Assessment of Acceptance Sampling Plans Using Posterior Distribution for a Dependent Process

Transcription

1 Rochester Institute of Technology RIT Scholar Works Articles Assessment of Acceptance Sampling Plans Using Posterior Distribution for a Depenent Process A. Erhan Mergen Rochester Institute of Technology Z. Seya Deligonul St. John Fisher College Follow this an aitional works at: Recommene Citation A. Erhan Mergen & Z. Seya Deligonul (2010) Assessment of acceptance sampling plans using posterior istribution for a epenent process, Journal of Applie Statistics, 37:2, , DOI: / This Article is brought to you for free an open access by RIT Scholar Works. It has been accepte for inclusion in Articles by an authorize aministrator of RIT Scholar Works. For more information, please contact ritscholarworks@rit.eu.

2 ASSESSMENT OF ACCEPTANCE SAMPLING PLANS USING POSTERIOR DISTRIBUTION FOR A DEPENDENT PROCESS A. Erhan Mergen (*) Z. Seya Deligonul Sauners College of Business Bittner School of Business 107 Lomb Memorial Drive St. John Fisher College Rochester Institute of Technology Rochester, NY Rochester, NY USA USA emergen@cob.rit.eu (*) corresponing author. 1

3 ASSESSMENT OF ACCEPTANCE SAMPLING PLANS USING POSTERIOR DISTRIBUTION FOR A DEPENDENT PROCESS ABSTRACT In this stuy, performance of single acceptance sampling plans by attribute is investigate by using the istribution of fraction nonconformance (i.e., lot quality istribution (LQD)) for a epenent prouction process. It is the aim of this stuy to emonstrate that, in orer to emphasize consumer risk (i.e., the risk of accepting a ba lot), it is better to evaluate a sampling plan base upon its performance as assesse by the posterior istribution of fractions nonconforming in accepte lots. Similarly, it is the esire posterior istribution that sets the basis for esigning a sampling plan. The prior istribution use in this stuy is erive from a Markovian moel of epenence. KEY WORDS: Acceptance sampling, epenent prouction processes, lot quality istribution (LQD), posterior istribution, mean square nonconformance. INTRODUCTION In prouction processes where the iniviual items are prouce an forme into lots sequentially, the epenence in quality of successive items significantly affects the variance measures. The resultant epenence bias often istorts the performance of the conventional acceptance sampling plans. In orer to prevent the failure of such plans, there is a nee to incorporate the serial epenence in the process as an integral part of the sampling plan. This problem is substantiate in a long list of stuies (Mergen [9], Holmes an Mergen [6], Chen an Chou [1], Tang an Cheong [11]). 2

4 To aress the outgoing quality in a epenent process, our stuy starts with a custom tailore epenence istribution as the prior. This istribution escribes a stochastic process that is representative of many real applications. Using such a prior we first evelop a posterior istribution an then a convenient inicator to assess the outgoing quality in the accepte lots uner a given acceptance sampling plan. Integrating the quality istribution into an acceptance ecision, our measure economically summarizes the information about outgoing quality levels. This moel is applicable to situations where quality is measure by ichotomous quality conformance. The posterior istribution is erive using a two-state Markovian process. By recognizing the serial nature of the prouction process, the Markovian moel irectly incorporates sequential epenence as an integral part of the acceptance sampling plan. In the following, first we summarize the backgroun stuies an the evelopment of our prior istribution. Then, we emonstrate the computation of the posterior istribution an outgoing quality inicator. In the subsequent section we provie three examples. One example is a synthetic application, the secon is base on ata from practice, an thir attempts to emonstrate the level of inaccuracy from erroneous inepenence assumption. Finally, we close the article with a conclusion. BACKGROUND The focus of this line of research has been capturing the epenence with a proper istribution or an ajustment to a selecte conventional istribution. For example, recognizing the sequentiality problem, Mergen [9] an Holmes an Mergen [6] iscuss a 3

