SERIE RESEARCH mEmORfMDnQUAIITATIVE .MULTIPLE CRITERIA CHOICE ANALYSIS THE DOMINANT REGIME METHOD Edwin Hinloopen P e t e r Nijkamp Researchmemorandum 1986-45 December 1986 VRIJE UNIVERSITEIT FACULTEIT DER ECONOMISCHE WETENSCHAPPEN A M S T E R D A M of Economics Amsterdam RE/192/PN/hl Edwin Hinloopen and Peter Nijkamp .1 QUALITATIVE MULTIPLE CRITERIA CHOICE ANALYSIS THE DOMINANT REGIME METHOD Paper presented at the Conference on Conflict Management. April 1986 Free University Dept. Amsterdam. . 3 c § S oq po o ct HO a o * •o T O o.O C PU M Hct PU ct HC D o T rawHJ -i C D C O C D •* C D C U C O ct HO B po cr1 B B C O O H' C D 3 pu 3 3" i-1 !->• ë H O 3" OH- 3 •• t. 3 P1 > H- 3 Oq ct C D T !-•• 3 VO —3 —k er o C O <<. ra PJ ct !-•• ra xD po C co s co !-•• H O M C O C O O H C O O ^/ HC O • M 3 O Cr co 3 O Hl PJ C O •a c t ^—N C D -i o !-. C D -3 C N N sC a H3 oq C D PJ ë co O Hl B C 3 <! PJ M ë 3 ra C D o •o ct H- C D ct «<! ra o (-•• 3 C D raHB 3 C O C O e C ffi 3 Cu C ct po C O O Hl C D P O C o O Hl PJ !-•• a C o 3 t-. C D «< C O H- co C D ct P O H <! DJ M C ct HO !-•• Ct DJ ct o t-> H- P O 3" C D O ct po co ra o ca C D »< C O v_x C D ra o 3 < C D 3 ct M- B PJ ct H- po t-> C D < (-•• • co o po H C D C O 3* co P> (-•• <i C D • « ë po h" C O HC O < C D B s C D ct C D Ct Ct i O 3 --—v e.3 P • HO vo -a co 3 PJ & ct H1 PJ ct ct B 3 C D PJ H" 3 H.1 B 3 3 < << ë 3 ë ?r o 3 C D c 3 o. 3 l CD O ra ct c H H•o & H" CD CD T T a ra >-•• O H-PJ oq 3 C D PJ M ra 3" o •o C a !3 c C E •<! C D !-•• 1 !-•• O C D co ra Ct H- 3 3 ra ct H. ct C ct H- B C D ct 3 Hl O T C O PJ <! ffi o B H•• ra C D < B !-•• N *t w ra po D. co 3 O C D t^. H- <! B 3 ct HO 3 n 2 B ct C O ra -i ct Hl O rato PJ Oq "5 C D PJ ct •• po 3 C D C D ct H- H. ra ~i B •a P O ra ct v_^ o.ra « PJ CD C O ra. er ct H- er H» c po ct O !-•• 3 co 3 3O er 3 C D C D H) t-1- o.O H l Hl 3- co HDPJ ct ct C D ct HO >-•• ra C O ct HO 3 O- <! < p G C c H O H> i-^ C S- C O O- oq B 3 3 co C D B C D ct B C u TT C D O O 3 B C C D C D 3 3 ct O H> S CD M C D 3 ct o 3 ct C D O T 3 3 C D PJ C O H- T 3 po C D Ct C D C O C O DJ B O 3 C D ct o >-1> ct C D •o T C U O ct (-•• 3 ct B ra po co co o 3 C O C C O 3 C U H •• o T 3 H 3 Ct CD ct !-'• cr O y—N 3 O Hl T C D (-• <<! B C D C O Q T B s~^ CD 3- 3" H- O. C O 3" 3* O co ^ co o.M ra co c 3. PJ (-<• O P O P O ct HO oq • • -• 3 c. ol * H ct a •» co cO P O T Hl M M l-' vj ë po a co v_^ o T ra T P O O 3 C u H C D M !-•• *-* £> !-•• po B C D C O C O Ct C D HP O O »-b ct- o *-4 P > ra 3" Hi H> •o T O T M er -i po fc a <! H. er 1 3 3 oq (-•• c+ 3" C D po 3 ë C D B o 3 -s C D O Ct HO a o Oi C O C D X3 a M o •< co o HOq co O P O H M T ë po ra T i-1ct C D O ct a i c o C D »t> i >-t <t> C D ra H* c t Ct H.• o p.3" C D <j C D T 3" !-•• ra C D o 3 C U C O 3 o *D C < C D T H- 3 oq ra H. C O s~^ o ct H•ii O- co ct H- H- po B P) O 3 s c o 3 C O h-> ct H. co -* * >-i 3" C D C O C D .co 1 HPJ O B C o.C D B C O H- •< B H C D H- 3 C O ct HCt •o e ct C D T **_^ < • ct C D O 3 O Hj ra 3- B O -5 C D ct & C D c H C o ct a C O O |_u 3" O !-•• < D C O v_^ C D e PJ ct HHl ct co !-•• C O •• o O 3 O O T C O H" •-•• fu 3 <! po H C D a o 3 po T P > O 3 C o o H- P O ct (-•• PJ er C D a H3 P O H" «< *-•• C O O co 3 B C D Ct SO C D C D er CT» o •v-*. C D ra H3 Oq 3" & B < WJ H) T ct C O ra 3- ë H-- O 3 ca & ct H- ra o T H- ct H' • ^ ^ PU M co 3 •-?> >-* C D T C D ra co a 3D C « a co C D ct 3 O <<: C O ct ct O -J O O o H> ci- H C D 3 PU 1 oq 3 O H) O "S 3 co 1 1 c ra 3PJ ct O 3* T 1 3 Ct ^^ o *s 3 «<i C •ö D o 3 a • ffi B H> O -5. co H. C D C O C O H O 3 O3J O Hl ct HO C O PJ O co !-•• B 3 & ct H1 (-> C D C U PJ HW C U <i O H. < o T 3" C D l-1 ct C D S C O B •ö po ty C D -i C D Q •* ra B •a1 O hH3" ct ra CD O ra 3" 3* C D o ra C-i.. Q T B PJ ct HO ^-^ « • fu 3 O.O c • o •* 3" B C D ct VD C» VJ1 v_^ —» 3 B C D ct (-" • •< » co C O 1 o po 3 3 O ct ë 3" 3* O o o. oq • ra H> c H.^ a < • c o ra c t (-•• er D co C HH- Hct C D PJ o 1 ra ct H- ra 3" O 3" a co v « Q.C O C O o C D B "O T C D H> O 3 o •a c t T Ct (-•• O I-". Oq C D s o c T M ct H" 3 H"- B C D P O C D C *D C 3" O a co x-^v C U M po T O C D C O -s % C U o. a 3 C D ct •a po • ü> H1 ICI- -s ol I-" H e C D Hl ra C D •o T o cr \ffi B co co C D C D •—V C D H- o * T B PJ ct 3" co M B po T 3 ct HC O ra1 Hl O T C O & C D ra f* t-" • oq • » • T C o ct H' O ra o H. CO C D er 1 ë O w *< C O HC O •< 1 er po co C D o •• ct po O i-1 C D < C U !-•• c ra •-b •-4 C O C D 3 <<: ct C D B 3 •o t-" C D O *1 H* ct C O T •"•• 3 O 3 C D H po er M ct M 3" <-=! ct C O HO OQ 0 3" p ct & CD 3" ra co T Hf- ra 3" PJ po H1 I- c po co ct co T —i 3 oq ct »< B PJ O "J Hl ct PJ ë 3 T C D Oq P O T cr Hct C D T H- C U < 3 ct »< er M co C D O Hl C O C D B PJ C i-* D ra po y— •o ra. C_|.