Construction and Selection of Single Sampling Quick Switching Variables System for given Control Limits Involving Minimum Sum of Risks

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1 Construction nd Selection of Single Smpling Quick Switching Vribles System for given Control Limits Involving Minimum Sum of Risks Dr. D. SENHILKUMAR *1 R. GANESAN B. ESHA RAFFIE 1 Associte Professor, Deprtment of Sttistics, PSG College of Arts & Science, Coimbtore Reserch Scholr, Deprtment of Sttistics, PSG College of Arts & Science, Coimbtore Abstrct- A tble nd the construction procedure is given for finding the single smpling quick switching vribles system involving minimum sum of risks for specified, Acceptble Qulity Level nd Limiting Qulity Level. Key Words: Quick Switching System, Vribles Smpling, Acceptble Qulity Level (AQL), Limiting Qulity Level (LQL) nd Minimum Sum of Risks. I.Introduction Acceptnce smpling is scrutiny procedure pplied in sttisticl qulity control for mnufcturing the products nd is used to mesure rndom smples of popultion considertion. his contins lot of products tht re ginst predetermined qulity. In order to chieve uniqueness of mnufcturing products with high qulity level, mny methodologies hve been implemented. Among them, Quick Switching Smpling System is regrded s high efficient protocol for cceptnce smpling. Previously, it ws proposed by Dodge (1967) nd it ws further investigted by vrious uthors, Romboski (1969) nd Govindrju(1991). It ws lso exmined by ylor (1996) to identify method to evlute nd select Quick Switching Smpling System. As the bse to the bove rguments, in 1957 Golub Abrhm described the designing of single smpling inspection pln for fixed smple size, which ws noticed s the smpling pln involving minimum sum of risks. Govindrju nd Subrmni (1990) hve constructed tbles for selection of single smpling quick switching systems (QSS-1) for given Acceptble Qulity Level nd Limiting Qulity Level. hey lso provided the tble tht is used to select n MDS-1 pln for given AQL (p 1 ) nd LQL (p ) with minimum sum of risks. Plnivel (1999) hs constructed the tble on Quick Switching Vribles Single Smpling (QSVSS) Systems. Senthilkumr nd Sivkumrn (004) hve constructed the tble for selection of minimum risk single smpling quick switching system QSS-1(n; c, c 1 ) by ttributes. his pper provides procedure for finding the single smpling quick switching vribles system for which the sum of producer's nd consumer's risk re minimum for specified Acceptble Qulity Level nd Limiting Qulity Level. Quick Switching Vribles Smpling System of type QSVSS (n;k N,k ) he conditions nd the ssumptions under which the QSVSS cn be pplied re s follows: Conditions for pplictions Production is stedy so tht results on current, preceding nd succeeding lots re brodly indictive of continuing process. Lots re submitted substntilly in the order of production. Normlly lots re expected to be essentilly of the sme qulity. Inspection is by vribles, with qulity defined s the frction of non-conforming. Bsic Assumptions he qulity chrcteristic is represented by rndom vrible X mesurble on continuous scle. Distribution of X is norml with men nd stndrd devition. An upper limit U, hs been specified nd product is qulified s defective when X>U. [when the lower limit L is specified, the product is defective one if X<L. he Purpose of inspection is to control the frction defective, p in the lot inspected. When the conditions listed bove re stisfied the frction defective in lot will be defined by p = 1-F (v) = F (-v) with v = (U-) / nd y 1 z F( y) exp( ) dz (1) provided tht the qulity chrcteristic of interest is normlly distributed with men µ nd stndrd devition σ, nd the unit is clssified s non-conforming if it exceeds the upper specifiction limit U. he operting procedure of QSVSS (n; k N, k ) is described below. 1789

