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... Need for Professional Development. Vol. 27, No. 1 (March 1995), 121.Subject Index 1992–1996Algorithms and Data StructuresGupta, Smolka, and Bhaskar March 1994Matousek ...

Author Name:
Subject index

Title:
Subject index

AKA names:
Subject Index

Title:
Fuzzy sets, uncertainty, and information

Title:
Fuzzy set theory: foundations and applications

Abstract:
<p>Fuzzy Set Theory - And Its Applications, Third Edition is a textbook ... Its Applications, Third Edition is a textbook for courses in fuzzy set theory. It can also be used as an introduction to ... It can also be used as an introduction to the subject. . The character of a textbook is balanced with the ... expert systems, fuzzy control, fuzzy data analysis, decision making and fuzzy set models in operations research. All chapters have been updated. Exercises ...

Title:
Fuzzy Set Theory - and Its Applications

Abstract:
... use of conditions involving an aggregate function applying to a fuzzy set is not yet possible except for the cardinality (count) in ... aggregate functions (such as the maximum...) could be applied to fuzzy sets

References:
De Luca, A., & Termini, S. (1972). A definition of non-probabilistic entropy in the setting of fuzzy sets theory. Information Control, 20, 301-312.
Dubois, D., & Prade, H. (1985). A review of fuzzy set aggregation connectives. Information Sciences, 36, 85-121.
Dubois, D., & Prade, H. (1990). Measuring properties of fuzzy sets: a general technique and its use in fuzzy query evaluation. Fuzzy Sets and Systems, 38, 137-152.
Dubois, D., Prade, H., & Testemale, C. (1988). Weighted fuzzy pattern matching. Fuzzy Sets and Systems, 28, 315-331.
Grabisch, M., Murofushi, T., & Sugeno, M. (1992). Fuzzy measure of fuzzy events defined by fuzzy integrals. Fuzzy Sets and Systems, 50, 293-313.
Murofushi, T., & Sugeno, M. (1989). An interpretation of fuzzy measure and the Choquet integral as an integral with respect to fuzzy measure. Fuzzy Sets and Systems, 29, 201-227.
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338-353.

Keywords:
fuzzy sets

Abstract:
... That forecaster's knowledge is expressed as a set of 'rules-of-thumb'. Fuzzy set methodologies have proved well suited for encoding the forecaster's knowledge, ... inherent in the specification of rules, as well as in subjective and objective input. MEDEX uses fuzzy set theory in two ways: as a fuzzy rule base in ...

References:
Dubois D, Prade H, Yager RR (eds) (1993) Readings in fuzzy sets for intelligent systems. Morgan Kaufmann, San Mateo, CA
Hansen BK (2000) Analog forecasting of ceiling and visibility using fuzzy sets. In Proceedings of the Second conference on artificial intelligence. American Meteorological Society, Long Beach, CA
Zadeh L (1965) Fuzzy sets. Information and Control 8: 338-353
Tag P, Hadjimichael M, Brody L, Kuciauskas A (1996) Automating the subjective recognition of 500 mb wind patterns as input to a meteorological forecasting system. In Proceedings, 15th conference on weather analysis and forecasting, Norfolk, VA. American Meteorological Society, Boston, MA, pp 347-350