• « X C D c 3 3 ra co c: P o e Ht < C c PJ 3 ct O 3" O B c 3 O O co C D ra 3" O H- ct Hct i. 3co C D T C D ct O 3 P tC H C D C U C O •a 1 ct o 3 !-•• po ct C D C_i.3 H- H.1 M H- 0 B C D <! ol PJ H 3 ct C D H- 3" C D C O *< C 3 O. H- O O ct B po 3 ct <! C O HC O ra ^H ct C D *S (-•• o C_i. 3" C O HM • < tr < P C O X) C D T i-1' er ct C D 3 y—v a c o ^. co K 3" < c c O 3 po M 3O HO co 3 ra o H h-1 CD PU • ct C D ct O ë C O •o C U -i C D C C T « • ct •a 3" H- < •a T O 3H- C D fu e C O M H- •1 C D O C U (-• T C D c+ C U ct H- er co P O C D H' ct B C o<! oC O o 3" 3 C D T C D H C D o C o N (-•• e N B co << •o (-"• (-•• 3 o OQ T ct ct co 3 *<. a co C U !-•• C O o. O 3* po •O 3* O H- o •o ct H- H C D !-•• te * ra o co O HO C D •a T O KC D •a M HC D ct B C po 3 O 3 B Hl O T a po co C D ct a PJ O C D H ct (-•• 3 •a i-1i-1S C O C O O- — . << er C D.o 3" O HO co TC H Ië •ö OJ g - O Hl C D £> I- co B C D Cf O £B B 3 ct 1 C P> 1 o 3" O !-•• P O 3 po O C_i.t PJ C !-•• co C ra 31 pu co o T C O C O a B C D ct M C U M er Po § O C U T •< C D C D ra C D Ct B Ht- B C D et O OC O 3 fu1 B co po 1 3" M C po O 3 <! po co 3" •ö -i C D C O C D a C D OC D C O H- •—^ 3 O C D ct O O H C O H.- i-"P U H1 C_i. !-•• < C D (-• C D C D ct !-•• «<i 1 i 3 oq 3 •-"• i •o s o 1 ct C D er1 h- T K^ po ffi B co •a C c c . § C D 1 !-•• ra T C (-•• f •< C O H- l. O H O O pi Oq c t HC D O oq (ö HO H O 't rt P) et HC D H- co c o •-4 P> 3 CL ct <i & Ct ^-^ •• 3 P) » <j . « H- ffi •o B T C D <! C D C D PJ M rt HO & O "-4 HT C D C O P > CT C D 3. *^N « *< < * •+ C D B HC D ct O VO -3 « & C D Ct B HM H * 3 ct C o C D •-4 H1 C D O rt H* 3" O o 3. (-• 3 a p> ë 3 PJ M «< P> O O 3 a • P) !-•• T3 rt C O H> C D O C 3 P) i CT I-" o C D p > c o B rt « C O C D 3" 3 CD — X * e <£> •o P) c o M H.— * H' O O C D H * ^ — C O !-•• w-A 3 T C O rt C O O (-•• B 3" a • c •< o o 3 et C D O ct ct H C O et O o pi a o H O ë O 3" O a 3" T C D C D C D H H3 p > CO <! C D CT C D O O O. O H" D C D C C O o B p) C D T HP> O 3" O HO C D o O a a 3- C O C D C D o a 3 C D rt O «< H- < • • c o c o /•—v < C D C D l-1 c o ct 3 ?r C O ct O ct C D — * vó 00 co v_^ ca C D C D ë C D M •-4 ct P> 3 Q. C O C D C D W P> P> ct P 3 > *-•• •o C D »-4 C D T C D o C D C O C O H<J C D <3 Pi M C CD O T Het C D T (-•• a * ^ N 3" ^ .Q c t p) H • p O C 3" r t O c t *r 3 ct H P CD ! . -s <! C D T P> H O >-3 .oq 3 3 3" H.*—*v ë PI M rt C O ffi 3 C D HO C C O O C D ffi 3 B O T C D ct O 3 P > ca 3 (-•• ct O 3 »-* 35 C D H3^ rt •o H^ CD et 3" C D M C D C O C O O O T -i C D O et et S C D C U rt 1 < O OQ V* H- • C D rt O O.O O 7\ C D P > 3 *3 C D C O O HC D rt H•"4 H- HT M >->> tu c 3 o c •o M HO C D p) T CT C D O C C O C D *< •a o T3 H P) 3 O H- •o (-< C D C O O •• O 3 P > et C D e P) !-• 0 w P > = s C D rt O P > 3 o C D C O C O H- *-* et H- co c •*-^ o 3 3" vO 00 — 3" * CT C D o et o 3C 3 C D 1 C D H" 3 CD rt a.c > o C D ct O 3 M> O T et HO 3 ot r cr C D ct C D c o et rt HC O ct C D C_i. Ct H" 3 ^-^ X o rt T -J C D Pi et H- C D t.>->• * o c o et P) et HHO H- M H H M C D -S C D P) 3 et oq 3" C D 1 o o ~J O < •o G co T rt C D C O C D M> T 3 oq O O O PJ H1 S H 3 et >-4 oq p) H.a H'S C D 3 ct C O P > 3.M B B C D C O HC D 3 ct 13 J V H HO O G Hrt B 3 et C D rt O 3" C D C O C D 3 O Hrt ct co O S C D 3 O 3" B £> e t cr (-•• (-•• CD 1 c t << C D G C O D Iet C D 3 C D << 3" 3 e t c o 3" O P> O O C D C O C O H- Oq C D O o a B £ 1 <! _» a CD CD C P) vO > c 3 Ct P T 00 o B •o •• c t 3* H H. — . 3* T3 C D C D C O O »-4 H>-4 a c o ^^ CT P > C O C D 0. 3 * 3" *• 3 C D e C D H . H>-4 •-4 C D T C D <l p) c o H) 3 c t1 CD P 3 -5 > C D 1 < H P) ct HC D C D s 3 rt < .3* B C D •—N 3 G B Pi et c o.• 3 W C • •*_• H c 3 o1 pi C O T C D C D C U H C D G O G C O 3. 3" c oq o • p) ^ . 3 et C D 1 *-•• 3 O ct G P) H o.n Q_ 9 rt C O O P> 3 CT C D P> C O C O C D C O C D G O H- < C D 1 C D et a B o e ha H3 Pi H H- 3" c o O o •< C U G H B C D 3 O ct 31 C D T3 C D T O 1 •-4 HO et !-•• Ct G < 3" ct T O "-b et O H" I-1 C u 3 p1 > i- a 3" P) C O C D C D T cr H- X 3- << et H-- O «< 3 rt •-•• O C D C O ct et H•o H C D O T HCt ^-^ C O C D C D H3 C D rt PJ H1 C C D <! N _ ^ 3 3- 3" C D C O *-—>* cr P > c o C D G O C D P> ct ct C D (-•• B C D ct 3" et C D T !-•• X •ö •a C D -i sD C •-4 C D T C O oq 3 C D ct 3" O a ct r P> rt H3 B Ct C D O 3O Q.H>-4 .P) O T o O N C D •^^ r t Ct 3 P > P > P £ 1 « CD r t > • et B I-" C a H- o.»-t> oo C D !-•• H.^ p> et C D C O C D * C O • f-"C_i. CT 3" s~^ et c r (-• ë pi et 1 c o o 3" P) P J a B 3 o CD •-!> •ö et 3" O ct T O P > 3* a r t 3 P c > o O. rt O c o C D O rt HO 3 •o -J C D C O C D oq V C D CX £ ffi 3D CD B C 3" 3 C O (-•• o p> H 1 rt rt C D C O C O 3" ct 3 H. s C O C D C D — * •o C D v-> C D !-•• O C D C D ct 3" O 3 3 CT C O C D G O 3 !-•• P) TT e 3" C D ct a C O H> 3 rt •o s-^ ro •O co lP) C D O -1 " *i I I • P > 3 3* P CD o > C D rt B 3 !