Designing of Skip-Lot Sampling Plan of Type (Sksp-3) For Life Tests Based On Percentiles of Exponentiated Rayleigh Distribution

Transcription

1 Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 Desgnng of Skp-Lot Samplng Plan of Type (Sksp-3) For Lfe Tests Based On Percentles of Exponentated Raylegh Dstrbuton Pradeepa Veerakumar K, Umamaheswar 2 P, Aruna H M 3, Suganya S 4 Emal: pradeepaveerakumar@buc.edu.n, uma_m2485@yahoo.com 2, aruna.rs@_buc.edu.n 3, suganstat@gmal.com 4 Abstract- Skp-lot samplng plans ads n reducng the nspecton cost. SkSP-3 plan s developed based on the prncples of CSP-2.In ths study, Skp lot samplng plan of type SkSP-3 wth Sngle Samplng plan as reference plan s developed for the lfe testng based on the percentles of Exponentated Raylegh Dstrbuton (ERD). Comparatve study on SkSP-3 wth SSP as reference plan s also made. The operatng characterstc values of the plan are tabulated and the curve s drawn. Numercal llustraton s provded to valdate the effcency of the developed plan. Keywords: Exponentated Raylegh Dstrbuton, Percentles, Lfe tests, Sngle Samplng Plan, SkSP -3.. INTRODUCTION The acceptance samplng plans procedure nvolves acceptng or rejectng a submtted lot based on the results of nspecton of sample nspected taken from the lot. Acceptance samplng plan requres mnmum sample sze to for testng. If the characterstc of the qualty s concernng the lfe tme of the product, testng s termed as lfe testng. Acceptance samplng plan based on lfe testng generally appled to decde whether to accept or reject the products where the lfe span of t s a vtal factor. Usually, when the results of the lfe testng shows that the mean lfe of the tested product exceeds pre-determned mean, the lot s accepted or else rejected. Lfe testng acceptance samplng plans are prmarly assumed lfe tme dstrbuton, truncated scheme and testng condtons. The drawback n the lfe testng plans based on the assumed lfe dstrbuton and testng condton s the determnaton of samples to be nspected. Truncated lfe test used n determnaton of smallest sample sze, thus reducng the testng tme and cost. Ths method nvolves the termnaton of lfe testng at a predetermned tme on the mean lfe of the tested products. The number of falures s observed, f t s above the specfed acceptance number c, the lot s rejected otherwse accepted. Lfe testng plans based on truncated tme provde better protecton to the consumer as the concluson to accept the lot s made only f specfed mean lfe can be reached wth a predetermned hgh probablty. The lfe testng plans based on tme truncated assumed under non-normal condtons are studed by many researchers for nstance, Epsten (954),, Goode and Kao (96),Gupta and Groll (96), Kantam et al. (2, 26), Baklz (23), Balakrshnan et al. (27), Aslam and Shahbaz (27), Aslam and Kantam (28), Rao etal.(28, 29a, 29b), Lo, Tsa and Wu (2), Srramachandran and Palanvel (24) and Pryah and Sudaman (25). Most of the researchers consder lfe testng based on the mean. The drawback of lfe testng plans based on mean s that t may not fulfll engneerng requrements on pre-determned strength or break stress. When applyng plans based on mean on nspectng t may result n acceptng bad qualty of the products. Ths results n ncreased decoratng of the products and wll lead to the reducton of the lfe tme of the products. Lfe testng plans based on medan s more effectve than the testng based on the lfe testng plans based on mean. So n ths study, lfe testng plans based on 5 th percentle (medan) s consdered. Many characterstcs of Exponentated Raylegh dstrbuton s smlar to that of gamma, Webull and exponenated exponental dstrbuton. The dstrbuton and densty functon of ERD are n close forms. As a result t s easly appled to the truncated plans. The cumulatve dstrbuton functon of ERD s gven by, F( t;, ) 2 2( t ) e, t /τ>, θ> () Where τ and θ are the scale and shape parameter respectvely. The probablty densty functon of ERD can be wrtten as, 223

