ON RANKING OF ALTERNATIVES IN UNCERTAIN GROUP DECISION MAKING MODEL

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1 IJRRAS (3) Ju _5.pdf ON RANKING OF ALRNAIVS IN UNCRAIN GROUP DCISION MAKING MODL Chao Wag * & Lag L Gul Uvrty of chology Gul 544 Cha * mal: wagchao244@63.com llag6666@26.com ABSRAC h wghtd gomtrc ma ordrg vctor ad logarthmc lat quar ar propod to aggrgat formato of dco mar to grat wght vctor from ucrtaty comparo matrc ucrta group AHP modl. pcally wh th judgmt matrc ar complt th lattr ca b dvlopd to drv prort udr th codto of mmal dvato. Som cocpt ucrtaty thory ar troducd ad th ucrtaty dtrbuto ar aumd to b lar. wo umrcal xampl ar xamd to how th applcato of th propod mthod. Kyword: Group dco mag AHP Ucrtaty thory Ucrta AHP Ucrta group AHP.. INRODUCION h Aaly Hrarchy Proc (AHP) a multcrtra dco mag (MCDM) tchqu propod by Saaty [] that grat th rlatv wght of crtra. h advatag of th MCDM tool t capablty of hadlg ubjctv crtra ad vagu judgmt th dco mag proc. Dco mag problm th proc of fdg th bt choc from all of th altratv. I almot all uch problm th multplcty of crtra for judgg th attrbut prvav. For may uch problm th xprt wat to olv a multpl crtra dco mag (MCDM) problm. A MCDM problm ca b cocly xprd matrx format a D A x x x A x x x A x x x whr A A2 A ar pobl altratv amog whch dco mar hav to choo 2 th dgr of prfrc of attrbut rato cal wth rpct to crtro C 2 m x j 9 9 h rarch wa upportd by th Guagx Natural Scc Foudato of Cha udr th Grat NO. 2GXNSFA849; Iovato Projct of Guagx Graduat ducato udr th Grat No M A ovr attrbut. A j whch ta from th A w all ow t mportat to gv th judgmt of o attrbut ovr aothr th applcato of AHP. I clacal MCDM mthod w alway codrd that thr wa o dco mar who gav th comparo matrc. Howvr ral-lf tuato vral or may dco mar wll partcpat th proc of dco mag ad occaoally th judgmt matrc gv by dco mar ar complt. W call th two ca group dco mag ad complt judgmt matrc. Morovr du to th complxty ad ucrtaty volvd ral-world dco problm ad hrt ubjctv atur of huma prfrc judgmt t alway uraltc ad fabl to obta xact rato x j 2. h th mthod ug fuzzy trval or ucrta judgmt for part or all of th judgmt a parw comparo matrx ar mployd aturally. A umbr of tchqu hav b dvlopd to u uch a fuzzy trval or ucrta comparo matrx to drv wght. I th followg cto w focu o th rag of altratv ucrta group dco mag modl udr complt ad complt formato vromt. I clacal AHP Wag ad Xu [2] propod vral mthod for group dco mag modl to obta rag vctor of altratv uch a gvalu mthod [3] logarthmc lat quar [4] ad mmum dvato mthod. Wth th mthod thy ca drv th wght vctor ad gav th codto of ra prrvato. Bcau of th complxty ad ucrtaty of ral-world dco aaly problm ad th ubjctvty of huma judgmt Ch [5] mployd xto of th OPSIS (tchqu for ordr prformac by mlarty to dal oluto) u a lgutc dco proc to olv th multpl crtra dco mag problm udr fuzzy vromt. Ucrtaty thory propod by Lu [6] 27 ad rfd by Lu [7] 2 provd a w approach to dal wth o-dtrmtc factor. Ad mor dtal ca b obtad h wor [8]. Wag [9 ] ubqutly

