New Approach to Fuzzy Decision Matrices

Transcription

1 Acta Polytecnca Hungarca Vol. 14 No New Approac to Fuzzy Decson Matrces Pavla Rotterová Ondře Pavlačka Departent of Mateatcal Analyss and Applcatons of Mateatcs Faculty of cence Palacký Unversty Oloouc 17. lstopadu 119/ Oloouc Czec Republc pavla.rotterova01@upol.cz ondre.pavlacka@upol.cz Abstract: Decson atrces represent a coon tool for odelng decson-akng probles under rsk. Tey descrbe ow te decson-aker's evaluatons of te consdered alternatves depend on te fact wc of te possble and utually dsont states of te world wll occur. Te probabltes of te states of te world are assued to be known. Te alternatves are usually copared on te bass of te expected values and te varances of ter evaluatons. However te states of te world as well as te alternatves evaluatons are often descrbed only vaguely. Terefore we consder te followng proble: te states of te world are odeled by fuzzy sets defned on te unversal set on wc te probablty dstrbuton s gven and te evaluatons of te alternatves are expressed by fuzzy nubers. We sow tat te coon approac to ts proble based on eployng crsp probabltes of te fuzzy states of te world coputed by te forula proposed by ade s not approprate. Terefore we ntroduce a new approac n wc a fuzzy decson atrx does not descrbe dscrete rando varables but fuzzy rule bases. Te proble s llustrated by an exaple. Keywords: decson atrces; fuzzy decson atrces; decson akng under rsk; fuzzy states of te world; fuzzy rule bases syste 1 Introducton A decson atrx s often used as a tool of rsk analyss n decson akng under rsk [3] [4] [7] [14]. It descrbes ow te decson-aker's evaluatons of te consdered alternatves depend on te fact wc of te possble and utually dsont states of te world wll occur. Te probabltes of occurrences of tese states of te world are supposed to be known. Tus te evaluatons of te alternatves are dscrete rando varables. Te alternatves are usually copared on te bass of te expected values and te varances of ter evaluatons. Te decson-aker selects te alternatve tat axzes s/er expected evaluaton or axzes te expected evaluaton and sultaneously nzes te varance. In practcal applcatons te states of te world as well as te evaluatons of te alternatves can be deterned vaguely. Te states of te world are ostly 85

2 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces descrbed verbally lke "te gross doestc product wll ncrease oderately durng next year". oetes t can be probleatc to express te evaluatons of alternatves precsely because we ay not ave enoug nforaton. For nstance te evaluaton under a certan state of te world can be descrbed as about 5%. In soe cases t s ore natural for a decson-aker to express te evaluatons by selectng a ter fro a gven lngustc scale. Te vaguely descrbed peces of nforaton can be ateatcally odeled by eans of tools of fuzzy sets teory. Dfferent vews of uncertanty and fuzzy decsons n a decson atrx are dscussed n [7]. Multple attrbute decson akng probles descrbed by a decson atrx wt crsp and fuzzy data are analyzed n [1]. In [] a fuzzy decson atrx s appled to a group decson akng. An applcaton of rsk analyss wt fuzzy sets eployng te decson atrx s presented n [3]. In [4] te autors consdered decson atrces wt fuzzy targets. In [5] te estant fuzzy decson atrx.e. a decson atrx contanng fuzzy sets wt a dfferent defnton of ebersp functon ten te orgnal one proposed by ade [15] s consdered. A decson atrx wt te fuzzy states of te world and te fuzzy evaluatons of te alternatves under te partcular fuzzy states of te world s called a fuzzy decson atrx. In [Error! Reference source not found.] te autors consdered a odel were te fuzzy states of te world are expressed by fuzzy sets on te unversal set on wc te probablty dstrbuton s gven. Tey proposed to proceed n te sae way as n te case of te crsp (.e. exactly descrbed) states of te world; tey set te probabltes of te fuzzy states of te world applyng te forula proposed by ade n [17]. Wtn ts approac te evaluatons of te alternatves are understood as dscrete rando varables takng on fuzzy values wt te probabltes of te fuzzy states of te world. In [10] te autors sowed tat te ade s probabltes of fuzzy events lack te coon nterpretaton of a probablty easure. Anoter proble s a precse defnton of "te occurrence of te partcular fuzzy state of te world" (see te dscusson n ecton 3.3). Terefore an alternatve to ow te nforaton contaned n a fuzzy decson atrx can be treated was proposed n [8]. Te way s based on te dea tat a fuzzy decson atrx does not deterne dscrete fuzzy rando varables but a syste of fuzzy rule bases (a fuzzy rule base was ntroduced n [16]). However only te crsp (.e. not fuzzy) evaluatons of alternatves were consdered n [8] wc akes te proble uc spler. Te an a of te paper s to extend ts approac to te case were te evaluatons of alternatves are expressed by fuzzy nubers and to derve te forulas for correct coputatons of fuzzy expected values and fuzzy varances of evaluatons of alternatves. Te obtaned fuzzy caracterstcs wll be copared wt tose obtaned by te approac consderd n [1]. Te paper s organzed as follows. A decson atrx tool s brefly recalled n ecton. In ecton 3 te coon approac to te fuzzfcaton of a decson 86

