THE TARGETING PROBLEM FOR S-TYPE QUALITY CHARACTERISTICS

Transcription

1 HE ARGEING PROBLEM FOR S-YPE QUALIY CHARACERISICS K. ahera, R. N. Ibrahm ad P. B. Lochert Departmet of Mechacal Egeerg, Moash Uversty, Wellgto Road, Clayto, Australa E-mal: ABSRAC he targetg problem geerally detfes the process parameters such as the tal process meas of a producto process. he ot determato of the tal process meas ad the producto ru has bee reported the lterature. Most of these studes cosdered a process wth multple depedet omal-the-better type (N-ype of qualty characterstcs. I a real dustral stuato, the qualty characterstcs may deped o each other. I ado, the process could have other types of qualty characterstcs. I ths paper, a mathematcal model s developed that otly determes the optmum tal process meas ad producto ru of a process wth multple depedet smaller-the-better type (S-ype of qualty characterstcs. A Numercal example s also provded to demostrate the applcato of the proposed model. Keywords: multple depedet S-type qualty characterstcs, process mea, producto ru. INRODUCION he targetg problem s a classcal problem whch deals wth the problem of selectg the optmal tal process meas settg for a gve product specfcato. he mproper selecto of ths settg affects the expected total proft (or cost, the umber of defectve tems, the specto cost ad the reprocessg cost. Geerally, a loss fucto s used to quatfy the qualty loss whe the qualty characterstcs devate from target values. hus t asssts maagers to evaluate the uobservable costs such as warraty cost, loss of maret share, sales of retur resultg from the customer s dssatsfacto. he traoal step loss fucto does ot corporate these uobservable costs to a qualty loss computato. hs approach assumes that the qualty cost does ot deped o the actual value of the qualty characterstc as log as t s wth the specfcato lmt. I other words, the step loss fucto treats a tem that falls exactly o the target value as the same as a tem that falls ust sde the specfcato lmt. herefore, ths approach favours maufacturers by allowg them to produce tems wth a large varace. Dr. J. aguch [] volves socety loss terms of customers satsfacto ad evromet cost to the qualty classfcato. He defed that qualty loss s what the product costs socety from the tme the product s released from shpmet. Afterwards, the aguch loss fucto s cosdered as a stadard qualty measuremet system. Other qualty loss fuctos [] have bee proposed whch have a smlar obectve as the aguch loss fucto. I realty, the process mea settg devates from ts tal value as a result of the presece of assgable causes whch evetually moves the process to a out-of-cotrol state. I ths state, the crease of the umber of defectve tems leads to a crease of qualty loss. hs stuato ca be accepted for a certa tme above whe t s ecessary to tae a restorato acto. hs acto ca clude readustg the process parameters to ts optmum tal values ad/or upgradg the tool. Such acto s geerally expesve ad should be doe from a vewpot of the ecoomc feasblty. herefore, determg the producto ru s aother crtcal factor to produce qualty tems. A ot determato of the optmum process mea ad producto ru helps the maufacturer to produce tems at a mmum producto cost. A few researchers vestgated the ot effect of process mea ad producto ru the process uder dfferet assumptos (for a revew see [3]. hese models ca be summarsed as follow. Frst, most models wored wth a omal-the-better type (N-ype of qualty characterstcs. Secod, oly a sgle qualty characterstc was assumed most of the studes ad they used step loss fucto or aguch loss fucto as a qualty measuremet system. hrd, few researchers [4-5] cosdered multple qualty characterstcs, but they assumed that the qualty characterstcs were depedet o each other. However, the determato of tal process meas settg ad producto ru of a process wth multple smaller-the-better type (S-ype ad larger-the-better type (L-ype qualty characterstcs has ot bee vestgated prevous research. here are may stuatos where qualty characterstcs may deped o each other. hese depedeces have ot bee explored developg the process target models. hs paper addresses these ssues ad proposes a mathematcal model to determe the optmum tal settg of process meas ad the optmum producto ru of a process wth multple smaller-the-better type (S-ype of qualty characterstcs that curs a mmum total cost. he depedecy of the qualty characterstcs s cosdered ths model. o llustrate the applcato of the proposed model, a umercal example s provded. MODEL DEVELOPMEN A mathematcal model for determg the optmal process meas ad the optmal producto ru s developed for a deteroratg process. he expected total producto cost ths study cossts of (a the off target cost, whch s modeled by the multvarate qualty loss fucto, (b the process adustmet cost ad (c the mateace cost durg the producto ru. he followg assumptos ad omeclatures are used the model: 7

