Acceptance sampling uses sampling procedure to determine whether to

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1 DOI: 0.545/mji Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope K.K. Sureh, S. Umamahewari and K. Pradeepa Veerakumari Department of Statitic, Bharathiar Univerity, Coimbatore, Tamilnadu, India. Abtract Thi paper deal with deigning of Bayeian Repetitive Deferred Sampling Plan (BRDS) indexed through incoming and outgoing quality level with their relative lope on the OC curve. The Repetitive Deferred Sampling (RDS) Plan ha been developed by Shankar and Mohapatra (99) and thi plan i an extenion of the Multiple Deferred Sampling Plan MDS - (c, c 2 ), which wa propoed by Rambert Vaert (98). Keyword: Bayeian Sampling Plan, Acceptable Quality Level (AQL), Limiting Quality Level (LQL), Indifference Quality Level (IQL), Relative Slope, Gamma- Poion Ditribution.. INTRODUCTION Acceptance ampling ue ampling procedure to determine whether to accept or reject a product or proce. It ha been a common quality control technique that ued in indutry and particularly in military for contract and procurement of product. It i uually done a product that leave the factory, or in ome cae even within the factory. Mot often a producer upplie number of item to conumer and deciion to accept or reject the lot i made through determining the number of defective item in a ample from that lot. The lot i accepted, if the number of defective fall below the acceptance number or otherwie, the lot i rejected. Acceptance ampling by attribute, each item i teted and claified a conforming or non-conforming. A ample i taken and contain too many non-conforming item, then the batch i rejected, otherwie it i accepted. For thi method to be effective, batche containing ome non-conforming item mut be acceptable. If the only acceptable percentage of non-conforming item i zero, thi can only be achieved through examing every item and removing the item which are non-conforming. Thi i known a 00% inpection. Effective acceptance ampling involve effective election and the application of pecific rule for lot inpection. The acceptance-ampling plan applied on a lot-by-lot bai become an element in the overall approach to maximize quality at minimum cot. Since different ampling plan may be tatitically valid at different time during the proce, therefore all ampling plan hould be periodically reviewed. Mathematical Journal of Interdiciplinary Science Vol., No. 2, March 203 pp by Chitkara Univerity. All Right Reerved.

2 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. 42 The general problem for proce control i one toward maintaining a production proce in uch a tate that the output from the proce confirm to deign pecification. A the proce operate it will be ubject to change, which caue the quality of the output to deteriorate. Some amount of deterioration can be tolerated but at ome point it become le cotly to top and overhaul the proce. The problem of etablihing control procedure to minimize long-run expected cot ha been approached by everal reearcher through Bayeian deciion theory. Claical analyi i directed toward the ue of ample information. In addition to the ample information, two other type of information are typically relevant. The firt i the knowledge of the poible conequence of the deciion and the econd ource of non ample information i prior information. Suppoe a proce a erie of lot i upplying product. Due to random fluctuation thee lot will be differing quality, even though the proce i table and incontrol. Thee fluctuation can be eparated into within lot variation of individual unit and between lot variation. 2. BAYESIAN ACCEPTANCE SAMPLING Bayeian Acceptance Sampling approach i aociated with utilization of prior proce hitory for the election of ditribution (viz., Gamma Poion, Beta Binomial) to decribe the random fluctuation involved in Acceptance Sampling. Bayeian ampling plan require the uer to pecify explicitly the ditribution of defective from lot to lot. The prior ditribution i the expected ditribution of a lot quality on which the ampling plan i going to operate. The ditribution i called prior becaue it i formulated prior to the taking of ample. The combination of prior knowledge, repreented with the prior ditribution, and the empirical knowledge baed on the ample which lead to the deciion on the lot. A complete tatitical model for Bayeian ampling inpection contain three component:. The prior ditribution (i.e.) the expected ditribution of ubmitted lot according to quality. 2. The cot of ampling inpection, acceptance and rejection. 3. A cla of ampling plan that uually defined by mean of a retriction deigned to give a protection againt accepting lot of poor quality. Rik-baed ampling plan are traditional in nature, drawing upon producer and conumer type of rik a depicted by the OC curve. Economically baed ampling plan explicitly conider certain factor a cot of inpection, accepting Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