2 Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 f ( t;, ) 2 2 2( t ) t 2( t ) e e. 2 (2) Pradeepa Veerakumar and Ponneeswar (26) proposed SSP for lfe testng based on the percentles of ERD. Later, Pradeepa Veerakumar and Ponneeswar (27) developed DSP for lfe testng based on the percentles of ERD. The major objectve of ths study s to mnmzng the sample sze at specfed qualty levels. To reduce the usage of large samples t s always preferred to the use acceptance samplng plans, whch safeguards consumer as well as producer from rsk. In order to meet the requrements of less sample sze wth protecton of consumer and producer, SkSP-3 wth SSP as reference plan for lfe tests based on percentles of ERD as reference plan s developed. 2. SKIP LOT SAMPLING PLAN Dodge (955) proposed skp-lot samplng plans based on the prncple of contnuous samplng plan of type CSP- for a seres of lots or consgnments of materal.ths plan s termed as SkSP- plan and t s applcable when there s a bulk materal or products manufactured n successve lots. Later Perry (973) ntroduced the concept of nspectng each lot accordng to the reference plan. Soundararajan and Vjayaraghavan (987) developed a new samplng system SkSP-2 based on the prncples of contnuous samplng plan of type-2 of Dodge & Terry (952) for the nspecton of bulk products when there s contnuous flow. SkSP-3 uses reference plan. Vjayaraghavan (2) developed SkSP-3 wth zero acceptance number SSP as reference plan usng Markov chan. The modus operand of the SkSP-3 plan s gven below:. At the outset, starts wth normally nspectng lot usng the reference plan. At ths level, the products are normally nspected accordng to the order of producton or n the order submtted for nspecton. 2. If consecutve lots are accepted durng normal nspecton dscontnue nspectng every lot and swtch to skppng nspecton. 3. Inspect only a fracton f of the submtted lots durng skppng nspecton level, untl a lot s rejected. 4. If a lot s rejected whle skppng nspecton, then nspect next k lots produced or submtted. 5. Swtch to the normal nspecton, f a lot s rejected whle nspectng k lots. 6. If all the k lots are accepted, contnue skppng nspecton. 7. Replace the non-conformng unts n the rejected lot wth conformng one. SkSP-3 s characterzed by three skppng parameters namely f, & k, the value of f les between t, &k are postve ntegers. It nvolves three phases of nspecton namely screenng phase, lmted samplng phase and unlmted samplng phase. The flow chart representng the modus operand of the SkSP-3 s gven below: Fgure. Flow chart for operatng procedure of SkSP-3 plan 3. Desgnng of Skp lot samplng plan of type SkSP-3 wth SSP for lfe test based on the percentle of ERD as reference plan Skp lot samplng plan of type 3(SkSP-3) has the provson to skp few lots when the qualty standards of the submtted products are good and thus reduces the nspecton cost. Ths plan makes use of basc attrbute lot-by-lot acceptance samplng plans to nspect the ndvdual lots, whch s desgnated as reference plan. SkSP-3 wth SSP as reference plan for lfe tests based on percentles of ERD s executed accordng to the procedure llustrated n.. Incorporatng the modus operand of SSP for lfe tests based on percentles of ERD to nspectng each lot. SSP for lfe tests based on the percentles of ERD s characterzed by sample sze n, acceptance number c and falure probablty p where p F t, ). ( The modus operand of SkSP-3 wth SSP as reference plan based on percentles of ERD are as follows: 224

