Designing an Optimum Acceptance Sampling Plan Using Bayesian Inferences and a Stochastic Dynamic Programming Approach

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1 Trasactio E: Idustrial Egieerig Vol. 16, No. 1, pp. 19{5 c Sharif Uiversity of Techology, Jue 9 Desigig a Optimum Acceptace Samplig la Usig Bayesia Ifereces ad a Stochastic Dyamic rogrammig Approach Abstract. S.T. Akhava Niaki 1; ad M.S. Fallah Nezhad 1 I this paper, we use both stochastic dyamic programmig ad Bayesia iferece cocepts to desig a optimum-acceptace-samplig-pla policy i quality cotrol eviromets. To determie the optimum policy, we employ a combiatio of costs ad risk fuctios i the objective fuctio. Ulike previous studies, acceptig or rejectig a batch are directly icluded i the actio space of the proposed dyamic programmig model. Usig the posterior probability of the batch beig i state p (the probability of o-coformig products), rst, we formulate the problem ito a stochastic dyamic programmig model. The, we derive some properties for the optimal value of the objective fuctio, which eable us to search for the optimal policy that miimizes the ratio of the total discouted system cost to the discouted system correct choice probability. Keywords: Quality ispectio; Acceptace samplig pla; Bayesia iferece; Stochastic dyamic programmig. INTRODUCTION AND LITERATURE REVIEW There are various methods of ispectio i quality cotrol for improvig the quality of products [1]. Acceptace samplig plas are practical tools for quality assurace applicatios ivolvig quality cotracts o product orders. The samplig plas provide the vedor ad buyer with decisio rules for product acceptace, i order to meet the preset product quality requiremet. With the rapid advacemet of maufacturig techology, suppliers require their products to be of high quality with very low fractio o-coformities, ofte measured i parts per millio. Ufortuately, traditioal methods of calculatig fractio o-coformities o loger work, sice some samples of reasoable size probably cotai o ocoformig product items []. ear ad Wub [] itroduced a eective samplig pla based o a process capability idex, C pk, to deal with product acceptace determiatio for 1. Departmet of Idustrial Egieerig, Sharif Uiversity of Techology, Tehra,.O. Box , Ira. *. Correspodig author. Niaki@Sharif.edu. Received 1 April 7; received i revised form 15 September 7; accepted 1 November 7 low fractio o-coformig products. The proposed ew samplig pla was developed based o a exact samplig distributio rather tha approximatio. ractitioers ca use this proposed method to determie the umber of required ispectio uits ad the critical acceptace value, ad make reliable decisios i product acceptace. Klasse [3] proposed a credit-based acceptace samplig system. The credit of the producer was deed as the total umber of items accepted sice the last rejectio. I this system, the sample size of a lot depeds o a simple fuctio of lot size ad credit, ad results i a guarateed upper limit o outgoig quality that is much smaller tha for a isolated lot ispectio. O the oe had, the classical model of acceptace samplig by variables for idividual lots, is ot suitable [4-6]. O the other had, it may be cosidered as a approximatio for large lots. Ivestigatig the approximatio error for small lots, Seidel [4], by use of risk fuctios, showed that models with ukow variaces perform better tha those with kow variaces. For very small lots, it is ofte better to use a test based o a ukow variace, eve if the variace is kow. Because, by estimatig the variace, uder certai coditios, oe obtais a test statistic that

