INCOMPLETE PAIRWISE COMPARISON MATRIX AND ITS APPLICATION TO RANKING OF ALTERNATIVES

Transcription

1 M U L T I P L E C R I T E R I A D E C I S I O N M A K I N G Vol Jaroslav Ramík INCOMPLETE PAIRWISE COMPARISON MATRIX AND ITS APPLICATION TO RANKING OF ALTERNATIVES Abstract A fuzzy preferece mrx s the result of parwse comparso - a powerful method mult-crtera optmzo. Whe comparg two elemets, the decso maker assgs a value betwee 0 ad 1 to ay par of alterves represetg the elemet of the fuzzy preferece mrx. Here, we vestge relos betwee trastvty ad cosstecy of fuzzy preferece mrces ad multplcve preferece oes. The results obtaed are appled to decso stuos where some elemets of the fuzzy preferece mrx are mssg. We propose a ew method for completg the fuzzy preferece mrx wth mssg elemets called the exteso of the fuzzy preferece mrx ad vestge a mportat partcular case of the fuzzy preferece mrx wth mssg elemets. Next, usg the egevector of the trasformed mrx we obta the correspodg prorty vector. Illustrve umercal examples are suppled. Keywords: parwse comparso mrx, fuzzy preferece mrx, recprocty, cosstecy, trastvty, fuzzy preferece mrx wth mssg elemets. 1 Itroducto I varous felds of evaluo, selecto, ad prortzo processes the decso makers (DM) try to fd the best alterve(s) from a feasble set of alterves. I may cases, the comparso of dfferet alterves accordg to ther desrablty decso problems caot be doe by oe perso or usg oly a sgle crtero. I may DM problems, procedures have bee establshed to combe opos about alterves reled to dfferet pots of vew. These procedures are ofte based o parwse comparsos, the sese th processes Jaroslav Ramík, Slesa Uversty Opava, School of Busess Admstro Karvá, Departmet of Mhemcal Methods Ecoomcs, e-mal: ramk@opf.slu.cz

2 Icomplete parwse comparso mrx ad ts applco are lked to some degree of preferece of oe alterve over aother. Accordg to the ure of the formo expressed by every DM, for every par of alterves dfferet represeto forms ca be used to express prefereces, e.g. multplcve preferece relos (see Aloso et al., 2008; Say, 1980; Say, 1991), fuzzy preferece relos (see Fodor, Roubes, 1994; Herrera-Vedma et al., 2004; Ramík, 2011), terval-valued preferece relos ad also lgustc preferece relos (see Aloso et al., 2008). Usually, experts are characterzed by ther ow persoal backgroud ad kowledge of the problem to be solved. Expert opos may dffer substatally, some of them caot effcetly express a preferece degree betwee two or more of the avalable optos. Ths may be true due to a mprecse or suffcet level of kowledge of the problem o the part of a expert, or because the expert s uable to determe the degree to whch some optos are better tha others. I such stuos the expert wll provde a complete fuzzy preferece relo (see Aloso et al., 2008; Herrera-Vedma et al., 2007). I ths paper, we preset a geeral method to estme the mssg formo the form of complete fuzzy preferece relos- multplcve or fuzzy. Our proposal s dfferet to the approach descrbed Herrera-Vedma, et al., (2007) ad Ma et al., (2006), where specal averages of expert evaluos ad cosstecy/trastvty propertes are appled. I the lterure (Xu ad Da, 2005), the problem s solved by the least devo method to obta a prorty vector of a fuzzy preferece relo. Here, we propose the classcal result of Perro-Frobeus theory to obta the prorty vector for a trasformed fuzzy preferece mrx. Moreover, our approach eables us to obta a prorty vector for addtve-trastve preferece relos ad also for addtve-cosstet oes,.e. addtve-recprocal ad multplcve-trastve oes. It also allows for completg a parwse comparso mrx wth mssg elemets ad for fdg out the closest cosstet/trastve mrx to the cosstet/trastve oe,.e. by reparg the cosstecy of fuzzy preferece relos. 2 Multplcve ad addtve prefereces The DM problem ca be formuled as follows. Let X={x 1, x 2,...,x } be a fte set of alterves. These alterves have to be ordered from best to worst, usg the formo gve by a DM the form of parwse comparso mrx. The prefereces over the set of alterves, X, may be represeted two ways: multplcve ad addtve (also called fuzzy preferece relos). Let us assume th the prefereces o X are descrbed by a preferece relo o X gve by a postve mrx A={a }, where a > 0 for all, dces the preferece testy for the alterve x to th of x. The elemets of A={a } ssfy the followg recprocty codto. A postve mrx A={a } s multplcve-recprocal (m-recprocal), f: a. a =1 for all,{1,2,,}. (1)

