A maths puzzle

[threebikesmcginty]

Normal service will resume shortly...

 

[DaveReading]

Weighting is only ever done in relation to other values, if it is ever entirely arbitrary it becomes meaningless.

In this case it is noise values. If the event is more noisy than the other six it has a multiplier applied - if all are equally noisy then no multiplier would be applied.

So to get your weighting multiplier you need to know the cda values for each event, average them, then determine each as a percentage of that average. Then apply that to the event.

I think you're misinterpreting the problem. Maybe we should stick with the sporting analogy, if it makes the problem more accessible than incorporating the somewhat techy subject matter. Think of it just as a maths puzzle

The weighting is applied to the relative rankings in each category. You don't need to know the actual category scores, only the placings. That's what the overall score is based on.

If there was no weighting, then one place higher in any category would cancel out one place lower in any other category.

If different categories are weighted differently, then it's not a one-for-one balance. So, as per my earlier example, if the relative weightings for two categories are 150% and 50%, then a ranking of one place higher in category X would cancel out 3 places lower in category Y.

It makes no difference whether you express the weightings as 150%:50%, or 3-to-1, or any other equivalent ratio, you'll get the same result. Bear in mind you're going to adjusting the final results so that they sit on the arbitrary 0-1000 points scale, so it doesn't matter what multiplier you use at the intermediate stage.

So, at the risk of repeating myself, knowing the rank per category, the weightings (however expressed) and the maximum/minimum achievable overall score range should be sufficient information to arrive at an answer.

Isn't it ?

 

[jefmcg]

I generally like maths puzzles, but it seems here you have taken something fairly dull and confusing and made it a bit duller and a lot more confusing.

HTH.

 

[DaveReading]

I generally like maths puzzles, but it seems here you have taken something fairly dull and confusing and made it a bit duller and a lot more confusing.

As indeed many practical applications of maths are ...

Though, as you have sussed despite my attempt to disguise the origin, there could be potential implications for an organisation that appears to have published stats based on flaky arithmetic.

I still live in hope that somebody on CC with a sharp enough pencil will provide those two numerical answers that I can then compare with my own.

Failing that, tomorrow I'll post a worked example and brace myself to have it torn to shreds in a wave of apathy.

 

[Tin Pot]

I think you're misinterpreting the problem. Maybe we should stick with the sporting analogy, if it makes the problem more accessible than incorporating the somewhat techy subject matter. Think of it just as a maths puzzle

The weighting is applied to the relative rankings in each category. You don't need to know the actual category scores, only the placings. That's what the overall score is based on.

If there was no weighting, then one place higher in any category would cancel out one place lower in any other category.

If different categories are weighted differently, then it's not a one-for-one balance. So, as per my earlier example, if the relative weightings for two categories are 150% and 50%, then a ranking of one place higher in category X would cancel out 3 places lower in category Y.

It makes no difference whether you express the weightings as 150%:50%, or 3-to-1, or any other equivalent ratio, you'll get the same result. Bear in mind you're going to adjusting the final results so that they sit on the arbitrary 0-1000 points scale, so it doesn't matter what multiplier you use at the intermediate stage.

So, at the risk of repeating myself, knowing the rank per category, the weightings (however expressed) and the maximum/minimum achievable overall score range should be sufficient information to arrive at an answer.

Isn't it ?

Well, no.

The problem is your 1000 pt absolute, and your relative weighting.

Using the racing analogy, you know the rank after one race, but you don't know whether first place gets 100ots or 300pts because the weighting isn't determined yet.

In racing this never happens, all races are equal so points can be awarded for each race position easily.

You would never want to tell an airline hey scored 300 last night, and then two days later say that score is only 100 because the subsequent flights were at dramatically different noise levels. No one would trust your points system.

What you can do is determine the ranking after each race, as this is allowed to flow as it does in a league.

 

[Tin Pot]

In your position I would use cumulative ranking on a daily basis, and publish the points according to the final rank at the end of a given reporting period.

 

[DaveReading]

You would never want to tell an airline hey scored 300 last night, and then two days later say that score is only 100 because the subsequent flights were at dramatically different noise levels. No one would trust your points system.

That doesn't happen. All the participants know from the outset which categories have higher/lower weightings. And the points scored in an individual category are irrelevant (we're not given that information and we don't need it) - it's just the ranking that matters in determining the final score.

 

[Tin Pot]

That doesn't happen. All the participants know from the outset which categories have higher/lower weightings. And the points scored in an individual category are irrelevant (we're not given that information and we don't need it) - it's just the ranking that matters in determining the final score.

