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Priyadharshini R & Shalini K RT&A, No 1 (82)
CONSTRUCTION OF GZIP DSP Volume 20, March 2025
CONSTRUCTION OF GAMMA ZERO-INFLATED POISSON DOUBLE SAMPLING PLANS
Priyadharshini R1 and Shalini K*
Department of Statistics, Salem Sowdeswari College, Salem - 636 010, Tamil Nadu, India. [email protected] and'[email protected] Correspondence Email: [email protected]
Abstract
In a well-supervised production framework, non-conformities occur seldom, resulting in a more number of zeros in the count of non-conformities. The zero-inflated Poisson (ZIP) distribution is a suitable model for handling zero inflation. Double sampling plan (DSP) is a precise quality inspection method where a decision on the approval or rejection of a lot is made after reviewing two samples, providing stronger conclusions than single sampling plan (SSP). In practice, decision-making for submitted lots requires a consistent assessment of both within-lot and between-lot variations, which can be addressed using Bayesian methodology. A Bayesian approach integrates prior knowledge and provides more information for making decisions about the approval or rejection of a lot. This article focuses on the designing of Bayesian DSPs; employing a Gamma prior to the parameter in the Poisson component of ZIP distribution the operating characteristic (OC) function is derived. Examples are provided to assess Gamma-ZIP (GZIP) DSPs. The significance of GZIP DSPs over conventional ZIP DSPs is also presented.
Keywords: Sampling inspection by attributes, Double sampling plan, Prior distribution, Zero-inflated Poisson distribution, Average quality level, Limiting quality level, Operating characteristics function.
I. Introduction
Sampling inspection is a method employed to assess the quality of items by examining a sample rather than inspecting every individual item. This approach is widely applied in manufacturing, service industries, and other sectors where full inspection would be too expensive or time-consuming. Acceptance sampling is a strategy that helps determine whether entire batches can be approved or declined based on sample inspection. Sampling inspection is categorized into two main types: sampling inspection by attributes and variables, both of which help to assess the quality standard of a batch.
In a SSP, the judgment to approve or decline a batch is based on inspecting only one sample. However, there are situations where a single sample may not provide sufficient information for a conclusive decision. In such cases, a DSP is implemented, where the decision is made based on the inspection of two samples. DSP functions as an extension of SSP, offering more reliable decisionmaking in quality control. Designing DSP parameters offers enhanced decision-making accuracy and provides better protection to both producer and consumer.
A Bayesian approach to acceptance sampling integrates prior knowledge with observed data to improve decision-making. When items are manufactured in lots, quality variations can occur due to within-lot and between-lot variation. Conventional acceptance sampling often assumes that between-lot variation is less significant than within-lot variation, leading to the assumption that the fraction of nonconforming items in a lot remains constant. In reality, decisions about submitted lots should consider both within-lot and between-lot variations. In such cases, Bayesian methods can be employed to design effective sampling plans based on predictive distributions.
The ZIP distribution is particularly effective in situations where non-conformities are rare. It is suitable for processes where there is a high occurrence of zero non-conformities, though occasional non-conformities are still possible. Loganathan and Shalini [5, 6] pioneered the determination of ZIP SSPs. Later, Uma and Ramya [17], Rao and Aslam [13], and Fu-Kwun and Sharew [3] discussed the construction of sampling plans in different perspectives. The Bayesian approach to developing ZIP SSPs has been explored by Suresh and Latha [15], Vijayaraghavan et «/.,[19], Shalini et al., [11], Palanisamy and Latha [7, 8] and Kaviyarasu and Sivakumar [4].
The designing of ZIP DSPs has been addressed by Shalini and Sheik [12], Pramote and Wimonmas [9], Wimonmas and Pramote [20]. The integration of Bayesian principles into the designing of DSPs have been discussed by Vijayaraghavan and Sakthivel [18], Balamurali et al.,[l], and Suresh and Usha [16].
According to the literature, there has been no research conducted on developing GZIP DSPs. This article focuses on the determination of GZIP DSPs. The OC function of the GZIP DSP is derived in Section 2. Designing GZIP DSPs is discussed in Section 3. Numerical examples and the significance of GZIP DSPs over the conventional ZIP DSPs are given in Section 4. Results are summarized in the concluding section.
DSPs offer more flexibility than SSPs by reducing the risk of making premature decisions. This approach is widely used in industries where cost and time efficiency are critical, providing a balance between minimizing inspection efforts and ensuring product quality.
A DSP is structured around five specific parameters: ni, rn, ci, C2 and cs, where ci < C2 and C2 < C3 (the third acceptance number). When C2 is taken in equal to C3 (i.e., C2 = cs), the DSP is described through its parameters ni, n2, ciand C2, which represent the sizes of the first and second samples and the first and second acceptance numbers, respectively (Duncan [2] and Stephens [14]).
