Acceptance sampling is used to decide either the whole lot will be accepted or rejected, based on inspection of randomly sampled items from the same lot. As an alternative to traditional sampling plans, it is possible to use Bayesian approaches using previous knowledge on process variation. This study presents a Bayesian two-sided group chain sampling plan (BTSGChSP) by using various combinations of design parameters. In BTSGChSP, inspection is based on preceding as well as succeeding lots. Poisson function is used to derive the probability of lot acceptance based on defective and non-defective products. Gamma distribution is considered as a suitable prior for Poisson distribution. Four quality regions are found, namely: (i) quality decision region (QDR), (ii) probabilistic quality region (PQR), (iii) limiting quality region (LQR) and (iv) indifference quality region (IQR). Producer’s risk and consumer’s risk are considered to estimate the quality regions, where acceptable quality level (AQL) is associated with producer’s risk and limiting quality level (LQL) is associated with consumer’s risk. Moreover, AQL and LQL are used in the selection of design parameters for BTSGChSP. The values based on all possible combinations of design parameters for BTSGChSP are presented and inflection points’ values are found. The finding exposes that BTSGChSP is a better substitute for the existing plan for industrial practitioners.
Acceptance sampling is a process of testing and deciding to accept or reject the lot as a quality standard. The main purpose of the acceptance sampling is to distinguish between good and poor lots. We have two methods one is 100 percent inspection and the other is sampling inspection. Sampling is more realistic, quicker and cheaper than 100 percent inspection. In sampling inspection, a lot is accepted or rejected based on the number of defective items in the random sample from the lot [
Bayesian sampling schemes require the user to specifically define the distribution from lot to lot of defects. The prior distribution of the sampling plan is the expected distribution of product quality [
Epstein [
Mughal et al. [
Mughal et al. [
Hafeez et al. [
Based on [
The operating procedure for TSGChSP is based on the following steps: Select an ideal number of Count the total number of defectives Accept the lot if no defective is found in total Reject the lot if more than one defective is found in the current lot immediately preceding If no defective is found in current sample (
All the above steps can be summarized in a flow chart presented in
For TSGChSP, the above procedure can also be shown through a tree diagram for
From the tree diagram in
For TSGChSP, the general expression of the probability of acceptance for
When developing the procedures,
For group chain sampling, replace mean
After replacing
For the equal number of preceding and succeeding lots