2 Operting Procedure he steps involved re s follows Step 1. Drw rndom smple of size n σ. Inspect nd record the mesurement of the qulity chrcteristic for ech unit of the smple. Compute the smple men x. Step. i) If x kn U or x kn L, ccept the lot nd repet step 1 for the next lot. ii) If x kn U or x kn L, reject the lot nd go to step 3. Step 3. Drw rndom smple of size n σ. Inspect nd record the mesurement of the qulity chrcteristic for ech unit of the smple. Compute the smple men x. x k U or x k L, ccept the Step 4. i) If lot nd repet step 1 for the next lot. ii) If x k U or lot nd repet step 3. x k L, reject the Where k N nd k re the cceptble criterion for norml nd tightened inspection, with x nd σ s the verge qulity chrcteristic nd stndrd devition respectively. IV. Operting Chrcteristic Function According to Romboski (1969), the OC function of QSS-1 is given by P P (p) 1 P P () N Bsed on the OC function of the QSS Romboski (1969) the OC function of QSVSS (n;k N,k ) cn be written s P (p) with w w N (w ) 1 (w ) (w ) (3) n n σ σ v (U μ)/σ N (U k (U k μ)/σ (v k μ)/σ (v k Under the ssumption of norml pproximtion to the non-centrl t distribution (Abrmowitz nd Stegun, 1964), the vlues of P N nd P re respectively given by P F(w ) pr[(u x)/ k (4) N N P F(w ) pr[(u x)/ k (5) ) ) n n σ σ Where, P N nd P re the proportion of lots expected to be ccepted using norml (n σ, k N ) nd tightened (n σ, k ) vrible single smpling plns respectively. hese two equtions re pplied in the OC function of QSVSS (n;k N,k ). We get the following P (p) Pr[(U x)/σ k 1 Pr[(U x)/σ k Pr[(U x)/σ k If P (p 1 ) nd P (p ) re Pr[(U x)/σ k P (p ) 1 α 1 1 Pr[(U x)/σ k Pr[(U x)/σ k Pr[(U x)/σ k P (p ) β 1 Pr[(U x)/σ k Pr[(U x)/σ k (6) (7) (8) he expression for the sum of producer s nd consumer s risks is given by α β 1 P (p 1 ) P (p ) (9) For given AQL nd LQL, the prmetric vlues of QSVSS nmely k N, k, the smple size n σ, α nd β re determined by using computer serch. A procedure for QSVSS (n ;k N,k ) System involving Minimum sum of risks From tble1, procedure nd the designing of Quick Switching Vribles Smpling Systems involving minimum sum of risks for the given vlues of AQL nd LQL is indicted below. ble1 is used to select the QSVSS for given vlues of (AQL, 1-α), (LQL, β). he plns given here hve the minimum sum of risks. Fix the vlues of p 1 nd p from which Quick Switching Vribles Smpling System cn be selected under known -method. Entering the row giving p 1 nd p, one gets the cceptnce criteri k, k N, α, β nd the smple size n of QSVSS (n ;k N,k ). For exmple, for given p 1 =0.005, p =0.009, n =139, k =.79, k N =.9, α=6% nd β=0%. Plotting the OC Curve he OC curve for the quick switching smpling system by vribles with n=44, k =3.0, k N =.99, α=8% nd β=1% nd Figure 1 shows the OC curve of quick switching smpling system by vribles involving minimum sum of risks. 1790

3 Probbility of Acceptnce P (p) ISSN: ightened OC Curve Composite OC Curve Norml OC Curve Frction Defective (p) Figure 1. Norml, Composite nd ightened OC Curves with minimum sum of risks for QSVSS, (n=44, k =3.0, k N =.99, α=8% nd β=1%) QSVSS with unknown vribles pln s the reference pln If the popultion stndrd devition σ is unknown, then it is estimted from the smple stndrd devition S (n-1 s the divisor). If the smple size of the unknown sigm vribles system (s method) is n s nd the cceptnce prmeters re k N nd k, then the operting procedure is s follows: Step 1. Drw rndom smple of size n σ. Inspect nd record the mesurement of the qulity chrcteristic for ech unit of the smple. Compute the smple men x nd ( x x) S. n 1 Step. i) If x kns U or x kns L, ccept the lot nd repet Step 1 for the next lot. ii) If x kns U or x kns L, reject the lot nd go to step 3. Step 3. Drw rndom smple of size n σ. Inspect nd record the mesurement of the qulity chrcteristic for ech unit of the smple. Compute the smple men x nd ( x x) S n 1 Step 4. i) If x ks U or x ks L ccept the lot nd repet Step 1 for the next lot. ii) If x ks U or x ks L reject the lot nd repet step 3. Here X nd S re the verge nd the stndrd devition of qulity chrcteristic respectively from the smple. Under the ssumptions for Quick Switching System stted, the probbility of cceptnce P (p) of lot is given in the eqution () nd P nd P N respectively re 1791

