A DEA Based MODM Model for Designing X Control Chart

Transcription

1 IANG Iteatoal Joal of Appled Mateatc, 38:3, IJAM_38_3_06 A A Baed MOM Model fo egg X Cotol Cat Sev Aadade, Fad Koala Abtact Cotol cat ae wdely pleeted f to etabl ad ata tattcal cotol of a poce wc lead to te poved qalty ad podctvty. eg of cotol cat eqe tat te egee elect a aple e, a aplg feqecy ad te cotol lt fo te cat. I t pape, a poble cobato of deg paaete codeed a a deco ag t wc detfed by tee attbte: oly expected cot, detecto powe of te cat ad -cotol aveage legt. Optal deg of cotol cat ca be folated a ltple obectve deco ag (MOM). Te cot fcto exteded fo gle to ltple agable cae becae tee ext ltple agable cae eal pactce. A algot g A appled to olve te MOM odel. A ecal exaple ed to lltate te algot pocede. Fally, etvty aaly a bee caed ot to vetgate te obte of te odel Idex Te Cotol cat deg, ata evelopet aaly, X cotol cat, Mltple-obectve deco ag (MOM) I. INTROUION If a podct to eet o exceed ctoe expectato, t old be podced by a poce tat table o epeatable. Stattcal poce cotol a powefl collecto of poble olvg tool efl acevg poce tablty ad povg capablte tog te edcto of vaablty. Te a tool of tattcal poce cotol te tattcal cotol cat. Te egeeg ad teccal pleetato of cotol cat etal electg aple e, aplg feqece ad te cotol lt fo te cat. Selecto of tee tee paaete called te deg of cotol cat. Tadtoally, cotol cat ave bee deged wt epect to tattcal ctea oly, bt te deg of a cotol cat a ecooc apect too. Te ft odel t cae wa popoed by ca []. Sce tat te, te ecooc appoac a eceved codeable atteto ad vao odel ave bee ggeted t aea. Bt a declaed by Woodall [2], cotol cat baed o optal ecooc deg, ave poo Macpt btted Jly 23,2008 Coepodg Ato: Sev Aadade, MS tdet, epatet of Idtal geeg, K.N.Too Uvety of Tecology. No.7, ad St., Molla-Sada St., Vaa Sq., Tea, Ia..O.BOX ; poe: ; fax: e_al: _aadade@yaoo.co. Fad Koala, Atat ofeo, epatet of Idtal geeg, K.N.Too Uvety of Tecology No.7, ad St., Molla-Sada St., Vaa Sq., Tea, Ia..O.BOX ; poe: ; fax: e_al: oala@t.ac. tattcal popete. To olve t poble, Saga [3] oted tat oe of te ctc of ecooc deg ca be ovecoe by todcg tattcal cotat te poble ad olvg te odel g olea optato tecqe. el Catllo, Motgoey ad Mac [] popoed a teactve lt obectve algot baed o t pocede. I addto, Ce ad Lao [5] folated optal deg of cotol cat a a ltple ctea deco ag wt epect to te cotat popoed by Saga. I te etoed atcle, a gle agable cae cot fcto wa ed. Howeve 97, ca [6] developed pevo odel ad peeted a ew odel te peece of ltple agable cae. Sce te, ay ted to opte t cot fcto. Cg [7] caed ot beqet wo o ca odel [6]. Ce ad Yag [8] codeed webll -cotol te wt ltple agable cae. Y ad Ho [9] opted te cotol cat paaete wt ltple agable cae ad vaable aplg teval. Alo, Y, To ad Hag [0] ed ca' odel ad te popoed cotat by aga [3] to vetgate ecooc-tattcal deg of X cotol cat. Table ow te copao of dffeet odel, etoed above. Howeve, ltple obectve deg of X cotol cat wt ltple agable cae a ot bee addeed te lteate. Teefoe, te ppoe of t pape to fd te opt deg paaete of X cotol cat g A, peece of ltple agable cae wc atfy all ecooc ad tattcal obectve. A ecal exaple gve to lltate te odel' wog. Setvty aaly a bee doe to ee te apte of te odel. II. CONOMIC COST FUNION WITH MULTIL ASSIGNABL CAUSS ca [6] geealed gle agable cae odel to ltple oe. I t odel, tee a -cotol tate, a agable cae of agtde (, 2..., ) wc occ at ado, elt a ft te ea to ete o ad o cage te tate tl te cae detected. Meawle, dg te eac fo te agable cae, te poce allowed to cote opeato. Te cycle cot of fo peod: ) I-cotol peod It aed tat agable cae occ accodg to oo poce wt occece pe o. So, ag (Advace ole pblcato: 2 Agt 2008)

