Fuzzy Multicriteria Decision Making 1

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1 БЪЛГАРСКА АКАДЕМИЯ НА НАУКИТЕ. BULGAIAN ACADEMY OF SCIENCES КИБЕРНЕТИКА И ИНФОРМАЦИОННИ ТЕХНОЛОГИИ, 1 CYBENETICS AND INFOMATION TECHNOLOGIES, 1 София Sofa Fuzzy Multcrtera Decson Makng 1 Vana Peneva, Ivan Popchev Insttute of Informaton Technologes, 1113 Sofa Abstract: The purpose of ths nvestgaton s drected towards summarzng and researchng of models for decson makng support n multcrtera problems under uncertantes from fuzzy type. The models smulate (approxmate) human decson makng by means of applyng of the fuzzy sets theory. Multcrtera fuzzy decson makng problems wll be consdered n cases of fuzzness present n ntal nformaton and the stages of problem s solutons, as well. Keywords: Multcrtera decson makng, fuzzy sets theory, fuzzy relatons, fuzzy numbers, aggregaton operators. I. Introducton Alternatves n decson makng problems are usually evaluated from dfferent ponts of vew, whch corresponds to partcular crtera. In real-lfe stuatons, evaluatons are nether certan nor precse. There are three man sources of uncertanty [50]: mprecson, because of the dffculty of determnng the scores of alternatves on partcular crtera; nterdetermnaton, snce the method of evaluaton results from a relatvely arbtrary choce from several possble defntons; uncertanty, snce the values nvolved vary n tme. Fuzzy sets and fuzzy logc are powerful mathematcal tools for modelng and controllng uncertan systems. They are facltators for approxmate reasonng n decson makng n the absence of complete and precse nformaton. Ther role s sgnfcant 1 Ths research work was supported by the project No Modelng of systems wth parameter uncertantes and analyss of multobjectve optmzaton problems of IIT BAS. 1 6

2 when appled to complex phenomena, whch are not easly descrbed by tradtonal mathematcs. Let A { a1, a2,..., a n } be a fnte set of alternatves, K j a crteron, c j the weght of the crteron j ( j 1,..., m). The purpose of the decson makng can be the choce, rankng or clusterng problem by comparng the alternatves. It has to take nto account the followng, when t makes ths: ther fuzzy performances on all crtera; the weghts attached to each crteron; the possble dffcultes of comparng two alternatves, when one s sgnfcantly better than the other on a subset of crtera, but much worse on at least one crteron from the complementary subset. The decson makng problems under uncertanty may be classfed nto two groups: ) The alternatve estmatons by crtera are crsp, but procedures for decson makng mtate the human behavor,.e. t uses the fuzzy sets theory; ) the crtera are fuzzy,.e. the alternatves estmatons are lngustcally varables and the decson makng may be realzed applyng tradtonal or fuzzy methods. By no doubt, the problems from the second group may be reduced to the ones from the frst group, f the lngustcally estmatons are transformed nto quanttatve ones. For example, the lngustcally varables may be represented as fuzzy numbers. Then a functon mappng of each fuzzy number on the real lne may be determned. 2. Fuzzy models of multcrtera decson makng by crsp crtera The problem s defned n the followng way: a fnte set of alternatves s evaluated from several nonfuzzy crtera (utlty functons, nonfuzzy orderngs). The alternatves have to be compared n such a way that solutons of the problems for: choce of a subset from the best, n some sense, alternatves; orderng over the whole set of alternatves; partton the set of alternatves of the subsets from the smlar, close ones,.e. partton from clusters, are obtaned. The nformaton about the alternatves can be suppled n dfferent scales. In ths case, t s requred to make the nformaton unform. One basc approach to make ths s to use a fuzzy relatons over the set