A New Method for Consistency Correction of Judgment Matrix in AHP

Transcription

1 Ne Method for Cosstecy Correcto of Judgmet Matrx HP Hao Zhag Haa Normal UverstyHakou 5758Cha Ygb We 2 Haa College of Softare TechologyQogha 57400Cha E-mal:W6337@63.com Gahua Yu 3 Haa Normal UverstyHakou 5758Cha E-mal: @qq.com I order to cotue to study the cosstecy correcto of the udgmet matrx e studed the relatoshp betee the cosstecy of the udgmet matrx ad the perturbato matrx proposed the cocept of the perturbato devato matrx. The to e methods for the cosstecy correcto of the udgmet matrx are preseted based o the covex combato of perturbato devato matrx. Fally the e methods ere valdated by examples ad compared th the tradtoal HP method. PoS(CENet207)086 CENet July 207 Shagha Cha Speaker 2 Ths study s supported by the Nato Natural Scece Foudato of Cha (NSFC) uder Grat No Scece ad Techology gecy of Haa Provce uder Grat No.DXM ad Key Scece ad Techology Pla Proects of Haa Provce uder Grat No.SQ204YJJC002 3 Correspodg uthor Copyrght oed by the author(s) uder the terms of the Creatve Commos ttrbuto-nocommercal-nodervatves 4.0 Iteratoal Lcese (CC BY-NC-ND 4.0).

2 . Itroducto The alytc Herarchy Process (HP) [] has bee dely appled as a qualtatve ad quattatve decso-makg tool. successful decso o makg process requres the udgmet matrx to be cosstet. If a udgmet matrx fals to fulfll the requremets of cosstecy the the eght obtaed from the udgmet matrx ca ot be utlzed as the bass for makg decso. I that case certa adustmets o the matrx ll be further requred; therefore the problem of cosstecy correcto becomes a mportat research cotet the HP. Wth the presetato of HP there have bee rch lterature focusg o the cosstecy correcto research. Ma ad Xu [2] proposed a eghted arthmetc mea correcto method ad to crtera for correcto valdty. Xu ad We [3] proposed a eghted geometrc mea method. The eghted arthmetc mea ad eghted geometrc mea method ere aalyzed ad compared [4]. Xu [5] proposed a eghted arthmetc mea ad eghted geometrc mea method by aalyzg the maxmum devato the udgmet matrx. Some authors proposed the vector correcto method ad perturbato matrx correcto method[6-9]. The cosstecy of the udgmet matrx the fuzzy udgmet matrx ad the tutostc fuzzy udgmet matrx are rectfed by the devato matrx ad the vector method [0]. Based o the accelerated geetc algorthm to kds of NLP model correcto methods ere proposed []. Bayesa correcto method Hadamard product duced bas matrx (HPIBM) method ad the graph theory correcto method ere proposed [2-3]. We have proposed a e algorthm for the cosstecy test of udgmet matrx based o probablstc statstcs ad hypothess testg [4]. Based o the sad lterature to e methods for the cosstecy correcto of the udgmet matrx based o the covex combato of perturbato devato matrx are preseted hereof to further verfy the e methods ad make comparso th the tradtoal HP. PoS(CENet207) Prelmares Defto 2. [4] Let ( a ) ad p q B ( b ) p q satsfed the product of c be a matrx f B B s called the Hadamard product of matrx here a b ad refer to the Hadamard product symbol. a Defto 2.2[4] Let ( ) correspodg to the largest egevalue max be a udgmet matrx ad the egevector λ s [... ] T the the matrx 2 ( ) s called the characterstc matrx of ( a ) here c s * ( ) p q { 2... }.

