4.2.2. RPN Evaluation—O Criteria

Table 3 shows the O criteria, one of the RPN evaluation factors for fuel cell-based hybrid power systems. To precisely evaluate O, the evaluation criteria were classified into 1 (failure occurrence frequency), 2 (possibility of occurrence), 3 (high occurrence rate), and 4 (Cpk value). In the third stage, the high incidence rate was evaluated by applying the PPM(Parts Per Million) index and the Cpk statistical tool was used, which measures the ability of the process to produce output within the required specifications. Cpk represents the capability of the process. If both sides have specifications (upper and lower limits) and the center of the distribution does not match the median of both specifications, bias occurs. In order to prepare and evaluate the incidence criteria of the entire system in detail, evaluation criteria were divided into three stages and four stages. In general, the O is considered good when Cpk is 1.33 or greater for a system or a process. The method for obtaining Cpk is as follows [40].



To get the value of Cpk, the capability index Cp is required. Cp is calculated to assess the degree of process capability. Cp can be obtained as Equation (1).

$$\text{Cp} = \frac{\text{LSL} - \text{LSL}}{6\sigma} = \frac{\text{Size of standard}}{\text{Actual Process Scale}}.\tag{1}$$

Here, USL: upper specification limit and LSL: lower specification limit.

The value of Cpk can be calculated from the measured data. If there is only an upper limit of the specification, if there is only a lower limit of the specification, it can be divided into a case where both the upper and lower limits of the specification, the calculation formula is as follows (2)–(4).

$$\text{Only upper limit of specification}: \text{ Cpk} = \frac{\text{USL} - \overline{X}}{3\sigma}, \tag{2}$$

$$\text{Only lower limit of specification}: \text{ Cpk} = \frac{\overline{X} - LSL}{3\sigma},\tag{3}$$

When both upper and lower limit are specified : Cpk = (1 − k) × Cp, (4)

where Cp is the capability index and K is the bias.

K is obtained as follows (5).

$$\mathbf{K} = \begin{array}{c} \frac{\left(\text{ISL} + \text{ISL}\right)}{2} - \overline{X} \\ \frac{\left(\text{ISL} - \text{ISL}\right)}{2} \end{array} \tag{5}$$

4.2.3. RPN Evaluation—D Criteria

Table 4 shows the D criteria, one of the RPN evaluation factors for fuel cell-based hybrid power systems. The evaluation criteria were divided into 1 (detectability), 2 (detection difficulty), and 3 (detailed description) to minimize ambiguity and ensure evaluation accuracy.



#### *4.3. Evaluation Method for RPN Evaluation Items Using Kendall's Concordance Coe*ffi*cient*

First, the research hypothesis was established for the RPN evaluation items S, O, and D, and the evaluation items reestablished within the FMEA team were evaluated. Based on the results of the internal evaluation, the significance probability was compared to confirm the validity of the research hypothesis for the evaluation items. The process returned to the potential effect evaluation step if the research hypothesis was rejected. Here, 'P' indicates the significance probability, i.e., the probability that the null hypothesis occurs. The probability that the research hypothesis occurs is set to '1-P'; if the significance probability is less than 5%, then the null hypothesis is rejected, and the research hypothesis is supported. Table 5 shows the null and research hypotheses of this study [35].



*H*<sup>0</sup> is the null hypothesis, which refers to the already established hypothesis. *H*<sup>1</sup> is the research hypothesis, which negates the null hypothesis; it refers to the method of validating the established research hypothesis.

There are many ways to find correlations, but the most common correlation coefficients are Pearson, Kendall, and Spearman. For the FMEA evaluation items, a non-parametric test was applied instead of a parametric test because an analysis method that directly calculates the probability and statistically tests the data is appropriate regardless of the shape of the population. Pearson is basically used for the correlation analysis, but since it is a parametric test that shows correlations when variables are continuous data, one of the Kendall and Spearman's methods was used to apply nonparametric tests without linear correlation. Spearman generally has higher values than Kendall's correlation coefficient, but is sensitive to deviations and errors in the data. Therefore, Kendall's correlation coefficient was applied in this study because the sample size was small and the data dynamics were large.

The internal evaluation of the FMEA team confirmed the validity of the research hypothesis on the reestablished evaluation items, after which the Kendall's concordance coefficient was compared to determine the reliability of the evaluation items for the individual evaluations. Kendall's concordance coefficient indicates a correlation between multiple evaluators assessing the same sample. The coefficient ranges from 0 to 1, with a higher value indicating stronger correlations. Coefficients above 0.9 are generally considered to indicate very high concordance, meaning that the evaluators apply essentially the same criteria when evaluating the samples, decreasing the ambiguity of the evaluation items, removing evaluation arbitrariness, and encouraging objectivity [36]. If the coefficients for each item deviate from the criteria, the process returns to the potential effect evaluation step.

This study calculated Kendall's concordance coefficient using Equations (6)–(8) and Statistical Package for the Social Sciences (SPSS), a widely used program in statistical analysis. The coefficient was calculated to analyze the concordance between the evaluators for the reestablished S, O, and D evaluation results.

$$\mathcal{W} = \left[\frac{12\,\text{\AA}\,\,T\_i^2}{K^2 \mathcal{N}(N^2 - 1)}\right] - \frac{\Im(N+1)}{N-1},\tag{6}$$

where *TI* is the sum of the classes assigned to each target item by the evaluators, *K* is the number of evaluators, and *N* is the number of target items.

The formula for calculating *Ti* finds the mean (*Ri*) for the sum of sequence scales.

$$
\overline{R\_i} = \frac{\sum R\_i}{N}.\tag{7}
$$

Then, the average deviation *Ti* for each item can be obtained as follows.

$$T\_i = \sum \left(\overline{\mathcal{R}\_i} - \overline{\mathcal{R}\_i}\right)^2. \tag{8}$$

When establishing the evaluation items, the reliability of the internal evaluation results is verified using the significance probability for the research hypothesis for S, O, and D. The Kendall concordance coefficient was applied to reestablish the evaluation items that satisfy the criteria.

Based on the confirmed evaluation items, the external evaluators were requested to simultaneously evaluate both the existing and reestablished evaluation items. The significance probability and Kendall's concordance coefficient could again be applied to the results of the existing and reestablished evaluation

items to judge the application of the same standard. Thus, using the reestablished evaluation items, it is possible to verify that the evaluators are making objective, rather than arbitrary, decisions.
