A RANKING PROCEDURE FOR FUZZY DECISION-MAKING IN PRODUCT DESIGN

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Isabel Caldwell
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1 Ornl rtl rons o IDMME - Vrtl Conpt 2008 jn Cn Otor RNKING ROCEDURE FOR FUZZY DECISION-MKING IN RODUCT DESIGN Mro rjs r r 2 ): Déprtmnt Mtémtqs t Gén Instrl Éol olytnq Montrél C S. Cntr-vll Montrél QC) H3C37 CND E-ml: mro.rjs@polymtl. 2): CIRRELT Déprtmnt Mtémtqs t Gén Instrl Éol olytnq Montrél C S. Cntr-vll Montrél QC) H3C37 CND E-ml: r.r@polymtl. strt: T son-mkn pross s ky ss or most mntrn n srvs ornztons n orms n ssntl prt o t prot sn pross. In snrs r lwys onront wt t lmm o oosn twn rnt ltrntvs wt rnt prmtrs otn xprss n rly v trms. For ts rson mny tools v n propos orn mor n ttr wys to mprov ompny son-mkn n ts r. T rnkn o zzy nmrs s on o t most mportnt o ts tools s zzy lo s powrl pty to mn t v prmtrs slly ssot wt t xprsson o prot rqrmnts y t mn prsonlty. Improvn t ry o s tools s wy to mprov prot sn. In ts vn n xtnson o prvos lortms s propos or t rnkn pror n zzy son-mkn pross. Ts xtnson s s on rml trpzol zzy nmrs n spports rtnlr n trnlr rml zzy nmrs s wll) nst o trtonl trnlr zzy nmrs. T rnkn pror s s on t zzy prrn rlton. prot sn xmpl s prov. Ky wors: prot slton zzy rnkn zzy son-mkn zzy prrn rlton mmn stn - Introton T slton o prot or sp stomr s possl tro son-mkn pross n w t stomr vlts t vnts n svnts o prot rltv to otrs. In t s wr stomrs v sp rqrmnts t stomrs vlt t prots wt t m o ntyn t prot wt t potntl to prov t st lvl o stston. It my tt prot orrspons xtly to ts rqrmnts t my orrspon t som lvl o stston. Fzzy nmrs v t pty to rprsnt s rs o stston. O two rnt prots provn rnt r o stston t stomr slts t prot tt omnts. orn to Tsn n Kln ) mny rnkn mtos or zzy nmrs v n vlop. Howvr tos mtos t onsr mny mportnt tors s s t sp rnkn orr n rltv prrn or omnn o zzy nmrs s wll s t s o omptton o t rnkn lortm. It s tror nssry to vlop nw rt tv n nt lortm l to rnk vros sps o zzy nmrs. L 2000) onsr tt t mtos or rnkn zzy nmrs n lss nto two tors. T rst s s on zzton n t son on t zzy prrn rlton. H mntns tt oo rnkn mto sol stsy t ollown or rtr: ) zzy prrn prsntton; 2) t rtonlty o prrn orrn; 3) rostnss; n 4) ny. T lortm or t rnkn pror n t zzy son-mkn pross prsnt n ts work xtns t lortm propos y Tsn n Kln 989) w onsrs omntons o trnlr zzy nmrs. T propos xtnson ss rml trpzol zzy nmrs s nrl mol mon w rtnlr n trnlr zzy nmrs r prtlr ss. ny prws omnton o tm s spport. prot sn xmpl provs ts prormn pty n t prot sn son-mkn pross. T ppr s ornz s ollows. Ston 2 prsnts ltrtr rvw. Ston 3 srs t propos rnkn pror. Ston 4 vs n xmpl ppl to prot sn. Ston 5 onls t ppr. 2- Ltrtr rvw 2.- Fzzy lo n prot sn rot sn nvolvs vros pss lk vlton omprson n son-mkn. T pplton o zzy pr Nmr -- Copyrt o IDMME - Vrtl Conpt

