Quick Switching Conditional RGS Plan-3 Indexed through Outgoing Quality Levels

Transcription

1 Mathematics and Statistics 2(): DOI: 0.389/ms Qick Switching Conditional RGS Plan-3 Indeed thogh Otgoing Qality Levels V. Kaviyaas Depatment of Statistics, Bhaathia Univesity, Coimbatoe, Tamil Nad, India *Coesponding Copyight 204 Hoizon Reseach Pblishing All ights eseved. Abstact This pape ties to stdy the designing of new attibte sampling plan towads Qick Switching Conditional Repetitive Gop Sampling System (QSCRGSS)-3 indeed thogh Aveage Otgoing Qality (AOQ), Aveage Otgoing Qality Limit (AOQL) and its Opeating Ratio (OR). Tables ae povided with nmeical illstations fo newly developed plan fo its vaios plan paametes. Keywod Qick Switching System, Conditional Repetitive Gop Sampling Plan, AOQ, AOQL and Opeating Ratio. Intodction Statistical Qality Contol is widely sed in indsty to ense cstome satisfaction de to mass podction of podcts and sevices. Acceptance sampling was widely sed to test the qality of the podct towads edction of vaiability in pocess and podct whee vaiation is meased by statistical methods. An impotant field of statistical qality contol is acceptance sampling which is eithe to accept o eject podcts of a lot nde sample inspection when 00% inspection is not possible. The pimay objective of sampling inspection is to edce the cost of inspection while at the same time assing the cstome to an adeqate level of qality fo the items being inspected nde identical conditions fo both the podce and consme. This pape pesents a new epetitive sampling plan fo attibtes which has consideed by base and efeence sampling plan. The Qick Switching System (QSS) as base plan and Conditional Repetitive Gop Sampling Plan as efeence plan to popose a new sampling plan called as Qick Switching Conditional Repetitive Gop Sampling System (QSCRGSS)-3. Sampling plan is widely sed in govenment secto and indsty fo contolling the qality of shipment of components, spplies and final podcts (e.g. life of electic blbs, tensil stength and hadness of steel casting etc.,). Simila sitations can be thoght ove the time whee the qality of the podct impoves with decease in the vale of inspection cost. 2. Qick Switching System Dodge (967) poposed a new sampling inspection plan involving nomal and tightened inspection which is sally efeed as two-plan system. Romboski (969) stdies a Qick Switching System, switching to tightened inspection when the ejection comes nde nomal inspection. De to instantaneos switching between nomal and tightened plans, this system is efeed as Qick Switching System (QSS). Romboski (969) has fthe stdied the meits and demeits of switching les of QSS when it is compaed with two-plan system (m, d). The le of QSS is etained at m= whee as tightened le is made when d>.two natal choice fo tightened le fo d=2, d=3 etc., which ae designed as QSS-2, QSS-3 and QSS-d. Schilling (98) has mentioned the switching les employed in the system ae simple and the condition fo application has evolved. 3. Conditional Repetitive Gop Sampling (CRGS) Plan The concept of Repetitive Gop Sampling (RGS) plan was intodced by Sheman (965) in which acceptance o ejection of a lot is based on the epeated sample eslts on the same lot. Sondaaajan and Ramasamy (986) has tablated the vales fo selection of RGS plan indeed thogh (AQL, AOQL), (p 0, h 0 ) and (p *, h * ). Gai shanka and Mohapata (993) has developed a new Repetitive Gop Sampling plan designated as Conditional Repetitive Gop Sampling plan in which disposal of lot on the basis of epeated sample eslts is dependent on the otcome of the inspection of the immediate peceding i lots. Fthe they deived the fomlae fo OC and ASN fnction. Devaj Amainayagam (99) has stdied QSS- with RGS plan as efeence plan. Fthe Sesh and Kaviyaas (2008) have stdied QSS- and QSS-2 with Conditional RGS Plan as Refeence plan indeed with Acceptable

