Design of Bayesian MDS Sampling Plan Based on the Process Capability Index

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1 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Desg of Byes MDS Splg Pl Bsed o the Process pblty Idex Dvood Shshebor, Mohd Sber Fllh Nezhd, S Sef Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ Abstrct I ths pper, vrble ultple depedet stte (MDS) splg pl s developed bsed o the process cpblty dex usg Byes pproch. The optl preters of the developed splg pl wth respect to costrts relted to the rsk of cosuer d producer re preseted. Two coprso studes hve bee doe. Frst, the ethods of double splg odel, splg pl for resubtted lots d repettve group splg (RGS) pl re elborted d verge sple ubers of the developed MDS pl d other clsscl ethods re copred. A coprso study betwee the developed MDS pl bsed o Byes pproch d the exct probblty dstrbuto s crred out. Keywords MDS splg pl, RGS pl, splg pl for resubtted lots, process cpblty dex, verge sple uber, Byes pproch. I. INTRODUTION TATISTIAL qulty cotrol s wdely used world Shgh-tech dustres; qulty cotrol ethods ttept to reduce the wste of producto s uch s possble. If ger wts to perfor qulty cotrol dfferet prts of orgzto, due to the hgh cost of cotrol ethods, he should use precse d sutble techques. I the cotext of products qulty cotrol ll producto stges, soe esures re ppled whch re referred to s process cpblty lyss. A dex clled PI s used for lyzg the process cpblty. By pyg ore tteto to the wde pplcto of these dces, selecto of ther esttors d probblty dstrbuto of these dces s very portt; thus, precse ethod lke Byes sttstcl techques s used to solve ths proble. Ths techque specfes pror dstrbuto fucto for the gve preters d the the ext stge fors posteror dstrbuto fucto for these preters usg the collected dt. Oe usge of PI s to desg cceptce splg pl to ke decsos bout receved lot fro supplers producto evroets, so tht the rsk of producer d cosuer dpts wth the specfed qulty stdrds. Aog the clsscl ethods of cceptce splg pl, the vrble splg pl s very portt due to the qutttve lyss of qulty chrcterstcs. Although the use of ths pl s ore dffcult copred to the ttrbute splg pl, t results ore precse esureets for decso kg bout lot d s less rsky. The MDS splg pl s oe of the effectve splg pls ethods Dvood Shshebor s wth the Yzd Uversty, Ir, Islc Republc Of (e-l: shshebor@yzd.c.r). whch s bsed o codtoed procedure d hs bee troduced by Worth d Bker []. Ths splg pl cosders ot oly sples tke fro the curret lot, but lso t cosders splg results of prevous or future lots. Wth tteto to the portce of the vrble splg pl bsed o PI, these types of splg pls hve bee dscussed recet yers. Exples clude []-[6] d y others. Recetly, ew MDS splg pl bsed o PI hs bee proposed by Asl et l. [7]. Also ew ethod of MDS splg pl s preseted by Blurl d Ju [8] for the lot cceptce proble. Soudrr d Vyrghv [9] developed procedure for scheg ultple depedet (deferred) stte splg pls. Verst [0] preseted pl to crete MDS splg pls. Wu et l. [] proposed ovel lot setecg ethod by vrbles specto bsed o MDS. I ddto, RGS pls bsed o the process cpblty re proposed by Sher [], Blurl d Ju [8], Asl et l. [7] d Asl et l. [3]. Also, ew ethods of splg pl for resubtted lots re proposed by Govdru d Geslg [4] d Asl et l. [7]. I ths pper, vrble MDS splg pl s developed bsed o PI usg Byes pproch. I ths pl, t s supposed tht the desred qulty chrcterstc follows orl dstrbuto fucto. I wht follows, double splg odel, splg pl for resubtted lots d RGS pl re developed bsed o Byes pproch d the coprso study s crred out betwee ASNs of troduced splg pl. The cotrbutos of ths reserch re