## 5 Multi-Criteria Decision Making
### SAW Concepts Applied to Tiling-Based Path Planning  
The energy consumption level is a key factor of any electromechanical system, and for cleaning applications, the amount of area coverage signifies another aspect of the performance. Here, we want to apply MCDM regarding these key factors (area and energy) as the decision-making criteria. As mentioned above, an area decomposition is defined based on the obstacle layout and the whole environment is divided into five sub-areas, as shown in Figure 5. Then, the four introduced tiling theorems are applied to each sub-area and five tiling sets are devised for each theorem in each sub-area. Regarding the given tiling sets for each sub-area, the results for the decision criteria are summarized within Tables 1-5. The final aim is to introduce a superior theorem for each sub-area; the theorem which could best tile the area in terms of area coverage and energy consumption. In order to do this, a fitness function for each given tiling set is described, as in Equation (2), in which \(c_{1}\) is a positive weighting factor and \(c_{2}\) is a negative weighting factor since we would like to increase the area coverage and decrease the energy utilization and create a balance between the two. For now, we assume that the area coverage and energy are of the same importance for the user and assume the importance \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Area Coverage**} & \multicolumn{2}{c}{**Energy**} \\ \cline{3-6}  & & **Area \%** & **Real Value (m\({}^{2}\))** & **Energy \%** & **Real Value (A)** \\ \hline \multirow{8}{*}{**Sub-Area 5**} & Tiling set 1 & 98.039 & 1.96 & 18.30 & 1.354 \\  & Tiling set 2 & 94.117 & 1.8816 & 18.68 & 1.382 \\  & Tiling set 3 & 90.196 & 1.8032 & 19.46 & 1.440 \\  & Tiling set 4 & 94.117 & 1.8816 & 20.04 & 1.483 \\  & Tiling set 5 & 94.117 & 1.8816 & 19.23 & 1.4237 \\ \hline \multirow{8}{*}{**Sub-Area 5**} & Tiling set 1 & 81.372 & 1.6268 & 19.09 & 1.489 \\  & Tiling set 2 & 86.27 & 1.7248 & 20.31 & 1.491 \\  & Tiling set 3 & 82.352 & 1.6364 & 20.80 & 1.511 \\
2.156 m\({}^{2}\) & Tiling set 4 & 82.352 & 1.6464 & 18.31 & 1.534 \\  & Tiling set 5 & 86.28 & 1.7248 & 21.03 & 1.487 \\ \hline \multirow{8}{*}{
\begin{tabular}{c} 1.9992 m\({}^{2}\) \\ (_obs-frac_{}) \\ \end{tabular} } & Tiling set 1 & 90.19 & 1.8032 & 20.53 & 1.512 \\  & Tiling set 2 & 94.117 & 1.8816 & 19.41 & 1.490 \\  & Tiling set 3 & 90.196 & 1.8032 & 19.05 & 1.506 \\  & Tiling set 4 & 86.27 & 1.7248 & 19.69 & 1.493 \\  & Tiling set 5 & 86.28 & 1.7248 & 21.10 & 1.510 \\ \hline \multirow{8}{*}{**Theorem 4**} & Tiling set 1 & 78.43 & 1.568 & 17.26 & 1.497 \\  & Tiling set 2 & 78.43 & 1.568 & 17.87 & 1.487 \\ \cline{1-1}  & Tiling set 3 & 86.274 & 1.7248 & 20.33 & 1.503 \\ \cline{1-1}  & Tiling set 4 & 78.431 & 1.568 & 17.65 & 1.480 \\ \cline{1-1}  & Tiling set 5 & 82.352 & 1.6464 & 19.77 & 1.501 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Data for area coverage and energy for sub-area 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sub-Area 2** & **Rank** & **Rank Sum** \\ \hline \multirow{6}{*}{**Theorem 1**} & Tiling set 1 & 5 & \\  & Tiling set 2 & 10 & \\  & Tiling set 3 & 8 & 30 \\  & Tiling set 4 & 6 & \\  & Tiling set 5 & 1 & \\ \hline \multirow{6}{*}{**Theorem 2**} & Tiling set 1 & 12 & \\  & Tiling set 2 & 14 & \\  & Tiling set 3 & 2 & 56 \\  & Tiling set 4 & 13 & \\  & Tiling set 5 & 15 & \\ \hline \multirow{6}{*}{**Theorem 3**} & Tiling set 1 & 18 & \\  & Tiling set 2 & 20 & \\  & Tiling set 3 & 16 & 90 \\  & Tiling set 4 & 17 & \\  & Tiling set 5 & 19 & \\ \hline \multirow{6}{*}{**Theorem 4**} & Tiling set 1 & 11 & \\  & Tiling set 2 & 3 & \\ \cline{1-1}  & Tiling set 3 & 9 & 34 \\ \cline{1-1}  & Tiling set 4 & 7 & \\ \cline{1-1}  & Tiling set 5 & 4 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Ranking of the tiling sets based on the fitness value for sub-area 2.