3 a Defto 2.3 [4] Let ( ) be a udgmet matrx f E the E s called perturbato matrx of here E ) (ε. Defto 2.4 Let E ) (ε be a perturbato matrx of the () If a elemetε E satsfesε 0 the{ ε } s called perturbato devato matrxdeoted as D ; () If a elemetε E satsfedε 0 the{ ε } s called perturbato zero devato matrxdeoted as D 0. Theorem 2. If the udgmet matrx satsfes E the s a completely cosstet matrx f ad oly f E D0. Proof. Suffcecy: s s the characterstc matrx of satsfes the complete cosstecy ad the perturbato matrx E D0 defto e ko satsfes the complete cosstecy. accordg to the Hadamard product Necessty: If satsfes the complete cosstecy ad also satsfes the complete cosstecy accordg to the Hadamard product deftoso e ko E D0. Defto 2.5 Let D ad D 0 be the perturbato devato matrx ad the perturbato PoS(CENet207)086 zero devato matrx respectvelyad the covex combato of D ad D 0 ca be expressed as D λd + ( λ) D0 the ε λε + ( λ) ε / ε here 0 λ. Theorem 2.2 [9] Let cosstecy udgmet matrx be rectfed as D f max ε λ max ( D) the λ ) < λ ( ). max ( max 2 2 Defto 2.6 [4] Let χ pσ 0 / ( ) be a crtcal value of the cosstecy dexhch s called the Ch- Square Cosstecy Idex(breflyCSCI ).

4 Order p p p Order p p p Order / p / p / p / Table : Crtcal Value of CSCI If ( λ ) /( ) 0. RI max < through the cosstecy test [].Where the the matrx through the cosstecy test;otherseot CI ( λ max ) /( ) ad 0.RI s a crtcal value for CI to test the cosstecy of the udgmet matrx.i ths papere preset CSCI as proposed [4] as a e crtcal value of ( ) /( ) udgmet matrx. Table 2: Crtcal Value of RI ad CI [] 3. e algorthm for cosstecy correcto to test the cosstecy of the Order RI CI0.RI Order / RI / CI0.RI / It s proposed to replace d the perturbato matrx th γd + ( γ ) ad the PoS(CENet207)086 Hadamard product ( / ) [ γd + ( γ )] s used as the correcto result of the elemet the orgal matrx[9].hoever cosderg there are too may elemets for each correcto [9]t s dffcult to mata the formato the orgal matrx. I ths paper e propose a cosstecy correcto method hch s easy to mata the formato the orgal matrx. Let a ) be a udgmet matrx k be the umber of teratve tmes ad ( the specfc steps are as follo: ( 0) (0) Step : Let ( a ) 0 θ ad k 0.

5 Step 2: Calculate the maxmum egevalue λ ( ) T max of ad the prorty vector Step 3: If Step 4: () (Method ) ( λ max ) /( ) < CSCI Normalze all colums of hch refers to the colum vector of the go to Step 6; otherse go to ext step. the get the ormalzed matrx agle betee ad k k a a amely cosθ. a here The determe t () (Method 2) Let a a a ) ( 2. Calculate the cose value of the cluded ( ) ( ) ( ) so that cos m{cos θ t θ } ad let ( ) ε θε t + ( θ ) t t θε t + ( θ ) ε t ε a / ad determe p q so that ( k + ) ( / ) ε here ε θε θε Step 5: Let k k + ad retur to Step 2. ( ) k Step 6: Output λ max CI k k ( ) + ( ) ( ) / ε { } max. Let ε pq ε pq + ( θ ) ( ) ( p q) ( ) ( ) ( q p) k pq + ( θ ) ε ( ) ( p q) ( q p) hch refers to the correcto matrx ad refers to PoS(CENet207)086 the vector of prortes. Step 7: Ed. The to crtera for measurg the proxmty of the orgal matrx to the correcto matrx are gve [2]as follos: δ { a a } N ( 0) max σ ( a a (0) 2 ) /

