Designing Bayesian Sampling Plans with Adaptive Progressive Hybrid Censored Samples

This paper studies the acceptance sampling for exponential distributions with type-I and type-II adaptive progressive hybrid censored samples. Algorithms are proposed for deriving Bayesian sampling plans. We compare the performance of the proposed sampling plans with the sampling plans of Lin and Huang (2012). The numerical results indicate that the proposed sampling plans outperform the sampling plans of Lin and Huang (2012). 1. Introduction Suppose we are given a batch of lifetime components for acceptance sampling. We let denote the lifetimes of these units. It is assumed that are mutually independent and follow an exponential distribution, having expected lifetime？？, and the parameter follows a gamma prior distribution. In the context of life test experiment, identical units are sampled from the batch and placed on life test without replacement with a suitable sampling scheme. At the end of the experiment, let denote the duration of the experiment, and let be the number of failures among the items put on life test. Let be an action regarding the acceptance sampling. When , it means accepting the batch, while when , it means rejecting the batch. Let denote the cost per unit inspected. Also, let be the loss of rejecting the batch and let be the loss of accepting the batch where , , , and . In many situations, the cost of time used for life test is an essential issue and should be taken as a part of the loss function. We let be the cost per unit time used for the life test experiment. When the life test experiment terminates, the unfailed components can be reused and thus have salvage value , where . Thus, many researchers including Chen et al. [1], Liang and Yang [2], and Lin and Huang [3] have considered using the loss function for the acceptance sampling, where Lin and Huang [3] have studied acceptance sampling for exponential distributions with the loss based on adaptive type-I and type-II progressive hybrid censored samples. For the adaptive progressive hybrid censoring (APHC), a positive time and progressive censoring scheme？？ should be determined before the life test experiment, where , , for , and . Let denote the observable variables obtained either by using type-I APHC or by using type-II APHC and an observed value of . A decision function is a function defined on the sample space of the observable variables such that is the probability of accepting the batch when is observed. The determination of the design parameters and the decision function is called a sampling plan, which is denoted by . With the loss of (1) and the prior distribution,