# On the Use of Data Envelopment Analysis for Multi-Criteria Decision Analysis
## 2 Materials and Methods
### Data and Analyses
#### 2.3.2 Waste Management Data  
The analysis was also applied using data from a previous MCDA analysis involving a range of different options for development of a waste management system. For each management alternative, a measure of the cost of its implementation and the outcomes in terms of environmental impacts and benefits were estimated. The data were originally produced by [PERSON] and [PERSON] [40], who applied ELECTRE III [41] to assess the different options, and more recently used by [PERSON] [42], who compared DEA to a range of other MCDA approaches. The data include several undesirable outputs. There are a number of ways of dealing with undesirable outputs, including ignoring them, including them as inputs, or transforming the variables [43]. [PERSON] [42] included these undesirable outputs as inputs, along with the costs of their implementation. In our study, we have applied the approach developed by [PERSON] and [PERSON] [44], where the undesirable outputs (\(y\)) are transformed as \(y^{\prime}=-y+D\), where D is a constant value that makes the resultant transformed output (\(y^{\prime}\)) positive for all \(y\). In this case, the maximum value of each of the undesirable variables plus one was added to the negative values. The 'plus one' was added such that the minimum value of the transformed variable was 1 (rather than zero). The higher the value of the transformed variable, the better the option compares to the worst performing option. This approach was adopted as the resultant data can be readily applied in both a radial and slacks-based DEA. The transformed data are presented in Table 2 for the 22 different management alternatives. A hierarchy containing three higher-level criteria (Environmental, Technical, and Economic) was assumed and the outputs allocated accordingly. All Environmental outputs were undesirable, with the values in Table 2 being transformed as indicated above. The cost of each management alternative was used as the input. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Structure** & **Variables in Each Group** \\ \hline All indicators & 9 individual indicators & V1–V9 \\ Hierarchy & & \\ Equal & \((3,3,3)\) & V1–V3; V4–V6; V7–V9 \\ Unequal & \((2,3,4)\) & V1–V2; V3–V5; V6–V9 \\ Very unequal & \((2,2,5)\) & V1–V2; V3–V4; V5–V9 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental design used in the hypothetical analysis. This is a relatively small data set for use in DEA. To ensure appropriate degrees of freedom in the analysis, the number of comparisons generally should be greater than \(max[3(m+n),\,m\times n]\), where \(m\) and \(n\) are the number of outputs and inputs respectively ([PERSON] et al., 2018), suggesting that 24 or more comparisons would produce more reliable results. However, fewer alternatives may incorrectly increase the share of fully efficient options (i.e., TE = 1). Unlike the hypothetical data, the waste treatment data are more highly correlated (Table 3), both positively and negatively. This indicates a greater degree of proportionality in outputs as well as explicit trade-offs (e.g., greenhouse gases compared with acidic gases). This may affect the efficiency scores derived using the different methods. As with the hypothetical data, the alternatives were examined using both the radial and slacks-based models, and with and without a hierarchical structure imposed. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Alternative**} & \multirow{2}{*}{\begin{tabular}{c} **Cost** \\ **(Input)** \\ \end{tabular} } & \multicolumn{4}{c}{**Environmental**} & \multirow{2}{*}{\
\begin{tabular}{c} **Technical** \\ **(R)** \\ \end{tabular} } & \multicolumn{2}{c}{**Economic**} \\ \cline{3-8}  & & & & **Acidic Gases** & & **Nitrogen** & **Emplonent** &
\begin{tabular}{c} **Waste Recovery** \\ **(R)** \\ \end{tabular} \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 2: Transformed data used in the waste management example. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{