## 3 Results and Discussion  
Once the numerical model was validated with experimental measurements, it was used to analyze the results of several pre-injection patterns. Five pre-injection rates were employed: 5%, 10%, 15%, 20% and 25%; five pre-injection durations: 1\({}^{\circ}\) Crank Angle (CA), 2\({}^{\circ}\) CA, 3\({}^{\circ}\) CA, 4\({}^{\circ}\) CA and 5\({}^{\circ}\) CA; and five pre-injection starting instants: \(-\)22\({}^{\circ}\) CA ATDC, \(-\)21\({}^{\circ}\) CA ATDC, \(-\)20\({}^{\circ}\) CA ATDC, \(-\)19\({}^{\circ}\) CA ATDC and \(-\)18\({}^{\circ}\) CA ATDC. These ranges were chosen according to a previous paper [43], in which it was obtained that wider ranges lead to excessive increments in NOx emissions or consumption. Taking into account these values of pre-injection rates, durations and starting instants, a total of  
Figure 1: SFC and emissions numerically and experimentally obtained at different loads. Figure 2: In-cylinder pressure numerically and experimentally obtained, 100% load. 125 cases were analyzed. These are summarized in Figure 3. SFC and emissions of NOx, CO and HC were characterized for these 125 cases. As indicated previously, a MCDM method was employed. The first step in MCDM approaches consists on defining the decision tree with the relative importance of each parameter, Table 1. Two requirements (general aspects) were considered: consumption and emissions. The second level represents the sub-requirements (specific aspects). The relative importance of each requirement and sub-requirement must be defined too. In the present work, this importance was distributed equally, i.e., the same importance was given to SFC and emissions: 50% (\(\alpha\) = 0.5 in per unit basis) for each one. Regarding emissions, the same importance was given to NOx, CO and HC: 33.3% (\(\beta\) = 0.333 in per unit basis) for each one. It is worth mentioning that other scenarios could also be possible in case it is necessary to provide other levels of importance. The next step was to calculate the variation of each sub-requirement respect to the case without pre-injection, in per unit basis, Equation (1). \[V_{i}=\frac{X_{i}-X_{ref}}{X_{ref}} \tag{1}\]  
where \(X_{i}\) is the value of the sub-requirement \(i\) and \(X_{ref}\) the value corresponding to the case without pre-injection, i.e., SFC = 172 g/kWh, NOx = 13.3 g/kWh, CO = 4.5 g/kWh and HC = 5.5 g/kWh. Once each indicator was transformed into its variation in per unit basis, the global adequacy index, \(AI\), was computed by Equation (2). \[AI=\sum_{i=1}^{125}\alpha_{i}\beta_{i}V_{i} \tag{2}\]  
where \(\alpha_{i}\) is the weight of each requirement and \(\beta_{i}\) the weight of each sub-requirement in per unit basis. The minimum value of the \(AI\) is the most adequate solution. It was determined using an in-house code programmed using the open software GNU Octave. For the parameters analyzed in the present \begin{table}
\begin{tabular}{c c} \hline \hline
**Requirement** & **Sub-Requirement** \\ \hline SFC (50\%) & SFC (100\%) \\  & NOx (33.3\%) \\ Emissions (50\%) & CO (33.3\%) \\  & HC (33.3\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Decision tree for the MCDM approach. Figure 3: Schematic representation of the cases analyzed. work, the minimum _AI_ is \(-0.0122\), corresponding to \(-19^{\circ}\) CA ATDC pre-injection starting instant, 20% pre-injection rate and \(1^{\circ}\) CA pre-injection duration. The results are detailed in Appendix A. As can be seen in this appendix, the minimum _AI_ value was obtained in the ninety-first case analyzed. On a per-unit basis, SFC varies by 0.067; NOx \(-0.3467\), CO 0.0347 and HC 0.0383, corresponding to an SFC increment of SFC 6.7%; NOx reduction of 34.7%, CO increment of 3.47% and HC increment of 3.83% in comparison with the case without pre-injection. Despite the increments in SFC, CO and HC, these solution results are the most appropriate because the increments are low and the NOx reduction is too significant.