-•• O •o1 hC D B 3 P > 3* C D 3 a. (-" (-» et << pj 3* C D T C D CT C rt OQ O O n CD HT »<: (-•• o O O (-• i-'- B 3 p1 > P l. rt et 3" C D HT et 1 C D C U rt B s PI ct o c C D 3 CT T O C D TT P> * -^ Q.3 C D et rt w • •a P! •* 3* P O > a 3" c o C D G et O ct •a1 ho *5 *-> ^ ^ co — P < ct > « rt P > !-•• O 3 C D O ct C D •o *D CD C H. ca H." • 3 > H O O. o. ^ — C O C D C D h* oq HO pi H C D ct *< T C D O C D rt l-1 n P> 3" o 3 1 O T C D B C D O T O T O.y •a s C D •-4 C D T C D 3 C D et O C D ca 3" C O ^-s o. O O < ?T 0} •-•• c o 31 ^ oq - a •o - 3 3 o.r t rt £1 *-•• 3r 3 CD e oq C D •o o T rt 3 s e C D H- oq 3" rt H" 3 < HO ff P > c oo C O •CT S B CD O C D C rt C D ë O C D Pi ct ct O oq B a » • G EC 31 C D •o ~i 3 M O O C O O o o 3 (-"• c o C U C D 3" P 3 G pi 3 c •o "S 3 B t-1- •o ^ C D 3 C D rt P> 3 C D §" O s C D HC O C O C O C D •o •o 3 rt T C O B c o p) ^_.B C H ct H- «D ~J -q v™> — * W HC D ct H-1 D. OP) ct HO T Hét B <! C C P > 3 B C D ct O (X C O 3* P> C O CT C D C D 3 G C D t-« O H1 et HC D co CD H.M) C D •^—' VO -^1 -3 •n C D H H- p > VO —» —» oo 0 0 oq — o CD * 3 « • <o C D H- C D 3 C O HM •9 O O HO CD et 3 B e C D 3" ct C D O C D H- < 1 M (-•• C O 3" B C D HX C D Q B 3 C D o »-• -i CT 3 ct C D C O C D £> o Hc o !-•• oq B 3O p > p) »-4 O T 3.% *• C D > -3 O N P) C O •—x B 3" P > a CD H3 c CT M o P •< > O CT P) C O C D G O C O H- 3* -i P) P 3 c 3 o 0. O 3 c o o •-4 CT C D G C D O ct C D G ct O • 3 oq ct 3" C D T C D rt 3" C D cr C rt «+ H1 PJ C D C O C O 3" ~i C D ct B g P > ct C D M sa HC_i. H3 P> M H3 »i> O -i ct ca H£1 C D Pi et HO P> C O B • a.1 O T HC O < C D C O K-*' B C D O >"4 et . » ë B ë f O O ct C U C O o. data regarding the ordinal value of all J judgement criteria for all I choice options. In case of weighting methods this information is given by means of preference weights attached to the successive criteria.. the better'.. the weights are represented .. • • f J) • The basic . then choice option i is preferabie i' for judgement criterion j . In the sequel Suppose we alternatives of this regime method should in principle lead to an so that always a dominant choice option is be as much accessible as possible to a choice section. identified.. Without loss of generality. in other words: if ejj > e i i j . summation or multiplication of ordinal numbers) .I. • . . If we deal with ordinal information.J) represents thus the rank order of alternative i according to judgement criterion j .the technique should be easily applicable on a computer .. j=1..the technique should maker . It will be concluded that the regime method is able to encapture a wide variety of qualitative multiple criteria choice problems based on both ordinal and mixed ordinal-cardinal data regarding both the characteristics (or impacts) of a choice option and the weights (or priorities) of a choice maker.the application of the unambiguous solution. As there is usually not a single dominating alternative. 2. so that the following <v effect E - • • • *u (2.the technique should not use methodologically unpermitted operations (for instance... have a discrete choice problem with I choice options or criteria j information we have is composed of qualitative ranking of all I i (i=1. characterized by J judgement (j»1 .1). we may assume a rank order characterized by the condition 'the higher...1) e ir e •• • • u The entry e^j (i=1. For pedagogical reasons our presentation will be based on some numerical examples..3 while next various new elements and extensions of this method will be developed (section 3 ) .. Principles of the Regime Method The regime method for qualitative multiple criteria choice analysis is based on the following considerations. we need additional information on the relative importance of (some of) the judgement criteria. the essence and structure of the regime method will be further described. In particular we assume a partial choice options for each matrix can be constructed: criterion j . 2) Clearly.signs. .ru r2-|. By making such a pairwise comparison for any two a l t e r n a t i v e s i and i ' for a l l c r i t e r i a j ( j = 1 . These regime vectors can be included in an Jxl(l-1) regime matrix R: R = 1 1 r 13 .wj)T (2.. .t would only oon^^ tain + signs. we may construct a Jxl regime vector r ^ i . L »j . . or (in case of t i e s ) 0ii>j»O. . rx1. (2. i ' .. Usually however a regime vector contains both + and . i t should be noted t h a t in case of ordinal Information.1 . s i . In order t o explain the mechanism of the regime method. the regime vector contains only + and . i'*i (2..r I( z^ ) *4 . <JÜ»J » . so t h a t then the mutual comparison of two choice options i s not influenced by the presence and effects of other a l t e r n a t i v e s . Consider two a l t e r n a t i v e choice options i and i ' .e j j . but only i t s sign. J ) in a weight vector w: j w . if a ü ' j = sign s ^ f j = +1. . . Next.. ..3) Thus. J ) . defined a s : r i i . the independence of i r r e l e v a n t a l t e r n a t i v e s problem). Of course. if a certain reg: ne vector r . we w i l l f i r s t define the concept of a regime. i t i s again assumed that W > wji implies that c r i t e r i j on j i s regarded as more important than j ' .signs (or in case of • ties also 0 signs). V i . and hence also 1(1-1) regime vectors. and reflects a certain degree of (pairwise) dominance of choice option i with respect to i' for the unweighted effects for all J judgement criteria.4) 1^1 P-2 It is evident that. = ! (»ii'1»•••»<Jii'j)T. so that then additional information in the form of the weights vector (2. .. Otherwise. . Consequently. e .(wlf. If for c r i t e r i o n j a c e r t a i n choice option i i s b e t t e r than i ' ( i . we have altogether 1(1-1) pairwise comparisons. the eventual rank order of any two a l t e r n a t i v e s i s co-determined by remaining a l t e r n a t i v e s (cf. .e i t j > 0 ) . . . then a l t e r n a t i v e i i s b e t t e r than i f for c r i t e r i o n j . alternative i would absolutely dominate i'. Clearly.4 by means of rank orders W ( j = 1 .2) is required. the regime method uses a pairwise comparison of a l l choice options. the order of magnitude of s i i ' j i s n o t r e l e v a n t . ... The ordi— —1 -J .5) -il' j-1 n'j-j If Xii' i s positive. This argument is essentially based on the 'principle of insufficiënt reason'. ¥.. Consequently. the weitonts vector w* can adopt with equal probability each value that is in agreement with the ordinal information implied by w. choice option i is dominant with respect to i'. This implies that we have to make an assumption about the probability distribution function of both the Wj*'s and of the SijM's. In view of the ordinal nature of the Wj's. . However. However...there is without any prior information no reason to assume that a certain numerical value of _w* has a higher probability than any other value. » . the crucial problem here is to assess p^i and p^.SiPü' (2 7) - Then it is easily seen that p _ is the average probability that j alternative i is higher valued than any other alternative. w* > 0. 1976). However. —j —j j nal ranking of the weights is thus supposed to be consistent with the quantitative information incorporated in an unknown cardinal vector w*. the eventual rank order of choice options is then determined by the rank order (or the order of magnitude) of the p^'s. if due to prior information in .. the assumption is now made here that the ordinal weights wj (j=1.. in other words: wj > wj i •• Wj* > Wjt*. if the ordinal weights vector w is interpreted as originating from a stochastic weight vector w*.. Next.5 In order to treat ordinal information on weights.6) i ïVi.1.w* r (2. In other words. in our case we do not have information on the cardinal value of Wj*. we assume * that the weighted dominance of choice option i with regard to i' can be represented by means of the following stochastic expression based on a weighted summation of cardinal entities (implying essentially a additive linear utility structure): J v. The motive is that. we introducé a certain probability p^i for the dominance of i with respect to i': Pii' • P^OD (vii' > °) and define as an aggregate probability measure: p = (2..w*) with max{w*}= 1. it is plausible to assume for the whole relevant area a uniform density function for the Wj*'s..j) are a rank order representation of an (unknown) underlying oardinal stochastic T weight vector w*= (w*.. . which also constitutes the foundation stone for the so-called Laplace criterion in case of decision-making under uncertainty (see Taha. but only on the ordinal value of Wj (which is assumed to be consistent with w-j*). Therefore. e . •—•\ H co •• H1 cr c et O l--- o H H X) T H H pi o PI O «! < ct O ct cr C D 3 C D Oq H H^- c » C D H- cr o 3 < < C D CL C D C U M et C O et e C U et (-•• 3" P> Ct O 3 H- C D C O C O / ** / |J" / P / <o cr »O < C! ffi B 0) ct 3 £> • p] • O 3 C O C 4= • CD -" Ó cr C O C 3 o O C O T D O CL et e t H - •* >-•• !-•• <. a H- • H 3* C D C O C D o « O O * a H- et 3ct H- 3 C O C D O •a T) T O § ^ 1 Q § 3 OH- 3 •o • M •-4 ct •o O C O 3 h* * >-ti H t->- •o T ct M p> 3 ct C a 3 *-* >-4 C D T C D •ë C O 3C D CL o C D "-4 *-4 !-•• o % o 3 a co C D CD C U C O C O CJ B C D CL « 3 3 3 CJ co fcr H> !-• !-•• O H (-• O *3 1 s 3 Oq C D C O cu cr o C D Cr O H- | 3* ^ » C D o >-•• H *.o <S T l. «<! co Hco (-• -s aH 3 C u H X •o1 "O H- P > C D P) o 3 M- 3 (-"• C O C D O 3 PJ cr CD cr co cr Hc CD o C O et -i pi • O PJ H •-4 O T P) < S C D P) M> P) aH Ca et H- et o -3 C O ^ C D O Ct o 3 O >-4 !