2 IDMME - Vrtl Conpt 2008 Sort rtl Ttl lo troot ll ts pss mks t possl to nl mor rt rmton rlt to stomr srs. Svrl ppros mtos n mols v n propos or t prot sn pross w mr rnt zzy mols s s t zzy ol prormmn mol to trmn t lvl o llmnt o t sn rqrmnts Cn n Wn 2006) Grn Fzzy Dsn nlyss w ss zzy lo to vlt prot sn ltrntvs s on nvronmntl onsrtons n Fzzy Mlt-ttrt Dson-Mkn to slt t most srl sn ltrntv Ko t l. 2006). Sow 2006) propos mo zzy mol to oornt n l wt t tnl zzy n n-zzy sss nvolv n prot sn. Morovr t Intrnt s w n s n ts l som xmpls o w nl Sq n Nnn 2005) wo prsnt n Intrnt-s rmwork w ss rmmtl ppro to rprsnt n vlop mols o stomz prots n orr to ntrt stomr srs nto t sn o mss-stomz prots. T Intrnt s lso n ppl to vlop W-s vrtl sn nvronmnt mto w llows stomrs to prtpt n prot sn n lp snrs onvnntly jst t strtr o tr prots. Ts mto ws pls y Sn t l. 2005). Drn t lst w yrs lr nmr o pplton opportnts v ppr to tk vnt o zzy lo n t prot sn pross. It sms tt prot sn s n r wt rt potntl or t pplton o zzy lo rjs n r 2008) Fzzy son-mkn n prot sn rot sn s n nnrn pross nvolvn trtv n omplx son-mkn. It slly strts wt t ntton o n pros tro sqn o tvts to n n optml solton to t prolm n ns wt tl srpton o t prot D t l. 2005). Vros ppros v n propos or t zzy mltrtr son-mkn pross s prnplly on t zzy prrn rlton Fn t l. 2002; üyüközkn n Fyzol 2005; Işıklr n üyüközkn 2006; Zn t l. 2007). Sn n W 2006) propos n ppro or t rnkn pross s on n sy n nttv zzy smlton nlyss mto. Smlrly som zzy rnkn mtos or t son-mkn pross v n vlop. Cn n Kln 994) ppl t α-t n zzy strton oprtons to llt t r nr t nw zzy nmr. Wn n rkn 2005) prsnt tr optmzton mols to ssss t rltv mportn wts o ttrts n mltpl-ttrt son-mkn prolm. In ll ts ppros t zzy rnkn o zzy nmrs plys n mportnt rol s prt o t son-mkn pross. pplton o zzy rnkn or mlt-rtr sonmkn s n Kwkrnk 977; Cn n Kln 994). T zzy prrn rlton s n wly s or t zzy rnkn pror Dlo t l. 988; L 2000; Morrs n S-Nz 200). Otr mtos nl t pplton o sp onpts lk trnlr mmrsp ntons Cn 98) n mxmzton n mnmzton o sts Cn 985). L n Yo 2003) prsnt zzy rnkn mto or zzy nmrs w onsrs vros ntrstn ntons n ns s s t zzy stston nton t zzy vlton vl r o zzton r o vlton n t rltv nx o zzton. vl mto rportn zzy prrns n rn rton tnqs ws propos y M n L 2008). Yn 99) prsnt or rtr or vltn zzy rnkn mtos n sst n mprov rnkn mto s on zzy prrn rprsntton t rtonlty o zzy orrn stnslty n rostnss. 3- Rnkn pror n zzy son-mkn s mnton n t prvos ston t propos lortm or t rnkn pror n zzy son-mkn s n xtnson o Tsn n Kln 989) n onssts o mor nrl mol w onsrs rnt sps o zzy nmrs or rnkn. Dntons o nrn n omnn r xpln n ston 3.; ston 3.2 xplns t zzy prrn rlton; n ston 3.3 prsnts t prrn rlton lortm. 