3 116 J. Ramík A postve mrx A={a } s multplcve-cosstet (or, m-cosstet), f: a = a k. a k for all,,k{1,2,,}. (2) Note th a = 1 for all, ad th a m-cosstet mrx s m-recprocal (however, ot vce versa). Here, a > 0 ad m-cosstecy s ot restrcted to the Say scale. I partcular, we exted ths scale to the closed terval [1/σ; σ], where σ>1. Sometmes t s more ural, whe comparg x to x, th the decso maker (DM) assgs oegve values b to x ad b to x, such th b + b = 1. Wth ths terpreto, the prefereces o X ca be uderstood as a fuzzy preferece relo, wth membershp fucto μ R : X X [0;1], where μ R (x, x ) = b deotes the preferece of the alterve x over x. The most mportat propertes of the above metoed mrx B ={b }, called here the fuzzy preferece mrx, ca be summarzed as follows. A mrx B ={b } wth 0 b 1 for all ad s addtve-recprocal (a-recprocal), f: b + b = 1 for all,{1,2,,}. (3) Evdetly, f (3) holds, the b = 0.5 for all {1,2,,}. To make a coheret choce of evaluos b (whe assumg fuzzy preferece mrx B ={b }), a set of propertes to be ssfed by such relos has bee suggested the lterure, the termology of propertes of relos s, however, ot establshed yet, compare e.g. Aloso et al. (2008); Fodor, Roubes (1994); Tao (1984). Here, we use the usual termology whch s as close as possble to the oe used the lterure. Trastvty s oe of the most mportat propertes of prefereces, ad t represets the dea th the preferece testy obtaed by comparg two alterves drectly should be equal to or greer tha the preferece testy betwee those two alterves obtaed usg a drect cha of alterves. Let B ={b } be a a-recprocal mrx wth 0 b 1 for all ad. We say th B ={b } s multplcve-trastve (m-trastve), f: b b b b k k b b k k for all,,k{1,2,,}. (4) Note th f B s m-cosstet the B s m-trastve. Moreover, f B s m- recprocal, the B s m-trastve ff B s m-cosstet. We say th B ={b } s addtve-trastve (a-trastve), f: (b - 0,5) = (b k - 0,5) + (b k - 0,5) for all,,k{1,2,,}. (5) Ths property s also called addtve cosstecy; here, we reserve, however, ths ame for a dfferet oto, see below. Now, we shall vestge some reloshps betwee a-recprocal ad m- recprocal parwse comparso mrces. We start wth a exteso of the result

