Need help solving a real-life logic problem

Factual Questions
1

Is anyone who’s will versed in Excel* willing to tackle the following calculations? I’ve been trying to do it on paper, and I’m just to the point of staring and mumbling to myself.

*I think Excel would be a good tool for this, but what do I know? Maybe some sort of programming is needed.

Feel free to tell me to go to hell and/or do my own math; I know this is a tall order (but I also know how math geeks like to tackle problems just for the exercise, and maybe what looks difficult to me would be easy for one of you). This is NOT a homework problem. It is a real race and I am one of the runners. This is partly out of personal curiosity, but also to see how our current leg assignments stack up against the possibilities.

Here’s the problem.

Six runners will be running a relay race that has ten legs, or segments. Runners can be assigned one or two legs each. I would like to know the max and min finish times, both considering and neglecting runner preferences (so four answers are needed), and neglecting time taken at the exchange points (runners must hand off a vest and a wristband). It would be especially helpful to know the specific leg assignments for the min time with runner preferences taken into account (one of the four answers).

Runners:

AM: pace 14:00 min/mile, prefers 1 long leg, no trail
JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs
AS: pace 11:30 min/mile, no trail
LA: pace 11:30 min/mile
KS: pace 10:30 min/mile, no trail
AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails

Legs:

Leg 1: 4.8 miles
Leg 2: 4.6 miles
Leg 3: 6.5 miles
Leg 4: 4.0 miles
Leg 5: 3.4 miles
Leg 6: 7.8 miles
Leg 7: 3.0 miles
Leg 8: 7.7 miles, trail
Leg 9: 7.7 miles, trail
Leg 10: 5.8 miles, trail

You win my effusive thanks should you choose to accept this challenge!

2

I should add that we know we are slow overall and we are NOT looking to win (as if). But we do want to get to the afterparty before everyone packs up and leaves. :smiley: Hence the desire to minimize our time so as to maximize our fun.

3

Do you have any requirement for trying to equalize the number of miles run by each participant? Otherwise, your two slowest runners would run legs 5 and 7, only 3 or so miles each, while your fastest runner runs close to 15 miles. Is that really what you’d want?

4

Excellent question.

I guess I am thinking of the extremes such as you suggest for illustrative purposes only. We would not really assign those distances, but if we want to say, “This is our slowest possible time and this is our fastest,” those would be the two answers NOT taking preferences into account. But they would be interesting to know.

I guess we need to add one overall qualification to the other two answers, in addition to individual runner preferences: yes, let’s try to have a reasonable equity of mileage, though of course with shorter distances given to slower runners and vice versa.

(You see why this was giving me a headache.)

5

This sounds like an Operations Research problem. Operations Research is about finding the optimum solution to a problem set. Unfortunately, I’m not sure how to setup or solve this problem.

From a practical perspective, you should probably get more details about how far each runner can run. If the farthest someone has run is 3 miles, they should not be assigned to the 7.8 mile leg. They probably won’t finish it.

6

The faster the runner, the longer leg(s) you assign.
Obviously, the more distance you cover fastest, the soonest you will finish.
First, by not allowing for preferences. Then, allowing for preferences.

i.e.
AT is fastest, give him legs 6 and 8 (the longest 2);
KS next fastest, give legs 9 and 3, next longest.
and so on…

Or, allowing for preference
AT gets the 2 longest without trail: Legs 6 and 3.
and so on…

Or did you mean the 2 legs have to be consecutive?
That makes it a little tougher.
AT would get 8 & 9, or with preference, 5 & 6 is 11.2 miles, longest non-trail.

make a first sheet in excel of leg lengths.
Make a second sheet in Excel of initials, speed.
Do a vlookup of initials to get speed, so you can put initials beside each leg and the next cell is the caluclated time based on that speed.
Tally the time column, while shuffling the initials.

7

If a runner is assigned 2 legs, they do not have to be consecutive. In fact, JF specifically WANTS 2 shorter nonconsecutive legs, in order to get a rest in between.

(We are all female, by the way.)

All of the runners are capable of running the distance needed. JF is the only one who has not completed a half marathon (13.1 miles) or longer, but she is OK with her current assignment of 9.8 miles in 2 legs. However, I feel that perhaps that distance is a little long for her relative to the other runners. And right now we have AT, our fastest runner, spending one of the shortest times on the course. That seems backward to me.

8

By assigning the longest legs to the fastest runners and the shortest legs to the slowest for a fast time and vice versa for a slow time I get a fast time of 10:11 and 11:28 for a slow time.

Looking at preferences for trails but ignoring AT’s preference for legs 6 and 7, she should be running legs 6 and 3, but that is a total of 14.3 miles. Can she maintain her 10 min/mile pace for that long?