3.- Inrn n omnn Lt n two zzy nmrs w r onvx n rml zzy ssts. I n r two zzy nmrs tn tr xst two tons nrn n omnn twn t zzy nmrs. - I tr xsts n r o ovrlp twn zzy nmrs n ntrston twn n ) tn t ovrlp r s n s nrn; tt s n r nrnt to on tr n tt r. 2- I tr xst on or mor n-ovrlp rs twn zzy nmrs n tn or n-ovrlp r tr omnts or omnts. Ts tons or t nrl trpzol zzy nmrs n n sn n Fr. In s ) omnts n n s ) omnts. Cs ) sows t ton o nrn rprsnt y t ntrston r twn zzy nmrs n Rnkn mtos T rnkn o zzy nmrs orms t ss o t sonmkn pross wt zzy lo L n Yo 2003). s o t mportn n ppllty o t rnkn pross svrl mtos v n vlop nln t ) ttp:/ ) ) Fr : Domnn n nrn twn n. pr Nmr -2- Copyrt IDMME - Vrtl Conpt

3 IDMME - Vrtl Conpt 2008 Sort rtl Ttl orn to t ov ntons o nrn n omnn t n-ovrlp rs rprsnt t omnton rs or or. T omnton twn zzy nmrs s vn y t rton o n Fzzy prrn rlton I n r two zzy nmrs tn t zzy prrn rltons R ) n R ) r n s ollows: R ) R ) rs wr omnts ) + r wr n r nrnt) r o )+r o ) rs wr omnts ) + r wr n r nrnt) r o )+r o ) It s tn ovos tt R) + R) wr R ) n R ) r ntrprt s t r to w s prrr to n s prrr to rsptvly. T rs wr omnts or omnts n otn sn t Hmmn stn. Fr 2 lso sows t posslty o omprn ny prws omnton o trpzol trnlr n rtnlr zzy nmrs.. trnlr-trnlr trnlr-trpzol trnlr-rtnlr n so on). Lt n two rml trpzol zzy nmrs wr t spport o s t ntrvl ) n t spport o s t ntrvl ). T trnlr zzy nmr s prtlr s wn or or or zzy nmrs rsptvly. In t sm wy Rtnlr zzy nmrs n possl wn n or zzy nmr n wn n or zzy nmr s Fr 2). Css r pnn on t zzy nmr s llstrt n Fr 3). Tr my rnt nmrs o ponts o ntrston pnn on t rltv poston o t zzy nmrs. ll ts ss r pt n Tl. For s t s tn nssry to onsr t rlvnt rs wn omptn. Lt S n ntrvl n t rl ln R. T Hmmn stn twn two zzy nmrs n on S s tn n y wr ) D S) ) S S R D R) D ) T Hmmn stn twn zzy nmrs n s tlly t n-ovrlp r o t two nmrs; tt s t sm o t rs wr tr omnts or omnts or ot. In ts ppr t Hmmn stn s s to otn t prrn rltons twn two zzy nmrs. W llstrt ts onpt onsrn rml trpzol zzy nmrs s wll s trnlr n rtnlr zzy nmrs s Fr 2). Fr 3: Som possl ntrton ponts twn zzy nmrs rrn rlton lortm s on t ntons o omnn n nrn t ollown lortm n s to trmn prrn rlton. Stp ) Fn t r wr n r nrnt ntrston r) Stp 2) Fn t rs wr omnts Stp 3) Fn t rs wr omnts Stp 4) Fn t rs o n Stp 5) Compt t zzy prrn rltons R) n R) Fr 2: Fzzy prrn rltons or trpzol trnlr n rtnlr zzy nmrs. Lt R) t zzy prrn rlton n R ) t mmrsp nton rprsntton o R). T orrn o zzy nmrs n s n s ollows. pr Nmr -3- Copyrt IDMME - Vrtl Conpt