3 Acta Polytecnca Hungarca Vol. 14 No atrx s analysed and te related probles are dscussed. Our new approac to ts proble s ntroduced and analysed n ecton 4. In ecton 5 bot approaces are copared by an llustratve exaple. Decson Matrces In ts secton let us descrbe a decson atrx as a tool for supportng a decson akng under rsk. Let us consder a probablty space (Ω A P) were Ω denotes a non-epty unversal set of all eleentary events A s a σ-algebra of subsets of Ω.e. A represents te set of all consdered rando events and P: A [01] denotes a probablty easure. Now let us descrbe a decson atrx under rsk consdered e.g. n [3] [4] [7] and [14]. Te decson atrx s sown n Table 1. In te atrx x 1 x x n represent te alternatves of a decson-aker 1 were A for 1 denote te utually dsont states of te world.e. k for any k {1 } k and p 1 p p stand for te 1 Ω probabltes of te states of te world 1.e. p P( ) and for any {1 n} and {1 } eans te decson-aker's evaluaton f e/se cooses te alternatve x and te state of te world occurs. Te evaluaton of te alternatve x s coonly understood as a dscrete rando varable H : { 1 } R wc takes on te values H ( ) wt te probabltes p 1. Table 1 Crsp decson atrx 1 p1 p p x EH1 var H1 x 1 EH var H xn n1 n n EHn var Hn Te alternatves are usually copared on te bass of te expected values and te varances of ter evaluatons (an overvew of decson akng rules can be found e.g. n [Error! Reference source not found.]). Te expected values of te decson-aker's evaluatons denoted by EH 1 EH EH n are gven for any {1 n} by: 87

4 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces EH p. (1) Te varances of te decson-aker's evaluatons denoted by var H 1 var H var H n are calculated for any {1 n} as follows: var H p EH ). ( () Te alternatve tat axzes te expected evaluaton and nzes te varance of te evaluaton s selected as te best one. 3 Fuzzy Decson Matrces Now let us descrbe te coon approac to te generalzaton of a decson atrx to te case were te states of te world and te evaluatons of te alternatves are expressed by fuzzy sets consdered e.g. n [Error! Reference source not found.]. Wtn ts approac te probabltes of te fuzzy states of te world coputed by te forula proposed by ade n [17] are used for coputatons of te caracterstcs of te evaluatons of te alternatves. 3.1 Fuzzy tates of te World Vaguely defned states of te world can be ateatcally expressed by fuzzy sets. A fuzzy set A on a non-epty set Ω s deterned by ts ebersp functon A: Ω [0 1]. Let us denote te faly of all fuzzy sets on Ω by F(Ω). A support of A and a core of A are gven as upp A : ω Ω A(ω) > 0 and Core A : ω Ω A (ω) 1 respectvely. A eans an -cut of A.e. A : ω Ω A(ω) for any (01]. Reark Any crsp set A Ω can be seen as a fuzzy set A F(Ω) of a specal knd were ts caracterstc functon χ A concdes wt te ebersp functon A of te fuzzy set. In fuzzy odels ts conventon allows us to consder also precsely descrbed events gven by crsp sets. In fuzzy decson atrces fuzzy states of te world are descrbed by te fuzzy events. Accordng to ade [17] a fuzzy event A F (Ω) s a fuzzy set wose - cuts are rando events.e. A A for all (01]. As an analogy to a decoposton of te unversal set Ω by crsp states of te world te fuzzy states of te world denoted by 1 ave to for a fuzzy partton of te unversal set Ω.e. for any ω Ω t as to old tat 88