of alternatves as the man element of unform representaton. Therefore, t needs some transformaton functons, whch defne the relatons between the couple of alternatves by each crteron. These functons defne relatons wth dfferent propertes, for example smlar or preference relaton. It s more realstc to use fuzzy relatons because they appear as a more convenent and adequate form for representng the relatonshp between the alternatves then crsp relatons. The fuzzy relatons may model stuatons, whenever nteractons between the alternatves are not exactly determned. Besdes that, they reflect the nterests of the experts or decson maker. The fuzzy relatons and ther propertes are nvestgated by many authors [18, 32, 37, 53, 58, 59, 61]. Takng ths nto account, the defned problem s solved n the followng three stages [11]: A. Unform stage. It derves an ndvdual fuzzy relaton for each crtera. Dfferent transformaton functons are used to do ths. B. Aggregaton stage. A purposeful approach for untng ndvdual fuzzy relatons s to use the aggregaton procedures that realze the dea of compensaton and compromse between conflctng crtera, when compensaton s allowed. Usng the concept of fuzzy majorty represented by a lngustc quantfer and applyng some 2 1 7

3 aggregaton procedure, an aggregated fuzzy relaton s obtaned from the ndvdual fuzzy relatons. C. Explotaton stage. The problems of choce, rankng or clusterng have to be solved n ths stage on the base of aggregated fuzzy relaton. A. Informaton unform stage A fuzzy multcrtera decson makng problem s nvestgated n [11], when the nformaton about the alternatves can be represented by means of nonfuzzy preference orderng, utlty functons and fuzzy preference relatons. The purpose s to establsh a general model for makng the nformaton unform, whch cover all possble representatons. Frstly, the relatonshp between the utlty values, gven on the base of a postve rato scale and fuzzy preference relatons s study. Let a k and a be the utlty values of the alternatves a and a j accordng to the crteron K k and these values belong to the nterval [0,1]. Then any possble transformaton functon h: [0, 1][0, 1] [0, 1], dependng only on the values a k and a presents a fuzzy k preference relaton,.e. pj h( ak, a ). Ths functon h must by a non-decreasng one n the frst argument and a non-ncreasng one n the second argument,.e. a k h(a k, a ) = l, where l s a nondecreasng functon. The functon h has to a verfy the followng propertes: (1) h(x, y) + h(y, x) = 1, x, y[0, 1], (2) h(x, x) = 0.5, h(x, 0) = 1, x[0, 1], (3) h(x, y) > 0.5 f x > y, x, y[0, 1]. The type of transformaton functons l are nvestgated and several examples of functons l are defned. Several fuzzy preference relatons derved from utlty values of the alternatves are gven n the sequel. Let [53] ak, a,, j 1,..., n, k 1,..., m, be postve ntegers. A fuzzy relaton on A for each crteron s defned as follows: 0 f ak > a ( ak, a ), 1 [ ak a ]/ a f ak a where ( a k, a ) s the membershp degree to the defned relaton and [ ak a ] s the remander, when a k s dvded by a. It s proved, that s reflexve, perfect antsymmetrcal and -transtve,.e. (4) ( x, x) 1, x [ 0, 1 ], (5) f ( x, y) 0, then ( y, x) 0, x, y [0,1], (6) ( x, z) max(0, ( x, y) ( y, z) 1), x, y, z [0,1]. A parwse preference approach whch permts a homogeneous treatment of dfferent knds of evaluatons s suggested n [49]. It supposes that greater a, k 1,..., n, corresponds to the better alternatve. The degree to whch the alternatve a s not worse then a j for the crteron k s defned wth the help of two thresholds: an ndfference 1 8