Upon inspecting all items in the sample, the number of nonconformities (di) is established. If di < ci, the lot is approved; if di > C2, it is declined. When ci < di < C2, the initial sample fails, a second sample of size m is taken and the nonconformities (jdi) are counted. The cumulative count, D = di + d2, is compared to C2\ the lot is approved if D < C2 and declined if D > C2. This two-stage process enhances the reliability of quality assessment, allowing for more effective decision-making regarding lot approval or rejection based on observed nonconformities (Schilling and Neubauer
The effectiveness of a sampling plan can be evaluated through its OC function. In various industrial environments, careful monitoring of production processes often results in the frequent occurrence of zero non-conformities. In such scenarios, the ZIP distribution is the fitting probability distribution for nonconformities. The probability mass function (pmf) of the ZIP distribution is defined as follows
II. OC function of GZIP DSPs
[10]).
—X
when d = 0 when d = 1,2,3,...
Priyadharshini R & Shalini K RT&A, No 1 (82) CONSTRUCTION OF GZIP DSP_Volume 20, March 2025
In this model, p and A are parameters, where p (0 < p < 1) denotes the mixing proportion, which is assumed to be known.
When the variation in lot quality is significant from lot-to-lot, it indicates an unstable production process. In such cases, the process parameter A is assumed to vary randomly between lots and follows Gamma (a, m) distribution. This gamma distribution is the natural conjugate prior to A= np in the Poisson component of the ZIP distribution.
Shalini et al. [11] derived the probability distribution of d under the conditions of a ZIP distribution and a gamma prior distribution for A, where the shape parameter of the gamma distribution is m.
( + n _ ) ( m )m
rr,. , I y cP-'\np+m) , whend = 0 .„.
f (d\ip, n,p,m) = j sn + d-u wh™ d = 1.2,3.... (2)
V V m — 1 / \np+mj \np+mj
The OC function of the GZIP DSP can be described as:
Pa(p) = ^ c1\n1) +Y,cd2i=Ci+1f(d1\n1p,p,m)F(d2 < c3 - d1\n2) (3)
As proposed by Vijayaraghavan and Sakthivel [18], the prior knowledge about the production
s2
process must be used to estimate the value of m. The moment estimator m = — can be used for m,
s2
where s1 = ^EiLi^i = X and s2 = ^£¿=1(^1 _ ; = 3,... ,k.
III. Designing GZIP DSPs
GZIP DSPs are designed by determining the optimum parameters ni, ni, ciand ci based on the prescribed points (pi, 1-a) and (pi, ft) on the OC curve so that the determined GZIP DSPs provide adequate protection to both producer and consumer.
The plan must satisfy the following requirements:
(i) Pa (pi) > 1- a
(ii) Pa (pi) < ft
The values of the plan parameters ni, ni,ci and ci can be derived for each set of p, m, pi, a, pi and |3 by applying the unity value approach. The values npi and npi satisfying respectively equations (i) and (ii) are termed as unity values (Schilling and Neubauer [10]). The plan parameters can be arranged in tables for different combinations of (pi, a, pi, ft). The use of an operating ratio R =
reduces the number of tables.
The plan parameters are determined for specific sets of values of p, m, pi, a, pi and ft under the condition of GZIP distribution. The unity values are computed for various values combinations of (p, m, c, Pa (p)) by solving the OC function of GZIP DSPs for each combination of ci and ci with ni= ni = n. The values taken for m are 5 and 10 and for Pa (p) are 0.99, 0.90, 0.50, 0.20 and 0.10. The value considered for ci and ci in these combinations are 1(1)9 and 2(1)10 respectively. The values taken for p are 0.03 and 0.07 and are given in Table 1. The operating ratio values calculated corresponding to (a = 0.05, ft = 0.10), (a = 0.10, ft = 0.20), p = 0.03 and 0.07, m = 5 and 10, ci = 1(1)9 and ci = 2(1)10 are listed in Table 2.
For specified strength (pi, a, pi, ft) and values of p and m these tables can be used to determine the plan parameters by implementing the following procedure:
First, we compute the operating ratio R = — .Next select the unity value npi and the acceptance
numbers (ci, ci) from Table 2 corresponding to the value of p, m, a, ft associated with an operating ratio closest to R. Then, determine the unity values npi from Table 1 and calculate the sample size n
as —-. Thus, the acceptance numbers and the sample size determined together with <p and m are the parameters of the desired plan.
