A new approach of chain sampling inspection plan
aa r X i v : . [ s t a t . O T ] S e p A new approach of chain sampling inspection plan
Harsh Tripathi a and Mahendra Saha a ∗ a Department of Statistics, Central University of Rajasthan, Rajasthan, India
Abstract
To develop decision rules regarding acceptance or rejection of productionlots based on sample data is the purpose of acceptance sampling inspectionplan. Dependent sampling procedures cumulate results from several precedingproduction lots when testing is expensive or destructive. This chaining ofpast lots reduce the sizes of the required samples, essential for acceptanceor rejection of production lots. In this article, a new approach for chainingthe past lot(s) results proposed, named as modified chain group acceptancesampling inspection plan, requires a smaller sample size than the commonlyused sampling inspection plan, such as group acceptance sampling inspectionplan and single acceptance sampling inspection plan. A comparison study hasbeen done between the proposed and group acceptance sampling inspectionplan as well as single acceptance sampling inspection plan. A example hasbeen given to illustrate the proposed plan in a good manner.
Keywords:
Consumer’s risk, life test, operating characteristic curve, producer’srisk. ∗ Corresponding author e-mail: [email protected] bbreviations and notations: ASIP : Acceptance sampling inspection plan.SASIP : single acceptance sampling inspection plan.DASIP : Double acceptance sampling inspection plan.GASIP : Group acceptance sampling inspection plan.SEASIP : Sequential acceptance sampling inspection plan.ChSP : Chain sampling plan.MChSP : Modified chain sampling plan.MChGSP : Modified chain group sampling plan.OC : Operating characteristic.P : OC function of SASIP. P a : OC function of GASIP. P ac : OC function of MChSP. P acg : OC function of MChGSP. A extremely important aspect about a product is quality. On the basis of lot quality,one may take a decision to accept or reject the lot. 100% inspection of whole lotis not considerable due to time, cost, risk regarding product liability. Sometimestesting is destructive and expensive, so it would be better to inspect the sample ofa lot and take a decision about lot acceptance or rejection on the basis of samplequality of submitted lot. Main aim of using ASIP is to reduce the cost of inspection,time of experimenter and provide protection to producer as well as consumer. ASIPis classifies in two broad areas: sampling inspection plan by attribute and samplinginspection plan by variable. In literature many attribute sampling inspection plansare available viz., SASIP, DASIP, GASIP, SESAIP etc., while variable sampling planuses the accurate measurements of quality characteristics for decision-making ratherthan classifying the products as conforming or non-conforming. Both types of sam-pling inspection plan (attribute and variable) are used for sentencing a lot based onsample of that lot and the parameters are determined with the help of two pointapproaches: AQL and LQL. 2any researcher have discussed the time truncated SASIP and some of them listedhere, namely, Gupta(1962), Gupta et al.(1961), Rosaiah et al. (2005), Tsai et al.(2006), Baklizi et al. (2004), Balakrishnan et al.(2007), Aslam et al. (2010) andAl-omari (2015). Many authors have discussed the time truncated DASIP namely,GS Rao (2011), Ramaswamy et al. (2012) and Gui (2014). Also, many researchershave studied GASIP with time truncated life test and readers may refer to Aslamet al. (2009) for gamma distribution, Aslam et al. (2009 , In this section, we have proposed a new sampling inspection plan, named as MChGSP.Plan parameters of MChGSP are the number of groups ( g ), acceptance number ( c )and number of chained sample results ( i ) respectively. A MChGSP plan is deter-mined by the triple of natural numbers ( g , c , i ).It is to be noted that the values of group size ( r ), truncation time ( t ), producer’srisk ( α ) and consumer’s risk ( β ) should be pre-fixed for this proposed plan. Now,the step by step procedure of MChGSP is follows:1. Select n items from a particular lot and allocate r items to g groups, i.e, n = r × g . Start with normal inspection for pre-fixed experiment time t .2. Inspect all the groups simultaneously and record the number of non-conformingunits (d) upto pre-fixed experiment time t .3. If d ≤ c the lot is accepted provided that there is at-most 1 lot among thepreceding i lots in which the number of defective units d exceeds the criterionc, otherwise reject the lot.Now, the probability of acceptance P a of GASIP is obtained by the following Equa-tion: P a = c X i =0 (cid:18) rgi (cid:19) p i (1 − p ) ( rg − i ) = c X i =0 f ( rg, p, i ) (2.1)4here, p is the probability that observed number of failures occurs before the exper-imental time t and p = F ( t ) , (2.2)where, F ( . ) is the CDF of the considered probability