Designing and Selection of Mixed Relational Complete Chain Sampling Plan Using Variance Principle (RCCSP-VP)

Abstract

There are many measurable dimensions for products in industries, and their quality should not deviate too much from the established norm. Therefore, a big part of the consignment’s acceptance is determined by the variance of distributions. In the production of surgical or critical care equipment, measurable dimensions are very important, since each part of the equipment should meet the indicated specification with tolerable variance. Therefore, a procedure for constructing a combined (mixed) sampling plan with a variance measure is provided in this module. This sampling plan supports the medical device engineer in establishing the target dimensions for the quality control of medical instruments because many variations in dimensions result in heavy losses for manufacturers. The relational complete chain sampling plan (RCCSP) is utilized in the II phase inspection to have a significant impact on production quality. The performance measures, algorithm, and design of the mixed sampling plan are provided. The plan is indexed through AQL. In order to choose the plan easily, the table is set up with an example.

1. Introduction

Several acceptance sampling inspections through variables are developed with the mean. However, medical device engineers feel that variance is the most powerful tool to improve variable sampling plans. Many quality control experts also argue that quality compromises are not desirable, as defects in many industries can lead to significant losses. Therefore, nonconformities are maintained in the manufacturing process to persist in competitive markets. Measuring and irradiating deviations help reduce production costs and improve the quality of the product. In mixed sampling inspections, there are two phases that consider both the quality of the variable and attribute factors when deciding whether a lot should be accepted or rejected. Mixed sample plans are frequently applied at many production phases due to contemporary quality control systems. Variable criteria play an important part in the control point in industries. Many items in industries have measurable dimensions, and their quality should not differ significantly from the specified standard. As a result, the process variation has a substantial impact on the decisions made by lots. Both the variability and the number of nonconformities must be decreased in several quality control areas. The combined variable-attribute sampling plan has two phases. The first phase is inspection through variables, and the second phase is through attributes. The test in the second phase plays an important role because a relational complete chain sampling plan correlates current consignment results with the results of previous consignment quantities. This type of inspection gives more security to the producer and the consumer, since the second-phase inspection depends on the current consignment results and the results of previous consignment quantities, which has a high effect on the quality of the product.

2. Literature Review

In [1], a chain sampling inspection plan is introduced. After that, Schilling [2] gave a common procedure for finding the OC of a mixed sampling plan. Govindaraju and Lai [3] made a modified chain sampling plan (MChsp) for a small sample size. Suresh and Devaarul [4, 5] constructed a mixed chain sampling plan, and they have made the plan for variance criterion also. To regulate multivariate quality features, Suresh and Devaarul [5] presented multidimensional mixed sampling plans. Radhakrishnan et al. [6] constructed a dependent mixed sampling plan using a single sampling plan as an attribute plan. Vijayaraghavan and Sakthivel [7] have shown Chsp inspection plans with the Bayesian methodology. Latha and Jeyabharathi [8] derived routine measures of Bayesian Chsp using the binomial distribution Devaarul and Moses [9] developed a relational chain sampling plan based on attribute quality features. For the continuous production process, Deva Arul and Senthil Kumar [10] created 2S-VSP with the mean and variance. Based on exponential distributions, Savage [11] proposed mixed variables and attributes sampling plans. Suresh and Devaarul [12] created a new algorithm that combined process and product control strategies to govern lot quality. A novel form of mixed sampling method has been developed by Devaarul and Jemmy Joyce [13]. To regulate a sequence of lots, Suresh and Devaarul [12] devised mixed chain sampling plans. To lower sampling costs, Suresh and Devaarul [14] devised a new form of mixed sampling strategy. Mixed sampling plans for second quality lots were created by Devaarul and Jemmy Joyce [13]. In IJSAM, Curtiss [15] looked upon acceptance sampling by variables, with a focus on the mean or variance. Devaarul and Senthil Kumar [16] proposed a new 3-stage mixed sample strategy for VAV quality features based on the mean and number of defects. Based on the mean and variance criteria, Devaarul and Senthil Kumar [10] created a new 2-stage variable sampling approach. Variable sampling plans include a few problems according to Collani [17]. Hamaker [18] described a method for creating single sampling variable inspection plans in which the OC curves of a single sampling attribute inspection plan and the OC curve of the variable plan have the same indifference quality level p_o and relative slope h₀. Fazal and Bashir [19] studied the Poisson distribution family and applications. A note on the philosophy of sampling inspection plans was written by Hamaker in 1950. Liebermann and Resnikoff [20] proposed sample plans for variable inspections. Edna and Joyce [21] designed mixed two-sided CChsp based on the variance principle. For linear profiles, Wang et al. [22] have recently created dependent mixed and mixed repeated sampling designs. The selection of a mixed sample plan with a double sampling plan as an attribute plan was explored by Sampath Kumar et al. [23] and was indexed using MAPD and AQL utilizing IRPD. A mixed repetitive sampling approach based on the process capability index was presented in 2013 [24].