2 S.T. Akhava Niaki ad M.S. Fallah Nezhad follows more closely the fractio ocoformities i the lot. Chu ad Riks [7] assumed that the fractio o-coformig product is a radom variable that follows a Beta distributio. They derived ot oly modied producer ad cosumer risks, but, also, the Bayes producer ad cosumer risks. Tagaras [] studied the joit process cotrol ad machie maiteace problem of a Markovia deterioratig machie. Assumig that samplig ad prevetive maiteace were performed at xed itervals, he searched the best X cotrol chart limits, prevetive maiteace itervals ad samplig itervals, i order to umerically miimize the average related maiteace ad quality cotrol costs. Kuo [9] developed a optimal adaptive cotrol policy for joit machie maiteace ad product quality cotrol. He icluded iteractios betwee machie maiteace ad product samplig i a search for the best machie maiteace ad quality cotrol strategy for a Markovia deterioratig, state uobservable, batch productio system. However, ulike previous studies i this area, he did ot impose madatory xed sample sizes ad xed samplig itervals o the system. Istead, he let the dyamic programmig mechaism dictate the best sample size ad samplig epoch based o the curret state of the system. He derived several properties of the optimal value fuctio, which helped to d the optimal value fuctio ad idetify the optimal policy more ecietly i the value iteratio algorithm of the dyamic programmig. I this paper, i order to reject or accept a batch i a quality ispectio problem, we propose a adaptive optimal policy. This policy is derived based upo stochastic dyamic programmig ad Bayesia estimatio approaches that develop a optimal framework for the decisio-makig process at had. The rest of the paper is orgaized as follows. First, the model is preseted. The, its applicatio by a umerical example is demeostrated. At the ed, the paper is cocluded. THE MODEL Stochastic dyamic programmig is oe of the most powerful techiques used to model the stochastic behavior of decisio-makig processes [1]. I acceptace samplig plas, i cases where we are to decide betwee acceptig ad rejectig a batch, we are i a stochastic state ad ca ever surely say that a batch is acceptable or should be rejected. Sice the stochastic state of the process may be dyamic, we may use the cocept of stochastic dyamic programmig to model a acceptace-samplig pla. Before doig so, we rst eed to have some otatios ad deitios. Notatios, Assumptios ad Deitios We will use the followig otatios ad deitios i the rest of the paper. `p' is deed as the fractio o-coformatio of the product. I situatios i which little is kow, priori, relative to what the data has to tell us about p, we employ the likelihood fuctio. Furthermore, it is possible to express the ukow parameter, p, i terms of a metric f(p), so that the correspodig likelihood is data traslated. This meas that the likelihood curve for f(p) is completely determied, a priori, except for its locatio, which depeds o the data yet to be observed. The, i order to express that we kow little a priori relative to what the data is goig to tell us, it may be said that we are almost equally willig to accept oe value of f(p) as aother. This state of idierece may be expressed by takig f(p) as locally uiform, ad the resultig priori distributio is called o-iformative for f(p), with respect to the data. Accordig to Jereye's rule [11], assumig (p) = E xjp log i for the observed data, x, the iformative for p should be chose, so that, locally, f(p) = :5 (p). Specically, for the case of Beta distributio, the iformatio measure is (p) p 1 (1 p) 1. Hece, f(p) p :5 (1 p) :5 ad we ca coclude that f(p), as a o-iformative prior distributio, is Beta (.5,.5) [1]. The, usig Bayesia iferece, we ca easily show that the posterior probability desity fuctio of p is: f(p) = ( + + 1) ( + :5) ( + :5) p :5 (1 p) :5 ; (1) where is the umber of o-coformig products ad is the umber of coformig products i all of the past stages of the decisio-makig process. m: the total umber of products i a batch, : the sample size at stage i, : the discout factor, R: deed as the cost of rejectig a batch, C: the cost of havig oe o-coformig product shipped to the customer, V (p): the cost associated with p whe there are remaiig stages to make the decisio, W (p): deed as the probability of correct decisio associated with p, whe there are remaiig stages to make the decisio, d : the upper threshold for p. If the probability of havig a o-coformig product is more tha d, the, we reject the batch,