s follows: () Itroducg vrble MDS splg pl bsed o PI d Byes pproch. () Presetg dfferet vrble splg pls (whch cludes SSP, DSP, RGS d Splg pl for resubtted lots) bsed o Byes pproch. () oprso of the developed vrble splg pls d deterg the best pl. (v) oprso of the MDS splg pl bsed o two dfferet pproches (whch cludes Byes pproch d exct probblty dstrbuto) d selectg the sutble pproch. II. PROESS APABILITY ANALYSIS A. Estto of PI Bsed o the Multple Sples Suppose tht the qulty chrcterstc follows orl dstrbuto wth e d vrce. PI s defed s: Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

2 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ USL LSL d M, where d ( USL LSL)/ d M ( USL LSL)/ d LSL, USL re lower d upper specfcto lts. The verge of the observtos ( X ) s used to estte the e vlue whe the e vlue of process s ukow. Assue k subsples hve bee gthered d the qulty chrcterstc of the process s orlly dstrbuted, so tht the sple sze of th subsple s equl to. We cosder x s the rdo observto fro orl dstrbuto wth the e d vrce,,,..., k d,,...,. We cosder tht the process s uder sttstcl cotrol by X d S cotrol chrts. Wth ths ssupto, the sttstcs X d S for ech subsple wll be defed s: ˆ x, x N S ( x x ), d N where N s the totl uber of observtos. Assug N k () () ( ), we use the verge of sples e s ubsed esttor of d lso ccuulto of the sples vrce s ubsed esttor of, the the estto of PI bsed o the observtos of ultple sples s s: ˆ x, x N ˆ ( ) S p S N d x M ˆ USL x x LSL, 3s 3s 3s p p p Ths esttor of PI s ostly ppled prctcl d dustrl pplctos. B. Posteror Probblty Dstrbuto Bsed o the Multple Sples A Byes ethod proposed by Per d L [5] s ppled to evlute the probblty dstrbuto fucto of (3) (4) PI, bsed o ultple sples. Accordg to ther Byes pproch, the posteror probblty dstrbuto fucto of for specfed costt vlues of w c be obted s: p Pr{the process s cpble X} Pr{ w X} (5) The, the probblty of hvg cpble process wll be s [6], [7]: p Pr{ w X } exp( ) ( ) y y ] } dy 0 { [ b ( y )] [ b ( y ) where ( u ) s the cuultve fucto of stdrd orl u / dstrbuto, ( u) ( ) exp( t / ) dt d the other preters re defed s []: (6) x M ( N )/, (7) s ( x x ) / Ns p ( x x) / Ns ( x x) p ˆ * b ( y ) 3 N w N y ˆ * b ( y) 3 N w 3 N y p (8) (9) (0) III. PROPOSED MDS SAMPLING PLAN MDS splg pl s type of codtoed splg ethods whch cosder splg results of pst or future lots. For pplcto of vrble MDS splg pl, the followg ssuptos should be vld. () Lots re subtted fro process whch hs costt proporto of o-coforg tes. () The qulty chrcterstc of terest follows orl dstrbuto. () There s o reso to beleve tht prtculr lot s poorer th the precedg lots. (v) Optl preters of the proposed splg pl re defed s follows: : Sple sze, : Nuber of precedg lots, k : The upper threshold of PI for cceptg the lot, k : The lower Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

3 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ threshold of PI for reectg the lot. I order to dsply the proposed splg pl for prctcl use, we expl the procedure of proposed MDS splg pl usg rel exple. I vrous dustres, the process cpblty of terls, products d systes eeds to be coptble wth the egeerg tolerces. Here we cosder copy wth che tools s prctcl cse. The etoed copy eeds to keep che tools wth the desred tolerces. The egeers of the copy cosder desol tolerce for dfferet portos of the tool d lso clculte the PI for the. Flly, ther decso bout receved lot bsed o MDS procedure d PI s defed s follows: Step. tke rdo sple of sze d copute PI, ˆ. Step. f ˆ k, the ccept the lot d f ˆ k, the reect t. If k ˆ k, the f the vlue of PI ech of the sples tke fro the precedg lots s ore th k the ccept the lot otherwse reect the lot. It s oted tht f k k, the proposed splg pl coverts to the sgle splg pl. The O fucto of vrble MDS splg pl s obted s [8]: P ( ) Pr{ ˆ k X } Pr{ k ˆ k X }.