Optimizing Medical Appointments

Introduction

In a previous post I asserted that because medical appointments were so frequently late the providers of those services were rude (or trying to maximize their own revenue) and that with better planning these appointments could be on-time way more often. Given the subject matter I should have anticipated a lot of resistance to these ideas and flushed out the statistics behind scheduling but I didn't. This post attempts to rectify that short-coming and will dig into the details of scheduling and why I believe the current methods of scheduling medical appointments is not optimal.

In a Perfect World...

Some of the criticism of my first approach was that "healthcare deals with people, not numbers", "each patient is different" and "there is no way to predict which patients will require more time". It seems I was being accused of thinking that medical appointments could be analyzed and suddenly they would all take exactly the same amount of time or somehow we could know exactly how long each of them would take. I am under no such illusions. That said, let's take a look at what scheduling would be like in a perfect world.

What do I mean by a perfect world? Well in a perfect world (for scheduling at least) all appointments would take exactly the same amount of time and that time would be known. For the sake of argument let's pretend that every medical appointment took 15 minutes and the standard deviation was 0 minutes (i.e. they are all exactly 15 minutes). If the world was this simple we wouldn't need spreadsheets because we'd simply make each appointment 15 minutes and there would be no patients waiting and no doctor downtime. That said, I've shown a summary of what this looks like in the table below.

Wouldn't this be great! Patients don't wait, service providers don't wait and there is absolutely no cost to society. Unfortunately this world doesn't exist...

My Commute

Now I got some flak for comparing patient scheduling with traffic but hopefully this section explains why I chose this analogy. The reason I talk about traffic is that everybody knows traffic is variable from day to day (i.e. it's not like our perfect world). For me personally, my commute averages about 30 minutes but it is not unusual for it to take 5 minutes more or less and occasionally it's really delayed (or fast). Based on this it probably isn't unreasonable to assume that the time it takes for me to commute to work looks something like the chart below.

So even though no two commutes end up being exactly the same they do form a probability distribution that I can use to make informed decisions. Now most people don't actually make a distribution like this and then analyze it in Excel but that's what we're going to do later just to show a methodology that can be applied to more complex problems. That said, intuitively you should be able to tell that if you only allow 30 minutes for your commute you will be late 50% of the time (you know who you are...). Furthermore, if you remember some statistics you could probably figure out that if you leave 5 minutes earlier you will be on time 84% of the time (68% of a sample is within 1 standard deviation of the mean so you get the right half of 68% and the whole left side of the spectrum too).

Beyond those two rules of thumb though we can use excel to calculate the probability of being on time no matter how long you leave for your commute. More to the point however we can also figure out how early we need to leave in order to be on time 95% or 99% of the time. The table below shows these calculations.

What you can see in this table is we took the properties of the distribution (mean of 30 and standard deviation of 10) and then calculated how much time I need to leave in order to be on time 95% and 99% of the time. Basically, if I want to be on time 95% of the time (late roughly one day a month) I should leave 43 minutes to get to work. If I have a very important meeting that I really don't want to miss I should probably leave a solid 50 minutes to make it (I call this buffering). The other item of note in this table is the cost of being on time. What I am getting at here is that I have to be at work early by an average of 13 minutes/day in order to have 95% confidence that I will be on time. My time isn't free so being on time has a cost to me and I've calculated this cost (13 min / 60 min/hour x $28/hour) to be about $6/day (PS - I do not earn $28/hour but it is the number I calculated for the average Canadian in the last post).

Increasing the Difficulty

Now that we've tackled the basic traffic problem I want to point out some changes in the probability distribution that would make it harder to be on time. It's probably intuitive for those who remembers their statistics classes but if we make the standard deviation a bigger number (i.e. widen the distribution) then I will need to add more buffer in order to be on time. So if the standard deviation on my commute goes from 10 minutes to 20 minutes I will have to leave 56 minutes before work in order to get there on time 95% of the time (this will also increase the cost of being on time to $12/day). You can see in the chart below that the 95% confidence level has moved substantially to the right (i.e. more time required) when the distribution gets wider.

Another wrinkle is when distributions aren't normal or are skewed right. If the underlying distribution is skewed right or is lognormal (first two images, respectively) then clearly we will need to increase the 'buffer' and will incur more costs to be on time. Also, if the distribution is uniform (the rectangle) it changes again and the math actually gets really easy! Suffice it to say however that the underlying distribution matters a lot.

So What Do Medical Professionals Have to Deal With?

Well they definitely don't live in the perfect world (or we wouldn't be having this conversation) but from what I can tell most medical professionals are dealing with a mostly normal distribution that has a slight skew right (I would also not be surprised to learn it was a Poisson distribution). I'm sure different specialties/professionals have different distributions but the data I got from the ASRN (American Society of Registered Nurses) shows a distribution that is more or less in line with what I would have expected. I think it's also consistent with the anecdotal evidence cited by medical professionals when describing their scheduling challenges (i.e some patients take a lot longer than budgeted because they need to be consoled, are more complicated, came in with an emergency, etc.).