3 a non-conforming unit and rejection a conforming unit, in an attempt to deign a cot-effective plan. Bayeian plan deign procedure take into account the pat hitory of imilar lot ubmitted previouly for the inpection purpoe. Non- Bayeian plan deign methodology i not explicitly baed upon the pat hitory. Cae and Keat (982) have examined the relationhip between defective in the ample and defective in the remaining lot for each of the five prior ditribution. They oberve that the ue of a binomial prior render ampling uele and inappropriate. Thee erve to make the deigner and uer of Bayeian ampling plan more aware of the conequence aociated with election of particular prior ditribution. Calvin (984) ha provided procedure and table for implementing Bayeian Sampling Plan. A et of table preented by Oliver and Springer(972)which are baed on aumption of Beta prior ditribution with pecific poterior rik to achieve minimum ample ize, which avoid the problem of etimating cot parameter. It i generally true that Bayeian Plan require a maller ample ize than a conventional ampling plan with the ame producer and conumer rik. Scafer (967) dicue ingle ampling by attribute uing three prior ditribution for lot quality. Hald (965) ha given a rather complete tabulation and dicued the propertie of a ytem of ingle ampling attribute plan obtained by minimizing average cot, under the aumption that the cot are linear with fraction defective p and that the ditribution of the quality i a double binomial ditribution. The optimum ampling plan (n, c) depend on ix parameter namely N,p r,p,p,p 2 and w 2 where N i the lot ize, p r, p are normalized cot parameter and p,p 2,w 2 are the parameter of prior ditribution. It may be hown, however that the weight combine with the p i uch a way that only five independent parameter are left out. The Repetitive Deferred Sampling Plan ha been developed by Shankar and Mohapatra (99) and thi plan i eentially an extenion of the Multiple Deferred Sampling Plan MDS - (c, c 2 ) which wa propoed by Rambert Vaert (98). In thi plan the acceptance or rejection of a lot in deferred tate i dependent on the inpection reult of the preceding or ucceeding lot under Repetitive Group Sampling (RGS) inpection. RGS i a particular cae of RDS plan. Vaert (98) ha modified the operating procedure of the MDS plan of Wortham and Baker (976) and deigned a MDS-.Wortham and Baker (976) have developed Multiple Deferred State Sampling (MDS) Plan and alo provided table for contruction of plan. Sureh (993) ha propoed procedure to elect Multiple Deferred State Plan of type MDS and MDS- indexed through producer and conumer quality level conidering filter and incentive effect. Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope 43 Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

4 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. 44 Vedaldi (986) ha tudied the two principal effect of ampling inpection which are filter and incentive effect for attribute Single Sampling Plan and alo propoed a new criterion baed on the (AQL, α ) point of the OC curve and an incentive index. Lilly Chritina (995) ha given the procedure for the election of RDS plan with given acceptable quality level and alo compared RDS plan with RGS plan with repect to operating ratio (OR) and ASN curve. Sureh and Pradeepa Veerakumari (2007) have tudied the contruction and evaluation of performance meaure for Bayeian Chain Sampling Plan (BChSP-). Sureh and Saminathan (200) have tudied the election of Repetitive Deferred Sampling Plan through acceptable and limiting quality level. Sureh and Latha (200) have tudied Bayeian Single Sampling Plan through Average Probability of Acceptance involving Gamma- Poion model. The operating ratio wa firt propoed by Peach (947) for meauring quantitatively the relative dicrimination power of ampling plan. Hamaker (950) ha tudied the election of Single Sampling Plan auming that the quality characteritic follow Poion model uch that the OC curve pae through indifference quality level and the relative lope of OC curve at that quality level. Thi paper related to Bayeian Repetitive Deferred Sampling Plan for Average Probability of Acceptance function for conumer and producer quality level and relative lope. 3. CONDITIONS FOR APPLICATION OF RDS PLAN. Production i teady o that reult of pat, current and future lot are broadly indicative of a continuing proce. 2. Lot are ubmitted ubtantially in the order of their production. 3. A fixed ample ize, n from each lot i aumed. 4 Inpection i by attribute with quality defined a fraction nonconforming. 4. OPERATING PROCEDURE FOR RDS PLAN. Draw a random ample of ize n from the lot and determine the number of defective (d) found therein. 2. Accept the lot if d c, Reject the lot if d > c If c < d < c 2, Accept the lot provided i proceeding or ucceeding lot are accepted under RGS inpection plan, otherwie reject the lot. Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