5 lot quality istribution for an item to item epenence in a prouction process. Specifically, they introuce a new measure for outgoing quality in accepte lots using a lot quality istribution. Chen an Chou [1] propose a linear cost moel for continuous sampling plans uner a epenent prouction process. Similarly, Tang an Cheong [11] evelop a control scheme for etecting changes in fraction nonconforming in processes with correlation. In aitional work, the significance of epenence in processes has been emonstrate in Deligonul an Mergen [4]. Also, Deligonul an Mergen [4] examine the impact of correlate ata on the control limits for p-chart. Along this stream of research a group of stuies has consiere a Bayesian approach to the epenence problem. Those stuies, however, start from generic priors (Wetherill [12], Hal [5], Chiu [2], Jaraiei et.al. [8], Jaraiei an Asouegi [7]). Although this venue is a welcome aition to the acceptance sampling literature, it oes not give the selection of prior the attention it evotes to sampling esign an its outcomes (e.g,chun an Rinks, [3]. Most priors in this line of research are selecte by criteria base on mathematical convenience. Typically, the question of suitability an convergence characteristic of the priors receive scant attention, if any at all. MODEL Prior istribution of fraction nonconformance: The first step in this stuy is the evelopment of a prior istribution for fraction nonconformance in epenent prouction processes. To this en, the following first orer Markovian moel, which is base on our prior work, is use to epict the epenence 4

6 behavior of the prouction process (see Mergen [9], Mergen an Holmes [10], an Deligonul an Mergen [4]). Let s suppose that large lots of size N are forme from a epenent prouction process. Further suppose that (, g) an (, b) are two states with number of nonconforming items given that the last item prouce (an lotte) was conforming an nonconforming, respectively. Let P(: g, N) an P(: b, N) enote the probabilities of number of nonconforming items in a lot of size N where the last item was conforming an nonconforming, respectively. At any time p inicates the fraction nonconforming (i.e., p =/N). Let x an y be the conitional process probabilities of having a conforming item given that the last item was conforming an nonconforming, respectively. Then the ifference equations associate with the Markov Process epicting the number of nonconforming items are given as follows: P ( : N) P( : b, N) P( : g, N) (1) where P ( : b, N) (1 x)p( 1: g, N 1) (1 y)p( 1: b, N 1) (2) P ( : g, N) xp( : g, N 1) yp( : b, N 1) (3) for = 1, 2,.., N an N > an P (0 : b, N) 0 P (0 : g, N) xp(0 : g, N 1) for N>1 5

7 The prior istribution giving the probabilities associate with the lot quality is obtaine from the solution of the above recursive equations, assuming that the Markov process starts, say, from the steay state probability of a conforming item, namely, y P (0 : g,1) (4) 1 x y This choice of the above initial conition is a reasonable start up for the Markovian process given that the process is convergent. In literature there are alternative suggestions, such as antitetehic initiation proceures, for effective start up options. However, for our purposes the initial point in (4) is sufficient as it is rapi in convergence an reliable in use 1. The solution of the ifference equation yiels the following istribution (see Mergen [9] an Mergen an Holmes [10] for etails): P() N Min(N, 1) i 1 N N 1 i x N i (1 x) i 1 i y (1 y) i 1 C x N 2 1 (1 x) 1 y D for = 1,2,.., N-1 (5) where R Q 0 if R 0 an Q < 0 1 We thank a referee for bringing this issue to our attention. 6

8 C 1 i 1 (1 x) i 1 (1 x x)(1 y y) 1 i 1 (1 y) 2 (6) N 1 N 1 N 1 2 x(1 x) ( 1) x(1 y) y(1 x) N 2 N 2 N 2 1 D (7) 1 x y P() N 1 yx 1 x y for = 0 (8) P() (1 x)(1 1 x y) y N 1 for = N (9) The above istribution epicts the probability of the number of nonconforming items. For the fractions nonconforming, associate probabilities are exactly the same if we switch from with the range 0,1, 2,, N to p with 0/N, 1/N, 2/N,, N/N. Therefore P(p : N) is obtaine irectly from the above istribution. NOTATION TABLE P() N N x y n c = probability istribution of nonconforming items in lots of size N = number of nonconforming items = lot size = probability that the next item will be conforming given that the last item was conforming = probability that the next item will be conforming given that the last item was nonconforming. = sample size = acceptance number 7

