7 out of 13 + 1 driver) at least 4 out of 5 days in the week. It’s worth noting that small vans are going to have more of a problem with this because they typically have more variability in who shows up.

Looking at the long-term, a van needs to have at least 8 riders 80% of the time. That implies that the average number of days until the van is not “half” full needs to be 5. If we assume that each day is independent of the others, then we can model this as a geometric distribution. To get the desired average of 5, we need a “success” probability of 1/5. That is, the probability of a van being less than half full on any single day needs to be 20%. Is that realistic?

Let’s look at the 14 riders. Let’s say they each ride each day with probability p and that each day is independent. Then we can model the number of riders each day R~ Binomial(14,p). We need P(R<8) < 0.2 to meet requirement #2 (from the last paragraph). But if p=0.5 (as required), then P(R<8) = 0.605.  We need p > 0.642 to get P(R<8) < 0.2.

Thus far, I’ve assumed that each day is independent. Making that assumption will yield optimistic results. The “optimistic” results suggest that even under idealistic assumptions, the new van pool requirements will be extremely difficult to meet.

But this assumption is not valid. Riders ride less on Mondays and Fridays (40% of the work week). In practice, this means a van is likely to have at least 8 riders on three days of the week (Tue-Thu), but not four as required. For simplicity, let’s say that a van always has 8 riders on Tue-Thu (an idealistic assumption, but not too unreasonable). Let’s say that each rider takes the van with probability p = 1/3 on Mondays and Fridays (a modest change), and let’s assume that each rider and day are independent. Then the van has 8 riders with probability 0.0576 on these days. The probability that the van covers at least one Monday or Friday per week is 0.111. Therefore, under more realistic assumptions, it is even more unlikely that a van can meet assumption #2.

Vans could meet these requirements if they “overbook” and allow more riders than there are seats in the van. They will surely need to risk turning riders away to meet their requirements if more than 14 show up some day.  This may happen when gas prices change, since ridership increases when gas prices shoot up. However, my husband has noted problems meeting requirement #2 in the past couple of months when gas has been ~$4 per gallon.

Van pools and (un)intended consequences

Van pools are intended to improve the environment by encouraging car pooling. Van pools ultimately create an incentive for people to live very, very far from where they work, thus leading to more fuel usage. In the end, I do not think they save much (or any) fuel from being consumed. But this is open for debate.

On the other hand, van pools are great if your significant other wants a career. It has made our “two body problem” quite manageable. Accommodating one’s spouse’s career is rarely cited as a benefit to van pools. I am grateful that van pools exist so that both my husband and I can have careers. I hope it’s still possible for him to carpool with new rules.

Final thoughts

I applaud the Federal government for trying to crack down on fraud. But I encourage them to create rules that make it possible for riders to follow the letter of the law. And maybe someone who has taken a course on probability should look over the next set of rules and crunches some numbers.

Do you ride a van? Please correct anything I got wrong in the comments and add your two cents about how they can be improved.

The short answer: it wasn’t easy.

Both my husband and I drive Honda Civics. We could easily fit two car seats in our Civics. The problem was when we were expecting child #3, we didn’t think it could happen. Here are the challenges:

(1) We clearly could not add our baby car seat to the existing two car seats (actually, one convertible car seat and a booster). We could visually see that this wouldn’t work. Clearly, we would have to invest in new car seats. Thus, we had to find the right mix of three car seats.

(2) It’s not a stationary problem. When the baby arrived, we would have one baby seat, one forward facing seat, and one booster. After a few months, we would need one rear-facing seat, one rear-facing seat, and one booster.

(3) Some seats have dual functionality. For example, convertible seats can work as front- or rear-facing. The seat would need to fit in both modes. And that depends on the other two seats. The combinatorics made my head spin.

(4) Most of the information is bogus. The width of the back seat on car web sites is a rough estimate. The contours on the seats can affect the fit. Car seat widths reported on baby sites are often the size of the box that the car seat comes in (see this seat on Babies R Us that lists dimensions of 28.0 x 22.2 x 20.7).

(5) Just because you can fit three across doesn’t mean you can expect to use seat belts. Amazingly, “family friendly” cars such as the Toyota RAV4 have weird seat belt configurations that don’t really let you put three in the back seat at the same time.

(6) I wish I had a nickel for every time we heard someone say, “We couldn’t fit two car seats in our [insanely huge sedan] so we had to upgrade to a [insanely huge SUV].” (here here are people who “cannot fit” three in a giant Honda Pilot or GMC Envoy!) I have no idea why the common wisdom is so off here.

These are the two things we had going for us:

(1) Car seats are enormous. They are made for people who drive mini-vans. But with high gas prices, there is a growing market for slim-fit car seats that fit in small, fuel-efficient cars (yeah!!) While it might have been impossible to fit three across a few years ago, we had a fighting chance in 2011.

(2) This thread on fitting three car seats in small cars was literally the only reason why we even tried to keep both Civics. With all the misinformation out there, we didn’t think it was possible.