5 Acta Polytecnca Hungarca Vol. 14 No (3) ade [17] extended te crsp probablty easure P to te case of fuzzy events. Let us denote ts extended easure by P. A probablty P (A) of a fuzzy event A s defned as follows: P A E ω A. : dp (4) A 3. Fuzzy Evaluatons of Alternatves under te Partcular Fuzzy tates of te World As was entoned n Introducton t can be dffcult for a decson-aker to evaluate eac alternatve under eac state of te world by a real nuber. One reason can be a lack of nforaton caused e.g. by naccuraces of easureents or a lower qualty of data transssons. Anoter reason can be tat t s ore natural for te decson-aker to descrbe te evaluatons lngustcally rater tan by nubers. Lngustc ters or uncertan quanttes can be ateatcally odeled by fuzzy nubers. A fuzzy nuber A s a fuzzy set on te set of all real nubers R suc tat ts core A s non-epty ts -cuts A are closed ntervals for any (0 1] and ts support upp A s bounded. Te faly of all fuzzy nubers on R wll be denoted by F N(R). In soe odels fuzzy evaluatons can be restrcted only to a closed nterval ostly [01]. A fuzzy nuber defned on te nterval [a b] s a fuzzy nuber wose -cuts belong to te nterval [a b] for all (01]. Te faly of all fuzzy nubers on te nterval [a b] wll be denoted by F N([a b]). Tus tere are two ways of expressng a fuzzy evaluaton of an alternatve. Te frst way s to specfy te evaluaton drectly by a fuzzy nuber. For nstance soe expert can evaluate te partcular alternatve drectly by te fuzzy nuber "about fve percent proft" wose ebersp functon s sown n Fgure 1. Te second possblty of expressng te fuzzy evaluaton of te alternatve conssts n te fact tat te evaluaton s odeled by a lngustc varable (lngustc varables were ntroduced n [16]). A decson-aker evaluates te alternatves under te partcular states of te world by approprate lngustc ters wose ateatcal eanngs are descrbed by fuzzy nubers. A set of lngustc ters T 1 T T r fors a lngustc scale on [a b] f T 1 T T r F N([a b]) representng ter ateatcal eanngs for a fuzzy partton of [a b]. 89

6 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces Fgure 1 Exaple of an expertly specfed evaluaton Exaple Let us consder a lngustc scale sown n Fgure. Ts scale s fored by te lngustc ters "a bg loss" (BL) "a sall loss" (L) "approxately wtout proft" (AWP) "a sall proft" (P) and "a bg proft" (BP). In soe cases a selecton of soe lngustcally descrbed value lke "a sall proft" fro te gven lngustc scale can be ore convenent for a decson-aker. Fgure Exaple of a lngustc scale 3.3 Coon Approac to a Fuzzy Decson Matrx Let us descrbe a coon approac to a fuzzy decson atrx tat was consdered e.g. n [Error! Reference source not found.]. In te fuzzy decson atrx gven n Table x 1 x x n denote te alternatves of te decson-aker and 1 stand for te fuzzy states of te world. Probabltes of te fuzzy states of te world 1 calculated accordng to (4) are denoted by p 1 p p.e. p P ( ). For any {1 n} and {1 } H expresses te fuzzy evaluaton of te alternatve x under te fuzzy state of te world. 90