-•• -i CD O 3 O C O et PJ 3 O.• Ho et cr o Cr et H. H- s: C D y->- P> ct H- p> O co PJ *-4 -i CD • < • -3 et H O o 3 et C O 3 < P) CT I-1 C D C O JC HO ca 3 31 Q "ï •o et H- C D C O C D V "-4 Oq C O e cr co C D ot e h- 3 B O HX H- et C D C D C O et C D !-•• 3 Oq C O O 3" C D « 3" •o o P) g ct 3C D P > cr ct co S •-4 HO H- !-•• -s co H e t P) -i H- * !-<• a h"' 3 P> (-• v^> O co T — (-•• C O C u -i € -s co O C D 3 C O 3 3 0 B w cr C et H- et o> P 3 s O O co c o ez B h* 3 TT O C D C O H- o B •o C U -J *-•• •o et O C D C O o H- • cr C D 3 et C O -i C D H a O C D P> 3 Q.co HX C D a O T a H" 3 CD •"4 »"4 CD / ƒ / ƒ e o o C D C O C O HC D M B P> et H- H- <5 "O C U 3" C D T 3 O T H' I- 3 O 3 C D 03 C O ca H- ct a *% et O ct X £ & et H- O ƒ rt / C ƒ O /O o 3 -i C D o o e B C D co C O !-•• 3 P) 1 o •t ct C O C D O ct H- O 3P> B H cr C D g T H B CL C Hu ct B C D C U C O H ctH- H' 3 a ^ C D (-"• H- <-4 •-4 o IS C D C D C O C D p. < ! < C D P>-4 o T 3 C u ct !-•• o. O pi H M Pi H l-"- C co H •a C t C D er oH p> H H *: < o 3 cr c t oU PJ C cr cr cr s H.pi P) Q. C D & o 3 ct 3 P> CJ H- ct C D T X) 3 S C D C D P> C O ct ct T 3 ^ et et Cr Oq p> H > H C D 3 B Cr C D cr !-•• co O T H- T co C D CL C D ct ct *->• O 3 i-"- C H O 3 C 0) D (-•• et 3* H- * N P) B H- • << (-•• C O et -i !->• << < B a> HB C X C D M ct H- X C D CL H" 3C D -3 PJ 3 C D ct C D H C D 3 C U !-•• ct Cr C D T O CL H3 P> M O T < o CL !-•• H^- C u er pi ct H- 3 * * Ha •a M H- •a p) !-•• •a CD cr "-(> a a •a cr O P) (-•• M> ^ CD T T o H H P) H C O H- ca et H- C D C D 3 PJ ct H- 3 3 3 Ó co e t e o o 3 •-•• ca C D H-1 »-•• O et I-1 O "3 HCL 3 >-4 o.1 - !-•• 3 e o 3* PJ et ct H1- O H- •o -J o cr -i 3 CD O C D C O B > T O ct O 3 • a *• Q 3 O << cr HHCJ. B o p> £. CD C O et- < o pi co C D »-4 tion tion CT -• o Cu O C O OJ a i-"- •a H C D O H"et n •a O tet 3(D et H-Cr 3 o "3 3 ct 3 pi t-> CL ro fu 3 3C D 't» O H H O T a C u et H- O 3 •—s ffi 3 ct H- <-t> 1 HCD ( . & C U CT C D B •o -i 3 " ca ct O C D cr co CD CD CO ot C !-<• e o. CD co S O C D C O 3 Cr C D co C D (-•• ct H-- C D 3 et co C o C D B <: < 3 • 3 1 C D T 1 . H et H- cr 3 «< H- *< cr a !B• • H•-4 x> >-4 H" C D T C D ct H" ct C D T HC D C O ct C O C D HH 3 C D Hct et O co O. O 3 -Cr P > 3 (U (-• & C CD < ! !-•• < ct P) C P) et H- H 3* C D ct T C D C O C D *3 H- *D C 3- 3 C D "3 C D O o 3 cr *ï 3 C D cr C D T C D (-• 3 O C O 3 et S C D • co c o HX C D B «<. C u co o s a Hl-" o h" O C D O « C D 3 H1 i- II O er C D CL H*-4 •-4 C D C D p) 3 P) co o 3 O a P) • H 3* •o pi HH- C D a j= • P) ct C D *3 O 3 i-* !->• • C D o cr C D «! < O. 3 <: C u 3 3 O ^ *D C 3" P > C D ^ et C D C O o et h* et HC D C O 3 o et HC D C O ƒ H/ et /** Ct ct a p> P) •o T O Oq PJ H- P) oq P> «<i P > 3 oq ffi T C D O C D C O 3 'S cu T HC D et C D C O . measured in a s t r i c t l y ordinal sense ( i . with a low number of c r i t e r i a ) . R = — + + + + + + + + - (3. or of the presence of mixed data. Structure of the Standard Regime Method The regime method was o r i g i n a l l y developed for purely q u a l i t a t i v e information on multidimensional choice problems. In our case we assume the following choice problem. For simple examples ( i . Then we may assume the following effect matrix E: eriterion alternative^ 1 E - 1 2 3 3 2 1 2 2 1 3 3 1 2 3 4 1 3 2 (3.4) and: P12 = prob ( _ ] > 0) v-2 = prob{(wi*+w?*-W3*-w4*)>0} (3.2) which of course also implies the following consistency condition for the cardinal weights: w-| * > W2* > W3* > W4*. .7 3. numerical i l l u s t r a t i o n s can without l o s s of generality d i r e c t l y be used t o explain the basic steps of the regime algorithm. of the presence of t i e s . computer assistance was necessary. e t c ) . e t c ) . plans. i t turned out that in case of many c r i t e r i a (7 or more). we assume the following weight vector w: w =* (wj = ( 4 wj? W wi})"^ 3 3 "2 1 )T (3. impacts. A decision-maker has to make a choice out of 3 a l t e r n a t i v e commodities (goods. without t i e s ) . . A pairwise comparison of the information in E leads to the following regime matrix R: V1X \!n \JU fe h\ ••!