## 2 Materials and Methods
### Methods  
The DEA is an efficiency measurement technique based on obtaining an efficiency frontier from the set of observations considered without the estimation of any production function, i.e., without the need to know any form of functional relationship between inputs and outputs. DEA attempts to optimise the efficiency measure of each unit analysed to create an efficient frontier based on the Pareto criterion, allowing the use of multiples inputs and outputs without imposing any functional form on the data or making assumptions of inefficiency. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Alternative** & **Kms Travelled** & **Hours in Service** & **Number of Vehicles** \\ \hline Diesel & 1,160,198 & 94,568 & 23 \\ GNC & 1,160,198 & 94,568 & 23 \\ Diesel hybrid & 1,160,198 & 94,568 & 23 \\ Plug-in electric & 1,187,888 & 96,001 & 31 \\ Induction electric & 1,172,226 & 112,873 & 27 \\ \hline \hline \end{tabular} Source: Compiled by the authors based on data from Madrid City Council [69,70].  
\end{table} 
Table 1: Data on requirements by type of fuel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Alternative** & **DMU** & **Output:** & **Input 1: NO\({}_{\text{x}}\)** & **Input 2: PM** & **Input 3: CO\({}_{\text{2}}\)** \\  & **Km/N\({}^{\circ}\) Bus** & **(kg)** & **(kg)** & **(kg)** & **(kg)** \\ \hline Diesel & DMU1 & 50,443.39 & 1138 & 3.9 & 1,589,471 \\ GNC & DMU2 & 50,443.39 & 1044 & 3.4 & 1,833,113 \\ Diesel hybrid & DMU3 & 50,443.39 & 910 & 3.1 & 1,264,616 \\ Plug-in electric & DMU4 & 38,318.97 & 0 & 0 & 724,612 \\ Induction electric & DMU5 & 43,415.78 & 0 & 0 & 926,059 \\ \hline \hline \end{tabular} Source: Compiled by the authors based on data from Madrid City Council [69,70].  
\end{table} 
Table 2: Data used in the DEA. Using DEA, the efficient frontier is the benchmark against which the relative performance of different types of engines is measured. The engines that form the efficient frontier use a minimum amount of inputs to produce the same amount of outputs. The distance to the efficient frontier represents the measure of efficiency or lack thereof. Basic DEA model, called CCR, was firstly applied in 1978 [73]. The relative efficiency of a decision making unit, from now on DMU, is calculated as the ratio between the weighted sum of outputs and the weighted sum of inputs. The weights are determined by linear programming so that the DMU maximises its efficiency value. Let \(E_{0}\) be the efficiency score of the observed \(DMU_{0}\). Let \(y_{j0}\), \(j=1,\ldots,s\) be the outputs and \(x_{i0}\), \(i=1,\ldots,m\) the inputs used to compute the efficiency, and \(n\) the total number of DMUs (\(r=1,\ldots,n\)). Then, the relative efficiency of \(DMU_{0}\) is calculated as:  
\[E_{0}=Max\frac{\sum_{j=1}^{s}u_{j}y_{j0}}{\sum_{i=1}^{m}v_{i}x_{i0}} \tag{1}\]  
where \(u_{j}\) and \(v_{i}\) are the non-negative weights. If the condition that the efficiency be less or equal than 1 is imposed, then it is obtained the classical CCR model under the Constant Return to Scale (CRS) assumption:  
\[E_{0}=Max\frac{\sum_{j=1}^{s}u_{j}y_{j0}}{\sum_{i=1}^{m}v_{i}x_{i0}} \tag{2}\]  
\[s.t.:\]  
\[\frac{\sum_{j=1}^{s}u_{j}y_{j0}}{\sum_{i=1}^{m}v_{i}x_{i0}}\leq 1\ \ for\ r=1,\ldots,n\]  
\[u_{j},\ v_{i}\geq 0\ \ \ \forall i,j\]  
The transformation developed by [73] for linear fractional programming selects a concrete weight solution for which \(\sum_{i=1}^{m}v_{i}x_{i0=1}\) and leads to the following linear problem:  
\[E_{0}=Max\sum_{j=1}^{s}u_{j}y_{j0} \tag{3}\]  
\[s.t.: \tag{4}\] \[\sum\limits_{j=1}^{s}u_{j}y_{jr}-\sum\limits_{i=1}^{m}v_{i}x_{i0}=1\] \[u_{j},\ v_{i}\geq 0\ \ \ \forall i,j\]  
DEA solves \(n\) different LP problems for a set of \(n\) DMUs. A DMU is efficient if its efficiency score is equal to one and there exists at least one optimal solution with non-zero weights. The work in [74] proposed a version of the CCR model with VRS, or variable returns to scale, which is commonly known as the BCC model, and use to be applied for situations where the CRS assumption is not met.