To find the point with the greatest convenience for students, a factor \(c\) was calculated, called the “convenience factor.” This convenience factor was defined as the overlaps of paths divided by the average distance between the dormitories, as given by the equation: \(c=\frac{t}{a}\). Greater intersection traffic is directly proportional and greater average distance indirectly proportional to the convenience factor. This equation controls for variations in modes of transportations and walking speeds: it determines the ease of traveling to vendor position in an absolute sense.
\(t\) was defined as intersection traffic, or the overlaps of the paths between the dorms, calculated as the number of times in which a point coincided with a path ( \(\overline{AB}\), \(\overline{CD}\), and so forth). \(a\) was defined as average distance from each dorm to that location, calculated as an arithmetic mean: the sum of the shortest distances between the dorms and a point, divided by the number of paths.
We were asked to determine the best location for the hot dog vendor on the map given. For this baseline, calculations for \(t\), \(a\), and \(c\) are shown in Tables 1, 2, and 3, respectively (Appendix). In this system, as \(E\) had the greatest convenience factor (\(c=1.5\)), \(E\) is recommended as the vendor location which optimizes student convenience.
If multiple vending locations are added, the efficiency of the system is improved, as students will eat at the nearest vendor location of two. To evaluate this system, the top four performing intersections (\(E\), \(D\), and \(B\)) from the baseline model were conjugated as a permutation: \(E\) with \(D\), \(D\) with \(B\), \(C\) with \(B\), \(E\) with \(B\), and so forth). Intersection traffic was determined as the number of non-redundant events during which a path between dormitories coincided with either position. Event were considered redundant if a path coincided with both positions. In this case, only one event would be counted. Table 4 summarizes these calculations (Appendix).
For the same reason, average distances were calculated as the distances from each dormitory to the corresponding closest vendor location. The resulting \(c\) values are shown in Table 5 (Appendix). In this system, \(B\) and \(D\) (\(c=3.2\)) were found to have the greatest convenience factor, and therefore these points are recommended as the optimal vendor positions.
It was assumed in the question that dormitories corresponding to points \(A\) and \(C\) were female, while \(D\), \(E\), and \(F\) were male dormitories. It was also given that \(80 \%\) of males and \(30 \%\) of females purchase hot dogs. In this scenario, several modifications to the baseline model is called for.
Intersection traffic was calculated as \(t=\sum\nolimits_{} g_i\), where \(g\) is gender value. The gender value is calculated as \(.8\) for a path from a male dormitory to another male dormitory and \(.3\) for female to female. Since traffic going both ways of a path is equal, the value was given as an arithmetic mean of purchase likelihoods: \(.55\) for male to female dormitory paths and vice versa. Since location \(B\) does not correspond to a dormitory, it was not included in these calculations. The gender values for each dormitory and the corresponding \(t\) values are given in Tables 4 and 5 (Appendix). As the average distances did not change, the baseline \(a\) values were obtained from Table 2. The resulting calculated \(c\) values are summarized in Table 6 (Appendix).
It was found that location \(E\) (\(c=.6750\)) was best suited in the new environment. This is intuitive given that it is nearest to a male dormitory. In addition, \(E\) is the point closest to the midpoint of the line \(\overline{DF}\), which approximates an alignment of the male dormitories. This is important because male students in this system are more than twice as likely as female dormitory to purchase hot dogs.
We were also tasked with accounting for a change in topography: how would optimal location change if the paths \(\overline{BC}\) and \(\overline{DE}\) go uphill? We found that a change in topography does not alter \(t\), \(a\), or, accordingly, \(c\) values. Rather, since students are equally likely to travel from one dormitory to the other, the net effect \(\Delta c\) is \(0\).
Therefore, the distribution of \(c\) values is identical to the baseline values calculated above, and \(E\) is recommended as the optimal vendor location in this system.