9 p = fraction nonconforming (=lot quality = /N) Posterior istribution conitione upon acceptance: After obtaining the unconitional istribution of fractions nonconforming, we obtain the posterior istribution of the same ranom variable conitione upon acceptance, that is P(p : A), where A enotes the event that the lot was accepte. Furthermore, we suppose the single acceptance sampling plan S(n, c) is aopte. Here n, c enotes sample size an acceptance number, respectively. If no more than c nonconforming items are foun in a sample size n ranomly rawn from a lot size N, the lot will be accepte as goo. Given the sampling plan S(n, c), probability of acceptance, P(A: p, S), can be calculate by using binomial istribution. 0 for =N P(A : p,s) P a i c 0 n i p i (1 p ) n i for p = (-1)/N >0, for =2,, N 1 for p = 0 (10) Then the posterior istribution P(p : A, S) is obtaine as, P(A : p,s) P(p : N) P (p : A,S) P(p ) for N P(A : p,s) P(p : N) j 1 j j p = (-1)/N>0 for = 2,, N (11) 8

10 The calculation implicitly assumes that uring an inspection process fractions nonconforming are not affecte. Although this assumption oes not hol for all cases, it is reasonable for large lot sizes. Computing the quality inicator MSNC: The average outgoing quality limit (AOQL), which is the maximum value of average outgoing quality (AOQ) values as p ranges from 0 to 1, is a possible single quantity of interest for assessing the outgoing quality (where AOQ can be compute as AOQ p P p a where P a is the probability of accepting a lot with p fraction nonconformance using a specific acceptance sampling plan). However, AOQL is a conservative measure in the sense that it inflates the maximum outgoing fraction nonconformance since it oes not take into account the lot quality istribution. This in turn causes sample sizes getting bigger to maintain a esire outgoing quality. In fact, Holmes an Mergen [6] 2, Jaraiei et al. [8], Jaraiei an Asouegi [7], an others enrich the iea by treating p as a ranom variable an calculate the expecte value of the average outgoing quality values (EAOQ). Although this seems to be an improvement over the AOQL, this inicator is not sensitive to the ispersion of the istribution. This is a significant efect against many fractions nonconformance istributions (i.e., lot quality istribution (LQD s)) observe from epenent prouction processes. Mergen [9] surprisingly reports that, in practice, 2 Mergen [9] an Holmes an Mergen [6] efine EAOQ as, 1 EAOQ p Pa P(p : N) if the P(p : N) is iscrete p 0 1 EAOQ p Pa P(p : N) p if the P(p : N) is continuous. 0 9

11 some LQD s exhibit higher right tails, which is significant for cases with strong epenence. In fact, this is one of the prime reasons ifferentiating the observe LQD s from Poisson or binomial types. This characteristic injects a ownwar bias in the expecte value of the AOQ computations. Here we propose an inicator combining the square of the mean with the variance of the posterior istribution. This quantity, terme mean square nonconformance (MSNC), aresses both the expecte value component an variability. In MSNC the expecte value of p is square to bring it to the same scale with the variance. With its encompassing nature this measure captures a wier view of the quality characteristics by imitating the mean square quantity: MSNC E 2 (p : A) Var(p : A) (12) MSNC, as compute from the posterior istribution, incorporates the epenence characteristics of the process an the ispersion of nonconformance fraction as an integral part of the sampling plan. EXAMPLES In orer to emonstrate the approach propose in this stuy, first a epenent process with x=0.95 an y=0.30 is utilize. Although this example is a synthetic situation, it escribes the mechanics of our approach well. To emonstrate the computational process, the lot quality istributions, both prior an posterior, have been calculate together with MSNC as the inicator for outgoing quality uner various inspection plans. The results are exhibite in the Table 1. 10