7 Acta Polytecnca Hungarca Vol. 14 No p 1 p Table Fuzzy decson atrx p x 1 H 11 H 1 H 1 EH 1 var H 1 x H 1 H H EH var H x n H n1 H n H n EH n var H n Tus te evaluaton of te alternatve x s understood as a dscrete fuzzy rando varable H : { 1 } F N(R) were H ( ) H for 1. Its fuzzy expected value denoted by EH s coputed accordng to te generalzed forula (1) were te probabltes p of te states of te world are replaced by te ade's probabltes p of te fuzzy states of te world and te crsp evaluatons are replaced by te fuzzy evaluatons H.e. EH (5) p H L U Te -cuts EH E E L U H by E and E L U p p. are obtaned for all (01] as follows: Let 1. Te boundary values of L EH are obtaned (6) U. (7) Coputaton of te fuzzy varance var H s ore coplex. It was sown n [9] tat te forulas proposed n [Error! Reference source not found.] were not correct because te relatonsps between te fuzzy evaluatons H 1 H H and te fuzzy expected evaluaton EH were not nvolved n te calculaton. Ts fact causes tat te uncertanty of te resultng fuzzy varance s falsely ncreased. Te proper forulas for te coputaton of te fuzzy varance were proposed n [9]. For any {1... n} and any (01] te -cut of te fuzzy varance L U var H var var as to be calculated as follows: Let us denote 1 k k. k 1... p p z (8) 91

8 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces Ten L var n z 1... H 1... and (9) U var ax z 1... H (10) As t s wrtten n secton te eleent of te atrx gven n table 1 descrbes te decson-aker's evaluaton of te alternatve x f te state of te world occurs. If we consder te fuzzy states of te world nstead of te crsp ones a natural queston arses: Wat does t ean to say "f te fuzzy state of te world occurs"? Let us suppose tat soe ω Ω as occurred. If 1 ten t s clear tat te evaluaton of te alternatve x s exactly. However wat s te 0 (wc also eans tat 0 1 for soe evaluaton of x f 1 k )? Tus peraps t s not approprate n te case of a decson atrx wt te fuzzy states of te world to treat te evaluaton of x as a dscrete rando varable H tat takes on te fuzzy values H 1 H H. Moreover t was ponted out by Rotterová and Pavlačka [10] tat te ade s probabltes p 1 p p of te fuzzy states of te world express te expected ebersp degrees n wc te partcular states of te world wll occur. Tus tey do not ave n general te coon probablstc nterpretaton - a easure of a cance tat a gven event wll occur n te future wc s desrable n te case of a decson atrx. Terefore we cannot say tat te values EH 1 EH... EH n gven by (6) and (7) and var H 1 var H... var H n gven by (9) and (10) express te expected values and varances of evaluatons of te alternatves respectvely. Orderng of te alternatves based on tese caracterstcs s questonable. k 4 Fuzzy Rule Bases yste Deterned by te Fuzzy Decson Matrx In ts secton let us ntroduce a dfferent approac to te odel of decson akng under rsk descrbed by te decson atrx wt fuzzy states of te world presented n Table. Takng te probles dscussed n te prevous secton nto account we suggest not to treat te evaluaton of te t alternatve x {1 n} as a dscrete rando varable H takng on te fuzzy values H 1 H... H wt te probabltes p 1 p p. Instead of ts we propose to understand te nforaton about te evaluaton of te alternatve x as te followng fuzzy rule base: 9

9 Acta Polytecnca Hungarca Vol. 14 No If te state of te world s 1 ten te evaluaton of x s H 1. If te state of te world s ten te evaluaton of x s H. (11) If te state of te world s ten te evaluaton of x s H. In [8] t was sown tat n te case of te fuzzy decson atrx wt crsp evaluatons under te partcular fuzzy states of te world t s approprate to use te ugeno s etod of fuzzy nference ntroduced n [11]. Te obtaned output fro te fuzzy rule base was expressed by a real nuber. In te paper we deal wt te fuzzy evaluatons of te alternatves under te fuzzy states of te world. Tus te so-called generalsed ugeno s etod of fuzzy nference ntroduced n [13] sould be appled for obtanng an output fro te fuzzy rule base (11). Accordng to ts etod te evaluaton of an alternatve x for a gven ω Ω s coputed n te followng way: H ω ω H ω ω H L U For any (01] let us denote H L U H. (1) 1 and H are. Ten te boundary values of coputed as follows: L and ω ω L U U ω ω. Reark In te forula (1) te denonator equals to one due to te assupton tat te fuzzy states of te world 1 for a fuzzy partton of Ω. It s wort to note tat n our approac ts assupton can be otted. nce we operate wtn te gven probablty space (Ω A P) H s a fuzzy rando varable suc tat H : Ω F N(R). Reark It can be easly seen fro (1) tat n te case of te crsp states of te world 1 and te crsp evaluatons 1 n under te partcular fuzzy states of te world te fuzzy rando varables H concde wt dscrete rando varables H takng on te values wt te probabltes p 93

