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Eur opean Jou r na l o f Ope r a t i ona l Resea r ch 48 (1990) 9 -26 9N o r t h - H o l l a n d

How to make a decis ion:The Analytic Hierarchy ProcessThomas L. SaatyJoseph M. K atz Graduate School of Business, University of Pittsburgh, Pittsburgh, PA 1526 0, USA

Abstract: This paper s e rves as an in t roduct ion to the Analy t i c H iera rchy Proces s - A mul t i c r i t e r ia dec i s ionmaking approach in which fac tor s a re a r r anged in a h ie rarch ic s t ruc ture . The pr inc ip les and the ph i losophyof the theory a re summar ized g iv ing genera l background in format ion of the type of measurement u t i l i zed ,i t s p roper t i es and appl ica t ions .Keywords: Decis ion, pr ior i ty , rank, cos t-benef i t , scales , rat ios

1. How to s tructure a dec is ion problemPerhaps the mos t c rea t ive t ask in making a

dec i s ion is to choose the f ac tor s tha t a re im por ta n tfor tha t dec i s ion . In the Analy t i c H ierarchy Proc-ess we arrange these factors , once selected, in ah ie rarch ic s t ruc ture descending f rom an overa l lgoal to cr i ter ia , subcr i ter ia and al ternat ives insuccessive levels.

To a per son unfami l i a r w i th the sub jec t therem ay b e s o m e co n ce r n ab o u t w h a t t o i n c l u d e an dwhere to inc lude i t . When cons t ruc t ing h ie rarch iesone mus t inc lude enough re levan t de ta i l to :

r epresen t the p rob lem as thoroughly as poss i -ble, but not so thoroughly as to lose sens i t ivi ty tochange in the e lements ;

cons ider the env i ronment sur rounding thep r o b l em ;

iden t i fy the i s sues o r a t t r ibu tes tha t con t r ibu teto the solut ion; and

iden t i fy the par t i c ipan t s as soc ia ted w i th thep r o b l em .Arranging the goals , at t r ibutes , issues , and s take-holders in a hierarchy serves two purposes . I tp rov ides an overa l l v iew of the complex r e la t ion-ships inherent in the s i tuat ion; and helps the deci-s ion maker assess whether the issues in each levela re o f the s ame order o f magni tude , so he canco m p ar e s u ch h o m o g en eo u s e l em en t s a ccu r a t e l y .Rece ived November 1989

One cer ta in ly cannot compare accord ing to s i ze afoo tba l l w i th Mr . Everes t and have any hope ofge t t ing a meaningfu l answer . The foo tba l l and Mt .Everes t mus t be compared in s e t s o f ob jec t s o ftheir class . Later we give a fundamental scale ofuse in making the compar i son . I t cons i st s o f verba ljudgments r ang ing f rom equal to ex t r eme (equal ,modera te ly more , s t rongly more , very s t ronglymore , ex t r emely more) cor responding to the verba ljudgm ents a re the numer ica l judgm ents (1, 3 , 5 , 7 ,9 ) and compromises be tween these va lues . Wehave comple ted compi l ing a d ic t ionary of h ie r -archies per taining to all sorts of proble ms, f rompersonal to corpora te to publ ic .

A h ie rarchy does no t need to be comple te , tha tis , an element in a given level does not have tofunc t ion as an a t t r ibu te (o r c r i t e r ion) fo r all theelements in the level below. A hierarchy is no t thet r ad i t iona l dec i s ion t r ee . Each l evel may represen ta d i f f e ren t cu t a t the p rob lem. One leve l mayrepresen t soc ia l f ac tor s and ano ther po l i t i ca l f ac-to r to be ev a lua ted in t e rms of the soc ia l f ac tor s o rvice versa. Fur ther , a decis ion maker can inser t ore l imina te l eve l s and e lements as neces sary toclar i fy the task of set t ing pr ior i t ies or to sharpenthe focus on one or more par t s o f the sys tem.Elements tha t have a g loba l charac te r can berepresen ted a t the h igher l eve l s o f the h ie rarchy ,o ther s tha t spec i fi ca l ly charac te r ize the p ro b lem a thand can be deve loped in g rea te r dep th . The t askof set t ing pr ior i t ies requires that the cr i ter ia , the

0377-22 17 / 90 / $0 3 . 50 1990 - E l sev i e r Sc i ence Pub l i she r s B. V. (Nor th -Ho l l and )

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10 T.L. Saaty / The AHP: How to make a decisionproper t ies or fea tures of the a l te rna t ives be ingcompared , and the a l t e rna t ive s themse lves a regradual ly layered in the hierarchy so tha t i t ismeaningfu l to compare them among themse lves inre la t ion to the e lements of the next higher leve l .

F ina lly , a f t er judgm ents have been made on theimpac t o f a ll the e lements and pr ior i t i e s have beencom puted for the h ie ra rchy a s a whole , some times ,and with care , the less important e lements can bedropped f rom fur the r cons ide ra t ion because ofthe ir re la t ive ly small impact on the overa l l objec-t ive . The pr ior i t i e s c an then be r ecomputedthroughout , e i the r w i th or w i thout changing therema in ing judgments .

2 . S c a le s o f me asur e me nt - Avoid ing me r e num-ber crunchingOne might a rgue the whole process of dec i s ion

making i s so uns t ruc tured and so amorphous tha ti t is no use t rying to be prec ise . One is thent e mp t e d t o g o f u r t h e r a n d c o n c l u d e t h a t " p a r -t ic ipant sa t is fac t ion" is the main objec t ive of dec i-s ion making. Were this the case , then mult ic r i te r iadec i s ion making would be a s imple ma t te r o fus ing ingenui ty , suppor ted wi th ma thema t ica lt e rminology , to improve numbers tha t p lea se peo-p le . But d i f fe ren t s e t s o f a rb i t r a ry numbers a rel ike ly to resul t , producing diffe rent dec is ions , andwe are r ight back where we s ta r ted. One se t ofnumbers p lea se s a g roup of people who might beequa l ly p lea sed wi th anothe r s e t o f numbers tha tcont rad ic t s the r ecommenda t ions of the f i r s t s e t .This i s me re number c runching . I f a dec i s ionsuppor t theory i s to be t rus twor thy the re mus t beuniqueness in the r epre sen ta t ion of judgm ents , thesca les der ived from these judg men ts , and the sca lessynthes ized from the der ived sca les .

Le t us cons ide r for a moment group in te rac t iontha t o f ten leads to ce r ta in expec ta t ions , some t imesa r is ing f rom the ve ry hea t o f the deba te . Such adeba te cannot a lways incorpora te in i t s a rgumentsthe r e f inements r e su l t ing f rom the ma thema t ica lt r adeof f s to ensure drawing va l id conc lus ions .Some t imes pa r t i c ipan t s a ccept a p rocess and i t sou tco me because the s i tua t ion i s so complex an dthe a rguments so convolu ted tha t wha teve rsurfaces in the end appears plaus ible . Althoughconvinc ing a group about i t s qua l i t a t ive pre fe r -ences involves the pol i t ics of persuas ion and of

wheel ing and dea l ing, i t is essent ia l tha t the dec i-s ion theory i tse lf used to ass is t the group in a rr iv-ing a t a dec is ion be invar iant to pol i t ics andbehavior . I t should be a sc ience of sca l ing basedon ma thema t ic s , ph i losophy and psychology .

A mo n g t h e v a r i o u s n u mb e r c r u n c h i n g p r a c -t ices , the most objec t ionable one is to ass ign anyse t o f num bers to judg men ts o n a l t e rna tive s unde ra pa r t i cu la r c r i t e r ion , and then norma l ize the senumbers (by mul t ip ly ing them by a cons tan t tha tis the rec iproca l of the ir sum). Genera l ly, dif fe rentset s o f numb ers a re used to scale the judgm entsfor the a l te rna t ives under dif fe rent c r i te r ia . All thenew normalized se ts now l ie in the inte rva l [0, 1] ,no ma t te r wha t s ca le they or ig ina l ly came f rom,and can be pa ssed of f to the un in i t i a ted a s com-pa rab le . The appea l ing pa r t o f th is p rac t ic e is tha tt h e n u mb e r s h a v e a n a p p a r e n t l y u n i f o r m u n d e r l y -ing s t ruc ture and look no t un l ike probabi l i t i e s .Thus they go uncha l lenged by the dec i s ion make rand a re then manipula ted by the consu l tan t o rfac i l i t a to r who we ights and adds them to f ind themos t p re fe r red a l t e rna t ive .

Note tha t when the in i t i a l numbers a s s ignedbe fore norma l iza t ion a re ord ina l s , a rb i t r a ry num-be rs tha t p re se rve orde r bu t c a r ry no in forma t ionabout d i f fe rences or r a t ios of r e la t ive magni tudes ,the r e su l t ing t r ans forma t ion produces a new se t o ford ina l s ly ing be tween ze ro and one and you canbe sure only tha t i t preserves the same order . Theope ra t ions of we ight ing and adding cannot thenbe meaningful because i t is very l ike ly tha t dif fe r-en t r e su l t s w i l l be produced for d i f fe ren t choice sof the ord ina l numbers . By a jud ic ious choice oford ina l s one can make an a l t e rna t ive tha t i s domi-nant on even one c r i t e r ion , no ma t te r how unim-por tan t tha t c r i t e r ion may be , have the l a rges tva lue a f te r we ight ing and adding and thus tu rnout to be the mos t p re fe r red . Any me thod such a sth i s i s no t a dec i s ion theory bu t an approach tha tcan mis lead people .

The Ana ly t ic H ie ra rchy Process i s r igorous lyconce rned wi th the sca l ing problem and wha t sor tof numbers to use , and ho w to cor rec t ly com binethe pr ior i t ies resul t ing f rom them. A sca le ofmeasurement cons i s t s o f th ree e lements : A se t o fob jec t s , a s e t o f numbers , and a mapping of theobjec ts to the numbers . In a s tandard sca le a uni ti s used to cons t ruc t the r e s t o f the numbers of thesca le . Examples of such a uni t a re the inch, thepound, the angs t rom, and the do l la r . A s tanda rd

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T.L. Saaty / The AH P: H ow to make a decision 11

sc a l e c a n be use d t o me a sure ob j e c t o r e ve n t s wi t hre spe c t t o t he p ro pe r t y fo r whi c h a s c a l e is de s ign-ed to measure . Since the uni t i s a rbi t ra ry, one canha ve d i f fe re n t numbe rs t o whi c h t he ob j e c t s a rema ppe d . Be c a use a s t a nda rd sc a l e i s no t un i que , i ti s i mpor t a n t t o i n t e rp re t t he me a n i ng o f t he num-bers used in the sca le . Thus , in genera l , the num-be rs ob t a i ne d f rom suc h a s c a l e a re me re l y s t i mul ifor the memory (what i t fe l t l ike the las t t ime thet e m p e r a t u r e w as - 1 5 C ) a n d h a v e n o i n tr i ns i cs i gn i f i c a nc e . Howe ve r , mos t c a re fu l l y de s i gne ds t a nda rd sc a l e s a re he l p fu l i n t ha t t he y p re se rvec e r t a i n nume r i c a l r e l a t i ons i n t he me a sure me nt s( t he ma ppi ng) o f t he ob j e c t s , g i v i ng us a be t t e rwa y t o i n t e rp re t t he s t i mul i t he y a re me a sur i ngthan a rbi t ra ry sca les .

