Development of multi stage multiple sampling plans for variable quality characteristics

Introduction

A variable sampling plan requires fewer samples than an attribute plan since more information is available during measurements. There are two methods available for a variable sampling plan, the first method is named the k-Method and the second method is named the M-Method. To describe a variable multiple sampling plan, the number of samples and variable criterion must be determined. The variable sampling systems are meant to be used for sampling a stream of lots from a supplier. The MIL-STD-414 and subsequent Sampling plan procedure for multiple sampling and its designing procedure fall short of an easy algorithm, designing the sampling plans through various quality levels and selection of the plans. Hence an attempt has been made to design and develop new multi stage multiple sampling plans for variable quality characteristics with known and unknown standard deviation. In the multi-stage multiple sampling plans for variables of unknown sigma, if the first stage does not lead to acceptance of the lot then the second stage of sampling continues, even in the second stage if the lot is not accepted then the third stage of variable sampling is continued and the procedure continues until m th stage. At m th stage final decision is taken about acceptance or rejection of the lot. The designing procedure is given and tables are constructed for easy selection of the plan. Grant Ireson and George Resnikoff (1952) [6] have studied the variables inspection plan based on the normal distribution for one sided and two-sided specification limit for unknown standard deviation. Schilling (1967) [13] has designed the operating characteristics of mixed variables and attributes sampling plans. Steven Sidik (1971) [16] has designed the acceptance sampling plan for variables based on the normal distribution of unknown standard deviation. Helmut Schneider (1989) [7] has analysed the variable sampling plan based on failure censored samples based on Weibull and log-normal distribution. Schilling and Neubauer [2008] have reported the variable sampling plans for different process parameters. Devaarul and Rebecca Jebaseeli Edna [2011] [2] have designed the operating characteristic function and related measures act of mixed sampling product control for costly and destructive items. Sankle and Singh [2012] [12] have studied Single Sampling Plans for variables under the measurement error. Rebecca Jebaseeli Edna et al.

[2013] [11] , have developed Multi-dimensional variable sampling plans.

[2014] [14] have developed multiple deferred state variable sampling plan by minimizing the sum of risks. Muhammad Aslam et al. [2015] [8] have studied skip-lot sampling plans for variable quality characteristics. Shih-Wen Liu and Chien-Wei Wu [2015] [15] have developed new variable sampling plan based on several quality characteristic. Geetha [2016] [4] has mentioned Inverse Gaussian distribution used in the single sampling plans by variables. Mujahid Sayyed [2016] [9] studied about the correlated data for variable sampling plan with known standard deviation. Chien-Wei-Wu et al. [2017] [1] have defined the lifetime performance index based on the variable sampling plan. Ramya and Devaarul [2019] [10] have studied the variable sampling plan for non-normal based on log-logistic failure distribution. Geetha and Pavithra [2019] [5] have studied single sampling plan by variables using range that is specified by two points on the operating characteristic curve.

Formulation of multi stage multiple sampling plans for variables with unknown sigma

Let, N = Lot size and U = Upper Specification 𝑛 1 = First stage sample size 𝑛 2 = Second stage sample size 𝑛 3 = Third stage sample size 𝑛 𝑗 = j th stage sample size 𝑘 1 = Variable factor such that lot is accepted, if Step 2: Determine the sample mean 𝑥̅ 1 and sample standard deviation 𝑠 1 .

Step 3: If 𝑥̅ 1 + k 1 s 1 ≤ U , accept the lot.

Step 4: If 𝑥̅ 1 + k 1 s 1 > U , go to second stage sampling.

Stage: II

Step 5: Draw a random sample of size 𝑛 2 .

Step 6: Determine the sample mean 𝑥̅ 2 and sample standard deviation 𝑠 2 .

Step 7: If 𝑥̅ 2 + k 2 s 2 ≤ U , accept the lot.

Step 8: If 𝑥̅ 2 + k 2 s 2 > U, go to stage three sampling.

Stage: III

Step 9: Draw a random sample of size 𝑛 3

Step 10: Determine the sample mean 𝑥̅ 3 and sample standard deviation 𝑠 3 .