10 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces 1. Tus ts new approac can be seen as an extenson of a decson atrx to te case of te fuzzy states of te world and te fuzzy evaluatons of alternatves were approprate. Analogously as n te coon approac to te fuzzy decson atrx te orderng of te alternatves x 1 x x n can be based on te fuzzy expected values and te fuzzy varances of te rando varables H 1 n. Let us ntroduce te forulas for coputatons of te -cuts of EH and var H. For any (01] te -cut of te fuzzy expected output fro te fuzzy rule base L U gven by (11) denoted by EH E E s obtaned as follows: L E n ω dp H 1... ω L dp (13) and E U ax U ω dp. ω dp L U Te -cut var var H 1... var H of te fuzzy varance of te output fro te fuzzy rule base s obtaned as follows: Let us denote s 1... (ω) t 1 k 1 Ten k (t) k dp dp. L var n s 1... H 1... and (14) (15) (16) U var ax s 1... H (17) Now let us copare te fuzzy expected values EH and EH and te fuzzy varances var H and var H. 94

11 Acta Polytecnca Hungarca Vol. 14 No Teore 1 For 1 n te expected fuzzy evaluaton EH and te expected output fro te fuzzy rule base EH concde. L U Proof For any (01] let EH E E L U expected output fro te fuzzy rule base and E E be te -cut of te EH of te fuzzy expected evaluaton. For te boundary values of E and E L U p p L ω dp L U E L U ω dp E U. be te -cut ω dp Tus all te -cuts are te sae. Terefore EH EH. ω dp EH L U t olds: In [8] te autors sowed tat n te case of a fuzzy decson atrx were te evaluatons under te partcular fuzzy states of te world are expressed by real nubers te varances var H and var H are real nubers as well and var H var H. Now let us copare te fuzzy varances var H and var H. Teore For 1 n te fuzzy varance var H of te fuzzy evaluaton s greater or equals to te fuzzy varance var H of te output fro te fuzzy rule base (11). Proof Let and z 1... s be te auxlary functons 1... defned by (8) and (15) respectvely. For te sake of splcty let us denote for a gven 1 H E p ω We can express te dfference of and follows: dp z s as 95

12 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces d +... z... s... 1 E E p + E E (ω) (ω) (ω) ( )dp ( ) ( )dp ( ) dp 1 E (ω) E dp E dp dp E (ω) + ( ) E ( ) dp E (ω) ( ) 1 dp E dp + dp ( ) ( ) dp E dp dp dp were relatons (3) (13) (14) and te followng relaton fro easure teory: dp P 1 were appled. Te ntegrand (ω) ( ) s clearly non-negatve (t represents te varance of a dscrete rando varable tat takes on te values 1 wt te "probabltes" () 1 ). It s equal to 96

13 Acta Polytecnca Hungarca Vol. 14 No zero f and only f k for any k suc tat bot p and p k are postve. Tus te functon... s always non-negatve. d 1 However d 1... s te auxlary functon for coputaton of te fuzzy dfference D between var H and var H. For any (01] te -cut of L U te fuzzy dfference d d s gven as follows: D L d n d 1... H 1... and U d ax d 1... H Due to te non-negatvty of te auxlary functon d te -cut of te fuzzy dfference D contans only non-negatve values.e. var H var H. Hence var H var H. Tus altoug te fuzzy expected values EH and EH concde te fuzzy varances var H and var H dffer n general. Ts can affect te rankng of te consdered alternatves wc s llustrated by te exaple n ecton 5. Now let us focus on te nterpretaton of EH and var H. Bot caracterstcs descrbe a rando varable tat explans outputs fro te fuzzy rule base (11). Tere are no suc nterpretatonal probles as tose dscussed n te prevous secton. o ts approac sees to be ore approprate for te practcal use. 5 Illustratve Exaple Let us llustrate te dfference between bot descrbed approaces on te slar proble as was consdered n [9]. Let us copare two stocks A and B wt respect to ter future yelds. We consder te followng states of te econoy: "great econoc drop" (GD) "econoc drop" (D) "econoc stagnaton" () "econoc growt" (G) and "great econoc growt" (GG). Let us assue tat te consdered states of te econoy are gven only by te developent of te gross doestc product abbrevated as GDP. Furter we assue tat te next year predcton of GDP developent [%] sows a norally dstrbuted growt of GDP wt paraeters µ 1.5 and σ. 97

