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I'm currently taking a mathematical modeling class and am slightly baffled by this example in my textbook. This example comes from chapter 3.2 (on multivariable optimization) of Mathematical Modeling, Fourth Edition, by Mark Meerschaert.

It starts with the following map/table of the number of emergency calls per year in each square mile of a particular city. Up is north and right is east. (So, for example, the most-northwest square mile of the city got 3 calls, and the square mile to the east of that one got 0).

$$\begin{matrix} 3 & 0 & 1 & 4 & 2 & 1 \\ 2 & 1 & 1 & 2 & 3 & 2 \\ 5 & 3 & 3 & 0 & 1 & 2 \\ 8 & 5 & 2 & 1 & 0 & 0\\ 10 & 6 & 3 & 1 & 3 & 1\\ 0 & 2 & 3 & 1 & 1 & 1 \end{matrix} $$

The example reads as follows:

A suburban community intends to replace its old fire station with a new facility. The old station was located at the historical city center. City planners intended to locate the new facility more scientifically. A statistical analysis of response-time data yielded an estimate of $3.2+1.7r^{0.91}$ minutes required to respond to a call $r$ miles away from the station. Estimates of the frequency of calls from different areas of the city were obtained from the fire chief. They are presented [in the above matrix]. Each block represents one square mile, and the numbers inside each block represent the number of emergency calls per year for that block. Find the best location for the new facility.

We will represent locations on the city map by coordinates $(x, y)$, where $x$ is the distance in miles to the west side of town and $y$ is the distance in miles to the south side. For example, $(0, 0)$ represents the lower left-hand corner of the map, and $(0, 6)$ are upper left-hand corner. For simplicity we will divide the city into nine 2 x 2 mile squares and assume that each emergency is located in the center of the square.

It then states that, if $(x, y)$ is the location of the new fire station, the average response time to a call is $z = f(x, y)$, where

$$ z = 3.2+1.7[6\sqrt{(x - 1)^2+(y - 5)^2}^{0.91} + 8\sqrt{(x-3)^2+(y-5)^2}^{0.91} + 8\sqrt{(x-5)^2+(y-5)^2}^{0.91} + 21\sqrt{(x-1)^2+(y-3)^2}^{0.91} + 6\sqrt{(x-3)^2+(y-3)^2}^{0.91} + 3\sqrt{(x-5)^2+(y-3)^2}^{0.91} + 18\sqrt{(x-5)^2+(y-3)^2}^{0.91} + 8\sqrt{(x-3)^2+(y-1)^2}^{0.91} + 6\sqrt{(x-5)^2+(y-1)^2}^{0.91}]/84 $$

Just in case I made a transcription error (which isn't at all unlikely), here's a picture of the equation as well: enter image description here

I'm slightly baffled as to how they got this equation. Take, for example, $6\sqrt{(x-1)^2 + (y - 5)^2}^{0.91}$ - I understand where they got $0.91$, but what about the rest of the equation? Where did they get 6, for example, or $(x - 1)$? Can someone explain why this equation actually predicts the average response time and how they came up with that?

TL;dr They also have the following graph, for what it's worth: enter image description here

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The last sentence in your example is important:

For simplicity we will divide the city into nine 2 x 2 mile squares and assume that each emergency is located in the center of the square.

This means that you should divide your array into 3 rows and 3 columns of $2 \times 2$ subarrays. The coordinates for the center of the subarray are given inside the distance functions and the coefficients of the distance are the sum of the entries in that particular subarray.

For instance, the upper left $2 \times 2$ block has center $(1,5)$ and the sum of the entries is $3+0+2+1 = 6$. So, this contributes $6 \sqrt{(x-1)^2 + (y-5)^2}$ to your function.

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  • $\begingroup$ That makes a lot more sense, then - they're basically looking at every subarray (hence the 9 different equations, since obviously 3x3 = 9)? $\endgroup$ – EJoshuaS - Reinstate Monica Mar 1 '17 at 20:51
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    $\begingroup$ That is exactly correct. $\endgroup$ – user197427 Mar 1 '17 at 22:26

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