_Table 1: Unity Value of GZIP DSPs_
Pa(p)
p m ci C2 -
0.99 0.95 0.9 0.5 0.2 0.1
2 0.3488 0.6455 0.8704 2.2795 4.2836 6.2789
3 0.5684 0.9343 1.1976 2.7349 4.7742 6.7438
4 0.791 1.2209 1.5219 3.2119 5.3513 7.3541
5 1.0182 1.5102 1.8489 3.7034 5.9797 8.0607
6 1.2504 1.8033 2.1795 4.2049 6.6399 8.8299
7 1.4872 2.1002 2.5138 4.7136 7.3208 9.6402
8 1.728 2.4004 2.8512 5.2275 8.0156 10.4782
9 1.9723 2.7034 3.1912 5.745 8.7201 11.3353
10 2.2194 3.0087 3.5333 6.2654 9.4313 12.2058
2 3 0.6147 1.0308 1.3368 3.2234 5.9143 8.5762
2 4 0.8351 1.3024 1.6345 3.5798 6.2375 8.8573
2 5 1.0569 1.5775 1.9398 3.9877 6.6701 9.2693
2 6 1.2833 1.8588 2.2535 4.4292 7.1839 9.7977
2 7 1.515 2.1461 2.5744 4.8934 7.755 10.4186
2 8 1.7515 2.4385 2.9012 5.3734 8.3665 11.1092
2 9 1.9921 2.7352 3.2328 5.8649 9.0067 11.8514
2 10 2.2362 3.0355 3.5682 6.3648 9.6678 12.6316
3 4 0.9132 1.4502 1.8397 4.2225 7.5954 10.8922
3 5 1.1254 1.6984 2.1037 4.4942 7.8127 11.082
3 6 1.3418 1.9581 2.3863 4.8329 8.1275 11.3654
3 7 1.5646 2.2284 2.6835 5.2197 8.5307 11.7496
3 8 1.7936 2.5073 2.9918 5.6407 9.006 12.2295
3 9 2.028 2.7932 3.3088 6.0863 9.5374 12.7925
3 10 2.2669 3.0847 3.6324 6.55 10.1113 13.4239
4 5 1.2365 1.8964 2.3715 5.2571 9.2958 13.2063
4 6 1.4357 2.1188 2.6012 5.4625 9.4506 13.3504
4 7 1.6438 2.3606 2.8589 5.7376 9.6809 13.5583
4 8 1.8607 2.6176 3.1373 6.0692 9.9911 13.8426
4 9 2.0853 2.8862 3.4308 6.4439 10.3759 14.2093
4 10 2.3161 3.1639 3.7359 6.8511 10.8249 14.6568
5 6 1.5793 2.364 2.9258 6.3131 11.0028 15.5146
5 7 1.7633 2.5604 3.123 6.4695 11.121 15.6345
5 8 1.961 2.7829 3.3551 6.6905 11.2948 15.7986
5 9 2.1704 3.0249 3.613 6.9699 11.5339 16.0185
5 10 2.3891 3.2817 3.8902 7.298 11.8413 16.3033
6 7 1.9377 2.8482 3.4973 7.3819 12.7115 17.8174
6 8 2.1057 3.02 3.6652 7.5033 12.808 17.9234
6 9 2.2918 3.2226 3.872 7.6804 12.9446 18.0614
6 10 2.4923 3.4487 4.1088 7.9135 13.1325 18.2403
7 8 2.3088 3.3455 4.0817 8.4583 14.4201 20.1156
Pa(p)
0.03
0.03 10
0.99 0.95 0.9 0.5 0.2 0.1
7 9 2.4608 3.4947 4.224 8.555 14.5031 20.2132
7 10 2.6346 3.6778 4.4068 8.6978 14.6156 20.3348
8 9 2.6901 3.8529 4.676 9.5394 16.128 22.4102
8 10 2.8268 3.9821 4.7965 9.6186 16.2024 22.5023
9 10 3.0796 4.3683 5.2775 10.6234 17.8349 24.7018
2 0.3675 0.6684 0.8907 2.192 3.8343 5.2986
3 0.6025 0.9714 1.2293 2.6331 4.2825 5.709
4 0.843 1.2746 1.568 3.0994 4.8159 6.2568
5 1.0908 1.5836 1.9123 3.5825 5.3982 6.8918
6 1.3466 1.8991 2.2628 4.0768 6.0096 7.5801
7 1.6097 2.2207 2.6189 4.5787 6.6386 8.3008
8 1.879 2.5474 2.9796 5.0858 7.2784 9.0415
9 2.1537 2.8783 3.3442 5.5963 7.9249 9.7945
10 2.4329 3.2129 3.7119 6.1093 8.5754 10.5551
2 3 0.6554 1.0761 1.3762 3.0882 5.2284 7.1066
2 4 0.8908 1.3588 1.6808 3.4237 5.5103 7.3413
2 5 1.1303 1.649 1.9979 3.817 5.9025 7.703
2 6 1.378 1.9494 2.3277 4.2483 6.3765 8.1781
2 7 1.6342 2.2591 2.6681 4.7053 6.9071 8.7405
2 8 1.8981 2.5768 3.017 5.18 7.4764 9.3656
2 9 2.1685 2.9009 3.3727 5.6669 8.0719 10.0347