4 Icomplete parwse comparso mrx ad ts applco publshed Herrera-Vedma et al. (2004). For ths purpose, gve σ > 1, we defe the followg fucto σ as: 2t1 ( t) for t[0;1]. (6) We obta the followg result, characterzg a-trastve ad m-cosstet mrces; for the proof see Ramk, Vlach (2013). Proposto 1. Let σ > 1, B ={b } be a mrx wth 0 b 1 for all, {1,2,,}. If B s a-trastve the A={ b ) } s m-cosstet. ( Now, let us defe the fucto as follows: t ( t) 1 t for 0 < t < 1. (7) We obta the followg result (see Ramk, Vlach, 2013). Proposto 2. Let B ={b } be a a-recprocal mrx wth 0 b 1 for all,{1,2,,}. If B s m-trastve the A={a }={ b ) } s m- cosstet. From Proposto 2 t s clear th the oto of m-trastvty plays a smlar role for a-recprocal fuzzy preferece mrces as the oto of m-cosstecy does for m-recprocal mrces. Th s why t s reasoable to troduce the followg defto: Ay a-recprocal m-trastve mrx B ={b } s called addtvely cosstet (a-cosstet). Accordg to ths defto Proposto 2 ca be reformuled as follows: Proposto 2. Let B ={b } be a mrx wth 0 b 1 for all,{1,2,,}. If B s a-cosstet the A ={a }={ b ) } s m-cosstet. By Proposto 1, resp. Proposto 2 we ca trasform a-trastve, resp. a- cosstet mrces to m-cosstet oes by a appropre trasformo fuctos σ, resp. Ф. I practce, perfect cosstecy/trastvty s dffcult to obta, partcularly whe measurg prefereces o a set wth a large umber of alterves. 3 Icosstecy of parwse comparso mrces, prorty vectors If for some postve mrx A ={a } ad for some,,k{1,2,...,}, the multplcve cosstecy codto (2) does ot hold, the A s sad to be multplcve-cosstet (or m-cosstet). If for some fuzzy preferece mrx B ={b } wth 0 b 1 for all ad, ad for some dces,,k{1,2,...,}, (4) does ot hold, the B s sad to be addtve-cosstet (or, a-cosstet). Fally, f for some fuzzy mrx B ={b } wth 0 b 1 for all ad, ad for some dces,,k{1,2,...,}, (5) does ot hold, the B s sad to be addtve-trastve (a-trastve). I order to measure the degree of cosstecy/trastvty of a gve mrx several measuremet methods have ( (

5 118 J. Ramík bee proposed the lterure (see e.g. Aloso et al., 2008). I AHP, multplcve recprocal mrces have bee cosdered (Say, 1991). As far as addtve-recprocal mrces are cocered, some methods for measurg a-cosstecy/a-trastvty are proposed here. We start, however, wth measurg the cosstecy of postve mrces whch s based o Perro- Frobeus theory (see e.g. Fedler, Nedoma, Ramík, Roh, 2006). Ler o, we shall deal wth measurg a-cosstet ad a-trastve mrces. The Perro-Frobeus theorem descrbes some of the remarkable propertes eoyed by the egevalues ad egevectors of rreducble oegve mrces (e.g. postve mrces). Theorem (Perro-Frobeus). Let A be a rreducble oegve mrx. The the spectral radus, ρ(a), s a postve (real) egevalue, wth a postve (real) egevector w such th Aw= ρ(a)w. I the decso makg cotext the above metoed egevalue ρ(a) s called the prcpal egevalue of A. It s a smple egevalue (.e. t s ot a multple root of the characterstc equo), ad ts egevector, called the prorty vector, s uque up to a multplcve costat. Now, let A be a oegve m-recprocal mrx. The m-cosstecy of A s characterzed by the m-cosstecy dex I mc (A) defed (Say, 1980) as: ( A) I mc (A) = 1, (8) where ρ(a) s the spectral radus of A ( partcular, the prcpal egevalue of A). Moreover, we suppose th A={a } s a parwse comparso mrx wth elemets a based o evaluo of alterves x ad x, for all ad. For the purpose of decso makg, the rak of the alterves X={x 1, x 2,...,x } s determed by the vector of weghts w = (w 1,w 2,...,w ), where w > 0, for all w 1 {1,2,...,}, such th 1, ssfyg the characterstc equo Aw = ρ(a)w. Ths vector w s the (ormalzed) prorty vector of A. Sce the elemet of the prorty vector w s terpreted as the relve mportace of the alterve x, the alterves x 1, x 2,...,x X are raked by ther relve mportace. The followg result has bee derved Say (1980). Proposto 3. If A ={a } s a postve m-recprocal mrx, the I mc (A) 0. Moreover, A s m-cosstet f ad oly f I mc (A) = 0. To provde a cosstecy measure depedetly of the dmeso of the mrx A, T. Say (1980) proposed the cosstecy ro. To dstgush t here from the other cosstecy measures, we shall call t m-cosstecy ro. Ths s obtaed by takg the ro of I mc to ts mea value R mc, estmed by a arthmetc average over a large umber of postve m-recprocal mrces of dmeso, whose etres are radomly ad uformly geered (see Say, 1980),.e.:

6 Icomplete parwse comparso mrx ad ts applco CR mc = I mc /R mc. (9) It was proposed th a parwse comparso mrx could be accepted ( a DM process) f ts m-cosstecy ro does ot exceed 0.1 (see Say, T.L., 1980). The m-cosstecy dex I mc has bee defed for m-recprocal mrces; here, we vestge the cosstecy/trastvty of a-recprocal mrces. For ths purpose we use relos betwee m-cosstet ad a-trastve/a-cosstet mrces derved Propostos 1 ad 2. Let B ={b } be a m-recprocal mrx wth 0 b 1 for all ad. We defe the a-cosstecy dex I ac (B) of B ={b } as: I ac (B) = I mc (A), where A={ b ) }. (10) From (10) we easly obta the followg result, whch s parallel to Proposto 3. Proposto 4. If B ={b } s a a-recprocal fuzzy mrx wth 0 b 1 for all ad, the I ac (B) 0. Moreover, B s a-cosstet f ad oly f I ac (B) = 0. Now, we shall deal wth measurg a-trastvty of a-recprocal mrces. Let σ > 1 be a gve value characterzg the scale. Let B ={b } be a a- recprocal fuzzy mrx wth 0 b 1 for all ad. We defe the a- trastvty dex I (B) of B ={b } as: where: I (B) = I mc (A σ ), (11) ( b ) A σ ={ }. (12) By applyg (8) ad (12) we obta the followg results correspodg to Propostos 3 ad 4. Proposto 5. If B ={b } s a a-recprocal mrx wth 0 b 1 for all ad, the I ( (B) 0. Moreover, B s a-trastve f ad oly f I (B) = 0. Let A ={a } be a a-recprocal mrx. I (12), the m-cosstecy ro of A deoted by CR mc (A) s obtaed by takg the ro of I mc (A) to ts mea value R mc (),.e.: CR mc (A) = I mc (A)/R mc (). The values of R mc () for =3,4,, ca be foud Say (1980). Smlarly, we defe the a-cosstecy ro CR ac (A) ad the a-trastvty ro CR (A). Deote B) { ( b )}, the the correspodg prorty vector w ac s gve ( by the characterstc equo (B) w ac =ρ( (B) )w ac. Gve > 1, let us deote B) { ( b )}, the the prorty vector w s ( defed by the characterstc equo (B) w =ρ( (B) )w. I practce, a-cosstecy of a postve a-recprocal fuzzy prorty mrx B s acceptable f CR ac (B) < 0.1. Also, a-trastvty of a postve a-recprocal

