Need formula - Brain teaser! But real situation

M

mark999

New member
Joined
Sep 5, 2022
Messages
5
I am setting up a road maintenance agreement on a private road with a mixture of full year and part year residents.
Each resident needs to pay a percentage of the amount dependant on two factors
1- the distance along the road so % of road used
2- if part year resident then only pay 50% of the cost a full year resident would pay.

Details
Road is 400m long
8 full time residents
11 part time residents

Residents live from 10m (2.5%) along the road to entire road 400m (100%)

What formula can I use to apply so all residents pay an amount that incorporates the 2 points above to reach say a figure of $10000 ( a random figure to keep it simple for me) per year.
Please help as I am now out of my depth.
 
If full time residents have road length fractions [imath]f_i[/imath] and part time residents have fractions [imath]p_i[/imath] and part time residents pay amount [imath]a[/imath] for the full-length then full time residents have to pay [imath]2a[/imath]. The total amount collected will be [imath]A = a\left(2\sum_i f_i + \sum_j p_j\right)[/imath]. Since you know that A=$10000 and know each [imath]p_i[/imath] and [imath]f_i[/imath] you can now figure out [imath]a[/imath]. Obviously full time residents will pay [imath]2af_i[/imath] and part time [imath]ap_i[/imath].
 
Looks awesome but need to breakdown further with actual example of one of the residents as formula too complicated for me to process.

full time resident is 350m along the road which is 87.5% (road length) can you add this to show how formula works?

part time resident is 160m along the road which is 40% (road length), can you do same again.

Thanks, appreciate the reply
 
mark999 said:
Looks awesome but need to breakdown further with actual example of one of the residents as formula too complicated for me to process.

full time resident is 350m along the road which is 87.5% (road length) can you add this to show how formula works?

part time resident is 160m along the road which is 40% (road length), can you do same again.

Thanks, appreciate the reply
Click to expand...
Exactly where you are getting lost? Do you understand the variables in:

A=a(2∑i fi+∑j pj)
 
I do not understand the variables, haven’t seen a symbol (backwards capital E before) so not sure what this represents
 
mark999 said:
Looks awesome but need to breakdown further with actual example of one of the residents as formula too complicated for me to process.

full time resident is 350m along the road which is 87.5% (road length) can you add this to show how formula works?

part time resident is 160m along the road which is 40% (road length), can you do same again.

Thanks, appreciate the reply
Click to expand...
The payment of each resident cannot be determined until [imath]a[/imath] is computed: obviously the payments depend on the total number of residents and their relative fractions. But we can say that the first (full time) resident's payment will be [imath]1.75a[/imath], and the second one [imath]0.4a[/imath].

We can further discuss this once you reply to @Subhotosh Khan's post.
 
@blamocur I'm not 100% sure about your method even though initially it seems very compelling (and it's clever). I can best explain my thoughts with a (perhaps extreme) example...

Consider the road having just 2 houses. 1 house at the end 400m. And 1 house near the start 40m. For ease of calculation they are all part time.

$10000 = a ( 1 + 0.1 ) therefore a ≈ 9090.91 and the house at 40m pays $909.09

Another way of thinking about this is that the first 40m of the road costs $1000 (one tenth of the total cost of $10000). Two houses need to use this section of road. Therefore the house at 40m should perhaps only pay $1000/2 = $500.00

What do you think?

But, perhaps the person at 40m should pay a bit more than $500 because the contractors might charge more than $1000 for surfacing a short (40m) section (if they weren't doing the whole length of the road). I'm just guessing here. It seems difficult to think of a reasonably fair method. :unsure::)
 
Cubist said:
@blamocur I'm not 100% sure about your method even though initially it seems very compelling (and it's clever). I can best explain my thoughts with a (perhaps extreme) example...

Consider the road having just 2 houses. 1 house at the end 400m. And 1 house near the start 40m. For ease of calculation they are all part time.

$10000 = a ( 1 + 0.1 ) therefore a ≈ 9090.91 and the house at 40m pays $909.09

Another way of thinking about this is that the first 40m of the road costs $1000 (one tenth of the total cost of $10000). Two houses need to use this section of road. Therefore the house at 40m should perhaps only pay $1000/2 = $500.00

What do you think?

