**Optimal Plan of Multi-Stress–Strength Reliability Bayesian and Non-Bayesian Methods for the Alpha Power Exponential Model Using Progressive First Failure**

**Ehab M. Almetwally 1,2, Refah Alotaibi 3, Aned Al Mutairi 3, Chanseok Park 4,\* and Hoda Rezk 5**



**Abstract:** It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little attention to. The goal of this paper is to develop inference for multi-reliability using unit alpha power exponential distributions for stress–strength variables based on the progressive first failure. As a result, the problem of estimating the stress–strength function R, where X, Y, and Z come from three separate alpha power exponential distributions, is addressed in this paper. The conventional methods, such as maximum likelihood for point estimation, Bayesian and asymptotic confidence, boot-p, and boot-t methods for interval estimation, are also examined. Various confidence intervals have been obtained. Monte Carlo simulations and real-world application examples are used to evaluate and compare the performance of the various proposed estimators.

**Keywords:** multi-stress–strength; progressive first failure censoring; balanced loss functions; Lindley's approximation; Markov Chain Monte Carlo; symmetric and asymmetric loss functions; bootstrap confidence intervals
