Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 307 https://internationalpubls.com Revolutionary Hybrid RDS, Big Data and AI Quality Control: Cpk- Based Analysis 1B.Esha Raffie, 2M. Vijayakumar, 3K.Kalaivani, 4Dr.Srimathi, 5G. Suganthi, 6Dr.B.Sivasankari 1Assistant Professor of Statistics, Department of Mathematics, Sri Krishna Arts and Science College 1Email ID: esharaffieb@skasc.ac.in 2Department of Computer Technology, Nandha Arts and Science College, Erode, Tamilnadu, India. 2Email ID:Vij370@gmail.com 3Assistant Professor in Computer Science and Business Systems, M.Kumarasamy College of Engineering, Karur 639119 4Associate Professor, KPR College of Arts Science and Research, Coimbatore.4Email ID: sriphd2020@gmail.com 5Assistant professor, Mathematics, Sona College of Technology, Salem,5Email ID:sugachandru@gmail.com 6Professor (Mathematics), Department of Agricultural Economics, Agricultural College and Research Institute, Madurai 6Corresponding author email ID:sankari83@tnau.ac.in Article History: Received: 14-07-2024 Revised: 29-08-2024 Accepted: 11-09-2024 Abstract: In this research, a combination of inspection and repeated deferred sampling is suggested utilizing the process capability index Cpk. This sampling approach can effectively evaluate the quality characteristics of normal distributions even when the mean and variance are unknown. Symmetrical and asymmetrical regions failing to meet optimal conditions for practical use are identified through specific tables Furthermore, the benefits of this proposed mixed sampling design are discussed. A comparison between the suggested policies and current practices is presented to illustrate the differences. A practical example is utilized to demonstrate the practicality of this hybrid sampling strategy. The optimal parameters can be determined using the two-point method of the operating characteristic curve. Keywords: Mixed sampling designs, process capability indices, decision-making, nonlinear programming, and quadratic programming. I. Introduction The assessment of features, variables, and other quality aspects is typically carried out acceptance sampling. Items are categorized as meeting or not meeting standards based on their characteristic quality, while the quality attributes of the item sample are assessed on a continuous scale in variable control. Acceptance criteria-related sampling regulations can be found in Acceptance Sampling Methods. Variable sampling plans generally offer more insight into a product compared to feature sampling methods. Furthermore, the same level of assurance can be achieved by using a variable sampling plan with a smaller sample size than that of a characteristic plan. Therefore, variable sampling strategies are employed in cases where testing is costly and risky. Various sampling schemes have been introduced in the literature, such as Single Sampling Scheme (SSP), Double Sampling Scheme, Multiple Sampling Scheme, Sequential Sampling, etc., tailored to different scenarios. These schemes, also known as standard sampling schemes, base all judgments on a given batch solely on the mailto:esharaffieb@skasc.ac.in mailto:Vij370@gmail.com mailto:sriphd2020@gmail.com mailto:sugachandru@gmail.com mailto:sankari83@tnau.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 308 https://internationalpubls.com data derived from the sampled items of the existing batch. Sherman's (1965) cluster repeat sampling (RGS) method, Wortham and Baker's (1976) multiple delayed (state-dependent) sampling, and Dodge's (1955) chain sampling method are examples of specialized sampling techniques. Chain sampling, MDS, and RGS schemes have been investigated by only a handful of authors, including Aslam, Azam, and Jun (2013), Aslam, Wu et al. (2013), Balamurali et al. (2017), Balamurali, Jeyadurga, Usha, and Venkatesulu (2018), Balamurali and June (2007), Fallahnejad and Seifi (2017), June et al. (2010), among others. Aslam et al. (2016) merged the characteristics of the MDS sampling plan with the RGS plan to develop a novel sampling plan for assessing feature quality traits, which they termed multi-dependency state recursive cluster sampling (MDSRGS) design. The MDSRGS method has demonstrated superior performance over traditional MDS and RGS methods. Furthermore, an extended ISWG variable testing procedure based on the Unidirectional Process Capability Index (2018) has been introduced, as well as the new strategy for sampling mixed repeat clusters for product approval utilizing process capability indices by Balamurali and Usha (2018). A hybrid RDS verification scheme based on the Cpk process capability index is proposed in this study. II. Mixed inspection RDS plan applications in big data, Neural Network and Artificial Intelligence Big Data: The mixed inspection RDS plan can be used in big data scenarios to ensure quality and reliability of the data. Some specific uses of mixed inspection RDS plan in big data include: 1. Evaluation of data quality: The mixed inspection RDS plan can be used to evaluate how well large data complies with predetermined quality requirements. 2. Assessment of huge data's compliance with established quality requirements can be done using the mixed inspection RDS approach. 3. Data reliability assessment: The mixed inspection RDS plan can be used to evaluate the reliability of big data by assessing the likelihood of non-conformance. 