2 IANG Iteatoal Joal of Appled Mateatc, 38:3, IJAM_38_3_06 tat poce beg te -cotol tate, te te teval tat te poce ea cotol a expoetal ado vaable wt ea o: 2) Ot of cotol peod We te poce goe to ot of cotol tate, te pobablty tat t wll be detected o ay beqet aple elated to te agable cae occed. If te t agable cae appe, te te detecto powe wll be: ( ) d + ( ) d (2) wee ( ), te pobablty dety fcto of tadaded oal dtbto. Teefoe, te aveage aple tae afte te t agable cae appe,. I addto, gve te occece of te t agable cae betwee te t ad +t aple, te expected te of occece wt t teval : ( ) ( ) ( )e ( e ) e t ( t ) dt e t dt Wee te aplg feqecy. Model/yea el Catllo, Motgoey ad Mac(996) Ce ad Lao (200) Cg (99) Ce ad Yag (2002) Y ad Ho (2006) Y, To ad Hag (2007) Table. Copao of te odel Apto Sgle agable cae xpoetal -cotol te Sgle agable cae xpoetal -cotol te Mltple agable cae xpoetal -cotol te Mltple agable cae Webll -cotol te Mltple agable cae vaable aplg teval Mltple agable cae xpoetal Otpt (3) Mlt-obectve deg (ecooc-tattcal) Mlt-obectve deg (ecooc-tattcal) cooc deg cooc deg cooc deg cooc-tattcal deg (ecooc obectve wt tattcal cotat) -cotol te Teefoe, te te eqed to obeve a ot of cotol ala we te t agable cae occ, wll be: () 3) Te te to tae aple ad tepet te elt a cotat g popotoal to te aple e, o tat g te legt of t pat of te cycle. ) Te te eqed to fd te agable cae. If t te fo te t agable cae, te te expected te a cycle fo detectg agable cae : Teefoe, te expected legt of a cycle : g g If te fxed copoet of aplg cot a ad te vaable oe a 2, te te cot of tag a aple of e wll be a a. Te cot of fdg a agable cae, 2 a ad te cot of vetgatg a fale ala a 3. Te expected be of fale ala geeated dg a cycle, te te expected be of aple tae befoe te ft o: e (7) e I wc calclated tog te below eqato: 2 ( ) d (8) ARL Sbeqetly, f oe defe a 5, a te oly pealty cot aocated wt podcto ot of cotol tate, te te expected cot pe cycle wll be: CC 5 3 a e ( a a ) 2 e Ad te expected cot pe o ca be dcated a: ( a a ) CC 2 ( a g a 5 3 a e e a g a / g ) ( ) () (5) (6) (9) (0) (Advace ole pblcato: 2 Agt 2008)

3 IANG Iteatoal Joal of Appled Mateatc, 38:3, IJAM_38_3_06 Teefoe, ecooc deg of X cotol cat volve deteato of optal paaete, ad wc e. III. MULTIL OBJIV OTIMAL SIGN OF X CONTROL CHART To etabl te ltple obectve deco ag odel, we old ft detee a et of coflctg obectve tat defe te poble fo te qalty cotol aage. e to te ate of te A etod ed algot, vao cobato of deg paaete, ad old alo be et advace. Tag to accot te Saga cotat [3], te lt-obectve odel : Max ARL Max M.t. () (, 2,..., ) l ATS ATS (, 2,..., ) ARL te -cotol aveage legt (ecpocal of fale ala ate ( )), te detecto powe of te cotol cat ad te expected cot pe o. ad ATS ae epectvely te detecto powe ad te aveage te to gal we te t agable cae occ ad a poble cobato of deg paaete tat a bee ow bacet fo te ete tee obectve fo epag o te fact tat t doe ave a pact o te vale of obectve. Te a of MOM odel ae to fd olto tat ca atfy ad et a balace aog all obectve. To olve MOM poble, te A etod oe of te ot powefl ad popla etod to opte te feable cobato of deg paaete pecfcally we eag te effcece of la t de codeato. IV. ATA NVLOMNT ANALYSIS (A) A te optato etod of lea pogag to geeale te Faell [] gle pt, gle otpt teccal effcecy eae to te ltple-pt, ltpleotpt cae by cotctg a elatve effcecy coe