7 120 J. Ramík parwse comparso mrx B s acceptable f CR (B) < 0.1. The fal rakg of alterves s gve by the correspodg prorty vector, see Example 1. The followg two results gve a characterzo of a m-cosstet mrx as well as a a-cosstet mrx by vectors of weghts,.e. postve vectors wth the sum of ther elemets equal to oe. The proofs are straghtforward ad ca be foud Ramk, Vlach (2013). Proposto 6. Let A ={a } be a postve mrx. A s m-cosstet f ad oly f there exsts a vector w = (w 1, w 2,...,w ) wth w > 0 for all {1,2,...,}, ad w 1 such th: 1 w a for all, {1,2,...,}. (13) w Proposto 7. Let A ={a } be a a-recprocal mrx wth 0 < a < 1 for all,{1,2,...,}. A s a-cosstet f ad oly f there exsts a vector v = (v 1, v 2,...,v ) wth v > 0 for all {1,2,...,}, ad v 1 such th: a v for all, {1,2,...,}. (14) v v A assoced result ca be derved also for a-trastve mrces. Proposto 8. Let A ={a } be a a-recprocal mrx wth 0 < a < 1 for all, {1,2,...,}. A s a-trastve f ad oly f there exsts a vector u = (u 1, u 2,...,u ) wth u > 0 for all {1,2,...,}, ad u 1 such th: a = 21 (1+u u ) for all,{1,2,...,}. (15) The proof of ths proposto s based o the observo th for a-trastve mrx A ={a } we have: Settg for all {1,2,...,}, we obta the requred result. Example 1 Let X={x 1, x 2, x 3, x 4 } be a set of 4 alterves. The prefereces o X are descrbed by a postve mrx B = {b }: B = (16) 1 1

8 Icomplete parwse comparso mrx ad ts applco Here, B s a-recprocal ad a-cosstet, whch may be drectly verfed by (7), e.g. b 12.b 23.b 31 b 21.b 32.b 13. At the same tme, B s a-trastve as b 12 +b 23 +b 31 = We cosder σ = 9 ad calcule: E = (B) =, (B) F = = We calcule ρ(e) = 4.29, ρ(f) = 4.15, the we obta CR ac (B) = 0.11>0.1 wth the prorty vector w ac = (0.47; 0.25; 0.18; 0.10), whch gves the rakg of alterves: x 1 > x 2 > x 3 > x 4. Smlarly, CR (B) = < 0.1 wth the prorty vector w =(0.44;0.27;0.18;0.12), wth the same rakg of alterves: x 1 > x 2 > x 3 > x 4. As t s evdet, a-cosstecy ro CR ac (B) s too hgh for the mrx B to be cosdered a-cosstet. O the other had, a-trastvty ro 9 CR 9 (B) s suffcetly low for the mrx B to be cosdered a-trastve. The rakg of the alterves gve by both methods remas, however, the same. I ths example we ca see th the values of cosstecy ro ad trastvty ro ca be dfferet for a a-recprocal mrx. I order to vestge a possble reloshp betwee the cosstecy/-trastvty dces, we performed a smulo expermet wth radomly geered 1000 a-recprocal mrces, (=4 ad =15). The we calculed the correspodg cosstecy ad trastvty dexes. Numercal expermets show th there s o strog reloshp betwee a-cosstecy ad a-trastvty. 4 Fuzzy preferece mrx wth mssg elemets I may decso-makg procedures we assume th experts are capable of provdg preferece degrees betwee ay par of possble alterves. However, ths may ot always be true, whch crees a mssg formo problem. A mssg value a fuzzy preferece mrx s ot equvalet to a lack of preferece of oe alterve over aother. A mssg value ca be the result of the capacty of a expert to quatfy the degree of preferece of oe alterve over aother. I ths case he/she may decde ot to guess the preferece degree betwee some pars of alterves. It must be clear th whe a expert s ot able to express a partcular value a, because he/she does ot