A sc a l e ma y o r ma y no t ha ve z e ro fo r a n o r i g in .For e xa mpl e , a s c a l e o f o rd i na l numbe rs c a n be g i nwi t h a ny numbe r . A ra t i o s c a l e , suc h a s a bso l u t et e mp e ra t u r e p re se rve s o r i g i n. In t e rva l s c a le s , whi chm e a s u r e t h e s a m e p h e n o m e n a l i k e t e m p e r a t u r e ,p re se rve l i ne a r r e l a t i ons , bu t ma y ha ve d i f fe re n tor igins . Zero on the Fahrenhe i t sca le i s a di ffe rentt e mpe ra t u re t ha n z e ro on t he Ce l s i us s c a l e . Bo t ha re i n t e rva l s c a l e s . Aga i n , t he numbe rs on t he sesc a l e s me a n no t h i ng un l e s s one c a n re c a l l s i t ua -t i ons a s soc i a t e d wi t h t he nume r i c a l r e a d i ngs be i ngc ons i de re d . The y a re j us t a c onve n i e n t me a n s o fc ommuni c a t i ng c ha ra c t e r i s t i c s o f ob j e c t s o r s i t ua -t i ons wi t hou t e ve ry bod y ha v i ng t o e xpe r i e nc et he m.

I t o f t e n ha ppe ns t ha t t he i n t e rp re t a t i ons o fnume r i c a l s t i mul i f rom a g i ve n s t a nda rd sc a l ed i f fe r de pe nd i ng on t he c i r c ums t a nc e s . The re i s nos i mpl e ru l e t ha t c a n be a pp l i e d t o i n t e rp re t r e a d-ings f rom even a s ingle sca le when i t i s appl ied toa na t u ra l phe nome non . In t e ns i t y o f sun l i gh t ha s ad i f fe re n t s i gn i f i c a nc e fo r d i f fe re n t purpose s . I tma y be use fu l fo r sunba t h i ng , bu t t oo b r i gh t fo rre a d i ng . S i mi l a r l y , a monot on i c r e l a t i on be t we e nsuc c e ss i ve r e a d i ngs f rom a s t a nda rd sc a l e do no ta s sure us t ha t e ve n h i ghe r r e a d i ngs wi l l be be t t e r(o r worse ) . More (o r l e s s ) t e mpe ra t u re doe s no tne c e s sa r i l y c or re spond t o more (o r l e s s ) use fu l -ne s s . Low t e mpe ra t u re s a re unc omfor t a b l e . As t here a d i ngs r i s e , t he y be c ome more c omfor t a b l e a nda s t he y r i s e h i ghe r t he y c a n a ga i n be c ome unc om-for t a b l e . On t he o t he r ha nd , t o p re se rve somefoods , low t e mpe r a t u re s a re ve ry de s ir a b l e a nd a st he r e a d i ngs r i s e , t he y be c ome unde s i ra b l e , a nd a sthey r i se s t i l l h igher they could become des i rable

a ga i n . Our va l ue s o f c omfor t a nd de s i r a b i l i t y a ndo t he r soc i a l e f fe c t s ha ve t o be a t t he bo t t om ofe ve ry i n t e rp re t a t i on a nd de p e nd on h i ghe r goa l st ha t we ma y ha ve i n mi nd .

For a l a rge numbe r o f s c a l e s use d i n phys i c s , i ti s impl ic i t ly assumed tha t the sca le can be ex-t e nde d ou t t o i n f i n it y a nd a pp l i e d t o e ve ry i ma gi n-a b l e c i r c ums t a nc e . In o t he r words , i n t e rp re t a t i oni n phys i c s a s sume s e ve n t s a s homoge ne ous , noma t t e r how ne a r o r f a r f rom t he o r i g i n t he y ma yfa l l . Wha t i s mos t a s t on i sh i ng i s t he a s sumpt i on i nphys i c s t ha t ob j e c t s ye t unknown bu t wi t h t hesa me d i me ns i ona l c ha ra c t e r i s t i c s o f t he knownobj e c t s be i ng me a sure d c a n i n fa c t be me a sure d i nt he sa me wa y . Re a l i z a t i on o f t he d i f fe re n t i n t e r -p r e t a t i o n s w e c a n m a k e o f t h e s a m e n u m b e r i nphys i c s woul d i nd i c a t e t ha t whe n numbe rs fa l louts ide the rea lm of exper ience i t i s logica l tosuspe nd t he e x t e ns i on o f t he t ru t h we c ons t ruc tf rom e xpe r i e nc e t o a doma i n fo r whi c h we ha ve noknowl e dge a nd fe e l i ng . I t woul d be mos t l y f i c t i vespe c u l a t i on .

In e c onomi c s , t he a r i t hme t i c va l ue o f a do l l a r i sa s s u m e d t o b e t h e s a m e n o m a t t e r w h e t h e r ape r son ha s on l y one o r a mi l l i on do l l a r s . Bu t i nrea l i ty i t i s not . To buy a new Mercedes , t endo l l a r s a nd one hundre d do l l a r s a re ne a r l y e qua l l yi na de qua t e o r use l e s s a s down pa yme nt s . On t heo t he r ha nd , fo r buy i ng g roc e r ie s , a hundre d do l -l a r s i s muc h more use fu l , p ra c t i c a l l y t e n t i me smore use fu l t ha n t e n do l l a r s . The f i r s t t housa ndd o l l a r s e a r n e d i s m u c h m o r e i m p o r t a n t t h a n t h ef i r s t t housa nd do l l a r s e a rne d a f t e r a mi l l i on .

We mus t be c ons t a n t l y a nd c a re fu l l y a t t e n t i vet o how we i n t e rp re t da t a f rom sc a l e s . S t a nda rdsca les force on us a way of thinking tha t i s not inc ompl e t e ha rmony wi t h t he wa y we re a l l y fe e la bou t wha t t he y a re me a sur i ng .

The re i s a more ge ne ra l me t hod o f me a sure -me nt t ha t doe s no t ma ke use o f s t a nda rd sc a l e s . I ti s t he me t hod o f r e l a t i ve me a sure me nt s use fu l fo rp rope r t i e s fo r whi c h t he re i s no s t a nda rd sc a l e o fme a sure me nt ( l ove , po l i t i c a l c l ouL s t r a i gh t ne s s ) .The se a re known a s i n t a ng i b l e p rope r t i e s . Thenumbe r o f suc h p rope r t i e s i s e x t re me l y l a rge . Wec a n sc a rc e l y hope t o de v i c e s t a nda rd sc a l e s fo rthem a l l . We are dr iven to re la t ive sca les , and asurpr i s ing thing i s tha t they can se rve as a s tand-a rd fo r how t o ha nd l e t he ve ry fe w s t a nda rd sc a l e swe ha ve , a nd no t t he o t he r wa y round . A re ma rka -ble aspec t of re la t ive sca les i s tha t they can use

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i n forma t i on f rom s t a nda rd sc a l e s whe n t he re i s apa r t i c u l a r ne e d t o do so . Me a sure me nt s i n as t a nda rd ra t i o sc a l e a re t r a ns forme d t o me a s -ure me nt s i n a r e l a t i ve ra t i o sc a l e by norma l i z i ngt he m.

Thi s c onve rs i on p roc e ss g i ve s us a h i n t a bou tt he d i f fe re nc e be t we e n t he t wo k i nds o f s c a l e s . Are l a t i ve sc a l e fo r a p rope r t y i s ge ne ra t e d fo r aspec i f ic se t of ent i t ies or objec ts . A s tandard sca lefor a p rope r t y i s a l wa ys ou t t he re re a dy t o beca l led into use . More s igni f icant ly , a re la t ive sca lei s e s se n t i a l t o r e pre se n t p r i o r i t y o r i mpor t a nc e i fone i s ge ne ra t i ng t he sc a l e by ma ki ng d i re c t ob-s e r v at i o ns a n d j u d g m e n t s a b o u t t h e p r o p e r t y u n -de r s t udy . I t i s a l so use fu l whe n one i s i n t e rpre t i ngwha t t he da t a f rom a s t a nda rd sc a l e r e a l l y s i gn i fy .R e l a t i ve sc a l e s a re a l wa ys ne e de d t o re pre se n tsub j e c t i ve unde rs t a nd i ng . More i s s a i d a bouta r i t hme t i c r e l a t i ons be t we e n t he t wo t ype s o f s c a l e sin Sec t ions 8 and 12.

3. Paired comparisons as rat iosW h e n w e m e a s u r e s o m e t h i n g w i t h r e s p e c t t o a

prope r t y , we usua l l y use some known sc a l e fo rt ha t purpose . A ba s i c c on t r i bu t i on t o t he sub j e c to f t h i s pa pe r , t he Ana l y t i c Hi e ra rc hy Proc e ss(AH P) i s how t o de r i ve re l a t ive sc a l es us i ng j udg-me nt o r da t a f rom a s t a nda rd sc a l e , a nd how t op e r f o r m t h e s u b s e q u e n t a r i t h m e t i c o p e r a t i o n o nsuc h sc a l e s a vo i d i ng use l e s s numbe r c runc h i ng .T h e j u d g m e n t s a r e g iv e n in t h e f o r m o f p a i r edcomparisons [6,7,8] One of the uses of a hie rarchyi s tha t i t a ll ows us t o foc us j ud gm e nt se pa ra t e l y one a c h of s e ve ra l p rope r t i e s e s se n t i a l fo r ma ki ng asound de ci s ion The m os t e f fe c t i ve wa y t o c on-c e n t ra t e j udg e me nt i s t o t a ke a pa i r o f e le me nt sa n d c o m p a r e t h e m o n a s i n g l e p r o p e r t y w i t h o u tc onc e rn fo r o t he r p rope r t i e s o r o t he r e l e me nt s .T h i s i s w h y p a i r e d c o m p a r i s o n s i n c o m b i n a t i o nwi th the hie rarchica l s t ruc ture a re so useful inde r i v i ng me a sure me nt . We a l so no t e t ha t some -t i me s c ompa r i sons a re ma de on t he ba s i s o f s t a nd-a rds e s t a b l i she d i n me mory t h rough e xpe r i e nc e o rt ra ining.

Assume t ha t we a re g i ve n n s t one s , A t , . . . , An,whose we i gh t s w I . . . . . w , , r e spe c t i ve ly , a re kno wnt o us . Le t us fo rm t he ma t r i x o f pa i rwi se ra t i oswhose rows g i ve t he ra t i os o f t he we i gh t s o f e a c hs t one wi t h re spe c t t o a l l o t he r s . He re t he sma l l e r

A 1AI wl/w~A 2 w 2 / w l

A , w n / w 1{Wwl2~ - n

of e a c h pa i r o f s t one s i s use d a s t he un i t , a nd t hel a rge r one i s me a sure d i n t e rms of mul t i p l e s o ft ha t un i t . I t i s d i f f i c u l t t o do t he i nve rse c ompa r i -son wi t hou t a ga i n us i ng t he sma l l e r s t one a s t heun i t . Th i s i s a so r t o f b i a s i n huma n t h i nk i ng ,whi c h l e a ds t o c ons i de r i ng a nonsymme t r i c a l ou t -c o m e a n d t h e i n c l i n a t i o n n o t t o f o r c e s y m m e t r yon i t . We ha ve t he ma t r i x e qua t i on :

A2 . . . An

wl/w2 "" Wl/W. I W 1IWa/W2 . . . w jw o l w?

w./w2 . . . w./wn ] w.

T h e f o r e g o i n g f o r m u l a t i o n h a s t h e a d v a n t a g e o fgiving us the solut ion But i t a l so gives r i se to a fa rre a c h i ng t he ore t i c a l i n t e rpre t a t i on . We ha ve mul t i -p l i e d A on t he r i gh t by t he ve c t o r o f we i gh t sw = ( % , w 2 . . . . w n ) x . The re su l t o f t h i s mul t i p l i -ca t ion i s n w . I f n i s an e igenvalue of A, then w i st he e i ge nve c t or a s soc i a t e d wi t h i t . Now A ha s ra nkone be c a use e ve ry row i s a c ons t a n t mul t i p l e o fthe f i rs t row. Thus a l l i t s e igenvalues except onea re z e ro . The sum of t he e i ge nva l ue s o f a ma t r i x i sequal to i t s t race , the sum of the diagonal e le -ments , and in thi s case , the t race of A i s equal ton. Therefore , n i s the la rges t , or pr inc ipa l , e igen-va l ue o f A .