3 Desigig a Optimum Acceptace Samplig la 1 d : deed as the lower threshold for p. If the probability of havig a o-coformig product is less tha d, we accept the batch, 1 : the miimum acceptable level of the batch quality (Accepted Quality Level (AQL)), : deed as the miimum rejectable level of the batch quality (Lot Tolerace roportio Defective (LTD)), CS: the evet of makig the correct decisio, " 1 : the probability of type-oe error i makig a decisio, " : the probability of type-two error i makig a decisio. Derivatios Cosider a process that is observed at the time poits of possible states that are coutable ad are labeled as oegative itegers ; 1; ;. After observig the state of the process, a actio must be take. We let A (assumed ite) represet the set of possible actios. If the process is i state i at stage of the decisio-makig process ad actio a is chose, the, idepedet of the past, two thigs occur: (i) We receive a expected reward; (ii) The ext state of the system, (j), is chose, accordig to the trasitio probabilities. Some authors have developed sequetial aalysis iferece, i combiatio with a optimal stoppig problem, to determie the probability of makig correct decisios. Oe such research is a ew approach i the probability distributio ttig of give statistical data that Eshragh ad Modarres [13] amed the Decisio O Belief (DOB). I this decisio-makig method, a sequetial aalysis approach is employed to d the best uderlyig probability distributio of the observed data. Moreover, Eshragh ad Niaki [14] applied the DOB cocept as a decisio-makig tool i the respose surface methodology. I this article, we use the cocept of DOB to model the problem. We may model a acceptace samplig process as a optimal stoppig problem, i which, at each stage of the decisio-makig process, we take a sample from a batch ad, based o the iformatio obtaied from the sample, we decide whether to accept or reject the batch or to cotiue to take more samples. We metioed that the probability distributio of the fractio o-coformities could be modeled by Bayesia iferece as a Beta distributio, with parameters + :5, + :5. Hece, (p d ) shows the probability of rejectig a batch ad (p d ) shows the probability of acceptig a batch. The, assumig d d (to esure that the probability is ot egative), by use of the total probability theorem, [1 (p d ) (p d )] shows the probability of either acceptig or rejectig a batch ad, hece, cotiuig to take more samples. If we dee to be the idex of the decisiomakig stage ad p to be the state variable, the, R (p d ) shows the cost whe we reject the batch, Cmp (p d ) represets the cost whe we accept the batch ad V 1 (p) shows the cost whe we cotiue to the ext stage. It is obvious that we eed the discout factor,, to evaluate the cost of the ext stage at the curret stage (accordig to the approach of stochastic dyamic programmig). Hece, we ca dee the stochastic dyamic equatio of the cost as follows: E(cost) = E(costjReject) (Reject) + E(costjAccept) (Accept) + E(costjNeither acceptig or rejectig) (Neither acceptig or rejectig): () The, the cost associated with p, whe there are remaiig stages to make the decisio, is: V (p) = R (p d ) + Cmp (p d ) + (1 (p d ) (p d ))V 1 (p); (3) ad we eed to miimize it over d ad d. I other words, the stochastic dyamic equatio of the cost is mi d ;d fv (p)g. However; (CS) = (CSjReject) (Reject) where: + (CSjAccept) (Accept) + (CSjNeither acceptig or rejectig) (CSjReject) = (CSjAccept) = (Neither acceptig or rejectig); (4) f(p)dp; f(p)dp: The, the probability of a correct decisio associated with p, whe there are remaiig stages to make the decisio, is: W (p)= (p d ) f(p)dp + (p d ) f(p)dp +(1 (p d ) (p d ))W 1 (p); (5)