4 IDMME - Vrtl Conpt 2008 Sort rtl Ttl pr Nmr -4- Copyrt IDMME - Vrtl Conpt Tl : Nmr o ponts o ntrston n rltv poston o t zzy nmrs. onts o ntrston Cs sttmnt s on rml trpzol zzy nmrs I t mmrsp r R ) s rtr tn 0.5 tn t orrn s n s s prrr to or. I t mmrsp r R ) s ql to 0.5 tn t orrn o n s n s s nrnt to or ~. I t mmrsp r R ) s lss tn 0.5 tn t orrn s n s s prrr to or. To xtn t rlr ntons Tsn n Kln 989) w pply r pso-orr prrn mol or t otrnkn o t zzy nmr pror Roy n Vnk 984; Wn 997; Gr n rkn 2000; n ykozkn n Fyzol 2004). Lt t zzy prrn rlton twn two s n or rtron otn y prws omprson o ) n ) w wll sow t lnst prormn o s n rsptvly. ) n ) r rprsnt y zzy nmrs. Tr typs o prrn rlton r n n trms o t zzy prrn rltons twn two ltrntvs. C : ) ) p ) ) p Q ) ) q I wr n Q pt strt n wk prrn rsptvly n I pts nrn. T prrn trsol p n nrn trsol q n y ommon sns Roy n Vnk 984) r s to srmnt twn t nrn strt prrn n wk prrn o two ltrntvs or rtron. 4- ppltons Ston 3 provs rnkn pror n zzy sonmkn w mks t possl to trmn prrn rltons twn two rml zzy nmrs trpzol trnlr rtnlr n mx o ts). Ts ol sl n t son-mkn pross. Consr t sn o prot to stsy sp mrkt sr. ompny s mrktn prtmnt rr ot n nlyss to vlt t stomr s rqrmnts or s potntl prot. Ts nlyss provs nswrs w sr t prot s ntonlts n rtr v srptv trms or xmpl: T prot sol ml-rn on. T pr sol twn x n x 2. Most stomrs prr rtrst C to smll n rtrst C 2 to. n so on Dln wt s rmton my lt. It s somtms possl to vlt som rtr rly prsly. n vlton o t pr or xmpl s rrnt ntor o prrn wn snn t prot. It ol mor omplt wn t rtrsts r lt to vlt owvr. C ol t prpton o prot sz n C 2 prot rllty. lso t trms n smll v to n. It s sl to onsr onrs n s ss. C my onsr ptl or xmpl t prot sz s lss tn 50 mm n t wt or mtr s Fr 4).

5 IDMME - Vrtl Conpt 2008 Sort rtl Ttl C s ptl) ys 50 prot sz mm) Fr 4: Dsrt vlton or stomr rqrmnt C. Wt s t st ltrntv or t sn tm to slt n wy? Ts qstons my lt to nswr wtot n pproprt mto. Ston 3 provs ll t lmnts o s mto w s stp-y-stp ppro. For ltrntv rnkn o rtrst wll prorm. On possl zzy molln o t rtrsts or ltrntv n ltrntv 2 s prov n Fr 7 rnt sps r possl). t wt ppns t prot s 5 mm or 52 mm or 55 mm n n sp sz. Som stomrs my wlln to pt t prot wt mor zzy onrs. My s t rows to 60 mm wr n wr stomrs wll onsr t prot to smll. zzy rprsntton o t rqrmnt s tn st pt s Fr 5). C s ptl) C ltrntv ltrntv 2 ys C prot sz mm) prot sz mm) Fr 5: Fzzy vlton or stomr rqrmnt C. T sm ppls or rtron C 2. Consr or xmpl tt rtron C 2 tks t ollown sp Fr 6) vlt on [0 0] sl 0 t prot wll nvr work 0 t prot wll lwys work!). C 2 s ptl) ys prot rllty Fr 7: Fzzy molln o t prot ltrntvs. For ltrntv n rtrst C t mto onrns t rnkn o t ollown two zzy nmrs Fr 8). C s ptl) prot rllty Fr 6: Fzzy vlton or stomr rqrmnt C 2. Consr w tt tr r two ltrntvs wn snn t prot n tt t sn tm s to w on to slt. ltrntv s prot wr C s 53 mm n t s rllty C 2 o 6.5 wl ltrntv 2 s prot wt C 47 mm n rllty C 2 4 s Tl 2). Tl 2: pproxmt rtrsts o t prot ltrntvs to ompr. C sz) C 2 rllty) ltrntv 53 mm 6.5 ltrntv 2 47 mm 4 ys ltrntv Cstomr prrn prot sz mm) Fr 8: Fzzy rnkn o ltrntv or rtrst C. Dt s t stomr prrn n s ltrntv. Tn Dom ) s t r wr zzy nmr omnts zzy nmr In-) s t r o nrn twn n r) s t r o zzy nmr n r) s t r o zzy nmr. To otn t zzy prrn rlton ston 3.2) w n: Dom ) 0 pr Nmr -5- Copyrt IDMME - Vrtl Conpt