2 10 2.4444 3.2302 3.7337 6.1625 8.6853 10.7341
3 4 0.9843 1.5265 1.9068 4.0436 6.6651 8.9182
3 5 1.209 1.7809 2.1715 4.2873 6.8423 9.0682
3 6 1.442 2.0528 2.4618 4.6048 7.115 9.3052
3 7 1.6857 2.3403 2.7723 4.977 7.4781 9.6411
3 8 1.9393 2.6406 3.0984 5.3883 7.9157 10.0719
3 9 2.2014 2.9512 3.4365 5.8276 8.41 10.5838
3 10 2.4706 3.27 3.7839 6.287 8.9462 11.1598
4 5 1.3465 2.0131 2.4756 5.0383 8.1145 10.7165
4 6 1.5546 2.2357 2.6989 5.2109 8.2333 10.8275
4 7 1.7771 2.4852 2.9588 5.4576 8.4208 10.9938
4 8 2.0134 2.7563 3.2463 5.7678 8.6878 11.2317
4 9 2.2617 3.0441 3.5545 6.1278 9.0324 11.5508
4 10 2.5197 3.3449 3.8786 6.5254 9.4443 11.9512
5 6 1.7368 2.5297 3.0754 6.0563 9.5646 12.5006
5 7 1.925 2.7199 3.2593 6.1786 9.6524 12.5929
5 8 2.1339 2.9448 3.487 6.3653 9.7858 12.7213
5 9 2.3604 3.1967 3.7488 6.616 9.98 12.8984
5 10 2.601 3.4693 4.0364 6.9226 10.243 13.1364
6 7 2.1506 3.0708 3.6995 7.0875 11.0118 14.2731
6 8 2.3177 3.2302 3.8481 7.1763 11.083 14.3555
6 9 2.511 3.4293 4.0438 7.3159 11.1839 14.4633
6 10 2.7254 3.6602 4.2784 7.5139 11.328 14.6044
V
m
Pa(p)
0.03 10
0.07
0.99 0.95 0.9 0.5 0.2 0.1
7 8 2.5841 3.6314 4.3422 8.126 12.4553 16.0363
7 9 2.7301 3.7629 4.4607 8.1931 12.5175 16.1132
7 10 2.9065 3.9365 4.6261 8.2982 12.5992 16.2091
8 9 3.0339 4.2073 4.9987 9.1684 13.8953 17.7925
8 10 3.1598 4.3147 5.0928 9.2219 13.9523 17.8657
9 10 3.4971 4.795 5.6656 10.213 15.3323 19.543
2 0.3554 0.66 0.8925 2.3944 4.8306 8.5599
3 0.5786 0.9547 1.2272 2.8707 5.3748 9.1613
4 0.8047 1.2465 1.5583 3.3685 6.013 9.946
5 1.0351 1.5407 1.8917 3.8806 6.7086 10.8654
6 1.2702 1.8385 2.2284 4.4028 7.4406 11.8792
7 1.5099 2.1399 2.5686 4.9321 8.1966 12.9589
8 1.7534 2.4444 2.9117 5.4664 8.9689 14.085
9 2.0003 2.7516 3.2573 6.0045 9.7525 15.2443
10 2.2501 3.0611 3.605 6.5453 10.5441 16.4278
2 3 0.6249 1.0517 1.3677 3.376 6.631 11.531
2 4 0.8491 1.3291 1.6727 3.7531 7.0123 11.9703
2 5 1.0741 1.6092 1.9844 4.1799 7.5013 12.556
2 6 1.3036 1.895 2.3041 4.64 8.0733 13.2756
2 7 1.5381 2.1866 2.6307 5.1228 8.7063 14.1084
2 8 1.7772 2.4833 2.963 5.6217 9.3835 15.0324
2 9 2.0205 2.7842 3.3 6.1323 10.093 16.0281
2 10 2.2671 3.0885 3.6408 6.6514 10.8262 17.0794
3 4 0.9273 1.4775 1.8794 4.4115 8.4709 14.4839
3 5 1.1431 1.7314 2.1507 4.7059 8.756 14.8475
3 6 1.3627 1.9959 2.4394 5.0632 9.1321 15.32
3 7 1.5883 2.2705 2.7422 5.4671 9.5924 15.9023
3 8 1.8199 2.5535 3.0558 5.905 10.1245 16.589
3 9 2.0569 2.8434 3.3779 6.3677 10.7143 17.3698
3 10 2.2983 3.1388 3.7067 6.8489 11.3496 18.2324
4 5 1.2543 1.9301 2.4198 5.4807 10.3229 17.4184
4 6 1.4573 2.1584 2.6573 5.712 10.5536 17.742
4 7 1.6684 2.4048 2.9209 6.0069 10.8534 18.1493
4 8 1.888 2.6658 3.2045 6.3553 11.226 18.6439
4 9 2.115 2.9383 3.503 6.7457 11.6683 19.2263
4 10 2.3483 3.2196 3.8129 7.1685 12.173 19.8938
5 6 1.6007 2.4037 2.9824 6.5697 12.1774 20.339
5 7 1.7887 2.6065 3.1881 6.7554 12.3767 20.6388