T h e s o l u t i o n o f A w = n w , c a l l e d t he p r i nc i pa lr i gh t e i ge nve c t or o f A , c ons i s t s o f pos i t i ve e n t r i e sa nd i s un i que t o wi t h i n a mul t i p l i c a t i ve c ons t a n t .To ma ke w un i que , we norma l i z e i t s e n t r i e s byd i v i d i ng by t he i r sum. I t i s c l e a r t ha t i f we a reg i v e n t h e c o m p a r i s o n m a t r i x A , w e c a n r e c o v e rthe sca le . In thi s case the solut ion i s the normal-i z e d v e r s i o n o f a n y c o l u m n o f A .

T h e m a t r i x A = ( a i j ) , a i j = w i / w j , i , j =1 . . . . . n , ha s pos i t i ve e n t r i es e ve ry whe re a nd sa ti s -f i e s t he re c i p roc a l p rope r t y a j i = 1 / a i j . A n y m a -t r i x wi t h t h i s p rope r t y i s c a l l e d a r e c i p roc a l ma -t r ix . In addi t io n, A is c o n s i s t e n t be c a use t he fo l -lowing condi t ion i s sa t i s f ied:a j ~ = a i k / a i j , i , j , k = 1 . . . . . n . (1)

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T.L. Saaty / The AHP : How to make a decision 13We se e t ha t t he e n t i r e ma t r i x c a n be c ons t ruc t e df rom a se t o f n e l e me nt s whi c h fo rm a c ha i n (o rmore ge ne ra l l y , a spa nn i ng t r e e , i n g ra ph- t he ore t i ct e rmi no l ogy) a c ros s t he rows a nd c o l umns .

I t is ea sy t o p rove t ha t a c ons i s te n t m a t r i x m us th a v e th e r at i o f o r m A = ( w J w j ) , i , j = 1 . . . . . n. Ane c e ssa ry c ond i t i on fo r c ons i s t e nc y i s t ha t A bere c i p roc a l . We show be l ow t ha t a ne c e s sa ry a ndsuf f i c i e n t c ond i t i on fo r c ons i s t e nc y i s t ha t t hepr i nc i pa l e i ge nva l ue o f A be e qua l t o n , t he o rde rof A . Whe n A i s i nc ons i s t e n t , t he se t wo obse rva -t ions se rve to he lp us der ive a ra t io sca le whosera t ios a re c lose to those of an under lying sca lew - - (w I . . . . w , ). The re c i p roc a l a x i om of t he AH Pe nsure s t ha t pe r t u rba t i ons o f a r a t i o s c a l e a ret he mse l ve s r e c i p roc a l . The homoge ne i t y a x i om e n-sure s fo r t he i nc ons i s t e n t c a se t ha t t he pe r t u rba -t i ons woul d be sma l l , a nd he nc e t ha t t he t wopr i nc i pa l e i ge nva l ue s a re c l ose , f rom whi c h i two uld fol low by an a rg um ent g iven in [14, p . 671tha t the der ived sca le i s c lose to an under lyingra t io sca le . In addi t ion, thi s second axiom enablesus t o e xp l o re t he i mprove me nt o f some of t hej u d g m e n t s , t h u s a l s o t h e i m p r o v e m e n t o f i n c o n -s i s t e nc y a nd t he sc a l e a pprox i ma t i on . Bu t t he res t i l l r e ma i ns t he que s t i on o f o rde r p re se rva t i on .The me t hod we use t o de r i ve t he sc a l e mus t no tonly yie ld a ra t io sca le , but a l so capture the orderi nhe re n t i n t he j udgm e nt s , a ve ry s t rong re qu i re -me nt i nde e d .

In a ge ne ra l de c i s i on-ma ki ng e nv i ronme nt , wec a nno t g i ve t he p re c i se va l ue s o f t he w , / w ) b u ton l y e s t i ma t e s o f t he m. Le t us c ons i de r e s t i ma t e sof t he se va l ue s g i ve n by a n e xpe r t who ma y ma kesma l l e r ro r s i n j udg me n t . I t is known f rom e i ge n-va l ue t he ory [14] , t ha t a sma l l pe r t u rba t i on a rounda s imple e igenva lue , as we have in n when A i sc ons i s t e n t , l e a ds t o a n e i ge nva l ue p rob l e m of t hef o r m Aw=)tmaxWwhere Xmax i s the pr inc ipa le i ge nva l ue o f A whe re A ma y no l onge r be c on-s i s tent but i s s t i l l rec iproca l . The problem now is :t o wha t e x t e n t doe s w re f l e c t t he e xpe r t ' s a c t ua lop i n i on? N ot e t ha t i f we ob t a i n w by so l v ing t h isp rob l e m, a nd t he n fo rm a ma t r i x wi t h t he e n t r i e s( w , / w j ) , w e o b t a i n a n a p p r o x i m a t i o n t o A b y ac ons i s t e n t ma t r i x .

We now show t he i n t e re s t i ng , a nd pe rha pssurpr i s i ng re su l t t ha t i nc ons i s t e nc y t h roughoutt he ma t r i x c a n be c a p t u re d by a s i ng l e numbe rX m~x - n , w hi c h me a su re s t he de v i a t i on o f t hej u d g m e n t s f r o m t h e c o n s i s t e n t a p p r o x i m a t i o n .

L e t a , j = ( 1 + 6 ~ j ) w J ~ * ) , 6~j> 1 , b e a p e r -t u r b a t i o n o f w y w j , where w i s the pr inc ipa l e igen-ve c t o r o f A .T h e o r e m 1. Xmax > / g/"Proof: Usi ng a j , = 1 / a j i , a n d A w - - X . . .. w , w eh a v e

I a,2- = - ~ l + 6 , j >~0" [] (2)~max n n l ~ i < j ~ n

T h e o r e m 2. A is consi stent i f an d only i f )~m a x = ?Z.Proof . I f A i s consis tent , then because of (1) , eachrow of A i s a c ons t a n t mul t i p l e o f a g i ve n row.Thi s i mpl i e s t ha t t he r a nk o f A i s one , a nd a l l bu tone of i t s e igenva lue s Xi, i = 1, 2 . . . . . n , a re zero.Howe ve r , i t fo l l ows f rom our e a r l i e r a rgume nt t ha tF~= iX, = Tr ace (A ) = n. Th erefo re , X m a x = n. Col3-verse ly, Xm,x = n imp lies 6,j = 0, and a~j = wywj .[]

For t he c ons i s t e nc y i nde x (CI ) , we a dop t t heva l ue (k i n , X n ) / (n - 1 ). I t i s t he ne ga t i ve a ve r -a ge o f t he o t he r roo t s o f t he c ha ra c t e r i s t i c po l y -nomi a l o f A . Th i s va l ue i s c ompa re d wi t h t hesa me i nde x ob t a i ne d a s a n a ve ra ge ove r a l a rgen u m b e r o f r e c i p r o c a l m a t r i c e s o f t h e s a m e o r d e rwhose e n t r i e s a re r a ndom. I f t he r a t i o ( c a l l e d t hec o n s i s t e n c y r a t i o C R ) o f C I t o t h a t f r o m r a n d o mmatr ices i s s igni f icant ly smal l (careful ly spec i f iedto be about 10% or less) , we accept the es t imate ofw. Ot he rwi se , we a t t e mpt t o i mprove c ons i s t e nc y .

T h e r e a d e r m a y k n o w a b o u t t h e e x p e r i m e n t a lf i nd i ngs o f t he psyc ho l og i s t Ge orge Mi l l e r i n t he1950's [4] . He found tha t in genera l , people (sucha s c he ss e xpe r t s ) c ou l d de a l wi t h i n fo rma t i on i n -vo l v i ng s i mul t a ne ou s l y on l y a fe w fa c t s, s e ve n p l usor mi nus t wo , he wro t e . Wi t h more , t he y be c omec o n f u s e d a n d c a n n o t h a n d l e t h e i n f o r m a t i o n . T h i si s i n ha rmony wi t h t he s t a b i l i t y o f t he p r i nc i pa le i ge nva l ue t o sma l l pe r t u rba t i ons whe n n i s sma l l[7 ,14] , and i t s cent ra l role in the measurement ofc ons i s t e nc y .

Vargas [12] s tudied the case where the coeff i -c i e n t s o f t he ma t r i x a re r a ndom va r i a b l e s . Hefoc use d h i s a t t e n t i on on ga mma d i s t r i bu t e d c oe f f i -c i e n t s a nd de r i ve d a Di r i c h l e t d i s t r i bu t i on fo r t hec o m p o n e n t s o f t h e e i g e n v e c t o r w h e n t h e m a t r i x i s

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14 T.L. Saaty / The AHP : How to make a decision

consistent. When the matrix is inconsistent, the10% consistency bound is a sufficient measure toensure that the eigenvector follows the Dirichietdistribution with given parameters which can becomputed from the corresponding consistent ma-trix. The gamma assumption is a powerful onebecause of the inherent density of linear combina-tions of these distributions.

4. Two examplesThe AHP is used with two types of measure-

ment, relative and absolute, the latter having to dowith memory standards mentioned above. In both,paired comparisons are performed to derive prior-ities for criteria with respect to the goal. In rela-tive measurement, paired comparisons are per-formed throughout the hierarchy including on thealternatives in the lowest level of the hierarchywith respect to the criteria in the level above. Inabsolute measurement, paired comparisons arealso performed through the hierarchy with theexceptions of the alternatives themselves. The leveljust above the alternatives consists of intensities orgrades which are refinements of the criteria orsubcriteria governing the alternatives. One pair-wise compares the grades themselves under eachcriterion by answering questions such as: Howmuch better is a student applicant with excellentgrades than one with very good grades? and howmuch better is a student applicant with averageletters of recommendation than one with poorones? and so on. The alternatives are not pairwisecompared, but simply rated as to what category inwhich they fall under each criterion. A weightingand summing process yields their overall ranks.

This will become clear in the second examplebelow. There is no reason why forcing standardson a problem should produce the same outcomeobtained through relative measurement. These aretwo different descriptive (what can be) andnormative (what should be) settings.

4.1. Relative mea surem ent: Choosing the best houseto buy

When advising a family of average income tobuy a house, the family identified eight criteriawhich they thought they had to look for in ahouse. These criteria fall into three categories:economic, geographic and physical. Although onemay have begun by examining the relative impor-tance of these clusters, the family felt they wantedto prioritize the relative importance of all thecriteria without working with clusters. The prob-lem was to decide which of three candidate housesto choose. The first s tep is the structuring of theproblem asa hierarchy.

In the first (or top) level is the overall goal of'Satisfaction with house'. In the second level arethe eight criteria which contribute to the goal, andthe third (or bottom) level are the three candidatehouses which are to be evaluated in terms of thecriteria in the second level. The definitions of thecriteria follow and the hierarchy is shown in Fig-ure 1.The criteria important to the individual familywere:(1) Size of house: Storage space; size of rooms,number of rooms; total area of house.(2) Location to bus lines: Convenient, close busservice.

SAT,SFACTIONWIT""OUSE]

Figure 1. Decomposition of the problem into a hierarchy

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T.L. Saaty / The AH P: How to mak e a dec ision 15

Tab le lT h e f u n d a m e n t a l s c a l eI n t e n s i t y o f i m p o r t a n c e De f i n i t i on E x p l a n a t i o non an abso lu te sca le1 E q u a l i m p o r t a n c e3 M o d e r a t e i m p o r t a n c e o fone ove r ano the r5 Essen t i a l o r s t rong impor -

t ance7 Very s t rong impo r tance

9

2 , 4 , 6 , 8

Rec ip roca l s

Ra t i o n a l s

E x t r e m e i m p o r t a n c e

I n t e r m e d i a t e v a l u e s b e -tween the two ad jacen tj u d g m e n t s

Two ac t iv i t i e s con t r ibu te equa l ly to the ob jec t iveExper ience and judg me n t s t ron g ly favo r one ac tiv i ty ove r ano the rExpe r ience and judge men t s t rong ly favo r one ac t iv ity ove r ano the r

An ac t iv i ty i s s t rong ly favo red and i t s dom inanc e dem ons t r a t ed inp rac t i ceThe ev idence favo r ing one ac t iv i ty ove r ano the r i s o f t i l e h ighes tposs ib le o rde r o f a f f i rma t ionW h e n c o m p r o m i s e i s n e e d e d

I f ac t iv i ty i ha s one o f the above numbers a s s igned to i t when compared wi th ac t iv i ty j , then j hasthe r ec ip roca l va lue when compared wi th iRa t io s a r i s ing f rom the sca le I f cons i s t ency were to be fo rced by ob ta in ing n numer ica l va lues to

s p a n t h e m a t r i x

(3) Neighborhood: Little traffic, secure, niceview, low taxes, good condition of neighborhood.