3 Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 Step : A random sample of sze n s drawn and put Draw a random sample of sze and t. placed on test for tme Step 2: The number of defectves d are counted and comparson s made wth the acceptance number c.. If d c, then reject the lot.. If d c, then accept the lot. Step 3: If d c, s obtaned before the specfed tme t, termnate the test and reject the lot. 3. Formaton of the Samplng Plan The crucal factor n desgnng a samplng plan s selecton of plan parameters satsfyng the consumer and producer s requrement. Two pont OC curve approach s appled to satsfy the requrements ARL (p ) and LRL (p ) such that L (p ) =-α & L (p ) =β n lfe testng procedure s to termnate the test at a specfed tme t. The probablty of rejectng bad lot s P* and the maxmum number of defectves accepted s c.the acceptance sngle samplng plan for percentles based on truncated lfe test s derved to get mnmum sample sze n for the gven acceptance number c so that, L (p ) =β does not exceed -P*s appled n SkSP- 3.A lot s sad to be bad lot f the true A bad lot means that the true q th percentle tq s less than the predetermned percentle t q. Therefore, the probablty P* s defned as the confdence lmt of rejectng a bad lot.e. acceptng a good lot wth t q < t q s at least equal to P*. The modus operand of the proposed plan s as follows: Step : fx 2 and generate the value of for the specfed percentle ( th ) from equaton, 2 ln( q ) (3) Step 2: Randomly determne the value of, f, specfed P*and the acceptance number c. Step 3: choose the smallest sample sze n from Table of Pradeepa Veerakumar (26). The value of P can be calculated from SkSP-3 from the OC functon of SSP for lfe test based on percentles of ERD. Hence, the procedure Hence SkSP-2 wth SSP for lfe test based on the percentles of ERD as reference plan s usually specfed by SSP for lfe tests based on the percentles of ERD characterzed by clearng nterval and samplng frequency f. 3.2 Operatng characterstc functon OC functon s the most appled technques to measure the effcency of the samplng plan and from where the probablty of acceptance s derved. It gves the probablty that the lot can be accepted. The OC functon of SSP for lfe tests based on the percentles of ERD s as follows, L( p) c n p ( p) Where F t, ) ( n (4) p represents the falure probablty at tme t gven a determned q th percentle of lfetme t and p depends only on t t. The OC values are q q tabulated n Table 3 of Pradeepa Veerakumar (26). The OC gven by, functon of SkSP-3 for the lot qualty p are [fp ( f )P (2 P )] L(P) [f ( f )P (2 P )] Then, the Average Sample number s (5) ASN (p) ASN(R)F (6) Where, ASN(R) represents the Average Sample number of the reference plan, P represents the probablty of acceptance of the reference plan. Fracton of lots nspecton s gven below: F Illustraton : f f (2 P )P ( P) (7) Engneers experenced that the lfe tme of the electrc goods follows ERD. Skp lot samplng plan of type 2 wth SSP as reference plan based on percentle s appled for testng. The parameters for the lfe testng s as follows: θ=2, t 4hrs, t. 2hrs,c=2, α=.5and β=. then η= from the equaton and the rato s found to be t / t. 2. By applyng Table of Pradeepa Veerakumar (26) the mnmum sample sze accordng to the requrements s n 7 and the correspondng OC values L ( p) for the Sngle Samplng plan for the lfe tests based on percentles of ERD ( n,c,t / t. ) (7,2,.5 ) wth P * =.9 under 225

4 t / t. Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 ERD from Table 3 of Pradeepa Veerakumar (26).are,. L ( p) L(p) s the P value for SkSP-3 wth SSP for lfe tests based on the percentles of ERD as reference plan. For =4 and f=/3,k=4; the probablty of acceptance L(p) values of SkSP-3 wth SSP for lfe tests based on percentles of ERD are found from eqn. 5 as,. / t. t L ( p) From the llustratons, t s ndcated that the actual th percentle s almost equal to the requred th percentle ( t / t.) the producer s rsk s.. approxmately.934 (-.865). Also the producer s rsk s nearly equal to.5 or less and the actual producer rsk s large or nearly equal to.75 tmes of the requred percentle. The OC curve s provded for the llustraton as fg. Fgure :.OC Curve for =4, f=/3,k=4, P*=.9, d=d. and θ=2 The fgure 2 clearly says that the plan attans ARL when the actual lfe tme percentle s n close proxmty to.75 tmes greater than the specfed th percentle and attans LRL when the actual lfe tme.99 8 percentle s approxmately equal to the specfed lfe tme percentle. For the purpose of convenence OC values of the table are constructed and tabulated wth parameters =4, f=/3,k=4 and c=2 n Table 4. Comparson of SkSP-3 wth SSP for lfe tests based on percentles of ERD as reference plan over SSP for lfe test based on percentles of ERD It s usual practce whle comparng two plans comparng t by ts OC curve and ASN values. The performance measures for nstance ARL and LRL of the correspondng samplng plans are also used to compare the plans. Illustratons are provded for the comparson of Skp lot samplng plan wth SSP for lfe tests based on percentles of ERD as reference plan wth SSP for lfe tests based on percentles of ERD wth the followng llustraton. Illustraton 2: Consder that θ=2, t 4hrs, t. 2hrs, c=2, α =.5, β=. then,η= s calculated from the equaton 3.3 and the rato, t t 2. and from Table of /. Pradeepa Veerakumar (26) the mnmum sample sze sutable for the gven nformaton s found to be as n 7. And Table 3 of Pradeepa Veerakumar (26) gves ther respectve OC values. Wth ths plan as the reference plan the Skp lot samplng plan of type SkSP-3 s desgned and ther respectve OC values are also tabulated n table wth =4 ;k=4and f=/3. The ASN s also found usng the relaton gven n equaton 6. For the comparatve purpose the values representng the llustraton are tabulated n table 3. The table 2 says that the OC values of SkSP -3 wth SSP for lfe tests based on percentles of ERD s slghtly ncreased from SSP for lfe tests based on percentles of ERD. The curves representng comparson of OC values for the defned parameters obtaned from the table 2 are gven n fgure 3 respectvely. 226