9 122 J. Ramík have a clear dea of how the alterve x s better tha the alterve x, ths does ot mea th he/she prefers both optos wth the same testy. The DM could be also bored by evalug too may pars of alterves. To model these stuos, the followg we troduce the complete preferece relo mrx. Here, we use a dfferet approach ad dfferet oto tha Aloso (2008). Let > 2, I ={1,2,...,} be a set of dces, I²=I I the Cartesa product of I,.e. I²={(,),I}. Here, we assume th the recprocty codto s ssfed. Therefore, we shall cosder oly a-recprocal fuzzy preferece mrces. Let L I², L={( 1, 1 ),( 2, 2 ),...,( q, q )} be the set of pars (,) of dces such th there exsts a parwse comparso value a, 0 a 1. By L we deote the symmetrc subset to L,.e. L = {( 1, 1 ),( 2, 2 ),...,( q, q )}. By recprocty, each subset K I² of the gve elemets ca be expressed as follows K = LL D, (17) where L s a set of pars of dces (,) of the evalued elemets a ad D s the dagoal of the fuzzy preferece mrx, D = {(1,1),(2,2),...,(,)}, here a = 0.5 for all. The elemets a wth (,)I²-K are called mssg elemets. Now we defe the fuzzy preferece mrx B(K) = {b } K wth mssg elemets by: b = a f (, ) K, f (, ) K. Here, the mssg elemets of the mrx B(K) are deoted by a dash -. O the other had, the elemets evalued by the experts are deoted by a where (,)K. By a-recprocty, t s suffcet th the expert quatfes oly elemets a where (,)L, such th K = LL D; the other elemets are calculed automcally by (3). I wh follows we shall vestge a partcular mportat case of L, amely, L={(1,2);(2,3);...,(-1,)}. 5 Exteso of fuzzy preferece mrx wth mssg elemets ad ts cosstecy/trastvty I ths secto we shall deal wth the problem of fdg the values of mssg elemets of a gve fuzzy preferece mrx so th the exteded mrx s as much a-cosstet/a-trastve as possble. I the deal case the exteded mrx wll become a-cosstet/a-trastve. We start wth the a-cosstecy property. Let K I², let B(K) = {b } K be a fuzzy preferece mrx wth mssg elemets. The mrx B ac (K)={ b ac } K called a ac-exteso of B(K) s defed as follows:

10 Icomplete parwse comparso mrx ad ts applco b ac f (, ) K, b = v (18) f (, ) K. v v Here, v ( v1, v2,..., v ), called the ac-prorty vector wth respect to K, s the optmal soluto of the followg optmzo problem: v (P ac ) d ac (v,k) = ( b ) 2 m; v v subect to: v 1 (, ) K 1, v ε > 0 for all =1,2,..., ( 19) (ε > 0 s a gve suffcetly small umber). Note th the a-cosstecy dex of the mrx B ac (K) = { b ac } K s defed by (15) as I ac (B ac (K)). The followg proposto follows drectly from Proposto 7. Proposto 9. B ac (K) = { b ac } K s a-cosstet,.e. I ac (B ac (K)) = 0 f ad oly f d ac (v,k) = 0. Now, we look for the values of mssg elemets of a gve fuzzy preferece mrx so th the exteded mrx s as much a-trastve as possble. I the deal case the exteded mrx wll become a-trastve. Aga, let K I², B(K) = {b } K be a fuzzy preferece mrx wth mssg elemets. The mrx B (K) = { b } K called a -exteso of B(K) s defed as follows: b = b f (, ) K, max{0, m{1, 1 (1+u -u )} f (, ) K (20). 2 Here, u ( u1, u2,..., u ) called the -prorty vector wth respect to K s the optmal soluto of the followg optmzo problem: 2 (P ) d (u,k) = ( b 1 (1 u u )) m; subect to: u 1 (, ) K 1, u ε > 0 for all =1,2,..., (ε > 0 s a gve suffcetly small umber). 2 ( 21)