9

Fastest speed, ignoring preferences:

Leg 6: 7.8 miles AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails
Leg 8: 7.7 miles, trail AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails
Leg 9: 7.7 miles, trail KS: pace 10:30 min/mile, no trail
Leg 3: 6.5 miles KS: pace 10:30 min/mile, no trail
Leg 10: 5.8 miles, trail LA: pace 11:30 min/mile
Leg 1: 4.8 miles LA: pace 11:30 min/mile
Leg 2: 4.6 miles AS: pace 11:30 min/mile, no trail
Leg 4: 4.0 miles AS: pace 11:30 min/mile, no trail
Leg 5: 3.4 miles JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs
Leg 7: 3.0 miles AM: pace 14:00 min/mile, prefers 1 long leg, no trail

So
15.5 * 10 = 155
14.2 * 10.5 = 149.1
10.6 * 11.5 = 121.9
8.6 * 11.5 = 98.9
3.4 * 13 = 44.2
3.0 * 14 = 42

Total = 611.1 minutes, or over 10 hours. Does that sound reasonable?

10

It’s much more complicated working out preferences, mainly because there are only two runners who don’t prefer no trails and one of them is JF who wants two separate, short legs. JF can’t have trails and two short legs. It’s best for a fast time if JF just does one short leg, the second shortest and AM does the shortest.

11

The total distance is 55.3 miles, or an average of 9.217 per person. I think that to most evenly spread the distances, you’d need to have leg 6, and either leg 8 or 9 as the two single legs. Your combinations that most evenly spread the distances, but still ignoring preferences, would be

Leg 9: 7.7 miles, trail AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails
Leg 7: 3.0 miles AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails

Leg 3: 6.5 miles KS: pace 10:30 min/mile, no trail
Leg 5: 3.4 miles KS: pace 10:30 min/mile, no trail

Leg 10: 5.8 miles, trail AS: pace 11:30 min/mile, no trail
Leg 4: 4.0 miles AS: pace 11:30 min/mile, no trail

Leg 1: 4.8 miles LA: pace 11:30 min/mile
Leg 2: 4.6 miles LA: pace 11:30 min/mile

Leg 6: 7.8 miles JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs

Leg 8: 7.7 miles, trail AM: pace 14:00 min/mile, prefers 1 long leg, no trail

7.714 + 7.813 + 9.211.5 + 9.811.5 + 9.910.5 + 10.710 = 638.65 minutes, or 10:39. So about a half-hour longer to evenly spread the distances.

12

Slowest paces, ignoring preferences:

Leg 1: 4.8 miles AS: pace 11:30 min/mile, no trail – 55 minutes, 12 seconds
Leg 2: 4.6 miles LA: pace 11:30 min/mile – 52 minutes, 54 seconds
Leg 3: 6.5 miles JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs – 84 minutes, 30 seconds
Leg 4: 4.0 miles LA: pace 11:30 min/mile – 46 minutes
Leg 5: 3.4 miles KS: pace 10:30 min/mile, no trail – 35 minutes, 42 seconds
Leg 6: 7.8 miles AM: pace 14:00 min/mile, prefers 1 long leg, no trail – 109 minutes, 12 seconds
Leg 7: 3.0 miles AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails – 30 minutes
Leg 8: 7.7 miles, trail JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs – 100 minutes, 6 seconds
Leg 9: 7.7 miles, trail AM: pace 14:00 min/mile, prefers 1 long leg, no trail – 107 minutes, 48 seconds
Leg 10: 5.8 miles, trail AS: pace 11:30 min/mile, no trail – 66 minutes, 42 seconds

Total: 11 hours, 28 minutes, 6 seconds.

Of course, this assumes that trails have no effect on a runner’s speed.

13

The current leg assignments we are using put us around 10:55. So it sounds like we’re roughly in the middle of the possible range. (And the range isn’t all that extreme.) Not bad.

AT recently ran a half marathon (13.1 miles) with an overall 10:00 pace. 14.3 miles with a break in between should be no problem.

Yup, that fits with Richard Pearse’s numbers and also our best rough guess.

To be clear, those who have stated no preference about trails are willing to run on either trails or pavement. The legs that are not labeled as trails are paved roads.

Hm. My rough layout with my little pieces of paper came out to 10:40! Different legs, though.

The other thing to consider, of course, is that feelings are going to be hurt if we stick our slow runners with only a tiny portion of the course, essentially sticking them way out in far left field. We are all friends, we know we’re not going to win anything, and we’re mostly in it to have fun (which includes the ride in the team vehicle from checkpoint to checkpoint along the course). So it seems like we’re pretty much doing OK. It’s just gonna be a long day, and hopefully a lot of fun.

Thank you so much to everyone for applying your brains to the problem!