2 Assumptos ( he producto process has a ow ad costat varace for each of the depedet qualty characterstcs. ( he qualty characterstcs are expoetally dstrbuted wth a mea of λ ( At a radom pot tme, τ, the process chages from a -cotrol state to a out-of-cotrol state as a result of a postve drft ad a postve shft. (v he producto ru s assumed to be the tme dfferece betwee the startg of the process to the tme whe the process meas are readusted to the tal values. Nomeclature y (t, τ he radom varable deotg the qualty measuremet for the -th qualty characterstc at tme t Loss coeffcet assocated wth qualty characterstc Loss coeffcet assocated wth qualty characterstcs ad Ital mea for -th qualty characterstc * ξ * Optmal tal mea for -th qualty characterstc Process varace for -th qualty characterstc Process covarace betwee -th ad -th qualty characterstc arget value for -th qualty characterstc Producto ru Optmal producto ru (t, τ he process mea for -th qualty characterstc at tme t ( t, τ ( t, τ W ( t, τ τ t τ < t W (t, τ Deteroratg fucto W t, τ τ t ( ( t, τ δ ( t τ θ W τ < t (For a shft ad a drft C A Adustmet cost ρ Producto rate per ut tme τ he elapsed tme utl the occurrece of the assgable cause. It s a radom varable ad s assumed to be expoetally dstrbuted wth a mea of λ (τ f he desty fucto of the occurrece tme of the e λ λτ assgable cause M (t Mateace cost at tme t We cosder a process wth multple smaller-the-better type (S-ype of qualty characterstcs. For such a process the am s to eep the qualty characterstcs as small possble, for stace: the surface roughess, the devato from a desg value, the emsso of Carbo Mooxde gas. A traoal step loss fucto or a smple aguch loss fucto ca ot capture the qualty loss case of multple qualty characterstcs. herefore, a multvarate qualty loss fucto should be appled to quatfy the qualty loss. For ths process where the qualty characterstcs are statstcally depedet, a multvarate qualty loss fucto [] ca be used: L(y y y yy ( Here, ad are the qualty loss coeffcets, whch ca be determed by usg the regresso method or solvg a system of smultaeous lear equatos. Whle the uvarate qualty loss fucto oly udges the dvdual qualty characterstcs, the multvarate approach cosders the customer s percepto of the qualty characterstcs. I ths case, the qualty characterstcs are terdepedet, leadg to a cost ( whch accouts for the customer s reactos. he qualty loss at tme t ca be expressed as: L(t, τ [ y(t, τ ] [ y (t, τ ] [ y (t, τ ][. (y (t, τ ] Whe the qualty characterstcs deped o each other, a covarace term s cluded the multvarate qualty loss fucto. Sce E [ X ] V[ X ] { E[ X ]}, where V[.] deotes varace, the expected qualty loss ca be gve by: E[L(t] ( t ( ( t ] ( t ( ( t ] ( t ( ( t ( ( t ] We assume that the chage mea does ot affect the process varablty. Cosder a deteroratg state where the effect of a postve shft ad a postve drft s based o ts radom occurrece tme τ (Fgure-. For example, a ( (3 8

3 mache ca experece wear out ad leaage at the same tme. herefore ths ot deterorato effect s mportat for some processes. Fgure-. Deteroratg process states. o represet ths ature of the process, we assume: If t < τ the (t (t (t (t (t If t > τ the (t (t (t (t W (t, τ (t W (t, τ W t, τ δ ( t τ θ W ( t, τ δ ( t τ ( θ hus, the expected loss at tme t: E[L(t] t ] ] ] t ( δ θ( t τ ] ( δ θ ( t τ ( δ θ( t τ ( δ θ ( t τ [ ] f ( τ dτ f ( τ dτ (4 We assume the deterorato follows a λτ expoetal dstrbuto.e. f ( τ λe. After performg tegrato by parts ad rearragg the terms, we fd the total expected qualty loss over the producto ru s: L(, E[ L( t] λ e A ( D D' λ λ e L(, ρ ( C E λ λ 3 e C'. 3 3 λ λ λ Where A B B' C C' E ] ] [ δ ] [ δ δ ] δ θ [ θ ] [ θ θ ] δ θ δ δ θ θ [ δ θ δ θ ] λ (5 Durg resettg of the process parameters, a adustmet cost, C A, wll be curred durg ths process. he mateace actos are carred out durg the producto perod, ad the cost of mateace accumulates up to tme. hus, the total expected cost s as follows: E[ C] L(, C A M ( t (6 he optmum process mea ad producto ru ca be foud by mmsg the expected cost per ut tme as: * * C[, ] m L(, C A M ( t (7 he total cost ca be computed by determg the optmum tal mea settgs of the two qualty characterstcs ad the optmum producto ru. heoretcally, the optmal process parameters ca be foud by settg the dervatves wth respect to each varable to 9