4 threshold IT[ a k ] and a preference one PT[ a ]. A membershp functon related to a fuzzy nterval (fuzzy number) a ~ k may be defned wth the help of these thresholds (PT>IT). By comparng of two fuzzy numbers the degree of credblty for the preference of a over a j for the crteron K k s obtaned. The related structure s called a fuzzy nterval order. Another transformaton functon s suggested n [41,43]. The degree of preference s defned as: k ( a, a j 1 ) a 0.5 2(max{ a k a } mn{ a }) k k f f The preference relaton gven by ths functon s reflexve (4), recprocal (1) and max-mn transtve,.e. (7) ( x, z) mn( ( x, y), ( y, z)), x, y, z [0,1]. Therefore, ths relaton s a fuzzy total orderng accordng to the defnton gven n [58]. In the case, when the nformaton about the alternatves by the crtera are nonfuzzy orderngs [11] O k, k 1,..., m, t s supposed that the lower the poston of an alternatve n a preference orderng s, the better alternatve and vce versa. It asserts the exstence of a transformaton functon f that assgns a credblty value of preference of any alternatve over any other one from any preference orderng, k k k pj f ( o ( ), o ( j)).ths transformaton functon must be a non-ncreasng one n the frst argument and a non-decreasng n the second one. The functon f has to satsfed the propertes (1), (2) and (3). The examples of transformaton functons f are gven as well. B. Aggregaton stage procedures The aggregaton of the ndvdual relatons at the second stage of the fuzzy models, consdered here, may be realzed wth the help of procedures, satsfyng the requrements mentoned above. These procedures may be performed by usng aggregaton (fuzzy logc) operators (FLOs). The frst ones ntroduced by L. Zadeh are for the logcal operatons AND, O and NOT as extensons of ther Boolean orgns. These operators are Mn, Max and 1 ( s the membershp degree to a gven fuzzy subset). Some research works reveal that the degree of compensaton through whch humans aggregate crtera s not expressed only by these operators. There exst some operators whch represent human decson makng more accurately. The Weghted Mean [3, 7, 11, 56, 57], Weghted Geometrc [12], Weghted MaxMn and Weghted MnMax [21, 49] operators e.g., use the mportance of the crtera, gven as weghts. In order to decde a varety of phenomena n decson stuatons, several operators wth parameters are ntroduced. For example, such operators are MaxMn, MnAvg and Gamma [54, 62]. These parametrcal operators gve the possblty to the decson maker by means of the parameters values changes to take part n the process of decson makng. There are aggregaton operators whch are sutable for combnng scores n multcrtera evaluaton problems. These averagng operators compensate a bad score j, j. 1 9

5 for one crteron by a good one for another crteron. All these operators represent partcular cases of the Generalzed Mean operator [17, 25, 51]. A very good overvew of the aggregaton operators, by presentng the characterstcs, the advantages and dsadvantages of each operator and the relatons between them, s avalable n [17, 25]. There s a large range of operators, whch can be advantageously used n the confluence of fuzzy crtera. The choce of an operator for specfc applcaton depends on varous factors. In fact, some choces have to be made accordng to, e.g.: the mathematcal model of the operators; the propertes of the operators for decdng problems of rankng or choce, or clusterng of the alternatves set; the senstvty of the operators for small varatons of ther arguments. The dependence between the propertes of the aggregated relaton and the propertes of the ndvdual relatons by each fuzzy crteron for the above operators are proved n [38, 39, 40, 42, 43, 44, 47]. In [47] these connectons are summarzed and presented n a table. The senstvty of the operators wth respect to varatons n ther arguments s defned and computed n [46]. The lst of aggregaton operators ncludes the followng ones besdes: the ordered weghted average (OWA) operator as a generalzaton