Keywords:
fuzzy sets
subjective variables

Title:
A meteorological fuzzy expert system incorporating subjective user input

Full Text:
... principle, where the sum of the sequence {ui }i?N of fuzzy sets is given by (??i=1 ui )(x) =sup{inf{ui (xi )}i?N , ... all seriesconverge, all three definitions are equivalent. The class of fuzzy sets with compact convex levels with some additionalrestriction (usually some form ... convex compact structures is involved [8,10,12,13]. Thew-sum is not the subject of our paper.In Section 4 we investigate the properties and ... the series. They can be usedin a wider class of fuzzy sets then the weak sum. Since there is no any convexity ... of nonempty (closed), (bounded), (weakly compact),M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 3117(compact), (convex) subsets of X ... 3115–3127 3117(compact), (convex) subsets of X ,F(X ) set of fuzzy sets defined on X with nonempty ?-levels,Fusc(X ) set of normal ... X with nonempty ?-levels,Fusc(X ) set of normal upper semicontinuous fuzzy sets defined on X ,F( f )(w f )(b)(wk)(k)(c)(X ) subset ... ? X ,IA characteristic (indicator) function of A,u? ?-level of fuzzy set u,? in index always denotes ?-level of fuzzy set, ,l1 l1 = {{xn}n?N ? RN :??n=1 |xn| < ?},l2 ...
... An).The mapping u : X ? [0, 1] is a fuzzy set on X . The set u? ? X , ? ... X , ? ? (0, 1], is the ?-level of fuzzy set u defined byu? = {x ? X : u(x)??}. Throughout ... index will always be used to denote the ?-level of fuzzyset. ... . By F(X ) we denote the set of all fuzzy sets with nonempty ?-levels defined on X . Let {A?}??(0,1] be ... P(X ). We say that the family {A?}??(0,1] defines a fuzzy set if there exists u ? F(X ) such that for ... Since the necessary and the sufficient condition for upper semicontinuityof fuzzy set u is the ?-levels u? to be closed, Fusc(X ) ... v ? F f,b(X ),i.e. u, v are upper semicontinuous fuzzy sets with bounded ?-levels, but in spite of that u + ... ?F(X ) and the family {cl(u? + v?)}??(0,1] defines a fuzzy set, , then we denote this fuzzy set by u.+ v.3118 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127In the next five lemmas we ... l-seriesLet {ui }i?N ? F(X ) be a sequence of fuzzy sets and let {sk}k?N ? F(X ) be such that sk ...
... of fuzzy setsFuzzy Sets and Systems 160 (2009) 3115–3127www.elsevier.com/locate/fssSeries of fuzzy sets? ?Mila Stojakovic a,?, Zoran Stojakovic baDepartment of Mathematics, Faculty of Engineering, University ... 2008; accepted 22 December 2008Available online 3 January 2009AbstractSeries of fuzzy sets with weakly closed, weakly compact or compact ?-levels are considered. ... Thesubject of this paper is investigation of infinite addition of fuzzy sets and condition when the sum of ?-levels is equal to ... the same authors in Stojakovic andStojakovic [Addition and series of fuzzy sets, , Fuzzy Sets and Systems 83 (1996) 341–346] for the series with compact ... ?-levels. 2009 Elsevier B.V. All rights reserved.Keywords: Addition; Series of fuzzy sets; ; Banach space1. IntroductionIn recent years there has been an ... sets.Addition is a basic operation defined on the set of fuzzy sets which plays a crucial role in measure theory, theory ofintegration, ... the extension principle given by Zadeh [15], the addition of fuzzy sets u, v : X ? [0, 1]is defined by (u ... (u + v)? = u? + v?,the functional approach in fuzzy set addition can be replaced by set addition. It makes possible ... valued calculus to solve the problems related to addition of fuzzy sets. . This important, very useful property ispreserved only for some ... important, very useful property ispreserved only for some classes of fuzzy sets. . Namely, in [9] it was proved that (u + ... applied in the set of fuzzy numbers (normal upper semicontinuous fuzzy sets u : R ? [0, 1] withcompact support), then, using ... all these papers the addition isapplied on simplest form of fuzzy sets— —fuzzy numbers. In Example 2 it is shown that ?-level ... Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2008.12.0133116 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127sets can be different from sum ... sum of ?-levels even when addends are upper semicontinuous normal fuzzy sets and havebounded, closed ?-levels. But in this example the basic ... is one more disadvantageof definition of addition—some important classes of fuzzy sets are not closed with respect to this operation. Example1 shows ... operation. Example1 shows that the sum of two upper semicontinuous fuzzy sets is not upper semicontinuous even when basic space isX = ... F(X ) and the family {cl(u? + v?)}??(0,1] defines a fuzzy set, , then we denote this fuzzy set by u.+ v and callit an l-sum of u and ... we show that they are equivalent for the set of fuzzy sets with compact or weakly compact?-levels. Also, some properties of both ... be developed without the infinite sum. The definition of additionof fuzzy sets can be generalized on infinite sum in different ways. Depending ...
... of the series?ui , denoted by??i=1 ui , is a fuzzy set defined by( ??i=1ui)(x) = sup{inf{ui (xi )} : x =??i=1xi ... =??i=1xi (uncond.)}, x ? X .M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 3119It is interesting to note ... {un}n?N ? F f (X ) be a sequence of fuzzy sets such that the sequence of the related partial l-sums.sk= ?kn=1 ... for every n?k, there exists3120 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127the sequence {xni }i?N,??i=1 xni = ...
... = ?ni=1 ui.+ ??i=n+1 ui .M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 3121Proof. To prove the first ... sequencewhich implies both relations. ?3122 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127The next two theorems have the ...
... x /???i=1(ui )?. By closeness ofM. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 3123??i=1(ui )?, d(x,??i=1(ui )?) = ... {??i=1(ui )??1/n}n?{k,k+1,. . .}. Since3124 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127{??i=1(ui )??1/n}n?{k,k+1,. . .} is a ...
... same as in Theorem 14. ?M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 31255. ExamplesExample 1. Let X ... /? P f (R) and u + v /?Fusc(R). The fuzzy set (u.+ v)(x) = 1, for all x ? cl(u1 + ... dimensional, separable reflexive Banach space (l2 is aHilbert space also). Fuzzy sets u and v are upper semicontinuous with bounded ?-levels (u, ... means that the family {cl(u? + v?)}??(0,1] does notdefines a fuzzy set. . So, u.+ v does not exist.Example 3. HereX = ... degeneracy of the infinite sum which will be a trivial fuzzy set. . In the first case ui (1) = 1 and ... l1, which is an infinite dimensional nonreflexive separable Banach space. Fuzzy sets uk , k ? N, with closed,bounded (but not weakly ... ? l1{2?n?1e1,?2?ne2,?2?ne3, . . .},3126 M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127u2n(x) =???????????1, x = 2?n?1e1,1 ? ...
... l2, which is an infinite dimensional reflexive separable Banach space. Fuzzy sets uk , k ? N, aredefined as in the last ... =??i=1sup??(0,1](1 ? ?)i =??i=11 = ?.M. Stojakovic , Z. Stojakovic / Fuzzy Sets and Systems 160 (2009) 3115–3127 3127Further, we shall show that ... note on the law of large numbers for fuzzy variables, Fuzzy Sets and Systems 55 (1993) 235–236.[4] D.H. Hong, Y.M. Kim, A ... of large numbers for fuzzy numbers in a Banach space, Fuzzy Sets and Systems 77 (1996) 349–354.[5] D.H. Hong, P.I. Ro, The ... law of large numbers for fuzzy numbers with unbounded supports, Fuzzy Sets and Systems 116 (2000) 269–274.[6] D.H. Hong, C.H. Ahn, Equivalent ... for laws of large numbers for T-related L-R fuzzy numbers, Fuzzy Sets and Systems 136 (2003)387–395.[7] E. Klein, A. Thompson, Theory of ... 331–346.[9] H.T. Nguyen, A note on the extension principle for fuzzy sets, , J. Math. Anal. Appl. 64 (1978) 369–380.[10] H. Roman-Flores, ... 64 (1978) 369–380.[10] H. Roman-Flores, M. Rojas-Medar, Embedding of level-continuous fuzzy sets on Banach spaces, Inform. Sci. 144 (2002) 227–247.[11] M. Stojakovic , ... (2002) 227–247.[11] M. Stojakovic , Z. Stojakovic , Addition and series of fuzzy sets, , Fuzzy Sets ... and Systems 83 (1996) 341–346.[12] M. Stojakovic , Fuzzy valued measure, Fuzzy Sets and Systems 65 (1994) 95–104.[13] P. Teran, An embedding theorem ... 65 (1994) 95–104.[13] P. Teran, An embedding theorem for convex fuzzy sets, , Fuzzy Sets and Systems 152 (2005) 191–208.[14] P. Teran, Strong law of ...
arithmetics, Fuzzy Sets and Systems 159 (2008) 343–360.[15] L.A. Zadeh, The concept of ...

Abstract:
Series of fuzzy sets with weakly closed, weakly compact or compact @a-levels are considered. ... @a-levels are considered. The basic space is Banach space. The subject of this paper is investigation of infinite addition of fuzzy sets and condition when the sum of @a-levels is equal to ... same authors in Stojakovic and Stojakovic [Addition and series of fuzzy sets, , Fuzzy Sets and Systems 83 (1996) 341-346] for the series with compact ...

References:
Fuller, R. and Triesch, E., A note on the law of large numbers for fuzzy variables. Fuzzy Sets and Systems. v55. 235-236.
Hong, D.H. and Kim, Y.M., A law of large numbers for fuzzy numbers in a Banach space. Fuzzy Sets and Systems. v77. 349-354.
Hong, D.H. and Ro, P.I., The law of large numbers for fuzzy numbers with unbounded supports. Fuzzy Sets and Systems. v116. 269-274.
Hong, D.H. and Ahn, C.H., Equivalent conditions for laws of large numbers for T-related L-R fuzzy numbers. Fuzzy Sets and Systems. v136. 387-395.
Nguyen, H.T., A note on the extension principle for fuzzy sets. J. Math. Anal. Appl. v64. 369-380.
Roman-Flores, H. and Rojas-Medar, M., Embedding of level-continuous fuzzy sets on Banach spaces. Inform. Sci. v144. 227-247.
Stojaković, M. and Stojaković, Z., Addition and series of fuzzy sets. Fuzzy Sets and Systems. v83. 341-346.
Stojaković, M., Fuzzy valued measure. Fuzzy Sets and Systems. v65. 95-104.
Teran, P., An embedding theorem for convex fuzzy sets. Fuzzy Sets and Systems. v152. 191-208.
Teran, P., Strong law of large numbers for t-normed arithmetics. Fuzzy Sets and Systems. v159. 343-360.

