## 6 Results, Discussion, and Findings  
The suggested Bayesian HEWMA control chart based on various RSS schemes with two different LFs is compared with the Bayesian HEMWA and Bayesian AEWMA control chart under simple random sampling (SRS) with the same value of the smoothing constants for posterior and posterior predictive distribution with different LFs is presented in Tables 1-6. Tables 1 and 2 indicate the ARL and SDRL results of the suggested Bayesian HEWMA control chart using RSS schemes with informative prior under SELF and Bayesian HEWMA, and Bayesian AEWMA control chart with SRS. The results indicate that the suggested Bayesian HEWMA control chart using RSS schemes more efficiently detects out-of-control signals then the existing Bayesian HEWMA and Bayesian AEWMA control chart under SRS. For example, the \(ARL\) results of the HEWMA control chart using the Bayesian approach under SRS with SELF for posterior and posterior predictive distribution at smoothing constants \(\lambda_{1}=0.10\), \(\lambda_{2}=0.05\), and different shifts, i.e., \(\sigma=0.0\), 0.30, 0.50, 0.80, 1.50, and 4 are 371.67, 45.28, 19.77, 9.08, and 1.02; under the same case the ARL values for the Bayesian AEWMA control chart with SRS are 370.98, 35.40, 13.55, 5.62, 2.25, and 1.01. In a similar situation, the ARL results of the suggested Bayesian HEWMA control chart using RSS are 370.17, 19.73, 8.51, 3.80, 1.47, and 1 and 368.36, 16.67, 7.04, 3.20, 1.29, and 1 for MRSS. ARL values for ERSS are 371.51, 22.04, 9.50, 4.34, 1.61, and 1. The results indicate that the suggested HEWMA control chart using RSS schemes at each shift gives smaller ARL values than the Bayesian HEWMA and Bayesian AEWMA control chart using SRS, which shows that the suggested Bayesian HEWMA control chart using RSS schemes is more sensitive than the Bayesian HEWMA using SRS. Similarly, under LLF, Table 6 gives the comparison of the suggested control chart with the Bayesian HEWMA control chart using SRS utilizing informative prior distribution. The ARL values for Bayesian HEWMA using SRS at \(\lambda_{1}=0.25\), \(\lambda_{2}=0.05\) and shift \(\sigma=0.0\), 0.30, 0.50, 0.80, 1.50, and 4 are 370.35, 48.07, 20.77, 9.58, 3.35, and 1.04. Further, 368.67, 54.92, 25.97, 12.79, 4.94, and 1.09 are the ARL values of the Bayesian AEWMA with SRS. The ARL results of the suggested Bayesian HEWMA control chart using RSS are 370.12, 20.89, 8.98, 4.13, 1.58, and 1; for MRSS, the ARL values are 369.24, 16.83, 7.31, 3.34, 1.36, and 1. The ARL values for ERSS are 369.89, 23.28, 10.20, 4.65, 1.71, and 1. The values for the suggested Bayesian HEWMA control chart under RSS schemes at the larger shifts rapidly decrease, which shows that the proposed control chart efficiently detects the out-of-control signals more than the Bayesian HEWMA and Bayesian AEWMA control charts under SRS. Tables 1-6 show the computed results of the suggested HEWMA control chart using Bayesian theory under RSS schemes with informative prior distribution utilizing different LFs such that SELF and LLF for posterior and posterior predictive distribution. The main findings of the suggested HEWMA control under the Bayesian approach are discussed below:* The performance of the suggested Bayesian HEWMA control chart using RSS schemes is observed with the changes in the values of the smoothing constants, i.e., \(\lambda_{1}\) and \(\lambda_{2}\). The effect of the proposed control chart takes the fixed value of the \(\lambda_{2}\) with different values of the observed. Tables 1 and 2 present the results of ARL and SDRL for the suggested Bayesian HEWMA control chart using informative prior under SELF for posterior and posterior predictive distribution; the results show that for a smaller value, the proposed Bayesian HEWMA control chart performed well. For example, at \(ARL_{0}=370\), \(\lambda_{1}=0.10\), \(\lambda_{2}=0.05\), and shift \(\delta=0.20\) the ARL value is 29.40, and for \(\lambda_{1}=0.25\) the ARL value is 38.87 using RSS. The values are 28.90 and 28.93 for MRSS, and for ERSS the ARL values are 29.78 and 33.75.