4. Process control: The mixed inspection RDS plan can be used in process control to monitor and improve the quality of big data in real-time. 5. Decision making: Mixed inspection RDS design can be used for decision making to support quality-related decisions based on big data evaluations. In general, the mixed inspection RDS plan is a useful tool for ensuring the quality and reliability of big data and can be used in various stages of the data life cycle to support data-driven decision making. Neural Network: Mixed inspection RDS plan can be used in neural network training as a strategy to sample the training data. In large datasets, it may not be feasible to use all the data for training the neural network. The mixed inspection RDS plan can be used to randomly sample a subset of the training data, which can then be used to train the neural network. By using a mixed inspection RDS plan, the training data can be efficiently sampled, while still ensuring that the sampling process is representative of the entire Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 309 https://internationalpubls.com dataset. This can improve the training efficiency, reduce the memory requirements, and help prevent overfitting. Artificial Intelligence: The mixed inspection RDS plan can be used in various applications of Artificial Intelligence (AI) such as training machine learning models, evaluating the performance of AI systems, and managing large datasets. The mixed inspection RDS plan provides a way to efficiently sample data while maintaining the representativeness of the sample. In AI, this can help reduce the computational costs of training models and evaluating their performance, as well as ensure that the training and evaluation processes are based on a representative subset of the available data. The mixed inspection RDS plan can also be used in AI applications to manage large datasets by randomly sampling subsets of the data for processing, which can help improve the scalability of the AI systems. Process Capability Indices A procedure is deemed satisfactory when the majority of measurements fall within the designated range. By analyzing the results of a procedure, you can evaluate its stability regarding specific process variability criteria. A capability index, known as PCI, is a statistical tool utilized to quantify this variability. Due to its potential in enhancing product quality, PCI has been extensively studied recently. PCI is employed to a manufacturing process's capability to manufacture batteries within specified tolerances. While various forms of PCI exist in literature, PCI Cp, Cpk, and Cpm are considered the primary metrics. References to PCI can be found in works by Kane (1986), Chan et al. (1988), Yum and Kim (2004), Spring et al. (2003), and Kotz and Johnson (2002, 2011). Let's suppose that USL and LSL represent the upper and lower limits of the specification, respectively, then introduce PCI Cp and Cpk. Cp = USLβˆ’LSL 6Οƒ (1) The Cpk indicator was developed and defined as follows because the Cp indicator does not accurately capture the effect of the location of the process mean. Where The terms upper specification limit and lower specification limit (USL and LSL, respectively) are used. If the process variance Οƒ2 is unknown, Kane (1986) uses the unbiased sampling variance S2 to calculate the likelihood index. The Cpk estimator is described as follows: πΆπ‘π‘˜ = min ( π‘ˆπ‘†πΏβˆ’ πœ‡ 3𝜎 , πœ‡βˆ’πΏπ‘†πΏ 3𝜎 ) (2) οΏ½Μ‚οΏ½π‘π‘˜ = min ( π‘ˆπ‘†πΏβˆ’ οΏ½Μ…οΏ½ 3𝑆 , π‘‹βˆ’πΏπ‘†πΏ 3𝑆 ) (3) The sample mean and standard deviation are denoted S in the literature (Kane, 1986). Balamurali and Kalyanasundaram (2002) utilized the bootstrap method to establish lower bounds of confidence for PCI Cpk, and Balamurali (2003) provided confidence levels for short-term manufacturing processes. From the existing knowledge of PCI Cpk and acceptance sampling, various researchers have designed acceptance sampling techniques. For instance, Aslam et al. (2013b) assessed the Resampling Group Scheme (RGS) for PCI Cpk testing, while Pearn and Wu (2007), Negrin et al. (2009), Balamurali and Usha (2014) introduced and analyzed a PCI Cpk-centered single sample design (SSP). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 310 https://internationalpubls.com Hybrid Acceptance Sampling Plan Hybrid acceptance sampling was first proposed by Schilling and Dodge (1969). The study of characteristics and variables is part of a mixed sampling approach. A mixed sampling plan is a good way to achieve this. The mixed sampling plan described in MIL-STD-414 has been studied by several authors, including Gregory and Reznikoff (1955), Bowker and Goode (1952), and Savage (1955) and (1957). Proposed Plans for Mixed Sampling Many authors have studied mixed sampling techniques in a variety of situations. Aslam and Lee are two examples (2012). Aslam et al. (2013a), Aslam et al. (2015a, 2015b) and Balamurali et al. All had composite sampling plans based on the Cpk capability index (2016a, 2016b). 2019 Balamurali and Aslam propose a multi-dependent state cluster resampling scheme for testing variables. III. The implementation and operation of a Repetitive Deferred Sampling Plan (RDSP) typically involves several key players: Quality control personnel - These are the individuals responsible for the implementation and operation of the RDSP, as well as the analysis of the data obtained from the samples. Production personnel - The production team handles the production of the items being sampled and must adhere to the quality requirements defined in the RDSP. Management - Management oversees allocating the resources required to set up and run the RDSP and of making choices in light of the data gathered from the samples. Consumers - Customers are the ultimate recipients of the quality control efforts inherent in the RDSP because they are the end users of the products being produced. Material