[Pr{ ˆ k X }] () where Pr{ ˆ k X } s the probblty of cceptg the lot bsed o sgle sple. Also, the ter Pr{ k ˆ k X }.[Pr{ ˆ k X }] s the probblty of cceptg the lot bsed o the qulty level of precedg lots. A optzto odel s forulted to ze ASN cosderg the costrts of frst d secod type error. Also the optl vlue of c be obted by sestvty lyss. The preseted odel s s: where Mze subect to : P ( ) A d P ( ) producer d LTPD A LTPD () s the u cceptble vlue of PI for the s the xu ucceptble vlue of PI for the cosuer. The preters d re the probbltes of frst d secod type error, respectvely. IV. DESIGNING A DOUBLE-SAMPLING PLAN A double-splg pl bsed o process cpblty s desged wth the followg preters: : Sple sze of the frst sple, : Sple sze of the secod sple, k : The lower threshold of PI for reectg the lot bsed o the frst sple, k : The upper threshold of PI for cceptg the lot bsed o the frst sple, k : The 3 upper threshold of PI for cceptg the lot bsed o the secod sple The decso-kg ethod s s follows: Step. Select observto the frst sple fro the lot d copute ˆ. If ˆ k the ccept the lot d reect the lot f ˆ k where k k.f k ˆ k, the obt secod sple wth observtos. Step. opute ˆ k 3 ˆ, otherwse reect the lot. the secod sple. Accept the lot f I double splg pls, the geerl forul for the ASN, c be obted s: ASN Pr ( k) - Pr ( k) (3) The costrt of producer rsk s s: Pr ( ˆ k ) + Pr ( ˆ k ) 3. Pr( k ˆ k ) - (4) The costrt of cosuer rsk s s follows: Pr ( ˆ k ) + Pr ( ˆ k ) 3. Pr ( k ˆ k ) (5) Therefore, by solvg the optzto odel for specfc vlues of, d the dfferet vlues of d, the decso preters of double splg pl c be obted. V. DESIGNING A VARIABLE RGS MODEL The preters of vrble RGS pl re s follows: : Sple sze, k : The lower threshold of PI for Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

4 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ reectg the lot bsed o the sgle sple, k : The upper threshold of PI for cceptg the lot bsed o the frst sple Ad ts procedure s surzed s follows: Step. ollect sple wth observto fro lot. Step. If k the ccept the lot d f ˆ the reect t. If The O fucto of the vrble RGS pl s s: r ˆ k k ˆ k, the repet step d step. ˆ ˆ P Pr( k ) P ( ) (6) P P Pr( ˆ k ) Pr( ˆ k ) The obectve fucto of the optzto odel s the ASN d ts costrts re producer rsk, d cosuer rsk, tht c be obted s [8]: MzeT subect to : Pr( ˆ k ) Pr( ˆ k ) Pr( ˆ k ) = Pr( ˆ k ) Pr( ˆ k ) d = Pr( ˆ k ) Pr( ˆ k ) Pr( ˆ k ) VI. DESIGNING A VARIABLE SAMPLING PLAN FOR RESUBMITTED LOTS (7) The vrble splg pl for resubtted lots s oe of the ost effectve splg pls. The preters of ths splg pl re s: : Nuber of resubssos, : Sple sze, k : The lower threshold of PI for cceptg the lot bsed o the sgle sple. Ad ts procedure s surzed s follows: Step. Tke sple wth observto d copute PI. If k ˆ the ccept the lot otherwse go to step. Step. Repet step for tes. If the lot ws ot ccepted fter repettos of step the reect the lot. The O fucto of the vrbles splg pl for resubtted lot wll be s [4]: ˆ P ( ) ( P ) P ˆ k (8) A The ASN of developed splg pl s s [4]: Therefore, the optzto odel s s: ( ( P ) ) Mze ASN P subect to : P ( ) A d P ( ) A (0) I order to obt the pl preters, we preset ethodology of how the etoed pl preters c be obted. The wth cosderg the decso vrbles d ssupto of pl, d wth usg grd serch, we c obt the u ASN pls serchg the ultdesol grd fored settg = d, = ()00, k =.4(0.00).4, k = 0.7(0.00).3. VII. ANALYSES AND DISUSSIONS A. Sulto Studes