5 8 1.9894 2.8335 3.4261 6.9985 12.6274 21.0059
5 9 2.2013 3.0794 3.6891 7.2957 12.9365 21.4438
5 10 2.4223 3.3398 3.9708 7.6391 13.3066 21.9543
6 7 1.9627 2.8938 3.5619 7.6702 14.0314 23.2495
6 8 2.1349 3.0725 3.7391 7.824 14.2118 23.5338
6 9 2.324 3.2798 3.9521 8.0262 14.4309 23.8743
V
m
Pa(p)
0.07
0.07 10
0.99 0.95 0.9 0.5 0.2 0.1
6 10 2.5269 3.5097 4.1939 8.2787 14.6957 24.2737
7 8 2.3372 3.3968 4.154 8.7776 15.8839 26.1528
7 9 2.4938 3.5536 4.3066 8.9092 16.0525 26.426
7 10 2.6706 3.7416 4.4961 9.08 16.2509 26.7478
8 9 2.7219 3.9098 4.7557 9.889 17.7348 29.0506
8 10 2.8635 4.0472 4.8875 10.0052 17.8956 29.3156
9 10 3.1147 4.4305 5.3644 11.0028 19.5843 31.9442
2 0.3744 0.6832 0.9129 2.2942 4.2581 6.8443
3 0.6134 0.9921 1.2588 2.7535 4.7481 7.3537
4 0.8574 1.3007 1.6042 3.2377 5.3299 8.0376
5 1.1086 1.6146 1.9546 3.7384 5.9662 8.8447
6 1.3675 1.9346 2.3108 4.2501 6.6352 9.734
7 1.6334 2.2605 2.6724 4.7692 7.3243 10.6776
8 1.9055 2.5913 3.0384 5.2933 8.0258 11.6572
9 2.1828 2.9262 3.4081 5.8206 8.7349 12.6611
10 2.4645 3.2646 3.7808 6.35 9.4486 13.6816
2 3 0.6663 1.0975 1.4071 3.2217 5.7651 9.0165
2 4 0.9057 1.386 1.7188 3.5751 6.0952 9.3774
2 5 1.1485 1.681 2.042 3.9842 6.5325 9.8823
2 6 1.3992 1.9858 2.3772 4.4308 7.0528 10.5184
2 7 1.6583 2.2997 2.7228 4.903 7.6328 11.2631
2 8 1.9249 2.6213 3.0767 5.393 8.2547 12.0916
2 9 2.1978 2.9492 3.4373 5.8954 8.9057 12.983
2 10 2.4762 3.2823 3.8031 6.4065 9.5767 13.9204
3 4 0.9993 1.5547 1.9465 4.2066 7.3015 11.1529
3 5 1.228 1.8147 2.2183 4.4706 7.5386 11.4466
3 6 1.4641 2.091 2.514 4.8033 7.8632 11.8479
3 7 1.7105 2.3824 2.8293 5.1888 8.2725 12.3593
3 8 1.9666 2.6864 3.1601 5.6132 8.7546 12.9752
3 9 2.2312 3.0006 3.5027 6.0657 9.2948 13.6839
3 10 2.5028 3.3229 3.8546 6.5388 9.8795 14.4708
4 5 1.3657 2.0478 2.5239 5.2283 8.8418 13.2582
4 6 1.5777 2.2761 2.7546 5.4251 9.0295 13.5176
4 7 1.803 2.5298 3.0196 5.6884 9.2796 13.8605
4 8 2.0419 2.8044 3.3113 6.0113 9.6 14.2917
4 9 2.2924 3.0954 3.6237 6.3826 9.9908 14.8122
4 10 2.5527 3.3994 3.9518 6.7914 10.4459 15.4192
5 6 1.7601 2.5708 3.132 6.2712 10.378 15.3406
5 7 1.9523 2.7671 3.324 6.4218 10.5396 15.5798
5 8 2.1639 2.9959 3.5567 6.6279 10.7447 15.8875
5 9 2.3926 3.2509 3.8223 6.8922 11.0024 16.2675
5 10 2.6353 3.5264 4.1135 7.2093 11.3191 16.722
6 7 2.1779 3.118 3.7639 7.3255 11.9085 17.4062
6 8 2.3492 3.2841 3.9216 7.4462 12.0559 17.6321
V
m
_P!(P)_
rn m c1 c2 -
0.99 0.95 0.9 0.5 0.2 0.1
6 9 2.545 3.4872 4.1225 7.609 12.2343 17.9166
6 10 2.7615 3.7209 4.3607 7.8226 12.4509 18.2623
7 8 2.6153 3.6844 4.4139 8.3859 13.4337 19.4595
7 0.07 10 7 9 10 2.7658 2.9447 3.8236 4.5429 4.0012 4.714 8.4879 8.6197 13.5728 13.7352 19.6757 19.9435
8 9 3.0688 4.2658 5.0773 9.4496 14.9545 21.5035
8 10 3.1997 4.3818 5.1832 9.5399 15.0883 21.7122
9 10 3.5356 4.8587 5.7507 10.515 16.4715 23.5402
Table 2: Operating Ratio of GZIP DSPs
V m c1 c2 R
a = 0.05, ß =0.10 a = 0.10, ß =0.20
1 2 9.7272 0.2222