4 S.T. Akhava Niaki ad M.S. Fallah Nezhad ad we eed to maximize it over d ad d. I other words, we ca dee the stochastic dyamic equatio of makig the correct decisio as max d ;d fw (p)g. Sice we are to miimize the objective fuctio give i Equatio 3 ad maximize the objective fuctio of Equatio 5 simultaeously, based o the ratio of the cost to the probability of a correct decisio criterio, we combie these two equatios i oe, as follows: H (p) = V (p) W (p) ; (6) ad it is obvious that this fuctio should be miimized. Theorem The optimal values of d ad d i Equatio 6 are i the boudary limits of d ad d. roof We take the rst derivatives of H (p) i Equatio 6, with respect to d ad d ad set them both equal to zero. That is, (a): = ) W (p)( f(d ))(R V 1 (p)) (W V (p)( f(d ))R 1 = ) W (p)(f(d )) f(p)dp W 1 (p) (W (p)) = ; m ++1 +:5 V 1(p) W (p)) V (p)(f(d ))R 1 I other words, (a) (b) R 1 W (p) V (p) = R 1 W (p) V (p) = f(p)dp W 1(p) (W (p)) = : f(p)dp W 1 (p) (R V 1 (p)) ; (7) f(p)dp W 1(p) : () ++1 C V 1(p) m +:5 As Equatios 7 ad share a uique left had side, their right had sides must be equal. However, we otice that, i geeral, they caot be equal, cocludig that, at most, oe of the derivatives i either (a) or (b) ca be equal to zero. Suppose the derivative i (a) is equal to zero. I this case, the derivative i (b) is ot equal to zero ad we coclude that the optimal values of d are at its boudary limits. However, if we expad Equatio 7, we will have: (p d ) = " W 1 (p)(r V 1 (p)) V 1 (p)r 1 # f(p)dp W 1 (p) R 4 (Cmp V 1 1(p)) R 1 f(p)dp W 1(p) f(p)dp W 1 (p) (R V 1 (p)) This is a cotradictio, because we showed that the optimal value of d is at its boudary limits. Therefore, we coclude that oe of the derivatives i Equatios 7 ad is equal to zero ad, hece, the optimal values of d ad d are at their boudary limits. For situatios i which the derivative i (b) is equal to zero, the reasoig is similar. I order to determie the boudary limits of d ad d, we use the cocepts of rst ad secod type errors. First type error shows the probability of rejectig the batch whe the fractio o-coformig of the batch is acceptable ad secod type error is the probability of acceptig the batch whe the fractio o-coformig of the batch is ot acceptable. The, o the oe had, if p 1, the probability of rejectig the batch will be smaller tha " 1 ad, o the other had, i cases where p, the probability of acceptig the batch will be smaller tha ". Hece, i cases where the expected value of p is 1, we will have: E(p) = + = 1 ) + = = 1 = (1 1 ) ad the probability of rejectig the batch is obtaied by: (p d ) = Z1 d p 1 1 (1 1 ) 1 (1 1 ) (1 p) dp "1 : : 1

5 Desigig a Optimum Acceptace Samplig la 3 I other words, F p (1) F p (d ) " 1, where F p (d ) is the cumulative probability distributio fuctio of p, evaluated at d. Kowig that F p (1) = 1 implies F p (d ) 1 " 1, if we dee t 1 to be the boudary limit of d, sice F p (d ) is a icreasig fuctio, we have: d F 1 p (1 " 1 ) = t 1 : (9) I a similar way, whe E(p) =, we have: E(p) = + = ) + = = = (1 ) ad the probability of rejectig the batch is obtaied by: (p d ) = Zd p ki (1 ) 1 (1 ) (1 p) ki 1 dp " : I other words, F p (d ) F p () ". I this case, F p () = implies that F p (d ) ". Accordigly, if we dee t to be the boudary limit of d, we have: d F 1 p (" ) = t : (1) Now, sice the optimal values of d ad d are at the boudary limits, i order to make the optimum decisio, we ca cosider the decisio tree give i Equatio 11 to make a decisio. I this equatio, +:5 ++1 is the mea of the Beta distributio for p, give i Equatio 1. 1: ; d = t ) d = 1; ( if +:5 ++1 t ) accept the batch else cotiue to the ext ) d = 1; d = ) cotiue to the ext stage: 3: ; d = t ) if t 1 +:5 if t ) d = t 1 ; ++1 t! impossible, because d d ++1 t 1! cotiue to the ext stage if t 1 ; t ++1 +:5! reject the batch if +:5 ++1 t ; t 1! accept the ; d = if +:5 ) d = t 1 ; ++1 t 1! reject the batch else cotiue to the ext stage (11) I order to idetify the sigs of the derivatives, the values for W (p) ad V (p) are required. Moreover, to obtai these fuctios, the values of d ad d are eeded. We showed that these values are at their boudaries, resultig i four cases, which are combiatios of the values for d ad d, as d = 1, d = t 1, d = ad d = t. The, we may evaluate the objective fuctio, give