## Tuesday, June 28, 2011

### How Do You Calculate An Optimal Pit Stop Strategy for a 2011 F1 Race?

In aid of National Motorsport Week (see GoMotorsport for more details), I thought I'd put a couple of questions out there to the F1 teams, and maybe some of the more academic minded F1 fans.

Here's the first question, lifted from a comment I posted earlier today on James Allen's F1 blog.

In response to a quip of the sort JA makes in many of his strategy briefings ("[m]any had planned to do the race on a two-stop strategy, which on paper was eight seconds faster than a three-stop, assuming you had a trouble-free run in traffic"), my question is this:

Is there anywhere that shows the working for this sort of calculation? I'm thinking of something like the refuelling strategy calculation that McLaren published as part of an outreach project a few years ago.

With the case of refuelling, it's easy enough to demonstrate that the optimal strategy when it comes to refuelling, all other things being equal, is not to start the race with a full fuel load, but instead incur a pitstop penalty in lieu of the time penalty incurred by lugging around a dead weight of fuel for much of the duration of the race.

With no refuelling, the time gains that are used to offset the pit stop losses must come from reducing some other time penalty, which can only leave the tyres as the dominant responsible factor (all other things being equal). In particular, there are presumably two tyre models to be taken into account, one for each sort of tyre that has to be used during the race.

Ian Horlock's Bristol University Masters thesis on Prediction of Formula One Results Using Driver Characteristics had something to say about tyre models I think, so I should probably read through that thesis again (and pay more attention this time!)

If you can describe the model that justifies James Allen's claim that the optimal strategy for Valencia was a two stopper, and also show that it was predicted at 8 seconds faster than a three stopper, either in the comments or via a link to worked description elsewhere, I'd be keen to see it...:-)

PS here are some possible relevant references (I haven't had chance to check them out yet, and apologise in advance for them being links to commercial publications (though you'll probably be able to get them free if you're a student or work for a university, and may even be able to get them though a public library; free downloads of the papers may also be available - I'll add links if I find them)

- The 2007 IEEE CEC simulated car racing competition, Julian Togelius, Simon Lucas, Ho Duc Thang, Jonathan M. Garibaldi, Tomoharu Nakashima, Chin Hiong Tan, Itamar Elhanany, Shay Berant, Philip Hingston and Robert M. MacCallum, et al. GENETIC PROGRAMMING AND EVOLVABLE MACHINES Volume 9, Number 4, 295-329, DOI: 10.1007/s10710-008-9063-0
- Planning Formula One race strategies using discrete-event simulation,Bekker, J; Lotz, W. Journal of the Operational Research Society, Volume 60, Number 7, May 2009 , pp. 952-961(10)

1. This is my comment from JA on F1, but shortened to be under 4096 characters:

They would calculate the rate of degradation of the tires in practice, and then simply add that to their ideal lap time, sum up all of those laps, plus however many stops (resetting the degradation at each stop), and you end up with a total race time.

Say the soft declined at 0.1 sec per lap for 15 laps (“Phase 1″), and then declined at 0.5 sec per lap for the next 5 laps (“Phase 2″). You would see that they’d lose 1.5 seconds per lap when on the 15th lap, and then the end of the next five lap window they’d lose a further 2.5 seconds per lap on the 20th lap, or 4 seconds per lap compared to a fresh set.

...

I compared a one-stop, two-stop and three-stop, since that is what seems to be the most popular today, with a spreadsheet I created while writing this comment.

Assumptions:
-Phase 1 degradation = 0.1 sec per lap
-Phase 2 degradation = 0.5 sec per lap
-Pitstop + Pitlane – Front straight = 23 sec
-Full fuel-load lap time = 100 sec (1m40.0)
-Lap-time gain from fuel consumption = 0.05 sec per lap
-Hard Tire is 1 sec slower per lap, has no Phase 2, degrades at 0.08 sec per lap, used after -first stop.
-One-stop = Lap 20
-Two-stop = Laps 19 & 39
-Three-stop = Laps 15, 28 & 43

Results:
Two-stop is faster than the three-stop by 5.32 seconds, for a total race time of 5633.18 sec. The one-stop is slower by 18.72 sec, but will lead by 9 seconds after the second stop for the two-stopper.

Notes:
-If hard tire degradation is less, it favours strategies with less stops
-If pitstop times are higher, it rewards stopping less
-If the hard tire is much slower than the soft tire, it rewards stopping more.

...

Trying to make the one-stop appeal:
-Pitstop time = 23 sec
-Hard tire 1 sec per lap slower than soft
-Hard tire degrades at 0.05 sec per lap
-One-stop @ Lap 20 to take advantage of the hard tire

Results:
Two stop is 4.95 sec faster than one stop, and 10.45 sec faster than three-stop; however, after the two-stopper’s second stop, they would be 20 seconds behind the one-stopper, necessitating an overtake in the last two laps, when their car would be only 0.25 sec faster. If the two-stopper drove smoothly, didn’t make any mistakes and extended phase one by two laps, they would be 1.2 sec faster and might have a chance at passing. But, if there was a three-stopper, they would hold up the two-stopper in their middle stint, potentially ruining the stint.

Therefore, the two-stopper is a risky strategy, necessitating easy overtakes and minimal lap-time lost, whereas the one-stopper is a safe strategy, where they would be leading after the two-stoppers stop the second time.

...

It’s not too difficult, but a lot can be understood from it. Of course, with any simulation, the more complex it is, the more accurate it will be. What I did would be good for an amateur race. ;-)

-----------------

Here is the Google document: http://bit.ly/ivntvH

It's only partially parametric... I just slapped it together to make something basic. :-)

2. @malcolm Wonderful - thanks for that; is it okay if build on what you've done as the basis for some quickstart "Armchair Strategist" posts?!;-)

I'll try to start digging around in the practice data to see if there is anything in there that might help identify ballpark figures for tyre degradation..

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