# On the Use of Data Envelopment Analysis for Multi-Criteria Decision Analysis
## 2 Materials and Methods  
### Data Envelopment Analysis  
DEA is a non-parametric (linear programming) frontier-based method that is used to assess the relative efficiency of a set of decision making units (DMUs). DEA is a well-established method for productivity analysis [29; 30; 31] that has been broadly applied to a wide range of industries. The efficiency of a DMU is determined by its level of outputs (e.g., level of production) and inputs (e.g., capital and labor) relative to other DMUs. The efficient frontier is defined by the data, given by the set of DMUs that produces the highest level of (different combinations of) outputs for a given set of inputs. A particular advantage of DEA is the ability to include multiple outputs and inputs in the analysis. A further advantage of DEA is that the underlying functional form of the production function (i.e., relationship between outputs and inputs) does not need to be specified, and essentially, a separate 'production function' is identified for each DMU. Outputs are represented by the extent to which objective or criterion outcomes are achieved under each management option. In this context, each management option is effectively treated as DMU-equivalent in a production context. In practice, these outputs are generally represented by indicators associated with the objective rather than the objective per se. There do not need to be any inputs in the traditional productivity analysis sense if each option involves the same level of resources to implement, although if different management options involve different resource inputs, these can be captured in the analysis. For example, if the different management options involve different costs, these can be included as an input, and the resultant measure is then similar to what may be produced in a cost effectiveness analysis. Otherwise, each DMU is assigned an equal 'input' with a value of 1 (one). We consider two approaches to the estimation of the management option 'efficiency': an output-oriented radial DEA model [24] and an output-oriented additive model (often referred to as a slacks-based model) [25; 32; 33]. A simplified comparison of the two approaches is given in Figure 1, where the radial model results in a radial expansion to the frontier, while the additive model estimates the horizontal or vertical distance to the frontier. The radially (or additive) expanded inefficient DMU at the frontier may fall between two (or more in the case of multiple outputs) other DMUs (as seen in Figure 1), the position determined by the weighted average of the outputs of the efficient DMUs. The radial DEA model has been applied in a wide range of contexts, including to assess different management options given multiple objectives ([PERSON], 1996). For the purposes of using DEA as an MCDA tool, we employ an output-oriented model as we seek to identify the management options that maximize the outcomes with respect to the multiple objectives or criteria. We can define the output-oriented radial model as:  
\[ \begin{array}{c}Max\ \Phi_{1}\ \text{Subject to}\\ \Phi_{1}y_{1,m}\leq\sum\limits_{j}z_{j}y_{j,m}\quad m\in M\\ \sum\limits_{j}z_{j}x_{j,n}\leq x_{1,n}\quad n\in N\end{array}  \tag{1}\]  
where \(\Phi_{1}\) is a scalar showing by how much the outcomes of each DMU can increase output, \(y_{j,m}\) is the outcome against objective \(m\) by DMU \(j\), \(x_{j,n}\) is amount of input \(n\) used by the DMU \(j\) (set at 1 for all DMUs) and \(z_{j}\) are weighting factors. In a productivity context, we can add an additional constraint, \(\sum\limits_{j}z_{j}=1\), to imposes variable returns to scale in the model; otherwise, the model (implicitly) imposes constant returns to scale. With a single (constant) input, there is no difference between imposing variable returns or constant returns to scale (Stein  
where the \(y\) and \(x\) variables are as defined previously, with \(s^{+}_{1,m}\) and \(s^{-}_{1,n}\) representing the slack variables required to reach the frontier.