# On the Use of Data Envelopment Analysis for Multi-Criteria Decision Analysis
## 3 Results  
### Using Hypothetical Data  
Using the radial model, 66 DMUs were found to be fully efficient when using all indicators together (Table 4). The number of efficient DMUs declined when the hierarchical structure was imposed. Overall, only 48 DMUs were estimated to be efficient across all scenarios, indicating that some treatments suggested that a DMU was efficient, while others suggested that it was not efficient. All three hierarchical treatments produced similar numbers of efficient DMUs, even though the number of indicators compared within each of the broader objectives varied. Forty-eight DMUs were found to be fully efficiency in all cases.  
In contrast, the slacks-based model found only 34 fully efficient DMUs, and this number decreased to between two and ten when a hierarchical structure was imposed. In contrast to the analysis using the radial model, no single DMU was found to be fully efficient under all treatments. That is, the results are sensitive to how the variables are distributed between each higher order criterion.  
The relative ranking of the outcomes was sensitive to the type of model used as well as how the model was applied (Table 5). The DMU rank using all the indicators was reasonably correlated between the outputs of the two different models, but the rank correlation declined when a hierarchy approach was adopted (and how many indicators were associated with each higher-level objective in the hierarchy). Ranks rather than efficiency scores were compared, as the ordering of the management options (DMUs) is of more relevance to decision analysis.  
The outcomes are also compared in Figures 4 and 5 for the radial and slacks-based model respectively. These figures present the distribution of efficiency scores of each treatment against each other treatment.  
\begin{table}
\begin{tabular}{c c c} \hline
**Scenario** & **Radial** & **Slacks-Based** \\ \hline All indicators & 66 & 34 \\ Hierarchy & & \\ Equal (3,3,3) & 49 & 6 \\ Unequal (2,3,4) & 49 & 2 \\ Very unequal (2,2,5) & 48 & 10 \\ All scenarios & 48 & 0 \\ \hline \end{tabular}
\end{table}
Table 4: Number of efficient (TE = 1) DMUs in each scenario.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & \multicolumn{4}{c}{**Radial Model**} & \multicolumn{4}{c}{**Slacks-Based Model**} \\ \cline{2-9}
**Model** & **All** & **3,3,3** & **2,3,4** & **2,2,5** & **All** & **3,3,3** & **2,3,4** & **2,2,5** \\ \hline Radial & & & & & & & \\ All & 1.000 & 0.821 & 0.858 & 0.841 & 0.785 & 0.446 & 0.441 & 0.450 \\
3,3,3 & 0.821 & 1.000 & 0.964 & 0.976 & 0.628 & 0.470 & 0.438 & 0.435 \\
2,3,4 & 0.858 & 0.964 & 1.000 & 0.975 & 0.687 & 0.439 & 0.379 & 0.378 \\
2,2,5 & 0.841 & 0.976 & 0.975 & 1.000 & 0.656 & 0.472 & 0.417 & 0.416 \\ Slacks-based & & & & & & & & \\ All & 0.785 & 0.628 & 0.687 & 0.656 & 1.000 & 0.422 & 0.395 & 0.430 \\
3,3,3 & 0.446 & 0.470 & 0.439 & 0.472 & 0.422 & 1.000 & 0.878 & 0.829 \\
2,3,4 & 0.441 & 0.438 & 0.379 & 0.417 & 0.395 & 0.878 & 1.000 & 0.884 \\
2,2,5 & 0.450 & 0.435 & 0.378 & 0.416 & 0.430 & 0.829 & 0.884 & 1.000 \\ \hline \end{tabular}
\end{table}
Table 5: Spearman’s rank correlations between different treatments.  
For both model types, using all indicators separately in the analysis (i.e., the 'All' category in Figures 4 and 5) results in a greater proportion of options being identified as  
Figure 4: Comparison of efficiency scores with different hierarchical structure, radial model.  
Figure 5: Comparison of efficiency scores with different hierarchical structure, slacks-based model.  
fully efficient (i.e., TE = 1), with many of these values being less than 1 when the indicators are split into sub-groups and assessed as a hierarchy.  
The value of the efficiency score is also dependent on how the hierarchy is structured, i.e., whether equally balanced across the objectives or unequally balanced. This is more pronounced for the slacks-based model (Figure 5), where the efficiency scores showed greater differences based on which hierarchy structure was imposed than for the radial DEA model.