2 Mathematics and Statistics 2(): 32-38, Qality Level (AQL), Limiting Qality Level (LQL), Indiffeence Qality Level (IQL) and AOQL. This pape povides a new pocede fo selection of QSS-3 with Conditional RGS plan indeed thogh otgoing qality levels, which ae tailo made fo indstial shop floo applications. 4. Designation QSCRGSS-3 (n;, 2 ; v, v 2 ) efes to a Qick Switching System of type 3 whee the nomal CRGS plan has a sample size n and acceptance nmbe, 2 ( < 2 ) and the tightened CRGS plan has a sample size n and acceptance nmbe v, v 2 (v < v 2, v and v 2 2 ). The plan has si paametes namely n,, 2 ; v, v 2 and i. 5. Opeating Pocede The Qick Switching Conditional Repetitive Gop Sampling plan-3 is caied ot thogh the following steps: Step : Daw a andom sample of size n and test each nit fo its confomation fo the specified eqiements. Step 2: Unde nomal inspection, inspect the plan nde Conditional Repetitive Gop Sampling plan with the paametes n, and 2. If lot is accepted, contine step 2 othewise step 3. Step3: Unde tightened inspection, inspect the plan nde Conditional Repetitive Gop Sampling plan with the paametes n, v and v2. If a lot is accepted, Contine step 3 fo net 3 lots and goes to step 2, othewise step 3. Ths, the Qick Switching Conditional Repetitive Gop Sampling Systems (QSCRGSS)-3 ae chaacteized with si paametes namely, n,, 2, v, v 2, and i. Hee, it may be obseved that when = 2 = v = v 2, and i = 0 the eslting plan is edced to Repetitive Gop Sampling plan de to Sheman (965). Fthe, the Qick Switching Conditional Repetitive Gop Sampling plan-3 is applicable to a steam of lots and not fo isolated lots. Whee, And P T = P N = P ( X ) [ p ( X ) + P ( X )] [ p ( X v ) + P ( X v )] P ( X 2 2 v ) Glossay N = lot size n = sample size p = incoming qality of the sbmitted lot P a (p) = Pobability of Acceptance P N = pobability of acceptance nde nomal inspection P T = pobability of acceptance nde tightened inspection and 2 = acceptance nmbes nde nomal inspection v and v 2 = acceptance nmbes nde tightened inspection α = Consme isk and β = podce isk P 0.95, P 0.50 = the lot o pocess qality fo which the pobability of acceptance is 0.95 and 0.50 etc., fo given sampling plan i = cleaance inteval P L = Aveage Otgoing Qality Limit (AOQL) and P m = Qality at which AOQL occs. 7. Designing QSCRGSS-3 fo Diffeent Paametes 7.. Designing Systems fo given AQL and AOQL (2) (3) 6. Opeating Chaacteistics Fnction Based on Sheman (965), Romboski (969), Amainayagam (99) and Jayalakshmi (2009) the epession fo OC fnction of QSCRGSS-3 is given by, Pa (p) = 3 T 3 T PN P P + + PT ( PN )( + PT ) ( P )( + P ) N T () Fige. AOQ cve fo QSCRGSS-3 Plan

3 34 Qick Switching Conditional RGS Plan-3 Indeed thogh Otgoing Qality Levels Table 2 can be sed to design a QSCRGSS-3 indeed by AQL and AOQL. Fo eample, given AQL = 5.% (α= 0.05) and AOQL = 7.04% one can compte AOQL / AQL = 7.04 / 5. = Fom the table the vale of AOQL / AQL closest to the desied vale is This vale coesponds to =, 2 = 4, v =, v 2 = 3, i = 2 and np = Table. AOQL vales fo QSCRGSS-3 when i = 2 v v 2 np m Pa(np m) naoql AOQL/p fo np fo

4 Mathematics and Statistics 2(): 32-38, Table 2. AOQL vales fo QSCRGSS-3 when i = 2 2 v v 2 np m Pa(np m) naoql AOQL/p fo np fo Detemining the vale of AOQL of a given system Table can be sed to obtain the vales of AOQL and p m fo a given system. Fo eample, QSCRGSS-3(63; 3, 8; 3, 6) when i =, fom table, coesponding to these enty plan paametes one can get naoql = and np m = So AOQL = naoql / n = / 63 = and p m = np m / n = 3.06 / 63 = Convesion fom one set of paametes to the othe Table can be sed to convet the given set of paametes to anothe familia eqivalent set. Fo eample, fo given AQL = 0.05, α = 0.05, LQL = 0.09 and β = 0.0, fom the Table one can fi the QSCRGSS-3 plan as n = 55, = 3, 2 = 4, v = 2, v 2 = 3 when i =. Coesponding to these enty paametes, one can find the following vales fom tables. np =.808 np m = naoql =.898 np 0 = h 0 =.846 Fom these vales, one can find the eqivalent paametes of QSCRGSS-3 as follows:

5 36 Qick Switching Conditional RGS Plan-3 Indeed thogh Otgoing Qality Levels AQL (α = 0.05) = 0.05, AOQL = naoql / n =.898 / 55 = IQL = np 0 / n = / 55 = Ths, fo given (AQL, -α) and (LQL, β), othe eqivalent paametes wee (AQL, AOQL) and (p 0, h 0 ) ae (0.05, ) and (0.0473,.846) Designing Systems given p, α, p 2 and β Table 3 can be sed to design Qick Switching Conditional Repetitive Gop Sampling System (QSCRGSS)-3, when two points on the OC cve (p, - α) and (p 2, β) ae given. To design a QSCRGSS-3 the opeating atio (OR) = p 2 / p. Fom table 3, one can detemine the vale of OR which is neae to the desied atio. Coesponding to the selected OR vales of, 2, v, v 2 and np when i =. The sample size is detemined by dividing np by p. Fo eample, let p = 0.05, α = 0.05, p 2 = and β = 0.0, calclate the Opeating Ratio (OR) = p 2 / p = / 0.05 = 7.9. Fom the table 3 the vale of OR fo α = 0.05 and β = 0.0 which is the neaest to the desied atio is Coesponding to this selected OR vales ae =, 2 = 2, v = 0, v 2 = when i = and np = The sample size is obtained as n = np / p = / 0.05 = 7.44 ~ 7. The desied system is QSCRGSS-3(7;, 2) fo nomal and (7; 0, ) fo tightened case when i =. 2 v v 2 Table 3. Opeating Ratio vales fo QSCRGSS-3 (n,, 2, v, v 2) when i= β=0.0 p 2/p fo α = 0.0 p 2/p fo α = 0.05 β=0.0 np fo β=0.0 β=0.0 np fo

6 Mathematics and Statistics 2(): 32-38, v v 2 Table 4. Opeating Ratio vales fo QSCRGSS-3 (n,, 2, v, v 2) when i=2 β =0.0 p 2/p fo α = 0.0 p 2/p fo α = 0.05 β=0.0 np fo β=0.0 β=0.0 np fo Opeating Ratio Designing Systems fo given Opeating Ratio Table 3 and 4 can be sed to design QSRGSS-3, when two points on the OC cve (p, - α) and (p 2, β) ae given. To design a QSRGSS-3 calclate the Opeating Ratio (OR) = p 2 / p. Fom table 3 and 4, one can detemine the vale of OR which is neae to the desied atio, coesponding to the selected OR vales of, 2, v, v 2, i and np. The sample size is detemined by dividing np by p. Fo eample, let p = 0.02, α = 0.05, p 2 = and β = 0.0, calclate the Opeating Ratio (OR) = p 2 / p = / 0.02 = 4.3. Fom table 2.5., the vale of OR fo α = 0.05 and β = 0.05 which is the neaest to the desied atio is Coesponding to this selected OR the paametes ae =, 2 = 3, v =, v 2 = 2 and np =.9. The sample size is obtained as n = np / p =.9 / 0.02 = The desied system is QSRGSS-3(56;, 3;, 2) Constction of Tables The epession fo pobability of acceptance of Qick Switching Conditional Repetitive Gop Sampling System (QSCRGSS)-3, nde the assmption of Poisson model, the