Keywords:
Series of fuzzy sets

Title:
Series of fuzzy sets

Title:
Calculation of subjective estimates

Full Text:
Categories with fuzzy sets and relationsAvailable online at www.sciencedirect.comFuzzy Sets and Systems 256 (2014) ... online at www.sciencedirect.comFuzzy Sets and Systems 256 (2014) 149–165www.elsevier.com/locate/fssCategories with fuzzy sets and relationsJohn Harding, Carol Walker?, Elbert WalkerDepartment of Mathematical Sciences, ... online 12 April 2013AbstractWe define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrichthis category with additional monoidal ... is made of extending theseresults to the setting of type-2 fuzzy sets. . 2013 Elsevier B.V. All rights reserved.Keywords: Fuzzy sets; ; Relations as morphisms; Biproducts; 2-Category; Category with involution; Symmetric ... morphisms; Biproducts; 2-Category; Category with involution; Symmetric monoidal tensor1. IntroductionA fuzzy set is a map A : X ? I from a ... reason, and its inherent interest, we widen thecategory FSet of fuzzy sets and functions to the category FRel of fuzzy sets and relations. Here the objects are fuzzysets as before, but a morphism between fuzzy sets A : X ? I and B : Y ? ... Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.fss.2013.04.004150 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165In particular, FRel has finite ... matter of extending the categorical setting to interval-valued and type-2 fuzzy sets. . The idea is toreplace I with the appropriate truth ... some other categorical generalizations of Rel.2. The category FRel of fuzzy sets and relationsIn this section, we define the categories of interest ... the categoryof primary interest, FRel.Definition 2.1. The category FRel of fuzzy sets and relations is defined as follows:1. An object is a ...
... FRel that play an interesting role.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 157Definition 4.9. Let Proj, ... Y ?X , and ?X : X ? A? I ,subject to certain coherence conditions [14, p. 157]. The bifunctor? is ... unit I = ({?}, 0).158 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165Proof. Let R : (X, ...
... We begin with the notion of a t-norm for type-2 fuzzysets. . The following is perhaps the most restrictive notion [19].Definition ... convex normal functions in M.162 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165In [19] it is shown ... it yields atensor on Rel(D). ?J. Harding et al. / Fuzzy Sets
... that FRel is a category.Definition 2.3. The category FSet of fuzzy sets and functions is the subcategory of FRel whose objects are ... the case R is a function.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 151Notation 2.4. We often ... to (Xi , Ai ).152 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165Suppose Ri : (Xi , ...
... X to be the converserelation R?.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 153Proposition 3.9. There is ...
... and epic are exactly the154 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165bijective correspondences, and for these ...
... replaced byepics (see [12] for details).J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 155Fig. 1. Functors relating ... : 2X ? 2Y .156 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165Theorem 4.3. The pair (F1, ...
... we come to our key notion.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 159Proposition 5.7. For a ...
... actual equalities in thiscase. ?160 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165We say t-norms T and ... related to FRel and FSet. These include extensions to interval-valued fuzzy sets [5]and type-2 fuzzy sets [19]. These can be viewed as extensions of our earlier ... consider instances of Rel(V ) arising from structures related to fuzzy sets. . We begin with interval-valuedfuzzy sets.Definition 6.3. Define I[2] = ... .For an account of the role of I[2] in interval-valued fuzzy sets see [5]. We only mention that it is a completelydistributive ... 1 ? a).Definition 6.4. Define the category IFRel of interval-valued fuzzy sets and relations to be Rel(I[2]), and let IFSet beSet(I[2]).A t-norm ... to be Rel(I[2]), and let IFSet beSet(I[2]).A t-norm for interval-valued fuzzy sets is a function T : I[2] I[2] ? I[2] ... and R‡ = R?.We next consider matters for the type-2 fuzzy sets introduced by Zadeh. While the reader should consult [19] for ... truth values for type-2 fuzzy sets.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 161Definition 6.6. The algebra ... –165 161Definition 6.6. The algebra of truth values for type-2 fuzzy sets isM = ([0, 1][0,1],unionsq,?,? , 0 , 1 )where the operations are ...
... appropriately modified domain and codomain.164 J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165So I provides access to ...
... (FRel,?T ) is not compact closed.J. Harding et al. / Fuzzy Sets and Systems 256 (2014) 149 –165 165Proof. Recall, the tensor ... IEEE Computer Society, New York, 2004, pp. 415–425.[2] M. Barr, Fuzzy set theory and topos theory, Bull. Can. Math. Soc. 29 (1986) ... Gehrke, C. Walker, E. Walker, Some comments on interval valued fuzzy sets, , Int. J. Intelligent Syst. 11 (1996) 751–759.[6] J. Goguen, ... , Int. J. Intelligent Syst. 11 (1996) 751–759.[6] J. Goguen, L-fuzzy sets, , J. Math. Anal. Appl. 18 (1) (1967) 145–174.[7] J. ... Harding, C. Walker, E. Walker, Lattices of convex normal functions, Fuzzy Sets Syst. 159 (2008) 1061–1071.[10] J. Harding, C. Walker, E. Walker, ... J. Harding, C. Walker, E. Walker, Convex normal functions revisited, Fuzzy Sets Syst. 161 (2010) 1343–1349.[11] J. Harding, C. Walker, E. Walker, ... (2010) 1343–1349.[11] J. Harding, C. Walker, E. Walker, Finite type-2 fuzzy sets, , manuscript.[12] H. Herrlich, G.E. Strecker, Category Theory, Sigma Series ... Heldermann Verlag, Berlin, 1979.[13] U. H hle, L. Stout, Foundations of fuzzy sets, , Fuzzy Sets Syst. 40 (1991) 257–296.[14] Saunders MacLane, Categories for the Working ... Dissertation, The University of Arkansas, 1969.[18] C. Walker, Categories of fuzzy sets, , Soft Comput. 8 (4) (2004) 299–304.[19] C. Walker, E. ... C. Walker, E. Walker, The algebra of fuzzy truth values, Fuzzy Sets Syst. 149 (2005) 309–347.[20] M. Winter, Goguen Categories, Springer, The ...