1 3 7.218 4.9214
1 4 6.0235 3.9865
1 5 5.3375 3.5162
1 6 4.8965 3.2342
1 7 4.5901 3.0465
1 1 8 9 4.3652 4.193 2.9122 2.8113
1 10 4.0568 2.7325
2 3 8.3199 2.6693
2 4 6.8008 4.4242
2 5 5.8759 3.8162
2 6 5.271 3.4386
2 7 4.8547 3.1879
2 8 4.5558 3.0124
0.03 5 2 9 2 10 4.3329 4.1613 2.8838 2.786
3 4 7.5108 2.7094
3 5 6.525 4.1286
3 6 5.8043 3.7138
3 7 5.2727 3.4059
3 8 4.8776 3.1789
3 9 4.5799 3.0102
3 10 4.3518 2.8824
4 5 6.9639 2.7836
4 6 6.3009 3.9198
4 7 5.7436 3.6332
4 8 5.2883 3.3862
4 9 4.9232 3.1846
4 10 4.6325 3.0243
5 6 6.5629 2.8975
5 7 6.1063 3.7606
R
Ф
0.03
0.03 10
VI 2 а = Ü.Ü5, ß =Ü.1Ü а = Ü.1Ü, ß =Ü.2Ü
5 8 5.677 3.561
5 9 5.2955 3.3665
5 10 4.9679 3.1923
6 7 6.2557 3.0439
6 8 5.9349 3.6347
6 9 5.6046 3.4945
6 10 5.289 3.3431
7 8 6.0127 3.1962
7 9 5.784 3.5329
7 10 5.5291 3.4335
8 9 5.8164 3.3166
8 10 5.6509 3.4491
9 10 5.6548 3.378
1 2 7.9273 3.3794
1 3 5.8771 4.3048
1 4 4.9088 3.4837
1 5 4.352 3.0714
1 6 3.9914 2.8229
1 7 3.7379 2.6558
1 8 3.5493 2.5349
1 9 3.4029 2.4427
1 10 3.2852 2.3697
2 3 6.604 2.3102
2 4 5.4028 3.7992
2 5 4.6713 3.2784
2 6 4.1952 2.9544
2 7 3.869 2.7394
2 8 3.6346 2.5888
2 9 3.4592 2.4781
2 10 3.323 2.3933
3 4 5.8423 2.3262
3 5 5.0919 3.4954
3 6 4.5329 3.151
3 7 4.1196 2.8902
3 8 3.8142 2.6974
3 9 3.5863 2.5548
3 10 3.4128 2.4473
4 5 5.3234 2.3643
4 6 4.843 3.2778
4 7 4.4237 3.0506
4 8 4.0749 2.846
4 9 3.7945 2.6762
4 10 3.573 2.5411
5 6 4.9415 2.435
m
R
Ф
0.07
0.03 10
i 2 а = 0.05, ß =0.10 а = 0.10, ß =0.20
5 7 4.6299 3.11
5 8 4.3199 2.9615
5 9 4.0349 2.8064
5 10 3.7865 2.6622
6 7 4.648 2.5377
6 8 4.4442 2.9766
6 9 4.2176 2.8801
6 10 3.9901 2.7657
7 8 4.416 2.6477
7 9 4.2821 2.8684
7 10 4.1176 2.8062
8 9 4.229 2.7235
8 10 4.1407 2.7798
9 10 4.0757 2.7396
1 2 12.9695 2.7062
1 3 9.596 5.4124
1 4 7.9791 4.3797
1 5 7.0522 3.8587
1 6 6.4614 3.5463
1 7 6.0558 3.339
1 8 5.7622 3.1911
1 9 5.5402 3.0803
1 10 5.3666 2.994
2 3 10.9642 2.9249
2 4 9.0063 4.8483
2 5 7.8026 4.1922
2 6 7.0056 3.7801
2 7 6.4522 3.5039
2 8 6.0534 3.3095
2 9 5.7568 3.1669
2 10 5.53 3.0585
3 4 9.803 2.9736
3 5 8.5754 4.5072
3 6 7.6757 4.0712
3 7 7.0039 3.7436
3 8 6.4966 3.4981
3 9 6.1088 3.3132
3 10 5.8087 3.1719
4 5 9.0246 3.0619
4 6 8.22 4.266
4 7 7.5471 3.9716
4 8 6.9937 3.7158
4 9 6.5433 3.5032
4 10 6.179 3.3309
m
R
Ф
0.07
0.07 10
VI 2 а = Ü.Ü5, ß =Ü.1Ü а = Ü.1Ü, ß =Ü.2Ü
5 6 8.4615 3.1926
5 7 7.9182 4.0831
5 8 7.4134 3.8822
5 9 6.9636 3.6856
5 10 6.5735 3.5067
6 7 8.0342 3.3511
6 8 7.6595 3.9393
6 9 7.2792 3.8009
6 10 6.9162 3.6515
7 8 7.6992 3.5041
7 9 7.4364 3.8238
7 10 7.1488 3.7274
8 9 7.4302 3.6144
8 10 7.2434 3.7292
9 10 7.2101 3.6615
1 2 10.018 3.6508
1 3 7.4123 4.6644
1 4 6.1794 3.7719
1 5 5.478 3.3225
1 6 5.0315 3.0524
1 7 4.7236 2.8714
1 8 4.4986 2.7407
1 9 4.3268 2.6415
1 10 4.1909 2.563
2 3 8.2155 2.4991
2 4 6.7658 4.0972
2 5 5.8788 3.5462
2 6 5.2968 3.1991
2 7 4.8976 2.9669
2 8 4.6128 2.8033
2 9 4.4022 2.683
2 10 4.2411 2.5909
3 4 7.1737 2.5181