5 Here c and c 2 are acceptance number uch that c < c 2. When i= thi plan reduce to RGS plan. The operating characteritic function P a (p) for Repetitive Deferred Sampling Plan i derived by Shankar and Mohapatra (99) uing the Poion Model a Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope Where p ( p) = a i p ( p ) pp a c i ( p ) c c i a (4.) 45 p = p[ d c ] = a c i x e x r r= 0! r (4.2) 2 x r c e x e x pc = p[ c< d< c2] = r! r! c x r r= 0 r= 0 (4.3) Where x = np The probability denity function for the Gamma ditribution with parameter a and β i Γ( p / αβ, ) = e 0, p β Γα pβ α α, p 0, α 0, β 0 otherwie (4.4) Suppoe that the defect per unit in the ubmitted lot p can be modeled with Gamma ditribution having parameter a and β. Let p ha a prior ditribution with denity function given a pt e p t w ( p) =,, t> 0 and p > 0 (4.5) Γ() With parameter and t and mean, p = =µ (ay) t The Average Probability of Acceptance (APA) i given a p= pa ( p) w( p) dp 0 (4.6) Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

6 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. 46 = c r = 0 r c c ri r2 nt r n t r ( r ) () * 2 Γ( ) Γ( ri r2) * i! Γ ( n t) ( r! )( r! ) Γ ( n ni t) c2 r = 0 in ( r ) r ri i! Γ( ri 2r2) * ri 2 ( 2n ni t) r = 0 t ( r ri) * Γ ( n ni t) r2 r= c c c2 ri 2r2 Γ in t r ri i 2 r = 0 r2= c ( ) ( 2) r! r! Γ c 2r ri in t 2 i r = r n n 0 ( ) () * Γ( )! Γ ( 2 ) ri r2 2r ri i t 2 r ri (4.7) In particular, the average probability of acceptance for c = 0, c 2 = i obtained a follow: P = ( nµ ) nµ () ( nµ inµ ) µ 3 i( i)( n ) () ( )( 2) 3 23 ( nµ inµ ) 2 in ( µ ) () ( ) 2 ( 2 nµ inµ ) (4.8) 5. DESIGNING PLANS FOR GIVEN AQL, LQL, a AND b Table (a) and (b) are ued to deign Bayeian Repetitive Deferred Sampling Plan (c = 0, c 2 = ) for given AQL, LQL, α and β. The tep utilized for electing Bayeian Repetitive Deferred Sampling Plan (BRDS) are a follow:. To deign a plan for given (AQL, -α) and (LQL, β) firt calculate the operating ratio μ 2 /μ. 2. Find the value in Table (b) under the column for the appropriate α and β, which i cloet to the deired ratio. 3. Correponding to the located value of μ 2 /μ the value of, i can be obtained. 4. The ample ize can be obtained a nμ /μ where nμ can be obtained againt the located value μ 2 /μ. 5.. EXtAMPLE Suppoe the value for μ i aumed a and value for μ 2 i aumed a then the operating ratio i calculated a Now the integer approximately equal to thi calculated operating ratio and their correponding parametric value are oberved from the table (b). The actual nμ = and nμ 2 = at (α = 0.05 and β = 0.0), Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

7 Table (a): Certain nμ value for pecified value of P (μ) Probability of Acceptance i Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope 47 Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

8 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. 48 Table (b): Value of μ 2 /μ tabulated againt and i for given α and β for Bayeian Repetitive Deferred Sampling Plan i μ 2 /μ for α=0.05 β=0.0 μ 2 /μ for α=0.05 β=0.05 μ 2 /μ for α=0.05 β=0.0 μ 2 /μ for α=0.0 β=0.0 μ 2 /μ for α=0.0 β=0.05 μ 2 /μ for α=0.0 β= Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