2 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag troducd th ucrta varabl to AHP ad provdd th UVM (ucrta varabl mthod) to grat th wght vctor wth th ucrta maur. It appld to o xprt ytm of AHP. I th proc of dco mag ot all th judgmt ca b gv actually om of thm mg bcau of lacg formato. So th ca th matrc gv by xprt ar complt. W call thm complt comparo matrc. Study o th rag of altratv bad o complt comparo matrx th problm of drvg rag ordr udr complt formato vromt. I th papr w furthr xtdd th UVM to dvlop a mthodology for olvg mult-pro mult-crtra dco mag problm ucrta vromt. Codrg th ucrtaty th dco data ad group dco mag proc wh xpctd ucrta varabl ar ud to a th mportac of o attrbut ovr aothr wth rpct to ach crtro w ca covrt th ucrtaty comparo matrx [8] to a clacal judgmt matrx ad th mthod gvalu mthod ad logarthmc lat quar mtod abov ca b ud. Fally wght vctor thr udr complt or complt formato vromt drvd aturally. h papr orgazd a follow. I Scto II ucrtaty thory frt troducd mpl word ad om bac cocpt ad proprt ar gv ad algorthm for clacal AHP ar dcud. Scto III th ma part of th papr ucrta group AHP olvd ug th propod mthod. Scto IV prt two umrcal xampl to how th applcato o ucrta group AHP. Fally th cocluo ar provdd Scto V. 2. PRLIMINARIS 2.. Ucrtaty hory ad Ucrta AHP I th followg w brfly rvw om bac dfto of ucrtaty thory ad ucrta AHP. h bac dfto ad otato blow wll b ud throughout th papr utl othrw tatd. Dfto. (Lu [6]) A ucrta varabl a maurabl fucto from a ucrtaty pac LM to th t of ral umbr.. for ay Borl t of ral umbr th t B B a vt. Dfto 2. (Lu [8]) h ucrtaty dtrbuto of a ucrta varabl dfd by for ay ral umbr x. Lar ucrtaty dtrbuto ab L gv by x M x f x a x x a b a f a x b f x b whr a ad b ar ral umbr wth a b ( Fgur.). Dfto 3. (Lu [6]) A ucrtaty dtrbuto ad to b rgular f t vr fucto xt ad uqu for ach. xpctd valu th avrag valu of ucrta varabl th of ucrta maur ad rprt th z of ucrta varabl. Dfto 4. (Lu [6]) Lt b a ucrta varabl. h th xpctd valu of dfd by M M provdd that at lat o of th two tgral ft. r dr r dr. Dfto 5. (Lu [6]) Lt b a ucrta varabl wth rgular ucrtaly dtrbuto. h th vr fucto calld th vr ucrtaty dtrbuto of. 387

3 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag (x) a b x Fgur. A Lar Ucrtaty Dtrbuto. horm. (Lu [8]) Lt b a ucrta varabl wth ucrtaty dtrbuto. If th xpctd valu xt th Lt ab x dx x dx. L b a lar ucrta varabl. h t vr ucrtaty dtrbuto a b ad t xpctd valu [8] a b a bd. () 2 Dfto 6. (Wag [9]) Lt A b a ucrtaty comparo matrx of ordr who tr ar ucrtaty dtrbuto 2 ; j of ucrtaty rato 2 ; j th th ucrtaty comparo dotd by Sc ucrtaty dtrbuto A ; j of vr ucrtaty rato j x j x 2 ; j caot b aly dtrmd th of Dfto 2. So th papr w jut d to obta th ucrtaty dtrbuto of uppr tragular ucrtaty rato th lowr tragular ucrtaty rato ca b obtad by j whr th ucrta maur (cofdc lvl) wth. 2 ; j 388

4 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag Dfto 7. W call a matrx A a complt comparo matrx f thr o rato a ca ot b dtrmd dotd a. Appartly a. j 2.2. Mthod for Group Dco Mag (Group AHP) hr ar may wor [4 2] o group AHP ad a umbr of tchqu hav b dvlopd to u uch a comparo matrx to drv wght vctor. h mthod of wghtd gomtrc ma ordrg vctor ad logarthmc lat quar ar troducd hr. Lt b th umbr of xprt (dco mar) ad D D2 D b th comparo matrc rpctvly D d. Aftr chcg th cotcy w drv th wght vctor ug th Rght whr 2 gvctor Mthod (VM uch a (2)) Dw w (2) whr D d th potv parw comparo matrx who tr ar cho th t of valu max ud to tt th cotcy of D ad w w w w 2 th wght vctor of th attrbut w d wh t ormalzd. For ach xprt th ordrg vctor 2 w w w w 2 ad th comprhv wghtd gomtrc ma ordrg vctor w w w2 w whr. h w ca calculat th tadard dvato ad th tadard dvato w w 2 w j j 2 (3) w w w 2 w 2 (4) j of w j w w 2 (5) j j j of th w comprhv comparo matrx D d w wj d d 2 (6) from whch th fdbac formato ca b obtad by xprt for modfcato. Lootma propod logarthmc lat quar to hadl th tuato of group judgmt ad complt judgmt D d mxd. If 2 wth a complt comparo matrx th j d 2 w wj m log log. w w w2 w atfd (7) 3. H PROPOSD MHOD A wll ow whatvr th mthod of wghtd gomtrc ma ordrg vctor or logarthmc lat quar thy apply to th matrc whch tr ar xact. Wh w cotruct th ucrtaty comparo matrc th of Dfto 6 th lmt ar ucrtaty dtrbuto dotd L ab. h rato ar ucrta varabl. 389