Looking at the graph of the data please excuse that it looks very normal (or even left skewed). I didn't get to pick the intervals and the researchers didn't use intervals of equal value (i.e. the first 3 buckets are 5 minute intervals and the next two are 15 and 30 minutes, respectively). If you use your imagination and break those 15 and 30 minute buckets into fives I hope you'll see the slightly right skewed distribution I do...

The Story of Today

Now that we have something of a distribution to work with let's see how the current scheduling system works out. Basically, we've got a slightly right skewed, normal distribution with a mean of 17.5 minutes (that's what the math in the chart works out to) and a standard deviation of 11.9 minutes. Then we're going to try and schedule 24 appointments a day into two groups of consecutive 15 minute time slots (i.e. 12 straight in the morning and 12 straight in the afternoon). Without doing the math though I think it's clear that this distribution is not going to fit well into these time slots because the average appointment is already longer than the time scheduled...

For those of you interested in the actual math for this it's a little complicated but in general terms I have added a Monte Carlo simulator to Excel and I supplied it with our chosen distribution and schedule. Then I ran the simulator 10,000 times which should work out to about 40 or 50 years' worth of days to come out with the on-time percentage, average patient wait times and average provider downtime between patients.

Now there is a lot to look at with this chart but the first thing I notice is that it feels a little aggressive. Other research has shown that the average time a patient spends in the waiting room is about 15 to 20 minutes and we're coming up with 25 in this analysis so our distribution might be a little off from reality. That said, it's still pretty close and matches my life experience reasonably well (for what that's worth). Next I can't help but look at the rapidly declining on-time percentage as the day goes by. Basically, if you don't have an appointment in the first hour of business or the first hour after lunch you've got a really bad chance of getting in on time.

The last thing I will comment on is the total cost to society of all this waiting. What I've done different from last time though is I broke it out by patient and provider to reflect the fact that downtime to medical professionals has a cost and patients waiting has a cost. Just like my last post I assumed the cost of a patient waiting is $28/hour and I estimated that the cost of downtime to medical professionals at $120/hour (I really just made this up but it feels close enough). Now the rules of society suggest that it is rude to value your time higher than someone else's so there is some merit to making these numbers the same. That said, the cold hard economic truth is that a medical professional's time is more valuable than average so I don't mind reflecting this in the analysis.

With that out of the way I want to draw your attention to how much impact this system has on patients ($280/day per provider!) and how little downtime/cost there was for the medical professionals (over the whole day these professionals are only idle for 7 minutes and lose a grand total of $13.93). This approach hasn't quite downloaded all of the cost of waiting onto the patient but it's pretty darn close! Said another way, if I was trying to optimize the profit for my medical practice this schedule is almost perfect (i.e. perfect is no downtime).

What If Appointments Were 20 Minutes?

If one of your friends had a commute that usually took 30 minute but they insisted on leaving only 25 minutes to get there you'd think they were nuts and you'd have no sympathy for them when they got fired for being late "all" the time. So why are we trying to jam a 17.5 minute average appointment into a 15 minute time slot? Clearly, that's going to result in us being constantly late. Shouldn't we at least leave a couple of minutes worth of 'buffer' to try and keep our schedule together? In fact, what would happen if we scheduled appointments every 20 minutes instead of every 15?

So it makes a little bit of a difference...in fact it reduced the average patient wait time to only 7.6 minutes (from 25 minutes!) and reduced the total cost to society by more than half (now only $125/day per provider). Furthermore, it didn't have all that much impact on the medical professionals that served these patients. Certainly there was more downtime between patients (now 20 minutes/day instead of 7 minutes/day) and the cost of this is a little higher ($39 instead of $14) but on the whole there is no denying that it's a net gain to society.

Now I know that one of the objections to this 20 minute schedule is going to be that it results in a longer day for the provider and/or less patients/day. While that's technically true it's only by the slightest of margins. If you look at the 15 minute table you'll find that the appointments for that day ended at 15:45 which means the provider should be done at 16:00 but the reality is that the provider is usually 48.8 minutes late for that appointment so they really aren't seeing the patient until 16:29 and don't likely finish their day until 16:46 anyway. Is 16:46 slightly earlier than in the 20 minute schedule? Yes. Is it enough to justify the economic loss to the patients? I don't think so.

In addition to the later finish to the schedule I also show the medical professional starting at 8:00 am. I did this so they could actually have lunch (despite the hour scheduled they would often have very little time for this in the 15 minute schedule) but the schedule could just as easily start at 8:30 am or 9:00 am and then have patients scheduled into lunch (which is happening now anyway) if the early start time is non-negotiable.

That said, the math shows that both of the schedules result in the exact same amount of time with the patient, 17.5 minutes x 24 patients = ~7 hours. All we've done differently is spread those patients out a little further so they don't get stuck waiting a crazy amount of time. Of course this results in a little more downtime between appointments for the providers but I'd like to think they could find something constructive to do with that time (although I have explicitly not included this in the model. As far as the model is concerned it's a total loss and it still recommends this schedule...). The original schedule assumed they would only see patients for 6 hours/day so I'm guessing there is some administrative tasks they were going to complete with the rest of their work day and maybe they can get some of this done between patients now instead of at the end of the day (thus making the work day almost exactly the same). Obviously that won't always work out but I would like to think it makes this schedule a little more palatable.