14 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces Fgure 3 Lngustc scale of te states of te econoy A consdered state of te econoy can be expressed by a trapezodal fuzzy nuber wc s deterned by ts sgnfcant values a 1 a a 3 and a 4 suc tat a 1 a a 3 a 4. Te ebersp functon of any trapezodal fuzzy nuber A F N(R) s for any x R n te for as follows: x a1 f x [ a1 a) a a1 1 f x a a3 A (x) a 4 x f x ( a3 a4 ] a4 a3 0 oterwse. Te trapezodal fuzzy nuber A deterned by ts sgnfcant values s denoted furter by (a 1 a a 3 a 4). Let us assue tat te states of te econoy are ateatcally expressed by trapezodal fuzzy nubers tat for a lngustc scale sown n Fgure 3. Moreover let us consder tat te predctons of future stock yelds (n %) are set expertly. gnfcant values of te fuzzy states of te econoy and of te fuzzy stock yelds are sown n Table 3. Te probabltes of te fuzzy states of te econoy were calculated accordng to te forula (4) and are used only n te calculaton of te caracterstcs of te output wt respect to te coon approac descrbed n ecton 3. 98

15 Acta Polytecnca Hungarca Vol. 14 No Table 3 Consdered fuzzy decson atrx Econoy states GD ( ) D ( ) Probabltes A yeld (%) B yeld (%) Econoy states ( ) G ( ) Probabltes A yeld (%) B yeld (%) Econoy states GG (3 4 ) Probabltes A yeld (%) B yeld (%) Te resultant fuzzy expected values and te fuzzy varances can be copared for nstance accordng to ter centers of gravty. Te center of gravty of a fuzzy nuber A F N(R) s a real nuber cog A gven as follows: cog A x A A x x dx. dx Te fuzzy expected values EA and EB coputed by te forulas (6) and (7) (or (13) and (14)) are trapezodal fuzzy nubers. Ter sgnfcant values are gven n Table 4. Te fuzzy varances var A and var B obtaned by te forulas (9) and (10) as well as var A and var B coputed by (16) and (17) are not trapezodal fuzzy nubers. Ter ebersp functons are sown n Fgures 4 and 5. Te sgnfcant values of te fuzzy varances are also gven n Table 4 (by tese sgnfcant values we understand end ponts of te core and of te closure of te support). We can see tat te fuzzy varances of te outputs fro te fuzzy rule bases reac lower values tan te varances obtaned by te coon approac. Fro te results gven n Table 4 t s obvous tat te center of gravty of te fuzzy expected value EA s greater tan te center of gravty of te fuzzy expected value EB. Terefore wtout consderng te varances te decson-aker sould prefer te stock A. 99

16 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces Table 4 Resultant stocks caracterstcs tock Caracterstc gnfcant Values (%) Centre of Gravty EA EB var A var B var A var B In ts exaple we can also see tat te cange n te fuzzy varance coputaton can cause a cange n te decson-aker s preferences. Based on var A and var B te decson-aker s not able to ake a decson on te bass of te rule of te expected value and te varance descrbed n ecton wle based on var A and var B te decson-aker sould prefer te stock A (te ger expected value and te lower varance tan te stock B copared on te bass of centers of gravty of varances approxated by trapezodal fuzzy nubers). Fgure 4 Mebersp functons of var A and var B Fgure 5 Mebersp functons of var A and var B 100

