Designing and construction of chain sampling plan indexed through indifference quality level using truncated binomial distribution

Introduction

Acceptance sampling remains a crucial component of statistical quality control, helping manufacturers to determine whether to accept or reject lots by inspection of few samples. Traditional single sampling and double sampling plans are no longer enough for modern highreliability operations which work at very low levels of defects. Therefore, more attention is being paid to truncated distributions and chain-based sampling methods which can provide a good risk balance. In their research, Al-Omari and Al-Nasser (2021) [1] created acceptance sampling plans based on truncated life tests under power Lomax distribution and showed the potential of using truncated distributions in quality assurance in high-reliability systems. In their study, Hafeez et al. (2024) [2] calculated the values of the Acceptable Quality Level (AQL), Limiting Quality Level (LQL) and quality regions for group chain sampling plans under the binomial distribution. As continuation of the previous study, Hafeez, Aziz, and Shabbir (2024) [3] proposed the calculation of AQL, LQL and quality regions in group chain sampling plans. As the next step in development of the field, Hafeez et al. (2025) [4] created a generalized Bayesian double group sampling plan which is especially aimed to be applied in manufacturing industries. Pratheeba and Radhakrishnan (2015) [5] proposed single sampling plans, indexed through the Limiting Quality Level, under the truncated binomial distribution and focused on increased protection of consumers' interests at high level of defects. Previously, Radhakrishnan and Pratheeba (2011) [6] introduced double sampling plans, indexed through the Average Quality Level, under the truncated binomial distribution, forming the basis for the development of this field. In the same year, Radhakrishnan and Pratheeba (2011) [7] created single sampling plans, indexed through the Indifference Quality Level, under the truncated binomial distribution and stressed improved modelling of zero-defective process. In (2016) [8] , Teh et al. Proposed time-truncated group chain sampling plans based on the Rayleigh distribution, demonstrating benefits of using truncation. The current research aims to develop a Chain Sampling Plan of Type -1 (ChSP-1) under the truncated binomial distribution and indexed through the Indifference Quality Level (IQL), where the probability of acceptance equals 0.50 approximately. This approach allows providing a balance between producer and consumer risks using historical lot performances and achieving the optimal trade-off between the efforts required to conduct inspections and discrimination of the plans. The paper calculates the acceptance probability function and produces the parameter tables and gives examples of application of ChSP-1.

Methodology

The study constructed parameters of the ChSP-1 model by varying sample size (n), acceptance number (c), and number of preceding lots (i). IQL values were derived using the distribution to capture the expected defect levels without zerodefect occurrences. Data were tabulated for sample sizes ranging from 320 to 1000 units. The Operating Characteristic (OC) curve was generated for sample size to illustrate the probability of lot acceptance Pa (p) at different defect levels p. Industrial scenarios, such as pharmaceutical quality testing and automotive defect inspections, were mapped to the proposed sampling plan to validate real-world applicability. The proposed statistical model is based on the Chain Sampling Plan of Type-1 (ChSP-1) integrated with the Truncated Binomial distribution. Unlike the conventional binomial, the distribution eliminates the zero-defect probability, which is unrealistic in industrial production where defects are rare but not absent.

Truncated Binomial Distribution

The Truncated Binomial Distribution is used to model the number of defectives x in a sample of size n when zero defectives are excluded: The probability Mass Function (pmf) of the distribution is given as Where, n = sample size; p = probability of a defect; x = number of defects in the sample; i = maximum number of defectives in total (current mean); c = acceptance number; k = number of previous lots to chain in memory; The denominator 1-(1-p) n adjusts the probability by excluding the zero-defect case. This adjustment is appropriate for high-capability processes where the probability of zero defects is negligible.

