**3. Results and Discussion**

Once validated, this CFD model was used to provide the data for the MCDM approach. The simulation calculation was carried out simply as a process and as sample results for applying the optimal selection method of multiple injection conditions. The 125 cases illustrated in Figure 3 were analyzed. As can be seen, five pre-injection rates (R): 5%, 10%, 15%, 20%, and 25%; five pre-injection durations (D): 1º, 2º, 3º, 4º, and 5º crank angle (CA); and five pre-injection starting instants (S): −22º, −21º, −20º, −19º, and −18º crank angle after top dead center (CA ATDC), were employed.

**Figure 3.** Illustration of the 125 cases analyzed.

Four criteria were analyzed: SFC, NOx, CO, and HC. It is worth mentioning that PM emissions should be included in this model. They were not included because current numerical methods do not provide enough accuracy regarding PM [40]. Nevertheless, continuous efforts are being made to develop more models which will provide proper accuracy in the near future.

Table 3 outlines the pre-injection rate, duration, and starting instant, as well as the values of SFC, NOx, CO, and HC provided by the CFD model for the 125 cases. According to this, the data matrix is composed of 125 rows and 4 columns. In the remaining of the present work, each datum of the decision matrix will be represented as *Xij*, where *i* is the case and *j* is the criteria considered.


**Table 3.** Data for the multiple-criteria decision-making (MCDM) problem.


**Table 3.** *Cont.*


**Table 3.** *Cont.*

Each indicator was transformed into its variation in per unit basis through Equation (1).

$$\mathbf{V}\_{ij} = \frac{\mathbf{X}\_{ij} - \mathbf{X}\_{jref}}{\mathbf{X}\_{jref}} \tag{1}$$

where *Xjref* is the value corresponding to criterion *j* in the case without pre-injection.

As indicated previously, an important step in a MCDM approach is to determine the criteria weights, and this issue will be treated in Sections 3.1–3.5. Once determined, the adequacy index for each *i*-th case, *AIi*, was calculated through the SAW (simple additive weighting) method, Equation (2). Obviously, the most appropriate solution corresponds to the minimum value of *AI*.

$$AI\_{\bar{i}} = \sum\_{j=1}^{n} w\_{j} V\_{i\bar{j}} \tag{2}$$

where *wj* is the weight of the *j*-th criterion and *n* the number of criteria, i.e., four (SFC, NOx, CO, and HC).
