Hot Dog Stand Investigation
By Angelina Katsanis
August 2017

Introduction:
In this problem, we are trying to find the most ideal location for a hot dog stand on the campus depicted below. On this campus, there are 8 pathways, all of which connect to at least one other pathway (at either point A, B, C, D, E, or F). Only one hot dog stand is allowed to be placed, and it has to be placed somewhere along one of the paths. Dormitories are located at 5 of the 8 points (A, C, D, E, and F), with a number of students in each dorm as 200, 300, 300, 100, and 100, respectively. Below are the methods that were used to calculate the best location for the hot dog stand, as well as explanations of how they were utilized.


Methods
Maximum distance
The longest distance one would have to travel to each point from each dorm

Distance from Dorm to Stand
Location of Stand
A
C
D
E
F
Maximum
A
0
7
8
7
8
8
B
3
4
5
4
5
5
C
7
0
1
6
8
8
D
8
1
0
5
7
8
E
7
6
5
0
2
7
F
8
8
7
2
0
8
G
7
8
8
3
1
8
Based solely on maximum distance, the best place to put the stand would be on point B.

Average distance
The average distance a person would have to walk from any dorm to each point

Distance from Dorm to Stand
Location of Stand
A
C
D
E
F
Average
A
0
7
8
7
8
6
B
3
4
5
4
5
4.2
C
7
0
1
6
8
4.4
D
8
1
0
5
7
4.2
E
7
6
5
0
2
4
F
8
8
7
2
0
5
G
7
8
8
3
1
5.4
Based solely on average distance, the best place to put the stand would be on point E.

Weighted distance
Multiplied each distance from dorm to stand by the number of students in the dorm/100
Changes the distances, and thus the maximums/averages
According to the averages of the weighted table below, the ideal point is the average of C and D (specifically, coordinate (8, 5.5), and the ideal location for the maximum distance is at point C

Distance from Dorm to Stand

Location of Stand
A
C
D
E
F
Average
Maximum
A
0
21
24
7
8
12
24
B
6
12
15
4
5
8.4
15
C
14
0
3
6
8
6.2
14
D
16
3
0
5
7
6.2
16
E
14
18
15
0
2
9.8
18
F
16
24
21
2
0
12.6
24
G
14
24
24
3
1
13.2
24

Central location
The central location of the whole campus is at coordinate (4.5, 4). However, since this point is not on a path, the closest point that is on a path is (4, 4) - halfway between points B and E

Sum of squares
Below is a table of the sum of squares for both the unweighted and the weighted distances (respectively)


Sum of Squares
A: 1130
B: 446
C: 305
D: 339
E: 749
F: 1277
G: 1358
Based on the sum of squares of the weighted distances, point C would be the ideal place to put the hot dog stand.

Weaknesses:
Are any of the points near bus stop locations?
Which of the paths are drivable? Accessible by foot? Accessible by bike?
If they were all walking paths, which places are higher up (and therefore less attractive to customers)
Where is the competition located?
Which locations are the most visually attractive/appealing?

Conclusion:
If all the methods are taken into account and weighted according to the number of students in each dorm room, the four most ideal locations in each category are: (4, 4), (8, 5.5), (8, 6), and (8, 6). The average of all the x-coordinates of these points is 7, and the average of all the y-coordinates is 5.375. This would make the absolute ideal location (assuming that average, weight, maximum, sum of squares, and central location are all of equal importance) to be at (7, 5.375). However, given that the hot dog stand must be placed on the path, the closest place on the path to that point is at (7, 6). This makes the point located on the coordinate (7, 6) to be the best location within the constraints to place the hot dog stand.