11 124 J. Ramík I geeral, the optmal soluto u ( u1, u2,..., u ) of (P ) does ot ssfy the followg codto: (1 u ) u 1, (22) 1.e. B = b } { (1 u u )} fals to be a fuzzy preferece mrx. Th { 2 s why the defto of the -exteso of B(K) we use formula (20) esurg th all the elemets b belog to the ut terval [0;1]. I the ext secto we shall derve the ecessary ad suffcet codtos for (22) to be ssfed. Note th the a-trastvty dex (for a gve σ > 1) of the mrx B (K) = { b } K s defed by (11) as I (B (K)). The ext proposto follows drectly from Proposto 8. Proposto 10. Let σ > 1. If B (K) = { b } K s a-trastve,.e. (B ac (K)) = 0, the d (v,k) = 0. 6 A partcular case of fuzzy preferece mrx wth mssg elemets For a complete defto of a recprocal fuzzy preferece mrx we eed ( 1) N pars of elemets to be evalued by a expert. For example, f 2 =10, the N=45, whch s a cosderable umber of parwse comparsos. I practce we ask th the expert evalues oly aroud parwse comparsos of alterves whch seems a reasoable umber. I ths secto we shall deal wth a mportat partcular case of fuzzy preferece mrx wth mssg elemets where the expert should evalue oly -1 parwse comparsos of elemets. Let K I² be a set of dexes gve by a expert, B(K) = {b } K be a fuzzy preferece mrx wth mssg elemets. Moreover, let K = LL D. I fact, t s suffcet th the expert evalues mrx elemets oly from L. Here, we assume th the expert evalues the followg -1 elemets of the fuzzy preferece mrx B(K): b 12, b 23,...,b -1,. Frst, we vestge the ac-exteso of B(K). We obta the followg result. Proposto 11. Let L={(1,2);(2,3);...,(-1,)}, 0 < b < 1 wth b + b = 1 for all (,) L, let K = LL D, ad L = {(2,1);(3,2);...,(, -1)}, D = {(1,1),...,(,)}. The the ac-prorty vector v ( v1, v2,..., v ) wth respect to K s gve as: 1 v1 S, (23) v 1 a, 1v, for =1,2,..,-1, (24) I

12 Icomplete parwse comparso mrx ad ts applco where: a b 1 for all (,) L ad: (25) b S a, 1a 1, 2... a 1,. (26) Remark. The proof of Proposto 11 s straghtforward by usg (25), (26) ad the optmal soluto of (19). By Proposto 7 t follows th the acexteso of B(K),.e. the mrx B ac (K)={ b ac } K s a-cosstet. Now, we vestge the -exteso B (K) of B(K). We obta the followg result. Proposto 12. Let L={(1,2);(2,3);...,(-1,)}, 0 < b < 1 wth b + b = 1 for all (,) L, let K = LL D. The the -prorty vector u ( u1, u2,..., u ) wth respect to K s gve as: u for =1,2,...,, (27) where: 0, 0 for =1,2,...,-1. (28) b, 1 1 Remark. The proof of Proposto 12 s straghtforward by usg (27), (28) ad the optmal soluto of (21). I geeral, the optmal soluto u ( u, u,..., ) of (P ) does ot ssfy the codto: 1 2 u 0 (1 u u ) 1, for all,=1,2,...,, (29) e. B = b } { (1 u u )} s ot a fuzzy preferece mrx. We ca easly { 2 prove the ecessary ad suffcet codto for ssfyg (29) based o evaluos b,+1. Proposto 13. Let L={(1,2);(2,3);...,(-1,)}, 0 b 1 wth b + b = 1 for all (,) L, let K = LL D. The the -exteso B (K)={ b } K s a- trastve f ad oly f: 1 1 bk, k1 for =1,2,...,-1, =+1,...,. (30) k 2 2 The proof of Proposto 13 follows drectly from Proposto 8 ad Proposto 12.

13 126 J. Ramík Example 2 Let L={(1,2);(2,3);(3,4)}, let expert evaluos be b 12 =0.9, b 23 =0.8, b 34 =0.6, wth b +b =1 for all (,) L, K = LL D. The B(K) = {b } K s a fuzzy preferece mrx wth mssg elemets as follows: B(K) = Solvg (P ac ) we obta the ac-prorty vector v wth respect to K, partcular, v (0,864; 0,096; 0,024; 0,016). By (20) we obta B ac (K) - the acexteso of B(K) as follows: B ac (K) = By Proposto 9, B ac (K) s a-cosstet, hece I ac (B ac (K)) = 0. Solvg (P ) we obta the -prorty vector u wth respect to K as follows: u (0,487; 0,287; 0,137; 0,088). By (27) we obta B (K) the exteso of B(K) as follows: B (K) = , where, by Proposto 10, B (K) s a-trastve, as d ac (v,k) > 0. I partcular, I (B ac (K))= Coclusos I ths paper we have dealt wth some propertes of fuzzy preferece relos, partcular wth recprocty, cosstecy ad trastvty of relos gve the form of square oegve mrces. We have show how to measure the degree of cosstecy ad/or trastvty, ad also how to exted crsp comparsos to fuzzy oes,.e. how to evalue pars of elemets by fuzzy values. Also, we have proposed a ew method for measurg cosstecy based o Say s prcpal egevector method. Moreover, we have dealt wth the problem of the complete fuzzy preferece mrx, where some elemets of parwse comparso are mssg. We have proposed a specal method for

