Performance Measures for BSkSP-3 with BMChSP-1 as a reference plan

Transcription

1 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN Performance Meaure for BSkSP-3 with BMChSP-1 a a reference plan 1 Dr. M. Rajewari & 2 Bennet Joe 1 Aitant Profeor, Department of Mathematic, Karpagam Academy of Higher Education, Tamil Nadu, India, 2 Reearch Scholar, Department of Mathematic, Karpagam Academy of Higher Education, Coimbatore, Tamil Nadu, India, Abtract To decribe the random fluctuation in the Acceptance Sampling, Bayeian Acceptance Sampling approach i aociated with the utilization of prior hitory for the election of ditribution (viz., Gamma Poion, Beta Binomial). Thi paper preent the table for the election of Bayeian Skip Lot ampling Plan (BSkSP-3) with Bayeian Modified Chain Sampling Plan (BMChSP-1) a reference plan. Gamma ditribution i conidered a the prior ditribution for BMChSP-1. Key Word: BSkSP-3, Bayeian Modified Chain Sampling, Acceptance Quality Level (AQL), Indifference Quality Level (IQL), Limiting Quality Level (LQL). Correponding Author: Bennet Joe 1. Introduction The concept of quality control in manufacturing wa firt advanced by Walter. A. Shewhart of the Bell Telephone Laboratorie in the year The firt verion of Skip Lot Sampling Plan (SKSP-1) for inpection of raw material purchaed from a common ource i introduced by H.F. Dodge and Perry, R.L. At the end of 20th century, Dodge and Roming both developed the Acceptance Sampling Plan a an alternative to 100 % inpection in their Bell Telephone Laboratorie. The methodology that deal with the procedure, by which the deciion to accept or reject the lot of item are baed on the reult of the inpection of ample i called Acceptance Sampling (AS). How and when to perform Bayeian Acceptance Sampling ha preented in a clear and concie treatment by Calvin (1984). Vijayaraghavan (2000) introduced a Deign and evaluation of kip-lot ampling plan of type SkSP-3 plan and the operating characteritic function are derived uing the Markov Chain approach. Vijayaraghavan, Rajagopal and Loganathan (2008) ha given a procedure and table for the election of parameter of ingle ampling plan for pecified requirement 2017 RS Publication, rpublicationhoue@gmail.com Page 187

2 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN (trength) under the condition of a gamma prior and Poion ampling ditribution. Latha M. and Rajewari M. (2013) ha derived minimum average regret function for Bayeian Chain Sampling Plan-1 (BChSP-1). Soundararajan V. (1978) ha deigned the procedure and table for contruction and election of chain ampling plan (ChSP-1). 2. Bayeian Skip-Lot Sampling Plan Thi paper ha given election procedure and table contructed for BSkSP-3 uing Bayeian Modified Chain Sampling Plan (which ue Gamma prior ditribution) a a reference plan. 2.1 The Reference Plan The reference plan conidered in developing Bayeian kip-lot ampling plan, namely, BSkSP-3, i Bayeian Modified chain ampling plan BMChSP-1. The OC function Pa(p) of the MChSp-1 plan wa derived by Govindaraju and Lai (1998) a i i1 P p) P ( P ip P ) 2.2 Operating procedure of MChSP-1 Plan a( o, n 0, n o, n 1, n 1. Draw a random ample of ize n from each ubmitted lot. If one or more nonconforming unit are found in the ample, reject the lot. 2. If no nonconforming unit are found in the ample, accept the lot, provided that the preceding i ample alo contained no nonconforming unit except in one ample, which may contain at mot one nonconforming unit. reject the lot, otherwie. np 1 t With gamma prior ditribution, w( P) e p, t, p > 0 with parameter, t. The average probability of acceptance of BMChSP-1 i given by Latha M. and Rajewari M. (2013). The average probability of acceptance i given a p Pa ( p) w( p) dp where / t the mean value of the product quality p. 3. Bayeian Skip-Lot Sampling Plan BSkSP-3 0 ( n(1 i)) ni ( n(1 i)) (2.1) 1 = 1 In thi ection, deigning of BSkSP-3 (the SkSP-3 plan with Bayeian Modified Chain Sampling BMchSP-1 plan a a reference plan) i provided. Table are contructed uing which one can elect BSkSP-3 for given parametric value. Vijayaraghavan (1990) ha developed kip-lot plan SkSP-3, imilar to SkSP-2 of Perry (1973) which ue certain baic procedure employed in the continuou ampling plan, CSP-2 of Dodge and Torrey (1951). The operating procedure of Skip lot ampling plan SkSP-3 i given a follow: 1. Uing the reference plan, tart with normal inpection RS Publication, rpublicationhoue@gmail.com Page 188

