## 4 Research Methodology
### Radiation Effects  
Following [PERSON], [PERSON], and [PERSON] (2016), we used a two-step method to verify the mediating role of perceived justice. First, a bootstrap analysis with 10,000 samples was performed for the model; the analysis also set lines connecting physical participation and satisfaction, and mental participation and satisfaction. Then, the results of total effect and direct effect were calculated using SmartPLS 3.2.6 software. For the direct effect of PP\(\rightarrow\)SAT is significant (\(t\) = 4.281, \(p\) = 0.000 \(<\) 0.001), and the indirect effect of PP \(\rightarrow\) SAT though PJ is significant (\(t\) = 2.002, \(p\) = 0.045 \(<\) 0.05), the 95% percentile CI of PP \(\rightarrow\) SAT through PJ ([0.004, 0.091]) does not include 0, thus PJ partially mediates PP \(\rightarrow\) SAT. The direct effect of MP \(\rightarrow\) SAT is significant (\(t\) = 5.200, \(p\) = 0.000 \(<\) 0.001), and the indirect effect of MP \(\rightarrow\) SAT though PJ is significant (\(t\) = 2.840, \(p\) = 0.005 \(<\) 0.01), the 95% percentile CI of MP \(\rightarrow\) SAT through PJ ([0.022, 0.109]) does not include 0, thus PJ partially mediates MP \(\rightarrow\) SAT. Partial Least Squares Multi Analysis (PLS-MGA): The Moderation Role of the Expressions of the Apology  
Our method of PLS-MGA followed that of [PERSON] (1986). First, the SRMR, d\({}_{\text{ULS}}\) and d\({}_{\text{G}}\) of group 1 are 0.025, 0.032, 0.057, and group 2 are 0.045, 0.056, and 0.045 which means the overall goodness of fit of two groups' model was verified. And the results of Table 4 also provide clear support for the measures' reliability and validity (1986). Second, we use the measurement invariance of composite models (MICOM) procedure to test the measurement invariance for all groups. Configural invariance was fulfilled since the measurement models for the two groups meet the three requirements of \"identical indicators per measurement model, identical data treatment, and identical algorithm settings criteria\" (1986). Then, a permutation test was executed to verify compositional invariance. A permutation test was set at 5000 samples, and the test type was set as two-tailed with a significance level of 0.05. As shown in Table 7, the original correlations are equal to or greater than the 5.00% quantile correlations (shown in the 5% column), indicating that compositional invariance has been demonstrated for all the constructs (1986). The data in the mean original difference column fall within the 95% CI, and the permutation \(p\)-values are greater than 0.05, indicating that the construct passes the variance test. The data in the variance original difference column fall within the 95% CI, and the permutation \(p\)-values are greater than 0.05, further indicating that the entire construct passes the measurement variance test (1986). Third, since invariance is verified, we first analyze the groups separately to determine whether there is a difference between the two groups. Table 8 shows the bootstrap results of each of the two groups with 5000 sub-samples. As the last column of Table 7 shows, the permutation \(p\)-values of the three types of customer participation and perceived justice are all less than 0.10, indicating that these three relationships have significant differences between the two groups (1986). To further verify the analysis results according to the permutation test, we use the MGA algorithm to check the differences between the two groups. Table 8 shows the results of the test of CIs, which is a non-parametric test. The criteria establish that if the parameter coefficient for a path relationship of one group (see Table 8) does not fall within the corresponding CI of another group (Table 8) and vice versa, then there exists no overlap, and we can assume that there is a significant difference between the two groups (1986). Thus, according to Table 8, H2b and H2c are verified; however, emotional participation does not influence perceived justice in either direct or indirect situations; thus, H2a is not verified.