4 C (, C (, zero (.e. ad ad by solvg the resultg equatos. However, these dervatves are complcated ad mathematcally cumbersome. herefore, patter search techques of Hooe ad Jeeves [3] or Geeralsed Reduced Gradet (GRG algorthms ca be employed to determe the optmal process parameters. NUMERICAL EXAMPLE o llustrate the potetal dustral applcato of the proposed model, let us cosder a umercal example. Assume that a process wth two depedet smaller-the-better type (S-ype of qualty characterstcs (x ad y ad they follow a bvarate ormal dstrbuto wth gve stadard devatos (.e..49,.5 ad.5. he process adustmet cost s $ ad the producto rate s uts/moth. Further, the loss coeffcets,, ad are 3, ad respectvely. Let us also assume the process moves to a out-of-cotrol state as a result of a postve shft ad a postve drft. he deterorato fucto s expoetally dstrbuted at a falure rate, λ.5. he shft rates are cosdered to be δ ad δ. he drft rates are cosdered to be θ.4 ad θ. he mateace cost s M ( t 3t. We solve the cost model (7 usg Geeralsed Reduced Gradet, GRG. able- summarses the results. able-. Result of the umercal example. A sestvty aalyss could also be carred out to show the effect of dfferet put parameters o the output. CONCLUSIONS argetg problem s oe of the most studed problems the feld of combatoral optmsato. I partcular, ths paper has made the followg cotrbutos. Frstly, the ot determato of the optmum tal settg of process meas ad the optmum producto ru of a deteroratg process wth multple smaller-the-better type (S-ype of qualty characterstcs s developed. Secodly, the depedecy of the multple qualty characterstcs s cosdered. ACKNOWLEDGMEN he authors wll le to acowledge the support receved from CRCIEAM ad the postgraduate scholarshp to carry out the research wor. REFERENCES [] aguch G Itroducto to off-le qualty cotrol. Cetral Japa Qualty Cotrol Assocato, pp. -5. [] Sprg F. A ad Yeug A. S A geeral class of loss fuctos wth dustral applcatos. It. J. of Qualty echology. Vol. 3, pp [3] ahera K., Cha W. M. ad Ibrahm R. N. 7. Jot determato of process meas ad ru legth: A Revew. It. J. of Adv. Maufac. ech. Uder revew. [4] eeravaraprug J. ad Cho B.R.. Desgg the optmal process target levels for multple qualty characterstcs. It. J. of Producto Res. Vol. 4, pp [5] Cha W. M., Ibrahm R. N. ad Lochert P. B. 6. Desgg the optmal process meas ad the optmal producto ru for a deteroratg process. It. J. of Adv. Maufac. ech. Vol. 3, pp [6] Ba D. S. ad Lee M. K Optmal target values for a fllg process whe specto s based o a correlated varable. It. J. of Prod. Ecoo. Vol. 3, pp [7] Che S. L ad Chug K. J Determg the optmal producto ru ad the most proftable process mea for a producto process. It. J. of Prod. Res. Vol. 34, pp [8] Al-Sulta K. S. ad Al-Fawza M.A Determato of the optmal process meas ad producto cycles for multstage producto systems subect to process deterorato. It. J. of Prod. Plag ad Cotrol. 9(:

5 [9] Roa J., Gog L. ad ag K.. Jot determato of process mea, producto ru sze ad materal order quatty for a cotaer fllg process. It. J. of Prod. Eco. Vol. 63, pp [] Rahm M. A. ad uffaha F. 4. Itegrated model for determg the optmal tal settgs of the process mea ad the optmal producto ru assumg quadratc loss fuctos. It. J. of Prod. Res. Vol. 4, pp [] Harga M. A. ad Al-Fawza M. A. 5. Jot determato of target value ad producto ru for a process wth multple marets. It. J. of Prod. Eco. 96(: -. [] Kapur K. C. ad Cho B. R Ecoomc desg of the specfcato rego for multple qualty characterstcs. IIE rasactos. Vol. 8, pp [3] Hooe R. ad Jeeves. A. 96. Drect search soluto for umercal ad statstcal problems. J. of Ass. of Computg Machery. Vol. 8, pp. -9.