International Journal of Mathematical Archive-8(6), 2017, Available online through ISSN

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1 International Journal of Mathematical Archive-8(6), 27, Available online through ISSN BAYESIAN SPECIAL TYPE DOUBLE SAMPLING PLAN WITH BETA PRIOR DISTRIBTUTION Dr. S. JEYABHARATHI* Aociate Profeor, Hinduthan Intitute of Technology, Coimbatore-32, Tamil Nadu, India. Received On: 3-5-7; Revied & Accepted On: 4-6-7) ABSTRACT An iterative procedure of finding the parameter of a Special type Double ampling plan by attribute under Binomial ditribution with Bea prior atifying given condition with repect to producer and conumer rik i preented. Table are contructed for electing the ampling parameter. The dicriminating power of the Bayeian Special Type Double ampling i alo dicued by determining the operating characteritic curve. INTRODUCTION TO SPECIAL TYPE DOUBLE SAMPLING PLAN WITH BINOMIAL DISTRIBUTION Single ampling plan are imple to ue. Very often the producer i at a 'pychological' diadvantage if a ingle ampling plan i applied to the lot, ince no econd chance i given for the lot not accepted. In uch ituation, taking a econd ample i preferable. The operating procedure of double ampling i given in following chematic diagram.. OPERATING PROCEDURE OF DOUBLE SAMPLING PLAN Draw n and oberve d d c d c Accept c < d < c 2 Reject d c 2 d c 2 Draw n 2 and oberve d 2 Let d = d d 2 Correponding Author: Dr. S. Jeyabharathi* Aociate Profeor, Hinduthan Intitute of Technology, Coimbatore-32, Tamil Nadu, India. International Journal of Mathematical Archive- 8(6), June 27 33

2 Dr. S. Jeyabharathi* / Bayeian Special Type Double Sampling Plan with Beta Prior Ditribtution / IJMA- 8(6), June-27. Rajagopal, et al. [6] developed an iterative procedure for finding the parameter of ingle ampling plan by attribute under Polya ditribution atifying the requirement to enure protection to the producer and conumer and dicued the dicriminating power of the plan through the aociated OC curve with beta prior ditribution From the. figure, we can oberve that, in uual double ampling, if there i not a clear accept/reject deciion from the inpection of the firt ample; a econd ample i taken and the deciion i made on the reult of combined ample in which deciion of acceptance can be made even before the inpection of econd ample. But in pecial Type double ampling plan the deciion of acceptance i made only after the econd ample. Conflicting interet arie between producer and conumer in the ue of ingle ampling plan either with c= or c= for the product characteritic involving cotly and detructive teting. Single ampling plan with c= favor the conumer and with c= favor the producer. To overcome the conflicting interet in ingle ampling and the diadvantage of double ampling Govindaraj [] ha developed a pecial type double ampling plan. In Special Type double ampling plan the deciion of acceptance i made only after the econd ample. Govindaraj and Balamurali [2] tudied the pecial type ampling plan under the aumption that the proce fraction defective a a contant. BAYESIAN SPECIAL TYPE DOUBLE SAMPLING PLAN Suppoe a proce in a erie of lot upplying i product. Due to random fluctuation, thee lot will differ in quality even though the proce i table and in control. Thee fluctuation can be eparated into within lot (ampling) variation of individual unit and between lot (ampling and proce) variation. Bayeian ingle ampling attribute plan are analyzed by Hald [3] for continuou prior ditribution. The author briefly reviewed an approach for chooing a prior ditribution for a Baye attribute (good/bad) acceptance ampling plan. A prior ditribution i choen from confidence level correponding to claical lower confidence bound, where a Baye plan i acceptable, the ample ize cane be reduced. If thee two ource of variation are equal implying more proce variation, the diperion of the proce about the proce average i zero, and each lot can be conidered a a random ample drawn from a proce with a contant probability of producing a non-conforming unit. Thi i the premie behind conventional acceptance ampling. Frequently, between lot variation i greater than the within lot variation, indicating that proce variation exit and that the probability of obtaining non-conforming unit varie continuouly. Proper aement of the ampling rik required evaluation of thi proce variation a well a of the ampling variation. Bayeian acceptance ampling conider both ource of variation. Thu the ditinction between the conventional and Bayeian approach i aociated with utilization of prior proce hitory or knowledge in election of ditribution to decribe the random fluctuation involved in acceptance ampling. Lauer [4] conidered Polya ditribution a the ampling ditribution of the number of non-conforming unit and found that the OC curve of the ampling plan are different from that of the OC curve of non-bayeian ingle ampling plan when lot-to-lot variation i preented. Bayeian Special Type Double ampling ha the pychological advantage of giving a econd chance to inpect a econd lot of item becaue to ome people, epecially to the producer, it may eem unfair to reject a lot in the bai of a ingle ample. Vijayaraghavan and Banumathi [7] propoed a procedure for deigning pecial type double ampling plan by attribute for pecified dicrimination which enure protection to the producer and conumer with mall acceptance number uing Bayeian methodology BETA-BINOMIAL DISTRIBUTION We make n independent Bernoulli trial ( to trial) with probability parameter p. It i well known that the number of uccee x ha the binomial ditribution. Conidering the Probability parameter p unknown (but of coure the ample-ize parameter n i known), we have x/p ~ bin (n; p) Thi i given by, p(x) = nc x p x ( p) n x, n =,,2,3,4 We aume the prior ditribution for p i Beta ditribution, i.e. p~β(, t) with denity function, w(p) = β(.t) p ( p) t 27, IJMA. All Right Reerved 34