(4) Age of house: Self-explanatory.(5) Yard space: Includes front, back and side,

and space from neighbors.(6 ) Mod ern facilities: Dishwashers, garbage dis-posals, air conditioning, alarm system, and othersuch items possessed by a house.

(7) General condition: Repairs needed, walls,carpet, drapes, cleanliness, wiring.

(8) Financing available: Assumable mortgage;seller financing available, or bank financing.

The second step is the elicitation of pairwisecomparison judgments . Arrange the e lements inthe second level into a m atrix and elicit judg men tsfrom the people who have the problem about therelative importance of the elements with respect tothe overall goal, Satisfaction with House. Thescale to use in making the judg me nts is given inTable 1. This scale has been validated for effec-tiveness, not only in many applications by a num-ber of people, but also through theoretical com-parisons with a large number of other scales.

The quest ions to ask when comparing two

T a b l e 2Pa i rwise compar i son ma t r ix fo r l eve l 1

1 2 3 4 5 6 7 8 Prior i tyvec to r

I I1 1 5 3 7 6 6 ~ a 0.173I 1 I2 ~ 1 ~ 5 3 3 t ~ 0.054I3 ~ 3 1 6 3 4 6 ~, 0.1 884 t i 1 i i ; i6 1 3 a v ~ 0.01 8

5 ~ ~ ~ ~ ;3 3 1 ~ 5 6 0.0316 ~ ~ ~ ~4 2 1 ~ 6 0.036

1 I7 3 5 ~ 7 5 5 1 ~ 0.1678 4 7 5 8 6 6 2 1 0.333

~'max = 9.669, CI = 0.238, CR ~ 0.169

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16 T.L. Saaty / The AHP: How to make a decision

cr i ter ia are of the fol lowing kind: of the twocr i ter ia being compared, which is considered m oreimpor tan t b y the f ami ly buying the house wi threspect to the overal l goal of family sat is fact ionwith the house?

When the elements being compared are closertogether than indicated by the scale, one can usethe scale 1.1, 1.2 . . . . . 1.9. If st i l l finer, one can usethe appropr iate percentage ref inement .

The matr ix of pairwise compar isons of thecr i ter ia given by the homebuyers in this case isshown in Table 2, along with the resul t ing vectorof priorit ies. The vector of priorit ies is the prin-cipal eigenv ector of the matrix. I t gives the relativepr ior i ty of the cr i ter ia measured on a rat io scale.In this case f inancing has the highes t pr ior i ty wi th33% of the influence.

In Table 2, ins tead of naming the cr i ter ia, weuse the number previously associated with each.Next we move to the pairwise compar isons of theelements in the lowest level. The elements to be

compared pairwise are the houses wi th respect tohow much bet ter one is than the other is sat is fyingeach criterion in level 2. Thus there will be eight3 x 3 matr ices of jud gm ent s s ince there are eightelements in level 2, and 3 houses to be pairwisecompared for each element . Again, the matr icesconta in the judgm ents of the f am i ly involved . Tounder s t and the judgm ents , a br i e f descr ip tion ofthe houses follows.

House A. This house is the larges t of them al l .I t i s located in a neighborhood with l i t t le t raf f icand low taxes . I t s yard space is comparably largerthan houses B and C. However , the general condi-t ion is not very good and i t needs cleaning andpaint ing. Also, the f ina ncing is unsat is factory be-cause i t would have to be bank- f inanced a t h ighinterest .

House B. This house is a l i t t le smaller thanHouse A and is not close to a bus route. Theneighborhood gives one the feel ing of insecur i tybecause of t raff ic condi t ions . The yard space is

T a b l e 3C o m p a r i s o n m a t r i c e s a n d l o c a l p r i o r it i e sS i z e o f h o u s e A B C P r i o r i t y Y a r d s p a c e A B C P r i o r i t y

v e c t o r v e c t o rABC

T r a n s p o r t a t i o n

1 6 8 0.75 4 A 1 5 4 0.67 4I 1 4 0.1 81 B -~ 1 0.101I 1 1z 1 0.065 C z 3 1 0.22 6

~ m a x = 3 .136 , CI = 0 .0 6 8 , CR = 0 .117 ~kma x = 3 .0 8 6 , CI = 0 .0 4 3 , CR = 0 .0 7 4A B C P r i o r i t y M o d e m A B C P r i o r i t y

v e c t o r f a c i l i t i e s v e c t o rABC

N e i g h b o r h o o d

1 7 0.233 A 1 8 6 0.74 71 1 1 11 ~ 0.005 B ~ 1 5 0.06 0

15 8 1 0.713 C ~ 5 1 0.193~kma x = 3.247, CI = 0 .124, C R = 0 .213 ~max = 3 .197, CI = 0 .099, C R = 0 .170

A B C P r i o r i t y G e n e r a l A B C P r i o r i t yv e c t o r c o n d i t i o n v e c t o r

ABC

A g e o f h o u s e

I 11 8 6 0.745 A 1 ~ ~ 0.20 01 1 1 0.065 B 2 1 1 0.40 0g1 4 1 0.181 C 2 1 1 0.40 0

h m a x = 3 .13 0 , CI = 0 .0 6 8 , CR = 0 .117 ~ m a x = 3 . 0 0 0 , C I = 0 . 0 0 0 , C R = 0 . 0 0 0A B C P r i o r i t y F i n a n c i n g A B C P r i o r i t y

v e c t o r v e c t o rABC

| 1 0 . 07 21 1 1 0.333 A 1 v1 1 1 0.333 B 7 1 3 0.65 0

11 1 1 0.333 C 5 3 1 0.278?~max = 3 .000, cf = 0 .000, C R = 0 .000 ~max = 3 .065, CI = 0 .032, C R = 0 .056

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18 T.L. Saaty / The AHP : How to make a decisiont o t he i r pa re n t c r i t e r i on . An a l t e rna t i ve i se va l ua t e d , fo r e a c h c r i t e r i on o r subc r i t e r i on , byident i fying the grade which bes t descr ibes i t . F i -na l ly , t he we i gh t e d o r g l oba l p r i o r i t ie s o f t he g ra de sa re a dde d t o p roduc e a r a t i o s c a l e fo r t he a l t e rna -t i ve . Abso l u t e me a sure me nt ne e ds s t a nda rd t oma ke i t pos s i b le t o j udge whe t he r t he a l t e rna t i ve i sa c c e p t a b l e o r no t . Abso l u t e me a sure me nt i s use fu li n s t ude n t a dmi s s i on , f a c u l t y t e nure a nd p romo-t i on , e mpl o ye e e va l ua ti on , a nd i n o t he r a re a s whe ret he re i s f a i r l y good a gre e me nt on s t a nda rd whi c hare then used to ra te a l te rna t ives one a t a t ime .

Le t us c ons i de r a n a bbre v i a t e d ve r s i on o f t hep r o b l e m o f e v a lu a t in g e m p l o y e e p e r f o r m a n c e . T h eh i e ra rc hy fo r t he e va l ua t i on a nd t he p r i o r i t i e sd e r i v e d t h r o u g h p a i r e d c o m p a r i s o n s a r e s h o w nbelow. I t i s then fol lowed by a ra t ing of eache m p l o y e e f o r t h e q u a l i t y o f p e r f o r m a n c e u n d e re a c h c r i t e r i on a nd summi ng t he r e su l t i ng sc ore s o fob t a i n h i s ove ra l l r a t i ng . The h i e ra rc hy i n Ta b l e 5c a n be more e l a bora t e , i nc l ud i ng subc r i t e r i a , fo l -lowed by the intens i t i es for express ing qua l i ty .

Le t us now show how t o ob t a i n t he t o t a l s c orefor Mr. X (see Ta ble 5) :0.061 X 0.604 + 0.1 96 X 0.731 + 0.043 0.199

+ 0.071 x 0 .750 + 0.162 X 0.188+ 0.466 x 0.750 = 0.623.

Simi la r ly the score for Ms. Y and M r. Z can beshown to be 0 .369 an 0.478 respec t ive ly. I t i s c leart h a t w e c a n r a n k a n y n u m b e r o f c a n d i d a t e s a l o n gthese l ines . Here the vec tor of pr ior i t i es of thec r i t e ri a ha s be e n w e i gh t e d b y t he ve c t o r o f r e l a t i venumbe r o f i n t e ns i t i e s unde r e a c h c r i t e r i on a ndthen renormal ized. We ca l l th i s a s t ruc tura l resca l -ing of the pr ior i t i es .

5. Theoretical considerations [7]The re i s a we l l known pr i nc i p l e i n ma t he ma t i c s

t ha t i s wi de l y p ra c t i c e d , bu t s e l dom e nunc i a t e dwi t h su f f i c i e n t fo rc e fu l ne s s t o i mpre s s i t s i mpor -t a nc e . A ne c e s sa ry c ond i t i on t ha t a p roc e dure fo rso l v i ng a p rob l e m be a good one i s t ha t i f i tp roduc e s de s i r e d re su l t s , a nd we pe r t u rb t he va r i a -bles of the problem in some smal l sense , i t g ives usresul t s tha t a re ' c lose ' to the or igina l ones . This i sp re c i se l y t he use o f c on t i nu i t y a nd un i fo rm c on-t i n u i t y - - t o a s s u r e t h a t a f t e r t r a n s f o r m i n g a v a r i a -b l e , o r i g i na l l y ne a rby va l ue s go ove r t o ne a rby

va l ue s . An e x t e ns i on o f t h i s ph i l osophy i n p rob-l e ms whe re o rde r r e l a t i ons be t we e n t he va r i a b l e sa re i mpor t a n t , i s t ha t on sma l l pe r t u rba t i ons o ft he va r i a b l e s , t he p roc e dure p roduc e s c l ose , o rde rpre se rv i ng re su l t s . The p roc e dure de sc r i be d he rehas thi s charac te r i s t ic .

Be c a use o f t he na t u ra l wa y i n whi c h a ma t r i x o fra t i os a nd sma l l pe r t u rba t i ons o f t ha t ma t r i x l e a dt o a p r i nc i pa l e i ge nva l ue p rob l e m, one ma y s t a r twi t h t h i s a nd ge ne ra l i z e t o pos i t i ve ma t r i c e s t ha tn e e d n o t b e r e c i p r o c a l . T h e G e r m a n m a t h e m a t i -c i a n Oska r Pe r ron p rove d i n 1907 t ha t , i f A- -( ai j ) , a u > O, i , j = 1 . . . . . n, t he n A ha s a s i mpl eposi t ive e igen va lue ~m~x (ca l led the pr inc ip a l e i -g e n v a l u e o f A ) a n d ~kma x ~> ]~k k I fo r t he r e ma i n i nge i ge nva l ue s o f A . Fur t he rmore , t he p r i nc i pa l e i -ge nve c t o r w = (w I . . . . . w , ) v t ha t i s a so l u t i on o fA w = ~m~xw has w~ > 0, i = 1 , . . . , n . We ca n w ri tet h e n o r m o f t h e v e c t o r w a s I l w l l = e T w w h e r ee = (1, 1 . . . . . 1 ) T a nd we c a n norm a l i z e w by d i v id -i ng i t by i t s norm. For un i que ne ss , whe n we re fe rt o w w e m e a n i t s n o r m a l i z e d f o r m . O u r p u r p o s ehe re i s t o show how i mpor t a n t t he p r i nc i pa l e i ge n-ve c t o r i s i n de t e rmi n i ng t he r a nk o f t he a l t e rna -t i ve s t h rough domi na nc e wa l ks .

