A comprehensive review on economic reliability test plan for various life time distributions

Introduction

In an era where product reliability is crucial for maintaining competitive advantage, economic reliability test plans have emerged as essential tools for ensuring both quality and costeffectiveness. These test plans enable businesses to optimize resource allocation while achieving desired reliability levels, thereby enhancing customer satisfaction and reducing costs. The development of Economic reliability test plans has been influenced by various statistical methodologies, each contributing to more precise and efficient reliability assessments. From their historical origins to recent advancements, these test plans have evolved to incorporate new acceptance sampling plans, especially those based on truncated life tests for different statistical distributions. Reliability testing fundamentally about understanding and mitigating the risk of product failure. The cost implications of such failures can be significant, making economic considerations a key aspect of reliability test planning. By integrating cost benefit analyses with reliability assessments, economic reliability test plans provide a comprehensive framework for decision-making. This includes determining the optimal balance between the costs of testing and costs associated with potential product failures. Historically, the field of reliability life testing has seen substantial contributions from researchers who have developed various models and methodologies. Among these are acceptance sampling plans that allow for efficient decision-making under uncertain conditions. These plans have been adapted to various distributions. Sobel and Tischendrof (1959) [2] developed reliability test plans for exponential distribution. Goode and Kao (1961) [3] constructed sampling plans for Weibull distribution. Gupta and Groll (1961) [4] constructed sampling plans for Gamma distribution. Sampling plans similar to those of Gupta and Groll (1961) [4] were developed by Kantam and Rosaiah (1998) for half-logistic distribution. Kantam et al. (2001) developed a detailed study on life tests based on Log-logistic distribution. Recently, Jayalakshmi and Vijilamery (2022) [10] developed acceptance sampling plan for truncated life test based on percentiles using Gompertz Frechet distribution. Jayalakshmi and Aleesha (2024) [11] using Exponentiated Generalized Inverse Rayleigh distribution.

Advancements in the field have introduced innovative approaches to economic reliability test plans. These include the development of new acceptance sampling plans based on truncated life tests, which provide more reliability assessments by focusing on the early stages of product life. Such methodologies are particularly useful in industries where early failure detection is critical to maintaining product quality and customer satisfaction. Statistical Quality Control (SQC) plays a pivotal role in the development and implementation of economic reliability test plans. SQC involves using statistical methods to monitor and control production processes, ensuring that products meet specified quality standards. Techniques such as control charts and process capability analysis are integral to SQC an provide a quantitative basis for making informed decisions in reliability testing. This review paper aims to synthesize the vast body of literature on economic reliability test plans, providing a comprehensive analysis of different approaches and their practical applications. By examining the effectiveness and economic implications of these test plans, this paper seeks to offer valuable insights for researches and industry professions alike. Through a thorough exploration of historical developments, current methodologies, and future trends, this review will highlight the critical role of economic reliability test plans and statistical quality control in modern industry.

Economic Reliability Test Plan

A methodology designed to balance the costs of reliability testing with the risks of product failures. It ensures Economic reliability is understood as, on the one hand, an ability of an object to be implemented under certain interaction with the environment and, on the other hand, as a quantitative estimate of an object, explicitly relating probability with time or other parameters of the implementation process. Some properties of economic reliability include: effectiveness, stability, selforganization, vitality, and communicativeness. Kantam et al. (2006) [1] introduced Economic reliability test plan for Log logistic distribution. In this paper focuses on the log logistic distribution, which is used to model the life data of products. The distribution is particularly useful due to its flexibility and ability to fit various types of reliability data. The authors propose test plans that incorporate economic considerations into the reliability testing process. The frame work includes methods for calculating the optimal sample size and termination time required to achieve desired reliability levels at minimum cost. The paper provides numerical examples to illustrate the application of the proposed economic reliability test plans, demonstrating their effectiveness in real-world scenarios. Conclusion of this study is the termination time of the economic reliability test plan for Log logistic distribution is uniformly smaller than the corresponding time of the plan in Kantam et al. (2001). The cumulative distribution function of log logistic distribution is [8] developed Odds Exponential log logistic distribution for an Economic reliability test plan. The cdf of this distribution is

Origin of the Economic Reliability test plan

Rosaiah et al. (2018) [8] developed an economic reliability test plans for Exponetial Pareto distribution and Ravikumar et al.