on RDSP can be obtained in a variety of places, includes journals, publications, and websites devoted to statistical process control and quality control. Douglas S. Montgomery of Statistical Process Control and Thomas P. Ryan and D.H. Quality Assurance and Industrial Statistics have two important tasks to consider in one day. IV. Hybrid Inspection RDSP Based on Cpk Statistical process control and quality control both employ the RDSP approach. It entails periodically sampling a production process and delaying the inspection of the finished products until a predetermined volume of products has been produced. This enables the effective use of resources and may lower the cost of inspections. The goods are then examined all at once, and any flaws are used to modify the procedure and raise the standard of the product. The purpose of RDSP is to preserve product quality while lowering inspection costs. Repeated Deferred Sampling, developed by Shankar and Mohapatra in 1991, is simply an extension of Rambert Verst’s Multiple Deferred Sampling method MDS-(c1, c2) (1981). This method assumes that the results of inspecting the lot before or after the deferred lot during Repeated Group Sampling (RGS) inspection will decide whether the lot is accepted or rejected. RGS is therefore a special case of the RDS scheme. In addition to developing schedules for Multiple Delayed State Sampling (MDS), Wortham and Baker (1976) also produced tables for constructing MDS schedules. Suresh created methods to select multiple MDS- and MDS-1-like deferred status regimes indexed by producer and consumer quality levels, taking both filtering and incentive effects (1993). Lilly Christina (1995) compared an RGS plan to an RDS plan in terms of Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 311 https://internationalpubls.com operating ratio (OR) and ASN curve and explained how to select an RDS plan with acceptable quality standards. In their 2010 study, Suresh and Saminathan investigated repetitive postponed sampling programmers that used Acceptable and limited quality standards. Senthilkumar et al proposed a recurrent delay sampling plan (RDSSP). (2015) Investigate quality measures for normal distributions. Under this procedure, deferred lots are accepted or rejected based on the inspection results of the lots that precede or follow it during the RGS inspection. Senthilkumar et el (2017) proposed a repeated delay variable sampling plan (RDVSP) indexed on Six Sigma quality levels to analyze the quality characteristics of a normal distribution. The MDSRGS scheme devised by Balamurali and Usha (2017) is a newly created sampling strategy for the specific purpose of examining diversity. Therefore, the study will consider two sample sizes for feature edits and variable edits, respectively. Additionally, if a lot fails during a variable inspection under a mixed inspection RDS scheme, the characteristic inspection and the variable inspection are repeated. Unlike the RDS system, repeat sampling is only allowed under the proposed system, when lots are rejected and only after variable inspection. Proposed and existing programs differ in performance measures in terms of operations and parameters. Let USL and LSL be the specification limits for the quality characteristic in question. Items that deviate from these limits in terms of quality characteristics are considered unqualified. It is also assumed that the studied quality characteristics are normally distributed with undetermined means and standard deviations. The study by Aslam et al. (2013b) and Schilling and Dodge (1969) can be used as the basis for the implementation of the proposed mixed test RDS scheme based on Cpk as follows. V. RDS planning procedure. The first step (sampling by attributes): randomly select n1 samples from many products and count the number of unqualified products. Accept the result. The quantity of non-eligible products under c is the eligible quantity. If the number of nonconforming products exceeds c, go to step 3 and continue the variable control according to Cpk. (sample of variables): Estimate the exponent Cpk(C Μ‚pk) using a second random sample of size n2 as follows: οΏ½Μ‚οΏ½π‘π‘˜ = min ( π‘ˆπ‘†πΏβˆ’ οΏ½Μ…οΏ½ 3𝑆 , π‘‹βˆ’πΏπ‘†πΏ 3𝑆 ), where οΏ½Μ…οΏ½ = 1 𝑛2 βˆ‘ 𝑋𝑖 𝑛2 𝑖=1 and 𝑆2 = βˆ‘ (π‘‹π‘–βˆ’ οΏ½Μ…οΏ½)2𝑛2 𝑖=1 𝑛2βˆ’1 . If fail to accept the lot οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2, return it. If find that π‘˜2 ≀ οΏ½Μ‚οΏ½π‘π‘˜ Λ‚ π‘˜1 , Do not reject the whole thing, but rather pick a new sample of size n2 and redefine the sampling variable. In accordance with the RDS control plan, when 'm' is authorized from the previous or subsequent batch, accept it; otherwise, reject it. The proposed mixed-test RDS method is fully described by five parameters n1, n2, c, k1 and k2, which are the characteristic SSP sample size, the characteristic SSP acceptance criterion and the Cpk-based Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 312 https://internationalpubls.com SSP sample size. RGS Schema variables and Cpk-based variables are respectively acceptance and rejection criteria for RGS proposals. V. Measurements of the Performance Operational Characteristic Function of the Suggested Mixed Inspection RDS Plan The OC function of a sampling strategy represents the percentage of batches that are accepted for a given expected product quality. Therefore, the probability of acceptance of the proposed mixed-control RDS system (also known as the OC function) is calculated as: Let us denote the probability that a certain number of masses p is taken as Pa(p). Pa(p) is currently expressed as: Pa(p) = P1 + (1βˆ’ P1) m P2 (4) The