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A PRIMER ON FUZZY OPTIMIZATION MODELS AND METHODS
A PRIMER ON FUZZY OPTIMIZATION MODELS AND METHODS
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2
Fuzzy Linear Programming models and methods has been one ofthe most and well studied topics inside the broad area of Soft Computing. Itsapplications as well as practical realizations can be found in all the real worldareas. In this paper a basic introduction to the main models and methods infuzzy mathematical programming, with special emphasis on those developedby the authors, is presented. As a whole, Linear Programming problems withfuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrixare analyzed. Finally, future research and development lines are also pointedout by focusing on fuzzy sets based heuristic algorithms.
1
Fuzzy Linear Programming models and methods has been one ofthe most and well studied topics inside the broad area of Soft Computing. Itsapplications as well as practical realizations can be found in all the real worldareas. In this paper a basic introduction to the main models and methods infuzzy mathematical programming, with special emphasis on those developedby the authors, is presented. As a whole, Linear Programming problems withfuzzy costs, fuzzy constraints and fuzzy coefficients in the technological matrixare analyzed. Finally, future research and development lines are also pointedout by focusing on fuzzy sets based heuristic algorithms.
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21
J. M.
Cadenas
J. M.
Cadenas
Departamento de Ingenier´ıa de la Informaci´on y las Comunicaciones.
Facultad de Inform´atica., Universidad de Murcia., Campus de Espinardo. 30071Espinardo.
Murcia, Spain
Departamento de Ingenier´ıa de la Informaci´on
Spain
jcadenas@dif.um.es
J. L.
Verdegay
J. L.
Verdegay
Departamento de Ciencias de la Computaci´on e Inteligencia Artificial.
E.T.S. de Ingenier´ıa Inform´atica, Universidad de Granada., 18071. Granada,
Spain
Departamento de Ciencias de la Computaci´on
Spain
verdegay@decsai.ugr.es
Fuzzy linear programming
Fuzzy optimization
Heuristics algorithms
Intelligent systems
Decision support systems
[[1] R. E. Bellman and L. A. Zadeh, Decision making in a fuzzy environment, Management##Science, 17 (B) 4 (1970), 141–164.##[2] A. Blanco, D. Pelta and J. L. Verdegay, Applying a fuzzy setsbased Heuristic to the protein##structure prediction problem, Int’l Journal of Intelligent Systems, 17 (2002), 629–643.##[3] A. Blanco, D. Pelta and J. L. Verdegay, A fuzzy valuationbased local search framework for##combinatorial problems, Fuzzy Optimization and Decision Making, 1 (2002), 177–193.##[4] J. M. Cadenas and J. L. Verdegay, Using fuzzy numbers in linear programming, IEEE Transactions##on Systems, Man, and Cybernetics, 27 (B) 6 (1997), 1017–1022.##[5] J. M. Cadenas and J. L. Verdegay, Modelos de optimizaci´on con datos i mprecisos, Univesidad##de Murcia, Servicio de Publicaciones, 1999.##[6] L. Campos, Modelos de la PLD para la resoluci´on de juegos matriciales imprecisos, Tesis##doctoral, Universidad de Granada, 1986.##[7] L. Campos and J. L. Verdegay, Linear programming problems and ranking of fyzzy numbers,##Fuzzy Sets and Systems, 32 (1989), 1–11.##[8] M. Delgado, J. L. Verdegay and M. A. Vila, Imprecise costs in mathematical programming##problems, Control and Cybernet, 16 (2)(1987), 113–121.##[9] M. Delgado, J. L. Verdegay and M. A. Vila, A general model for fuzzy linear programming,##Fuzzy Sets and systems, 29 (1989), 21–29.##[10] M. Delgado, J. L. Verdegay and M. A. Vila, Relating different approaches to solve linear##programming problems with imprecise costs, Fuzzy Sets and Systems, 37 (1990), 33–42.##[11] M. Delgado, F. Herrera, J. L. Verdegay and M. A. Vila, Postoptimality analisys on the##membership function of a fuzzy linear programming problema, Fuzzy Sets and Systems, 53##(1993), 289–297.##[12] D. Dubois and H. Prade, Operations on fuzzy numbers, International Journal Systems Science,##9 (1978), 613–626.