of a gop of copetg deco ag t (MU). Applcato ad pleetato of A odelg pefoace eaeet ave gaed a lot of atteto ecet yea [2]. I t pape we ave ed te CCR odel (Cae, Coope ad Rode [3]). Te obectve CRR odel to axe te elatve effcecy vale of eac of MU fo aog a efeece et of deg, by electg te optal wegt aocated wt te pt ad otpt. Te algebac odel a follow: Max = U Y.t. (2) U Y fo ote deg Wee U : te wegt gve to otpt Y : aot of otpt fo t V : wegt gve to pt X : aot of pt fo t To olve te odel, t eceay to covet t to lea fo o tat etod of lea pogag ca be appled. T olea pogag eqvalet to two lea pogag: ) ettg t deoato to oe ad axg t eato (otpt axato) 2) ettg t eato to oe ad g deoato (pt ato). Becae CCR odel code cotat ete to cale, tee ext o dffeece wc oe to cooe ad CCR yeld te ae effcecy coe. Teefoe, te lea pogag wll be: Max = U Y.t. (3) U Y fo ote deg U,V If * 0, tat ea o ote deg oe effcet ta * deg de t ow wegt. If, te tee at leat oe ote deg tat oe effcet de optal et of wegt deteed. Calclato old be doe fo eac MU to fd te elatve effcecy of eac oe. V. SOLUTION ALGORITHM Ule ay ltple-obectve odel tat te M a a plct ow vale fcto, ee te vale of, ad ARL t be calclated fo eac potetal cobato accodg to fola to 0 advace. e to te coplcated lt-agable cae cot fcto, all calclato ave bee facltated by xcel oftwae. I addto, to evalate ad copae te effcece of MU, Mcooft xcel wt XlA a bee pleeted. Ce ad Lao [5] popoed a olto pocede fo te lt-ctea deco ag odel. I t pape, we ave eployed te -tep algot to olve o lt-obectve odel. Tey appled t pocede fo te odel wt oe agable cae cot fcto. Te pocede appoxately te ae except tep ad 2 wc ave bee odfed to t o popoed MOM odel. Te fo-tep pocede wll be a follow: ) eteg all poble olto by pttg bod o eac paaete. I t pape te cope of aple e () et fo to 35, ceaed by. Scope of aplg feqecy () cofed fo 0. to ceaed by 0. ad fally te cope of cotol lt wdt () codeed fo 0. to 3 te of tadad devato ceaed by (Advace ole pblcato: 2 Agt 2008)

4 IANG Iteatoal Joal of Appled Mateatc, 38:3, IJAM_38_3_ Coteplatg all poble cobato te be of potetal olto wll be 35*0*30=2000 2) I t tep we ave added aote cotat too, ode to tae to accot te vale of paaete, becae te two pevo cotat Ce ad Lao [5] ed, tee wa o g of paaete. beqetly, we elate feable olto by te followg cotat: ATS ATS l 3) atal optato. Rea te eleet wt aeto optalty fo eac bet Q. A olto "" wt aeto optato a et Q ea tat tee o ote olto te ae et c tat "" doated te of tattcal popete ad cot. ) Global optato. Mege all te eade to a et W ad elect te eleet wt get elatve effcecy aog W. Te elected eleet wll affod to M to ae fal deco. VI. NUMRICAL XAML I t ecto, ca [6] data wee eployed to lltate te e of te popoed odel ad algot. Te be of agable cae ae aed to be 2. We a agable cae wt te aveage occece of occ, t podce a ft of e te ea. Te cot of tag a aple tat depedet of aplg $ ad te vaable cot pe te of aplg, tetg ad plottg $0.. A aveage te of 0.05 eeded to tet ad aalye a aple te ad te cot of loog fo toble we oe ext, etated $25. Vale of ote paaete ave bee tablated table Table 2. Ipt vale of paaete a a Moeove, o tattcal cotat fale ala ate, detecto powe ad aveage te to gal ATS ae: ATS Te optato pocede ca be caed ot a decbed pevo ecto. Table3 lltate te elt. A dcated by, two deg paaete cobato ave eceved coe ad teefoe offeed to te M fo fal electo. Te te M ay cooe te ft cobato f low cot of paaot potace fo /e. Slaly f e/e c oe teeted te otgog qalty, te te ecod cobato wt lage aveage legt ad detecto powe ay be te