Let us consider gamma distribution as a suitable prior for the Poisson distribution, with PDF:
After replacing
Upon Replace mean
Now simplifying
To estimate the quality regions for BTSGChSP, Newton’s approximation is used in
1 | 2 | 1 | 0.0049 | 0.0229 | 0.0446 | 0.1165 | 0.3114 | 0.8732 | 2.5428 | 5.3214 | 27.5443 |
2 | 0.0045 | 0.0183 | 0.0333 | 0.0811 | 0.2081 | 0.5724 | 1.6544 | 3.455 | 17.8555 | ||
3 | 0.004 | 0.0148 | 0.0262 | 0.0616 | 0.1553 | 0.4239 | 1.2213 | 2.5482 | 13.1608 | ||
4 | 0.0036 | 0.0124 | 0.0215 | 0.0496 | 0.1237 | 0.3362 | 0.967 | 2.0167 | 10.4121 | ||
3 | 1 | 0.0032 | 0.0153 | 0.0297 | 0.0777 | 0.2076 | 0.5821 | 1.6952 | 3.5476 | 18.3629 | |
2 | 0.003 | 0.0122 | 0.0222 | 0.0541 | 0.1387 | 0.3816 | 1.1029 | 2.3033 | 11.9036 | ||
3 | 0.0027 | 0.0099 | 0.0175 | 0.0411 | 0.1035 | 0.2826 | 0.8142 | 1.6989 | 8.7739 | ||
4 | 0.0024 | 0.0083 | 0.0143 | 0.0331 | 0.0825 | 0.2241 | 0.6447 | 1.3445 | 6.9414 | ||
4 | 1 | 0.0024 | 0.0114 | 0.0223 | 0.0583 | 0.1557 | 0.4366 | 1.2714 | 2.6607 | 13.7721 | |
2 | 0.0023 | 0.0091 | 0.0167 | 0.0406 | 0.104 | 0.2862 | 0.8272 | 1.7275 | 8.9277 | ||
3 | 0.002 | 0.0074 | 0.0131 | 0.0308 | 0.0777 | 0.2119 | 0.6106 | 1.2741 | 6.5804 | ||
4 | 0.0018 | 0.0062 | 0.0107 | 0.0248 | 0.0618 | 0.1681 | 0.4835 | 1.0084 | 5.206 | ||
2 | 2 | 1 | 0.0049 | 0.0231 | 0.0447 | 0.1111 | 0.26 | 0.5715 | 1.1714 | 1.8418 | 4.6604 |
2 | 0.0046 | 0.0189 | 0.0341 | 0.0779 | 0.1734 | 0.3713 | 0.7516 | 1.1762 | 2.9615 | ||
3 | 0.0041 | 0.0156 | 0.027 | 0.0594 | 0.1293 | 0.2738 | 0.5514 | 0.8613 | 2.1642 | ||
4 | 0.0038 | 0.0132 | 0.0223 | 0.0479 | 0.1029 | 0.2166 | 0.435 | 0.6789 | 1.7041 | ||
3 | 1 | 0.0033 | 0.0155 | 0.0298 | 0.0741 | 0.1733 | 0.381 | 0.781 | 1.2279 | 3.1069 | |
2 | 0.0031 | 0.0126 | 0.0227 | 0.052 | 0.1156 | 0.2475 | 0.501 | 0.7841 | 1.9743 | ||
3 | 0.0028 | 0.0104 | 0.018 | 0.0396 | 0.0862 | 0.1825 | 0.3676 | 0.5742 | 1.4428 | ||
4 | 0.0025 | 0.0087 | 0.0148 | 0.0319 | 0.0686 | 0.1444 | 0.29 | 0.4526 | 1.1361 | ||
4 | 1 | 0.0024 | 0.0116 | 0.0224 | 0.0556 | 0.13 | 0.2858 | 0.5857 | 0.9209 | 2.3302 | |
2 | 0.0023 | 0.0095 | 0.017 | 0.039 | 0.0867 | 0.1857 | 0.3758 | 0.5881 | 1.4808 | ||
3 | 0.0021 | 0.0078 | 0.0135 | 0.0297 | 0.0646 | 0.1369 | 0.2757 | 0.4306 | 1.0821 | ||
4 | 0.0019 | 0.0066 | 0.0111 | 0.0239 | 0.0514 | 0.1083 | 0.2175 | 0.3395 | 0.8521 | ||
3 | 2 | 1 | 0.0049 | 0.0233 | 0.0449 | 0.1097 | 0.2458 | 0.5 | 0.9212 | 1.3306 | 2.7277 |
2 | 0.0046 | 0.0192 | 0.0345 | 0.0773 | 0.1638 | 0.3236 | 0.5874 | 0.8437 | 1.7179 | ||
3 | 0.0042 | 0.0159 | 0.0274 | 0.0589 | 0.122 | 0.2382 | 0.4298 | 0.6159 | 1.2507 | ||
4 | 0.0038 | 0.0135 | 0.0226 | 0.0475 | 0.0971 | 0.1883 | 0.3387 | 0.4847 | 0.9827 | ||
3 | 1 | 0.0033 | 0.0155 | 0.0299 | 0.0731 | 0.1638 | 0.3333 | 0.6141 | 0.8871 | 1.8185 | |
2 | 0.0031 | 0.0128 | 0.023 | 0.0515 | 0.1092 | 0.2158 | 0.3916 | 0.5625 | 1.1453 | ||
