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EUROPE NJ O U R N L

OF OPER TION LRESE RCHE L S E V I E R European Journal of O perational Research 89 (1996) 457-461

T h e o r y a n d M e t h o d o l o g yn o t e o n t h e u s e o f P R O M E T H E E m u l t i c r i t e r i a m e t h o d s

W i m D e K e y s e r , P e t e r P e e t e r s *Vrije Universiteit Brussel BE IE Pleinlaan 2 B-105 0 Brussels Be lgium

Received October 1993; revised March 1994

Ab st rac tT h e P R O M E T H E E m e t h o d s a r e r a t h e r p o p u l a r i n t he w o r ld o f o u t ra n k i n g m e t h od s . O n e o f th e r e a s o n s fo r th i spopu la r i t y is t he e x i s te nc e o f t he ve r y u se r f ri e nd ly so f tw a re , c a l le d P R O M C A LC - P R O M e th e e C A L C ula t i on . M or ea nd m or e p r a c t i t ione r s a r e u s ing P R O M C A LC to ha nd le t he i r m u l t i p l e c r i t e ri a p r ob l e m s . H ow e ve r , no t a l l u se r s a r ea w a r e o f th e c o n s e q u e n c es o f t h e m o d e l a s s u m p t io n s m a d e i n P R O M E T H E E m e t h o d s . A s h o r t o v e rv i ew o f s o m edr a w ba c ks o f P R O M ETH EE m e thods t ha t u se r s m us t know a nd a vo id a r e de sc r ibe d i n t h i s pa pe r .

Keywords: M ul t ip l e c r i t e r i a ; O u t r a nk ing m e thod

1 I n t r o d u c t i o nP R O M E T H E E m e t h o d s h a ve t a k en an i m p o r-t a n t p l a c e a m o n g t h e e x i st in g o u t r a n k i n g m u l t i p le

c r i te r i a m e t h o d s . T h e n u m b e r o f p r a c t i t io n e r sw h i c h a r e a p p l y i n g t h e s e m e t h o d s t o p r a c t i c alm u l t i p l e c r i t e r i a d e c i s i o n p r o b l e m s , a n d r e -s e a rc h e r s w h o a r e fu r t h e r d e v el o p in g a n d / o r a r ei n t e r e s t e d i n s e n si t iv i t y a s p e c t s o f t h e s e m e t h o d s ,i n c r e a s e s y e a r b y y e a r a s c a n b e i l l u s t r a t e d b y t h ei n c r e a si n g n u m b e r o f p a p e r s ( s e e r e f e r e n c e s ) a n dc o n f e r e n c e p r e s e n t a t i o n s u s i n g o n e o r m o r e o ft he P R O M E T H E E m e th o ds .

T h e m o r e a m e t h o d i s u s e d , t h e m o r e c r e d i ta n d t h e l e s s d o u b t s i t g e t s . N e v e r t h e l e s s e v e r ym e t h o d h a s it s r e s t ri c t i o n s , m o s t l y d u e t o m o d e la s s u m p t i o n s , w h i c h s h o u l d a lw a y s b e c o n s i d e r e dw h e n t h e m e t h o d is u s e d .

T h e o b s e r v a t i o n t h a t m o s t p r a c t it i o n e r s a n ds t u d e nt s d o n o t h e s it a te t o a p p l y P R O M E T H E E

* Corresponding author.

m e t h o d s o n a l m o s t a n y m u l t i p le c r i te r i a p r o b l e m ,e n c o u r a g e d t h e a u t h o r s t o f o r m u l a t e s o m e o f t h ed r a w b a c k s o f t h e s e m e t h o d s , e v e n t h o u g h m o s ta c a d e m i c r e s e a r c h e r s i n th e f i e l d o f m u l t i p l e c r i-t e r i a m e t h o d s a r e w e ll a w a r e o f t h e s e d r a w b a c k s .

S e v e r a l c o n s e q u e n c e s o f th e m o d e l a s s u m p -t io n s o f t h e P R O M E T H E E m e t h o d s a re fo r m u -l a t e d i n S e c t i o n 2 . I n S e c t i o n 3 , t h e b e h a v i o u r o ft h e P R O M E T H E E m e t h o d s is d i sc u ss e d w h e n a na c t i o n i s a d d e d o r d e l e t e d . T h e l a s t s e c t i o n c o n -t a i n s t h e c o n c l u s i o n s o f t h i s p a p e r .