probability of lot acceptance under the attribute control plan is represented by P1 whereas the corresponding probability based on variable RDS plan for PCI Cpk is denoted as P2. P1 therefore can be written in binomial probability model as follows; 𝑃1 = βˆ‘ ( 𝑛1 𝑑 ) 𝑝𝑑(1 βˆ’ 𝑝)𝑛1βˆ’π‘‘π‘ 𝑑=0 (5) Probability P2 can be calculated using Cpk in the path as: For the Cpk-based RDS method, Pa and Pc represent the probability of acceptance and probability of rejection of a batch based on a single sample, respectively. what's the matter. Pa = P(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) and Pc= 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2) The acceptance probability of the lot is then calculated according to the RDS plan using the acceptance probability P2 and Cpk. The probabilities Pa and Pc can be obtained by: 𝑃2 = π‘ƒπ‘Ž(1βˆ’π‘ƒπ‘)π‘š+π‘ƒπ‘π‘ƒπ‘Ž π‘š (1βˆ’π‘ƒπ‘)π‘š (6) P2 = P(CΜ‚pk β‰₯ k1) + P(k2 ≀ CΜ‚pk Λ‚ k1)[P(CΜ‚pk β‰₯ k1)] m 1 βˆ’ P(k2 ≀ CΜ‚pk Λ‚ k1){1 βˆ’ [P(CΜ‚pk β‰₯ k1)]m} where π‘ƒπ‘Ž = 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜π‘Ž) = 𝑃 { π‘ˆπ‘†πΏβˆ’ οΏ½Μ…οΏ½ 3𝑆 β‰₯ π‘˜π‘Ž, οΏ½Μ…οΏ½βˆ’ 𝐿𝑆𝐿 3𝑆 β‰₯ π‘˜π‘Ž} or 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) = 𝑃(οΏ½Μ…οΏ½ + 3π‘˜1𝑆 ≀ π‘ˆπ‘†πΏ) βˆ’ 𝑃(οΏ½Μ…οΏ½ βˆ’ 3π‘˜1𝑆 < 𝐿𝑆𝐿 and Pc= 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2) = 1 βˆ’ 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜2) = 1 βˆ’ 𝑃 { π‘ˆπ‘†πΏβˆ’ οΏ½Μ…οΏ½ 3𝑆 β‰₯ π‘˜2, οΏ½Μ…οΏ½βˆ’ 𝐿𝑆𝐿 3𝑆 β‰₯ π‘˜2} or 1 βˆ’ 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜2) = 1 βˆ’ 𝑃(οΏ½Μ…οΏ½ + 3π‘˜2𝑆 ≀ π‘ˆπ‘†πΏ) βˆ’ 𝑃(οΏ½Μ…οΏ½ βˆ’ 3π‘˜2𝑆 < 𝐿𝑆𝐿) For large samples, (οΏ½Μ…οΏ½ Β± 𝑐𝑆) is known to have an approximately normally distributed mean πœ‡ Β±cE(S) and variance 𝜎2 𝑛 + 𝑐2 Var (S) that is approximately normally distributed (see Duncan (1986), Balamurali et al (2005). i.e., ( οΏ½Μ…οΏ½ Β± 𝑐𝑆 ) based on this 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) can be approximately obeys N[πœ‡ Β± cΟƒ, 𝜎2 𝑛 + 𝑐2𝜎2 2𝑛 ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 313 https://internationalpubls.com written as 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) = πœ‘ [ π‘ˆπ‘†πΏβˆ’ πœ‡βˆ’3π‘˜1𝜎 𝜎 βˆšπ‘›2 √1+ 9π‘˜1 2 2 ] βˆ’ πœ‘ [ πΏπ‘†πΏβˆ’ πœ‡+3π‘˜1𝜎 𝜎 βˆšπ‘›2 √1+ 9π‘˜1 2 2 ] (7) 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2) = 1 βˆ’ πœ‘ [ π‘ˆπ‘†πΏβˆ’ πœ‡βˆ’3π‘˜2𝜎 𝜎 βˆšπ‘›2 √1+ 9π‘˜2 2 2 ] + πœ‘ [ πΏπ‘†πΏβˆ’ πœ‡+3π‘˜2𝜎 𝜎 βˆšπ‘›2 √1+ 9π‘˜2 2 2 ] (8) If pU and pL are defined as the fraction of items that do not conform when they are outside of the USL and LSL, respectively. Then P {X < LSL} = pL and P {X > USL}= pU, such that ( pL + pU = p). If we let zpU = π‘ˆπ‘†πΏ βˆ’ Β΅ Οƒ and -zpL = 𝐿𝑆𝐿 βˆ’ Β΅ Οƒ then the probabilities 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) can be determined as P(CΜ‚pk β‰₯ k1) = Ο† [(zpU βˆ’ 3k1) √ n2 (1+ 9k1 2 2 ) ] βˆ’ Ο† [(zpL βˆ’ 3k1) √ n2 (1+ 9k1 2 2 ) ] (9) 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2)=1 βˆ’ πœ‘ [(π‘§π‘π‘ˆ βˆ’ 3π‘˜2) √ 𝑛2 (1+ 9π‘˜2 2 2 ) ] + πœ‘ [𝑧𝑝𝐿 βˆ’ 3π‘˜2) √ 𝑛2 (1+ 9π‘˜2 2 2 ) ] (10) By replacing (9) and (10) with (9) and (10), P2, the probability of accepting the PCI-based RDS scheme variable Cpk (6) can be obtained. By substituting (5) and (6) into the equation, we can determine the required acceptance probability from the proposed RDS and the Cpk(4) mixed test. VII. Function of average sample size Generally, the ASN is the expected number of sample units needed to figure out the fate. ASN is a convenient concept in the context of Type B sampling, and sampling plans are often chosen when ASN is low. ASN is the objective function to be minimized in this work due to the constraints on the acceptance probabilities and the associated risks for producers and consumers. ASN is based on variable and attribute authentication in the proposed hybrid RDS authentication scheme. The amount of information needed to choose a batch is n1, and the amount of information required to make decisions based on RDS planning variables is n2, and it is likely that multiple samples will get (1-P1) to become. Therefore, using the ASN clustered quality function, we can implement the recommended RDS add-on test method as follows: ASN(p) = 𝑛1 + (1 βˆ’ 𝑃1) [ 𝑛2 π‘ƒπ‘Ž+𝑃𝑐 ] (11) VIII. Create a hybrid inspection RDS Plan Using Cpk The supply contract will itemize the requirements and potential risks for both the producer and the buyer. Manufacturers usually aim at a certain level of product quality which is called acceptant quality level. If p1 will be added, there is a high likelihood of being received. The p2 represents a threshold beyond which consumers decide to focus on the product quality due to its non-universality. This score Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 314 https://internationalpubls.com is often called Consumer Risk or LQL. Therefore, targeted sampling methods should consider threats either from producers’ or customers’ side. Thus our objective is to suggest an optimal selection parameters that minimize ASN and maintain trade-off between two risks – producer’s and consumer’s ones. And the nonlinear optimization problem based on this can be handled in order to find out best set of mixed sample parameter values. In formulating these planning parameters, two situations were considered; asymmetric failure rate and asymmetric failure rate. IX. The fraction when contravention is Symmetric. Assume first that symmetric fraction non-conforming. P{X < LSL} = P{X > USL} = p /2 i.e. pU = pL = p/2 In the event that the symmetric fractions do not match, the suggested system's function OC is provided by (4). The likelihood of acceptance as a variable