of the weghted mean and whch has as partcular case the operators Mn and Max; the dscrete fuzzy ntegrals Choquet and Sugeno. The Choquet ntegral generalzes the OWA operator, whle the Sugeno one generalzes the weghted maxmum and the weghted mnmum operators; the t-norms and the t-conorms, whch compute the ntersecton and unon (respectvely) of fuzzy sets; the unnorms, whch solve another problem connected wth the lack of full (downwards and upwards) renforcement. The complete references for these operators s gven n [17]. Contnuty of an operator s a genune property of practcal applcaton procedures n whch a small dfference n nput values cannot cause a bg dfference n the output values. The assocatvty on the other hand, models the ndependence of the aggregaton on the groupng of the nput values. An nvestgaton of contnuous assocatve aggregaton operators s made n [6]. One dfferent approach of aggregaton s nvestgated n [22]. Four classes of aggregaton procedures of ndvdual relatons wth weghts are examned. These procedures are based on the use of a valued mplcaton or complcaton. The exstence of a herarchy among them s showed. These results are appled for the defnton of two classes of smlarty measures between fuzzy sets, provdng n general a pessmstc degree and an optmstc degree of smlarty. C. Explotaton stage The aggregated degree to whch a s not worse then b obtaned at the end of the aggregaton process does not always present any orderng propertes (except reflexvty), havng n mnd mostly the max-mn transtvty (7). Then the aggregated graph G(A, ) (vertces correspond to the alternatves set A and valued arcs support the aggregated relaton ) cannot be nterpreted n terms of rankng or choce [23]. But the relaton may be transformed n such a way to obtan a modfed max-mn transtve 2 0

6 relaton. For example, every cut of s transtve n the crsp sense and corresponds to a quasorder, whch can be represented by Hasse dagram [18]. Other transtve relatons close to a gven preference relaton are gven n [49]. The general expresson for a reflexve (4) and transtve relaton (7) s defned n [52]. The man contrbuton of the noton of fuzzy preorder [18],.e. the relaton whch posses the propertes (4) and (6), conssts n a membershp degree proposal for a preference relaton wthout volatng the choce problem tself. Any antsymmetrzed fuzzy preorder relaton ',.e. the relaton wth propertes (4), (5), (6), s a fuzzy partal orderng. It s possble to represent ' as a trangular matrx. Due to perfect antsymmetry and transtvty, the graph correspondng to ths matrx has no cycle and cut of ' s a nonfuzzy partal orderng. The ways for solvng the rankng and choce problems are ntroduced n [49]. Fuzzy strct order relatons and the noton of ther reducton are defned n [8]. A necessary and suffcent condton s obtaned for the transtve closure of the reducton and some possble graph-theoretc sgnfcance s of the results are dscussed. Orlovsky s concept of decson makng on a fnte set of alternatves wth a fuzzy preference relaton s analyzed n [20, 29]. The applcaton of that concept for optmzaton of many decson problems s formulated and proved. Two quantfer guded choce degrees of alternatves are used n [11]: a domnance degree used to quantfy the domnance that one alternatve has over all the others, n fuzzy majorty sense, and a nondomnance degree, whch generalzes Orlovsky s nondomnated alternatve concept. The applcaton of the two choce degree can be carred out accordng to two dfferent selecton processes, a sequental selecton process and a conjuncton selecton process. A systematc study of fuzzy ordered sets and an ntrnsc fuzzy topology on them s gven n [53]. If the aggregated relaton s a smlarty or lkeness one, the problem of clusterng may be solved. A smlarty relaton over a fnte set of alternatves can be represented as a smlarty tree of a dendogram type [60], where each tree level represents an -cut of ths relaton. The set of elements on a specfc -level can be consdered as smlarty classes (fuzzy clusters) of -level. A method for comparson of fuzzy clusters s gven n [41]. The proposed algorthm s based on the asserton that the comparson between two fuzzy clusters can be made of comparng only fuzzy clusters -scores as t s proved n [48]. 