##[13] D. Dubois and H. Prade, Fuzzy sets and systems. theory and applications, Academic Press,##[14] D. Dubois and H. Prade, Ranking of fuzzy numbers in the setting of possibility theory, Information##Science, 30 (1983), 183–244.##[15] D. Dubois and H. Prade, The mean value of a fuzzy number, Fuzzy Sets and Systems, 24##(1987), 279–300.##[16] M. Fedrizzi, J. Kacprzyk and J. L. Verdegay, A survey of fuzzy optimization and fuzzy##mathematical programming, In: Fedrizzi M, Kacprzyk J, Roubens M (eds) Interactive Fuzzy##Optimization, Springer Verlag, Berlin, 1991.##[17] A. Gonz´alez, M´etodos subjetivos para la comparaci´on de N´umeros difusos, Tesis doctoral,##Universidad de Granada, 1988.##[18] A. Gonz´alez, A studing of the ranking function approach through mean values, Fuzzy Sets##and Systems, 35 (1990), 29–41.##[19] F. Herrera, M. Kovacs and J. L. Verdegay, Fuzzy linear programming problems with homogeneous##linear fuzzy functions, Presented to IPMU’92, 1992.##[20] H. Ishibuchi and H. Tanaka, Multipleobjective programming in optimization of the interval##objective function, EJOR, 48 (1990), 219–225.##[21] D. Pelta, A. Blanco and J. L. Verdegay, Fuzzy adaptive neighborhood search: Examples of##application, In J. L. Verdegay, editor, Fuzzy Sets based Heuristics for Optimization, Studies##in Fuzziness and Soft Computing ,PhysicaVerlag, 2003. ##[22] I. Requena, Redes neuronales en problemas de decisi´on con ambiente difuso, Tesis doctoral,##Universidad de Granada, 1992.##[23] H. Rommelfanger, R. Hanuscheck and J. Wolf, Linear programming with fuzzy objectives##Fuzzy Sets and Systems, 29 (1990), 31–48.##[24] A. SanchoRoyo, J. L. Verdegay and E. Vergara, Some practical problems in fuzzy setsbased##DSS, MathWare and Soft Computing VI, 23 (1999), 173–187.##[25] H. Tanaka, T. Okuda and K. Asai, On fuzzy mathematical programming, Journal of Cybernetics,##3 (4) (1974), 37–46.##[26] H. Tanaka, H. Ichihashi and F. Asai, A formulation of fuzzy linear programming problems##based a comparison of fuzzy numbers, Control and Cybernet, 13 (1984), 185–194.##[27] J. L. Verdegay, Fuzzy mathematical programming In: Gupta MM , Sanchez E (eds) Fuzzy##Information and Decisi´on Processes, 1982.##[28] J. L. Verdegay and E. Vergara, Fuzzy termination criteria in knapsack problem algorithms,##Mathware and Soft Computing, 23 (2000), 89–97.##[29] J. L. Verdegay and E. Vergara, Fuzzy setsbased control rules for terminating algorithms,##Comp. Science Journal, 10 (1) (2002), 9–27.##[30] J. L. Verdegay, Fuzzy sets based heuristics for optimization, Studies in Fuzziness and Soft##Computing, Springer Verlag, 2003.##[31] X. Wang and E. Kerre, On the classification and the dependencies of the ordering methods,##In: Ruan D (ed) Fuzzy Logic Foundation and Industrial Applications, International Series in##Intelligent Technologies. Kluwer, 1996.##[32] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.##[33] L. A. Zadeh, The concept of a linguistic variable and its applications to approximate reasoning,##Part I, Information Sciences, 8 (1975), 199–249. Part II, Information Sciences, 8 (1975),##301–357. Part III, Information Sciences, 9 (1975), 43–80.##[34] Q. Zhu and E. S. Lee, Comparison and ranking of fuzzy numbers, In: Kacprzyk J, Fedrizzi##M (eds) Fuzzy Regression Analysis, Onmitech Press Warsan and PhysicaVerlag, 1992.##[35] H. J. Zimmermann, Optimization in fuzzy environments. Presentado al XXI International##TIMS and 46th ORSA Conference, San Juan, Puerto Rico, 1974.##[36] H. J. Zimmermann, Description and optimization of fuzzy systems, International Journal of##General Systems, 2(1976), 209–215.##[37] H. J. Zimmermann, Fuzzy sets, decision making and expert systems, Kluwer Academic Publishers,##Boston, 1987.##]
FIXED POINT THEOREM ON INTUITIONISTIC FUZZY METRIC SPACES
FIXED POINT THEOREM ON INTUITIONISTIC FUZZY METRIC SPACES
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2
In this paper, we introduce intuitionistic fuzzy contraction mappingand prove a fixed point theorem in intuitionistic fuzzy metric spaces.