12 (Approximate location for Table 1) In Table 1, the ifference between prior an posterior istribution inicates the improvement in the outgoing lot quality through the use of a given sampling plan. As in this example, uner epenence, EAOQ an MSNC converge in performance for lower values of variance. As the variance gets bigger, especially inflate by epenence, then MSNC more accurately represents the average outgoing quality. This happens because at higher variance levels, inicators that are base on expecte value of the outgoing quality, such as EAOQ, will lag behin other inicators that incorporate variance. The secon example is a real case from inustry. The ata gathere from the manufacturing of a subassembly of an aircraft engine. Further etails about this ata can be foun in Mergen an Holmes [10]. In Table 2 we exhibit the pertinent lot quality istribution obtaine from a 100% inspection of 1487 lots of size 8 of the subassembly. MSNC values are calculate from the posterior istribution given various ifferent sampling plans. In orer to better show the utility of the propose metho we provie an application of the metho on the same ata, this time uner the inepenence assumption. The inepenence is ensure by assigning the steay state value of the fraction nonconforming to x an y. (Approximate location for Table 2) Results in Table 2 show the results uner the erroneous assumption of inepenence when the process is epenent. As can be seen, uner the assumption of inepenence, MSNC values show that we ten to overestimate the outgoing fraction nonconformance when the acceptance numbers are small an unerestimate it when the 11

13 acceptance numbers are bigger. Either way woul result in an increase cost to the proucer an/or the user. CONCLUDING REMARKS This stuy proposes a new inicator, MSNC, to gauge the performance of single acceptance sampling plans for attributes by using the istribution of fraction nonconformance (i.e., lot quality istribution (LQD). The context of the stuy is a sequential sampling from a epenent prouction process by a moel base on a two-state epenent process. It is wiely ocumente that the quality epenence of a stream of items from a manufacturing process will frequently violate the ranomness of the binomial process. In the presence of autocorrelation, back to back items will istort the acceptance sampling plan. The averse effect of this is that it obviously elevates the risk for both proucer an/or consumer. Moreover, frequent misspecification of outgoing quality will also iscreit quality programs in the eye of personnel an evalue the acceptance sampling plan. Our recommenation of a special inicator, MSNC, for the outgoing quality of the lots provies a simple approach for a single sampling inspection. By incorporating epenence as an integral part of its measure, MSNC helps the efficiency of sampling plan. First, it improves the measure by incorporating more information by the use of a custom tailore prior istribution which in turn improves precision. Secon, it accounts for variance, Var(p : A) for both small an large acceptance numbers (c values). This is 12

14 particularly important when the epenence variance is substantial in the original stream of items.. The outgoing quality inicator, MSNC, will rise with increasing acceptance numbers an eflate with sample size for a given acceptance number in the plan. This is expecte since the information content will be epresse in smaller sample sizes. Therefore MSNC reacts to the uncertainty in the anticipate way. Also, there might be other uses for MSNC. For example, MSNC, as calculate over the prior istribution, is the maximum value that can be observe with any sampling inspection plan. The ifference between a MSNC-prior an a MSNC-posterior is a measure of the contribution of the inspection plan in quality assessment. This proceure exhibits an inirect value of the use of alternative acceptance plans. As future research venues, we fin value in the performance simulation of MSNC in wie range of conitions. This will clarify its applicability an test its conformance to various contexts. Secon, it may be a useful irection to consier a MSNC for multiattribute ata. Alternatively, the concept can be use for acceptance sampling by variable measure as oppose to attribute ata. Despite its potential complexity, this will broaen the implementation potential of the inicator. Lastly, the performance of MSNC uner various sampling plans will be a valuable contribution to the practice. In short, MSNC may be useful in esigning acceptance sampling plans or evaluating existing ones from the point of view of consumer risk, an it iffers from the existing quality level oriente measures on two accounts. First, MSNC takes into account 13