When we put all of this together, we learned that it was possible to get three across, but the information to do so was “expensive.” The slim-fit car seats were available by online purchase only — they were not available in box stores.  Therefore, we couldn’t pull up to the local Babies R Us and pop a few floor model car seats into our cars until we found something that worked. We were going to have to do the best research we could and take a chance. Still, buying a few car seats was cheaper than buying a new car. I have have even built a decision tree to guide the process.

We ended up putting two Radian65 car seats in my husband’s Civic and a Combi Coccoro in mine along with older seats and boosters.  We ended up spending ~$600 on new car seats, and we are happy with our decision.

We have since learned that a colleague of my husband fits three across in an even smaller Honda Fit. Every article I have seen suggests that only two car seats are possible. A little persistence goes a long way when it comes to fitting car seats in cars.

What surprised me is that this process was in stark contrast to just about every other decision we make. Online information was plentiful and accurate when we were researching a new washing machine. Decisions regarding kids are entirely another matter.

We usually have a wealth of information/data in operations research. But not always. For example, problems in homeland security and natural disasters are hard to come by, and the data is often perishable. The car seat saga was a reminder on how good we have it (in terms of data) most of the time.

Question: There is a zombie outbreak in Richmond. The zombie population can be modeled as a linear growth birth death process.  Each zombie independently reproduces at a rate of λ = 2/hour and is killed by resourceful Virginians at a rate of μ = 0.5/hour.  If the population started with a pack of two zombies, find the average size of the zombie population after 24 hours.

Answer: The average size of the population can be modeled using a linear growth birth death process. Let Ei denote the expected size of the zombie population after 24 hours given that there are initially i zombies. Then Ei = i * E1.

The expected size of the zombie population is given by

Ei = ie^((λ-μ)t) = 2*e^36 after t=24 hours. That is a lot of zombies!

Comments: Is this model appropriate? Well, let’s take a look at the assumptions. They key assumption here is that there are an unlimited number of humans that can serve as zombie fodder. Clearly there are not  e^26 humans on the planet. More generally, as the zombie population grows, they will eventually run out of food (humans!), so their growth will slow down. That is, the linear growth assumption doesn’t make sense. It might work for the first hour or two, however. Likewise, the linear death assumption is not realistic, because after the zombie population explodes, there are no more resourceful Virginians left to kill zombies. Even when there are Virginians around, the rate that they can kill the zombies is probably not proportional to the size of the zombie population. But the students didn’t need to assess model realism for full credit.

This post expands upon a great post in The Operations Room blog on passport/immigration waiting times at Heathrow. The queues to clear passport control at Heathrow have become quite political (see this article). The reason is that the UK have increased security (read: slower server service rates) and are anticipating a large number of new passengers when they host the Olympics this summer (read: larger customers arrival rate).

I had the misfortune to travel to and from London via Heathrow in mid-April. (London was a pleasure, the queue at Heathrow was a misfortune). It took me nearly two hours to get through  immigration. I had plenty of time to think about queuing and logistics while I waited. Here are some of my observations.

• There were two large queues: one for EU residents and one for the rest of us. The wait at Heathrow was much shorter for EU residents. Residents should have a shorter wait. The Operations Room notes that the goal is to get EU residents through the queue in 25 minutes and non-EU residents through in 90 minutes.
• The problem at Heathrow was staffing too few people. There were 3-6 booths open for non-EU residents at any given time.  They could have maxed out to ~15 booths. There were two people at the front of the queue that would tell us when a booth became available.
• My service time was about 30 seconds, but 15 seconds to walk over to the booth. Some of the open booths were a long walk from the front of the queue. There wasn’t much inefficiency in the system except for the long walks to open booths.
• The system is not at steady state. International arrivals come in bunches. It was very busy when I arrived at immigration and less so later.  This makes it challenging to staff the booths. If you are a passenger, your wait is a bit of the luck of the draw. If you arrive at the wrong time, you may have to wait for awhile. Clearly, scheduling >6 people to work the non-EU booths is a bare minimum. I’m not sure if it was possible. One cranky member of the Heathrow staff was unsympathetic because she claimed to have been working for 14 hours at that point.

How could queue waits become shorter? It’s easy: open more booths! In reality, that may be easier said than done: manpower was recently reduced by 18%.  Here are some other suggestions.

• Having an “on deck” area near each immigration booth where the next person in line could wait would eliminate most of the walking time from the front of the queue to the booth (15 seconds per passenger). The walking time was large relative to the service time, since most service times are short.
• Some passengers, however, require a long service time. The problem with an “on deck” area is that passengers would get irate if they get stuck behind someone who takes a long time. It’s all about managing expectations.
• Heathrow in particular needs to find the right mix of staff in the booths vs. directing queue traffic. For the Olympics, it’s probably a good idea to have lots of people directing traffic since they’ll have fewer passengers who are familiar with the airport. People in the UK do this very well, so I expect that those who come for the Olympics will not be lost in the airport.
• When I entered the queue, there was a sign that said “45 minute wait from here.” The sign was clearly bogus, since it depends on how many booths were open. My wait was more than an hour longer than the posted sign. I would recommend against signs that raise service expectations. People will get even more impatient if they are promised a shorter wait.
• Centralized immigration booths are efficient (if run properly!). Therefore, I would advise against something like mobile agents who could do the passport checks at the gates. This would only introduce new inefficiencies. The agents would need extra time to move from gate to gate and then set up. Passengers may view a less efficient mobile agent system as “fair” or “preferable” since they are first-come-first-serve and they have shorter lines. But I’m not so sure about this.
• When all else fails, they can take a cue from Disney and install video projectors where they can show Beatles footage and play famous movies set in London. Or if they can’t find the overhead to install new equipment, they could hire street performers like jugglers to perform for those waiting in line (although I’m not sure if there will be enough room for juggling).