We ha ve se e n t ha t r a t i o s c a l e e s t i ma t i on ha s ana t u ra l s e t t i ng i n p r i nc i pa l e i ge nva l ue fo rmul a -t i on . We wi l l now show t ha t i n a dd i t i on , t hepr i nc i pa l e i ge nve c t o r a l so ha s t he o rde r p re se rv i ngprope r t i e s we se e k .

Whe n A i s c ons i s t e n t , one wa y t o de f i ne t heorde r o f t he a l t e rna t i ve s i s t o r e qu i re t ha t one rowo f t h e m a t r i x d o m i n a t e e l e m e n t w i s e a n o t h e r r o w .But whe n A i s i nc ons i s t e n t i t i s no l onge r poss i b l et o de f i ne domi na nc e i n t h i s ma nne r . Ins t e a d , web o r r o w t h e c o n c e p t o f d o m i n a n c e f r o m g r a p ht he ory whe re t he sum of t he c oe f f i c i e n t s i n e a c hrow o f A i s use d . Th i s c o nc e p t c a r r i e s ove r i n ana t u ra l wa y t o t he i nc ons i s t e n t c a se . Bu t we mus tl o o k a t a d i f f e r e n t w a y t o c a p t u r e d o m i n a n c e b yc ons i de r i ng fu r t he r poss i b i l i t i e s no t s i mpl y f romt he ma t r i x i t s e l f o r some a rb i t r a ry powe r o f i t , bu tf rom a l l i t s powe rs .

T h e m a t r i x A c a p t u r e s o n l y t h e d o m i n a n c e o fone a l t e rna t i ve ove r e ve ry o t he r i n one s t e p . Bu ta n a l t e rna t i ve c a n domi na t e a s e c ond by f i r s tdomi na t i ng a t h i rd a l t e rna t i ve a nd t he n t he t h i rddomi na t e s t he se c ond . Thus , t he f i r s t a l t e rna t i vedomi na t e s t he se c ond i n t wo s t e ps . I t i s knownt ha t t he r e su l t fo r domi na nc e i n t wo s t e ps i so b t a i n e d b y s q u a r i n g t h e p a i r w i s e c o m p a r i s o n m a -

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t r ix . S i mi la r l y , dom i na n c e c a n o c c ur i n t h re e s t e ps ,four s t e ps a nd so on , t he va l ue o f e a c h i s ob t a i ne dby ra i s i ng t he ma t r i x t o t he c or re spondi ng powe r .The ra nk o rde r o f a n a l t e rna t i ve i s t he sum of t here l a t i ve va l ue s fo r domi na nc e i n i t s row, i n ones t e p , t wo s t e ps a nd so on a ve ra ge d ove r t he num-be r o f s t e ps . The que s t i on i s whe t he r t h i s a ve ra getends to a meaningful l imi t . I t i s easy to see tha t i tdoes whe n A i s consis tent beca use A k = n k- 1A.

We c a n t h i nk o f t he a l t e rna t i ve s a s t he node s o fa d i re c t e d g ra ph . Wi t h e ve ry d i re c t e d a rc f romno de i to nod e j (w hich need not be di s tinc t ) , isa s soc i a t e d a nonne ga t i ve numbe r a i j o f t he domi -na nc e ma t r i x . In g ra ph- t he ore t i c t e rms t h i s i s t hei n t e ns i t y o f t he a rc . De f i ne a k -wa l k t o be ase que nc e o f k a rc s suc h t ha t t he t e rmi na t i ng nodeof e a c h a rc e xc e p t t he l a s t i s t he sourc e node o fthe a rc which succeeds i t . The intens i ty of a k-w alki s the produc t of the intens i t i es of the a rcs in thewa l k . Wi t h t he se i de as , we c a n i n t e rp re t t he ma t r i xAk: the ( i , j ) ent ry of A k i s the su m of thei n t e ns i t i e s o f a l l k -wa l ks f rom node i t o node j .De f i n i t i on . The domi na nc e o f a n a l t e rna t i ve a l ongall walks o f len gth k ~< m is given b y1 m A%

m ~-" e'rA% " (3 )k= lObse rve t ha t t he e n t r i e s o f A % a re t he row sumsof A k a nd t ha t eVA% i s the sum of a l l the ent r iesof A .T h e o r e m 3. The dominance of each alternative alongall walks k, is given by the solution o f the eigenvalueproblem A w = X maxW.P r o o f . Le t

A k es k eTA% (4 )a n d

1 mtm = m E sk. (5 )

k= l

The c onve rge nc e o f t he c ompone n t s o f t m t o t hesa me l i mi t a s t he c ompone n t s o f s , , i s t he s t a nd-a rd Ce sa ro summa bi l i t y . S i nc e

Ake ' w as k---, m , (6)s k - e T A k e

whe re w i s t he norma l i z e d p r i nc i pa l r i gh t e i ge n-ve c t o r o f A , we ha ve

1 ~ A%tm = -~ eVAk------ -~ w a s m ~ ~ . (7 )k= l

The so l u t i on o f t he e i ge nva l ue p rob l e m i s ob-ta ined by ra i s ing the mat r ix A to a suff ic ient lyl a r g e p o w e r t h e n s u m m i n g o v e r t h e r o w s a n dnorm a l i z i ng t o ob t a i n t he p r i o r i t y ve c t o r w =(w 1 . . . . wn)~ . The p roc e s s i s s t oppe d whe n t hed i f f e r e n c e b e t w e e n c o m p o n e n t s o f t h e p r i o r i t yve c t o r ob t a i ne d a t t he k - t h powe r a nd a t t he(k + 1 ) st pow e r i s l e ss t ha n some p re de t e rm i ne dsmal l va lue .

In re fe rence [7] we gave a t l eas t f ive di ffe rentwa ys o f de r i v i ng t he p r i o r i t i e s f rom t he ma t r i x o fpa i re d c ompa r i sons . Be s i de s t he e i ge nve c t o r so l u -t i on , t he se i nc l ude t he d i re c t row sum a ve ra ge , t hen o r m a l i z e d c o l u m n a v e r a g e , a n d m e t h o d s w h i c hmi n i mi z e t he sum of t he e r ro r s o f t he d i f fe re nc e sbe t we e n t he j udg me nt s a nd t he i r de r i ved va l ue ssuc h a s t he me t hods o f l e a s t squa re s a nd l oga r i t h -mi c l e a s t squa re s . We po i n t e d ou t t ha t t he l oga -r i t hmi c l e a s t squa re s so l u t i on c o i nc i de s wi t h t hepr i nc i pa l r i gh t e i ge nve c t o r so l u t i on fo r ma t r i c e s o forder n = 3, which i s the f i rs t va lue of n for whichi nc ons i s t e nc y i s poss i b l e a nd l e f t a nd r i gh t e i ge n-ve c t o r s a re r e c i p roc a l s o f e a c h o t he r whi c h i s no ta lways the case for l a rger va lues of n . Since thea p p e a r a n c e o f t h e A n a l y t i c H i e r a r c h y P r o c e s s i nt he l i t e ra t u re , o t he r me t hods ha ve be e n p ropose d[7,15] . Al l methods yie ld the same answer whent he ma t r i x i s c ons i s t e n t . The c ombi ne d use o f ame a sure o f i nc ons i s t e nc y whi c h c a n be de r i ve d i nt e rms o f bo t h l e f t a nd r i gh t e i ge nve c t o r s , a l ongwi t h t he r i gh t e i ge nve c t o r so l u t i on whi c h c a p t u re st he domi n a nc e e x pre s se d i n the j udg me n t s , i s a ne f fe c ti ve wa y t o l ook a t t he p ro b l e m. W e a rguet ha t so l ong a s i nc ons i s t e nc y i s t o l e ra t e d , domi -na nc e i s t he ba s i c t he ore t i c a l c onc e p t fo r de r i v i nga sc a l e a nd no o t he r me t hod qua l i f i e s . In a dd i t i on ,a c o u n t e r e x a m p l e h a s b e e n p r o v i d e d i n w h i c h t h em e t h o d n o t o n l y d o e s n o t g e n e r a t e a g o o d a p -prox i ma t i on , bu t a l so r e ve r se s r a nk . Some ha vee ve n re sor t e d t o a r t i f i c i a l a x i oma t i z a t i on t h i nk i ngt ha t i t g i ve s a me t hod t he a ppe a ra nc e o f r i gor ,a l t h o u g h a x i o m s a r e a s s u m p t i o n s n o t p r o o f s . I tma y be t ha t t he a r i t hme t i c o f a me t hod i s s i mpl e rt ha n t ha t use d t o ob t a i n t he e i ge nve c t o r , bu t t ha tno l onge r ma t t e r s , be c a use o f t he wi de spre a d use

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20 T.L. Saaty / The AH P: How to make a decisionof t he c omput e r . I t i s r e a sona b l e t o a rgue t ha t at h e o r y f o r m a k i n g s o u n d d e c i s i o n s m u s t s t a n d o nc l e a r ly j us ti f i a b l e g rounds .

The so f t wa re pa c ka ge Expe r t Choi c e , use fu l i nt e a c h i ng a nd i n r e a l a pp l i c a t i ons , c a n ha nd l e bo t hre l a t i ve a nd a bso l u t e me a sure me nt , a s we l l a s ha v-ing spec ia l capabi l i t i es such as s t ruc tura l resca l ing,c om bi n i ng g ro up j udgm e nt s , s e ns i ti v i t y a na l ys i sa nd de pe nde nc e a mong t he de c i s i on a l t e rna t i ve s[2] . The reader inte res ted in pursuing the subjec tfur ther should consul t re fe rences [3 ,7 ,8 ,13,15] .

6. Norm al izat ion - Scarc i ty and abundanceN o r m a l i z a t i o n i n t h e A H P i s n o t j u s t a m e c h a -

n i c a l ope ra t i on . I t c on t a i ns i n fo rma t i on on t het o t a l domi na nc e o f t he a l t e rna t i ve s be i ng c om-pa re d whi c h e na b l e s us t o a ppor t i on t he p r i o r i t yof t he c r i t e r i on t o e a c h a l t e rna t i ve a c c ord i ng t ot he r e l a t i ve domi na nc e o f t he a l t e rna t i ve .

Norma l i z a t i on c a n a l so be a s soc i a t e d wi t h t hei de a o f s c a rc i t y a nd a bunda nc e o f t he p re se nc e o fa c r i t e r ion in the a l te rna t ives such as redness inf ru i t . Too ma ny f ru i t s t ha t a re r e d ma ke re da b u n d a n t a n d u n i m p o r t a n t i n d i f f e r e n t i a t i n g b e -t we e n i nd i v i dua l f ru i t s . Conve rse l y , i f r e dne ss oc -curs intense ly in some frui t s but not in others , i t i sthus scarce and can be used as a c r i t e r ion tod i f fe re n t i a t e i n ma ki ng a de c i s i on a mong f ru i t s .Thus , t he g re a t e r t he c on t ra s t a mong t he a l t e rna -t ives , the more useful i s the pr ior i ty va lue of thec r i t e r i on a l l o t t e d t o e a c h . Conve rse l y , whe n fo re xa mpl e , t he c on t ra s t o f t he a l t e rna t i ve s i s sma l -les t (when the tota l do min anc e i s equa l to n 2 , seebe low) and hence a l l the a l te rna t ives a re a l ike , thepor t i o n o f t he c r i te r i on p r i o r i t y a s s i gne d t o e a c h i sequa l . Thus a c r i t e r ion wi th respec t to which therei s g r e a t e r c o n t r a s t a n d d o m i n a n c e a m o n g t h e a l -t e rna t i ve s i s ' s c a rc e ' a nd c onse que n t l y more i n -f l ue n t i a l i n de t e rmi n i ng ra nk t ha n a no t he r c r i t e -r i on on whi c h t he re i s no d i s t i nc t i on a mong t hea l t e rna t i ve s , i . e . , t he i r ma t r i x o f pa i re d c ompa r i -sons ha s more l ' s i n i t . In be i ng sc a rc e , more o ft he c r i t e r i on i s a l l o t t e d t o t he more domi na n ta l t e rna t i ve a nd he nc e i t a f fe c ts t he f ina l r a nk i ng o ft ha t a l t e rna t i ve more .