1
In this paper, we introduce intuitionistic fuzzy contraction mappingand prove a fixed point theorem in intuitionistic fuzzy metric spaces.
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29
Mohd.
Rafi Segi Rahmat
Mohd.
Rafi Segi Rahmat
School of Mathematical Science, Faculty of Science and
Technology, Universiti kebangsaan Malaysia, 43600 Bangi, Selangor D.E., Malaysia
School of Mathematical Science, Faculty of
Malaysia
mdrafzi@yahoo.com
Mohd.
Salmi Md. Noorani
Mohd.
Salmi Md. Noorani
School of Mathematical Science, Faculty of Science and
Technology, Universiti kebangsaan Malaysia, 43600 Bangi, Selangor D.E., Malaysia
School of Mathematical Science, Faculty of
Malaysia
msn@pkrisc.cc.ukm.my
Intuitionistic fuzzy metric spaces
Fuzzy metric spaces
Fixed point theorem
[[1] V. Gregory and A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and##Systems, 125 (2002), 245253.##[2] H. P. A. Kunzi and S. Romaguera, Quasimetric spaces, quasimetric hyperspaces and uniform##local compactness, Rend. Istit. Univ. Trieste, 30 (1999), 133144.##[3] R. Lowen, Fuzzy set theory, Kluwer Academic Publisher, Dordrecht, 1996.##[4] D. Mihet, A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets and Systems,##144 (2004), 431439.##[5] J. H. Park, Intutionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004), 1039##[6] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and##Fractals, 27 (2006), 331344.##[7] P. Vijayaraju and M. Marudai, Fixed point theorem for fuzzy mappings, Fuzzy Sets and##Systems, 135 (2003), 401408.##]
FUZZY CONTROL CHARTS FOR VARIABLE AND ATTRIBUTE QUALITY CHARACTERISTICS
FUZZY CONTROL CHARTS FOR VARIABLE AND ATTRIBUTE QUALITY CHARACTERISTICS
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2
This paper addresses the design of control charts for both variable ( x chart) andattribute (u and c charts) quality characteristics, when there is uncertainty about the processparameters or sample data. Derived control charts are more flexible than the strict crisp case, dueto the ability of encompassing the effects of vagueness in form of the degree of expert’spresumption. We extend the use of proposed fuzzy control charts in case of linguistic data using adeveloped defuzzifier index, which is based on the metric distance between fuzzy sets.
1
This paper addresses the design of control charts for both variable ( x chart) andattribute (u and c charts) quality characteristics, when there is uncertainty about the processparameters or sample data. Derived control charts are more flexible than the strict crisp case, dueto the ability of encompassing the effects of vagueness in form of the degree of expert’spresumption. We extend the use of proposed fuzzy control charts in case of linguistic data using adeveloped defuzzifier index, which is based on the metric distance between fuzzy sets.