9 Table (c): Value of tabulated nμ 0, nμ, nμ 2 and nμ 2 /nμ againt and i for given P (μ) for Bayeian Repetitive Deferred Sampling Plan. i nμ 0 nμ nμ 2 OR Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope 49 Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

10 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. 50 nµ Now n = = = µ The parameter required for the plan i n = 68 with i = 4 and = Deigning of Bayeian Repetitive Deferred Sampling Plan (Brd) Indexed with Relative Slope of Acceptable and Limiting Quality Level 6.. Selection of Parameter With Relative Slope h At The Acceptable Quality Level Table 2(a) i ued to elect the parameter for Bayeian Repetitive Deferred Sampling Plan indexed with μ and h.for example, for given μ = 0.0 and h = 0.07 from Table 2(a) under the column headed h, locate the value i equal to or jut greater than the deired value h. Correponding to thi h, the value of parameter aociated with the relative lope are nμ = 0.457, = and i = 5. From thi one can obtain the ample ize a n = nμ /μ 4.57.Thu the parameter are n = 5, = and i = Selection of Parameter with Relative Slope h 2 at The Limiting Quality Level Table 2(a) i ued to elect the parameter for Bayeian Repetitive Deferred Sampling Plan indexed with μ 2 and h 2.For example, for given μ 2 = 0.2 and h 2 =.7 from Table 2(a) under the column headed h 2, locate the value i equal to or jut greater than the deired value h 2. Correponding to thi h 2, the value of parameter aociated with the relative lope are nμ 2 = , =3 and i = 4. From thi one can obtain the ample ize a n = nμ 2 /μ Thu the parameter are n = 8, = 3 and i = Selection of Parameter with Relative Slope h 0 at The Inflection Point Table 2(a) i ued to elect the parameter for Bayeian Repetitive Deferred Sampling Plan indexed with μ 0 and h 0.For example, for given μ 0 = 0.05 and h 0 = 0.86 from Table 2(a) under the column headed h 0, locate the value i equal to or jut greater than the deired value h 0. Correponding to thi h 0, the value of parameter aociated with the relative lope are nμ 0 = , = 5 and i = 5. From thi one can obtain the ample ize a n = nμ 0 /μ Thu the parameter are n = 7, = 5 and i = 5. Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

11 Table 2(a): Relative lope for Acceptable, Indifference and Limiting Quality Level i nμ 0 nμ nμ 2 h 0 h h 2 h 2 /h h 2 /h 0 h 0 /h Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

12 Sureh, K. K. Umamahewari, S. Veerakumari, K. P CONSTRUCTION OF TABLES: The expreion for APA function for Bayeian Repetitive Deferred Sampling Plan p i given in equation p= p w ( p) dp a n p = n µ () ( µ ) ( nµ inµ ) 2 in ( µ ) () ( ) 2 ( 2nµ inµ ) i i n 3 ( )( µ ) () ( )( 2) (7.) 23 ( nµ inµ ) 3 Where μ = /t, i mean value of the product quality p. Differentiating the APA function with repect to μ give dp n = () dµ ( nµ ) n () ( nµ inµ ) 2 ( nµ inµ ) 2 2 ni( ) ( )( nµ )( 2 2( nµ ) i( nµ )) 3 ( 2nµ inµ ) 2 2 ni( i)() ( )( 2)( nµ )( 3 3( nµ ) in ( µ )) n in 4 ( 3 µ µ ) (7.2) The relative lope h at μ i, h = µ dp p d µ Differentiating the APA function with repect to μ and equating at μ we get variou value of (, i) and their correponding nμ, nμ 0, nμ 2 value are ubtituted in the equation and the relative lope at μ = μ 0, μ, μ 2 the value h h h 0 are obtained and tabulated in Table 2(a).,, 2 8. CONCLUSION There are many way to determine an appropriate ampling plan. However, all of them are either ettled on a non- economic bai or do not take into conideration the producer and conumer quality and rik requirement. Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