i Equatio 6, by these cases ad pick the oe with the lowest value. I the ext sectio, we provide a umerical example to illustrate the applicatio of the proposed methodology. NUMERICAL EXAMLE Suppose = 1. This meas that we are dealig with a oe-stage decisio-makig process. For the ext stages, we may use the recursive properties of W (p) ad V (p). Moreover, suppose = 1, = 4, R = 1, m = 5, = :9 ad C = 1. The, k 1 = + = 5 ad, usig Equatio 1, we have: f(p) = (6) (1:5) (4:5) p:5 (1 p) 3:5 = = 56 7 p:5 (1 p) 3:5 : I this case, assumig " 1 = :5, " = :, 1 = :1 ad = :, at stage oe, we have: H1(p) = 1 (p d1) + 15 (p d 1 ) + (1 (p d 1) (p d 1 ))V (p) :55 (p d1) + : (p d 1 ) + (1 (p d1) (p d 1 ))W(p) :

6 4 S.T. Akhava Niaki ad M.S. Fallah Nezhad Usig Equatios 9 ad 1 Z1 (p d )= (p d ) = d Zd (5) (:5) (4:5) p :5 (1 p) 3:5 dp:5; (1) (5) (1) (4) p (1 p) 3 dp :: (13) I order to calculate t 1 ad t, we eed to evaluate the itegrals of Equatio 1 umerically. This evaluatio leads us to t 1 = :36 ad t = :5. We will calculate the objective fuctio for dieret possible values of d 1, d 1 ad the, we choose d 1, d 1 that miimizes the objective fuctio: d 1 = :36; d 1 = ) H 1 (:5) = 3:4 :19 = 11:39; d 1 = 1; d 1 = :5 ) H 1 (:5) = 1:9 :17 = 64:35; d 1 = 1; d 1 = ) H 1 (:5) = ) o aswer; d 1 = :36; d 1 = :5 ) H 1 (:5) = 34:34 :146 = 35:: Hece, the optimum values for d 1, d 1 are d 1 = :36, d 1 = ad, sice the expected fractio o-coformig is equal to the mea of the Beta distributio with parameters = 1:5, = 4:5, that is.5, we are i state four of the decisio tree i Equatio 11 ad should cotiue samplig. At stage two of the samplig process, let k = 7, = 4 ad =. The, t 1 = :66 ad t = :11 ad: d 1 =:66; d 1 =:11)H 1 (:34)= 73:4 :6 = 11:13; d 1 =1; d 1 =:11 ) H 1 (:34)= 1:97 :11 = 1799:1; d 1 = 1; d 1 = ) H 1 (:34) = ) o aswer; d 1 = :66; d 1 = ) H 1 (:34) = 71:33 :6 = 115:5: Sice the miimum of the objective fuctio occurs at d 1 = :66, d 1 = poits, havig V 1 (:34) = 71:33 ad W 1 (:34) = :6, rst, we obtai V (:34) ad W (:34), based o Equatios 3 ad 5, respectively. The, we calculate H (:34), usig Equatio 6 for dieret corer poits as follows: d =:66; d =:11)H (:34)= 9:94 :774 =117:49; d = 1; d = :11)H (:34)= 65:45 :555 = 11:6; d = :66; d = ) H (:34) = 9:71 :7 ) 114:96; d = 1; d = ) H (:34) = 64:197 :55 = 115:4: Sice at the miimum value of the objective fuctio (d = :66, d = ) d = :66 < :34, we are i state four of the decisio makig tree ad reject the batch. CONCLUSION AND RECOMMENDATIONS FOR FUTURE RESEARCH I this article, we applied a stochastic dyamic programmig model to desig a acceptace pla i quality ispectio eviromets. I order to determie the optimal policy, we cosidered a cost fuctio, i combiatio with a correct decisio probability fuctio, ad tried to optimize the combied fuctio. The importat result of this method is that the boudary limits for decisio-makig thresholds, based o the system state that arises from the dyamic deitio of the system, would chage. For further research, we propose either to cosider other objective fuctios or to employ other fuctios to model the state of the system. Moreover, we ca employ the other fuctios to model the probability of correct choice whe the batch is accepted or rejected. REFERENCES 1. Li, T.F. ad Chag, S. \Classicatio o defective items usig uidetied samples", atter Recogitio, 3, pp (5).. ear, W.L. ad Wub, C.W. \A eective decisio makig method for product acceptace", Omega, 35, pp. 1-1 (7). 3. Klasse, C.A.J. \Credit i acceptace samplig o attributes", Techometrics, 43, pp. 1- (1). 4. Seidel, W. \A possible way out of the pitfall of acceptace samplig by variables: treatig variaces as ukow", Computatioal Statistics & Data Aalysis, 5, pp (1997). 5. Collai, E.V. \A ote o acceptace samplig for variables", Metrika, 3, pp (1991). 6. Collai, E.V. \The pitfall of acceptace samplig by variables", i Frotiers i Statistical Quality Cotrol, H.-J. Lez, G.B. Wetherill ad.-th. Wilrich, Eds., 4 (hysica, Heidelberg), pp (199).

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