-.w\* (3. e . however. e . p r o j e c t s .3) For instance. if we take regime r-|2» it is easily seen that: y_i2=wj*+W2*-W3*. which are characterized by 4 a t t r ï b u t e s (features.5) . However. and aimed even at designing a ohoice evaluation method which did not need the use of a computer.1) Furthermore. .6) holds. W2*£0. Thus it can directly be derived that P12 • 1. we can e a s i l y derive t h a t : P23 = 1/2 Now we can directly derive the total dominance by means of (2. W3*.3).T) Finally.8) (3. w-| > W 2 > «3 > W4 > 0. If we next take the choice alternatives 1 and 3» we can easily derive v^-j3 by means of (3. The r e l a t i v e s i z e of the various hyperplanes which make up the envelopes of the information embodied in the weight vector may thus be regarded as a p r o b a b i l i t y measure for the dominance of the a l t e r n a t i v e concerned. P1 = 1 / 2 ( 1 + 1/6) =7/1 2 p 2 . given the information implied by (3-2).10) Thus. i. one has t o take i n t o account the standardization condition that max{w-|*. Of course. y . W3%0.. we have to identify the r e l a t i v e s i z e of the four-dimensional hyperplane for which condition (3. If we thus assume t h a t a l l Wj*»s are uniformly d i s t r i b u t e d . w e will compare choice options 2 and 3.e. and Hinloopen and Smyth.3 =w-| *-W2*-W3 * -wij * _] (3 • 6) In t h i s case.8 The question is now whether we can make any valid statement regarding the value of p-|2> In this case the previous question is easy to answer. 1985). a p r i o r i no unambiguous statement regarding the value of p-j3 can be made. we can e a s i l y derive the value P13 in case of a uniform d i s t r i b u t i o n : p 1 3 = 1/6 (3. W2*.7). W4*}=1 and w-|*£0.9) of each choice option 1 final (3.1/2(ïï + f / 2 ) = 1 / 4 P3 « 1/2(5/6+1Y2)=2/3 (3. viz.Then w e have: V23=w-| *-W2*-W3 *+wi( * In t h i s case. unless we use the p r o b a b i l i t y approach outlined in the previous section. Then by using conditions (3-2) and (3.e. 1985. in our illustrative example the following alternatives results: alternative 3 > alternative 1 > alternative 2 ranking of .6) in addition to the standardization condition. W4*S0 (see for further d e t a i l s Hinloopen. i. but as this notation is more cumbersome than illustrative we suffice to conclude here that the regime method provides a fairly direct and unambiguous solution to a strictly qualitative multiple criteria choice problem. +.that: P12 .1/2 (4.. e . ejj = ei'j» i n other words: equal rank orders of two alternatives i and i' for a specific criterion j ) .4) The existence of t i e s has c l e a r l y consequences order of plans.3). Ties in the effect matrix If the effect matrix contains ties (i.9 It is evident that the foregoing example can easily be generalized in a formal notation. r 1 2 .5) .e. 4. .) T so that: Z. . Ties and Mixed-Data in the Regime Analysis additional problems caused by the presence of for both the effects e^j and the weights WJ In this section the ties and mixed data will be dealt with. we can easily derive uniform probability distribution . 1 1 2 2 2 T 2 2 1 3 3 1 2 4 1 3 2 (4.1): —-*^gri t er i on a l t er na'ErT'e--—. i .. .2) (4.3) by assuming again a (4. as in the present case we have: P1 = 1 / 2 ( 1 / 2 + 1/6)= P2 = 1 / 2 ( 1 / 2 + 1/2)= P3 = 1 / 2 ( 5 / 6 + 1 / 2 ) = 1/3 1/2 2/3 for the final rank (4. then the additional problems can easily be solved.1) 3 3 In t h i s case only the regime r-| 2 w i l l change.12=W2*-w_2*-v^4* On the basis of (4. 4.1.(0. This can easily be illustrated by including ties for the first criterion in the effect matrix (3. that also the existence of ties in the weight vector will not affect the regime vector. In this case the regime matrix is still equal to (3. as condition (4.1).10) is always satisfied.6). as can be seen from (3.2W3* By pursuing next the same stochastic distribution. Consequently.w 2 * (4.6).2.9) 2 and 3. i.W2* .( 4 3 2 2 )! (4. Clearly.e. in this case we may substitute the fact that W3* = W4* into (3. This will be illustrated by using again the same effect matrix (3.10) Then it is evident that P23=1. It is clear that in this case the same result emanates. given the initial condition (3-2).6) The treatment of ties will first be illustrated by comparing alternatives 1 and 2.. we may find the following final results: . whereas the weight vector is assumed to be equal to: w. Ties in the weights vector Ties in the weight vector (i.3). P12 . 4. we will compare alternatives becomes: v (4.wi» . Wj .e. without any difficulty this procedure can be directly generalized for the existence of multiple ties. It is evident.4). we find the result: P13 .w-j') imply a different situation regarding the evaluation of values of the choice criteria. so that the new condition becomes: V13 • w | * .1 (4. . However.8) analysis by means of a uniform (4.1/3 Finally. but no doubt the probabilities P ü ' will alter.7) If we next compare choice options 1 and 3. In this case V23 23 ™ w 1 * ~ w 2 * ~ w 3 * + w 4 * . whilst V12 remains also unchanged. vf 3 and p-j3 do not _ change either in comparison with (3..10 so that choice options 1 and 2 have changed position in the eventual rank order. either on the interval (0. so that part of the weights is ordinal and another part cardinal in nature. ' .e.11 p-l = P2 P3 = 1/2(1 + 1/3).3. 0) (if iij^i'j)- . 1) (if lijSeitj) or on the interval (-1. . 