2 . C o n s e q u e n c e s o f m o d e l a s s u m p t i o n s o fP R O M E T H E E m e t h o d s2 .1 . P R O M E T H E E m e t h o d s m a y o n ly be a p pl ie d i ft h e d e c is i o n m a k e r c a n e x p re s s h is p r e f e r e n c e b e -t w e e n t w o a c t i o n s o n a g i v e n c r i t e r i o n o n a r a t i os c a l e

T h e p r e f e r e n c e f u n c t i o n f j f o r a c r i t e r i o n jr e t u r n s , f o r a d i f f e r e n c e d b e t w e e n t w o e v a l u a -

0377-2217/96/ 15.00 1996 Elsevier Science B.V. All rights reservedS S D I 0 3 7 7 - 2 2 1 7 ( 9 4 ) 0 0 3 0 7 - 6

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458 w. De Keyser, P. Peeters European Journal of Operational Research 89 (1 996) 457 -461t i o n s o n t h a t c r i t e r i o n , a v a l u e f j ( d ) ~ [0 ,1] . Thisv a l u e i s a r e a l v a l u e o n a r a t i o s c a l e . A s a c o n s e -q u e n c e o f th is , th e d e c i s i o n m a k e r m u s t b e a b l et o e x p re s s th e m a g n i t u d e o f h is p r e f e r e n c e b e -t w e e n a c t i o n s o n a c r i t e r i o n a c c u r a t e l y , b y a c -c o r d i n g real v a l u e s - i n t h e i n t e r v a l [ 0, 1] - t o t h ed i f f e r e n c e s b e t w e e n e v a l u a t i o n s o n t h a t c r i t e r i o n .2 .2 . P R O M E T H E E m e t h o d s m a y o n ly b e a p pl ie d ift h e d e c i s i o n m a k e r c a n e x p r e s s t h e i m p o r t a n c e h eat taches to the cr i ter ia on a ra t io scale

m e t h o d s , e x c e p t f o r t h e s p e c i a l c a s e o f a c r i t e r i o nw i t h a n o r d in a l s c a l e w h e r e t h e p r e f e r e n c e o f t h ed e c i s i o n m a k e r i s s u c h t h a t h e p r e f e r s t h e b e s t o ft w o d i f f e r e n t e v a l u a t i o n s w i t h a p r e f e r e n c e f u n c -t i o n v a l u e o f 1 . T h i s m u s t b e d o n e b y a c a r d i n a l i -s a t io n o f t h e e v a l u a t i o n s i n c o m b i n a t i o n w i t h t h eu s e o f th e t y p e I p r e f e r e n c e f u n c t i o n .

I f o n e d o e s n o t p a y a t t e n t i o n t o t h i s a s p e c t o ft h e P R O M E T H E E m e t h o d s , o n e c a n f al l i n to th et r a p o f t r a n s f o r m i n g v a l u e s o n a n o r d i n a l s c a l ei n t o v a l u e s o n a r a t i o s c a l e .

P R O M E T H E E m e t h o d s ne e d q ua n ti f i a bl e c ri -t e r i o n i m p o r t a n c e s - w e i g h t s - o n a r a t i o sc a l e .T h e r e f o r e a d e c i s i o n m a k e r m u s t b e a b l e t o s u p -p l y s u c h q u a n t i t a t i v e c r i t e r i o n i m p o r t a n c e s w i t ht h e n e c e s s a r y a c c u r a t e n e s s . T h e d e c i s io n m a k e rm u s t f u r t h e r m o r e b e a w a r e t h a t a c r i t e r i o n w i t h ,e . g . w e i g h t 1 . 8 i s t w i c e a s i m p o r t a n t a s a c r i t e r i o nw i t h w e i g h t 0 . 9 , i n o r d e r t o c a l c u l a t e a v a l u ew h i c h e x p r e s s e s t h e o u t r a n k i n g r e l a t i o n .2.3 . The weights o f the cr i ter ia express t rade-o f f sbetween the cr i ter ia

T h e w e i g h t s , as d e f i n e d a n d u s e d i nP R O M E T H E E m e t h o d s, i nd ic a te t ra d e -o f fs b e -t w e e n t h e c r i t e r i a . L e t c r i t e r i o n i a n d c r i t e r i o n jh a v e r e s p e c t i v e l y a w e i g h t o f ~ i a n d o f A j . T h i sm e a n s t h e d e c i s io n m a k e r is p r e p a r e d t o e x-c h a n g e a d e c r e a s e o f A i o n t h e v a l u e o f t h ep r e f e r e n c e f u n c t i o n o f c r it e r i o n i f o r a n in c r e a s eo f ( d i l ~ i ) / / A j o n t h e v a l u e o f t h e p r e f e r e n c ef u n c t i o n o f c r i t e r io n j . T h e d e c i s io n m a k e r m u s ta g r e e w i t h t h is m o d e l o f h is p r e f e r e n c e s t r u c t u re .2 .4 . P R O M E T H E E m e t h o d s m a y o n ly b e u s ed w i thcr i ter ia where the d i f f erences between evaluat ions~ re m e a n i n g f u l

A p r e f e r e n c e f u n c t i o n , a s d e f i n e d inP R O M E T H E E m e t h o d s , is t u rn i n g a d i f fe r e n c eb e t w e e n t w o e v a l u a t i o n s o n a c r i t e r i o n i n t o a r e a lv a l ue - b e t w e e n 0 a n d 1 . T h i s f o r m s n o p r o b l e mf o r c r i t e r i a w i t h i n t e r v a l o r r a t i o s c a l e . F o r c r i t e -r i a w i t h a n o r d i n a l ( o r n o m i n a l ) s c a l e , t h e d i f f e r -e n c e h a s m a t h e m a t i c a l l y s p e a k i n g n o m e a n i n g .S u c h c r i t e r i a c a n n o t b e u s e d i n P R O M E T H E E

