Density & distance

[mummymonkey]

Is there a way to calculate the mean distance between points as the density of the points is changed.
I need to work out what the increase in mileage driven by our field workforce would be if the jobs were done over 6 days instead of 5.

 

[rppa]

Is there a way to calculate the mean distance between points as the density of the points is changed.

Yes.

I need to work out what the increase in mileage driven by our field workforce would be if the jobs were done over 6 days instead of 5.

I'm not sure exactly what the model is and how this relates to density, but given a model the mean distance can be estimated from it.

 

[Crossbow]

Is there a way to calculate the mean distance between points as the density of the points is changed.
I need to work out what the increase in mileage driven by our field workforce would be if the jobs were done over 6 days instead of 5.

Um, I am not really sure that you are defining the problem all that well: the density of the points does not affect there respective distance(s).

For example, if one has two ice cubes with an edge length of say 1 inch that are separated by say 10 feet. Such a short edge length compared to the distance between the object should enable one to treat them as a point.

Anyway, these objects would still be 10 feet apart even if the ice cubes were somehow transformed (without moving them) into wood cubes of the same size, or lead cubes of the same size, or foam cubes of the same size, and so on.

 

[mummymonkey]

I'm not explaining myself very well, sorry.

The proposal is that we move from 5 day working to 6 day working. The number of jobs and the geographical location of the jobs we do, will not alter. (We visit customers in their homes)

On each given day therefore there will be less jobs. Which means they will be further apart (less dense).

I've been asked to estimate the increase in miles/job over the week.

 

[T'ai Chi]

The proposal is that we move from 5 day working to 6 day working. The number of jobs and the geographical location of the jobs we do, will not alter. (We visit customers in their homes)

On each given day therefore there will be less jobs. Which means they will be further apart (less dense).

I've been asked to estimate the increase in miles/job over the week.

Interesting problem. Thinking about this some more, I'm trying to get some math notation going, but some more questions:

Do you always serve the same customers? So if you have 10 points on a grid, representing the locations of customers you serve, those points will not change once you go to working 6 days?

And, for a job, do you always drive from headquarters to the job, or, in addition, do you drive from one job to another job?

 

[mummymonkey]

Do you always serve the same customers? So if you have 10 points on a grid, representing the locations of customers you serve, those points will not change once you go to working 6 days?

Moving to 6 working days won't change the locations of the customers. They are randomly located within a fixed area.
Imagine a town with 100,000 houses, we might visit 12 houses each day, but the next day it will be 12 different houses. It's random. Over the week we visit 60 houses. If we go to 6 day working we would visit the same 60 houses, but 10 a day.

And, for a job, do you always drive from headquarters to the job, or, in addition, do you drive from one job to another job?

The technicians work from home, drive to each customer in turn, then go home again.

 

[Crossbow]

Moving to 6 working days won't change the locations of the customers. They are randomly located within a fixed area.
Imagine a town with 100,000 houses, we might visit 12 houses each day, but the next day it will be 12 different houses. It's random. Over the week we visit 60 houses. If we go to 6 day working we would visit the same 60 houses, but 10 a day.

The technicians work from home, drive to each customer in turn, then go home again.

Thanks for the clarification! It really helps.

Well now, it sounds to me as if the total number of miles covered per week will still be the same since your firm will still be making the same number of stops with the same number of employees using the same methodology.

However, the average miles traveled per workday per employee will decrease by 16.7%.

Example (using some numbers I just made up, but the results should be valid for your situation as well):

Case I --
Total Distance Traveled by all Employees: 1500
Total Number of Employees: 5
Total Number of Workdays/Week: 5

Average Distance Traveled per Employee Per Day =
Total Distance Traveled by all Employees/ Total Number of Employees/ Total Number of Workdays/Week
1500/5/5
Average Distance Traveled per Employee Per Workday = 60

Case II --
Total Distance Traveled by all Employees: 1500
Total Number of Employees: 5
Total Number of Workdays/Week: 6

Average Distance Traveled per Employee Per Day =
Total Distance Traveled by all Employees/ Total Number of Employees/ Total Number of Workdays/Week
1500/5/6
Average Distance Traveled per Employee Per Workday = 50

Conclusion --
Increasing the ‘Total Number of Workdays/Week’ from 5 to 6 and leaving everything else constant will result in the ‘Average Distance Traveled per Employee Per Workday’ being decreased by 16.7%.

I hope this helps!

 

[T'ai Chi]

Moving to 6 working days won't change the locations of the customers. They are randomly located within a fixed area.
Imagine a town with 100,000 houses, we might visit 12 houses each day, but the next day it will be 12 different houses. It's random. Over the week we visit 60 houses. If we go to 6 day working we would visit the same 60 houses, but 10 a day.

The technicians work from home, drive to each customer in turn, then go home again.

Ok. Thanks.

Do you have past data, and if so, what is the variation in the total number of jobs worked per day? What is the range of it (min, max)?

I put up a quick page here.

Please disregard my thinking out loud about the linear regression stuff. It may be totally bogus, not sure right now, haven't thought it through fully yet, and it is a mistake to get a theory going before having data... but I thought I'd type it up anyway because it is what came to mind first, and then follow it with this lengthy disclaimer. I just want to make sure the first pic, and the description, is correct.