5 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag 3.. Complt Iformato vromt h propod th mthod ca ot b ud o th ucrtaty comparo matrc. Howvr from th Dfto 4 th xpctd valu of ucrta varabl a xact umbr ad ca b aly calculatd ug th horm. Subqutly th w comparo matrx xprd a whr th xpctd rato A A D A j ad I ordr to obta th approxmat valu of gvctor (wght vctor) w ormalz th comparo matrx ug th formula j w 2. j Accordg to q. (3) ad (9) th compot wght vctor (8) D (9) w w w2 w ca b computd aturally. What mor cotcy rato CR.. ad th q. (5) ad (6) wll rflct th formato of comparo matrx Icomplt Iformato vromt I ucrtaty thory vr ucrtaty dtrbuto ca traform a ucrta varabl to a xact umbr wth ucrta maur. W ow that a xact umbr ad.5 wll ta. Lt.5 f thr ar lmt.5 d D th mot pobl valu 2 th complt comparo matrx udr th complt formato vromt. So q. (7) xprd a j whr a b.5 If th x w w w j x x 2 w wj m log.5 log. D th () a b of horm. 2 x y l.5 ad x j. h formula (7) rprtd by From how abov w drv th quato t j f d f d y x x 2 m. j 39

6 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag It ca b mplfd a x j j j j x y 2. I x I x y 2 () j j whr I th umbr of lmt whch ca b judgd th th row whch gv ad y th logarthmc um of judgd lmt th th row. I rprt th umbr of d Accordg to q. () w ca obta th oluto of x 2 2 drvd from x. hrfor accordg to th wght vctor w ca dtrm th rag ordr of all altratv ad lct th bt o from amog a t of fabl altratv. I a word a algorthm of th mult-pro multcrtra dco mag wth ucrta varabl approach gv th followg. Stp : Form a group of dco mar ad th dtfy th valuato crtra. Stp 2: Judg th rlatv mportac attrbut A ovr attrbut A ad cotruct th complt or complt ad ubqutly comparo matrc. Stp 3: Comput th xpctd valu of ach ucrta varabl ad cotruct th matrx a (8). j w w w w Stp 4: If t udr complt formato vromt calculat w w w ug (9) (4) ad (3) rpctvly ad th th wght vctor w obtad. Stp 5: Solv th quato t ug () f t udr complt formato vromt. Stp 6: Calculat th wght vctor Stp 7: Comput th tadard dvato w w w2 w th rag ordr of all altratv ca b dtrmd. of th w comprhv comparo matrx D d w wj. If t accptabl th w lct th bt o from amog a t of fabl altratv. Othrw th dco mar hav to modfy th matrc. h proc how Fgur 2. Ucrtaty Comparo Matrc Y Complt Iformato? No Wghtd Gomtrc Ma Ordrg Vctor Logarthmc Lat Squar Y Wght Vctor Accptabl? No Modfcato Fgur 2. Proc for gratg prort from group dco modl. 39

7 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag 4. NUMRICAL XAMPLS I th cto w offr two umrcal xampl that ar xamd ug th propod mthod ad how thr applcato. h ucrtaty dtrbuto ar obtad by Dlph mthod [8]. xampl. I a proc of dco mag thr dco mar gav th ucrtaty comparo matrc blow D L 3 L L D3 L D 2 L 2 L L L. 23 L W ow that t udr complt formato vromt bcau thr o lmt. Followg th tp w ca gt th w comparo matrc a (8) ug (). Accordg to q. (9) thr group of wght vctor ar obtad w w w Aum that a dco mar a mportat a aothr 2 3 wght vctor w. h tadard dvato comparo matrx D d w wj max.875. ug (4) ad (5) th fal compot of th w comprhv whr 2 Appartly th bt opto th frt attrbut. Ad w codr t accptabl du to th maxmum dvato mall. It wll play a mportat rol wh th rag ordr gv by vry xprt ar dffrt. xampl 2. hr wr thr xprt lctg th bt opto from fv altratv ad provdg th ucrtaty comparo matrc 3 L L L3 L23 24 D L 34 L3 4 2 D2 L