3 Dr. S. Jeyabharathi* / Bayeian Special Type Double Sampling Plan with Beta Prior Ditribtution / IJMA- 8(6), June-27. The emerged ditribution i called beta-binomial ditribution and can be written a x~betabin(n,, t) Operating Procedure of Special Type Double Sampling Plan (BSTDSP) From the given lot elect a random ample of ize n and count the number of defectived. If d reject the lot, if d = elect a econd ample of ize n 2 and count the number of defectived 2. If d 2, accept the lot otherwie reject the lot. OC FUNCTION OF BSPDSP The OC function of the plan i given by P a = P P The probability of acceptance Bayeian Special Type Double ampling plan baed on Binomial ditribution i given by P(x, n 2, p) = ( p) n n 2 p( p) n In Special Type double ampling plan we conider n = n n 2 Baed on the pat hitory of inpection, it i oberved that product quality p follow the Beta ditribution w(p)dp = ( p) t dp β(,t) p The emerged Probability of Acceptance i given by P = P(x, n 2, p) w(p)dp can be written in term of Beta ditribution a P= ( p)n n 2 p( p) n p ( p) t dp Where t = µ µ β(,t) P = {β(, n t) n β(,t) 2β(, n t )} (.) and µ = = Product quality t CONSTRUCTION OF TABLES For =, the expreion mentioned in (.) can be reduced into P = µ n 2 µ( µ) nµ µ (nµ µ)(nµ 2µ) (.2) For = 2, then P = (2 2µ)(2 µ) 2n 2 µ(2 2µ)(2 µ) (nµ2 µ)(nµ2 2µ) (nµ2 µ)(nµ2 2µ)(nµ2 3µ) (.3) For = 3, P = (3 3µ)(3 2µ)(3 µ) 3n 2 µ(3 3µ)(3 2µ)(3 µ) (nµ3 µ)(nµ3 2µ)(nµ3 3µ) (nµ3 µ)(nµ3 2µ)((nµ3 3µ)(nµ3 4µ) (.4) In general, for = r, the expreion mentioned in (4..) can be reduced a P = (r rµ)[r (r )µ][r (r 2)µ][r (r 3)µ] (r µ) [nµr rµ][nµr (r )µ][nµr (r 2)µ][nµr (r 3)µ].[nµr µ] n 2 rµ(r rµ)[r (r )µ][r (r 2)µ][r (r 3)µ] (r µ) [nµr (r)µ][nµr rµ][nµr (r )µ][nµr (r 2)µ][nµr (r 3)µ].[nµr µ] (.5) Illutration:. From the Table., it i oberved that for pecified value = 2, n = n 2 =5, µ =., the Probability of Acceptance i given by.59. Illutration:.2 For pecified value = 4, n = n 2 =5, µ =.8 the Probability of Acceptance i given by.256 From the Illutration. and.2, it i oberved that the probability acceptance i very leer for the greater value of and µ. 27, IJMA. All Right Reerved 35