distribution. Now, OC functionof MChSP is given by [see, Luca (2018)] P ac ( p ) = P (cid:8) P i + iP i − (1 − P ) (cid:9) . (2.3)where, P = P ( d n.p ≤ c ) is given by the probability that the observed number ofdefective units found in a lot is less than the criterion c [see, Luca (2018)]. Now, byusing the Equation (2 . P acg ( p ) = c X i =0 f ( rg, p, i ) ( c X i =0 f ( rg, p, i ) ) i + i ( c X i =0 f ( rg, p, i ) ) ( i − − ( c X i =0 f ( rg, p, i ) )! (2.4) Now we are interested to determine the parameters of proposed plan, which are men-tioned by the triple of natural numbers ( g , c , i ). Plan parameters of MChGSP planare determine with the help of two point approach. In order to determine the param-eters of the proposed sampling plan, we use producer’s risk (probability of rejectionof a good lot), denoted by α and consumer’s risk (probability of acceptance of a badlot), denoted by β . The objective of the producer is that sampling plan which min-imizes the chance of rejection of a good lot at acceptable quality level (AQL) whileconsumer wants to minimize the chance of accepting a bad lot at limiting qualitylevel (LQL). We will determine the plan parameters of the approximated approachof the proposed plan in such a way that the lot acceptance probability of a good lotis larger than the producer’s confidence level (1 − α ) and that the lot acceptanceprobability of a bad lot is smaller than consumer’s risk ( β ). Therefore, we can usetwo-point approach (at AQL and LQL) to determine the plan parameters of theproposed plan by using the following non-linear optimization problem:5inimize, ASN: n = g × r (2.5)subject to P acg ( p ) ≥ (1 − α ) (2.6) P acg ( p ) ≤ β (2.7)where, P acg ( p ) is defined in Equation (2 . p and p are the AQL and LQL respec-tively. In above optimizing problem, our aim is to minimize sample size n , where, n depends on number of groups g with each of group size r , i.e., we have to minimizethe number of groups g in such a way that g satisfies above optimization problemfor given r . Table 1 represents the plan parameters of the proposed plan for the group size ( r = 5)when consumer’s risk α = 0 .
05 and producer’s risk β = 0 .
10 are pre-fixed and forthe given value of AQL ( p ) and LQL ( p ). We have observed that the number ofgroups decreases for the fixed AQL and varying values of LQL. Table 2 indicates thecomparison study among the proposed plan with GASIP and SASIP. This compari-son study shows that the proposed plan MChGSP perform better than GASIP andSASIP in terms of group size. In some cases, proposed plan required same number ofgroups as in GASIP and SASIP for the same set of values of AQL and LQL for sen-tencing the lot. We would prefer to use proposed plan than the GASIP and SASIPfor the reason that past information plays significant role to take a decision aboutthe lot in proposed MChGSP rather than to take decision on the basis of currentsample. 6 Example
Suppose that the producer’s risk α and consumer’s risk β are assumed to be 0 . .
10 respectively. Also, the values of AQL ( p ) and LQL ( p ) are 0 .
05 and0 .
14 respectively and these values are known to experimenter to apply the two pointapproach for the estimation of the plan parameters of proposed plan. From Table 1,plan parameters are g = 13, c = 6 and i = 2 for the prefixed group size r = 5. Basedon theses obtained plan parameters, MChGSP is: • Select a sample of size 65 from a submitted lot. Allocate 5 items to 13 groups,i.e, n = r × g and start with normal inspection. • Inspect all the groups simultaneously and record the number of non-conformingunits (d). • Go to MChSP inspection plan, If d ≤ In this article, we have introduced a new sampling inspection plan, name as MChGSP.We have compared the proposed plan with existing GASIP and SASIP in terms ofrequired number of groups. Thus, MChGSP provides flexibility to reach a decisionregarding lot acceptance or rejection with the minimum number of groups by usingpast lot results.
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05 and β = 0 . p ) LQL ( p ) g c i P a ( p ) P a ( p )0 .
01 0 .
02 120 10 3 0 . . .
03 66 7 2 0 . . .
04 44 6 2 0 . . .
05 32 4 1 0 . . .
06 22 3 1 0 . . .
07 15 2 1 0 . . .
05 0 .
10 24 10 3 0 . . .
12 18 8 2 0 . . .
14 13 6 2 0 . . .
18 10 5 1 0 . . .
20 8 4 1 0 . . .
10 0 .
20 12 10 3 0 . . .
25 9 8 3 0 . . .
30 7 7 2 0 . . .
35 5 5 2 0 . . .
38 4 4 1 0 . . .
15 0 .
30 8 10 3 0 . . .
40 5 8 3 0 . . .
50 3 5 2 0 . . .
55 2 3 1 0 . . p ) LQL ( p ) MCh-GSP GASIP SASIP n = g × r n = g × r n = g × r .
01 0 .
02 600 = 120 × −− —0 .
03 330 = 66 × × .
04 220 = 44 × × .
05 160 = 32 × × .
06 110 = 22 × × .
07 75 = 15 × × .
05 0 .
10 120 = 24 × −− —0 .
12 90 = 18 × −− —0 .
14 65 = 13 × −− —0 .
18 50 = 10 × × .
20 40 = 8 × × .
10 0 .
20 60 = 12 × −− —0 .
25 45 = 9 × −− —0 .
30 35 = 7 × × .
35 25 = 5 × × .
38 20 = 4 × × .
15 0 .
30 40 = 8 × −− —0 .
40 25 = 5 × × .
50 15 = 3 × −− .
55 10 = 2 × −−−−