4. Then we impose the condition that all stochastic weights Wj* are standardized as follows: w* i—r-r X. The motive for this specific way of standardizing the weights is that (since Wj* are uniformly distributed) also the vector _ = (Xj .X_j)T X is uniformly distributed. Mixed data In case of mixed data in either the effect matrix or the weight vector.11) so that now the f i n a l ranking of a l t e r n a t i v e s becomes: alternative 1 > alternative 2 > alternative 3 Thus. dilM s*. First we will consider a situation of mixed data in the weight vector. .1/2 1/2(2/3+ 0 )= 1/3 ï (4. s*ii'j is also standardized.2/3 1/2(0 + 1 ) . the regime method has to be significantly adjusted. i t i s worth mentioning that also a s i t u a t i o n of combined t i e s in both the effect matrix and the weight vector can be handled in the same way.^ max{e l j f e l f J } (4. irrespective of the presence of ties. we will consider the presence of mixed data in the effect matrix..*Ü'J are assumed to be uniformly distributed. value of Xj is always equal to 1.. t i e s may exert a s i g n i f i c a n t impact on the eventual rank order of choice options..12) It is easily seen that in this case Xj < 1 . Next.13) Note that d ^ i j is also uniformly distributed. = i (4. . Also in this case the (stochastic qualitative) differences S. while the highest .. Finally. In order to be able to compare the differences across different criteria. i. Therefore. d.16) The r e s u l t i n g regime matrix R i s : (4.w2*) > 0} =1 (4. . 1 ) .19) Consequently..12 In order to compare now 2 alternatives i and i'. we define . we find: X12 =H 1 * ~ w 2 * . 1 ) . we find .. -ii' j=1 -II'J -j while next p^r is according to (2. we know that jd-j 2 1 and -dj 2 2 a r e uniformly d i s t r i b u t e d on the i n t e r v a l ( 0 . Note that X.6) defined as: Püt = prob (zjj_i>0) (4.21) uniform densities. Assume the following effect matrix: while w is assumed to be equal to: w . (4.5) .( 2 1 )T (4.20) since we know from (4.=1 _ ] and X2 i s _ uniformly d i s t r i b u t e d on ( 0 .15) The remaining part of the procedure i s then simular t o t h a t described in section 2. The various steps of the abovementioned exposition w i l l now be i l l u s t r a t e d by means of a simple numerical example.17) that w-j > w2 and hence also w-| * > w_2*. 2 12. we derive t h a t : i l 2 = l l 2 . .18) If we want to compare choice option 1 with 2.17) -D 2 1 2 (4.instead of Vj^» from (2.a new stochastic variable z^i as follows: z.14) (4. we may derive: P12 = prob (yj2 > 0) = prob {(w-j* . 1 + l l 2 .. A . = 1 . In addition. By using next our Standard procedure ultimately: for (4. 1 .23) As both _ . 2 ) J = = p r o b { .2 Consequently. interval (0. (4.1 (4.2 a r e d|.2)=Pr.1).ob(d12Ji>-d12>2) +prob{ (zj 2 >01 ( d ! 2 .l is uniformly (0. _3i2. 1 " i 2 ) > 0 } = 1/2 Thus.13 p 1 2 = prob ( z 1 2 > 0) = 3/4 The proof of (4.26) (^-27) on the interval (4-28) prob{z-|2 > 0 | ( d j 2 .i > -di2.d i 2 .30) .25) 2^» Next.2 1 and -^12.A2) This i m p l i e s that: distributed (4.2.2>J =1 (4. 1 < _ ^ 1 2 .22) runs as f o l l o w s .2/012.23) while also the following condition holds: prob{z_12 > 0|(d12. in order to calculate prob{z_-]2>0 | di 2<-^12 _ we define: 112.1 + ^12. 2 ( 1 1 2 .2 It is also easily seen that: ^12.2 .prob(dj 2.-012. 1 <~±\2.2)J-Pr. 2 . 2 ( i l .29) (4.22) <-di 2 .12.d i 2 .1). l>-di2.D. i n c o n c l u s i o n : prob(z 1 2 > 0) = 1/2 + 1/2 .2 A2 = . prob (zj2>0) = = prob{(z12>0|(d12.E.1 < "412. 1/2 = 3/4 Q.2)=1 /2 (4.l<-di2.1>-dl2.2)}. Next we may c a l c u l a t e £12= 112= -É12.ob(di2. we know that: uniformly distributed on the prob(d12.3. 2 ) (4. It is already known that A 2 and cii2.6 in 0.0.6.4 .6 In 0.6 + d-|2.6 + d-|2. 6 ..31) = 0.6 = 0.9 (4. say criterion 1. housing market.6 1/x 5. we will assume that we have cardinal information on'one of the criteria.like the concordance method -. not only in the field of public choice theory but also in the field of consumer theory and marketing analysis. Various empirical applications (e.1).2 lZ > °) = Pi"ob(di2. Let us assume then the following effect matrix: E= 20 8 2 1 2 °2J 8 (4.0.2 a r e independently _ uniformly distributed on (0. .1). make? it a powerful vehicle for evaluation analysis. Concluding Remarks The foregoing analysis has demonstrated that for discrete choice problems which are marked by complete (or partial) uncertainty in the form of ordinal (or mixed) information the dominant regime method may be an operational tooi.g. It leads to a probability statement regarding the choice of alternatives.2 hZ > ^2 are independent uniformly °-6) = } } dy dx= 1.2 *2) > °* = 0 . we find: 1. which often do not lead to an unambiguous solution).prob(-^2.2 lZ > 0 > 6 ) Since we know that ~di2.32) Then d 1 2 J so t h a t : prob(z 1 2 > 0) = prob {(0. Also its ability to deal with both qualitative information (including ties) and mixed information.6 = 0. transportation and physical planning) have also demonstrated its usefulness in practical choice situations.6) = 1-prob(-d-|2.