*3.5. Standard Deviation Weighting Method*

The standard deviation method determines the criteria weights in terms of their standard deviations through the following equation:

$$w\_j = \frac{\sigma\_j}{\sum\_{j=1}^n \sigma\_j} \tag{10}$$

To summarize, Table 4 summarizes the criteria weights obtained using these weighting methods. As can be seen, the entropy method assigns an important weight to SFC, while the CRITIC, variance, and standard deviation methods assign NOx as the most relevant criterion. Both variance and standard deviation procedures measure the spread, i.e., the degree to which each sample is different from the mean. As can be seen in Table 3 shown above, CO and HC emissions remain practically constant and thus lead to low values of both variance and standard deviation. On the other hand, NOx and, to a lesser extent, SFC present more spread and thus higher values of variance and standard deviation. For this reason, the variance and standard deviation methods provide a significant weight to NOx and low weights to CO and HC. Since the standard deviation is the square root of the variance, the variance method assigns a higher weight to NOx than the standard deviation method. The CRITIC method assigns high values of weights to those criteria with high standard deviation and low correlation with other responses. According to this, the results obtained through the CRITIC method are very similar to those obtained through the standard deviation method, but with less differences between the criteria weights. The entropy method also takes into account the uncertainty in the information and thus assigns low weights to CO and HC. Besides uncertainty, the entropy is based on the degree of disorder and thus assigns an important weight to SFC.


**Table 4.** Criteria weights according to the subjective, entropy, CRITIC, variance, and standard deviation weighting methods.

Table 5 outlines the results of the 125 cases analyzed using these procedures. As can be seen, the subjective weighting method provides case 91, with an adequacy index of *AI91* = −0.122, as the most appropriate injection pattern. This case corresponds to the −19º CA ATDC pre-injection starting instant, 20% pre-injection rate, and 1º CA pre-injection duration. Since the subjective method assigns an important weight to NOx, this 91st solution provides a significant NOx reduction with a low increment of SFC, CO, and HC. This solution provides an important pre-injection rate, 20%, due to its importance on NOx reduction. Retarding the pre-injection instant also reduces NOx noticeably but at expenses of important increments on consumption. This reason leads to the CRITIC, variance, and standard deviation weighting methods to provide case 105, corresponding to the −18º CA ATDC pre-injection starting instant, 5% pre-injection rate, and 5º CA pre-injection duration, as the most appropriate injection pattern, mainly due to the important weight of NOx over the other criteria and lower weight of SFC in comparison with the subjective weighting method. A value of −18º CA ATDC leads to important NOx reduction with a noticeable SFC penalty. Basically, the NOx reduction achieved with a high pre-injection rate or by a late pre-injection rate is reached through a reduction in the combustion temperature, since the high combustion temperatures reached in the combustion chamber are responsible for most NOx emitted to the atmosphere [41,42]. On the other hand, the entropy method provides case 25 as the most appropriate injection pattern, with a −22º CA ATDC pre-injection starting instant, 25% pre-injection rate, and 5º CA pre-injection duration. Since the entropy method assigns more weight to SFC and, to a lesser extent, to NOx, it provides an earlier pre-injection starting instant, which leads to a reduction in SFC and a higher pre-injection rate, which leads to a reduction in NOx.


**Table 5.** Adequacy index according to subjective, entropy, CRITIC, variance, and standard deviation weighting methods.




**Table 5.** *Cont.*