The Optimal Schedule for Society

The optimal schedule for society would be the one that minimizes the combined cost of patient waiting and medical provider downtime. Because our distribution has a standard deviation the best we can ever do here is a low number and not $0 (just like I can't be on time for work 95% of the time without being early on average). Just because that's true however does not mean that the current level of patient waiting is optimal. In fact, based on the analysis of the 15 minute schedule above it's clear to me that the only thing being optimized in the current schedule is patients/day. This could be because medical professionals are genuinely trying to see as many patients as possible or it could be because the business manager at the practice is trying to optimize their profit but either way the result is the same - no downtime for medical professionals and lots of waiting for patients.

So what would the optimal schedule be? Honestly, I don't know. I remember a lot from my undergraduate course in management science but I can no longer remember how to do an optimization calculation on top of a Monte Carlo simulation. I also didn't bother to try because it wouldn't really matter. The probability distribution I used for these examples seems pretty good but in reality every specialty and every medical professional will have a different underlying distribution so there would be no point in actually optimizing a schedule for this particular distribution. Furthermore, nobody is actually going to study their practice close enough to come up with a distribution and then learn how to perform this analysis anyway.

So What Do I Want to See?

If the purpose of this exercise wasn't to develop an optimal schedule then what was it? Well, the point was to demonstrate that a schedule that minimizes the downtime of medical professionals is not optimal for society. We pay doctors a lot of money so the optimal solution will result in more waiting for patients than downtime for doctors but when it's taken to the extreme (like the 15 minute schedule above) it breaks down. It doesn't matter how valuable a doctor is, a schedule that results in almost 3 extra hours of lost patient time to recoup 13 minutes for the doctor will never be optimal.

What I would really like to see is a recognition that the way we are scheduling patients today is not optimal and I would like to see medical professionals take a few steps in the right direction. I don't expect them to do an analysis like this one but I would like them to realize that when they have consistently long wait times it has a detrimental effect on the economy and hopefully they will take a few steps to decrease these times. I will also point out that the optimal schedule does not mean the medical professional is never late. In fact, depending on the underlying probability distribution of their appointments the optimal solution could even mean pretty long wait times. However, I think that is more of an exception than the norm and my guess is that most specialties/professionals could have average wait times less than 10 minutes if they were willing to forego a few minutes of productivity so the rest of us could gain a few hours...

A Few More Thoughts

I want to be really clear that I don't think medical professionals are bad people. On average I actually believe they are some of the most compassionate and caring people we have in all of society. Furthermore, I can see how spending your workdays living out that 15 minute schedule would make you feel like you are doing everything you can do to be on time and help your patients.

Think about it this way. These people get into work on time and start their first patient, then they see patient after patient with virtually no breaks but they keep getting further and further behind as the day goes on. Then to make up the time they lost in the morning they selflessly give up their lunch hour to keep seeing patients. Then in the afternoon they repeat the whole process over again and end up seeing patients for almost a whole extra hour there too. Then they probably stay late to finish up the administrative stuff they couldn't get to during the day. They aren't chatting with co-workers by the water cooler, they aren't taking long coffee breaks, they are barely even feeding themselves with the ultimate goal of helping as many people as possible throughout the day...and then Brandon complains because his appointment is 30 minutes late!

I get it. If I lived that day over and over again I would probably believe there is no way to see the same amount of patients while simultaneously reducing wait times. I mean look at the day I just described. There was absolutely no downtime and there is simply no more of that person to go around...but then we have that 20 minute schedule I just showed. Without all that much lost time to the medical professionals (and no drop in patients) we can cut the wait times by over 70% and reduce the cost of waiting and downtime to society by more than 50%. Honestly, if I lived that day over and over I probably wouldn't believe these numbers. Fortunately, I have a much easier job where I get to spend a lot of time working with statistics and I know that it's actually right despite how crazy it might feel.

The last thing I want to say is that I don't think most patient-facing medical professionals actually think about maximizing patients/day in order to maximize profit. There are undoubtedly a few business people in some of these practices that do think that way and they are contributing to the problem but I'm almost certain my ultrasound tech isn't thinking this way. In fact, I think most people in medicine are trying to maximize the patients they see and minimize their own downtime so they can help as many people as possible. The problem is that by martyring themselves in this way they are actually causing a very high cost to society via patients waiting. Furthermore, I’d even be okay if some medical professionals optimized their schedules by seeing a few less patients/day. The truth is that policymakers track how long it takes to get medical appointments and this helps them allocate future resources. So if the honest truth is there are just not enough of a specific type of professional I’d rather the policymakers get that signal by seeing those wait times increase than to have the problem masked by medical professionals that are trying to go above and beyond.