Step 11: If 𝑥̅ 3 + k 3 s 3 ≤ U , accept the lot, otherwise go to j th Stage

Stage: J

Step 12: Draw a random sample of size 𝑛 𝑗

Step 13: Determine the sample mean 𝑥 𝑗 ̅ and sample standard deviation 𝑠 𝑗 Step 14: If 𝑥 𝑗 ̅ + k j 𝑠 𝑗 ≤ U , accept the lot, otherwise reject the lot.

Operating Characteristic and associated measures of MSMSP Theorem 1 Probability of acceptance

The probability of acceptance of multi stage multiple sampling plans for variables is as follows:

Proof

Case 1: In the first stage, if 𝑥̅ 1 + k 1 s 1 ≤ U the lot is accepted otherwise, case 2 is implemented Case 2: From the first stage, if 𝑥̅ 1 + k 1 s 1 > U and in the second stage of 𝑥̅ 2 + k 2 s 2 ≤ U the lot is accepted, otherwise case 3 is implemented. Case 3: From the second stage, if 𝑥̅ 1 + k 1 s 1 > U and 𝑥̅ 2 + k 2 s > U and in the third stage if 𝑥̅ 3 + k 3 s 3 ≤ U the lot is accepted, otherwise the next case is implemented Case J: From the j-1 th stage, if 𝑥̅ 1 + k 1 s 1 > U and 𝑥̅ 2 + k 2 s 2 > U…and 𝑥̅ 𝑗-1 + k j-1 s j-1 >U and in the j th stage of 𝑥̅ 𝑗 + k j s j ≤ U the lot is accepted, and if 𝑥̅ 𝑗 + k j s j > U the lot is rejected.

Here Cases 1 to j are mutually exclusive cases, hence by the law of addition, We get, P a (p) = P(i) + P (ii) + P (iii)+………. P(j) Hence the Probability of Acceptance of the Lot is

Theorem 2 Average Outgoing Quality [AOQ]

The average outgoing quality of multi stage multiple sampling plans for variables is as follows.

Theorem 3 Average Sample Number (ASN)

The Average Sample Number of multi stage multiple sampling plans for variables is given ASN = n 1 P I + (n 1 + n 2 ) P II + (n 1 + n 2 + n 3 ) P III ++ (n 1 + n 2 + n 3 + …… n j ) P J Where, P II = 1-P I P III = 1-P II ………. P J = 1-P j-1 P I, P II, P III,… P J are probability of a decision [acceptance or rejection of the lot] on the basis of the samples.

Theorem 4 Average Total Inspection

The average total inspection of multi stage multiple sampling plan for variables is as follows: ATI = n 1 P(i) + (n 1 +n 2 )(P a (p) -P(i)) + (n 1 +n 2 +n 3 ) (P a (p) -P(ii)) + (n 1 +n 2 +n 3 +……….+n j )(P a (p) -P(j-1) + N(1-P a (p))

Designing of multi stage multiple sampling plan inspection by variables for unknown sigma

In the case of multi stage multiple sampling plan by variables under assumptions of normal distributions for a quality characteristic. The relationship between the fraction defective p and the probability of acceptance P a (p) is given by

This relationship is used to determine the parameters for various values of the fractional defective. The acceptance probabilities are decreasing starting from 1 to 0. This means that Pa(p) is decreasing function of p. In order to determine the procedure value n j and k j, we require the points (𝑝 1 , 1-𝛼 𝑗 ) and (𝑝 2 , 𝛽 𝑗 ) on the operating characteristic function. In the multi stage multiple sampling plans the criterion are P a (𝑝 1 ) ≥ (1-α) = β 1 and P a (𝑝 2 ) ≤β = β 2.

Designing multi stage multiple variable sampling plan through AQL

Given probability of acceptance of the lot is P a (𝑝 1 ) ≥ (1-𝛼)=β 1 , Let β 1 = 0.95

Step 1: Let the 1 st stage of probability of acceptance at AQL be

Step 2: Let the second stage of probability of acceptance at AQL be

Step 3: Let the third stage of probability of acceptance be

Step j: Let the j th stage of probability of acceptance be

The j th stage sampling parameters for n j and k j for the known AQL are determined as follows:

Based on the above equations, the parameters are determined and given in table (1).

