Question

Monroe County is trying to determine where to place the county fire station. The locations of the county’s four major towns are as follows: (10, 20), (60, 20), (40, 30), and (80, 60) (see Figure 7.50). Town 1 averages 40 fires per year; town 2, 25 fires; town 3, 20 fires; and town 4, 30 fires. The county wants to build the fire station in a location that minimizes the average distance that a fire engine must travel to respond to a fire. Because most roads run in either an east-west or a north-south direction, the fire engine must do the same. For example, if the fire station is located at (30, 40) and a fire occurs at town 4, the fire engine has to travel  miles to the fire.

1. Determine where the fire station should be located.

2. Use SolverTable to see how the optimal location of the fire station changes as the number of fires at town 3 changes.

Answer 1

The locations of the four major towns are:

Town 1: (10,20)

Town 2: (60,20)

Town 3: (40,30)

Town 4: (80,60)

Number of fires in the four major towns are:

Town 1: 40 per year

Town 2: 25 per year

Town 3: 20 per year

Town 4: 30 per year.

The x co-ordinate and y-coordinate of the facility can be determined by the number of fires in towns and their location.

We know that x-coordinate of the facility location can be mentioned as

(T1x*T1f + T2x*T2f + T3x*T3f + T4x*T4f)/(T1f+T2f+T3f+T4f)

Where Tix is the x coordinate of ith town

Tif is the number of fires in ith town

Similarly y-coordinate of the facility location can be calculated by:

(T1y*T1f + T2y*T2f + T3y*T3f + T4y*T4f)/(T1f+T2f+T3f+T4f)

Where Tiy is the y coordinate of ith town

Tif is the number of fires in ith town

Hence finding x and y coordinate

x = (10*40+60*25+40*20+80*30)/(40+25+20+30) = (5100/115) = 44.34

y = (20*40+20*25+30*20+60*30)/(40+25+20+30) = (3700/115) = 32.17

Hence the location of the fire station must be: (44,32)

If we change the number of fires in town 3, we will get different values of x and y coordinate.

Example:

With original given data

Town	x	y	Number of Fire		Fire Station
1	10	20	40		x	y
2	60	20	25		44.34783	32.17391
3	40	30	20
4	80	60	30

If we make number of fires in town 3 to 30

Town	x	y	Number of Fire			Fire Station
1	10	20	40		x		y
2	60	20	25		44		32
3	40	30	30
4	80	60	30

If we make number of fires in town 3 to 40

Town	x	y	Number of Fire		Fire Station
1	10	20	40		x	y
2	60	20	25		43.7037	31.85185
3	40	30	40
4	80	60	30