3 5 6.3077 3.7511
3 6 5.6661 3.3984
3 7 5.1878 3.1278
3 8 4.83 2.9239
3 9 4.5604 2.7704
3 10 4.3549 2.6536
4 5 6.4744 2.563
4 6 5.9389 3.5032
4 7 5.4789 3.278
4 8 5.0962 3.0731
4 9 4.7852 2.8992
m
R
p m ci C2 -
a = 0.05, $ =0.10 a = 0.10, $ =0.20
4 10 4.5359 2.7571
5 6 5.9672 2.6433
5 7 5.6304 3.3135
5 8 5.3031 3.1708
5 9 5.004 3.021
5 10 4.7419 2.8785
6 7 5.5825 2.7517
6 8 5.3689 3.1639
0.07 10
6 9 5.1378 3.0742
6 10 4.908 2.9677
7 8 5.2816 2.8553
7 9 5.1459 3.0435
7 10 4.9844 2.9877
8 9 5.0409 2.9137
8 10 4.9551 2.9454
9 10 4.845 2.911
IV. Numerical Examples
This section outlines the process for choosing GZIP DSPs for a defined strength, along with numerical examples.
When solving the OC function of the GZIP DSPs conditions, the unity values for various combinations of (p, m, c, Pa(p)) are calculated by considering each combination of ci and C2 with the condition ni = n2 = n. Then, the plan parameters for specific values of p, m, pi, a, p2 and $ are determined GZIP DSPs.
I. Example 1
Suppose that p = 0.07, m = 10, and the strength of the plan is specified as pi = 0.01, a = 0.05, p2 = 0.06, $ = 0.10, the operating ratio R corresponding to these specifications is computed as 6. The acceptance number can be determined from Table 2 as ci = 5 and C2 =6 with an R value of 5.9672, which is close to 6. The unity values corresponding to the p, m, pi, a, p2and $ parameters are obtained from Table 1 as npi = 2.5708.
Since n = = 2 5708 « 257. Based on these calculations, the optimum DSP is ni = 257, ci = 5, p1 0.01 r
n2 = 257 and c2 = 6.
II. Example 2
Assuming p = 0.03, m = 10, and the plan's strength is set at pi = 0.01, a = 0.05, p2 = 0.06, $ = 0.10, the operating ratio R can be calculated as 6. The acceptance number can be determined from Table 2 as ci = 1 and c2 = 3 with an R value of 5.8771, which is close to 6. Consequently, the unity values for p, m, pi, a, p2and $ from Table 1 are npi = 0.9714.
Since, n = = 0 9714 «97. the optimal GZIP DSPs for the specified specifications is ni = 97,
pt 0.01 r r r
ci =1, n2 = 97, c2 =3,p = 0.03 and m = 10.
The ZIP DSP for the specified strength is ni = 102, ci = 1, rn = 102, c2 =3,p = 0.03.
This indicates that GZIP DSP requires smaller sample compared to ZIP DSP. III. Significance of GZIP DSPs over non-Bayesian ZIP DSPs.
The producer's risk for the ZIP DSP is 4.63% when the lot fraction non-conforming is p = 0.009. On the other hand, the producer's risk for the GZIP DSPs for the given values of m = 5 and 10 are 3.32% and 3.56%, respectively. When the lot fraction non-conforming is p = 0.059, the consumer's risk for the ZIP DSP is 10.35%. In contrast, the consumer's risk for the GZIP DSPs for the specified values of m = 5 and 10 is 10.36% and 9.94%, respectively. For the ZIP DSP, the combined producer and consumer risk is 14.98%. On the other hand, the GZIP DSP has a total risk for m = 5 and 10 are 13.68% and 13.50% respectively.