15 that the process has certain epenency characteristics, an secon, it aopts a Bayesian approach by employing a prior istribution for computing outgoing quality. 14

16 REFERENCES [1] C.H. Chen an C.Y. Chou, Economic esign of CSP-1 plan uner the epenent prouction process an linear inspection cost, Quality Engineering, 16(2), (2004), pp [2] W.K. Chiu, A new prior istribution for attribute sampling, Technometrics, 16 (1974), pp [3] Y.H. Chun an D.B. Rinks, Three types of Proucer s an Consumer s Risks in the Single Sampling Plan, Journal of Quality Technology, 30(3), (1998), pp [4] Z.S. Deligonul an A.E. Mergen, Depenence bias in conventional p-charts an its correction with an approximate lot quality istribution. Journal of Applie Statistics, 14(1), (1987), pp [5] A.Z. Hal, The compoun hypergeometric istribution an a system of singlesampling inspection plans base on prior istribution an cost, Technometrics, 2(3), (1960), pp [6] D.S. Holmes an A.E. Mergen, Selecting acceptance sampling plans by expecte average outgoing quality, Proceeings of the North East Decision Sciences Institute Annual Meeting, New Port, RI, (1988), pp [7] M. Jaraiei an E. Asouegi, Computing AOQ uner ranom variations, ASQC Annual Quality Congress Transactions, Baltimore, MD., (1985), pp [8] M. Jaraiei, R. Segal, P. Khalili, an E. Asouegi, Effect of ranom variations in incoming quality on the average outgoing quality of acceptance sampling plans, 15

17 Proceeings of the Annual Meeting of the Northeast Decision Sciences Institute, Williamsburg, VA, (1986), pp [9] A.E. Mergen, The lot Quality Distribution for a Depenent Prouction Process an Its Impact on Quality Assurance Plans, Ph.D. Thesis, 1981, University Microfilms International, Michigan, U.S.A. [10] A.E. Mergen an D.S. Holmes, Lot Quality Distribution for Depenent Process, ASQC Annual Quality Congress Transactions, Anaheim, CA, (1986), pp [11] L.C. Tang an W.T. Cheong, A control scheme for high-yiel correlate prouction uner group inspection, Journal of Quality Technology, 38(1), (2006), pp [12] G.B. Wetherill, Some remarks on the Bayesian solution of the single sampling inspection scheme, Technometrics, 2(3), (1960), pp

18 Table 1. Prior an Posterior Distribution an MSNC s for a Depenent Process for Lot sizes 20, 30, 40 for various sampling plans. POSTERIOR PRIOR c = 1 N = 20 N = 20 p n = EAOQ MEAN VAR MSNC For all cases x = 0.95, y =

19 Table 1 (cont.) POSTERIOR PRIOR c = 1 c = 1 N = 30 N = 30 N = 30 p n = 2 n = EAOQ MEAN VAR MSNC For all cases x = 0.95, y =

20 Table 1 (cont.) POSTERIOR PRIOR c = 1 c = 1 c = 1 N = 40 N = 40 N = 40 N = 40 p n = 2 n = 3 n = EAOQ MEAN VAR MSNC For all cases x = 0.95, y =

21 Table 2. Comparison of the Posterior Distributions for Inepenent an Depenent Case. INDEPENDENT DEPENDENT N c E(p ) Var(p ) MSNC E(p ) Var(p ) MSNC x=0.95 y=0.274 Prior istribution: E(p )=0.154, Var(p )=

22 TABLE CAPTIONS (Mergen an Deligonul) Table 1. Table 2. Prior an Posterior Distribution an MSNC s for a Depenent Process for Lot sizes 20, 30, 40 for various sampling plans. Comparison of the Posterior Distributions for Inepenent an Depenent Case. 21