14

For a fast run accounting for preferences as well as I could:



Leg 1:  4.8 miles:        KS
Leg 2:  4.6 miles:        LA
Leg 3:  6.5 miles:        KS
Leg 4:  4.0 miles:        LA
Leg 5:  3.4 miles:        AM
Leg 6:  7.8 miles:        AT
Leg 7:  3.0 miles:        AT
Leg 8:  7.7 miles trail:  AS
Leg 9:  7.7 miles trail:  AS
Leg 10: 10.0 miles trail: JF

Starting with the fastest runners and working down:

AT gets her preferences.

KS runs the two longest non-trail legs left after AT has been assigned hers.

LA gets the next longest non-trail legs.

AS is the fastest runner who doesn’t mind running trails, she gets the two longest trail legs.

JF is the only other runner who doesn’t mind trails but is also inexperienced, one of the slowest, and wants two short legs. If we don’t have her run the 5.8 trail leg, one of the other runners won’t get their preference for no trails. Although you don’t say it, I’m considering JF as having a preference for trails or at least more of a preference for trails than those who’ve specifically said they don’t want trails. She gets the 5.7 trail leg but doesn’t get the two separate short legs she wanted.

AM is slowest and gets the shortest leg.

Total time is 10.43 hours or about 10:25

15

That should probably be counted as a preference as well “we want to have a roughly equal share of the workload.” As it is, my solution above has AS running about 5 times as far as AM which is not really what you want.

The difficulty is in assigning a value to preferences. How much does it matter to AT that she runs 6 and 7? How much does it matter to the no-trail runners that they run on a paved surface? (Rhetorical questions, I don’t expect you to try and quantify these things, just pointing out that at some point the numbers give way to gut feeling.)

ZenBeam has answered a question you didn’t ask (evenly spread distances), but I think what you are ultimately going for is just that, the most evenly spread workload across the runners taking their preferences into account.

16

I agree with Richard Pearse’s comment about it being hard to satisfy preference, because too many runners don’t want trails. Starting with my last post, and swapping to get the preferences a little closer, but still keeping the miles/person almost as close as before, I get:

Leg 6: 7.8 miles AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails
Leg 7: 3.0 miles AT: pace 10:00 min/mile, likes Legs 6 and 7, no trails

Leg 3: 6.5 miles KS: pace 10:30 min/mile, no trail
Leg 5: 3.4 miles KS: pace 10:30 min/mile, no trail

Leg 10: 5.8 miles, trail LA: pace 11:30 min/mile
Leg 4: 4.0 miles LA: pace 11:30 min/mile

Leg 1: 4.8 miles AS: pace 11:30 min/mile, no trail
Leg 2: 4.6 miles AS: pace 11:30 min/mile, no trail

Leg 9: 7.7 miles, trail JF: pace 13:00 min/mile, does not want Leg 1, prefers 2 separate legs

Leg 8: 7.7 miles, trail AM: pace 14:00 min/mile, prefers 1 long leg, no trail

7.714 + 7.713 + 9.211.5 + 9.811.5 + 9.910.5 + 10.810 = 638.35 minutes, or 10:38.

Richard Pearse’s assignments look good, but if you decide evenly spreading the mileage is more important, this would be better.

17

Thinking about that further, you might not want the most even distribution. Whatever competitiveness you have as a group will outweigh the desire for the slower runners to “do their share” and skew the groups desire away from a balanced run and towards a fast run. How far you get from a balanced run to the optimum fast run would depend on the value the group places on winning* as opposed to the value they place on participating and the value they place on each runner getting their stated preferences.

*Edit: Or placing as high as possible.

18

Er, um, OK, AS is me, and I double-checked to see that yes, I did say that I would prefer to avoid trails. So of course you give me 15 miles of trails!! :smiley: (But maybe I could swap some or all with LA, as we have the same pace (we train together) and she doesn’t mind trails.) This isn’t bad, except once again, I would feel bad about giving AM practically nothing to run.

Yeah, preferences make it more difficult, and some are not as strong as others. (For example, I say “no trails” for myself, but that’s just because I don’t like them; I would run one if necessary. KS, on the other hand, has a medical issue that would make a trail difficult for her, so I say she absolutely does not get one.)

And as I’ve said, we have zero chance of winning anything, and we want everyone to have a good and fun experience. That won’t happen if somebody feels slighted just so the team can gain a few minutes.

It’s looking to me like there isn’t really a LOT of difference timewise. I had thought the range would be wider.

It would have been nice if they could have just made all the legs equal! But the exchange points have to be located at safe places with room for parking. It’s not a closed course.

19

AM and AS don’t want to run on trails.

20

Oops. That’s what happens when you put data in Excel, then sort it alphabetically but forget to include one of the columns. My trail preferences were wrong.