Is there any defined mathematical parameter that can give me the average distance of a volume to a point?

Question

I'm currently trying to design a bulk-handling system which minimizes the distance from the bulk material pile(s) where the material is loaded (the volume) to the unloading platform(s) (the point), but I have many possible combinations on the volume shape and the number of unloading point. I would ideally like to iterate through various designs in order to find the most feasible one, and a useful parameter to get from each design would be to get this average travel distance from the pile to the platform.

I have toyed around with the idea of using the centroid as the average distance to a point, but the problem is that the directional component of the centroid makes it such that it doesn't truly represent the average distance. For example, a disk with a radius of 10 meters might have the same centroid as a disk with a radius of 100 meters, but the average distance from the center would be different.

$$\int_{V}r\text{ d}V$$

A slightly better representation would be the moment of inertia, since the squared component better represents average distance (so that opposite directions don't cancel out).

$$\int_V r^2\text{ d}V$$

However, to truly get the average distance, I would need something that looks more like this:

$$\int_V \lvert r\rvert\text{ d}V$$

Is there any simple way of getting this value from a CAD software package, or is it derivable from any other existing mathematical definitions like the centroid or the moment of inertia? Similarly, if this average distance is obtained only relative to the centroid, how could I extrapolate it to another arbitrary point?

Answer 1

First, let's consider a Cartesian plane with the loading platform at the origin. It is obvious that the distance to any point is on the plane is: $l = \sqrt{x^2+y^2}$.

If you extend this to the $3^{rd}$ dimension by plotting $l$ on the $z$-axis, you get an inverted cone with a pitch of 90 degrees with its tip at the origin. Your material piles will be projected onto this cone.

The total distance to all points on the pile from the loading bay will be the volume under the cone bounded by the area of the piles:

$$D=\int\int{\sqrt{x^2+y^2}}dydx$$

To calculate the average distance per trip, you need to divide the total distance by the number of trips. The non-discreet version of this equates to the volume under the cone divided by the plan area of the pile:

$$d=\frac{D}{A}$$

P.S.

To solve your real problem, the obvious answer is to have as much material as possible as close as possible to your loading platforms.

Just some additional info that might help with your problem

Obviously you can translate the functions to other points than the origin if you have to plot multiple loading platforms. Multiple cones will intersect each other at some point in space, and viewed in plan, these intersections will form lines enclosing the platforms in "cells", with any point within a cell being closest to the origin of the cell. This is called a Voronoi Diagram:

Image from Mathworks

You can use this as a graphic aid to assist with the placement and shape of the piles. I'll leave the determination of the ideal shapes and placements as an exercise to the reader.

Given that we don't know any of the restrictions, there won't be one optimal solution. (For all intents and purposes, with the info we have we could just as well put the stockpile right on the loading platform in a column of infinite height, thus the average distance would be 0).

Answer 2

Having been involved with a similar situation many, many years ago, I'm not aware of a simple equation that will solve the problem.

If you are good, or you know someone who good, at computer programming, or you could use a spreadsheet, you can use an imaginary grid. Fix either the volume location or loading location in the grid and then iteratively place the other location at points on the grid and calculate the distance between the two.

From this you will get a 2D array of distance data from which you can then develop contours of distance between the two locations.

You then have to repeat the procedure for different loading location or volume locations.

Answer 3

Based off of ChP and Fred's answers, I ended up coding the 2-D solution for the problem. I might eventually want to extrapolate it to 3-D, but it shouldn't be too complicated.

I used a "clipped" Voronoi diagram, code examples of which are readily available online, which I used to separate an arbitrary polygon (representing the pile) into different sections, each representing the area closest to one of the given points (representing the unloading/loading points). Then, I placed a uniform point grid inside of the polygon, and solved for the distance of each of these with respect to the closest loading point. After getting an array of these values, I averaged them to get the average distance. It's rudimentary, but it's accurate enough for my particular purpose.