Abstract:
We define a 2-category whose objects are fuzzy sets and whose maps are relations subject to certain natural conditions. We enrich this category with additional ... made of extending these results to the setting of type-2 fuzzy sets.

References:
M. Barr, Fuzzy set theory and topos theory, Bull. Can. Math. Soc., 29 (1986) 501-508.
M. Gehrke, C. Walker, E. Walker, Some comments on interval valued fuzzy sets, Int. J. Intelligent Syst., 11 (1996) 751-759.
J. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967) 145-174.
J. Harding, C. Walker, E. Walker, Lattices of convex normal functions, Fuzzy Sets Syst., 159 (2008) 1061-1071.
J. Harding, C. Walker, E. Walker, Convex normal functions revisited, Fuzzy Sets Syst., 161 (2010) 1343-1349.
J. Harding, C. Walker, E. Walker, Finite type-2 fuzzy sets, manuscript.
U. Höhle, L. Stout, Foundations of fuzzy sets, Fuzzy Sets Syst., 40 (1991) 257-296.
C. Walker, Categories of fuzzy sets, Soft Comput., 8 (2004) 299-304.
C. Walker, E. Walker, The algebra of fuzzy truth values, Fuzzy Sets Syst., 149 (2005) 309-347.

Keywords:
Fuzzy sets

Title:
Categories with fuzzy sets and relations

Full Text:
... these uncertainties translate into uncertainties aboutfuzzy set membership functions. Type-1 fuzzy sets are not able toy more to modre the two set-2 ... and institutions.aper, a new multicriteria decision making (MCDM)d on Type-2 fuzzy sets is proposed to the current projectystem of regional development agencies ... a numerical value is not possible, linguis-s are used. The fuzzy set theory enables comparisonves by digitizing linguistic variables. A major contri-zzy ... to the increasing of the socio-economic environment and the vagueness subjective nature of human thinking. This fact has ledthors to apply ... of human thinking. This fact has ledthors to apply the fuzzy set theory [5] to model the and vagueness in decision processes ...
... because their membership func-tally crisp. On the other hand, type-2 fuzzy sets areel such uncertainties because their membership func-emselves fuzzy. Membership functions ... degrees of freedomt possible to directly model uncertainties. Membershipf Type-1 fuzzy sets are crisp sets. For this reason, in caseseanings of criteria ... type-1 fuzzyables convenient modeling of problem. If we can usee-2 fuzzy sets [11] for handling fuzzy group decision-blems, then there is room ... there is room for more ?exibility due tot interval type-2 fuzzy sets are more suitable to repre-ainties than type-1 fuzzy sets [2]. Type-2 fuzzy sets havesfully applied in decision making process. Erdogan androposed an integrated ... process. Erdogan androposed an integrated multi-criteria decision-makingethodology based on type-2 fuzzy sets for selectionrgy alternatives. Then they tried to de?ne a roadmap ... [14] proposed a multi criteria decision makingethodology based on type-2 fuzzy sets to manage quali-uantitative criteria with uncertainties for selecting thetive fuel ... on the ranking values and the arith-ations of interval type-2 fuzzy sets (IT2 FSs) and used to illustrate the fuzzy GDM process ... literature, in our study, AHP and TOP-s based on type-2 fuzzy sets due to nature of problembject of our study. Besides, in ... of this paper is organized as follows. In Section 2, inter-fuzzy sets are brie?y introduced. In Section 3, type-2ethod is brie?y reviewed. ... in Section 4. The proposed decision making method- on type-2 fuzzy sets is detailed in Section 5. A real caseM. Kilic , ... aL1The footprint of uncertainty (FOU) of the type-2 fuzzy set [6]. related with evolution of investment project for a devel-ncy ... in Section 7. type 2 fuzzy setsarison to a classical fuzzy set that named type-1 fuzzys membership function that is crisp, a ... result, we can say that type-2 fuzzy setspable than ordinary fuzzy sets in dealing with problemsdgments that are more subjective and more impre-eneral type-2 fuzzy sets are computationally intensive,e-2 fuzzy sets more used into applications. In this sec-ic concepts and operations ... operations of interval type-2 fuzzy setsced below [2,6,11,16,19,20]:2.1. A type-2 fuzzy set?
... the same opinions and the setting of evaluation isnoisy, type-1 fuzzy sets cannot offer effective decision support.In such cases, type-2 fuzzy sets whose membership functions are fuzzy sets too enables convenient modeling of problem. havn thinto hav real ... for the evaluation of an invest-t. These criteria are generally subjective and extremelyexpress in numbers. However, using the fuzzy setsge facilities ... measurement challenges.tudy, a hybrid MCDM method has been utilized along fuzzy sets and crisp sets to be used in determining projects supported ... in determining projects supported by development agencies. Member-ns of type-1 fuzzy sets are crisp sets. For this reason, ine the meanings of ... fuzzy setsts have been simultaneously used. Using the both crispe-2 fuzzy sets methods in the same evaluation modelcompared to the models used ...
... address ambiguities and relativitiesin real world scenarios more conveniently, type-2 fuzzy sets and crisp sets have been simultaneouslyused. The proposed model for ... projects. It is necessproject. These criteria are generally susing the fuzzy sets provide huge faciling methodology the state are extremely important in ...
... over all admissible x and u.2.2. Let?A be a type-2 fuzzy set in the universe of dis-resented by the type-2 membership function ... a type-2presented as follows: ? Jx1(x, u)(3)0, 1]oidal interval type-2 fuzzy set is shown in Fig. 1upper membership function and the lower ... membership function and the lower member-n of an interval type-2 fuzzy set are type-1 membershipespectively. The footprint of uncertainty (FOU) is shown3. ... anothions on type-2 fuzzy setsbsection, the main operations on type-2 fuzzy sets are [2,6,11,16,19,20]:2.3. The addition operation between the trape-rval type-2 fuzzy set? ?A1 = (A?U1 , A?L1) = ((aU11, aU12, aU13,U1 ), ... min(H2(A?L1), H2(A?L2))). (4)2.4. The multiplication operation between the trape-val type-2 fuzzy sets, , A?L1) = ((aU11, aU12, aU13, aU14, ; H1(A?U1 ), ... min(H2(A?L1), H2(A?L2))). (5)2.5. The arithmetic operations between the trape-val type-2 fuzzy sets, , A?L1) = ((aU11, aU12, aU13, aU14, ; H1(A?U1 ), ... average decisionach element of the pairwise comparison matrix is ane-2 fuzzy set, , representing which is the more importantria. The pairwise comparison ...
... 1/?ak2n . . . 1?????????(7), (8)ric mean of k type-2 fuzzy sets is calculated as follows:?a2ij ? . . . ??akij]1/k=k??a1ij ??a2ij ... used concurrently. In this project where invest-cts are appraised, type-2 fuzzy sets are used for someing evaluation while for some criteria crisp ... investmentria which wTable 2Linguistic terms and their corresponding interval type-2 fuzzy sets [2].Linguistic terms Interval type-2 fuzzy setsVery low (VL) ((0, 0, ...
... operating in Turkey., a new MCDM methodology based on type-2 fuzzy sets by integrating crisp and linguistic evaluation together.dology brings some advantages ... To address ambiguities and relativities in projectcases more conveniently, type-2 fuzzy sets and crispeen simultaneously used. The proposed methodology?ne their evaluations by ... of criteria and project alternatives.e both crisp sets and type-2 fuzzy sets methods in theation model is superior compared to the models ...
... c32, c33, c34, c35, c42 and c43 how-ave used type-2 fuzzy set for the rest of the evaluation hierarchical structure of the ...
... possible to use a new methodMETHEE or ANP under type-2 fuzzy sets to calculateriteria. By the way, other fuzzy MCDM techniques thated ... calculateriteria. By the way, other fuzzy MCDM techniques thated type-2 fuzzy sets can be used and obtained resultsssed.dle Black Sea Development Agency, ... fuzzy axiomatic design and fuzzyierarchy process, Energy 34 (2009) 1603–1616.h, Fuzzy sets, , Inf. Control 8 (1965) 338–353.g, X. Liu, Y. Qin, ... cash ?ows, Inf. Sci. 142 (2002) 57–76.el, R.I.B. John, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst.02) 117–127.el, R.I. John, F.L. ... (2006) 808–821.n, I?. Kaya, An integrated multi-criteria decision-making methodology type-2 fuzzy sets for selection among energy alternatives in Turkey,zzy Syst. (2014) (in ... J.M. Mendel, Aggregation using the linguistic weighted average andinterval type-2 fuzzy sets, , IEEE Trans. Fuzzy Syst. 15 (6) (2007) 1145–1161.[16] S.M. ... onthe ranking values and the arithmetic operations of interval type-2 fuzzy sets, ,Expert Syst. Appl. 37 (1) (2010) 824–833.[17] J.M. Mendel, D. ... Making SubjectiveJudgments, IEEE Press, Piscataway, NJ, 2010.[18] J.M. Mendel, Type-2 fuzzy sets and systems: an overview, Comput. Intell. Mag.IEEE 2 (2007) 20–29.[19] ... group decision-making based on the arithmetic operations of interval type-2 fuzzy sets, , in:Proceedings of 2008 International Conference on Machine Learning and ... Yang, Fuzzy multiple attributes groupdecision-making based on ranking interval type-2 fuzzy sets, , Expert Syst. Appl.39 (2012) 5295–5308.[21] T.L. Saaty, Multicriteria decision ... Sar?, B. Oztaysi, C. Kahraman, Fuzzy analytic hierarchy process usingtype-2 fuzzy sets: : An application to warehouse location selection, in: Multi-criteria Decision ... Wiley & Sons, Ltd., 2013.[23] J.J. Buckley, Fuzzy hierarchical analysis, Fuzzy Set. . Syst. 17 (1985) 233–247.Investment project evaluation by a decision ...