4 w 1 z/ P e dz nd π wn 1 z/ P e dz with N π Uks μ 1 w N. S 1 k N n n s s Uks μ 1 w. S 1 k n n s s Designing QSVSS (n S ;k N,k ) System with unknown involving minimum sum of risks ble1 cn be used to determine QSVSS (n s ;k,k N ) for specified vlues of p 1 nd p. For exmple, if it is desired to hve QSVSS (n s ;k s,k Ns ) for given p 1 =0.005, p =0.006,ble1 gives n s =795, k s =.8, k Ns =.3 α=7% nd β=% s desired pln prmeters. Construction of the ble he expression for the probbility of cceptnce P (p) nd minimum sum of risks of QSVSS (n, k, k N ) under norml distribution re given by eqution () nd (9), respectively with the following expressions: P P (p) 1 P P nd N α β 1 P (p1) P (p Where w 1 z/ P e dz π nd wn 1 z/ P e dz N π Now, for given p 1, p, α nd β, i.e. the two points (p 1, 1- α ) nd (p, β) on the operting chrcteristic curve, one my write the following expressions: Pr[(U x)/σ k P (p ) 1 α 1 1 Pr[(U x)/σ k Pr[(U x)/σ k Pr[(U x)/σ k P (p ) β 1 Pr[(U x)/σ k Pr[(U x)/σ k ) nd Here, the vlues of n, k, k N, α nd β re obtined by using computer serch routine. he vlues of n, k, k N for the QSVSS stisfies the eqution () s well s given α nd β for minimizing the sum of risks in eqution (9). By using Hmker (1979) pproximtion, for finding prmeters of s- method scheme from σ method scheme with prmeters (n s ;k, k N ) were s follows: ns n σ(1k σ/), where k (k k )/ σ k k (4n 4)/(4n 5) nd s s s k k (4n 4)/(4n 5) Ns s s ble1 provides the vlues of n, k, k N, n s, k s nd k Ns which stisfies the equtions () nd (9). Conclusion he Quick Switching Vribles Smpling pln presented in this pper is prticulrly useful for testing the qulity of finished products t shop floor situtions. Such tht the Producer nd the consumer represents the sme prty. So, the sum of these two risks should be minimized wheres t tht sitution this pln is commendble. Reference [1. M,Abrmowitz, nd I.A.Stegun, Hndbook of Mthemticl Functions. Ntionl Bureu of Stndrds, Applied Mthemticl Series No.55, (1964). [. H.F.Dodge, A new Dul System of Acceptnce Smpling echnicl Report No. 16, he Sttistics Center, Rutgers-he Stte University, New Brunswick, NJ, 1967). [3. K. Govindrju, Procedures nd bles for the selection of Zero Acceptnce Number Quick Switching System for Complince esting. Communictions in Sttistics- Simultion nd computtion, vol.0, No.1 pp , (1991). [4. K.Govindrju, nd K.Subrmni,. ( Selection of single smpling ttributes pln for given cceptnce qulity level nd limiting qulity level involving minimum risks, Communiction is Sttistics- Simultion nd Computtion, 19, pp , 1990 [5. K.Govindrju, nd K.Subrmni, Selection of Multiple deferred stte MDS-1 smpling pln for given cceptble qulity level nd limiting qulity level involving minimum risks. Journl of Applied Sttistics, 17, pp , (1990b). [6. K.Govindrju, nd K.Subrmni, Selection of single smpling quick switching system for given cceptble qulity level nd limiting qulity level involving minimum risks, Interntionl Journl of Qulity nd Relibility Mngement, 18, pp.45-5,

5 [7. K.Govindrju, nd K.Subrmni, Selection of tightened-norml-tightened system for given vlues of the cceptble qulity level nd limiting qulity level. Journl of Applied Sttistics Volume 19, Issue. Pges 41-50,199. [8. H.C.Hmker Attribute System in Opertion. Bulletin of the Interntionl Sttisticl Institute, vol.37, No., pp , [9. M.Plnivel, Contributions to the Study of Designing of Quick Switching Vrible System nd Other Plns, PhD thesis, Bhrthir University, 1999). [10. L.D.Romboski, An Investigtion of Quick Switching Acceptnce Smpling Systems, Ph.D. hesis, Rutgers he Stte University, New Brunswick, New Jersey, [11. D.Senthilkumr, nd Sivkumrn, P. K Selection of Single Smpling Quick Switching System involving the Minimum Sum of Risks, Ntionl seminr on Recent Advnces in Sttisticl Methodologies nd Applictions, Deprtment of Sttistics, Bhrthir University, Coimbtore, mil Ndu, Indi, 004. [1. W.A.ylor, Guide to Acceptnce Smpling Plns. ylor Enterprises, Lke Vill, IL, 199. ble

6 he vlues of n σ, k, k, n s, k s, k Ns, α, β for given AQL nd LQL Involving Minimum Sum of Risks. p 1 p n σ k k α β n s k s k Ns ble 1 (Continued...) 1794

7 p 1 p n σ k k α β n s k s k Ns

8 ble 1 (Continued...) p 1 p n σ k k α β n s k s k Ns

9 ble 1 (Continued...) p 1 p n σ k k α β n s k s k Ns

10 ble 1 (Continued...) p 1 p n σ k k α β n s k s k Ns

11 ble 1 (Continued...) p 1 p n σ k k α β n s k s k Ns

12 ble 1 (Continued...) p 1 p n σ k k α β n s k s k Ns