Theorem 1. Let r₁ be the first-phase sample size, r₂ be the second-phase sample size, and be the sample variance ratio. The probability of acceptance is

Proof. First-phase testing is performed using variable inspections, and second-stage testing is performed with attribute inspections in mixed sampling schemes. If first-phase inspection fails to approve the lot, the second-stage attribute inspection becomes more crucial in order to distinguish the lot.
Either the first phase or the second phase will accept the lot. However, due to the mixed plan sampling technique, it will only be rejected at the second phase. The possible combinations for the acceptance of the lot in mixed RCCSP-VP plans are as follows:
(E) The lot will be accepted in the first stage if the sample variance ratio . (D) The lot is not accepted in the first stage if , and then, the number of defectives is counted. (i) When there is no defective lot, the current lot is accepted. (ii) When there is one defective lot, the existing lot is accepted if the earlier lot is admitted. Otherwise, it should be rejected. (iii) When there are two defective lots, the existing lot is accepted if the earlier two lots are admitted. Otherwise, it should be rejected. (iv) When there are three defective lots, the existing lot is accepted if the earlier three lots are admitted. Otherwise, it should be rejected. (v) In general, when there are “defective lots, the existing lot is accepted provided earlier “i” lots are admitted. Otherwise, it should be rejected.
The probability of the first-phase inspection isThe probability of the first-phase inspection is The two events E and D are mutually exclusive. Therefore, the probability of acceptance is given as

Theorem 2. The ASN (average sample number) function of the mixed RCCSP-VP plan is .

Proof. Let P(E) be the probability that in the variable inspection with the sample size r₁.
Event D with the sample size r₂ is defined as in Theorem 1.
Therefore,For event E, the expected sample size for decision isFor event D, the expected sample size for decision isSince E and D are mutually exclusive,

3. Combined Sampling Plan with the Variance Principle

The combined sampling plan with a variance criterion can be made with four parameters r₁, r₂, λ, and i.

3.1. First-Phase Variable Inspection

 Step 1: we review a random sample of size r₁ for the first-phase inspection Step 2: the consignment is accepted if  < λ, where is the sample ratio Step 3: if the ratio , we review another sample of size r₂ in the second-phase inspection

3.2. Second-Phase Attribute Inspection

We count the number of defective items in the second-phase inspection:(i)When there is no defective lot, the current lot is accepted.(ii)When there is one defective lot, the existing lot is accepted if the earlier lot is admitted. Otherwise, it should be rejected.(iii)When there are two defective lots, the existing lot is accepted if the earlier two lots are admitted. Otherwise, it should be rejected.(iv)When there are three defective lots, the existing lot is accepted if the earlier three lots are admitted. Otherwise, it should be rejected.(v)In general, when there are “i” defective lots, the existing lot is accepted provided earlier “i” lots are admitted. Otherwise, it should be rejected.

3.2.1. Measures of the Mixed Sampling Plan

3.2.2. Construction of the Mixed Plan Indexed through AQL for the First-Phase Sample Size r₁ Procedure

(1)We assign the possibility (probability) of acceptance of the first stage as .(2)We calculate S², the sum of the squares of the sample values for the identified sample size r₁.(3)We calculate the variable acceptance value λ from the equation:(4)We define the second-level possibility of acceptance as(5)We obtain the second-phase sample of size r₂ for the given index i using the equation:

The values of r₂ are estimated for the given i values by using a python program.

Table1 shows the values for n₁, n₂, k, and i for the mixed sampling plan through AQL and LQL when (p₁, β₁) and (p₂, β₂) are known. Assuming that β₁ = 0.99, β₂ = 0.01, β₁′ = 0.80, and β₂′ = 0.0055.

Table 1 

Comparison of RCCSP-VP with mixed two-sided CChsp based on maximum allowable variance.

4. Results and Discussion

It is found that as the number of lots to be chained increases, the second-phase sample size decreases. Also it is found that RCCSP-VP plan yields small sample size in the second phase for high probability of acceptance. The decision of accepting or rejecting the lot is based on the second-phase inspection when the variance criteria are more than the tolerable limit. This method of inspection may satisfy customers because both variation and nonconformities are tracked and managed. In the quality control area, the new algorithm has been made simple to use. Tables are built to aid quality control engineers based on the designing procedure. The two-stage RCCSP-VP sampling plan has been proven to be more sensitive to quality degradation.