A n ' a b u n d a n t ' c r it e ri o n c o n t r i b u t e s l es s to r a n kde t e rmi na t i on be c a use i t c on t r i bu t e s a n e qua l o rne a r l y e qua l p r i o r i t y t o e a c h a l t e rna t i ve a nd whe nthe sum is t aken over the c r i t e r ia i t has l i t t l e e ffec t

i n de t e r mi n i ng ra nk ( re c al l t ha t i f a > b t he na + c > b + c and c do es no t reverse ord er) . A na l t e rna t i ve t ha t i s a c opy o f a no t he r c a n d i l u t e t hepr ior i ty of a dec is ive c r i t e r ion so tha t i t i s nol onge r t he c on t ro l l i ng one i n de t e rmi n i ng t he f i na lra nk . We ha ve :T h e o r e m 4. ~ i j a i j = n 2, i f an d on ly i f a l l a l te rna-t ive s have equa l do min anc e , i .e . , a i j = 1 , i , j =1 . . . . , n , wi th re spec t to a c r i te r ion .

I t i s e a sy t o s e e t ha t n 2 i s t he mi n i mu m va l ueo f p o s si b l e to t a l d o m i n a n c e b y a p a ir e d c o m p a r i -son o f n a l t e rna t i ve s . The ma xi mum va l ue i s ,us i ng 9 a s t he uppe r r a nge o f t he sc a l e o f pa i re dc o m p a r i s o n s , [ n ( n - 1)](9 + ~) + n = ~(41 n 2 -32n) .

Our d i sc uss i on o f a bso l u t e a nd re l a t i ve me a s -u r e m e n t m a y b e f r a m e d i n e c o n o m i c t e r m s . I na b s o l u t e m e a s u r e m e n t , v a l u e a n d n e e d are ident i -c a l ; ' t he more va l ue , t he be t t e r t he ne e d i s s a t i s -f i e d ' . In r e l a t i ve me a sure me nt , v a l u e i s assessed int e r m s o f n e e d . H e r e s u r p l u s v a l u e m a y o r m a y n o tsa t i s fy more ne e d . In fa c t , t he re a re i ns t a nc e sw h e r e s a t i a t i o n t a k e s p l a c e a n d a b u n d a n c e c a nlead to a dec l ine in the sa t i s fac t ion of need.

7. ClusteringC o m p a r i s o n s o f e l e m e n t s i n p a i r s r e q u i r e s t h a t

t he y be homoge ne ous o r c l ose wi t h r e spe c t t o t hec ommon a t t r i bu t e ; o t he rwi se s i gn i f i c a n t e r ro r sm a y b e i n t r o d u c e d i n t o t h e p r o c e s s o f m e a s u r e -m e n t . I n a d d i t i o n , t h e n u m b e r o f e l e m e n t s b e i n gc o m p a r e d m u s t b e s m a l l ( n o t m o r e t h a n 9 ) t oi m p r o v e c o n s is t e n c y a n d t h e c o r r e s p o n d i n g a c c u -r a c y o f m e a s u r e m e n t . F o r e x a m p l e , w e m a y c l u s -t e r a pp l e s i n one wa y a c c ord i ng t o s i z e , i n a no t he rwa y a c c ord i ng t o c o l o r a nd i n s t i l l a no t he r wa ya c c ord i ng t o a ge . The que s t i on t he n i s how t op e r f o r m c l u s te r i ng o f h o m o g e n e o u s e l e m e n t s i n a ne f f i c i e n t wa y t o fa c i l i t a t e pa i re d c ompa r i sons .Cl us t e r i ng i s a p roc e s s o f g roup i ng e l e me nt s wi t hr e s p e c t t o a c o m m o n p r o p e r t y . O n e c a n t h e nde c ompose t he se t o f o rde re d e l e me nt s wi t h r e -spe c t t o a n a t t r i bu t e i n t o c l us t e r s o f , fo r e xa mpl e ,se ve n e l e me nt s e a c h , f rom l a rge s t t o sma l l e s t . Thesma l l e s t e l e me nt o f t he l a rge s t c l us t e r i s i nc l ude da s one o f t he se ve n e l e me nt s o f t he ne x t c l us t e r .The re l a t i ve we i gh t s o f a l l t he e l e me nt s i n t h i s

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se c ond c l us t e r a re d i v i de d by t he we i gh t o f t hec o m m o n e l e m e n t a n d t h e n m u l t ip l i e d b y i ts w e ig h ti n t he f i r s t c l us t e r i n t h i s ma nne r bo t h c l us t e r sb e c o m e c o m m e n s u r a t e a n d a r e p o o l e d t o g e t h e r .The p roc e ss i s t he n re pe a t e d t o t he re ma i n i ngc l us t e r s . Some t i me s one ma y ne e d t o i n t roduc ehypot he t i c a l e l e me nt s i n o rde r t o p re se rve t hegra dua l de sc e n t f rom l a rge t o sma l l .

We d i sc uss t h re e wa ys a s t o how t o pe r fo rmc l us t e r i ng on t he a l t e rna t i ve s o f a de c i s i on p rob-l e m w h o s e n u m b e r m a y b e v e r y l a rg e a n d n e e d s ad i f fe re n t so r t i ng fo r e a c h o f s e ve ra l a t t r i bu t e s .The y a re o rde re d i n t he fo l l owi ng d i sc uss i on f romthe leas t to the most e ff ic ient way.7.1. The elementary approach

orde r on t ha t a t t r i bu t e . The y c a n t he n be c l us t e re di n t o sma l l g roups a s de sc r i be d a bove a nd pa i rwi sec o m p a r e d .

O n e r e a s o n w h y a b s o l u t e m e a s u r e m e n t m a y n o tbe des i rable i s tha t i t i s s t rongly subjec t ive . Inp a i r e d c o m p a r i s o n s , m e a s u r e m e n t i s b a s e d o n o b -se rva t i on o f t he re l a t i ve i n t e ns i t y o f a p rope r t yb e t w e e n t w o e l e m e n t s . A b s o l u t e m e a s u r e m e n t i sb a s e d o n o b s e r v a t i o n s s t o r e d in m e m o r y w h i c hde pe nd on e xpe r i e nc e a nd on t he a b i l i t y t o r e c a l li t . For ma ny prob l e ms , i t i s use fu l t o f i r s t c a r ryo u t a b s o l u t e m e a s u r e m e n t t o s o r t a n d c l u s t e r t h ee l e me nt s , a nd t he n fo l l ow t ha t wi t h re l a t i ve me a s -ure me nt fo r g re a t e r a c c ura c y . Th i s i s pa r t i c u l a r l yre l e va n t i n p re d i c t i ng mos t l i ke l y ou t c ome s whi c hi nvo l ve syne rgy a mong t he a l t e rna t i ve s .

Given n e lements in a leve l of the hie rarchy,o n e m a y f i r s t m a k e a p a s s t h r o u g h t h e m b y c o m -pa r i ng one e l e me nt wi t h a no t he r , d ropp i ng i t a ndp i c k i ng a no t he r i f t ha t one i s pe rc e i ve d t o bel a rge r a nd c on t i nu i ng t he c ompa r i son . Thus , t hela rges t e lemen t is se lec ted in n - 1 such co mp ari -sons . The p roc e ss i s r e pe a t e d fo r t he re ma i n i ngn - 1 e l e me nt s t o i de n t i fy the se c ond l a rge s t e l e-me nt a nd so on . In t he e nd , t he e l e me nt s woul d bea r ra nge d i n de sc e nd i ng o rde r o f s i z e o r i n t e ns i t ya c c ord i ng t o a n a t t r i bu t e a nd a re se que n t i a l l yc l us t e re d i n t o g roupi ngs o f a fe w e l e me nt s e a c hfrom the la rges t to the smal les t . This process i sh i gh l y i ne f f i c i e n t a nd re qu i re s t he a s t ronomi c a ln u m b e r o f ( n - 1 )! c o m p a r i s o n s .7.2. Trial and error clustering

T h e a l t e r n a t i v e s m a y b e p u t i n t o g r o u p s o fl a rge , me di um a nd sma l l . The n t he e l e me nt s i ne a c h g roup a re pu t i n t o se ve ra l c l us t e r s o f a fe we l e me nt s e a c h , a nd a f i r s t pa s s a t c ompa r i sons i suse d t o i de n t i fy mi s f i t s whi c h a re t he n t a ke n ou ta n d p u t i n t o t h e a p p r o p r i a t e o n e o f t h e o t h e r t w oc a t e gor i e s . R e c l us t e r i ng i s t he n pe r fo rme d a ndc om pa r i so ns a re c a r r ie d ou t . I f e l e me nt s a re foundnot t o f i t , t he y a re a ga i n move d t o t he a ppropr i a t ec a t e gory . Th i s p roc e ss i s r e pe a t e d fo r e a c h a t t r i -bu t e .7.3. Clustering by absolute measurement

Ea c h a l t e rna t i ve i s e va l ua t e d by a bso l u t e me a s -ure me nt fo r a n a t t r i bu t e , a nd t hus i n de sc e nd i ng

8 . Comb ining re la tive and abso lute measuremen t -C o s t -bene f i t a na l y s i sAn e a sy p i tfa l l fo r a n i nd i v i dua l wh o ha s j us t

l e a r n e d a b o u t t h e A H P i s t o t a k e t w o o r m o r ec r i t e r i a on whi c h a l t e rna t i ve s a re me a sure d on t hesa me e x i s t i ng s t a nda rd sc a l e , suc h a s do l l a r s o rk i l o g r a m s , n o r m a l i z e e a c h s e t a n d t h e n c o m p o s ewi t h re spe c t t o t he c r i t e r i a . One qu i c k l y d i sc ove rst ha t t he a nswe r i s no t t he sa me a s t ha t ob t a i ne dt hrough t he usua l a r i t hme t i c a nd ha s t i l y c onc l ude st ha t t he AHP i s a t f a u l t . To a vo i d t h i s p rob l e m,one mus t e xe rc i se c a u t i on i n c onve r t i ng me a sure -me nt s on a s t a nda rd sc a l e t o r e l a t i ve va l ue s whe nsevera l such c r i te r ia a re involved [9] .