Abstract:
... to consider many criteria for evaluation of an investment project.The fuzzy sets provide huge facilities to decision makers in project evaluation.To address ... makers in project evaluation.To address ambiguities and relativities conveniently, type-2 fuzzy sets are used. Although investment projects supported by the state are ... the evaluation of an investment project. These criteria are generally subjective ... and extremely difficult to express in numbers. However, using the fuzzy sets provide huge facilities to decision makers in project evaluation process ... ambiguities and relativities in real world scenarios more conveniently, type-2 fuzzy sets and crisp sets have been simultaneously used. The proposed model ...

References:
L.A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965) 338-353.
J.M. Mendel, R.I.B. John, Type-2 fuzzy sets made simple, IEEE Trans. Fuzzy Syst., 10 (2002) 117-127.
M. Erdogan, İ. Kaya, An integrated multi-criteria decision-making methodology based on type-2 fuzzy sets for selection among energy alternatives in Turkey, Iran. J. Fuzzy Syst. (2014).
D.R. Wu, J.M. Mendel, Aggregation using the linguistic weighted average and interval type-2 fuzzy sets, IEEE Trans. Fuzzy Syst., 15 (2007) 1145-1161.
S.M. Chen, L.W. Lee, Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets, Expert Syst. Appl., 37 (2010) 824-833.
J.M. Mendel, Type-2 fuzzy sets and systems: an overview, Comput. Intell. Mag. IEEE, 2 (2007) 20-29.
L.W. Lee, S.M. Chen, A new method for fuzzy multiple attributes group decision-making based on the arithmetic operations of interval type-2 fuzzy sets, in: Proceedings of 2008 International Conference on Machine Learning and Cybernetics, vols. 1-7, IEEE, New York, 2008, pp. 3084-3089.
S.M. Chen, M.W. Yang, L.W. Lee, S.W. Yang, Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets, Expert Syst. Appl., 39 (2012) 5295-5308.
I. Uçal Sarı, B. Oztaysi, C. Kahraman, Fuzzy analytic hierarchy process using type-2 fuzzy sets: An application to warehouse location selection, John Wiley & Sons, Ltd., 2013.
J.J. Buckley, Fuzzy hierarchical analysis, Fuzzy Set. Syst., 17 (1985) 233-247.
J.M. Mendel, D. Wu, Perceptual Computing: Aiding People in Making Subjective Judgments, IEEE Press, Piscataway, NJ, 2010.

Keywords:
Fuzzy sets
Type-2 fuzzy sets

Title:
Investment project evaluation by a decision making methodology based on type-2 fuzzy sets

Abstract:
... in the entire system is a significant study of the subject. . Due to achieve the best compromise, the proposed method ...

Keywords:
Fuzzy Sets, Relational Database Estimation

Abstract:
... such as the impact ratios of the journals, and on subjective information, such as experts' judgments about the journals. Also, the ... and complexity of the decision process. This paper proposes a fuzzy set approach that integrates objective and subjective information for evaluating grades of journals. It provides a comprehensive ... Web-based decision support system that is based on the proposed fuzzy set

References:
{2} R. Biswas, An Application of fuzzy sets in students' evaluation, Fuzzy Sets and Systems 74 (1995) 187-194.]]
{6} D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic, New York, 1980.]]
{16} L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1965) 338-353.]]

