Designing of Combined Continuous Lot By Lot Acceptance Sampling Plan

Transcription

1 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Desgnng o Combned Contnuous Lot By Lot Acceptance Samplng Plan S. Subhalakshm 1 Dr. S. Muthulakshm 2 1 Research Scholar, 2 Proessor Department o Mathematcs, Avnashlngam Insttute or Home Scence and Hgher Educaton or Women Combatore-64 Abstract Ths paper analyses the combned contnuous lot by lot acceptance samplng plan whch combnes the eatures o contnuous samplng plan and lot by lot samplng plan. The operatng procedure o the combned plan and the dervaton o plan perormance measures usng the Markov-chan approach are gven. Desgnng o the combned contnuous lot by lot acceptance samplng plan ndexed by (, AQL) and (, LQL) are presented. and are provded to the desgned plans or derent choces o the parameters. Keywords: Combned Contnuous lot by lot acceptance samplng plan, Average Outgong Qualty Lmt (), Acceptable Qualty Level (AQL), Lmtng Qualty Level (LQL), Average Fracton Inspected (). I. INTRODUCTION Contnuous samplng plans are most suted to processes where lot ormaton s dcult. However, ther applcaton s not lmted and may also be appled to contnuous producton processes nvolvng lot ormaton. Combnng the best eatures o contnuous samplng plan and lot by lot nspecton plan s desrable n modern producton processes where the racton o producng nonconormtes s n the range o parts per mllon (ppm), snce the contnuous samplng plan or low levels o racton nonconormng requres ether a large clearance nterval or large racton to be sampled. Pesotchnsky [1] proposed a scheme or low racton nonconormng whch combned the strateges o contnuous samplng plan CSP-1 o Dodge [2] and the lot by lot nspecton plan. Bebbngton and Govndaraju [3] provded exact mathematcal results and new tables o perormance measures or Pesotchnsky scheme n order to overcome the complex and dcult procedure o mplementaton n shop loor. Usng the Markovchan approach o Stephens [4], Govndaraju and Bebbngton [5] proposed a smpled scheme and provded perormance measures. In ths paper, usng Roberts [6] Markov-chan approach varous perormance measures or the combned plan s derved and the method o desgnng the plan s gven. The plans ndexed by AQL wth producer rsk o 0.95 and, LQL wth consumer rsk o 0. and are desgned. The prmary ndex and are evaluated or the desgned plan wth varous choces o parameters. The llustraton o the mplementaton o the plan s gven. The operatng procedure o combned contnuous lot by lot acceptance samplng plan has the ollowng steps Step 1: Inspect % o the unts consecutvely untl unts n successon are ound to be deect ree. Step 2: When unts n successon are ound conormng dscontnue cent percent nspecton and start ormng lots o desred sze. Apply any lot by lot samplng nspecton plan as reerence plan. Step 3: I a lot s rejected, go to Step 1; otherwse, contnue wth lot by lot nspecton. In the lot by lot samplng nspecton consder (=ASN/N) as the racton o samplng where ASN s the average sample number o the lot by lot samplng plan consdered and N s the lot sze. Ths combned contnuous lot by lot acceptance samplng plan s ndexed by the parameter and the parameters o the reerence samplng plan (N, n, c). A low dagram s gven or the proposed plan n Fg.1.

2 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Start Inspect all the unts n the order o producton Are consecutve unts ound conormng? No Replace all the nonconormtes wth the standard ones Yes Contnue wth lot by lot nspecton usng the reerence plan No Is a lot rejected? Yes Fg.1 Flow Dagram o the Operatng procedure o combned contnuous lot by lot plan II. DERIVATION OF PERFORMANCE MEASURES Varous perormance measures o combned contnuous lot by lot acceptance samplng plan are derved usng Markov-chan approach due to Roberts (65). Let [X n ], n=1, 2, denote the dscrete parameter Markov-chan wth nte state space (S k ), k=1, 2,,(+3). The states o the process are dened as S k = A (k-1), k=1,2,3, +1 = percent nspecton s beng conducted and ncludng the latest artcle nspected and the last (k-1) consecutve artcles were ound to be conormng. S +2 = SA =Lot by lot samplng s n eect and the last lot submtted was nspected and accepted S +3 = SR = Lot by lot samplng s n eect and the last lot submtted was nspected and rejected These set o (+3) states dened above completely descrbes the mutually exclusve phases o nspecton or combned contnuous lot by lot acceptance samplng plan. The transtonal probablty matrx s presented n Table 1. The combned contnuous lot by lot acceptance samplng plan appled to an nspecton process whch s n statstcal control may be vewed as a dscrete parameter Markov-chan whch s nte, rreducble and aperodc. The vector o lmtng probabltes π can be determned usng the steady state equatons.