3 | 0.0028 | 0.0106 | 0.0183 | 0.0393 | 0.0814 | 0.1588 | 0.2865 | 0.4106 | 0.8338 | ||
4 | 0.0026 | 0.009 | 0.0151 | 0.0316 | 0.0647 | 0.1255 | 0.2258 | 0.3231 | 0.6552 | ||
4 | 1 | 0.0025 | 0.0117 | 0.0224 | 0.0548 | 0.1229 | 0.25 | 0.4606 | 0.6653 | 1.3639 | |
2 | 0.0023 | 0.0096 | 0.0172 | 0.0386 | 0.0819 | 0.1618 | 0.2937 | 0.4219 | 0.8589 | ||
3 | 0.0021 | 0.0079 | 0.0137 | 0.0295 | 0.061 | 0.1191 | 0.2149 | 0.308 | 0.6253 | ||
4 | 0.0019 | 0.0067 | 0.0113 | 0.0237 | 0.0485 | 0.0941 | 0.1693 | 0.2423 | 0.4913 |
In this quality region, the product is accepted with the specified quality average by the engineer. Quality is reliably maintained up to
Therefore, gamma is prior distribution with the mean
In PQR the product is accepted with a minimum probability of 0.10 and a maximum probability of 0.95. PQR is defined as (
The product is accepted with a minimum and maximum probability of 0.1 and 0.9. LQR is defined as an interval like
In this quality region, the product is accepted with a minimum probability 0.50 and a maximum of 0.9. IQR is described as (
In
1 | 2 | 1 | 0.0229 | 0.0446 | 0.3114 | 2.5428 | 0.0217 | 2.5199 | 2.4982 | 0.2885 | 0.0086 | 0.00867 | 0.0751 |
2 | 0.0182 | 0.0333 | 0.2081 | 1.6544 | 0.0151 | 1.6362 | 1.6211 | 0.1898 | 0.00922 | 0.00931 | 0.07947 | ||
3 | 0.0149 | 0.0262 | 0.1553 | 1.2213 | 0.0113 | 1.2064 | 1.1951 | 0.1404 | 0.00939 | 0.00948 | 0.08067 | ||
4 | 0.0124 | 0.0215 | 0.1237 | 0.967 | 0.009 | 0.9546 | 0.9455 | 0.1113 | 0.00947 | 0.00956 | 0.08127 | ||
3 | 1 | 0.0153 | 0.0297 | 0.2076 | 1.6952 | 0.0144 | 1.6799 | 1.6655 | 0.1924 | 0.0086 | 0.00867 | 0.07509 | |
2 | 0.0122 | 0.0222 | 0.1387 | 1.1029 | 0.01 | 1.0908 | 1.0807 | 0.1265 | 0.0092 | 0.00929 | 0.07935 | ||
3 | 0.0099 | 0.0175 | 0.1035 | 0.8142 | 0.0076 | 0.8043 | 0.7967 | 0.0936 | 0.0094 | 0.00948 | 0.08069 | ||
4 | 0.0083 | 0.0143 | 0.0825 | 0.6447 | 0.006 | 0.6364 | 0.6304 | 0.0742 | 0.00948 | 0.00957 | 0.08133 | ||
4 | 1 | 0.0115 | 0.0223 | 0.1557 | 1.2714 | 0.0108 | 1.2599 | 1.2491 | 0.1442 | 0.00859 | 0.00867 | 0.07504 | |
2 | 0.0091 | 0.0167 | 0.104 | 0.8272 | 0.0075 | 0.8181 | 0.8106 | 0.0949 | 0.00921 | 0.00929 | 0.07936 | ||
3 | 0.0074 | 0.0131 | 0.0776 | 0.6106 | 0.0057 | 0.6032 | 0.5975 | 0.0702 | 0.0094 | 0.00949 | 0.08077 | ||
4 | 0.0062 | 0.0107 | 0.0619 | 0.4835 | 0.0045 | 0.4773 | 0.4728 | 0.0557 | 0.0095 | 0.00959 | 0.08143 | ||
2 | 2 | 1 | 0.0232 | 0.0447 | 0.26 | 1.1714 | 0.0215 | 1.1482 | 1.1267 | 0.2368 | 0.01876 | 0.01912 | 0.09099 |
2 | 0.0189 | 0.0341 | 0.1734 | 0.7516 | 0.0152 | 0.7326 | 0.7175 | 0.1545 | 0.02071 | 0.02115 | 0.09821 | ||
3 | 0.0156 | 0.027 | 0.1293 | 0.5514 | 0.0115 | 0.5358 | 0.5244 | 0.1137 | 0.02138 | 0.02184 | 0.1007 | ||
4 | 0.0132 | 0.0223 | 0.1029 | 0.435 | 0.0091 | 0.4219 | 0.4128 | 0.0897 | 0.02158 | 0.02205 | 0.10146 | ||
3 | 1 | 0.0155 | 0.0298 | 0.1733 | 0.781 | 0.0144 | 0.7655 | 0.7511 | 0.1579 | 0.01876 | 0.01912 | 0.09096 | |