3. Fuzzy models of multcrtera decson makng by fuzzy crtera The problem under consderaton s the followng: a fnte set of alternatves s evaluated by several fuzzy crtera, the estmatons beng fuzzy numbers or fuzzy relatons. The alternatves have to be compared n a way to have the problems of rankng, choce or clusterng solved. The case of fuzzy relatons s already consdered n secton 2, usng the stages B and C. That s why, only the case of fuzzy numbers wll be presented n ths secton. In ths case, the comparson between the alternatves conssts n the comparson of fuzzy numbers or m-tuple of them Monocrteral comparson of fuzzy numbers Dfferent methods for comparng or orderng of fuzzy numbers exst. They can be classfed accordng to two dfferent approaches: 2 1

7 usng a crsp relaton wth the help of rankng functon; usng a fuzzy relaton on the set of fuzzy number, computng a comparson ndex for each par of them Methods wth crsp relatons. Let a ~, 1,..., n, be n normal convex fuzzy subsets (fuzzy numbers),.e. a ~ { x, ( x)}, x I I, I [0,1], where ( x) s the membershp functon of the fuzzy number a ~. A smple method for rankng a ~ conssts n the defnng of a rankng functon F, mappng each fuzzy number onto the real lne, where a natural order exsts. Ths functon s such that f F( a~ ) ( ~ F a j ), then a~ ~ a j. Ths approach has been followed by several authors, e.g. n [1, 9, 10, 19, 20, 24, 28, 30, 41, 55]. It s a relatvely smple and easy one for applcaton, but t reduces the whole nformaton about the fuzzy number nto a real one. Several well known rankng functons are tested on selected examples of fuzzy numbers n [1, 28, 41] Methods wth fuzzy relatons. These methods are based on the dea, that the property to be greater (less) than a fuzzy number s a lngustc property and every decson maker handles such a property n a personal way, measurng t accordng to nternal and external factors. Many methods for comparng fuzzy numbers wth the help of fuzzy relatons have been proposed n the lterature [14, 15, 23, 29, 31, 32, 34, 58, and etc.]. Each method has ts own advantages and dsadvantages, hence t should be chosen for each partcular problem. An attempt for evaluatng these rankng methods s proposed n [58]. Four crtera for ths evaluaton are suggested: fuzzy preference representaton, ratonalty of fuzzy orderng, dstngushablty between fuzzy numbers and robustness by small changes n the membershp functon of the fuzzy number. Based on these crtera, two exstng rankng methods (Baas and Kwakernaak s, [28] and Nakamuras [32]) are evaluated Comparson of m-tuple of fuzzy numbers Let a m-tuple of fuzzy numbers corresponds to gven alternatve from the set A, whch s the evaluaton of ths alternatve by the m fuzzy crtera. These m-tuple have to be compared to solve the problems of rankng, choce or clusterng of the set of alternatves. The exstng approaches to solvng ths problem may by classfed nto two groups: usng dstances between the m-tuples of fuzzy numbers; aggregatng every m-tuple of fuzzy numbers by each alternatve to a fuzzy number and then comparng the aggregated fuzzy numbers Comparng by dstances. The decson maker often has suppostons about the best or the worst alternatve,.e. the upper or lower horzon. Hence, t seems to be qute natural to order alternatves accordng to the dstance between each alternatve and the fxed horzon. The famly of dstances on the space of all trapezodal fuzzy numbers treated as elements of four-dmensonal space are ntroduced and nvestgated n [27]. These dstances, dependng on a parameter, generate the class of lnear orderngs under fxed upper horzon. It s proved, that such an orderng does not depend on the chosen horzon. 2 2