Note

Table (1) is developed for the given probability of acceptance (1-α)= 𝛽 1 = 0.95, the first stage of probability of acceptance is 𝛽 1 1 = 0.65, the second stage of probability of acceptance is

94 and the third stage of probability of acceptance is

Illustration 1

In a production process, let the incoming lot quality be 𝑝 1 = 0.001 (AQL) and the given probability of acceptance is (1-α) = β 1 = 0.95. Obtain the multi stage multiple sampling plan by variables indexed through AQL.

Solution

It's given that value of p 1 = 0.001 and the probability of acceptance is β 1 = 0.95 (β 1 = 1-α). Let the first stage of assuming probability of acceptance be 𝛽 1 1 = 0.65, the second stage of probability of acceptance is derived from this formula i.e., 𝛽 1 2 = 𝛽 1 -𝛽 1 1 1-𝛽 1 1 = 0.94 and the third stage of probability of acceptance is derived from this formula i.e., 𝛽 1

= 0.97.Then from the table (1), one can get the parameters of multi stage multiple sampling plans for variables as [𝑛 1 =356, 𝑛 2 =474, 𝑛 3 =608 and 𝑘 1 =3.1400, 𝑘 2 =3.2684, 𝑘 3 =3.2853].

Designing multi stage multiple sampling plans by variables through AQL and LQL Stage: I

To determine the first stage sampling parameters𝑛 1 and 𝑘 1 . It is known that for given 𝑝 1 and 𝑝 2 . Then consider

It is known that for given p 1 and p 2

From 1, substitute p = 𝑝 1

Based on the above designing procedure table (2) is constructed. The above sample size is the sample size of single sampling plans by variables with unknown sigma. Hence the proof.

Property 2: When j=1, MSMSP reduces to single sampling plans for variables Proof: We know that the variable factor for MSMSP is

The above variable factor is the variable factor for single sampling plans by variables with unknown sigma. Hence the proof.

Conclusion

In this research paper, a new multi-stage multiple sampling plan is developed with a final decision based on the results of each stage of the sampling plan, which will lead to accurate and effective decisions on the lots. The Operating characteristic function, ASN, ATI and AOQ are derived and tables are constructed for the multi-stage multiple sampling plans. The new algorithm helps to operate in the quality control section especially when the decision is based on several stages. Hence in the manufacturing industries one can maintain the standard quality of the lot or process by implementing multi-stage multiple sampling plans. It is found that when the index j=1, MSMSP converges to ordinary Single Sampling Plans.

References

1. Schneider H. Failure-Censored Variables-Sampling plans for Lognormal and Weibull Distributions, Technometrics. 1989;31(2):199-206.
2. Aslam M, Balamurali S, Jun CH, Hussain B. Design of SkSP-R Variables Sampling plans. Revista Colombiana de Estadística. 2015;38(2):413-429.
3. Singh JR, Sankle R, Khanday MA. Variables Sampling Plan For Correlated Data. Journal of Modern Applied Statistical Methods. 2013;12(2): 184-190.
4. Ramya D, Devaarul S. Determination of variable sampling plans for non-normal process through skewness and kurtosis. International Journal of Emerging Trends and Technology in Computer Science. 2019;8(4):15-20.
5. Edna RJK. Multi-dimensional mixed sampling plans for maximum allowable variance. International Journals of Mathematics and Computer Applications Research. 2013;3(1):223-228.
6. Sankle R, Singh JR. Single Sampling Plans for variables Indexed by AQL and AOQL with Measurement Error, Journal of Modern Applied Statistical Methods. Vikram University, India, Ujjain (M.P). 2012;11(2):396-406.
7. Senthilkumar D. Construct and selection of Multiple Deferred State (MSD-1) variable sampling plan Involving minimum sum of risks. International Journal of Innovative Research in Science, Engineering and Technology 2014;3(7):14232-14241