2 | 0.0126 | 0.0227 | 0.1156 | 0.501 | 0.0101 | 0.4884 | 0.4783 | 0.103 | 0.0207 | 0.02113 | 0.09814 | ||
3 | 0.0104 | 0.018 | 0.0862 | 0.3676 | 0.0077 | 0.3572 | 0.3495 | 0.0758 | 0.02144 | 0.02191 | 0.10101 | ||
4 | 0.0088 | 0.0148 | 0.0686 | 0.29 | 0.0061 | 0.2812 | 0.2752 | 0.0598 | 0.02151 | 0.02199 | 0.10116 | ||
4 | 1 | 0.0116 | 0.0224 | 0.13 | 0.5857 | 0.0108 | 0.5741 | 0.5634 | 0.1184 | 0.01874 | 0.0191 | 0.09086 | |
2 | 0.0095 | 0.017 | 0.0867 | 0.3758 | 0.0076 | 0.3663 | 0.3587 | 0.0773 | 0.02072 | 0.02116 | 0.09824 | ||
3 | 0.0078 | 0.0135 | 0.0646 | 0.2757 | 0.0057 | 0.2679 | 0.2622 | 0.0569 | 0.02137 | 0.02184 | 0.10069 | ||
4 | 0.0066 | 0.0111 | 0.0514 | 0.2175 | 0.0046 | 0.211 | 0.2064 | 0.0449 | 0.02164 | 0.02212 | 0.10172 | ||
3 | 2 | 1 | 0.0233 | 0.0449 | 0.2458 | 0.9212 | 0.0215 | 0.8979 | 0.8763 | 0.2225 | 0.024 | 0.02459 | 0.09686 |
2 | 0.0192 | 0.0345 | 0.1638 | 0.5874 | 0.0153 | 0.5682 | 0.5529 | 0.1446 | 0.02689 | 0.02764 | 0.10569 | ||
3 | 0.0159 | 0.0274 | 0.122 | 0.4298 | 0.0115 | 0.4139 | 0.4024 | 0.1061 | 0.02774 | 0.02854 | 0.10826 | ||
4 | 0.0135 | 0.0226 | 0.0971 | 0.3387 | 0.0092 | 0.3252 | 0.316 | 0.0836 | 0.02815 | 0.02896 | 0.10948 | ||
3 | 1 | 0.0155 | 0.0299 | 0.1638 | 0.6141 | 0.0144 | 0.5986 | 0.5842 | 0.1483 | 0.02399 | 0.02458 | 0.09685 | |
2 | 0.0128 | 0.023 | 0.1092 | 0.3916 | 0.0102 | 0.3788 | 0.3686 | 0.0964 | 0.02691 | 0.02765 | 0.10573 | ||
3 | 0.0106 | 0.0183 | 0.0814 | 0.2865 | 0.0077 | 0.2759 | 0.2682 | 0.0707 | 0.02774 | 0.02853 | 0.10821 | ||
4 | 0.009 | 0.0151 | 0.0647 | 0.2258 | 0.0061 | 0.2168 | 0.2107 | 0.0557 | 0.02809 | 0.0289 | 0.10931 | ||
4 | 1 | 0.0117 | 0.0224 | 0.1229 | 0.4606 | 0.0108 | 0.4489 | 0.4382 | 0.1112 | 0.024 | 0.02459 | 0.09687 | |
2 | 0.0096 | 0.0172 | 0.0819 | 0.2937 | 0.0076 | 0.2841 | 0.2765 | 0.0723 | 0.02683 | 0.02757 | 0.10542 | ||
3 | 0.0079 | 0.0137 | 0.061 | 0.2149 | 0.0058 | 0.207 | 0.2012 | 0.0531 | 0.02795 | 0.02875 | 0.10894 | ||
4 | 0.0067 | 0.0113 | 0.0486 | 0.1693 | 0.0046 | 0.1626 | 0.158 | 0.0418 | 0.02812 | 0.02893 | 0.10929 |
Given that
When QDR and PQR are specified, then
Suppose a manufacturing company required QDR
Let in a manufacturer company required QDR
Let in a manufacturer company required QDR
Consider shape parameter
Consider shape parameter
When the number of testers
From
For comparison purposes, BTSGChSP is compared with the existing BGChSP [
From
The presented work in this paper is limited to BTSGChSP and four quality regions are estimated for the specified producer’s and consumer’s risks. This plan gives protection to both producer and consumer. Many electronic components such as transport electronics systems, wireless systems, global positioning systems, and computer-supported and integrated manufacturing systems can be evaluated by using the proposed plan. Many other distributions and other quality and reliability characteristics can be explored in the future.