Table 3: Values of OC function of DSP ZIP and GZIP DSPs for (pi = 0.005, a = 0.05, pi = 0.05, § = 0.10)
Model
Parameters
Lot Fraction non-conforming (p)
9
0
0.009
0.059
Producer's Risk (%)
Consumer's Risk (%)
Total
Risk (%)
ZIP DSP
GZIP DSP
0.03
0.03
- 1 2 75 1 5 7 8 335 1
10
97
0.9537 0.1035 0.9668 0.1036 0.9643 0.0994
4.63 3.32 3.56
10.35 14.98
10.36 13.68 9.94 13.50
m
n
V. Conclusion
The fundamental assumption in the theory of sampling inspection procedures by attributes is that the fraction of nonconforming items in a lot remains constant. However, in real-world scenarios, lots produced from a process often exhibit quality variations due to random fluctuations, leading to a random variation in the fraction of nonconforming units across lots. In such situations, Bayesian acceptance sampling plans (BASP), which incorporate prior information about process variability when making decisions on submitted lots, can offer an advantage over conventional plans. In this paper, Bayesian double sampling plans by attributes are determined for the two specified points on the OC curve based on gamma-ZIP distribution. The GZIP DSPs requires fewer sample units for inspection compared to non-Bayesian ZIP DSPs. As a result, GZIP DSPs effectively lower both producer's and consumer's risks, offering better protection for both parties by minimizing the chances of rejecting good-quality lots and accepting poor-quality ones. By implementing GZIP DSPs, optimal sample sizes and acceptance numbers are achieved, reducing overall risk and delivering benefits such as enhanced customer satisfaction, increased productivity, and sustained market competitiveness.
References
[1] Balamurali, S., Aslam, M. and Jun, C.H. (2012). Bayesian double sampling plan under gamma-Poisson distribution. Research Journal of Applied Sciences, Engineering and Technology, 4(8): 949956.
[2] Duncan, A.J. Quality Control and Industrial Statistics, Homewood: Richard D. Irwin, Inc,
1986.
[3] Fu-Kwun, W. and Shalemu Sharew, H. (2018). Sampling plans for the zero-inflated Poisson distribution in the food industry, Food Control 85:359-368.
[4] Kaviyarasu, V. and Sivakumar, P. (2021). Optimization of Bayesian single sampling plan for the zero inflated Poisson distribution involving risk minimization using tangent angle method, International Journal of Scientific Research in Mathematical and Statistical Sciences, 8(3):01-11.
[5] Loganathan, A. and Shalini, K. (2014). Determination of single sampling plans by attributes under the conditions of zero-inflated Poisson distribution. Communications in Statistics- Simulation and Computation, 43(3):538-548.
[6] Loganathan, A. and Shalini, K. (2014). Selection of single sampling plans by attributes under the conditions of zero-inflated Poisson distribution. International Journal of Quality & Reliability Management, 31(9):1002-1011.
[7] Palanisamy, A. and Latha, M. (2018). Construction of Bayesian single sampling plan by attributes under the conditions of gamma zero - inflated Poisson distribution. International Research Journal of Advanced Engineering and Science, 3(1):67-71.
[8] Palanisamy, A. and Latha, M. (2018). Selection of Bayesian single sampling plan with zero-inflated Poisson distribution based on quality region. International Journal of Scientific Research Mathematical and Statistical Sciences, 5(6):313-20.
[9] Pramote, C. and Wimonmas, B. (2021). Designing of optimal required sample sizes for double acceptance sampling plans under the zero-inflated defective data. Current Applied Science and Technology, 21(2):227-239.
[10] Schilling, E.G. and Neubauer, D.V. Acceptance Sampling in Quality Control. Boca Raton: CRC Press, 2009.
[11] Shalini, K, Loganathan, A and Kavitha, N. (2014). Bayesian single sampling plans under the conditions of zero-inflated Poisson distribution. Research and Reviews Journal of Statistics, Special Issue 1:92-98.
[12] Shalini, K., Sheik Abdullah, A. (2018). Designing double sampling plans under the conditions of zero-inflated Poisson distribution. Journal of Emerging Technologies and Innovative Research, 5(12):529-534.
[13] Srinivasa Rao, G. and Muhammad Aslam. (2017). Resubmitted lots with single sampling plans by attributes under the conditions of zero-inflated Poisson distribution. Communications in Statistics - Simulation and Computation, 46(3):1814-1824.
[14] Stephens K.S. The Hand book of Applied Acceptance Sampling: Plans, Procedures and Principles, Wisconsin: ASQ Quality Press, 2001.
[15] Suresh, K.K. and Latha, M. (2001). Bayesian single sampling plan for a gamma prior. Economic Quality Control 16(1):93-107.
[16] Suresh. K. K. and Usha, K. (2016). Construction of Bayesian double sampling plan using minimum angle method. Journal of Statistics and Management Systems, 19(3):473-89,
[17] Uma, G. and Ramya, K. (2016). Determination of quick switching system by attributes under the conditions of zero-inflated Poisson distribution. International Journal of Statistics and Systems, ll(2):157-65.
[18] Vijayaraghavan, R. and Sakthivel, K.M. (2010). Selection of Bayesian double sampling inspection plans by attributes with small acceptance numbers. Economic Quality Control, 25:207-20.
[19] Vijayaraghavan, R., Rajagopal, K. and Loganathan, A. (2008). A procedure for selection of a Gamma-Poisson single sampling plan by attributes. Journal of Applied Statistics, 35(2):149-60.
[20] Wimonmas, B. and Pramote, C. (2021). Designing of double acceptance sampling plan for zero inflated and over-dispersed data using multi-objective optimization. Applied Science and Engineering Progress, 14(3):338-347.