Keywords:
subjective evaluation

Title:
A fuzzy set approach to the evaluation of journal grades

Full Text:
... evaluating the suitabil-atives, quantitative/qualitative assessments are often deal with uncertainty, subjectiveness and imprecise are best represented with fuzzy numbers. There-recision-based multicriteria ... on synthesized rankings. These studies concerninty/imprecise numeric values of decision/performancee subjective nature of human behavior. As a result,tended the TOPSIS method ... the concepts of TOPSIS and entropyweighting. Liang [25] incorporated the fuzzy set theory based onthe concepts of positive ideal and negative ideal ... [43] presented a fuzzy group MCDMbased on TOPSIS concepts with fuzzy sets to evaluate alternatives.Although the concepts of positive ideal and negative ... (MCDM) problems based on cevaluation problem often exists in criteria, subjective/ /objective assessmetaneously in management. Here, the are modeled as fuzzy ... fuzzy numbers by metool to deal with such uncertainties. ordinary fuzzy sets is not clear enoum/l ocate /asoction with MCDM based onaoyuan ...
strated that interval-valued fuzzy sets can provide moreThese fuzzy sets can effective represent the impre-information that results, and then one ... method can handle complexocesses, which often teem with vagueness: impre-ite, subjective and vague data and/or information. Theploys the major technique of ... vague data and/or information. Theploys the major technique of interval-valued fuzzy sets vague information and/or data, as interval-valued fuzzyrovide the ?exibility to ...
... Journal of Approximate Reasoning 29 (3) (2002). Y.H. Chang, Modeling subjective evaluation for fuzzy group multicri-sion making, European Journal of Operational ... (1975) 43–58., Multiple Criteria Decision Making, McGraw-Hill, New York, 1982.ermann, Fuzzy Set Theory and its Applications, Kluwer Academics, Boston, 1996.A soft computing ...
VIKOR and interval-valued fuzzy sets. . The performancelex administrative processes in which multiple evaluationnd fuzzy ... the presentation of linguistic expressions in the form of5,21]. Interval-valued fuzzy sets can provide more ?exibilityM.-S. Kuo, G.-S. Liang / Applied Soft ... often disagree on the method of de?ning linguisticsed on the fuzzy sets theory, (2) the method may gen-er-intuitive ranking outcomes for similar ... enough [15,21]. Bigand and Colot [4] and [14] presented interval-valued fuzzy sets to representise/vague information that results, as interval-valuedcan provide more ?exibility ...
... VIKOR and fuzzy numbersection, the basic de?nition of VIKOR [26,29–32,36,38]val-valued fuzzy sets are brie?y introduced2,33–35,39,41]. Based on these basic concepts, aCDM will ... , aL1ordinumbU1 , bL1betwditioB? ==tracB? ==ltiplB? ==? ?Fig. 1. Interval-valued fuzzy set A?.t:inifij, i = 1, 2, 3, . . . , ... rule.l-valued fuzzy numbersubsection, this paper considers the fuzzy demand byal-valued fuzzy sets. . Based on the de?nition of interval-y sets in Gorzalezany ...
... Mendel that the presentation of a linguistic expression in theinary fuzzy sets
... G. Montazer, Extension of fuzzy TOP-SIS method based on interval-valued fuzzy sets, , Applied Soft Computing 9 (2)(2009) 457–461.[2] C. B y k zkan, D. ...
... (1) (1985) 1–19., O. Colot, Fuzzy ?lter based on interval-valued fuzzy sets for imageFuzzy Sets and Systems 161 (1) (2010) 96–117.g, C.H. ... 65–73., Extensions of the TOPSIS for group decision-making under fuzzyent, Fuzzy Sets and Systems 114 (1) (2000) 1–9., C.H. Hsieh, A model ... 197–225.alczany, A method of inference in approximate reasoning based onalued fuzzy sets, , Fuzzy Sets and Systems 21 (1) (1987) 1–17.an-Guinness, Fuzzy membership mapped onto ... Math Logik Grundlag Mathe 22 (1975) 149–160.rzewski, Distances between intuitionistic fuzzy sets and/or interval-zzy sets based on the Hausdorff metric, Fuzzy Sets and Systems 148) 319–328.g, S. Lee, Some algebraic properties and ... Setsms 18 (2) (1986) 105–108.ik, J.M. Mendel, Operations on type-2 fuzzy sets, , Fuzzy Sets and Sys- (2) (2001) 327–348.ann, M.M. Gupta, Introduction to Fuzzy ... preference with Zadeh triples, FuzzySystems 78 (2) (1996) 183–195.en, Interval-valued fuzzy sets and compensatory, Fuzzy Sets and Sys-3) (1992) 295–307.en, Interval-valued fuzzy sets based on normal forms, Fuzzy Sets and20 (2) (1986) 191–210.g, C.W. Lin, S. Opricovic, Multi-criteria analysis ... Hadipour, J.S. Sadaghiani, M. Amiri, Extension of VIKOR method interval-valued fuzzy sets, , International Journal of Advanced Manu- Technology 47 (9–12) (2010) ...
... to solve and the computational procedure is summarized as judges’ subjective judgments use the linguistic termsthe importance weights (as shown in ...

Abstract:
... multicriteria decision-making (MCDM) problems based on combining VIKOR and interval-valued fuzzy sets. . The performance evaluation problem often exists in complex administrative ... to be taken into consideration simultaneously in management. Here, the subjective, , imprecise, inexact and uncertain evaluation processes are modeled as ... the presentation of linguistic expressions in the form of ordinary fuzzy sets is not clear enough [15,21]. Interval-valued fuzzy sets can provide more flexibility [4,14] to represent the imprecise/vague information ... of criteria are unequal by using the concepts of interval-valued fuzzy sets. . A case study for evaluating the performances of three ...

References:
Ashtiani, B., Haghighirad, F., Makui, A. and Montazer, G., Extension of fuzzy TOPSIS method based on interval-valued fuzzy sets. Applied Soft Computing. v9 i2. 457-461.
Bortolan, G. and Degani, R., A review of some methods for ranking fuzzy subsets. Fuzzy Sets and Systems. v15 i1. 1-19.
Bigand, A. and Colot, O., Fuzzy filter based on interval-valued fuzzy sets for image filtering. Fuzzy Sets and Systems. v161 i1. 96-117.
Chen, T.Y. and Tsao, C.Y., The interval-valued fuzzy TOPSIS method and experimental analysis. Fuzzy Sets and Systems. v159 i11. 1410-1428.
Chen, C.T., A fuzzy approach to select the location of the distribution center. Fuzzy Sets and Systems. v118 i1. 65-73.
Chen, C.T., Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems. v114 i1. 1-9.
Chen, S.H., Ranking fuzzy numbers with maximizing set and minimizing set. Fuzzy Sets and Systems. v17 i1. 113-129.
Gorzalczany, M.B., A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets and Systems. v21 i1. 1-17.
Grzegorzewski, P., Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric. Fuzzy Sets and Systems. v148 i2. 319-328.
Kacprzyk, J., Group decision making with a fuzzy linguistic majority. Fuzzy Sets and Systems. v18 i2. 105-108.
Karnik, N.N. and Mendel, J.M., Operations on type-2 fuzzy sets. Fuzzy Sets and Systems. v122 i2. 327-348.
Türken, I.B. and Bilgiç, T., Interval valued strict preference with Zadeh triples. Fuzzy Sets and Systems. v78 i2. 183-195.
Türksen, I.B., Interval-valued fuzzy sets and compensatory. Fuzzy Sets and Systems. v51 i3. 295-307.
Turksen, I.B., Interval-valued fuzzy sets based on normal forms. Fuzzy Sets and Systems. v20 i2. 191-210.
Vahdani, B., Hadipour, H., Sadaghiani, J.S. and Amiri, M., Extension of VIKOR method based on interval-valued fuzzy sets. International Journal of Advanced Manufacturing Technology. v47 i9-12. 1231-1239.
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Keywords:
Interval-valued fuzzy sets
Fuzzy sets