But, perhaps the person at 40m should pay a bit more than $500 because the contractors might charge more than $1000 for surfacing a short (40m) section (if they weren't doing the whole length of the road). I'm just guessing here. It seems difficult to think of a reasonably fair method. :unsure::)
Click to expand...
I haven't thought much about which way is fair, I simply assumed that the payments should be proportional to the length of the road used. Under such assumption the first house in your example should pay 10 times more than the second one -- shouldn't it?

But the assumption of linearity on my part is just an assumption. I agree that in real life the costs are rarely linear functions of the length. E.g., the contractor still has to bring her/his machines, for example, and pay for their transportation whether she/he is paving all 400m or just 40m. Both linearity, just as sum of squares, is a good illustration of streetlight effect.
 
Thank you both for your input. I am arranging the agreement to be fair and need it mathematically accurate otherwise will get pushback I’m sure.
The 2 house example above is fairly straight forward including part time resident paying 50%, unfortunately with 19 homes (11 part time and 8 full time) all with different distances I just can’t figure out how to work it out equally. Once I have the math or percentage each pays then can adjust for fairness if needed.

I will add each house distance and add either PT - part time or FT - full time, I appreciate everyone’s time is precious so I didn’t previously include full details of each house as wasn't expecting anyone to work it out for me but for ease of understanding or certainly helping me out, here is what I have:

1- PT 10m
2- FT 60m
3- FT 100m
4&5- PT 140m
6- FT 160m
7&8 - PT 160m
9- FT 170m
10- PT 190m
11- FT 190m
12&13- PT 230m
14- FT 250m
15- PT 280m
16- PT 350m
17- FT 350m
18- FT 400m
19- PT 400m
 
blamocur said:
I haven't thought much about which way is fair, I simply assumed that the payments should be proportional to the length of the road used. Under such assumption the first house in your example should pay 10 times more than the second one -- shouldn't it?
Click to expand...
I personally think that if a section of the road is most often used by a particular set of people, then the cost of that section ought to be shared between that set. EDIT: so the house at the far end should pay the full amount for the 9/10 of the road that only they will use. The cost for the remaining 1/10 of the road, the part that is shared, ought to be split between them.

Of course it all comes down to what this group of residents will collectively perceive as fair (what they think is "correct" or easiest/ least bother for them).

I just know that if my nearest #*@! neighbours were suddenly 360m away from me then I'd have a massive smile from ear to ear and I'd happily bear the whole cost of any private road maintenance myself (not caring which end of the street I was at) :ROFLMAO:
 
Last edited:
mark999 said:
... here is what I have:

1- PT 10m
2- FT 60m
...
Click to expand...

Thanks. I recommend that you put this data into a spreadsheet.
Initially, I'd aim to perform the calculation suggested by @blamocur in post#2. Here's a suggested layout...
Code: 
FT dist   |    PT dist   |   payment due
          |      10      |     ?
   60     |              |     ?
  100     |              |     ?
  ...
  400     |              |     ?
          |     400      |     ?
==========================================
? SUM f ? |  ? SUM p ?   |
------------------------------------------
A         |   10000      |
a         |     ???      |
...would you be able to calculate the cells with question marks? Start with the sums and then the "a" amount

EDIT: You could show your spreadsheet here by posting a screenshot image
 
Since the full-time residents pay twice as much, then assume that they have two houses at their location and the part-time residents have one house at their location.

Break down the 400m into 10m segments. Assuming that the total amount to maintain the road is $10,000, then each 10m cost $10,000/40 = $250

Assume that there are 2*8 + 11 =27 homes


1- PT 10m---Each homeowner (remember that full-time residents are assumed to own two homes!) will pay $250/27=$9.26 for the first 10m
2- FT 60m--Total for the next 50m is $1250--Each of the 26 remaining home owners will pay $1250/26 for the road from 10m to 60m
3- FT 100m--Total for the next 40m is $1000-Each of the 24 remaining home owners will pay $1000/24
4&5- PT 140m--Total for the next 40m is $1000-Each of the 22 remaining home owners will pay $1000/22
6- FT 160m
7&8 - PT 160m
9- FT 170m
10- PT 190m
11- FT 190m
12&13- PT 230m
14- FT 250m
15- PT 280m
16- PT 350m
17- FT 350m
18- FT 400m
19- PT 400m
 
The backwards E is a summation sign (it is actually a capital Greek sigma standing for sum).