based on Cpk inside the RDS framework was calculated using the formula below. The SSP function's (P1) estimation of the probability of acceptance is consistent with the description in (5). Schedule hybrid ASN inspections according the RDS policy described in (10). To select the ideal scheduling parameters, take into account the tracking optimization problem. The projected hybrid inspection RDS policy's ASN can be found in (10). Determine the ideal scheduling parameters while keeping in mind the tracking optimization challenge. P2 = P(CΜ‚pk β‰₯ k1)+P(k2 ≀ CΜ‚pk Λ‚ k1)[P(CΜ‚pk β‰₯ k1)] m 1βˆ’P(k2 ≀ CΜ‚pk Λ‚ k1){1βˆ’[P(CΜ‚pk β‰₯ k1)]m} (12) where 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) = 2πœ‘ [(𝑧𝑝 2 βˆ’ 3π‘˜1) √ 𝑛2 (1+ 9π‘˜1 2 2 ) ] βˆ’ 1 (13) and 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2) = 2 βˆ’ 2πœ‘ [(𝑧𝑝 2 βˆ’ 3π‘˜2) √ 𝑛2 (1+ 9π‘˜2 2 2 ) ] (14) where p12 represents the likelihood that the SSP attribute will be accepted in LQL, Pa(p1) represents the likelihood that the hybrid test RDS plan will be accepted in AQL, Pa(p2) represents the likelihood that the mixed test RDS plan will be accepted in LQL, Pa(p1) represents the likelihood that the RDS will be tested at LQL, Pa(p2) represents the likelihood based on fate, and Pa(p2) represents the likelihood. Minimize { 1 2 [𝐴𝑆𝑁(𝑝1) + 𝐴𝑆𝑁(𝑝2)]} = 1 2 {2𝑛1 + (1 βˆ’ 𝑃11) [ 𝑛2 π‘ƒπ‘Ž1+𝑃𝑐1 ] + (1 βˆ’ 𝑃12) [ 𝑛2 π‘ƒπ‘Ž2+𝑃𝑐2 ]} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 315 https://internationalpubls.com Subject to Pa(p1) β‰₯ 1βˆ’Ξ± Pa(p2) ≀ Ξ² n1 > n2 > 1, c β‰₯0, k1 > k2 > 0 (15) The fraction when contravention is Asymmetric. Assume that asymmetric fraction non-conforming. P{X < LSL} = pL and P{X > USL} = pU i.e.( pU = pL = p) The proposed system's operating characteristics satisfy the threshold for unacceptable asymmetry (4). If the skewness measurements do not agree, the acceptance probabilities of the Cpk-based variable RDS approach are used instead. The acceptance probabilities of the SSP and P1 features are comparable to equation (5). P2 = P(CΜ‚pk β‰₯ k1)+P(k2 ≀ CΜ‚pk Λ‚ k1)[P(CΜ‚pk β‰₯ k1)] m 1βˆ’P(k2 ≀ CΜ‚pk Λ‚ k1){1βˆ’[P(CΜ‚pk β‰₯ k1)]m} (16) where 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ β‰₯ π‘˜1) and 𝑃(οΏ½Μ‚οΏ½π‘π‘˜ < π‘˜2) is a factor depending on a strategy Using (9) and (10), Cpk RDS computes the probability of acceptance in the proper manner. Therefore, the optimization problem in the case of an incoherent asymmetric break Minimize { 1 2 [𝐴𝑆𝑁(𝑝1) + 𝐴𝑆𝑁(𝑝2)]} = 1 2 {2𝑛1 + (1 βˆ’ 𝑃11) [ 𝑛2 π‘ƒπ‘Ž1+𝑃𝑐1 ] + (1 βˆ’ 𝑃12) [ 𝑛2 π‘ƒπ‘Ž2+𝑃𝑐2 ]} Subject to Pa(p1) β‰₯ 1βˆ’Ξ± Pa(p2) ≀ Ξ² n1 > n2 > 1, c β‰₯0, k1 > k2 > 0 (17) The probability that the mixed test RDS scheme will be approved in the AQL is represented by Pa, the same probability in the LQL by Pa, and the corresponding probability of the SSP feature in the LQL by P12. PC1 and PC2 show the likelihood of being turned down at AQL and LQL, respectively, based on the planning variable RDS, whereas Pa1 and Pa2 show the likelihood of being accepted at AQL and LQL. The optimal settings for the Cpk-based hybrid control RDS technique can be found by solving the nonlinear equations for symmetric and asymmetric disjoint fractures presented in (15) and (17). There are numerous solutions because there are only two equations and five unknowns.A sample plan is generally chosen if the ASN is small and the required precautions are implemented. Thus, the constraints on the objective function that need to be reduced in this study are risks and the probability of producer and customer acceptability. By solving the nonlinear problem, the parameters (n1, n2, c, k1, and k2) can be determined. Two asymmetric failure scenarios that have been studied are pL=p/3, pU=2p/3, pL=p/4, and pU=3p/4, according to Aslam et al. (2013a). Consider the symmetric scenario in which pU=pL=p/2. The ideal values for the proposed schemes with pL=p/3, pU=2p/3 and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 316 https://internationalpubls.com pL=p/4, pU=3p/4, respectively, are given in Tables 2 and 3. However, Table 1 contains the ideal parameters for the symmetric failure of the proposed system. The recommended system is displayed. X. Illustrative example This part explains how to choose the best parameters based on predetermined standards for the suggested mixed sampling plan. To bolster this, we offer two instances. RDS plan for hybrid inspections Case of non-conforming symmetric fraction To ensure they fulfil the necessary criteria, auto parts go through several inspection phases during the manufacturing process a hybrid inspection plan can be used to assess part quality for unacceptable symmetrical breaks. After the quality inspection counts the number of unqualified products in the sample, a decision is made based on whether the rate of unqualified products is within the acceptable limit. For example, if the acceptance criteria are set at 5% non-conformists, and the sample of 100 parts has 5 or fewer non-conformists, the sample is considered acceptable. Variable inspection is used to measure the continuous characteristic of the non-conforming parts, such as size or weight. For example, the diameter of the non-conforming parts may be measured to ensure it falls within a certain tolerance limit. The combination of these two inspection methods provides a more comprehensive assessment of the quality of the parts and helps to ensure that both the number and the severity of non- conformists are within acceptable limits. This helps to improve the reliability and safety of the finished products and ensures that the customers receive products that meet their expectations. Based