6 Usually e thk that the smaller the value of δ σ s the more formato ll be retaed from the orgal matrx ad the better correcto ll be acheved. 4. Case aalyss Ths paper chooses the Matrx [2] ad rectfes t accordg to the above to methods as follos: / 3 / 4 / 5 / / 5 / 7 / / 5 / 7 / 5 / 6 / 3 / 4 / 7 / 8. / 6 / 3 / 3 3 / 2 / 5 / 6 / 6 / 3 / / 5 / / / The perturbato matrx of the orgal udgmet matrx s obtaed as follos: λ max > CSCI ( ). Whe p0.0 θ 0. 5 the correcto result Method s: λ max < 0.2 CSCI ) δ σ ( Whe p0.0 θ 0. 5 the correcto result Method 2 s: PoS(CENet207)086

7 λ max < 0.2 CSCI ( ) δ σ If 0.RI the tradtoal HP s used as the crtcal value of correcto result Method s: the the λ max <0.40.RI ( ) δ σ If 0.RI the tradtoal HP s used as the crtcal value of correcto result Method 2 s: λ max <0.4 0.RI ) δ σ the the ( PoS(CENet207)086

8 By cotrast t s foud that the value of δ ad σ the e correcto method s smaller tha the correspodg tradtoal correcto method so the e correcto method s better. 5. Cocluso I ths paper e studed the relatoshp betee the perturbato matrx ad the udgmet matrx cosstecy proposed to e methods of cosstecy correcto based o the theory of perturbato devato matrx. The elemet to be rectfed are determed by the sze of the dsturbace elemet the perturbato matrx ad the sze of the cose betee the vectors. To e methods have bee verfed by the satsfed results. Refereces [] Saaty T L. The aalytc herarchy process[m].ne York: McGra-Hll980. [2] Ma WeyeXu JagyueWe Yxag. practcal approach to modfyg parse comparso matrces ad to crtera of modfcatory effectveess[j].joural of systems scece & systems geerg9932(4): [3] Xu ZeshuWe Cupg. cosstecy mprovg method the aalytc herarchy process[j]. Europea oural of operatoal research9996: [4] Xu ZeshuDa Qgl.alyss ad comparso of to methods for mprovg cosstecy of udgmet matrx[j]. Joural of Southeast uversty(natural Scece Edto) (6): [5] Xu Zeshu. Practcal method for mprovg cosstecy of udgmet matrx the HP[J]. Joural of systems scece ad complexty20047(2): [6] Lu WalLe Zhu.Study o Rectfcato method for the udgmet matrx HP[J].Systems Egeerg-Theory&Practce997(6): (I Chese) [7] Xu Zeshu.To approaches to mprovg the cosstecy of complemetary udgemet matrx [J]. pp. Math.J.Chese Uv.Ser.B20027(2): PoS(CENet207)086 [8] Zhag QuhuLog Xhua. teratve algorthm for mprovg the cosstecy of udgmet matrx HP[J]. Mathematcs practce ad theory2003(5): (I Chese) [9] Dog CaoLarece C.LeugJaphet S.La. Modfyg cosstet comparso matrx aalytc herarchy process: heurstc approach[j].decso support systems200844: [0] Wexa LChegy Zhag.Decso-Makg Iteractve ad Iteratve pproaches[m]. Stzerlad: Sprger teratoal publshg [] Yucheg DogYhua CheJa Xao.To e method for mprovg the cosstecy of the udgemet matrx HP[J].Joural of systems scece ad formato20053(3): [2] Chagsheg LGag Kou.Bayesa revso of the dvdual parse comparso matrces uder cosesus HP-GDM[J].ppled soft computg20535:802-8.

9 [3] Gag KouDa ErguJefer Shag.Ehacg data cosstecy decso matrx:daptg Hadamard model to mtgate udgemet cotradcto[j].europea oural of operatoal research204236: [4] Gahua Yu Yae ZhagHag HuagChegy Zhag. Ne Method for Cosstecy Test of Judgmet Matrx HP[J].Mathematcs practce ad theory(ccepted).(i Chese) PoS(CENet207)086