(2019) developed Economic reliability test plan for Burr Type X distribution. The authors aim to identify the best termination time considering a specific sample size, producer's risk and termination number. By comparing the proposed plan with existing ones, the authors highlight its effectiveness in terms of reducing the duration and cost of experiments. The Operatic Characteristic (OC) curve is a graphical representation that shows the probability of accepting a lot given various levels of defect rates. In the context of these papers, the OC curve helps to visualize the performance of the proposed truncated life test plan. It illustrates how the probability of accepting a product changes with different defects rates, providing insights into the plan's effectiveness in distinguishing between acceptable and defective products. The cumulative distribution function of these distributions are given below.

] , 𝑡 > 0; 𝜆, 𝜎, 𝜃 > 0 𝐹(𝑥; 𝑘, 𝜆) = (1 -𝑒 -(𝑥/𝜆) 2 ) 𝑘 ; 𝑥 > 0, 𝑘 > 0, 𝜆 > 0

Gilliarose et al. (2022) [12] Jayalakshmi and Sivasanthiya (2025) [13] By focusing on these areas, future research can significantly advance the field of economic reliability test plans, leading to more efficient, cost-effective and reliable solutions for ensuring product quality and performance.

Conclusion

In this review, we have explored various economic reliability test plans across different statistical distributions, including the Marshall-Olkin Extended Lomax distribution, the Hybrid Exponential distribution, the Odds Exponential Log Logistic distribution etc., The significance of these test plans lies in their ability to optimize the duration of reliability testing, ensuring that products meet the desired quality and reliability standards. Despite the advancements, several gaps remain in the current research. In conclusion, Economic reliability test plans play a crucial role in ensuring product quality and reliability while maintaining cost-effectiveness. Continued research and innovation in this field will contribute significantly to the development of efficient, reliable and economically viable testing methodologies, ultimately benefiting both producers and consumers.

References

1. Kantam RRL, Rao GS, Sriram B. An economic reliability test plan: log-logistic distribution. Journal of Applied Statistics. 2006;33(3):291-6.
2. Sobel M, Tischendrof JA. Acceptance sampling with new life test objectives. In: Proceedings of fifth national symposium on reliability and quality control. 1959. p. 108, 118.
3. Goode HP, KAO JH. Sampling plans based on the Weibull distribution. 1961.
4. Gupta SS, Gupta SS. Gamma distribution in acceptance sampling based on life tests. Journal of the American Statistical Association. 1961;56(296):942-70.
5. Ravikumar MS, Durgamamba AN, Kantam RRL. Economic reliability test plan for Burr Type X distribution. International Journal of Advanced Engineering Research and Applications. 2019;5(3):56-63.
6. Balan R, Kunjunni S. Truncated Life Test Plans for Economic Reliability Based on Four-Parametric Burr Distribution. Journal of Industrial Mathematics. 2013;2013(1):489285.
7. Rosaiah K, Kantam RRL, Kumar CS. Exponentiated loglogistic distribution-an economic reliability test plan. PAKISTAN JOURNAL OF STATISTICS-ALL SERIES-. 2007;23(2):147.
8. Rosaiah K, Rao GS, Kalyani K, Sivakumar DCU. Odds exponential log logistic distribution-An economic reliability test plan. International Journal of Science and Research. 2018;7(11):1653-60.
9. Mughal AR, Aslam M, Hussain J, Rehman A. Economic reliability group acceptance sampling plans for lifetimes following a Marshall-Olkin extended distribution. Middle Eastern Finance and Economics. 2010;7:87-93.
10. Jayalakshmi S, Vijilamery S. Study on Acceptance Sampling Plan For Truncate Life Tests Based on Percentiles Using Gompertz Frechet Distribution. Reliability: Theory & Applications. 2022;17(1 (67)):31624.
11. Jayalakshmi S, Aleesha A. Study on acceptance sampling plan based on percentiles for exponentiated generalized inverse rayleigh distribution. Reliability: Theory & Applications. 2024;19(2 (78)):202-8.
12. Gilliarose et al. (2022) has developed an economic reliability test plan based on truncated life tests for Marshall-Olkin Power Lomax distribution with applications. Reliability: Theory & Applications. 2022;17(1(67)): 316-24.
13. Jayalakshmi S, Sivasanthiya R. An Economic Reliability Test Plan for Weighted Weibull Distribution. Indian Journal of Science and Technology. 2025;18(11):870-6.
14. Rao GS, Ghitany ME, Kantam RRL. Marshall-Olkin extended Lomax distribution: an economic reliability test plan. International Journal of Applied Mathematics. 2009;22(1):139-48.