8 A development and a generalzaton of the above method s presented n [26]. The two classes of metrcs ntroduced provde a possblty to extend the applcaton of ths approach not only to trapezodal fuzzy numbers, but to left-sded or rght-sded fuzzy numbers Aggregatng of m-tuple of fuzzy numbers. The dea for aggregatng a m-tuple of fuzzy numbers correspondng to an alternatve s appled n [5]. These m fuzzy numbers are aggregated usng fuzzy arthmetc nto fuzzy weghts ~ w, where ~ w s the fuzzy rankng assgned to alternatve a. In [45] the aggregaton of the sequences of fuzzy numbers, representng the alternatves, s done wth the help of fuzzy logc (aggregaton) operators n such a way that the aggregated evaluatons be fuzzy numbers, as well. The methods for gettng the aggregated fuzzy numbers are dfferent dependng on the selected operator and the type of fuzzy numbers. The arthmetc operatons between fuzzy numbers and operatons Max and Mn are used for the computatons of the aggregated fuzzy numbers. A new method mplementng the operatons Mn and Max over sets of fuzzy number s proposed n [13]. The methods from secton 3.1 may be used for comparson of the aggregated fuzzy numbers to solve the choce, rankng or clusterng problems of the alternatves after that. 4. Concludng remarks The purpose of ths nvestgaton s drected towards researchng the models for decson makng support n multcrtera problems under uncertantes from fuzzy type. The models have to smulate (approxmate) human decson makng by means of applyng one of the basc elements of soft computng fuzzy logc and more precsely the fuzzy sets theory. Multcrtera fuzzy decson makng problems are consdered n cases of fuzzness present n ntal nformaton and at the stages of problem s solutons, as well. Our nvestgatons and results solvng these problems are presented, too. Some numercal examples decdng the suggested here problems and usng the methods proposed from us, are gven n [41, 43, 48, 47]. e f e r e n c e s 1. B o r t o l a n, G.,. D e g a n. A revew of some methods for rankng fuzzy subsets. Fuzzy sets and systems, 15, 1985, B o u c h o n-m e u n e r, B. Aggregaton and fuson of mperfect nformaton. Hedelberg, Germany, Physca Verlag, B r o w n, T. N., E. H. M a m d a n. A new fuzzy weghted average algorthm. In: Proc. of QUADET 93, Barcelona, June 16-18, 1993, B u c k l e y, J. J. ankng alternatves usng fuzzy numbers. Fuzzy Sets and Systems, 15, 1985, B u c k l y, J. J. A fuzzy rankng of fuzzy numbers. Fuzzy Sets and Systems, 33, 1989, C a l v o, T.,. M e s a r. Contnuous generated assocatve aggregaton operators. Fuzzy Sets and Systems, 126, 2002, No 2, C a r l s o n C.,. F l l e r. A new look at lngustc mportance weghted aggregaton. In: Proc. of XIV EMCS 98, 1 (.Trappl, ed. ). Venna, Apr.14-17, 1998, C h a k r a b o r t y, M. K., M. D a s. educton of fuzzy strct order relaton. Fuzzy Sets and Systems, 15, 1985,

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10 34. O r l o v s k, S. A. Fuzzy relatons on a set of objects evaluated n dfferent scales. In: Proc. of the 11th EMCS 92 (.Trappl, ed.). Vena, Austra, Apr , 1992, O v c h n n k o v, S., M. o u b e n s. On strct preference relaton. Int. J. Fuzzy Sets and Systems, 43, 1991, O v c h n n k o v, S., M. o u b e n s. On fuzzy strct preference, ndfference and ncomparablty relatons. Int. J. Fuzzy Sets and Systems, 17, 1992, O v c h n n k o v, S. Numercal representaton of transtve fuzzy relatons. Fuzzy Sets and Systems, 126, 2002, No 