3 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN Switch to kipping inpection (inpect a fraction f of the lot), of i conecutive lot are accepted on normal inpection. 3. When a lot i rejected on kipping inpection, inpect next k lot produced. 4. Switch to normal inpection, when a lot i rejected while inpecting k lot. 5. Proceed to tep 2, When all k lot are accepted. 6. Screening of each rejected lot i carried out and correct or replace all the nonconforming unit found. Vijayaraghavan (1990) in a imilar approach ued by Brugger (1972) ha derived a formula for the OC function of a Skip-lot ampling plan SkSP-3 and i given by i i fp (1 f ) P (2 P ) Pa ( f, i) i i f (1 f ) P (2 P ) (2.2) where P i the OC function of the reference ampling plan. Here BMChSP-1 i conidered a the reference ampling plan. 3.1 Selection of Plan The election procedure of BSkSP-3 plan with BMChSP-1 a reference plan will now be indicated, under the condition for application of Poion model for OC curve, with Gamma Prior ditribution. Table 2 ha been contructed for variou parametric value indexed by, i and µ which can be ued for the election of uch plan. 3.2 The Average Probability of Acceptance of BSkSP-3 with BMChSP-1 a a reference plan The Probability of acceptance of Bayeian Skip lot Plan 3 i given by ubtituting the P of equation 2.1 in place of P of equation 2.2 will get the new P of BSkSP-3 a follow: P a f, i f ( n(1 i)) 1 ni ( n(1 i)) 1 f (1 f ) ( n(1 i)) (1 f ) ( n(1 i)) 1 ni ( n(1 i)) 1 1 i ni 2 1 ( n(1 i)) ( n(1 i)) i 1 ni 2 ( n(1 i)) ( n(1 i)) 1 1 ni ( n(1 i)) i 1 i where / t the mean value of the product quality p. (2.3) 3.3 Contruction of Table The Indifference Quality Level (IQL) or point of control µ 0 can be calculated by equating the above equation (2.3) to 0.50 for value of = 1, 2, f = 2/3, 1/3, 1/2 & 1/5, alo for i = 4, 6, 8, 10, 12 uing Newton method of approximation and thoe value are preented in Table 1. Similarly, we can find µ 1 and µ 2 after equating the ame equation (2.3) to 0.95 and It value are alo given in Table 1 and it parametric value are given in Table 2. Illutration RS Publication, rpublicationhoue@gmail.com Page 189

4 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN For = 1, f = 1/5, i = 4 and of P = 0.95, the value of nµ 1 = For =1 and different value of i and f (f = 2/3,1/2 & 1/3 and i = 6,8,10,12) uing Newton Method of Approximation and the nµ 1, nµ 2 & nµ 0 value are tabulated in table 1. Illutration 2 For = 2, f = 2/3, i = 4 and of P = 0.95, the value of nµ 1 = For = 2 and different value of i and f (f = 1/2, 1/3 & 1/5 and i = 6,8,10,12) uing Newton Method of Approximation and the nµ 1, nµ 2 & nµ 0 value are tabulated in table Concluion Controlling the quality of a product i neceary in the indutry. The work preented in thi paper mainly related to the performance meaure of Bayeian Skip Lot Plan BSkSP-3 with gamma ditribution a the prior ditribution. The table hown in the work are helpful and ueful for the quality control practitioner. Reference [1] A. Hald (1981), " Statitical theory of ampling inpection by Attribute, Academic Pre, Inc.(London) Ltd. [2] Brugger, R.M., (1975) A Simplification of Skip-Lot Procedure Formulation, Journal of Quality Technology, 7 (4), pp: [3] Calvin, T.W. (1984): How and When to perform Bayeian Acceptance Sampling. Vol.7, American Society for Quality Control, Milwaukee, Wiconin. [4] Dodge, H.F. (1955), Chain Sampling Inpection Plan, Indutrial Quality Control, Vol - 11, No.4, pp [5] M. Latha and M. Rajewari (2013), "Aymptotic property for Bayeian Modified Chain Sampling Plan". International Journal of Recent Scientific Reearch Vol.4, Iue,6, pp: , July [6] M. Latha and M. Rajewari (2013): Aymptotic property for Bayeian Chain Sampling Plan-1 (BChSP-1), International Journal of Advanced Scientific and Technical Reearch.Vol. 4, Iue 3, pp [7] Parker RD, Keler L (1981), A modified kip-lot ampling plan. Journal of Quality Technology 13(1): [8] Perry, R.L A Sytem of Skip-lot Sampling Plan for lot inpection, Ph.D. Thei, Rutger, The State Univerity, New Brunwick, New Jerey. [9] Soundararajan V (1978), Procedure and table for contruction and election of Chain Sampling plan (ChSP-1) part-1, Journal of Quality and Technology 10(2), pp: (56-60). [10] Vijayaraghavan (2000), Deign and evaluation of kip-lot ampling plan of type SkSP-3 Journal of Applied Statitic, Vol.27, iue 7, pp: RS Publication, rpublicationhoue@gmail.com Page 190

5 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN [11] Vijayaraghavan, Rajagopal and Loganathan (2008), A procedure for election of a gamma - poion ingle ampling plan by attribute. Journal of Applied Statitic, Vol. 35, iue 2, pp: Table 1 - µ value for Bayeian Skip Lot ampling plan (BSkSP-3) for given average probability of acceptance =1 f i nµ 1 nµ 2 nµ 0 2/ / / / = 2 f i nµ 1 nµ 2 nµ 0 2/ / / / RS Publication, rpublicationhoue@gmail.com Page 191

6 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN = 3 f i nµ 1 nµ 2 nµ 0 2/ / / / = 4 f i nµ 1 nµ 2 nµ 0 2/ / / / RS Publication, rpublicationhoue@gmail.com Page 192

7 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN Table 2 - Parametric Value of Bayeian Skip Lot Sampling Plan 3 (BSkSP-3) =1 f i nµ 1 nµ 2 nµ 0 µ 2 /µ 1 µ 0 /µ 1 2/ / / / = 2 f i nµ 1 nµ 2 nµ 0 µ 2 /µ 1 µ 0 /µ 1 2/ / / / RS Publication, rpublicationhoue@gmail.com Page 193

8 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN = 3 f i nµ 1 nµ 2 nµ 0 µ 2 /µ 1 µ 0 /µ 1 2/ / / / = 4 f i nµ 1 nµ 2 nµ 0 µ 2 /µ 1 µ 0 /µ 1 2/ / / / RS Publication, rpublicationhoue@gmail.com Page 194

9 International Journal of Advanced Scientific and Technical Reearch Iue 7 volume 4 July-Aug 2017 Available online on ISSN RS Publication, rpublicationhoue@gmail.com Page 195