fal coce. Table 3. No-doated olto wt laget effcece (,,) Cot ARL (27,2.9,2.6) * (28,2.9,2.6) (29,2.9,2.6) (29,2.9,2.7) (30,3,2.6) (30,3,2.7) (30,2.9,2.8) (3,3,2.6) (3,3,2.7) (3,3,2.8) (32,3.,2.6) (32,3,2.7) (32,3,2.8) (32,3,2.9) (33,3.,2.6) (33,3.,2.7) (33,3.,2.8) (33,3,2.9) * (3,3.,2.6) (3,3.,2.7) (3,3.,2.8) (3,3.,2.9) (3,3.,3) (35,3.2,2.6) (35,3.,2.7) (35,3.,2.8) (35,3.,2.9) (35,3.,3) VII. SNSITIVITY ANALYSIS Setvty aaly a bee doe to tdy te effect of odel paaete o te olto ad to dc te obte of te popoed odel. Fxed paaete effect c a cotat aplg cot, vaable aplg cot, cot of vetgatg te fale ala ate ad fally te te fo aplg ad aalyg, ave bee tded. Table lltate te effect of tee paaete o opt olto. Soe ctcal obevato aceved fo etvty aaly ae a follow. ) eceag fxed ad vaable aplg cot elt edcto aplg teval () ad oly expected cot. (Advace ole pblcato: 2 Agt 2008)

5 IANG Iteatoal Joal of Appled Mateatc, 38:3, IJAM_38_3_06 2) Te cot of eacg fo toble we oe ext ( a ), elatvely obt to te optal olto ad expected cot. 3) Te te to tet ad aalye a aple te (g) a eglgble effect o opt olto bt ceag t, ay lead to codeable ceae oly expected cot. Table. ffect of paaete o opt olto (,, ) Cot ARL a 0. (27,2.5,2.6) (33,2.7,3) (27,2.9,2.6) (27,3.6,2.6) (33,3.6,3) a2 0.0 (27,.8,2.6) (33,.7,3) (27,2.9,2.6) (27,3.6,2.6) (33,3.6,3) a 0 (27,2.8,2.6) (33,3,3) (27,2.9,2.6) (27,3,2.6) (33,3.,3) g 0.0 (27,2.9,2.6) (33,3,3) (27,2.9,2.6) (27,2.9,2.6) (33,3.,3) A lt-obectve odel fo degg X cotol cat peece of ltple agable cae popoed. Fo t odel, vao cobato of, ad ae coteplated a deco ag t (MU). A etod eployed to ae te effcecy of MU ad to elect te opt deg wt lage aveage legt, g detecto powe ad low expected cot. Necal exaple gve baed o te ca [6] data to lltate te olto pocede. Setvty aaly caed ot wc dcate tat optal deg of aplg feqecy paaete affected by fxed ad vaable aplg cot. Cot of eacg fo fale ala a appoxately gfcat effect o deg paaete ad oly expected cot ad fally, ceag te te of tet ad aaly of a aple ay lead to g expected cot. Ote teetg eeac aea fo fte eeac volve ltobectve deg of X cotol cat de webll oc ad lt-obectve deg of adaptve X cotol cat. RFRNCS [] ca A.J (956) Te ecooc deg of x cat ed to ata cet cotol of a poce.joal of te Aeca Stattcal Aocato 5(27): [2] Woodall W.H (986) Weae of te ecooc deg of cotol cat. Tecoetc 28(): [3] Saga. M (989) cooc tattcal cotol cat deg wt a applcato to X ad R cat. Tecoetc 3(3): [] el Catllo, Mac, Motgoey. C (996) Mltple-ctea optal deg of X cotol cat. II Taacto 28(6): 67-7 [5] Ce Y.K, Lao H.C (200) Mlt-ctea deg of a X cotol cat. Copte ad Idtal geeg 6(): [6] ca A.J (97) Te ecooc deg of x cat we tee a ltplcty of agable cae Joal of te Aeca Stattcal Aocato 66(333): 07-2 [7] Cg. (99) A algot fo coptg te ecoocally optal X cotol cat fo a poce wt ltple agable cae. opea Joal of Opeato Reeac 72(2): [8] CeY.S, Yag Y.M (2002) cooc deg of X cotol cat wt webll -cotol te we tee ae ltple agable cae. Iteatoal Joal of odcto cooc 77(): 7-23 [9] Y F.J, Ho J.L (2006) Optato of deg paaete fo cotol cat wt ltple agable cae. Joal of Appled Stattc 33(3): [0] Y F.J, To C.S, Hag K.I (2007) A cooc-stattcal eg of X Cotol Cat wt Mltple Agable Cae. Te 22 d opea Cofeece o Opeatoal Reeac. [] Faell M.J (957) Te eaeet of podctve effcecy. J.R. Stat. Soc. See A 20(3): [2] Coo W., Z J (2005) Modelg efoace Meaeet, Spge [3] Cae A, Coope W.W, Rode (978) Meag te effcecy of deco ag t. opea Joal of Opeatoal Reeac 2(6): 29- VIII. CONCLUSION (Advace ole pblcato: 2 Agt 2008)