F o r t h e a r i t h m e t i c t o c o n f o r m w i t h w h a t o n eord i na r i l y doe s , a s s i gn e a c h c r i t e r i on a p r i o r i t yt ha t i s t he sum of t he me a sure me nt s o f t he a l t e r -na t i ve s wi t h re spe c t t o i t , d i v i de d by t he sum oft he me a sure me nt s o f a l l t he a l t e rna t i ve s unde r a l lt he c r i t e r i a me a sure d wi t h t ha t un i t . To f i nd t hec ompos i t e p r i o r i t y fo r e a c h a l t e rna t i ve mul t i p l yt he c r i t e r i on we i gh t by i t s c or re spondi ng norma l -i z e d we i gh t unde r t ha t c r i t e r i on a nd sum ove r t hec r i te r i a. The re su l t ma y b e c ons i de re d a s t he we i gh tof t he a l t e rna t i ve wi t h re spe c t t o one supe r c r i t e -r i on c ompose d of a l l t he c r i t e r i a wi t h t he un i t o fm e a s u r e m e n t . T h a t s u p e r c r i t e r i o n m a y t h e n b ec o m p a r e d w i t h o t h e r i n t a n g i b l e c r i t e r i a a n d o t h e rsupe r c r i t e r i a wi t h d i f fe re n t un i t s o f me a sure me nt .A l t e r n a t i v e l y , o n e c a n p e r f o r m p r i o r i t y c o m p a r i -sons on a l l the da t a a va i l a b l e wi t hou t a sc r i b i ngl i ne a r i t y t o t he m a s one o rd i na r i l y doe s wi t h num-

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22 T.L. Saaty / The AHP: How to make a decisionbe rs . Tha t i s t o s a y , one i n t e rp re t s wha t t he num-b e r s m e a n a n d u s e s j u d g m e n t r a t h e r t h a n c r a n k i n gt he numbe r me c ha n i c a l l y . Wi t h r a re e xc e p t i on , t hej u d g e m e n t a p p r o a c h i s b y f a r t h e m o r e e f f e c t i v eproc e dure us i ng t he AHP t o de a l wi t h t he unde r l y -i ng c ompl e x i t y o f a de c i s i on p rob l e m, a nd onene e d no t be a f ra i d t o do i t . I t i s known t ha t s c a l e sa re i nve n t e d t o fa c i l i t a t e c ommuni c a t i on , a nd mus tbe e xpe r i e nc e d fo r a l ong t i me be fore t he y c a n bea s soc i a t e d wi t h our va l ue sys t e m, dur i ng whi c ht i me t he y a l mos t a l wa ys a re modi f i e d .

The AHP i s a de sc r i p t i ve t he ory . The re fore , i ti s n o t a n a u t o m a t i c s e t- u p f o r a c c o m m o d a t i n g a n yn o r m a t i v e a p p r o a c h s u c h a s u t i l i t y m a x i m i z a t i o n .I t ne e ds t o be i n t e rp re t e d a nd a da p t e d fo r t ha tpurpose . I t i s e ve n more d i f f i c u l t whe n a norma -t i ve t he ory ha s ma ny e xc e p t i ons so t he AHP i n t e r -p re t a t i on woul d no t be un i ve r sa l . In u t i l i t y ma xi -miza t ion i t i s assumed tha t i t i s a lways the caset ha t t he more u t i l i t y o r more mone y t he be t t e r ,bu t t he re a re ma ny i ns t a nc e s whe n t h i s i s no t t rue .For e xa mpl e , a gove rnme nt a ge nc y t ha t i s l e f twi t h more mone y a t t he e nd o f t he ye a r , ge t s asmal le r a l loca t ion in a future budget . A r ich indi -v i d u a l i n a p o o r c o u n t r y u n d e r r e v o l t b y t h e p o o ri s l ike ly to ge t ki l led. The ava i labi l i ty of money i so f t e n a n i nc e n t i ve t o use mone y whe re o t he r a l t e r -na t i ve s c ou l d be more e f fe c t i ve . Thus , AHP a da p-t a t i on fo r u t i l i t y c a l c u l a t i on purpose s ne e ds t ot a ke suc h c a ve a t s i n t o c ons i de ra t i on . One i de a t ok e e p i n m i n d w i t h t h e A H P i s t h a t t h e m o n t o n i c -i t y o f u t i l i t i e s ne e d no t be p re se rve d a nd ma y bec on t ra d i c t e d .

T h e f o r e g o i n g h a s b e a r i n g o n b e n e f i t - c o s tana lyses . Expec ted ut i l i t i es a re used for repea tedde c i s i on ma ki ng . In t ha t c a se, be ne f i t c os t a na l ys i sf rom t wo h i e ra rc h i e s i s a pp l i e d , a nd f rom i t onec a n a l so c a l c u l a t e ma rg i na l be ne f i t t o c os t r a t i os .R e s o u r c e a l l o c a ti o n m a y b e m a d e b y u s i n g b e n e f i tt o c os t r a t i os t hus de r i ve d . Shor t r a nge de c i s i onsoften inc lude low cos t s as benefi t s in a s ingleh i e ra rc hy . Th i s i s a use fu l a pproa c h whe n a de c i -s i o n t o s p e n d m o n e y o n o n e o f s e ve r al o p t io n s h a sa l r e a d y b e e n m a d e . I t i s o n e w a y t o c o m b i n ebe ne f i t s a nd c os t s a s one doe s wi t h do l l a r s byt a k i ng d i f fe re nc e s . In t he AHP one doe s no t used i f fe re nc e s . In suc h one - t i me de c i s i ons , c os t s ma ybe re ga rde d a s i nve r se be ne f i t s so t ha t be ne f i t s ,l ow c os t s a nd o t he r i nc onve n i e nc e s a re use d t oes tabl i sh pr ior i t i es for the a l te rna t ives .

I f , on t he o t he r ha nd , be ne f i t s a nd c os t s a re

me a sure d i n do l l a r s a l ong wi t h i n t a ng i b l e fa c t o r s ,t h e y m u s t f i r s t b e c o m p o s e d a c c o r d i n g t o t h edo l l a r s a nd t he n c ombi ne d wi t h t he i n t a ng i b l ec r i t e r i a a s de sc r i be d e a r l i e r . Ma rg i na l be ne f i t c os ta na l ys i s a l ong t r a d i t i ona l l i ne s c a n be c a r r i e d ou tby a r ra ng i ng t he c os t s i n i nc re a s i ng o rde r , a ndt he n fo rmi ng ra t i os o f suc c e s s i ve d i f fe re nc e s o fbe ne f i t s a nd c os t s i n t ha t o rde r . The ve ry f i r s tr a t i o i s t ha t o f t he a l t e rna t i ve wi t h t he sma l l e s tc os t . The n one fo rms t he r a t i o o f d i f fe re nc e s wi t ht ha t be t we e n t he ne x t h i ghe s t c os t a nd t he sma l l e s to n e i n t h e d e n o m i n a t o r , a n d t h e c o r r e s p o n d i n gd i f fe re nc e i n be ne f i t s i n t he nume ra t o r . Whe ne ve ra d i f fe re nc e i n a nume ra t o r i s ne ga t i ve , t ha t suc -c e e d i ng a l t e rna t i ve i s d roppe d f rom c ons i de ra t i on .In t h i s ma nne r , t he a l t e rna t i ve y i e l d i ng t he h i ghe s tma rg i na l r a t i o i s c hose n .

The ne x t que s t i on i s how t o f i nd a r e a sona b l ew a y t o c o m b i n e b e n e f i t s ( B ) a n d c o s t ( C ) , w h e n ara t io ra ther than a s ingle hie ra rchy a re used [10] .Th i s i s use fu l i n c ons i de r i ng c onf l i c t p rob l e msi nvo l v i ng more t ha n one va l ue sys t e m. S i nc e t heh i e ra rc hy o f c os t s l e a ds t o a ve c t o r o f va l ue si nd i c a t i ng re l a t i ve ma xi mum c os t s , t he r e c i p roc a l sof t he e n t r i e s o f t h i s ve c t o r c ou l d be t hought o f a st h e m i n i m u m c o s ts i n c u r r e d i n j o i n t l y m a x i m i z i n gbe ne f i t s a nd mi n i mi z i ng c os t s . In ge ne ra l , t he ou t -c ome s f rom a h i e ra rc hy o f be ne f i t s a nd a h i e ra rc hyof c os t s a re t wo ra t i o s c a l e s whose c or re spondi ngra t i os l e a d t o a me a n i ngfu l r a t i o s c a l e . The i r d i f -fe re nc e s woul d no t be me a n i ngfu l .

T o m a x i m i z e B / C i s e qu i va l e n t t o ma xi mi z i nglo g B / C , whi c h i f B a nd C a re c lose , ma y bea p p r o x i m a t e d b y ( B / C ) - 1 o r ( B - C ) / Ck n o w n a s r et u r n o n i n v e s t m e n t , R O I . I f B a n d Ca re no t c ompa ra b l e , i t woul d be i n i t i a l l y c l e a r t ha ton l y t he be ne f i t s o r on l y t he c os t s de t e rmi newhe t he r t he a l l oc a t i on shou l d be ma de o r no t .

S o m e p e o p l e h a v e a t t e m p t e d t o c o m p a r e t h efore go i ng wi t h wha t i s t r a d i t i ona l l y done i n as i ng l e c r i t e r i on c ho i c e p rob l e m. An e xa mpl e whe reB- C a l one g i ve s r i s e t o mi s l e a d i ng re su l t s i sg i ve n by my c o l l e a gue L .G. Va rga s : You ha ve $1mi l l ion to inves t to ge t $1.101 mi l l ion or you have$500000 t o i nve s t t o ge t $600000 . Whi c h i s ab e t t e r i n v e s t m e n t ?B - C ana lysi s gives : 1.101 - 1 = 0.101 ,

0.6 - 0.5 = 0.1,a nd t he f i r s t a l t e rna t i ve woul d be c hose n . ROI

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T.L. Saaty / The AH P: How to mak e a dec ision 23

analys is yields 10.1% and 20%, respect ively, andthe second a l t e rna t ive which m in imizes r isk wouldbe c or rec t ly chosen .

9. The semiot ic connect ionThe Analy t i c H ierarchy Proces s i s a too l o f

in format ion communica t ion and s ign i f i ca t ion . In ageneral sense, i t belongs to the s tudy of languageor semiot ics . Semiot ics or semiology is a coding inwhich one considers s ignals related by a set ofrules or syntax; a set of s tates or contents cal led asemant ic sys tem, a set of poss ible behavioral re-sponses independent o f the con ten t sys tem, and arule associat ing s ignals with contents or with be-havioral responses . The idea of a code covers al lfour phenomena ment ioned above . Eco [1] usess -code for the f i r s t th ree and code for the four th .

According to Morr is [5] , the creator of semi-ot ics , semios is is the process in which somethingfunct ions as a s ign . He ment ions tha t bo th manand animals do respond to cer tain things as s ignsof someth ing e l se , bu t the i r complex i ty in man i sfound in speech, ar t , wr i t ing, and even medicaldiagnoses . Science and s igns are inseparably inter -connected. Semiot ics , a s tep in the unif icat ion ofscience, suppl ies the foundat ions for any scienceof s igns ; l inguis t ics , logic, mathematics , rhetor icand aes thet ics .

In the Analy t i c H ierarchy Proces s , words a reused for concepts involved in decis ions . In a sense,the purpose of a l l in format ion i s to dec ide onsometh ing- -even i f i t i s an imaginary hypothes i s .The hierarchy is the syntax, the subject of a deci-s ion is the semant ic, pr ior i t izat ion is the rule asso-c ia t ing s igna l s w i th con ten t , and behavior com-pr ises the poss ible decis ion al ternat ives . This wayof looking a t the AHP needs to be h igh l igh ted andemphas ized to d r ive home the un iver sa l i ty o f thedecis ion process .

10. Catastrophe and the AHPOne of the mos t burn ing ques t s we under take

us ing the knowledge we acqui re i s exp la in ing andforecas t ing, and in par t icular , forecas t ing catas-t rophes , o r , mathemat ica l ly speak ing , sudden d i s -con t inu i t i es tha t a f fec t the o rder o f the a l t e rna-t ives . This is a s i tuat ion that is at odds with the

ear l i e r mathemat ica l p r inc ip le o f smal l per tu rba-t ions we gave in which smal l changes in inpu t l eadto smal l changes in ou tcomes . In ca tas t rophes ,there i s a par t o f the p rob lem in which smal lchanges in condi t ions a t some cr i t i ca l va lue cangive r ise to very large changes in outcome. All wecan do i s e i ther to an t i c ipa te and t ake s t rongprevent ive ac t ion or p repare our se lves in advancewi th con t ingenc y p lans toge ther w i th emot ion a laccep tance fo r how to dea l w i th the emergency topay a smal le r pena l ty i f i t occur s , o r be p reparedto pay the fu l l pena l ty hav ing accep ted tha t i tcou ld happen . Because ca tas t rophes a re usua l lyunseen surpr ises , i t gives us a feel ing of control toth ink we can account fo r ca tas t rophes in our da i lyl ives . Accord ing to the d ic t ionary , a ca tas t rophe i sa sudden d i sas te r o r happening tha t causes g rea tharm or damage; a ca lami ty . Even the idea ofwhat cons t i tu tes a ca tas t rophe i s r e la t ive . Some-o n e w h o s e l i f e i s t o r m en t ed b y an o p p o n en t m ayregard i t as a bless ing i f a catas t rophe befal ls thetormentor . In o ther t e rms , a ca tas t rophe may beregarded as a very s t rong d i scont inu i ty in th ink-ing . How do we represen t a ca tas t rophe in t e rmsof the AHP?