Full Text:
... 421–430, 2008. Springer-Verlag Berlin Heidelberg 2008 Agvs Distributed Control Subject to Imprecise Operation Times Grzegorz Bocewicz1, Robert W jcik2, and Zbigniew ... Engineering, Control and Robotics, 50-372 Wroc?aw, Poland [email protected] Abstract. The subject matter of the study are the automated guided vehicle (AGV) ... multi-agent coordination, and defined as the constraint satisfaction problem (CSP) subject to precise and imprecise operation times, and then solved with ... programming techniques. Keywords: knowledge engineering, constraints logic programming, deadlock avoidance, fuzzy set, , scheduling. 1 Introduction A considered class of objects covers ...
... The set {df1,1,...,df1,n1,df2,1,...,df2,n2,...,dfr,1,...,dfr,nr} provides any combination of distinct moments for fuzzy sets Tt. Since dfi,j can be selected in different ways, hence ... of triples (?hmin(h), ?hmax(h), h) can be treated as a fuzzy set determining the cycle time corresponding to (S0, ?) for the ... the given fuzzy operation times Tf : Agvs Distributed Control Subject to Imprecise Operation Times 429 H = {(?hmin(h), ?hmax(h), h)} ...
... LNCS (LNAI), vol. 4496, Springer, Heidelberg (2007) 13. Zadeh, L.A.: Fuzzy sets.
... ?6 Fig. 1. Graphical representation of CCPS Agvs Distributed Control Subject to Imprecise Operation Times 423 The processes cooperation is determined ...
... tj-1): xj = xj-1 + tj-1 (2) Agvs Distributed Control Subject to Imprecise Operation Times 425 • Constraints regarding processes servicing ...
... R13 and the following parameters are known: Agvs Distributed Control Subject to Imprecise Operation Times 427 P1 = (R1, R2, R5), ...

Abstract:
<p>The subject matter of the study are the automated guided vehicle (AGV) ... multi-agent coordination, and defined as the constraint satisfaction problem (CSP) subject to precise and imprecise operation times, and then solved with ...

References:
Zadeh, L.A.: Fuzzy sets. Information and Control 8, 338-353 (1965).

Keywords:
fuzzy set

Title:
Agvs distributed control subject to imprecise operation times

Abstract:
... attention. This article introduces a trust management model based on fuzzy set theory and named TMFC including direct trust measurement and computing, ...

Keywords:
subjective trust, fuzzy set, trust management, associated cheat, cloud security

Abstract:
... is analyzed, additionally, the tracking control performance of the manipulator subject to uncertainties is studied. Numerical simulations illustrate the proposed methodology ...

Keywords:
Manipulator dynamics, Uncertainty, Fuzzy sets

Full Text:
... a fuzzy constraint, i.e. possibility distribution of alternatives, because the subjective boundary between the ‘possible’ and the ‘non-possible’ meanings of a ... computer programs to the system DELTA. References [l] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Memo. No. ... UCB/ERL M77/24 (University of California, Berkeley, 1977). [3] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28. [4] A.N. Borisov and O.A. ... Systems (Riga Polytechnical Institute, Riga, 1979) 30-37. [5] O.A. Krumberg, Subjective possibility theory for decision making under uncertainty, in,: Methods of ...
... distribution to the valuation of decision alternatives is outlined. Keywords: Subjective possibility, Possibility distribution, Level constraints, Decision making. 1. Introduction The ... opportunity to extend the set of decision-making problems which are subjected to modelling [4, 51. If we have a universe of ... be noted that the concept of possibility concerns not only subjective correspondence of generic elements and variables, but can be extended ... processes usually refers to a process where an object is subjected to qualitative transformation. If we are able to consider alternatives ...
... noteworthy that the meanings of parameter Z have often no subjective utility for the decision maker. Therefore it is necessary to ...
... particular case for the task 0222 the alternative’s quality indicator ‘subjective power of alternative’ may be used: M3 = 71.(k)V(k)pa,(k) dk, ... - Computer’. The elaborated interactive procedures for assessing utility functions, subjective probability distributions, level constraints and some computing algorithms form the ...

References:
Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility. In: Memo. No. UCB/ERL M77/12, University of California.
Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems. v1. 3-28.
Krumberg, O.A., Subjective possibility theory for decision making under uncertainty. In: Methods of Decision Making Under Uncertainty, Riga Polytechnical Institute, Riga. pp. 47-52.