8 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag D 3 34 L L. Obvouly t udr th complt formato vromt bcau thr ar lmt. Ug th formula () th thr matrc ca b xpr a Accordg to () th quato t ad t umrcal oluto Fally bcau of x w D 2 7x x2 2x3 3x4 x5.328 x 3x2 x3 x x x2 5x3 2x x x 2x 7x x x x4 x x w j x x j x x th wght vctor w h rag of altratv ucrta group dco mag udr complt vromt problm olvd. 5. CONCLUSIONS I multpl crtra dco aaly problm huma judgmt ar rqurd ordr to grat rlatv wght of crtra. Du to ucrta ad mprc dcrpto of ral world dco problm ad ubjctv atur of huma judgmt ucrtaty thory ca provd a mor raltc framwor to xpla uch ucrtaty. Aftr bg llutratd by a proal lcto problm [9] th mthod to olv th mult- pro multcrtra dco mag problm udr complt formato vromt ar propod th papr. I group AHP proc vry oft th amt of altratv wth rpct to crtra ad th mportac wght ar utabl to u th ucrta varabl tad of prc umbr. I fact th ucrta varabl ad ucrtaty dtrbuto ca b ud to dcrb judgmt aly. h y that aggrgat ach dco mar formato to drv th bt lcto. Hr w propo wghtd gomtrc ma ordrg vctor whch ffctv ad mpl to grat wght vctor aggrgat formato ad calculat dvato udr complt formato vromt. I addto th logarthmc lat quar xtdd to ucrta group AHP udr th codto of complt formato vromt. I ordr to apply th mthod xpctd valu of ucrta varabl ad vr ucrtaty dtrbuto ar ud to traform a ucrtaty comparo matrx to a xact matrx. Fally th wght vctor obtad aturally. Howvr how to aggrgat th formato of mult-hrarchy tll ubjct to furthr vtgato.. 393

9 IJRRAS (3) Ju 22 Wag & L Altratv Ucrta Group Dco Mag 6. ACKNOWLDGMNS h author wh to tha Lag L th profor of Gul Uvrty of chology Cha for hr tructo ad upport; ad Huad Sh th matr of Uvrty of Scc ad chology Bg for hr couragmt ad upport. h wor ha b upportd by Gul Uvrty of chology. 7. RFRNCS []..L. Saaty A calg mthod for prort hrarchcal tructur Joural of Mathmatcal Pychology 5 pp (977). [2]. L.F. Wag S.B. Xu h Itroducto about th Aalytc Hrarchy Proc Cha Rm Uvrty Pr pp (99). [3]. P.. Harr L.G. Varga hory of rato cal tmato Saaty aalytc hrarchy Proc Dpartmt of Dco Scc (985). [4]. F.A. Lootma Prformac valuato of o-lar optmzato mthod va multcrtra dco aaly ad va lar modl aaly Acadmc Pr Lodo pp (98). [5]. C.. Ch xto of th OPSIS for group dco-mag udr fuzzy vromt Fuzzy St ad Sytm 4 pp. -9 (2). [6]. B. Lu Ucrtaty hory 2d d. Sprgr-Vrlag Brl (27). [7]. B. Lu Ucrtaty hory: A Brach of Mathmatc for Modlg Huma Ucrtaty Sprgr-Vrlag Brl (2). [8]. B. Lu Ucrtaty hory 4d d. Sprgr-Vrlag Brl (2). [9]. C. Wag L. L S. Wu A ucrta dco mag modl: A applcato to AHP Itratoal Joural of Advacd Rarch Computr Scc Vol. 3 No. 2 Aprl (22). []. C. Wag L. L J.J. Lu Ucrtaty wght grato approach bad o ucrtaty comparo matrc Appld Mathmatc Vol. 3 No. 5 May (22). []..L. Saaty h Aalytc Hrarchy Proc (988). [2]. J.M.D.S. Nv Motorg Cotcy Group Dco Mag: A mprcal Study of th Aalytc Hrarchy Proc (gvctor Prortzato Modl) Uvrty of Pylvaa pp. 233 (984). 394