By category you mean event, as per the original parameters in the OP?

You still have a cumulative weighted ranking problem. I don't think any standard functions handle the weighted part.

 

[nickyboy]

By category you mean event, as per the original parameters in the OP?

You still have a cumulative weighted ranking problem. I don't think any standard functions handle the weighted part.

You're right, they don't

That's why I haven't spent time wrestling with this. The question, as couched by the OP, isn't solvable

 

[DaveReading]

That's why I haven't spent time wrestling with this. The question, as couched by the OP, isn't solvable

OK, if that's your view.

My only comment at this stage is that you are implying that not only is my description of the problem incomplete (which is fair enough) but also the originator's description that I reproduced in post #26.

Anyway, as I promised, I'll post a worked example tomorrow and see if I can change your mind.

 

[ohnovino]

a) 676
b) 628

Start by working out the points available in each event, with the constraints that we know the relative weightings and the total points must sum to 1000. The sum of the weightings comes to 560, so if you multiply each event's weighting by 1000/560 you'll get the points per event e.g. Event 1 has 89.2857 points up for grabs.

Then for each event, divide the points available by 49 to get a figure for points per place; you divide by 49 rather than 50 because you need last place to score zero (i.e. there are 49 scoring positions). So again, Event 1 earns 1.822 points per position.

Then the score for each event is: Points per place * (50-rank). So coming 5th in Event 1 would earn you 1.822*45 = 81.99

ETA: I've ignored tied places because I think that would make it unsolvable

 

[DaveReading]

a) 676
b) 628

Start by working out the points available in each event, with the constraints that we know the relative weightings and the total points must sum to 1000. The sum of the weightings comes to 560, so if you multiply each event's weighting by 1000/560 you'll get the points per event e.g. Event 1 has 89.2857 points up for grabs.

Then for each event, divide the points available by 49 to get a figure for points per place; you divide by 49 rather than 50 because you need last place to score zero (i.e. there are 49 scoring positions). So again, Event 1 earns 1.822 points per position.

Then the score for each event is: Points per place * (50-rank). So coming 5th in Event 1 would earn you 1.822*45 = 81.99

ETA: I've ignored tied places because I think that would make it unsolvable

That's interesting. I came up with 757 and 628 points, respectively, so we're agreed on one team's score but not the other. I'll double-check my arithmetic.

Incidentally the original real-life example does appear to allow for tied places, I don't think there's anything in the algorithm that blows up as a result.

 

[ohnovino]

That's interesting. I came up with 757 and 628 points, respectively, so we're agreed on one team's score but not the other. I'll double-check my arithmetic.

Incidentally the original real-life example does appear to allow for tied places, I don't think there's anything in the algorithm that blows up as a result.

You're right with 757 - I tried to speed things up with an Excel sheet and I must have crtl-clicked the wrong box.

To solve for tied places you'd need to know how it affects the scores. For example, do two people tied at the top both score points equivalent to 1st place, or do they both score the average of 1st and 2nd.

BTW I looked at the real-world figures and can't for the life of me work out how the scores are so high, and how BA can finish almost bottom of one ranking and still score 95% overall.

 

[DaveReading]

You're right with 757 - I tried to speed things up with an Excel sheet and I must have crtl-clicked the wrong box.

To solve for tied places you'd need to know how it affects the scores. For example, do two people tied at the top both score points equivalent to 1st place, or do they both score the average of 1st and 2nd.

BTW I looked at the real-world figures and can't for the life of me work out how the scores are so high, and how BA can finish almost bottom of one ranking and still score 95% overall.

Thanks for the confirmation. Yes, all the scores appear to be wrong, but not by the same amount, so the resulting ranking is wrong, too.

Oh, and it's traditional that BA come top. They're actually fifth when calculated correctly, I think.

Joint firsts (or any other joint place) should get the same points as if they were the only occupant of that slot. It just bumps up the overall average score to around 525 instead of the theoretical 500.

Thanks to all who participated and humoured my mathematical foibles.

 

[srw]

They're actually fifth when calculated correctly, I think.

The chances of a publicly available ranking system calculated using a publicly available methodology and published by a major international airport waiting for a government decision and public support to approve a major investment being calculated wrong are approximately zero.

It's not impossible that the ranking has been designed to favour a particular airline, and at least one of the metrics depends directly on decisions made by ATC, so in theory could be frigged by favouring the planes of a particular airline, but it's extremely unlikely that the calculations are flawed. It's more likely that you've cocked up somewhere.

If you're sure you haven't, there are plenty of pressure groups and investigative journalists who have the time and resources to take you up.