In a ca tas t rophe the e lement o f surpr i s e a r i s esf rom a sh i f t in r ank ing of the ou tcomes . Thus , i f as i tua t ion i s ongoing a cer t a in cho ice of a l t e rna t ivesmay be ind ica ted , bu t a s l igh t change in the s i tua-t ion may cause a sudden sh i f t to ano ther cho icetha t would have been prev ious ly very undes i r ab le .How can we a l low for ca tas t rophic occur rences int h e A H P ?

One wa y to a l low for a ca tas t rophe is to a lwaysinc lude a c r i t e r ion for the unknown tha t r epre-sents a clus ter of unforseen threats . I t may i tselfhave subcr i t e r i a . The a l t e rna t ives a re carefu l lypr ior i t ized with respect to this cr i ter ion for theunexpec ted and i t s descendant s . The c r i t e r ion i t -self is ass igned a low pr ior i ty . Now we can imag-ine tha t a t each ins tan t the judgm ents in thecr i ter ia are f lashed in their total i ty on a screenfo l lowed by the bes t cho ice of a l t e rna t ives . Wi thsome changes in judgm ents , in genera l, cho ice iss t ab le and the changes in r e la t ive p r io r i ty o f theal ternat ive s l ight . After a cer tain lapse of t ime,there is a sudden c hange in judg me nt in the d ir ec-t ion of th i s c r i t e r ion making i t more impor tan t .The cor responding choice of a l t e rna t ives i s a l sosuddenly changed sur fac ing a very undes i r ab lea l t e rna t ive . Thus , one might inc lude among the

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2 4 T.L. Saaty / The AHP : How to make a decision

G O A L _ _

- ~ O L D W R E C K( 0 . 0 0 2 )

CH EV Y(0.009)-- SIZE >--> H O N D A

(0.0Sl) (0 .013)--> TOYOTA

(0.026)

i---~ OLDWRECK(0.005)CH EV Y_ O PERCO ST > (0.019)(0 .135) -> HOND A(0.044)"> TO Y O TA(0.067)

__> NORMAL(0.99O)

- - S T Y L E(0.269)

_ LO W PRICE(0.535)

-- SIZE(o.oos)

_ OPERCOST(0.001)L~ ACCIDENT ~(oolo) ---3

-- STYLE(0.0o2)

LOWPRICE(0.001)

II

O LD W RECK

I- '~ (0 .010)

CH EV Y(0.086)--~ HONDA(0.086)TOYOTA

( 0 . 0 ~ )O LD W RECKI (0.026)CH EV Y(0.081)H O N D A(0.201)TO Y O TA

(0.227)OLDWRECK(0.004)CHEVY(0.001)HONDA(0.000)TOYOTA(0.001)OLDWRECK(o.001)CHEVY(0.00o)HONDA(0.00o)TOYOTA(0.00o)

--~ OLDWRECKI (o.001)--> CHEVY(0.000)> --> HON DA(o.000

---> TOYOTA(0.00o)OLDWRECKI (0.o00)

--> CHEVY(0.o00)-~ HONDA(0.001)--~ TOYOTA(o.ool)

PRIORITIESTOYOTA~.142)HONDA(0.115)CHEVY(0.166)OLDWRECK~.576)

PRIORITIES

TO Y O TA(0.408)H O N D A(0.345)CH EV Y(0.197)O L D W R E C K(0.049)

F i g u r e 2 . T h e c a t a s t r o p h e h i e r a r c h y

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T.L. Saaty / The AH P: H ow to mak e a decision 25a l t e rna t i ve s some c a t a s t roph i c one s . Th i s t ype o ft h i nk i ng wou l d a p p l y i n h i e ra rc h ie s whe re a r e a lwrong c ho i c e i s e x t re me l y unde s i ra b l e a nd c os t l y .

We c a n a l so spe a k o f c ha os i n t he AH P: as i t ua t i on whe re t he poss i b i l i t y o f e xe r t i ng c on t ro lby c re a t i ng a h i e ra rc hy a nd se t t ing j ud gm e nt nolonger exis t s , for i t i s not c lear what t akes prece-d e n c e o v e r w h a t. U s u a l l y c a t a s t r o p h e p r o d u c e sc ha os . I t i s poss i b l e t o r e pre se n t bo t h c a t a s t rophea nd c ha os i n t he c on t i nuous se t t i ng o f t he AHP.

Bot h c a t a s t rophe a nd c ha os a re r e l a t i ve . A re v-o l u t i on ma y be r e ga rde d a s a n unde s i ra b l ec a t a s t rophe t ha t uproo t s a soc i e t y o r a s a ne c e s -sa ry k i nd o f a c t i on t o p roduc e a de s i r e d e nd . Inp l a n n i n g , s t r o n g o u t - o f - t h e - o r d i n a r y a c t i o n m a ybe c ruc i a l fo r a f fe c t i ng c ha nge .

A s i mpl e i l l us t r a t i on o f how t o r e pre se n t ac a t a s t rophe i s g i ve n i n F i gure 2 . An i nd i v i dua lb u y i n g a c a r m a k e s a n i n n o c e n t c h o i c e b y a s s u m -i ng t ha t norma l c ond i t i ons wi l l p re va i l ove r a c c i -dents in the ra t io of 0 .99 to 0 .01. He chooses an i c e ne w Toyot a , a l t hough he knows t ha t t he re i sa c ha nc e t ha t a n a c c i de n t c a n ha ppe n . I f he we ret o buy t he c a r unde r t he a s sumpt i on o f a h i ghl ike l ihood of an acc ident , 0 .99 to 0 .01, and i f hewere to ac t in hi s bes t se l f in te res t , he wouldc hoose a n o l d wre c k t o buy . One a nswe r t o t h i sd i l e mma i s ne i t he r t o buy t he o l d wre c k nor t heToyot a , bu t t o buy a fa i r l y good use d c a r . Bu tmo s t o f us a re i dea l i st s who do no t t h i nk t ha ta c c i de n t s wi l l ha ppe n t o us , a nd so we buy ne wcars .

11. What affects rank [lllAl t hough i n c a t a s t rophe s t he re i s a sudde n sh i f t

i n r a nk , due t o a c ha nge i n t he i mpor t a nc e o f t hec r i t e r i a , wha t ha ppe ns t o t he r a nk o f t he a l t e rna -t i ve s i n t he more munda ne e ve n t t ha t t he i r num-be r i s c ha nge d by a dd i ng ne w one s o r de l e t i ng o l done s? The t r a d i t i ona l ru l e i s t ha t r a nk re ve r sa l i sno t a c c e p t a b l e i f , g i ve n t ha t t he a l t e rna t i ve s t he m-se l ve s a re i nde pe nde n t o f e a c h o t he r , a ne w a l t e r -na t i ve doe s no t i n t roduc e a ne w c r it e r i on o r c ha ngethe weights of the exis t ing c r i t e r ia . We tend tot r e a t r a nk i n a posse s s i ve ma nn e r by s ome t i me si ns i s t i ng t ha t i t s t a y t he sa me no ma t t e r wha t l og i cs a y s . T h e a b s o l u t e m o d e o f m e a s u r e m e n t o f t h eA H P c om pl i e s wi t h t h is norma t i ve inc l i na ti on , bu tt h e r e l a t i v e m o d e o f m e a s u r e m e n t d o e s n o t , be -

c a u s e o f t h e d e p e n d e n c e o f t h e m e a s u r e m e n t s o ft he a l t e rna t i ve s on e a c h o t he r . Howe ve r , r e l a t i vem e a s u r e m e n t w i l l pre se rve r a nk wi t h r e spe c t t o as i n g l e c r i t e r i o n w h e n t h e c o m p a r i s o n s a r e c o n -s i s t e n t . M o s t p e o p l e u n d e r s t a n d t h e d e p e n d e n c eof a l t e rna t i ve s i n l i gh t o f t he no t i ons o f s c a rc i t ya nd a bunda nc e d i sc usse d e a r l i e r . The re i s no ne e dt o i mprov i se no t i ons o f r e l e va n t a nd i r r e l e va n ta l te rna t ives , as i s done in ut i l i ty theory, becausew i t h r e l a t i v e m e a s u r e m e n t e v e r y t h i n g b e i n g c o m -pa re d i s by de f i n i t i on re l e va n t , U t i l i t y t he ory i sobse s se d wi t h t he r e ve r sa l o f r a nk be c a use i n t ha tt he ory i t c a n ha ppe n e ve n wi t h r e spe c t t o a s i ng l ec r i t e r i on . Th i s i s a phe nome non t ha t i s s t rong l yc o u n t e r in t u i t iv e a n d c a n n e v e r b e m a d e m a t h e -mat ica l ly r ight . Ut i l i ty theor i s t s , however , t ry tom a k e i t r i g h t b y p h i l o s o p h i c a l a r g u m e n t s a b o u twhat i s or i s not a re levant a l t e rna t ive .

12. Summary of principlesThe AHP ge ne ra t e s r e l a t i ve r a t i o s c a l e s o f

m e a s u r e m e n t . T h e m e a s u r e m e n t s o f a s et o f o b -j e c t s on a s t a nda rd sc a l e c a n be c onve r t e d t ore l a t i ve sc a l e me a sure me nt s t h rough norma l i z a -t ion. Only in a very loca l ized way can a re la t ives e t o f m e a s u r e m e n t s h a v e a u n i t , o b t a i n e d b yd i v i d i ng t he e n t i r e s e t by t he sma l l e s t me a sure -m e n t . T h e n o r m a l i z a t i o n a n d c o m p o s i t i o n o fwe i gh t s o f a l t e rna t i ve wi t h r e spe c t t o more t ha n as i ng l e c r i t e r i on me a sure d on t he sa me s t a nda rdsc a l e l e a ds t o nonse ns i c a l numbe rs , be c a use nor -ma l i z i ng se pa ra t e s e t s o f numbe rs de s t roys t hel i ne a r r e l a t i on a mong t he m. The we i gh t s mus t f i r s tbe c ompose d wi t h r e spe c t t o a l l suc h c r i t e r i a a ndt h e n n o r m a l i z e d f o r A H P u s e . W e c a n i n t e r p r e tsuc h c ompos i t i on a s we d i d i n Se c t i on 8 a s aspe c i al k i nd o f we i gh t i ng o f t he pa r t i c u l a r c r i te r i a.Thus , t he AHP, wi t h i t s r e l a t i ve me a sure me nto f f e r s n o g u i d e o n t h e o u t c o m e o f m a n i p u l a t i o n sb a s e d o n c o m b i n i n g d i f f e r e n t m e a s u r e m e n t s f r o ma s t a nda rd sc a le suc h a s a c r i te r i on o f be ne f i t s a nda c r i t e r i on o f c os t s , bo t h me a sure d i n do l l a r s , a ndused to se lec t a bes t a l t e rna t ive .

I f we do n ot ins i s t tha t the l inear i ty o f a sca len e e d s t o b e p r e s e r v e d (a n o l d h a b i t f r o m w h e n w edi d no t ha ve a n e f fe c t ive wa y t o i n t e rp re t t hei n f o r m a t i o n c o n t e n t o f r e a d i n g s f r o m a s t a n d a r dsca le ) , we can then t rea t every c r i t e r ion as a