2, P e n e v a, V. G., I. P o p c h e v. Fuzzy relatons n decson makng. In: Proc. of FUZZ- IEEE 96, Int. Conf. on Fuzzy Systems, New Orleans, LA, Sept. 8-11, 1996, P e n e v a, V. G., I. P o p c h e v. Fuzzy orderng on the base of multcrtera aggregaton. Cybernetcs and Systems, 29, 1998, No 6, P e n e v a, V., I. P o p c h e v. Aggregaton of fuzzy relatons. Compt. end. Acad. Bulg. Sc., 51, 1998, No 9-10, P e n e v a, V., I. P o p c h e v. Comparson of cluster from fuzzy numbers. Fuzzy Sets and Systems, 97, 1998, No 1, P e n e v a, V., I. P o p c h e v. Decson makng wth fuzzy relatons. In: Proc. of the XIV EMCS 98 (.Trappl, ed.), Venna, Austra, Apr , I, 1998, P e n e v a, V. G., I. P o p c h e v. Fuzzy logc operators n decson makng. Cybernetcs and Systems, 30, 1999, No 8, P e n e v a, V., I. P o p c h e v. Aggregaton of fuzzy relatons n multcrtera decson makng. Compt. end. Acad. Bulg. Sc., 54, 2001, No 4, P e n e v a, V., I. P o p c h e v. Aggregaton of fuzzy numbers n a decson makng stuaton. Cybernetcs and Systems, 32, 2001, No 8, P e n e v a, V., I. P o p c h e v. Senstvty of fuzzy logc operators. Compt. end. Acad. Bulg. Sc., 55, 2001, No 3, P e n e v a, V., I. P o p c h e v. Propertes of the aggregaton operators related wth fuzzy relatons. Fuzzy Sets and Systems (submtted). 48. P o p c h e v, I., V. P e n e v a. An algorthm for comparson of fuzzy sets. Fuzzy Sets and Systems, 60, 1993, No 1, o u b e n s, M. Fuzzy sets n preference modelng n decson analyss. In: Proc. of VI IFSA World Congress, Sao Paulo, Brazl, 1995, o y, B. Man sources of naccurate determnaton, uncertanty and mprecson n decson models. Mathematcal and Computer Modelng, 12, 1989, No 10-11, S o u s a, J., U. K a y m a k. Model predcatve control usng fuzzy decson functon. IEEE Trans. on SMC-Part B: Cybernetcs, 31, 2001, No 1, V a l v e r d e, L. On the structure of F-ndstngushablty operators. Fuzzy Sets and Systems, 17, 1985, V e n u g o p a l a n, P. Fuzzy ordered sets. Fuzzy Sets and Systems, 46, 1992, V o n A l t r o c k, C. Fuzzy Logc and Neuro Fuzzy Applcatons n Busness and Fnance. Englewood Clffs, New Jork, Prentce Hall, Y a g e r,. A procedure for orderng fuzzy subsets on unt nterval. Inf. Sc., 24, 1981, Y a g e r,. On weghted medan aggregaton. Uncertanty, Fuzzness and Knowledge-Based Systems, 2, 1994, No 1, Y a g e r,. On mean type aggregaton. IEEE Trans. on SMC. Part B, 26, 1996, No 2, Y u a n, Y. Crtera for evaluatng fuzzy rankng methods. Fuzzy Sets and Systems, 44, 1991, Z a d e h, L. A. Smlarty relatons and fuzzy orderngs. Inform. Sc., 3, 1971, Z m m e r m a n n, H. -J. Fuzzy Sets, Decson Makng and Expert Systems. Boston, Kluwer Academc Publshers, Z m m e r m a n n, H. -J. Fuzzy Set Theory and Its Applcatons. Noewell, MA, Kluwer Academc Publshers, Z m m e r m a n n, H. -J., P. Z y s n o. Decsons and evaluatons by herarchcal aggregaton of nformaton. Fuzzy Sets and Systems, 10, 1983,

11 Размити многокритериални задачи за вземане на решения Ваня Пенева, Иван Попчев Институт по информационни технологии, 1113 София (Р е з ю м е) Целта на изследването е свързана с моделите за подпомагане вземането на решения при многокритериални задачи в условия на неопределеност от размит вид. Тези модели симулират (апроксимират) вземането на решения от човека, като за целта се използва един от основните елементи на soft computng размитата логика, а в по-широк аспект теорията на размитите множества. Изследвани са многокритериални задачи за вземане на решения, при които размитостта е както в началната информация, така и при етапите от решаването на задачите. Моделите са класифицирани в зависимост от началната информация и от методите за тяхното решаване. Показан е и приносът на авторите при разработване на модели за размито многокритериално вземане на решения. 2 6