(4. Then we have: prob (0.2 a n d distributed on (0.2 h2> -0. and in so doing it leads usually to a unique solution (which is a major advantage compared to other 'soft' multiple criteria choice methods .14 Next.33) The proof of the latter calculation can easily be given.9 0. Hinloopen. pp. R.. P. Gower. Free University. Voogd. Pion. Delft (1985b). and A. 1977.P. Co. J. Z. H. and H. Multicriteria Evaluation for Urban and Regional Planning. _. Decisions with Multiple Objectives. London. Smyth. 1985.. Amsterdam. Multiple Objective Decision Methods and Regional Planning . Decision Knowledge. Martinus Nijhoff. A Description of the Principles of a New Multicriteria Evaluation Technique. 1985 (mimeographed). pp..15 References Delft. 1981. North-Holland Publ. 9 (8). 1981. A.W. E. Rietveld.. .. New York. 380-388. De Regime Methode. Qualitative Discrete Multiple Criteria Choice Models in Regional Planning. 1980. pp.. Keeney. and H.. Martinus Nijhoff. Dordrecht. T. Rietveld. 4. Preferences and Value Tradeoffs. Measuring surable. Journal of Mathematical Psychology. 1976. H. Aldershot. 'E. Oxford.). Mastenbroek.. 1976. Pergamon Press. and Kindier.). MacMillan. Saaty's Priority Theory and the Nomination of a Senior Professor in Operation Research. Theory and Incomplete University Press. Consumer Demand. (eds. H. vol. 1983. Saaty. Wrigley (eds. 1980. and A. Water and Related Land Resource Systems. 1977.H. 19-30. P. Leitner and N. and J. Wiley. A Scaling Method for Priorities in Hierarchical Structures.A. Voogd. Taha. 883-895. New York. Regional Science and Urban Economios. Amsterdam. The Regime Method. pp. The Hague/Boston. Interfaculty Actuariat and Econometrics. Hinloopen. Columbia 1971. Nijkamp. P.D... Nijkamp. New Multicriteria Methods for Physical Planning by means of Multidimensional Scaling Techniques. pp. T.W. the Unmea- Nijkamp. 77-102. Multicriteria Analysis and Regional Decision-Making. Y.A. Lootsma.. Thesis. 1983. . Qualitative Multicriteria Analysis . New York. and P..A.. 13. Kmietowicz. 1977... Raiffa. 15. European Journal of Operational Research. 234-281. Lancaster.Applications to Airport Location. Operations Research. Pearman.. van. in: Haimes.L. pp. P. Environment and Planning A. Nijkamp and P. Hinloopen. in: Proceedings Colloquium Vervoersplanologisch Speurwerk 1985. ---• .. P. Paelinck. K. M. E. Brouwer and P.A.E. deel I Dlsaggregate Context Models of Cholse In a Spatial 1982-5 Peter Nljkamp Jaap Spronk Ruerd Ruben (ed. Klaassen and H. Smit F.M. Vos 1981-19 The political Economy of the Republic of Korea. Verbruggen A Con3taht-Market-Shares Analysls of ASEAN Manufactured Exports to the European Communlty 1982-3 1981-10 P. Nljkamp A. v. Kodde Het genereren en evalueren van voorspellingen van omzet en netto winst: een toegepast kwantitatieve benadering Financiële vraagstukken onder onzekerheid 1982-2 1981-9 P. Janssen R. Out J.P.J. Mathot S. Palm 1981-21 1981-22 1981-5 P. Nijkamp P. Sigar International Conflict Analysls 1981-11 1981-12 1981-13 P. van Duin en P. Jansen mei 1982 1981-11 W. Nljkamp Htdde P.M. Nljkamp and P. with H. Nijkamp .) H. Ruben 1981-6 Hldde P. Handenhoven and R.ing A quarterly econometrie model for the Price Formution of Coffee on the World Market DeiMdd i i Supply of Natural Rubber. Blommestein and P. A. de Graaff and E.1981-1 E. Part I id 1981-18 F.C. van Lierop and P.E. Nijkamp A.M.W. Schreudèr 1981-17 F. Nljkamp in co-op. Sneek 1981-2 1981-3 H. A proposal for a model framework of an open economy in the ESCAP-region.C. Palm and Th.C. Smit R.C. Rietveld F. Cornelis L'Utili3ation une Voiture du Crédit lors de 1'Achat d' 1982-1 A Multldlmenslonal Analysls of Regional Infrastructure and Economie Development P. H. Nijman F. Smit 1982-1 Peter Nljkamp 1981-8 B. Palm and J. Klelnknecht 1981-20 1981-1 F. Palm 1982-6 1982-7 1981-15 1981-16 The World Vehlcle Market Structural Econometrie Model ing and Time Series Analysls: An Integrated Approaoh Linear Regression Using Both Temporally Aggregated and Temporally Disaggregated Data 1982-9 1982-8 mei J. van Dijck and H. Schippers Piet van Helsdingen maart 1982 Onderzoek naar levensomstandigheden en opvattingen over arbeid bij mensen zonder werk. Rietveld H. Brouwer and P.Tówards an Integrated Ap—oaoh Urban Impact Analysls in a Spatial Context: Methodologie and Case Study 1981-23 Primaire ling exporten en ekonomiese ontwikke1981-21 1981-7 D. Nljkamp. Vogel v. Nijkamp and P. with emphasis on the Role of the State Structural Econometrie Model ing and Time Series Analysls .P. Sneek F.
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