on the information provided, The following parameters make up the ideal mixed inspection RDS plan for the given AQL and LQL: n1 = 24, n2 = 18, c = 0, k1 = 0.818, k2 = 0.808. At AQL (p1 = 0.01), the probability of acceptance is Pa(p1) = 0.950. At LQL (p2 = 0.04), the probability of acceptance is Pa(p2) = 0.098. At AQL and LQL, the average ASN is 45.029. The sample size n1 is utilized for attribute inspection in this mixed inspection RDS design, and n2 is used for variable inspection. The parameters c, k1 and k2 are used to calculate the acceptance/rejection criteria for both types of inspection. The probability of acceptance at AQL (p1 = 0.01) is 0.950, which means that with this mixed plan, there is a 95.0% chance of accepting a population with a non-conformity rate of 0.01 or less. The probability of acceptance at LQL (p2 = 0.04) is 0.098, which means that with this mixed plan, there is only a 9.8% chance of accepting a population with a non-conformity rate of 0.04 or less. The average ASN (average sample number) is 45.029, Specifies the average number of units that will be checked before deciding overall acceptability. Given the level of quality control required and the acceptable risk of accepting an unqualified population, this information can be used to decide whether this hybrid test RDS scheme is appropriate for the quality attribute under study. RDS plan for hybrid inspections Asymmetric Fraction Case Non-conforming According to the information given, the ideal mixed inspection RDS plan has the following characteristics For AQL and LQL presented with asymmetric percent mismatch (pL=p/3 and pU=2p/3): n1=58, n=52, c=0, k1=1.048, k2=1.038. At the AQL (p1 = 0.001), the probability of acceptance is Pa(p1) = 0.950. At LQL (p2 = 0.008), the acceptance probability is Pa(p2) = 0.098. At AQL and LQL, the average ASN is 68.824. Sample size n1 is used for testing features in this RDS mixed test plan, and n2 is used for testing variables. For both control methods, the pass/fail criteria are calculated using the variables c, k1 and k2. AQL (p1 = 0.001) has an acceptance probability of 0.950, which means there is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 317 https://internationalpubls.com a 95.0% chance of accepting a population with a failure rate of 0.001 or less in this blended design. The probability of acceptance of the LQL (p2 = 0.008) is 0.09929, which means that with this mixed design, there is only a 9.8% chance of accepting a population with a non-compliance rate of 0.008 or less. The average ASN (average sample size) is 68,824, which stands for the average number of units that will be reviewed before deciding on overall acceptability. Given the level of quality control required and the acceptable risk of accepting an unqualified population, this information can be used to decide whether this hybrid test RDS scheme is appropriate for the quality attribute under study. Implementation of the Suggested Mixed Inspection RDS Strategy in Industry Suppose a manufacturing company produces a component that has a critical dimension which is critical to the component's function. The dimension is required to be between limits of 10mm and 11mm. The company wants to implement a mixed inspection RDS plan to ensure that the component's critical dimension falls within the specified limits. The LQL is p2 = 0.05 and the AQL is p1 = 0.01 according to the company. Additionally, the corporation states that the acceptance probability at LQL is Ξ² = 0.1 and at AQL is Ξ± = 0.05.Using tables and calculations, the optimal mixed Check the RDS plan to be sure. Assume the following parameters: n1 = 15, n2 = 14, c = 0, k1 = 0.701, k2 = 0.691 The acceptance probability for AQL (p1 = 0.01) is Pa(p1) = 0.95600. The acceptance probability for LQL (p2 = 0.05) is Pa(p2) = 0.098. The average ASN for AQL and LQL is 2.612 In this mixed inspection RDS plan, 15 units are selected for attribute inspection and 14 units are selected for variable inspection. The parameters c, k1 and k2 set the acceptance/rejection standards for both types of inspection. Based on AQL (p1 = 0.01) probability of admission of 0.950, the mixed plan has a 95.0% chance of admitting a population with a non-conformity rate of 0.01 or below. There is only a 9.8% possibility of admitting a population with a non-conformity rate of 0 or below with this mixed plan, according to the probability of acceptance at LQL (p2 = 0.05) of 0.098. The average ASN (average sample number) is 2.612, which indicates the average number of units that will be inspected before deciding on the acceptability of the population. This information can be used to decide if this mixed inspection RDS plan is suitable for the company's quality control requirements and acceptable risk of accepting a non-conforming population. 1) Take 15 random samples from the product lot and count the number of failures in step-1 (characteristic check). 2) If you see no errors, accept the batch. If one or more fail, go to step 3 for RDS variable verification. 3) In the third step (validation of the variables), an estimate of the exponent Cpk (οΏ½Μ‚οΏ½π‘π‘˜) is obtained from A random sample of size 14 was selected from the population, with the measured data being reported by Wu and Pearn (2008). Mean (xΜ„) = 0.8557, Standard deviation (SD) = 0.038 οΏ½Μ‚οΏ½π‘π‘˜ = min ( π‘ˆπ‘†πΏβˆ’ πœ‡ 3𝜎 , πœ‡βˆ’πΏπ‘†πΏ 3𝜎 ) = min( 25 – 0.856 3 X 0.0381 , 0.856– 15 3 X 0.0381 ) = min (3.85, 3.88) = 3.85 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 318 https://internationalpubls.com Note: Here, we have assumed LSL = 15 and USL = 25 as given in the problem. 