 

[mummymonkey]

Do you have past data, and if so, what is the variation in the total number of jobs worked per day? What is the range of it (min, max)?

I put up a quick page here.

Please disregard my thinking out loud about the linear regression stuff. It may be totally bogus, not sure right now, haven't thought it through fully yet, and it is a mistake to get a theory going before having data... but I thought I'd type it up anyway because it is what came to mind first, and then follow it with this lengthy disclaimer. I just want to make sure the first pic, and the description, is correct.

Isn't the internet great!
We do between 6000 and 7000 jobs each week. Miles/job runs at about 11. Your picture of the house and the route is spot on.

 

[rppa]

Well now, it sounds to me as if the total number of miles covered per week will still be the same since your firm will still be making the same number of stops with the same number of employees using the same methodology.

However, the average miles traveled per workday per employee will decrease by 16.7%.

I agree with both predictions. Also the number OP is asking for, the number of miles per job, won't change. That's just the average distance between successive customers. If they're truly randomly selected from anywhere around town, then it doesn't matter whether you service them on a 5-day or 6-day schedule.

 

[T'ai Chi]

Isn't the internet great!
We do between 6000 and 7000 jobs each week. Miles/job runs at about 11. Your picture of the house and the route is spot on.

Ok great.

And would the company have kept past data (say two columns in a spreadsheet) like

total distance per day, total number of jobs per day

?

 

[69dodge]

I agree with both predictions. Also the number OP is asking for, the number of miles per job, won't change. That's just the average distance between successive customers. If they're truly randomly selected from anywhere around town, then it doesn't matter whether you service them on a 5-day or 6-day schedule.

On a 6-day schedule, there's one extra trip home at the end of a day.

mummymonkey, are the trips really random? Is any effort made to give people jobs that are close to their home or close to each other? Do people get their list of jobs at the beginning of the day so they can plan an efficient route?

 

[mummymonkey]

On a 6-day schedule, there's one extra trip home at the end of a day.

Having slept on it I'm coming round to the view that a close approximation would be to add 2 trips to the weekly load. An extra trip to the first job and a further extra trip back from the last one.

mummymonkey, are the trips really random? Is any effort made to give people jobs that are close to their home or close to each other? Do people get their list of jobs at the beginning of the day so they can plan an efficient route?

The jobs are allocated to each employee by some routing software but the location of each job is pretty random. Their workload is sent each night.

 

[mummymonkey]

Ok great.

And would the company have kept past data (say two columns in a spreadsheet) like

total distance per day, total number of jobs per day

?

I have data for the past few weeks. Mileage is collected weekly. Just divide by 5 for the daily split, there are no days that are significantly busier than the rest.

data

Edit: Changed data to a link

 

[69dodge]

Originally posted by mummymonkey
Having slept on it I'm coming round to the view that a close approximation would be to add 2 trips to the weekly load. An extra trip to the first job and a further extra trip back from the last one.

The trip to the first job of day 6 isn't really extra, because you have to get there in any case. It's just a question of whether you'll go there from home or from a previous job. If trips from home to a first job are about the same length as trips between jobs, then it doesn't matter at all. If on average there's some difference in length between the two types of trip, you'd need to take that difference into account, but counting the entire trip as extra is too much.

The jobs are allocated to each employee by some routing software but the location of each job is pretty random. Their workload is sent each night.

How flexible is this routing software? Can you rerun it on previous jobs, telling it to assume a 6 day week, and see what it does? I guess that would be the best thing to do, because that's what you actually would be doing in the future if you did lengthen the work week.

 

[Mason]

My old job had me driving crazy routes, so this is more based on anecdotal experience than the math...

Assuming the set of points has been selected for relative closeness before being assigned, and assuming the driver has mapped them for his personal best efficiency, and assuming a rough average (just for this example) of roughly one mile...

Say from point A to point B is one mile. If the driver has one extra day (day six) he has eleven miles to drive on that day. If, however, he has those points divided throughout the week, he may find that point Q is 3/4 mile from A and 3/4 mile from B, changing his route from A-B of one mile to A-Q-B of 1 1/2 mile.

Spacing this average throughout the week, he ends up with one extra mile per day for five days (2 1/2 miles) vs eleven miles on the sixth day.

Individual results will vary, of course. The drivers may find that they prefer their work spread out to give them more time to complete individual jobs, or they might prefer to be done in five days and rest on the sixth. Driving can take a lot out of a body, so a sixth day may be more stress and fatigue than it's worth.

Could it be worth looking into setting up a weekly workload so the drivers can map their weeks in advance? Perhaps mapping efficiency would allow them to do something like 9-10-9-12-10-10 or 9-13-11-14-13, depending on their own talents and preferences.

Just one angle.

 

[T'ai Chi]

It's not exactly what you are looking for, but hope it helps.

 

[mummymonkey]

It's not exactly what you are looking for, but hope it helps.

I appreciate all your efforts. I'm going to say 5% increase in fuel costs and hope they don't question me too much.

Many thanks to all who contributed.