3 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Table 1 Transton Probablty Matrx O Combned Contnuous Lot By Lot Plan States at (t+1) st tral States at t th tral A 0 A 1 A 2 A -1 A SA SR A 0 p q A 1 p A 2 p A -1 p q 0 0 A P a 1-P a SA P a 1-P a SR p q Here p the process racton o nonconormng tems q 1-p, the process racton conormng P a Operatng characterstc uncton o the reerence plan The steady state probabltes π j satsy the ollowng condtons π j > 0 or j=1,2,, (+3) π j = k=1 πk p kj such that k=1 πk= 1 These condtons result n the ollowng equatons π 0 = p(π 0 +π 1 +π π -1 +π SR ); π 1 = q(π 0 +π SR ); π 2 = qπ 1 ; π 3 = qπ 2 = q(qπ 1 ) = q 2 π 1 ; π = qπ -1 = q -1 π 1 ; or = 2,3,4,, π SA = P a (π +π SA ), π SR = (1 P a )(π +π SA ) On smplyng we get π 0 = p(1 q )(1 P a )/D; π k = q k p(1 P a )/D, where k = 1,2,, π SA = pq P a /D; π SR = pq (1 P a )/D, where D = (1 P a )(1 q ) + pq () () () (v) (v) The average number o unts nspected under the screenng nspecton s u = (1 q )/pq The average number o lots passed under the samplng nspecton s v = 1/(1 P a ) The OC uncton o the complex contnuous lot by lot plan s P A = pq /D I s the racton o the lot sampled,then AOQ s AOQ = 1 p 2 q /D s the maxmum o AOQ uncton. The average racton nspected () n the samplng nspecton s = ((1 P a )(1 q ) + pq )/D Tables are constructed or, and values o the combned contnuous lot by lot acceptance samplng plan ndexed by certan selected values o AQL wth P A =0.95, LQL wth P A =0., sample sze n and sample racton =n/n usng the search procedure. Numercal values n the tables reveal the ollowng eatures

4 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Increase n AQL and the sample sze decrease the clearance number.. Increase n LQL and the sample sze decrease the clearance number.. Increase n clearance number decreases v. Increase n clearance number decreases () () III. SELECTION OF PLANS Suppose that one requres combned contnuous lot by lot acceptance samplng plan wth AQL o 0.0% wth probablty o acceptance 0.95, the sample racton o 5% and acceptance number c=2. From the constructed table one obtans the requred plan as(,, n, c) =(326, 1/20, 400,2). For ths plan s whch s the worst qualty receved by the consumer by the applcaton o the constructed plan. At ths worst qualty the amount o nspecton s 40.75%. Suppose that one requres combned contnuous lot by lot acceptance samplng plan wth LQL o 0.% wth probablty o acceptance 0., the sample racton o % and acceptance number c=1. From the constructed table one obtans the requred plan as (,, n, c) =(, 1/,,1). For ths plan s whch s the worst qualty receved by the consumer by the applcaton o the constructed plan. At ths worst qualty the amount o nspecton s 58.26%. REFERENCES [1]. Pesotchnsky, L (87) Plans or low racton nonconormng, Journal o qualty technology,, pp [2]. Dodge, H. F. () A Samplng plan or contnuous producton, Annals o Mathematcal Statstcs,, pp, [3]. Bebbnton, M and Govndaraju, K (98) On Pesotchnsky s Scheme or very low racton nonconormng, Journal o qualty technology, 30, pp [4]. Stephens, K. S. (95) How to perorm Contnuous Samplng, Vol. 2, Second edton, ASQC Basc Reerences n Qualty Control (Wnconsn, Amercan Socety or Qualty control). [5].K. Govndaraju and M. Bebbngton (0) Combned contnuous lot by lot acceptance samplng plan, Journal o appled scences, Vol. 27, No.6, pp [6]. Roberts, S.W. (65) States o Markov chans or Evaluatng Contnuous Samplng plans, Transactons o the 17 th Annual All Day Conerence on Qualty control, Metropoltan Secton, ASQC and Rutgers Unversty, New Brunswck, N.J., pp

5 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Table 2 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=1 n AQL n AQL Table 3 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=2 n AQL n LQL

6 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Table 4 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=3 n AQL n LQL Table 5 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=1 n LQL n LQL

7 Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN Table 6 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=2 n LQL n LQL Table 7 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=3 n LQL n LQL