14 Icomplete parwse comparso mrx ad ts applco dealg wth th case. Some llustrg examples have bee preseted to clarfy the theory proposed. Ackowledgemets The research work was supported by the Czech Scece Foudo (GACR), grat No. 402/09/0405. Refereces Aloso S., Chclaa F., Herrera F., Herrera-Vedma E., Alcala-Fdes J., Porcel C. (2008), A Cosstecy-based Procedure to Estme Mssg Parwse Preferece Values, Iter. J. Itellget Syst. 23, p Chclaa F. Herrera F., Herrera-Vedma E. (2001), Itegrg Multplcve Preferece Relos a Multpurpose Decso Makg Model Based o Fuzzy Preferece Relo, Fuzzy Sets ad Systems 112, p Fedler M., Nedoma J., Ramík J., Roh J., Zmmerma K. (2006), Lear Optmzo Problems wth Iexact Da, Sprger, Berl-Hedelberg-New York-Hog Kog-Lodo-Mla, Tokyo. Fodor J., Roubes M. (1994), Fuzzy Preferece Modellg ad Multcrtera Decso Support, Kluwer, Dordrecht. Herrera-Vedma E., Herrera F., Chclaa F., Luque M. (2004), Some Issues o Cosstecy of Fuzzy Preferece Relos, Europea Joural of Oper. Res. 154, p Herrera-Vedma E., Chclaa F., Herrera F., Aloso S. (2007), A Group Decso-makg Model wth Icomplete Fuzzy Preferece Relos Based o Addtve Cosstecy, IEEE Tras. Syst, Ma Cyberetcs, Part B, 37(1), p Ma, J. et al. (2006), A Method for Reparg the Icosstecy of Fuzzy Preferece Relos, Fuzzy Sets ad Systems 157, p Ramík J., Vlach M. (2013), Measurg Cosstecy ad Icosstecy of Par Comparso Systems, Kyberetka 49(3), p Ramík J., Vlach M. (2001), Geeralzed Cocavty Optmzo ad Decso Makg, Kluwer Publ. Comp., Bosto-Dordrecht-Lodo. Say T. L. (1980), The Aalytc Herarchy Process, McGraw-Hll, New York. Say T. L. (1994), Fudametals of Decso Makg ad Prorty Theory wth the AHP, RWS Publcos, Pttsburgh. Swtalsk Z. (2003), Geeral Trastvty Codtos for Fuzzy Recprocal Preferece Mrces, Fuzzy Sets ad Systems,137, p

15 128 J. Ramík Tao T. (1984), Fuzzy Preferece Ordergs Group Decso Makg, Fuzzy Sets ad Systems 12, p Tao T.(1988), Fuzzy Preferece Relos Group Decso Makg, : No-Covetoal Preferece Relos Decso Makg, Ed. J. Kacprzyk, M. Roubes, Sprger-Verlag, Berl, p Xu Z. S.(2004), O Compblty of Iterval Fuzzy Preferece Relos, Fuzzy Optm. Dec. Makg 3, p Xu Z.S., Da Q.L. (2005), A Least Devo Method to Obta a Prorty Vector of a Fuzzy Preferece Relo, Europea Joural of Operoal Research 164, p Xu Z.S., Che J. (2008), Some Models for Dervg the Prorty Weghts from Iterval Fuzzy Preferece Relos, Europea Joural of Operoal Research 184, p