Keywords:
Subjective possibility

Full Text:
Aggregation of subjective evaluations based on discrete fuzzy numbersAvailable online at www.sciencedirect.comFuzzy Sets ... online at www.sciencedirect.comFuzzy Sets and Systems 191 (2012) 21–40www.elsevier.com/locate/fssAggregation of subjective evaluations based ondiscrete fuzzy numbersJ. Vicente Riera, Joan Torrens?Department of ... the aggregation of subjectiveevaluations. 2011 Elsevier B.V. All rights reserved.Keywords: Subjective evaluation; Discrete fuzzy number; Discrete aggregation function; t-Norm; Uninorm; Nullnorm; ... aggregation functions have a great numberof applications which includes many subjects not only from mathematics and computer sciences, but also from ... fields which is most closely related to aggregation functions is fuzzy set theory. Not only because manyof the usual fuzzy connectives like ... the use of fuzzy mathematics [14,17] allows to model the subjective and qualitativenature of such information. Thus, fuzzy sets have been used in aggregation applications to social sciences like ... B.V. All rights reserved.doi:10.1016/j.fss.2011.10.00422 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40However, the most usual way ... of the graded scale L and the use of the fuzzy set theory. In these cases, usually the fuzzy numbers usedare not ... paper,we wish to study the possibility of aggregating directly the subjective information which will be expressed as discretefuzzy numbers whose support ... in the discrete case, this process can yield to a fuzzy set that does not satisfy the conditions to be adiscrete fuzzy ...
... approach in aggregating subjectiveevaluations. After recalling some methods based on fuzzy set theory used in students’ evaluations, we present ourapproach. Specifically, we ... Bad (case n=8) and we present some examples of aggregationof subjective ... evaluations where evaluators or experts give their opinions as discrete fuzzy sets on L being the resultanother discrete fuzzy set on L.2. PreliminariesIn this section, we recall some definitions and ... t-norms and t-conorms as follows.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 23Definition 2.3. A uninorm ...
... ? U (max A?,max B?)}.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 33As A, B ? ... [0,3] to A[0,3]1 .34 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –403.2. Aggregation based on nullnormsLet ... 0.8/5} ? AL1 . Then F(A, B) = {0.3/2, 1/3}.4. Subjective evaluationsIn recent years, a number of researchers [2,10,24,25] have focused ... In [10], Chen and Lee provided two methods for applying fuzzy sets in students’ answerscripts evaluation.Recently, in [24], Wang and Chen presented ... us recall this last method.J. Vicente Riera, J. Torrens / Fuzzy Sets
... us to aggregate directly36 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40Fig. 4. Interval of confidence.subjective ... of these parameters can bemodelled using common concepts of the fuzzy set theory. In Section 4 we have reviewed some well known ... Section 4 we have reviewed some well known methodsof students’ subjective evaluation. In this way, we will use the extension of ...
... in Umin and Umax.24 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40Definition 2.9 (Mas et al. ... forany ? ? [?′0, ?0]).J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 25Theorem 2.14 (Wang et ...
... L(x, y)? O(x, y).26 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40We will denote as well ... the case of smooth t-norms.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 27Let us suppose that ...
... ?? AL1(A, B)?F(A, B)28 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40will be called the extension ... by 10 and 1n, respectively.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 29Proposition 3.8. Let F ...
... nullnorms on AL1 .30 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40Fig. 1. The dfn A ... 2, 3, 4, 5, 6}.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 31Fig. 2. The dfn ... . Then, U(A, B) = {0.3/0, 0.5/1, 1/2,0.8/3, 0.8/4, 0.8/5}. Fuzzy sets A and B can be viewed in Figs. 1 and ... supp(A) ? [e, n].32 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40Proof. Let A ? AL1 ...
... “j” to question Pi .J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 37Table 1Fuzzy grade sheet.Teacher’s ... {0.1/0, 0.6/1, 1/2, 0.4/3},38 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40Fig. 5. Evaluations of three ...
... corresponding to the linguistic terms.J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40 39Fig. 6. The final ... respectively. We have also40 J. Vicente Riera, J. Torrens / Fuzzy Sets and Systems 191 (2012) 21 –40studied some properties of these ... vol. 221, Springer,Berlin, Heidelberg, 2007.[2] R. Biswas, An application of fuzzy sets in students’ evaluation, Fuzzy Sets Syst. 74 (2) (1995) 187–194.[3] T. Calvo, B. De Baets, ... functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets Syst. 120 (2001)385–394.[4] J. Casasnovas, J.V. Riera, On the addition ... Extension of discrete t-norms and t-conorms to discrete fuzzy numbers, Fuzzy Sets Syst. 167 (1) (2011) 65–81.[9] S.J. Chen, C.L. Hwang, Fuzzy ... S.M. Chen, C.H. Lee, New methods for students’ evaluation using fuzzy sets, , Fuzzy Sets Syst. 104 (2) (1999) 209–218.[11] B. De Baets, R. Mesiar, ... B. De Baets, R. Mesiar, Triangular norms on product lattices, Fuzzy Sets Syst. 104 (1999) 61–75.[12] B. DeBaets, J. Fodor, D. Ruiz-Aguilera, ... H. Prade, Fuzzy cardinality and the modeling of imprecise quantification, Fuzzy Sets ... Syst. 16 (1985) 199–230.[14] D. Dubois, H. Prade, Fundamentals of Fuzzy Sets, , Kluwer, Boston, 2000.[15] L. Godo, V. Torra, On aggregation ...
... 127, CambridgeUniversity Press, 2009, pp. 5–9.[17] G. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic (Theory and Applications), Prentice Hall, New Jersey, ... pp. 189–230.[22] W. Voxman, Canonical representations of discrete fuzzy numbers, Fuzzy Sets Syst. 54 (2001) 457–466.[23] G. Wang, C. Wu, C. Zhao, ... fuzzy number arithmetic operations, Soft Comput. 12 (2008)919–934.[26] M. Wygralak, Fuzzy sets with triangular norms and their cardinality theory, Fuzzy Sets Syst. 124 (2001) 1–24.[27] R.R. Yager, Aggregation of ordinal information, ... pp. 177–191.[30] D. Zhang, Triangular norms on partially ordered sets, Fuzzy Sets

Abstract:
... finite chain) and they are applied to the aggregation of subjective

References:
Biswas, R., An application of fuzzy sets in students' evaluation. Fuzzy Sets Syst. v74 i2. 187-194.
Calvo, T., De Baets, B. and Fodor, J., The functional equations of Frank and Alsina for uninorms and nullnorms. Fuzzy Sets Syst. v120. 385-394.
Casasnovas, J. and Riera, J.V., Extension of discrete t-norms and t-conorms to discrete fuzzy numbers. Fuzzy Sets Syst. v167 i1. 65-81.
Chen, S.M. and Lee, C.H., New methods for students' evaluation using fuzzy sets. Fuzzy Sets Syst. v104 i2. 209-218.
De Baets, B. and Mesiar, R., Triangular norms on product lattices. Fuzzy Sets Syst. v104. 61-75.
Dubois, D. and Prade, H., Fuzzy cardinality and the modeling of imprecise quantification. Fuzzy Sets Syst. v16. 199-230.
Dubois, D. and Prade, H., Fundamentals of Fuzzy Sets. 2000. Kluwer, Boston.
Klir, G. and Yuan, B., Fuzzy Sets and Fuzzy Logic (Theory and Applications). 1995. Prentice Hall, New Jersey.
Voxman, W., Canonical representations of discrete fuzzy numbers. Fuzzy Sets Syst. v54. 457-466.
Wygralak, M., Fuzzy sets with triangular norms and their cardinality theory. Fuzzy Sets Syst. v124. 1-24.
Zhang, D., Triangular norms on partially ordered sets. Fuzzy Sets Syst. v153. 195-209.

Keywords:
Subjective evaluation

Title:
Aggregation of subjective evaluations based on discrete fuzzy numbers

References:
Buckles, B. P., & Petry, F. E. (1982). A fuzzy representation of data for relational databases. Fuzzy Sets and Systems, 7, 213-226.]]
Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338-353.]]

Keywords:
fuzzy sets

Title:
Processing subjective information for diagnostic assistance