Figure 1 shows the diagrammatic representation of the entire combined sampling plan with a variance principle. First-phase and second-phase inspections are clearly shown in Figure 1. Figures2 and 3 indicate that when the sample size increases, the acceptability variance measure increases as well. Figure 4 and 5 show the second-phase sample size values for the known AQL. Table 1 shows the values of the second-phase sample size n₂ for the given AQL and LQL (using two-sided CChsp as an attribute plan).

Figure 1 

Diagrammatic representation of the plan.

Figure 2 

Interpretation of variable measures (using Table 2).

Figure 3 

Interpretation of variable measures (using Table 3).

Figure 4 

RCCSP second-phase sample size (using Table 4).

Figure 5 

RCCSP second-phase sample size (using Table 5).

Table 2 indicates the values of the acceptance measure λ and the first-level sample size r1 for given  = 0.99 and  = 0.90. Table 3 indicates the values of the acceptance measure λ and the first-level sample size r1 for given  = 0.99 and  = 0.95. When the second-phase sample size values from Tables 4, 5, and 6 are compared to the n₂ values from Table 1, it is clear that the proposed mixed RCCSP-VP produces a tiny sample size so that the cost and duration of the inspection can be reduced.

Table 2 

Values of the acceptance measure λ and the first-level sample size r₁ for given  = 0.99 and  = 0.90.

Table 3 

Values of the acceptance measure λ and the first-level sample size r₁ for given  = 0.99 and  = 0.95.

Table 4 

Values of the second-phase sample size r₂ for the given AQL (using RCCSP as an attribute plan) ( = 0.99 and  = 0.85).

Table 5 

Values of the second-phase sample size r₂ for the given AQL (using RCCSP as an attribute plan) ( = 0.99 and  = 0.95).

Table 6 

Values of the second-phase sample size r₂ for the given AQL (using RCCSP as an attribute plan) = 0.99 and  = 0.90).

5. Application Example

In the production process of medical stretchers, the variance of the critical dimension of a sample of 15 medical stretchers is 21.064, with a fraction of nonconformities of 0.3%. We comment on the combined sampling plan based on the variance measure (RCCSP-VP), with a 90% chance of accepting the consignment, and i = 3.

5.1. Solution

 Step 1: from the consignment, we review a random sample of size r₁ = 15. Step 2: the consignment is accepted if <21.064, where is the sample ratio Step 3: if the ratio , we review another sample of size r₂ = 17 in the second-phase inspection Step 4: we count the number of defective lots in the second-phase inspection(i)When there is no defective lot, the current lot is accepted.(ii)When there is one defective lot, the existing lot is accepted if the earlier lot is admitted. Otherwise, it should be rejected.(iii)When there are two defective lots, the existing lot is accepted if the earlier two lots are admitted. Otherwise, it should be rejected.(iv)When there are three defective lots, the existing lot is accepted if the earlier three lots are admitted. Otherwise, it should be rejected.

6. Conclusion

In order to apply quality control to medical devices with critical dimensions, RCCSP-VP is designed in this module as an attribute plan using variance measurement by comparing the results of the current consignment with those of the amounts of past consignments. This sampling approach helps the medical device engineer in determining target dimensions because large fluctuations in dimension result in a significant loss to the manufacturer. The sample plan’s second stage is finished if the consignment quality is sufficient, in which case the lot is accepted; otherwise, an attribute sampling plan is used to assess the lot’s quality. The producer must therefore maintain variation in the production processes. The performance measures, methodology, and design of the mixed sampling plan are described. The parameters of the plan can be simply selected by using the above-framed Tables (2–6).

7. Future Research Work

Nowadays, many authors extended their research to neutrosophic statistics, which is the extension of classical statistics and is applied when the data are coming from a complex process or from an uncertain environment. In 1998, Florentin Smarandache introduced neutrosophy/neutrosophic probability, set, and logic, and in 2013, he has contributed to neutrosophic measure, neutrosophic integral, and neutrosophic probability. In 2019, Jana et al. worked on bipolar fuzzy Dombi aggregation operators and their application in the multiple-attribute decision-making process. The current study can be extended using neutrosophic statistics as future research.

8. Limitations of the Study

(i)It is challenging to determine whether the procedure is acceptable for the situation because the sample size is so small in comparison to the lot size.(ii)Additionally, there is a possibility of accepting inferior batches because there is no assurance that every batch will meet requirements.(iii)There is a possibility of rejecting good lots while accepting bad lots.

Data Availability

All the data used in this work is included in the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.