ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng Collg of Industral Tchnology, Nhon Unvrsty -- Izum-chou, Narashno, Chba -8, Japan msnoha@ct.nhon-u.ac.jp Kyords: complx numbr pars comparson, complx numbr AHP, ambguty Summary: N mod of ambguty xprsson, complx numbr, s ntroducd nto pars comparson and AHP. Th magnary part of judgmnt rflcts som typ of ambguty. Thr applcablty s xamnd through som xampls.. Introducton Thr can b many knds of ambguty mods. Som of thm ar probablty[kito0], fuzznss[buck8], ntrval xprsson[arbe9], and so on. As on of ambguty xprssons, propos to us complx numbr and to ncorporat t nto pars comparson and AHP, hch ar calld complx numbr pars comparson C-comparson) and complx numbr AHPC-AHP), rspctvly.. Complx numbr pars comparson C-comparson) Th j, k)th lmnt of pars comparson matrx A,, ndcats th domnanc of tm j ovr tm k, or ho many tms mor mportant tm j s than tm k. Ths a data ar usually masurd n th ral scal. Evn n cas of ambguty xstnc, thy hav bn xprssd n th forms of probablty dstrbuton[kito0], mmbrshp functon[buck8] and ntrval[arbe9]. Ths ambguty xprssons hav thr mrts and dmrts, hch ar summarzd n Tabl. As a n mod of ambguty xprsson for th pars comparson masurmnt ll propos to us complx numbr, nstad of ral numbr. Although ts masurmnt may not b so asy compard to th probablstc comparson or th ntrval comparson, th C-comparson has ts advantags n transtvty satsfacton n consstncy cas and undrstandng ts procss and rsult, hos xplanaton ll b trd. Masurng pars comparson judgmnt n complx numbr Wth th convntonal fundamntal scal, th ntnsty of mportanc rangs from to 9. Say th mportanc ntnsty strong mportanc) mans that xprnc and judgmnt strongly favor on actvty ovr anothr. Mathmatcally spakng t mans that xprnc and judgmnt tms mor favor on actvty ovr anothr. Hr all th pars comparson judgmnts ar masurd n th ral. If hav som knd of ambguty th ths ntnsty masurmnt, ho do dstngush among thm. Thn, complx numbr a ll b proposd to dstngush among th masurmnts th th sam ntnsty but dffrnt ambguty dgrs. a Procdngs th ISAHP 00 Bal, Indonsa 6
a r xp θ ) ) r cosθ + sn θ ) ) r : ntnsty or ampltud of j, k) pars comparson judgmnt θ :dgr of nclnaton angl from th ral toard th magnary magnary unt Tabl Mrts and dmrts of ambguty xprssons Probablstc Fuzzy Intrval comparson comparson comparson Masurmnt Easy Modrat Easy Wght Modrat Dffcult Dffcult stmaton Procss Modrat Qustonabl Qustonabl acountablty Valdty of Modrat Qustonabl Qustonabl rsult Transtvty Modrat Dffcult Dffcult satsfacton n consstncy cas. Estmatng complx prorty ght C-ght) from C-comparson matrx To C-ght stmaton mthods ar ntroducd n ths scton. Thy ar th por mthod and th gomtrc man mthod. Wth th por mthod, C-ght s calculatd by Eq.). p A lm ) p T p A Hr, A s a C-comparson matrx and s an all- column vctor of approprat sz. Wth th gomtrc man mthod, C-ght s calculatd by Eq.). Π r j n k j n n Π r k θ ) θ n ) n Nxt ll sho to numrcal xampls. ) Procdngs th ISAHP 00 Bal, Indonsa 6
Procdngs th ISAHP 00 Bal, Indonsa 6 Exampl Consdr a C-comparson matrx of sz gvn by Eq.). A 9 8 9 8 ) Thn, ts C-ght vctor stmatd by th por mthod s gvn by Eq.6). 0.9) 0.06) 0.08 0.096 0.06 0.99 6) Its C-ght vctor stmatd by th gomtrc man mthod s gvn by Eq.). 0.9) 0.06) 0.08 0.096 0.06 0.99 ) As can b sn from Eqs.6) and ), th to stmatd C-ght vctors concd xactly. Exampl Consdr a C-comparson matrx of sz gvn by Eq.8). A 6 6 8)
Its C-ght vctor stmatd by th por mthod s gvn by Eq.9). 0.606 0.88 0.8 0.09 0.8 0.) 0.). ) 9) Its C-ght vctor stmatd by th gomtrc man mthod s gvn by Eq.0). 0.68 0.8 0.68 0.0 0.68 0.) 0.).69 ) 0) As can bn sn from Eqs.9) and 0), th to stmatd C-ght vctors ar slghtly dffrnt. Thy do not concd n ths cas of sz. Basd on ths numrcal xampls, can say that, for a gnral, not ncssarly consstnt, C-pars matrx, th to stmatd C-ght vctors ar th sam n th cas of sz and thy can b dffrnt n th cas of sz or mor.. Complx AHP vs Ral AHP Through a car-slcton AHP xampl of Fg., ll llustrat ho C-comparson s ncorporatd n Complx AHP. Thr ral-valud comparson matrcs ar gvn n Tabl, hr th judgmnt s consdrd crsp th no ambguty. Th prorty ght vctor for th crtra s gvn by Eq.). c 0. 0.09 0.09 0.09 ) Th prorty ght vctor for th altrnatvs, A, B and C, s gvn by Eq.). A 0.88 0.80 0.86 ) Procdngs th ISAHP 00 Bal, Indonsa 6
Fg. AHP dagram of car slcton problm Nxt, th valu of a s changd from to xp ) n th pars comparson matrx A o among th crtra Tabl ). Ths mans that som ambguty of nclnd angl s consdrd n th judgmnt of a. Thn, th prorty ght vctor for th crtra s gvn by Eq.). * c 0. 0. 0.096 0.089 0. 0.09 0.6 0.0 ) Th prorty ght vctor for th altrnatvs s gvn by Eq.). * A 0.8698 0.0908 0.8 0.0 0.0 0.0606 ). Concluson In ordr to xprss som knd of ambguty n th dcson makng, propos to us complx numbr pars comparson C-comparson) and complx numbr AHP C-AHP). C-comparson and C-AHP ar llustratd through som xampls. Valdty and applcablty of C-comparson and C-AHP stll rman complx and uncrtan, hch ar futur rsarch subjcts. Procdngs th ISAHP 00 Bal, Indonsa 6
Tabl : Pars Comparson Matrcs Tabl : C-comparson Matrx Rfrncs Masahro KITO and Masaak SHINOHARA : Proposal of Probablstc AHP, Procdng of th Acadmc Confrnc of Nhon Unvrsty, Mathmatcal Informaton Engnrng Dpartmnt, CIT, -9, pp.-800.). J.J.Buckl :Fuzzy Hrarchcal Analyss, Fuzzy Sts and Systms,Vol.,pp.- Procdngs th ISAHP 00 Bal, Indonsa 66
98). A.Arbl and L.G.Vargas : Th Analytc Hrarchy Procss th Intrval Judgmnts, Multpl Crtra Dcson Makng, Sprngr Vrlag, pp.6-099). Procdngs th ISAHP 00 Bal, Indonsa 6