# Adaptive Behaviour for a Self-Organising Video Surveillance System Using a Genetic Algorithm
## 5 Limit-Cases and Benchmarking  
Limit-cases correspond to situations in which the value of one of the three weighting parameters \(\alpha\), \(\beta\) and \(\gamma\) is set to 1, which, according to the conservation law expressed by Equation (2), means that the other two global variables are not taken into account when computing fitness. As a benchmarking exercise, we investigated how our proposed genetic algorithm framework would deal with each of the three limit-cases, and for both reproduction methods. Figures 3-5 show the resulting values for \(A\), \(B\) and \(C\) when \(\alpha\), \(\beta\) and \(\gamma\) are evaluated in isolation, respectively.  
Figure 3: Benchmarking tests: \(\alpha=1\) We show the evolution over time of variables \(A\), \(B\) and \(C\) for the fittest individual of each generation (the plotted values are averaged over 20 independent realisations). If \(\alpha=1\) then \(\beta,\gamma=0\); and \(F\) is equal to \(A\). Consequently, the GA is optimising \(F\) by reducing the value of \(A\) to (eventually) zero. The convergence speed of the algorithm when using mutation only (**left** panel) is significantly slower than with sexual reproduction (**right** panel).  
Figure 2: A typical environment semi-randomly generated for the numerical experiment setup. Colour as in Figure 1: the zoomed-in picture on the right shows overlap (orange area) if all cameras are turned on. Note the “blind spots” (grey area) that are permanently outside coverage.  
If \(\alpha=1\), the optimal solution is to permanently switch off all cameras (\(A=0\)).  
Symmetrically (see Figure 4), if \(\beta=1\), then one optimal solution is to never switch any of them off (although there will usually be others, in which some cameras, the coverage area of which is 100% overlapping with others, are permanently deactivated).  
The third limit-case (\(\gamma=1\)) has no such obvious optimum. Even though the optimal solution for \(\alpha=1\) is also one for \(\gamma=1\), depending on cameras' locations and orientations, there will usually be many other, less trivial schedules for which there is no overlapping coverage (C = 0).  
The conclusion is twofold. First, the proposed framework is evidently capable of converging on the best schedule for each separate objective, namely, \"lowest resource consumption\" (\(\alpha=1\)), \"fewest blind-spots\" (\(\beta=1\)) and \"minimal overlap\" (\(\gamma=1\)). Second, the mixing of schedules/genes through sexual reproduction has, unsurprisingly, a very positive influence on the adaptation rate, especially when the objective is the conservation of resource or the avoidance of overlapping coverage.  
Figure 4: Benchmarking tests: \(\beta=1\), i.e., the value of \(F\) is only based on blind spots. Both variations (**left**: mutation only; **right**: with sexual reproduction) perform comparable, the value of \(B\) may never drop entirely to zero due to unavoidable blind spots caused by the camera placement.  
Figure 5: Benchmarking tests: setting \(\gamma=1\) implies (as above) that only one value contributes to \(F\), namely, \(C\), the _overlap_. In both variations of the GA C is being reduced but the GA with sexual reproduction (**right** panel) significantly outperforms the variation that uses only mutation (**left**).