Chain Sampling Plan Structure

The Truncated Binomial Distribution based Chain Sampling Plan (ChSP-1) utilizes the inspection results of the current lot in conjunction with information from i immediately preceding lots. The decision procedure is defined as follows  Accept the current lot if the observed number of defectives ≤ c.  If not, examine the records of the i preceding lots. If all of these lots were accepted under the same acceptance criterion (defectives ≤ c), then the current lot shall also be accepted.  If any of the i preceding lots were rejected, the current lot shall be rejected.

This dependency creates a cumulative inspection system, providing enhanced quality control by linking acceptance decisions across successive lots while utilizing the truncated binomial distribution to estimate the probability of acceptance more efficiently.

Glossary of Symbols

 n = sample size  c = acceptance number  i = number of immediately preceding lots considered  p = Proportion Defective / Lot Quality  q = 1p  β = Consumer's Risk  P a (p) = Probability of Acceptance of the Lot Quality p.

Conditions for Application

 The production process must be in a state of statistical control, ensuring that the probability of defect occurrence remains constant over time.  The lot size should be large, and the sampling fraction relatively small, justifying the use of the truncated binomial model as an approximation.  Reliable inspection and record-keeping of the preceding i lots is essential to maintain the chain structure.  The plan is suitable where inspection costs are high or destructive testing is involved, necessitating reduced sample sizes without compromising decision reliability.  The method assumes independent defect occurrences and random distribution of defects within lots.

It should not be applied when the production process is unstable, lots are non-homogeneous, or when preceding lot information is incomplete or unavailable.

Construction of

Operating Characteristic (OC) Curve

The probability of acceptance, Pa(p), under ChSP-1 is given as,

Where P(X=x) follows the distribution. For example, probabilities are computed at varying p (defect levels) to construct the OC curve. 1), the values of the IQL are obtained for the various combination of n, c and i using an excel package and are presented in Table 1. The parameters of chain sampling plan-1, n, c, and i are recorded for various combination of IQL. For a given IQL = 0.0838, the values of n, c and i are obtained from Table 1 as n = 600, c = 4 and i = 3 which are associated in parameters of chain sampling plan with the specified IQL = 0.0838. The OC curve for this plan is presented in Figure 1. 1), the values of the IQL are obtained for the various combination of n, c and i using an excel package and are presented in Table 1. The parameters of chain sampling plan-1, n, c, and i are recorded for various combination of IQL. For a given IQL = 0.0149, the values of n, c and i are obtained from Table 1 as n = 1000, c = 5 and i = 4 which are associated in parameters of chain sampling plan with the specified IQL = 0.0149. The OC curve for this plan is presented in Figure 2.

References

1. Al-Omari AI, Al-Nasser AD. Acceptance sampling plans from a truncated life test based on the power Lomax distribution. Statistica. 2021;81(1):45-58.
2. Hafeez W, et al. Estimation of AQL, LQL and quality regions for group chain sampling plan with binomial distribution. Thailand Statistician. 2024;22(1):1-15.
3. Hafeez W, Aziz N, Shabbir J. Estimation of AQL, LQL and quality regions for group chain sampling plan with binomial distribution. Thailand Statistician. 2024;22(3):674-687.
4. Hafeez W, et al. Generalized Bayesian double group sampling plan for manufacturing industry. Pakistan Journal of Statistics and Operation Research. 2025.
5. Pratheeba S, Radhakrishnan R. Construction of single sampling plan indexed through limiting quality level using truncated binomial distribution. International Journal of Management and Social Sciences. 2015;3(2):273-278.
6. Radhakrishnan R, Pratheeba S. Construction of double sampling plan indexed through average quality level using truncated binomial distribution. International Journal of Computational and Mathematical Sciences. 2011;4(1):63-68.
7. Radhakrishnan R, Pratheeba S. Construction of single sampling plan indexed through indifference quality level using truncated binomial distribution. Research Journal of Applied Sciences, Engineering and Technology. 2011;3(5):456-460.
8. Teh MAP, et al. Time truncated group chain sampling plans for Rayleigh distribution. Global Journal of Pure and Applied Mathematics. 2016;12(4):3693-3699.