13 Uing the Bayeian Sampling Attribute plan without a cot function for a prior ditribution can reduce the ample ize, while if producer rik and conumer rik are appropriate. The work preented in thi paper mainly relate to the procedure for deigning Bayeian Repetitive Deferred Sampling Plan indexed with relative lope at Acceptable, Limiting and Indifference Quality Level. Table are provided here which are tailor-made, handy and ready-made ue to the indutrial hop-floor condition. Bayeian Repetitive Deferred Sampling Plan Indexed Through Relative Slope REFERENCES 53 American National Standard Intitute/American Society for Quality Control (ANSI / ASQC) Standard A2 (987). Term, Symbol and Definition for Acceptance Sampling, American Society for Quality Control, Milwaukee, Wiconin, USA. Calvin, T.W., (984). How and When to perform Bayeian Acceptance Sampling, American Society for quality Control, Vol.7. Cae, K.E. and Keat, J.B., (982). On the Selection of a Prior Ditribution in Bayeian Acceptance Sampling, Journal of Quality Technology, Vol.4, No., pp.0-8. Hamaker, H.C., (950). The Theory of Sampling Inpection Plan, Philip Technical Review, Vol.4, No.9, pp Hald, A., (965). Bayeian Single Sampling Plan for Dicrete Prior Ditribution, Mat. Fy. Skr. Dan. Vid. Selk., Vol.3, No.2, pp.88, Copenhagen, Munkgaard. Lilly Chritina, A., (995). Contribution to the tudy of Deign and Analyi of Supenion Sytem and Some other Sampling Plan, Ph.D Thei, Bharathiar Univerity, Tamilnadu, India. Oliver, L.R. and Springer, M.D., (972). A General Set of Bayeian Attribute Acceptance Plan, American Intitute of Indutrial Engineer, Norcro, GA. Peach, P., (947). An Introduction to Indutrial Statitic and quality Control, 2 nd Edition, Edward and Broughton, Raleigh, North Carolina. Scafer, R.E., (967). Baye Single Sampling Plan by Attribute Baed on the Poterior Rik, Navel Reearch Logitic Quarterly, Vol. 4, No., March, pp.8-88, Shankar, G. and Mohapatra, B.N., (99). GERT Analyi of Repetitive Deferred Sampling Plan, IAPQR Tranaction, Vol.6, No. 2, pp Sureh, K.K., (993). A Study on Acceptance Sampling uing Acceptable and Limiting Quality Level, Ph.D. Thei, Bharathiar Univerity, Tamilnadu. Sureh, K.K. and Latha, M. (200). Bayeian Single Sampling Plan for a Gamma Prior, Economic Quality Control, Vol.6, No., pp Sureh, K.K. and Pradeepa Veerakumari, K., (2007). Contruction and Evaluation of Performance Meaure for Bayeian Chain Sampling Plan (BChSP-), Acta Ciencia Indica, Vol. 33, No. 4, pp Sureh, K.K. and Saminathan, R., (200). Contruction and Selection of Repetitive Deferred Sampling (RDS) Plan through Acceptable and Limiting Quality Level, International Journal of Statitic and Sytem, Vol. 65, No.3, pp Vaert, R., (98). A Procedure to Contruct Multiple Deferred State Sampling Plan, Method of Operation Reearch, Vol.37, pp Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203

14 Sureh, K. K. Umamahewari, S. Veerakumari, K. P. Vedaldi, R., (986). A new Criterion for the Contruction of Single Sampling Inpection Plan by Attribute, Rivita di Statitica Applicata, Vol.9, No.3, pp Wortham, A.W. and Baker, R.C., (976). Multiple Deferred State Sampling Inpection, The International Journal of Production Reearch, Vol.4, No.6, pp Prof. Dr. K.K. Sureh, Profeor and Head, Department of Statitic, Bharathiar Univerity, Coimbatore M. S. Uma Mahewari, Reearch Scholar, Department of Statitic, Bharathiar Univerity, Coimbatore Dr. K. Pradeepa Veerakumari, Aitant Profeor, Department of Statitic, Bharathiar Univerity, Coimbatore Mathematical Journal of Interdiciplinary Science, Volume, Number 2, March 203