6 IDMME - Vrtl Conpt 2008 Sort rtl Ttl In-) r) 55 r) 30 Tn: R ) For ltrntv 2 n rtrst C t sm ppls: Dom ).5 In-) 30 r) 55 r) 30 Tn: R ) For ltrntv n rtrst C 2 w otn: Dom ).7500 In-) 3.5 r) 5.5 r) 3.5 Tn: R ) For ltrntv 2 n rtrst C 2 w otn: Dom ) In-) r) 5.5 r) 3.5 Tn: R ) Tl 3 smmrzs tos rslts. Tl 3: Comprtv zzy prrns.. C sz) C2 rllty) ltrntv ltrntv I rtrst o n ltrntv mts t stomr rqrmnt xtly w wll v: r) r) In-) n Dom ) 0 Tn: R ) 0.5. For rtrst t st ltrntv s tn t on losst to 0.5. Tl 4: Evlton o t st ltrntv or rtrst. C sz) C 2 rllty) ltrntv ltrntv Tl 4 sows tt or t rst rtrst C ) ltrntv 2 s t st o. For t son rtrst C 2 ) ltrntv s t st o. t ts pont t s t possl to trmn t st ltrntv. T sn tm s to prortz rtrsts C n C 2. Consr t ollows wt o mportn w ) or rtrst: w 0.4 or rtrst C w or rtrst C 2 Tn t wt vr or ltrntv s s ollows: R n j j n R ) * w w For ltrntv : R [0.336) 0.4) ) 0.6)]/ ) For ltrntv 2: R 2 [0.3706) 0.4) ) 0.6)]/ ) Ts s to ompr to 0.5. W n w nlly onl tt ltrntv s ttr tn ltrntv 2 or mtn t stomr rqrmnt. 5- Conlsons In ts ppr xtn sop or t rnkn pror s n propos s on t zzy prrn rlton n t zzy son-mkn pross. n xmpl o ow prot sn son s r sows ts ppllty to t solton o rl prolms. Wt or pror t s possl to otn t prrn rlton strt wk or nrn) or ny prws omnton o rml zzy nmrs. For t prot sn pross ts ontrton s ly mportnt s mny prmtrs rom rl ppltons r t strt n molln wt zzy nmrs w r l to rprsnt v prmtrs s xprss y mn) wll nr t son-mkn pross. Ts powrl pty s rtl spt o prot sn w ol snntly mprov t sn pross. T xmpl prov sows possl pplton wr rnt prot ltrntvs r ompr wr t prmtrs r vlt wt v prmtrs or prmtrs xprss n t orm o prrns). Evn wt mltpl vrls n rnt prrn rltons or vros prmtrs t mtooloy s l to pont to t st omproms. tr xtnson o or rslt r wll rss prot onrton n t sn o prot mls. 6- kwlmnts Ts rsr s n spport y nn rom t Ntrl Sns n Ennrn Rsr Conl o Cn NSERC) n y t Fons Qééos l Rr sr l Ntr t ls Tlos FQRNT). 7- Rrns [K] s S. M. n Kwkrnk H. 977). Rtn n rnkn o mltpl-spt ltrntvs sn zzy sts tomt pr Nmr -6- Copyrt IDMME - Vrtl Conpt