Do you fantasize about optimizing queues while waiting in a queue?

I started to use google docs two years ago when a student put together a shared google spreadsheet for a multiobjective decision analysis project that we were performing as a class. It was a fantastic way to collaborate.

It’s worth noting that there are several ways to share a document in google docs: you can share the document with certain users, anyone with the link can see the document, or google can search and find your document. You can allow view access or modify access. You can give ownership to someone else. Therefore, you can be protection or open with google docs. As I said, it’s versatile.

I have found many ways to use google docs when teaching.

1. I created simple class calendars that are easy to update and change (see one here). I ended up changing the schedule quite a bit during the semester this semester, and it only took a minute to change the official class schedule on google.
2. I created a spreadsheet for my multiobjective decision analysis class project. Our project was to select the best restaurant for celebrating the end of the semester. We all participated in collecting and anayzing the data, and a common spreadsheet eliminated emailing of the “hot copy.” It was superior to DropBox in that we could modify the same document at the same time. If someone else modifies the spreadsheet, it shows up as new in my document list (its name is boldfaced), so I didn’t have to spend time figuring out if anything in it had changed since the last time I looked at it.
3. My stochastic processes grader and I came up with homework assignments together via google document. He would propose homework assignments, and I would modify the assignments according to what I covered in class. Then I posted the assignment to BlackBoard as usual.
4. My grader wrote homework solutions in google documents. It became easy to share the solutions with students.
5. I wrote exams with google docs so that my grader could offer feedback at his leisure.

I use BlackBoard to host all of my course materials. I have a love-hate relationship with BlackBoard. It provides many course materials, including a course calendar. But BlackBoard is cumbersome to use–I could spend all day in BlackBoard updating course materials. Google docs saves time.

I should note that at my university, student emails are run by google, so it’s extra easy to use google tools with students. I would imagine that all students have gmail accounts, but it might be hard to collect emails if you want to only share your docs with your students.

I’ve used google docs for other professional tasks. Recently, I set up a colleague’s visit with google docs. I shared the link with my colleagues here at VCU and gave them view access only. They were able to see the available times in the schedule and sign up for a meeting. They could always see the most recent version of the schedule, saving me the time of emailing to and from everyone. Then I emailed the link to my colleague.

My department ran faculty searches via google docs. It was an efficient way to keep track of the ~200 people who applied to the open positions and to manage campus visits.

The INFORMS IT committee used google docs extensively. It was the most impressive use of google docs I have ever seen. We wrote and edited reports. We went over them in meetings. We compared document histories.You can view how to find document histories in the image below.

Here are two things you might not know.

1. You can leave comments in a google doc. With multiple users, this can become a conversation.

2. You can chat in a doc in real-time. My grader and I would often do this before class as I was going over what I would assign to students.

You can view histories in google docs.

I’m looking forward to doing even more with google docs, such as writing collaborative proposals and replacing documents I would normally upload in BlackBoard.

How have you used google docs in (or out of) the classroom?

I put shuffled in quotes because I’m not sure how the shuffling algorithm works and if it is really a good pseudo-random ordering of the songs. But at the end of the day, I don’t want the songs to be random. I want them to be random-ish.

My mp3 player has a couple of shuffle varieties. One shuffle variety appears to play songs that were recently uploaded more frequently than old songs. I like this, but I change the songs so rarely that it isn’t very useful any more.

Another mode shuffles through all the songs, not repeating a song until all are played in a session (a run starts a new session starts). I never run long enough to repeat any of the songs. Therefore, I hear each song no more than once per run. But the memory resets every time I turn off the mp3 player, so I hear some songs during every run and rarely hear other songs.  I just updated my playlist and found a handful of songs that I have not heard this year.

I’m sure everyone has that song that always plays every time you turn on your mp3 player. For me, that song is Bruce Springsteen’s “Born in the USA” (at least it was until I removed it).

I know this is what “random” looks like, but I don’t like it: I don’t want to hear some songs too much and others not enough. I conclude that my “optimal” randomly ordered playlist is a very fake looking random sequence of songs to ensure that all songs are not over- or under-played. I feel like a fraud just admitting that!

This article suggests that playlists advertised as random may in fact be less random than we are led to believe. It discusses playlists that an mp3 player may create (kinda like Pandora) rather than a preset playlist. They tested iTunes’s Smart playlists and found that iTunes preferred Lionel Richie songs. Interesting. My mp3 player clearly prefers Bruce Springsteen.

How do you feel about shuffled playlists?