17 Acta Polytecnca Hungarca Vol. 14 No Conclusons We ave dealt wt te proble of te extenson of a decson atrx for te case of te fuzzy states of te world and te fuzzy evaluatons of te alternatves. We ave analyzed te coon approac to ts proble proposed n [Error! Reference source not found.] tat s based on applyng te ade's probabltes of te fuzzy states of te world. We ave found out tat te eanng of te obtaned caracterstcs of te evaluatons of te alternatves naely te fuzzy expected values and te fuzzy varances s questonable. Terefore we ave ntroduced a new approac tat s based on te dea tat a fuzzy decson atrx does not deterne dscrete fuzzy rando varables but fuzzy rule bases. In suc a case te obtaned caracterstcs of te evaluatons based on wc te alternatves are copared are clearly nterpretable. We ave proved tat te resultng expected values of te evaluatons are for bot approaces te sae wereas te varances generally dffer. In te nuercal exaple we ave sown tat te fnal orderng of te alternatves accordng to bot approaces can be dfferent. Future work n ts feld wll be focused on te case were te underlyng probablty easure s fuzzy. For nstance te paraeters of te underlyng probablty dstrbuton lke µ and σ n te case of te noral dstrbuton consdered n te nuercal exaple n ecton 5 could be expertly set wt fuzzy nubers. Acknowledgeent Ts work was supported by te proect No. GA of te Grant Agency of te Czec Republc Metods of Operatons Researc for Decson upport under Uncertanty and by te grant IGA_PrF_016_05 Mateatcal Models of te Internal Grant Agency of Palacký Unversty Oloouc. References [1] Cen. Hwang C.: Fuzzy Multple Attrbute Decson Makng Metods: Metods and Applcatons. prnger Berln Hedelberg 199 [] Ceng C.: A sple fuzzy group decson akng etod. In: Fuzzy ystes Conference Proceedngs IEEE Internatonal eoul out Korea [3] Ganouls J.: Engneerng Rsk Analyss of Water Polluton: Probabltes and Fuzzy ets. VCH Wene 008 [4] Huyn V. et. al.: A Fuzzy Target Based Model for Decson Makng Under Uncertanty. Fuzzy Optzaton and Decson Makng 6 3 (007) [5] Lao H. Xu.: Fuzzy Decson Makng Metodologes and Applcatons. prnger ngapore

18 P. Rotterová et al. New Approac How to Treat Fuzzy Decson Matrces [6] Matos M. A.: Decson under rsk as a ultcrtera proble. European Journal of Operatonal Researc 181 (007) [7] Özkan I. Türken I. B.: Uncertanty and fuzzy decsons. In: Caos Teory n Poltcs. Understandng Coplex ystes. (. Banereeet al. eds.) prnger Neterlands Dordrect [8] Pavlačka O. Rotterová P.: Fuzzy Decson Matrces Vewed as Fuzzy Rule-Based ystes. In: Proceedngs of te 34 t Internatonal Conference Mateatcal Metods n Econocs. (A. Kocourek M. Vavroušek eds.) Tecncal Unversty of Lberec Lberec [9] Rotterová P. Pavlačka O.: Coputng Fuzzy Varances of Evaluatons of Alternatves n Fuzzy Decson Matrces. In: Proceedngs of te 34 t Internatonal Conference Mateatcal Metods n Econocs. (A. Kocourek M. Vavroušek eds.) Tecncal Unversty of Lberec Lberec [10] Rotterová P. Pavlačka O.: Probabltes of Fuzzy Events and Ter Use n Decson Matrces. Internatonal Journal of Mateatcs n Operatonal Researc 9 4 (016) [11] ugeno M.: Industral applcatons of fuzzy control. Elsever cence Pub. Co. New York 1985 [1] Talašová J. Pavlačka O.: Fuzzy Probablty paces and Ter Applcatons n Decson Makng. Austran Journal of tatstcs 35 &3 (006) [13] Talašová J.: Fuzzy etods of ultple crtera evaluaton and decson akng (n Czec) Publsng House of Palacký Unversty Oloouc 003 [14] Yoon K. P. Hwang C.: Multple Attrbute Decson Makng: An Introducton. AGE Publcatons Calforna 1995 [15] ade L. A.: Fuzzy sets Inforaton and Control 8 3 (1965) [16] ade L. A.: Te Concept of a Lngustc Varable and ts Applcaton to Approxate Reasonng - I. Inforaton cences 8 (1975) [17] ade L. A.: Probablty Measures of Fuzzy Events. Journal of Mateatcal Analyss and Applcatons 3 (1968)