5 Probably of acceptance Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 Table 2: OC values of SSP for lfe tests based on ERD percentles and SkSP-3 wth SSP for lfe tests based on ERD percentles as reference plan OC Values ASN Values SkSP-3 wth SSP SkSP-3 wth SSP t. / t. SSP ERD Percentles SSP ERD Percentles ERD Percentles ERD Percentles Fgure 3. OC Curve of SkSP -3 wth SSP for ERD percentles and SSP for ERD percentles 4. CONCLUSION In ths study Skp-lot Samplng plan of type-3 wth SSP as reference plan based on the percentles of ERD. SkSP-3 plan requres less ASN than the SSP results n reducton of nspecton cost. The OC value of SkSP-3 wth SSP as reference plan s lttle hgher than the SSP. REFERENCES SkSP-3 SSP d [] Balakrshnan. N, Leva. V and Lopez. J, (27). Acceptance samplng plans from truncated lfe tests based on the generalzed Brnbaum- Saunders dstrbuton, Communcatons n Statstcs: Smulaton and Computaton, 36, pp: [2] Dodge H F, Perry R L. (97). A system of skplot plans for lot-by-lot nspecton. Amercan Socety for Qualty Control Techncal Conference Transactons, Chcago, Illnos, pp [3] Dodge H F. (955). Skp-lot samplng plan, Industral Qualty Control, (5):3-5 [4] Dodge H. F., Romg H G. (929). A method of samplng nspecton, Bell system Techncal Journal, 8: [5] Epsten B, (954). Truncated lfe tests n the exponental case, Annals of Mathematcal Statstcs, 25: pp [6] Goode. H P and Kao. J. H. K, (96). Samplng plans based on the Webull dstrbuton, Proceedngs n the 7 th Natonal Symposum on Relablty and Qualty Control, pp.24-4, Phladelpha, USA. [7] Gupta S. S and Groll. P. A, (96). Gamma dstrbuton n acceptance samplng based on lfe tests, Journal of the Amercan Statstcal Assocaton, 56, pp [8] Kantam. R.R. L, Rosaah. K and Rao G. S, (2). Acceptance Samplng based on lfe tests: Log-logstc model, Journal of Appled Statstcs, 28, pp [9] Lo Y L, Tsa T. R and Wu S. J (2). Acceptance samplng plans from truncated lfe tests based on the Brnbaum- Saunders dstrbuton for percentles, Communcatons n Statstcs: Smulaton and Computaton, 39, pp:

6 Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August 28 [] Perry R L. (973a). Skp-lot samplng plans. Journal of Qualty Technology, 5(3): [] Perry R L. (973b). Two-Level Skp-Lot Samplng Plans - Operatng Characterstc Propertes. Journal of Qualty Technology, 5(4): [2] Pradeepa Veerakumar K and Ponneeswar P(26), Desgnng of Acceptance Samplng Plan for lfe tests based on Percentles of Exponentated Raylegh Dstrbuton, Internatonal Journal of Current Engneerng and Technology,6(4): [3] Pradeepa Veerakumar K and Ponneeswar P(27), Desgnng of Acceptance Double Samplng Plan for Lfe Test Based on Percentles of Exponentated Raylegh Dstrbuton. Internatonal Journal of Statstcs and Systems, 2(3): [4] Rao. G. S, (2). Double acceptance samplng plans based on truncated lfe tests for the Marshall- Olkn extended expontal dstrbuton, Austran Journal of Statstcs, 4, pp: [5] Soundararajan V and Vjayaraghavan R.(989) A new system of skp-lot samplng nspecton plans of type SkSP-3, Qualty for Progress and Development (Inda, Wley Eastern). [6] Vjayaraghavan R (2). Desgnng and evaluaton of skp-lot samplng plan of type 3, Journal of Appled Statstcs, 27(7):9-98. Table : OC values for SkSP- 3 wth Sngle Samplng Plan (n, c=2, t/t. ) as reference plan for a gven P * under ERD when θ = 2 P* n t/t q t q/t q

7 Internatonal Journal of Research n Advent Technology, Vol.6, No.8, August