0.717 0.698 0.726 0.684 0.727 0.688 0.708 0.703 0.694 0.713 0.730 0.699 0.710 0.688 4) Since οΏ½Μ‚οΏ½π‘π‘˜ = 3.84 > π‘˜π‘Ž = 0.701, the batch is accepted in the first instance of the variable test itself. XI. Conclusions Since PCI is becoming a more popular strategy for achieving and preserving product quality, developing a sample plan based on it is essential. When the quality feature of interest follows a normal distribution with an unknown mean and an unknown standard deviation, the suggested hybrid sampling strategy can be utilized. By lowering the ASN under two key restrictions that offer the necessary protections for producers and consumers, the sample size needed for inspection and the related accept and reject criteria are proven. The study shows that when inspections are based on many characteristics and quality parameters, the suggested hybrid inspection RDS system requires less sample for batch evaluation. References [1] Aslam, M. , Azam, M, and Jun C.-H, A Mixed Repetitive Sampling Plan Based on Process Capability Index, Applied Mathematical Modelling, Vol..37, No.24, pp.10027-10035, 2013. [2] Aslam, M. , Azam, M, Balamurali, S, and Jun C.-H, A New Mixed Acceptance Sampling Plan based on Sudden Death Testing under the Weibull Distribution, Journal of the Chinese Institute of Industrial Engineers, Vol. 29, No.6, 427- 433, 2012. [3] Aslam, M, Wu, C-W, Azam, M, and Jun C.-H, Mixed Acceptance Sampling Plans for Product Inspection using Process Capability Index, Quality Engineering, Vol.26, No.4, pp.450-459, 2014. [4] Aslam, M, Wu, C-W, Jun C.-H, Azam, M, and Itay, N, Developing a Variables Repetitive Sampling Plan Based on Cpk with Unknown Mean and Variance, Journal of Statistical Computation and Simulation, Vol.83, No.8, pp.1507- 1517, 2013. [5] Aslam, M, Balamurali, S, and Jun C.-H. A New Mixed Variable Lot-Size Sampling Plan Based on Process Capability Index, Journal of Testing and Evaluation, Vol.43, No.6, pp.1456-1461, 2015. [6] Aslam, M, Balamurali, S, Azam, M, and Srinivasa Rao. G, Mixed Multiple Dependent State Sampling Plans Based on Process Capability Index, Journal of Testing and Evaluation, Vol.43, No.1, pp. 171-178, 2015. [7] Balamurali, S, Aslam, M, and Jun C.-H, Designing of Two Mixed Variable Lot-Size Sampling Plans Using Repetitive Sampling and Resampling Based on the Process Capability Index, Sequential Analysis, Vol.35, No.4, pp.413-422, 2016. [8] Balamurali, S, Azam, M, and Aslam, M, Attribute-Variable Inspection Lots Policy using Resampling Based on EWMA, Communications in Statistics- Simulation and Computation, Vol.45, No.8, pp.3014-3035, 2016. [9] Balamurali, S, and Usha, M, A New Mixed Repetitive Group Sampling Plan Based on the Process Capability Index for Product Acceptance, International Journal of Quality & Reliability Management, Vol.35, No.2, pp.463-480, 2018. [10] Balamurali, S, Bootstrap Confidence Limits for Short-Run Capability Indices, Quality Engineering, Vol.15, No.4, pp.643-648, 2003. [11] Balamurali, S, and Jun,C-H, Repetitive Group Sampling Procedure for Variables Inspection, Journal of Applied Statistics, Vol.33, No.3, pp.327-338, 2006. [12] Balamurali, S, and Kalyanasundaram, M, Bootstrap Lower Confidence Limits for the Process Capability Indices Cp, Cpk and Cpm, International Journal of Quality and Reliability Management, Vol.19, No.8/9, pp. 1088-1097, 2002. [13] Balamurali, S, and Usha, M, Optimal Designing of Variables Quick Switching System Based on the Process Capability Index Cpk, Journal of Industrial and Production Engineering, Vol.31, No.2, pp.85-94, 2014. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 319 https://internationalpubls.com [14] Balamurali, S, Park, H, Jun,C-H, Kim, K.J, and Lee, J, Designing of Variables Repetitive Group Sampling Plan Involving Minimum Average Sample Number, Communicationsin Statistics-Simulation and Computation, Vol.34, No.3, pp.799-809, 2005. [15] Suresh, K.K. and Saminathan, R. Construction and selection of Repetitive Deferred Sampling plan through AQL and LQL, Int. J. Pure Appl. Math. 65(3): pp.257-264, 2010. Table 1: Hybrid Inspection RDS Plan Optimal Parameters Based on Cpk (Symmetric Fraction Non-conforming) p1 p2 n1 n2 c k1 k2 Pa(p1) Pa(p2) ASN 0.001 0.002 419 416 0 1.037 1.027 0.950 0.098 617.415 0.003 194 191 0 1.030 1.020 0.950 0.099 321.174 0.004 131 130 0 1.030 1.020 0.950 0.095 212.672 0.006 80 79 0 1.036 1.026 0.950 0.096 128.233 0.008 57 50 0 1.046 1.036 0.950 0.096 89.756 0.01 44 42 0 1.065 1.055 0.950 0.098 69.924 0.015 29 28 0 1.131 1.121 0.950 0.098 43.079 0.02 23 10 0 1.326 1.316 0.950 0.098 28.625 0.0025 0.005 231 225 0 0.932 0.922 0.950 0.098 483.437 0.01 36 32 0 0.913 0.903 0.950 0.098 103.242 0.015 13 10 0 0.919 0.909 0.950 0.098 56.051 0.02 9 5 0 0.933 0.923 0.950 0.098 35.827 0.025 5 2 0 0.958 0.948 0.950 0.095 26.353 0.03 5 2 0 0.980 0.970 0.950 0.096 19.805 0.05 1 2 0 1.330 1.320 0.950 0.096 8.706 0.005 0.01 163 158 0 0.848 0.838 0.950 0.098 373.805 0.015 46 42 0 0.823 0.813 0.950 0.098 115.868 0.02 24 21 0 0.815 0.805 0.950 0.098 62.999 0.03 7 4 0 0.812 0.802 0.950 0.098 29.503 0.04 3 2 0 0.829 0.819 0.950 0.098 17.923 0.05 3 2 0 0.843 0.833 0.950 0.096 11.827 0.1 3 2 0 1.391 1.381 0.950 0.095 3.948 0.01 0.02 121 118 0 0.771 0.761 0.951 0.096 239.274 0.03 40 36 0 0.782 0.772 0.950 0.096 74.978 0.04 24 18 0 0.818 0.808 0.950 0.098 45.029 0.05 10 7 0 0.703 0.693 0.950 0.098 23.326 0.1 7 5 0 0.741 0.731 0.950 0.098 7.012 0.15 3 2 0 0.856 0.846 0.950 0.098 3.578 0.2 3 2 0 1.483 1.473 0.950 0.098 2.242 0.03 0.05 56 55 1 0.622 0.612 0.950 0.095 113.826 0.1 19 13 1 0.687 0.677 0.950 0.096 30.554 0.15 6 6 0 0.650 0.640 0.950 0.096 12.599 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 