7 IDMME - Vrtl Conpt 2008 Sort rtl Ttl [] rjs M. n r. 2008). T s o zzy lo n prot mly vlopmnt: ltrtr rvw n opportnts smtt. [F] üyüközkn G. n Fyzoğl O. 2004). zzylo-s son-mkn ppro or nw prot vlopmnt Intrntonl Jornl o roton Eoms [F2] üyüközkn G. n Fyzol O. 2005). Grop son mkn to ttr rspon stomr ns n sotwr vlopmnt. Comptrs & Instrl Ennrn [C] Cn W. 98). Rnkn o zzy tlts wt trnlr mmrsp ntons ro. Intrntonl Conrn on oly nlyss n Irmton Systms [CK] Cn C.-. n Kln C. M. 994). Fzzy Rnkn Mtos or Mlt-ttrt Dson Mkn. IEEE Con. on SMC Sn nto US [CW] Cn L. H. n Wn M. C. 2006). n vlton ppro to nnrn sn n QFD prosss sn zzy ol prormmn mols. Eropn Jornl o Oprtonl Rsr [C2] Cn S. 985). Rnkn zzy nmrs wt mxmzn st n mnmzn st Fzzy Sts n Systms [DO] D E.. Ostros E. Frny M. n Gor M. 2005). Conrl prot sn sn mltpl zzy mols. Jornl o Ennrn Dsn [DV] Dlo M. Vry J. L. n Vl M.. 988). pror or rnkn zzy nmrs sn zzy rltons Fzzy Sts n Systms [FM] Fn Z.-. M J. n Zn Q. 2002). n ppro to mltpl ttrt son mkn s on zzy prrn rmton on ltrntvs. Fzzy Sts n Systms [G] Günör Z. n rkn F. 2000). zzy otrnkn mto n nry poly plnnn. Fzzy Sts n Systms [I] Işıklr G. n üyüközkn G. 2006). Usn mltrtr son mkn ppro to vlt mol pon ltrntvs. Comptr Stnrs & Intrs [KC] Ko T. C. Cn S.-H. n Hn S. H. 2006). Envronmntlly onsos sn y sn zzy mltttrt son-mkn. Intrntonl Jornl o vn Mntrn Tloy [L] L H. S. 2000). nw zzy rnkn mto s on zzy prrn Rlton IEEE Intrntonl Conrn on Systms [LY] L J.-H. n Yo K.-H. 2003). zzy Rnkn Mto or Fzzy Nmrs IEICE Trns Fnmntls E [ML] M L.-C. n L H.-L. 2008). zzy rnkn mto wt rn rton tnqs Eropn Jornl o Oprtonl Rsr [MS] Morrs M. n S-Nz S. 200). Rnkn zzy nmrs y prrn rto Fzzy sts n Systms [RV] Roy. n Vnk. 984). Rltonl Systms o rrns wt On or Mor so-crtr: Som Nw Conpts n Rslts. Mnmnt Sn [S] Sow G. 2006). Dssson o t sn plosopy n mo n-xprt zzy st mol or ttr prot sn. Jornl o Ennrn Dsn [SZ] Sn L. Zo Z. L M. Zo W. L Y. W M. n Zn J. J. 2005). Costmr ornt vrtl ooprtv prot sn. 9t Intrntonl Conrn on Comptr Spport Cooprtv Work n Dsn IEEE [SN] Sq Z. n Nnn J ). rmmtl ppro to spport rl-tm sn o stomz prots. DETC2005: SME Intrntonl Dsn Ennrn Tnl Conrns n Comptrs n Irmton n Ennrn Conrn Lon C Unt Stts: mrn Soty o Mnl Ennrs Nw York NY. DETC [SW] Sn H. n W J. 2006). nw ppro or rnkn zzy nmrs s on zzy smlton nlyss mto ppl Mtmts n Comptton [TK] Tsn T..Y. n Kln C. M. 988). srvy n omprtv sty o rnkn prors n zzy son mkn: Dpt. In. En. Unv. Mssor-Colm Workn pr [TK2] Tsn T. Y. n Kln C. M. 989). Nw lortm or t rnkn pror n zzy son mkn IEEE Trnston Systms Mn n Cyrnts [W] Wn J. 997). zzy otrnkn mto or onptl sn vlton. Intrntonl Jornl o roton Rsr [W] Wn Y.-M. rkn D. 2005). Mltpl ttrt son mkn s on zzy prrn rmton on ltrntvs: Rnkn n wtn. Fzzy Sts n Systms [Y] Yn Y. 99). Crtr or vltn zzy rnkn mtos Fzzy Sts n Systms [ZW] Zn Q. Wn Y. n Yn Y. 2007). Fzzy mltpl ttrt son mkn wt t typs o prrn rmton on ltrntvs. IEEE Symposm on Compttonl Intlln n Mltrtr Dson Mkn. MCDM pr Nmr -7- Copyrt IDMME - Vrtl Conpt

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5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our