2 .5. I t i s no t pos s i b le t o t a ke d i s cor d ance i n t oa c c o u n t w h e n c o n s t r u c t i n g t h e o u t r a n k r e l a t i o n s o fP R O M E T H E E m e t h od s

D i s c o r d a n c e is h o w e v e r c o n s i d e r e d a s a r e a li s -t ic c o n c e p t i n m u l t i p l e c r i t e r i a d e c i s i o n a i d a n d i sc o n s i d e r e d a s o n e o f t h e r e a s o n s f o r d e v e l o p i n go u t r a n k i n g m e t h o d s ( s e e V i n c k e , 1 9 9 2 ) .

T h e p a r ti a l P R O M E T H E E I p r e o r d e r (P ~ , I [,R ) i s t r a n s i t i v e , w h i c h i s a c o n s e q u e n c e o f t h ea b s e n c e o f d i sc o r d a n c e . S i n c e t h e p a r t i a l p r e -o r d e r s h o u l d m o d e l / r e f l e c t t h e p r e f e r e n c e o f th ed e c i s i o n m a k e r t o w a r d s t h e d i f f e r e n t a c t i o n s ,t r a ns i ti v it y c a n n o t a l w a y s b e t a k e n f o r g r a n t e d .

3 A d d i n g o r d e le t i n g a n a c t io nT h e pa r ti a l P R O M E T H E E I p r e o r d e r a nd th e

c o m p le t e P R O M E T H E E I I p r e o rd e r a re b a s edo n a n a l l - t o - a l l c o m p a r i s o n b e t w e e n t h e a c t i o n s ;a d d i n g o r d e l e t i n g a n a c t i o n c a n p u t t h e s e p r e -o r d e r s u p s i d e d o w n : T h e o u t r a n k r e l a t i o n b e -t w e e n t w o a c t i o n s c a n s w i tc h (s e e 3 .1 ) a n d / o r t h er e s u l t o f e x p l o r i n g th e o u t r a n k i n g r e l a t io n s b e -t w e e n t h e a c t i o n s c a n b e a f f e c t e d ( s e e 3 . 2 ) .3 .1. T he r e l a t i ve pos i t i on i n t he p r eor der be t w eent w o a c t i o n s c a n c h a n g e b y a d d i n g o r d e l e t i n g a nac t i on

A c t i o n A i c o m e s b e f o r e a c t i o n A j i n th ep r e o r d e r m e a n s t h a t t h e d e c i s i o n m a k e r p r e f e r sa c t i o n A i a b o v e a c ti o n A j . T h e pr e f er ence o f ad e c i s i o n m a k e r b e t w e e n t w o a c t i o n s m u s t i n t h ea u t h o r s o p i n i o n o n l y b e b a s e d o n t h o s e tw o

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460 W. De Keyser, P. Peeters European Journal of Operational Research 89 (1996) 457-461T h e d e c i s i o n m a k e r c a n e x p r e s s h i s p r e f e r -

e n c e s b e t w e e n t w o a c t i o n s o n a l l t h e c r i t e r i a o nr a t i o s c a l e s .

T h e d e c i s i o n m a k e r c a n e x p r e s s t h e i m p o r -t a n c e h e a t t a c h e s t o t h e c r i t e r i a o n a r a t i o s c a l e .

T h e d e c i s io n m a k e r w a n t s t o t a k e a l l c r i te r i ai n to a c c o u n t a n d is a w a r e o f th e f a c t t h a t t h ew e i g h t s a r e r e p r e s e n t i n g t r a d e - o f f s .

F o r a l l c r i t e r i a t h e d i f f e r e n c e b e t w e e n e v a l u -a t i o n s m u s t b e m e a n i n g f u l .

N o n e o f t h e p o s s i b l e d i f f e r e n c e s o n a n y o ft h e c r i t e r i a c a n g i v e r i s e to d i s c o r d a n c e .

T h e d e c i s io n m a k e r k n o w s e x a c tl y w h a t c a nh a p p e n i f o n e o r m o r e a c t io n s a r e a d d e d o rd e l e t e d a n d i s f u ll y a w a r e o f t h e i n f l u e n c e s o n t h ef i n a l d e c i s i o n .

W e h o p e b y u n d e r li n in g t h e a b o v e r e m a r k s o nP R O M E T H E E m e t h o d s t o s t i m u la t e t he u s er st o a p p ly in g P R O M E T H E E m e t h o d s o n p r o b le m so f w h i c h d a t a a n d p r e f e r e n c e s t r u c tu r e o f th ei n v o lv e d d e c i si o n m a k e r c o m p l y w i t h t h e r e q u i r e -m e n t s a nd m o d e l a s su m p t i on s o f P R O M E T H E Em e t h o d s .

R e f e r e n c e s

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