I think cubist is right that eventually the computations should be done with a spread sheet.

But let’s think about what will seem fair. We are assuming the full cost is 10000. We shall assume 5 houses with three full timers and 2 part timers.For the road to the first house is 20 meters, to the second is 140, to the third is 210, to the fourth is 300, and to the last is 400. We assume only the odd numbered houses are full time.

What is the linear cost per meter? [imath]10000 \div 400 = 25.[/imath]

Counting each full timer as equivalent to two part timers, we have [imath]3 \times 2 + 2 = 8[/imath] part time equivalents.

Everyone uses the first 20 meters. The cost is [imath]20 \times 25 = 500.[/imath]

Dividing that cost among the eight part time equivalents gives [imath]500 \div 8 = 62.50[/imath].

So, FOR THAT PART OF THE ROAD, the resident of houses 1, 3, and 5 (full timers) are each charged 125, and the residents of houses 2 and 4 are each charged 62.50.

The resident of house 1 does not benefit from the rest of road. The part time equivalents who do number
[imath]2 \times 2 + 2 = 6.[/imath] The length of road between house 1 and house 2 is
[imath]140 - 20 = 120.[/imath] The cost is [imath]120 \times 25 = 3000.[/imath]
But that gets divied up into 6 part time equivalents of 500 each. So houses 2 and 4 are each charged 500 for that part of the road and houses 3 and 5 are charged 1000 each.

The residents of houses 1 and 2 do not benefit from the rest of the road. The part time equivalents who do number
[imath]6-1=5.[/imath] The length of road between house 2 and 3 is
[imath]210 - 140 = 70.[/imath] The cost is [imath]70 \times 25 = 1750.[/imath]
Divided 5 ways gives 350. So, for the third portion of the road, houses 3 and 5 (full timers) are each charged 700, and house 4 is charged 350.

The residents of houses 1, 2, and 3 do not benefit from the rest of the road. The part time equivalents who do number
[imath]5 - 2 = 3.[/imath] The length of road between house 3 and 4 is
[imath]300 - 210 = 90.[/imath] The cost is [imath]90 \times 25 = 2250[/imath]. Divided by 3 gives 750. So, for this part of the road, house 4 (part timer) is charged 750 and house 5 is charged 1500.

Only 1 resident benefits from house 5. The length of road between from house 4 to house 5 is 100. So the cost is 2500.

Resident 5 is charged 2500 + 1500 + 700 + 1000 + 125 = 5825.
Resident 4 is charged 750 + 350 + 500 + 62.50 = 1662.50.
Resident 3 is charged 700 + 1000 + 125 = 1825.
Resident 2 is charged 500 + 62.50 = 562.50.
Resident 1 is charged 125.

To check, add them up.
5825 + 1662.50 + 1825 + 562.50 + 125 = 10000.
 
When I get home from work I will look in depth but thanks everyone for breaking down for me. From my initial read I should be good, once I work it all out will share feedback.

A spreadsheet will definitely be applied here.
This is math being used for real situations! Love it!
 
mark999 said:
When I get home from work I will look in depth but thanks everyone for breaking down for me. From my initial read I should be good, once I work it all out will share feedback.

A spreadsheet will definitely be applied here.
This is math being used for real situations! Love it!
Click to expand...
What the most recent posts have done is to provide an algorithm rather than a formula. Algorithms are useful because they are procedures that are easily implemented on a computer.

The point about non-linearity is a good one. But it may be impossible to get a bid that separates fixed and variable costs. Moreover, if fixed costs are sizable, allocating them in a way that generates opting in by all parties may be very difficult. Thus, I ignored fixed costs, not because I believe the issue to be unreal, but rather because I suspect it is unmanageable.
 
You must log in or register to reply here.
Top