Theater auditorium solution

In our next case study, we look at an anonymous auditorium. The layout of this auditorium with 318 seats is depicted in the figure below. Unlike in the case of the open spaces, the mesh generation for the auditorium is rather straightforward. It consists of finding the center of each seat and denoting it as a point for possible location selection. This is one of the reasons the computations and processing of auditorium solutions is a bit cheaper than in the open space case.

In this study, instead of focusing our efforts primarily on the different numbers of locations, we focused on the different minimum distances that were achievable. We set the range on the number of locations between 10 and 160 and run our optimization algorithm. In what turned out to be good 2 days of computation, we have found 25 unique minimum distance values and their corresponding location ranges. The results are summarized in the tables below, with the distances measured in centimeters.

location range[10][11][12][13][14][15][16][17][18-19][20][21][22-23][24-25][26]
min distance566528496465444422400398371360338328324309
location range[27-29][30-32][33-40][41-43][44-47][48-54][55][56-80][81-82][83-159][160+]
min distance28027824021821120114616312912064

Unsurprisingly, for a smaller number of locations to find (up to roughly 26), the minimum distance decreases steadily with almost every increase in the number of locations (below are results for 15 and 20 locations).

After this initial instability, we get much more interesting results – the separation of the location numbers into larger intervals, where the minimum distance does not change. What this implies is that we can, for example, use the optimal design for 47 locations to seat between 44 and 47 persons, and be optimal in all these instances. Or, to put it in a different way, we know that there is no better way to seat 83 persons, which does not employ the zigzag pattern for the optimal layout for 159 persons. The optimal layouts for 29, 32, 40, 43, 47, 54, 55, 80, 82, and 159 locations are shown in the figures below.

In contrast to the open space case, these designs are optimal in the full sense of the word, without any additional specifications needed.