Quality control testing under acceptance sampling plans involves inspecting a representative sample of products or materials from a larger lot or batch to determine whether the lot meets predetermined quality standards. In this research, the modified Lindley distribution is used as a model for lifetime study. When a life test is amputated at a pre-appropriated time to decide on the admission or refusal of the submitted batches, the problems of the single, double, and multiple (three and four stages) acceptance sampling strategies are introduced. The optimal sample sizes are computed for single, double, and multiple acceptance sampling plans to ensure that the veritable mean life is greater than the prescribed mean life at the stipulated consumer’s risk. The operating characteristic functions are investigated at diverse quality levels. For single, double, and multiple acceptance sampling plans, the minimal ratios of the veritable mean life to the prescribed mean life at the established percent of the producer’s risk are obtained. To demonstrate the uses of single, double, and multiple, some numerical experiments are presented.

When a life test is terminated at a predetermined time to decide whether to accept or refuse the submitted batches, the types of group sampling inspection plans (single, two, and multiple-stages) are introduced. The tables in this study give the optimal number of groups for various confidence levels, examination limits, and values of the ratio of the determined experiment time to the fixed percentile life. At various quality levels, the operating characteristic functions and accompanying producer's risk are derived for various types of group sampling inspection plans. At the determined producer's risk, the optimal ratios of real percentile life to a fixed percentile life are obtained. Three case studies are provided to illustrate the processes described here. Comparisons of single-stage and iterative group sampling plans are introduced. The first, second, and third sample minimums must be used to guarantee that the product's stipulated mean and median lifetimes are reached at a certain degree of customer trust. The suggested sample plans' operational characteristic values and the producer's risk are given. In order to show how the suggested approaches based on the mean life span and median life span of the product may function in reality, certain real-world examples are examined.

It is often desired for practical reasons to cease a life test at a predetermined period $ {{

This study develops double and multiple (three-stage) acceptance sampling plans based on the X-gamma distribution, a flexible model widely used in reliability engineering. The plans are built around truncated life tests, meaning testing stops either after a predetermined time or once a set number of failures occur. For the two-stage (double) sampling approach, we determine the smallest practical sample sizes and calculate the average sample number (ASN). We then expand the framework to a three-stage plan, outlining how it works and specifying the sample sizes needed at each step. To evaluate performance, we derive operating characteristic (OC) curves for all plan types, showing how well they distinguish between good and poor-quality lots across different quality levels. We also identify the minimum ratio of actual to specified mean life required to keep producer’s risk within acceptable limits. Additional efficiency metrics like average total inspection (ATI) and average outgoing quality (AOQ) are calculated to give a complete picture of how each plan performs in practice. Real-life numerical examples are included to walk through how these plans guide lot-acceptance decisions using the X-gamma model. Ultimately, this work gives quality control professionals reliable, distribution-specific tools for designing smarter, more efficient acceptance sampling procedures.