7 38 Qick Switching Conditional RGS Plan-3 Indeed thogh Otgoing Qality Levels composite OC fnction is given by eqation () with P N = e ( np) P T = = o + 2 e = 0 = 0 = 0 ( np) /! e ( np) /! e ( np) v = o ( np) /! v v v + 2 e = 0 = 0 = 0 e ( np) /! e ( np) /! e ( np) /! i /! (4) i /! (5) Fo vaios assmed vales of, 2, v, v 2, i and P a (p) the eqation () is solved with the eqation (4) and (5) one can get the np sing iteation techniqes fom c++ compte pogam and vales ae obtained. Utilizing the np vales tablated fo diffeent vales of, 2, v, v 2 and i. Fom table and 2 the vaios otgoing qality levels and Opeating Ratio vales ae calclated fo diffeent α and β vales ae given in table 3 and 4. Assming naoq = np * Pa(p), vale of np which maimizes naoq was obtained by the method of sccessive appoimation and these vales (np m ) togethe with naoql( = np m *Pa(p m )) appea in table to Conclsion Acceptance sampling is a techniqe which gives a bette soltion to the poblems faced by the indsty. Acceptance sampling itself does not impove qality, bt wheneve the lot is ejected it indicates the instability of the podction pocess. Acceptance sampling is a cost efficient one and only an admissible method which is sed fo destctive and costly tests which povides qick eslts. Sampling plans ae necessay to povide the disposal of defective podcts made, while effots ae activated to contol the pocess. In that Qick Switching System and Conditional RGS Plan have wide potential applicability in indsties to ense a high standad of qality attainment and inceased cstome satisfaction. Hee, an attempt is made towads the concept of Qick Switching Conditional Repetitive Gop Sampling System (QSCRGSS)-3 in which disposal of a lot is on the basis of nomal and tightened plans. Poisson nity vales have been tablated fo a wide ange of plan paametes. The pesent development wold be valable addition in the liteate and a sefl device to the qality pactitiones. The concept of this aticle may be of assistance to qality contol enginees and plan designes in the development of fthe plans. REFERENCES [] AMERICAN NATIONAL STANDARDS, Institte/Ameican Society fo Qality Contol (ANSI/ASQC) STANDARD A2 (987): Tems, symbols and Definition fo Acceptance Sampling, Ameican Society fo Qality Contol, Milwakee, Wisconsin. USA. [2] ARUMAINAYAGAM, S. D. (99): Contibtion to the Stdy of Qick Switching System and its Application, Ph.D., Thesis, Depatment of Statistics, Bhaathia Univesity Coimbatoe, Tamilnad, India. [3] DODGE, H. F. (967): Notes on the Evoltion of Acceptance Sampling Plans, Jonal of Qality Technology. Vol., No.2, pp [4] GAURI SANKAR and MOHAPATRA, B. N. (993): GERT Analysis of Conditional Repetitive Gop Sampling Plan, Intenational Jonal of Qality & Reliability Management, Vol.0, No.2, pp [5] GAURI SANKAR and JOSEPH, S. (994): GERT Analysis of Mltiple Repetitive Gop Sampling Plans, IAPQR Tansactions, Vol.9, No.2, pp [6] JAYALAKSHMI, S. (2009): Contibtion to the Selection of Qick Switching System and elated sampling plans, Ph.D Thesis, Depatment of Statistics, Bhaathia Univesity, Coimbatoe, Tamil Nad, India. [7] KAVIYARASU, V. and SURESH, K. K. (20): Cetain Reslts and Tables Relating to QSS- with Mltiple RGS Plan, Jonal of Mathematical Reseach, Canadian Cente of Science and Edcation, Canada Vol.3, No.4, pp.58-67, Nov 20. [8] ROMBOSKI, L. D. (969): An Investigation of Qick Switching Acceptance Sampling Systems, Ph.D Thesis, Rtges- The State Univesity, New Bnswick, New Jesey. [9] SCHILLING, E. G. (98): A modified geneal pocede fo sampling inspection, Fonties in Statistical Qality Contol, Edited Lenz, H.J., Wetheill, G.B., Wilich, P.Th. Wilich. [0] SHERMAN, R. E. (965): Design and Evalation of Repetitive Gop Sampling Plans, Technometics, Vol.7, No., pp.-2. [] SOUNDARARAJAN, V. and RAMASAMY, M. M. (986): Pocedes and Tables fo Constction and Selection of Repetitive Gop Sampling (RGS) Plan, the QR Jonal, Vol.3, No.3, pp.9-2. [2] SURESH, K. K. and KAVIYARASU, V. (2008): Cetain Reslts and Tables Relating to QSS- with Conditional RGS Plan, IAPQR Tansactions, India, Vol.33, No., pp