4 Dr. S. Jeyabharathi* / Bayeian Special Type Double Sampling Plan with Beta Prior Ditribtution / IJMA- 8(6), June-27. ANALYSIS OF OC CURVE Figure.2 diplay comparative OC curve for variou value of µ with fixed value of n, n 2 and, and demontrated that Bayeian Special Type Double ampling plan decreae it probability of acceptance for maller change in µ and larger value of. At the ame time, the maller value of do not provide protection to the conumer againt unatifactory quality level. A the value of increae, the correponding OC curve exhibit the decreaing trend in the acceptance probabilitie for each value of µ. It can be oberved, further, from the figure the larger value of provide better protection to the conumer with leer rik of accepting the lot of unatifactory quality. It appear that if increae, the OC curve will tend toward the curve correponding to the larget value of. PERFORMANCE MEASURES OF BAYESIAN SPECIAL TYPE DOUBLE SAMPLING PLAN Sureh and Latha [5] derived the OC function and Performance meaure of Bayeian ingle ampling and chain ampling with Gamma ditribution a prior ditribution. She alo obtained the performance meaure of chain ampling and other plan parameter. If the Probability of Acceptance obtained in the expreion (.2), (.3) and (.4) can be equated to.5, the Indifference Quality level of the given plan will be obtained and diplayed jn the Table.2 In the imilar way, the expreion mentioned in (.2), (.3) and (.4) are equated to the given probability of acceptance, the AQL and LQL value of thi plan can be obtained and diplayed in the Table.3 Illutration:.3 From the Table.2, for pecified value of =, n =, n 2 =5, α=.5 and β=. the AQL =. and LQL =.266. The correponding point of control i given by IQL=.62 Illutration:.4 It i oberved that, for pecified value of =3, n =, n 2 =5, α=.5 and β=. the AQL =. and LQL =.55 and the point of control i given by IQL=.28. SELECTION BAYESIAN TYPE DOUBLE SAMPLING PLAN FOR GIVEN AQL AND LQL From Table.3, it i oberved that for a pecified µ, and µ 2, the operating ratio i contructed for varying value of and for fixed n= and n 2 =5. The operating ratio of µ, and µ 2 with beta prior ditribution can be ued a the meaure of dicrimination while deigning the Bayeian Special Type Double ampling plan. Illutration:.5 From the table.3, when =2, for the given value of nµ =. and nµ 2 =.632, the operating ratio i calculated by and the plan parameter correponding to thi ratio i given a = 2 and n =, n 2 =5 SELECTION OF PRODUCT QUALITY µ FOR THE GIVEN PROBABILITY OF ACCEPTANCE The µ value are obtained by equating the expreion mentioned in.2 to.4 to the required probability of acceptance by uing Method of approximation and value are diplayed in Table.3 Illutration:.6 For pecified = 3, n =, n 2 = 5 and P =.99 the average product quality µ =.3 For pecified = 4, n =, n 2 = 5 and P =. the product quality µ =.453. CONCLUSION To overcome the conflicting interet in ingle ampling and the diadvantage of double ampling Special Type double ampling plan ha developed. In Special Type Double Sampling plan the deciion of acceptance i made only after the econd ample. Bayeian ampling plan i a bet technique to evaluate quality from the conformance view. Thi paper i related to Bayeian Special Type Double ampling plan and it variou parameter with beta ditribution a prior ditribution. Thi plan will be more ueful to the quality control practitioner to meet out the conumer requirement. 27, IJMA. All Right Reerved 36

5 Dr. S. Jeyabharathi* / Bayeian Special Type Double Sampling Plan with Beta Prior Ditribtution / IJMA- 8(6), June-27. Table-.: Probability of Acceptance for variou value µ Table-.2: µ Value of BSTDS for the given Probability of Acceptance with and i. P Table-.3: Certain parametric value of BSTDS Plan with Binomial Ditribution P nµ nµ 2 nµ µ 2 / µ , IJMA. All Right Reerved 37

6 Dr. S. Jeyabharathi* / Bayeian Special Type Double Sampling Plan with Beta Prior Ditribtution / IJMA- 8(6), June = =2 =3 =4.7.6 Pa u Figure-.2: Comparative OC Curve for BSTDS with =, 2, 3, 4 REFERENCES. Govindaraju, K., Contribution to the Study of Certain Special Purpoe Plan, Doctoral Diertation, Bharathiar Univerity, Coimbatore, India. (984). 2. Govindaraju, K and Balamurali.S, Chain Sampling Plan for Variable Inpection, Journal of Applied Statitic, 25(), 3 to 9 (997). 3. Hald, A., Statitical Theory of Sampling Inpection by Attribution, Academic Pre Inc. (London) Ltd. (98). 4. Lauer, N.G., Acceptance Probabilitie for Sampling Plan when the Proportion Defective ha Beta Ditribution, Journal of Quality Technology, Vol., No.2, pp.52-55(978). 5. Latha.M, Certain Studie related to Bayeian Acceptance Sampling Plan, Unpublihed PhD Thei, Bharathiar Univerity, Coimbatore, Tamil Nadu, India. (23). 6. K. Rajaopal, A. Logan than and R. Vijayaraghavan, Selection of Bayeian Single Sampling Attribute Plan Baed on Polya ditribution, Economic Quality Control Vol. (24) No: 2, P.P No (29). 7. Vijayaragavan, R, and Banumathi S., Bayeian Deign of Special Type of Double Inpection Plan for Compliance Teting, Communication in Statitic-Simulation and Computation, Vol, 44, Accepted. (24). Source of upport: Nil, Conflict of interet: None Declared. [Copy right 27. Thi i an Open Acce article ditributed under the term of the International Journal of Mathematical Archive (IJMA), which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited.] 27, IJMA. All Right Reerved 38