320 https://internationalpubls.com 0.2 3 3 0 0.774 0.764 0.950 0.098 9.646 0.25 2 2 0 0.516 0.506 0.950 0.095 3.554 0.05 0.1 31 29 1 0.558 0.548 0.950 0.096 65.093 0.15 12 10 1 1.018 1.008 0.950 0.096 20.049 0.2 4 4 1 0.698 0.688 0.971 0.098 8.518 0.25 2 2 0 0.695 0.685 0.950 0.098 4.609 0.3 2 2 0 0.434 0.424 0.950 0.098 2.258 Table 2: Hybrid RDS Plan's Ideal Parameters Based on Cpk (Asymmetric Fraction) With pL= p/3 and pU= 2p/3) non-conforming p1 p2 n1 n2 c k1 k2 Pa(p1) Pa(p2) ASN 0.001 0.002 816 807 0 1.030 1.020 0.950 0.099 1416.75 0.003 243 238 0 1.024 1.014 0.950 0.095 405.937 0.004 146 143 0 1.026 1.016 0.950 0.096 236.371 0.006 84 78 0 1.035 1.025 0.950 0.096 132.688 0.008 58 52 0 1.048 1.038 0.950 0.098 68.824 0.01 45 39 0 1.067 1.057 0.950 0.098 70.35 0.015 30 16 0 1.141 1.131 0.950 0.098 41.239 0.02 23 11 0 1.327 1.317 0.950 0.098 28.868 0.0025 0.005 482 481 0 0.928 0.918 0.950 0.098 909.513 0.01 45 43 0 0.910 0.900 0.950 0.091 115.527 0.015 15 13 0 0.917 0.907 0.950 0.098 56.37 0.02 10 7 0 0.933 0.923 0.950 0.098 37.049 0.025 7 1 0 0.955 0.945 0.950 0.095 18.705 0.03 5 1 0 0.986 0.976 0.950 0.096 20.356 0.05 5 2 0 1.339 1.329 0.950 0.096 7.607 0.005 0.01 339 332 0 0.853 0.843 0.950 0.098 567.28 0.015 64 63 0 0.820 0.810 0.950 0.098 148.882 0.02 30 28 0 0.812 0.802 0.950 0.098 70.898 0.03 8 6 0 0.815 0.805 0.950 0.098 31.486 0.04 3 3 0 0.826 0.816 0.950 0.098 10.727 0.05 3 3 0 0.851 0.841 0.950 0.093 12.139 0.1 3 3 0 1.402 1.392 0.951 0.095 4.648 0.01 0.02 229 228 0 0.786 0.776 0.950 0.096 382.924 0.03 46 45 0 0.722 0.712 0.950 0.096 105.257 0.04 18 16 0 0.706 0.696 0.950 0.098 45.794 0.05 12 10 0 0.700 0.690 0.950 0.098 2.612 0.1 7 7 0 0.745 0.735 0.950 0.098 3.713 0.15 3 3 0 0.862 0.852 0.950 0.098 3.317 0.2 3 3 0 1.776 1.766 0.950 0.098 2.342 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 321 https://internationalpubls.com 0.03 0.05 86 84 1 0.676 0.666 0.950 0.095 132.702 0.1 18 16 1 0.590 0.580 0.950 0.096 36.243 0.15 6 6 0 0.635 0.625 0.950 0.096 13.565 0.2 3 3 0 0.503 0.493 0.950 0.098 11.693 0.25 2 2 0 0.520 0.510 0.950 0.095 3.554 0.05 0.1 47 45 1 0.594 0.584 0.950 0.096 80.704 0.15 8 5 1 0.604 0.594 0.971 0.096 17.59 0.2 4 4 1 0.542 0.532 0.950 0.098 5.965 0.25 2 2 0 0.703 0.693 0.950 0.098 4.609 0.3 2 2 0 0.369 0.359 0.950 0.098 2.021 Table 3: Cpk (Asymmetric Fraction)-Based Hybrid Inspection RDS Plan's Ideal Parameters Inconsistent with pU=3p/4 and pL=p/4 p1 p2 n1 n2 c k1 k2 Pa(p1) Pa(p2) ASN 0.001 0.002 1214 1209 0 1.022 1.012 0.950 0.099 2234.175 0.003 428 427 0 1.014 1.004 0.950 0.095 750.916 0.004 192 185 0 1.016 1.006 0.950 0.096 312.682 0.006 92 86 0 1.031 1.021 0.950 0.096 145.247 0.008 60 59 0 1.051 1.041 0.950 0.098 68.824 0.01 46 42 0 1.071 1.061 0.950 0.098 72.379 0.015 30 19 0 1.147 1.137 0.950 0.098 42.002 0.02 23 13 0 1.325 1.315 0.950 0.098 29.354 0.0025 0.005 650 650 0 0.920 0.910 0.950 0.098 1256.359 0.01 66 63 0 0.903 0.893 0.950 0.091 151.818 0.015 19 17 0 0.915 0.905 0.950 0.098 62.770 0.02 11 10 0 0.936 0.926 0.950 0.098 39.206 0.025 7 1 0 0.955 0.945 0.950 0.095 18.705 0.03 5 3 0 0.994 0.984 0.950 0.096 20.910 0.05 5 3 0 1.331 1.321 0.950 0.096 7.852 0.005 0.01 466 465 0 0.840 0.830 0.950 0.098 905.925 0.015 113 108 0 0.815 0.805 0.950 0.098 243.544 0.02 42 41 0 0.808 0.798 0.950 0.098 93.135 0.03 10 9 0 0.815 0.805 0.950 0.098 35.148 0.04 3 3 0 0.835 0.825 0.950 0.098 10.727 0.05 3 3 0 0.862 0.852 0.950 0.093 12.765 0.1 3 3 0 1.416 1.406 0.951 0.095 4.648 0.01 0.02 337 336 0 0.759 0.749 0.950 0.096 669.669 0.03 70 68 0 0.753 0.743 0.950 0.096 128.268 0.04 25 24 0 0.704 0.694 0.950 0.098 59.488 0.05 15 14 0 0.701 0.691 0.950 0.098 2.612 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 322 https://internationalpubls.com 0.1 7 7 0 0.721 0.711 0.950 0.098 4.556 0.15 3 3 0 0.871 0.861 0.950 0.098 3.317 0.2 3 3 0 1.794 1.784 0.950 0.098 2.342 0.03 0.05 163 162 1 0.614 0.604 0.950 0.095 305.107 0.1 22 21 1 0.673 0.663 0.950 0.096 37.404 0.15 6 6 0 0.631 0.621 0.950 0.096 15.583 0.2 3 3 0 0.787 0.777 0.950 0.098 9.927 0.25 2 2 0 0.557 0.547 0.950 0.095 3.554 0.05 0.1 107 106 1 0.528 0.518 0.950 0.096 218.450 0.15 9 8 1 0.596 0.586 0.971 0.096 20.037 0.2 4 4 1 0.556 0.546 0.950 0.098 6.701 0.25 2 2 0 0.714 0.704 0.950 0.098 5.383 0.3 2 2 0 0.443 0.433 0.950 0.098 2.021 Table 4: ASN(p2) of the hybrid inspection RDS Plan is Comparable with Other Attributes Currently Available Plans for a Single Sampling Characteristics of the RGS Plan and the RGS Plan (Symmetric Fraction Non-Conforming Situation) p1 p2 ASN(p2) Attributes SSP Attributes RGS Plan Attributes RGS Plan Mixed inspection RDS Plan 0.001 0.002 12376 8719.27 821.886 611.874 0.003 3922 2788.22 372.847 271.746 0.004 2317 1560.16 250.889 199.379 0.006 1112 794.84 154.971 109.431 0.008 664 593.07 93.006 71.886 0.010 531 315.88 88.278 76.066 0.0025 0.005 4948 3487.12 515.682 305.670 0.010 926 626.39 129.486 28.385 0.015 444 317.14 73.725 22.215 0.020 265 236.42 30.288 14.748 0.005 0.010 2473 1850.73 426.683 375.173 0.015 783 557.81 142.358 96.818 0.020 462 311.98 52.124 31.004 0.030 221 157.28 32.778 20.566 0.040 132 117.71 24.531 12.319 0.050 105 64.20 23.202 10.990 0.010 0.040 198 154.36 52.701 40.489 0.050 132 95.67 35.599 23.387 0.100 52 31.23 12.643 8.431 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1 (2025) 323 https://internationalpubls.com 0.150 25 27.00 7.804 5.592 0.030 0.060 410 297.63 141.285 95.745 0.090 129 99.68 42.935 21.815 0.120 65 61.18 26.385 14.173 0.150 43 40.84 16.241 4.029 0.300 12 15.59 5.004 2.792 0.050 0.100 233 183.58 56.473 35.353 0.150 77 55.33 28.177 15.965 0.200 38 30.94 11.224 9.012 0.250 25 24.30 6.234 4.022 0.500 7 9.31 4.502 2.290