# A Simplified Algorithm for Dealing with Inconsistencies Using the Analytic Hierarchy Process
## 3 Results
### Application to Coastal Preference Study  
The approach was applied to a large study on preferences for different coastal features. Data for the study were derived from an online survey of 1414 coastal residents in NSW, Australia [16]. The full hierarchy involved a comparison of 12 different coastal features, broken into four main categories, i.e., shoreline (sandy beach, headlands, rocky shoreline); backshore (sand dunes, adjacent scrubland, coastal freshwater lakes and rivers); intertidal (estuaries, mangroves, saltmarshes), and marine (seagrass, rocky reef, sandy seabed). For the purposes of illustrating the approach to correct for inconsistencies, only the top level of the hierarchy is considered here (i.e., the four main categories). The data and R code used in the analysis are provided in the Supplementary Materials. With only three comparisons, as in the previous example, convergence to an acceptably consistent matrix occurs in a single iteration. However, as the number of comparisons increases, the number of iterations required to achieve an acceptable consistency index also increases. In this example, we specified 20 iterations as a maximum number of iterations to prevent potential indefinite cycling (as in the case of the extreme cyclic triad above). The effect of the iterative approach on the level of inconsistency for the four broad coastal attribute groups can be seen in Figure 1. The critical value in this case (i.e., \(n=4\)) is 0.353 [43]. From the initial comparisons, only around 45% of the observations fell below the critical value. After the adjustments, all observations were below the critical value. The effect of the adjustment on the distribution of the attribute weights can be seen in Figure 2 and Table 2, which summarizes the weights derived from all observations, including those with a GCI value above the critical value. The overall distributions varied as a result of the adjustment, but median scores were relatively constant, and the correlation between the scores before and after adjustment were high (Table 2). \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Shoreline**} & \multicolumn{2}{c}{**Backshore**} & \multicolumn{2}{c}{**Intertidal**} & \multicolumn{2}{c}{**Aquatic**} \\ \cline{2-9}  & **Initial** & **Adjusted** & **Initial** & **Adjusted** & **Initial** & **Adjusted** & **Initial** & **Adjusted** \\ \hline Min. & 0.025 & 0.035 & 0.025 & 0.034 & 0.025 & 0.037 & 0.025 & 0.027 \\
1 st Quartile & 0.194 & 0.203 & 0.144 & 0.142 & 0.092 & 0.106 & 0.138 & 0.099 \\ Median & 0.250 & 0.250 & 0.241 & 0.234 & 0.186 & 0.193 & 0.250 & 0.250 \\ Mean & 0.317 & 0.330 & 0.232 & 0.230 & 0.186 & 0.201 & 0.266 & 0.239 \\
3 rd Quartile & 0.470 & 0.472 & 0.256 & 0.250 & 0.250 & 0.250 & 0.341 & 0.316 \\ Max. & 0.750 & 0.750 & 0.700 & 0.709 & 0.679 & 0.664 & 0.700 & 0.745 \\ Correlation & & 0.861 & & 0.814 & & 0.656 & & 0.895 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial and revised relative weights from the example coastal case study. Figure 1: Initial and final distribution of inconsistency scores. Figure 2: Initial and final distribution of the attribute weights. The less-than-perfect correlations indicate that some scores were substantially affected. This was particularly the case for the weights associated with the intertidal coastal assets (i.e., estuaries, mangroves, saltmarshes). Respondents may have been less familiar with the ecosystem services produced by these assets compared with more familiar assets, such as beaches (shoreline) and sand dunes (backshore), and hence there may have been greater \"error\" in scoring these assets against others. In terms of the ordinal ranking of the attribute weights, around 58% of the observations were unchanged, with around % changing one position (i.e., requiring two adjacent attributes to change position), and a further 14% changing two positions. The incidence of total rank reversal (i.e., changing from top to bottom and vice versa) was small, less than 0.5% (i.e., around seven of the 1414 observations).