The solutos of optzto odel of the MDS pl for gve vlues of producer rsk 0.05 d cosuer rsk 0.0 d fxed vlues of, d for the dfferet vlues of, re detered. The results re deoted Tbles I d II. I Tbles I d II, the vlue of the optl preters of the proposed MDS pl, cludg k,, k for dfferet vlues of, re deoted. For,, exple, Tble I, f.0,.3 d 0.5, 0.8,, the the vlues of 3, k.857, k.4 re obted s the optl soluto d the procedure of proposed MDS pl wll be s follows: Step. tke rdo sple of sze 3d copute ˆ. Step. f ˆ.857, the ccept the lot d f ˆ.4, the reect t. If.4 ˆ.857, the f the vlue of PI of precedg lot s ore th k ( ˆ.857 ) the ccept the lot, otherwse reect the lot. Also, t s observed tht the cse hs less sple sze ost of the sulted cses, thus t s preferred to pply the cse. ASN ( ( P ) ) (9) P Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

5 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ TABLE I OPTIMAL VALUES OF PARAMETERS FOR DIFFERENT VALUES OF, AND 0.5, 0.8, 0., 0.9, k k k k TABLE II OPTIMAL VALUES OF PARAMETERS FOR DIFFERENT VALUES OF, AND 0.5; 0.8;.0; 0.9; k k k k B. oprsos of Pls I ths secto, we copre the verge sple uber of dfferet cceptce splg pls. The results re deoted Tble III. For exple, f.8,.4 d 0.5, 0.8, the ASN of the sgle d double d the MDS splg pl ( ) re respectvely, 3, 0.86 d 5. Thus, ccordg to the results Tble III, t s observed tht the ASN of the vrble MDS pl s less th the ASN of other splg pls. The vrbles RGS pl perfors better th the splg pl for the resubtted lots d the double splg pl. The vrbles sgle splg pl hs the worst perforce coprso wth the other ethods. Sce the MDS splg pl eeds the hstorcl dt of precedg lots, t y be pplcble soe cses. Other Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

6 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 splg ethods do ot eed such dt, d thus, the RGS splg pl y be ore pproprte soe prctcl probles. Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ TABLE III RESULTS OF OMPARISON STUDY 0.5; 0.8 ASN Vrble sgle Vrble double Vrble Vrble MDS splg pl splg pl (RGS) pl pl ( ) splg pl for resubtted lots TABLE IV OPTIMAL VALUES OF PARAMETERS FOR DIFFERENT VALUES OF 0.5, 0.8,, 0.05, 0.0 ASN ( ) ASN ( ) ASN ( 3) ASN ( 4) Sestvty Alyss Bsed o the Preter I order to vestgte the effect of the preter o the proposed splg pl, sestvty lyss s crred out. The results re deoted Tble IV. It s observed tht the results of the MDS pl the cse of perfors better th the cses, =3, =4. For exple, where.8,., the ASN of the MDS splg pl the cse of s equl to 5 d the ASN for the cses of, 3 d 4, s equl to 6, 4 d 7, respectvely. VIII. DESIGNING AN MDS SAMPLING PLAN BASED ON EXAT PROBABILITY DISTRIBUTION The exct probblty dstrbuto of PI c be detered usg sttstcl techques. The we c desg the cceptce splg pl bsed o exct probblty dstrbuto of PI. Wth cosderg the turl esttor of tht s, for orlly dstrbuted process, sple ˆ d exct for of the cuultve dstrbuto fucto of the estted preter, ˆ, c be detered. The cuultve dstrbuto fucto s s follows (Per d L [5]). b (-)(b -t) F ˆ ( ) G Y 0 9..y () [ (t + ξ ) + (t - ξ )] dt where d b d b vlues c be rewrtte s: b 3 d M. Also G (.) s the DF of the ch-squre dstrbuto, wth - degrees of freedo d (.) s the PDF of the stdrd orl dstrbuto N (0,). Wth regrds to the O fucto of the developed vrble MDS splg pl d exct probblty dstrbuto, the requred sple sze c be obted usg the optzto odel (). IX. OMPARISON OF THE MDS SAMPLING PLAN BASED ON DIFFERENT APPROAHES Now, we lyze the ASN of MDS splg pl bsed o the Byes pproch d exct probblty dstrbuto (exct pproch). Wth cosderg the specfed vlues of 0.5, 0.8 d for Byes pproch d exct probblty dstrbuto fucto (PDF), t s observed tht the ASN of the vrble MDS pl uder Byes PDF pproch s less th the ASN of MDS splg pl uder the exct PDF pproch. The results re deoted Tble V. TABLE V RESULTS OF A OMPARISON STUDY MDS splg pl (Byes pproch) MDS splg pl (exct pproch) 0.5, 0.8,, Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /

7 World Acdey of Scece, Egeerg d Techology Vol:, No:0, 07 Itertol Scece Idex, Idustrl d Mufcturg Egeerg Vol:, No:0, 07 wset.org/publcto/ Whe soe hstorcl dt of the process s vlble, pplyg the Byes ferece pproch s preferred; however, whe we do ot hve ccess to such dt we ust pply exct probblty dstrbuto. X. ONLUSION I ths pper, vrble MDS splg pl bsed o PI hs bee developed where the requred probbltes re obted by the Byes pproch d exct probblty dstrbuto. The optl preters of the developed splg pl re obted bsed o the costrts of cosuer rsk d producer rsk. Also, the procedure of MDS splg pl hs bee expled wth rel exple. The two coprso studes re perfored; frst, we copre the ASN of the developed splg pl wth other clsscl splg ethods. Secod, we copred the ASN of MDS pl bsed o the Byes pproch d exct probblty dstrbuto. It ws cocluded tht the developed MDS pl s ore ecoocl th other splg pls. Also, the Byes MDS pl perfors better th MDS pl uder exct probblty dstrbuto. REFERENES [] Worth, A. W., Bker, R.. (976). Multple deferred stte splg specto. Itertol Jourl of Producto Reserch. 4: [] Wu,. W. (006). Assessg process cpblty bsed o Byes pproch wth subsples. Europe Jourl of Opertol reserch. 84 (): [3] Blurl, S., Asl, M., h, H. J. (03). A New Syste of Skp-Lot Splg Pls cludg Resplg. Scetfc World Jourl. Artcle ID 94, 6 pges. [4] Wu,. W., Asl, M., h, H. J. (0). Vrbles splg specto schee for resubtted lots bsed o the process cpblty dex, Europe Jourl of Opertol Reserch. 7: [5] Per, W. L., Wu,. W. (005). A Byes pproch for ssessg process precso bsed o ultple sples. Europe ourl of Opertol Reserch 65 (3): [6] Per, W. L., Wu,.W. (006). Vrbles splg pls wth PPM frcto of defectves d process loss cosderto, J. Oper. Res. Soc. 57 (4): [7] Asl, M., Az, M., h, H. J. (03). A xed repettve splg pl bsed o process cpblty dex, Appled Mthetcl Modellg. 37(4): [8] Blurl, S., Ju,. H. (007). Multple depedet stte splg pl for lot cceptce bsed o esureet dt. Europe Jourl of Opertol Reserch. 80: 30. [9] Soudrr, V., Vyrghv, R. (990). ostructo d selecto of ultple depedet (deferred) stte splg pl. Jourl of Appled Sttstcs 7, [0] Verst, R. (98). A procedure to costruct ultple deferred stte splg pl. Methods of Opertos Reserch. 37: [] Wu,. W., Lee, A. H. I., he, Y. W. (05). A Novel Lot Setecg Method by Vrbles Ispecto osderg Multple Depedet Stte. Qul. Relb. Egg. It. DOI: 0.00/qre.808. [] Sher, R. E., (965), Desg d Evluto of Repettve Group Splg Pls, Techoetrcs. 7: -. [3] Asl, M., Az, M., h, H. J. (05). Vrous repettve splg pls usg process cpblty dex of ultple qulty chrcterstcs. Appled Stochstc Models Busess d Idustry. [4] Govdru, K., Geslg, S. (997). Splg specto for resubtted lots, oucto Sttstcs Sulto. 6 (3): [5] Per, W. L., L, P.. (004). Testg process perforce bsed o cpblty dex wth crtcl vlue. oputers d Idustrl Egeerg. 47: [6] Asl, M., Wu,. W., Az, M., h, H. J. (03). Vrble splg specto for resubtted lots bsed o process cpblty dex for orlly dstrbuted tes. Appled Mthetcl Modellg. 37: [7] Asl, M., h, H. J., Az, M. (03). Multple Depedet Stte Splg Pl Bsed O Process pblty Idex, Jourl of Testg d Evluto. 4(): [8] Blurl, S., Ju,. H. (006). Repettve group splg procedure for vrbles specto. Jourl of Appled Sttstcs. 33: Itertol Scholrly d Scetfc Reserch & Iovto (0) scholr.wset.org/ /