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MOHAMMAD HASSAN
FAZEL ZARANDI
MOHAMMAD HASSAN
FAZEL ZARANDI
DEPARTMENT OF INDUSTRIAL ENGINEERING, AMIRKABIR
UNIVERSITY OF TECHNOLOGY, TEHRAN, IRAN
DEPARTMENT OF INDUSTRIAL ENGINEERING, AMIRKABIR
UN
Iran
zarandi@aut.ac.ir
ISMAIL BURHAN
TURKSEN
ISMAIL BURHAN
TURKSEN
DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, UNIVERSITY
OF TORONTO, TORONTO, ON, CANADA, M5S2H8
DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERIN
Iran
turksen@mie.utoronto.ca
ALI
HUSSEINIZADEH KASHAN
ALI
HUSSEINIZADEH KASHAN
DEPARTMENT OF INDUSTRIAL ENGINEERING, AMIRKABIR UNIVERSITY OF
TECHNOLOGY, P. O. BOX: 158754413, TEHRAN, IRAN
DEPARTMENT OF INDUSTRIAL ENGINEERING, AMIRKABIR
Iran
a.kashani@aut.ac.ir
Process control
Control chart
Quality characteristics
Fuzzy numbers
[[1] D. Dubois and H. Prade, Theory and applications, Fuzzy Sets and Systems, Academic Press, New York,##[2] F. Franceschini and D. Romano, Control chart for linguistic variables: a method based on the use of##linguistic quantifiers, INT. J. PROD. RES., 37 (16 ) (1999), 3791–3801.##[3] C. Kahraman, E. Tolga and Z. Ulukan, Justification of manufacturing technologies using fuzzy##benefit/cost ratio analysis, International Journal of Production Economics, 1 (2000), 45–52.##[4] C. Kahraman , E. Tolga and Z. Ulukan , Using triangular fuzzy numbers in the tests of control charts for##unnatural patterns. In: Proc INRIA/IEEE Conf Emerging Technologies and Factory Automation,##Vol 3, October, 10–13, Paris, France, (1995), 129–298.##[5] M. Ma, A. Kandel and M. Friedman, A new approach for defuzzification, Fuzzy Sets and Systems,##111 (2000), 351–356. ##[6] W. A. Shewhart, Economic control of quality of manufactured product. Princeton, NJ: D. Van Nostrand,##Inc., 1931.##[7] J. D .T . Tannock, A fuzzy control charting method for individuals, INT. J. PROD. RES., 41 ( 5) (2003),##1017–1032.##[8] J. H. Wang and T. Raz , On the construction of control charts using linguistic variables, INT. J. PROD.##RES., 28 (1990), 477– 487.##[9] L. R. Wang and H. Rowlands, A fuzzy logic application in SPC evaluation and control, IEEE Symposium##on Emerging Technologies and Factory Automation, ETFA, 1 (1999), 679684.##[10] G. Yongting, Fuzzy quality and analysis on fuzzy probability, Fuzzy Sets and Systems, 83 (1996),##[11] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338353.##]
SOME INTUITIONISTIC FUZZY CONGRUENCES
SOME INTUITIONISTIC FUZZY CONGRUENCES
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2
First, we introduce the concept of intuitionistic fuzzy group congruenceand we obtain the characterizations of intuitionistic fuzzy group congruenceson an inverse semigroup and a T^{*}pure semigroup, respectively. Also,we study some properties of intuitionistic fuzzy group congruence. Next, weintroduce the notion of intuitionistic fuzzy semilattice congruence and we givethe characterization of intuitionistic fuzzy semilattice congruence on a T^{*}puresemigroup. Finally, we introduce the concept of intuitionistic fuzzy normalcongruence and we prove that (IFNC(E_{S}), $cap$, $vee$) is a complete lattice. Andwe find the greatest intuitionistic fuzzy normal congruence containing an intuitionisticfuzzy congruence on E_{S}.
1
First, we introduce the concept of intuitionistic fuzzy group congruenceand we obtain the characterizations of intuitionistic fuzzy group congruenceson an inverse semigroup and a T^{*}pure semigroup, respectively. Also,we study some properties of intuitionistic fuzzy group congruence. Next, weintroduce the notion of intuitionistic fuzzy semilattice congruence and we givethe characterization of intuitionistic fuzzy semilattice congruence on a T^{*}puresemigroup. Finally, we introduce the concept of intuitionistic fuzzy normalcongruence and we prove that (IFNC(E_{S}), $cap$, $vee$) is a complete lattice. Andwe find the greatest intuitionistic fuzzy normal congruence containing an intuitionisticfuzzy congruence on E_{S}.
45
57
Kul
Hur
Kul
Hur
Division of Mathematics and Informational Statistics, and
Institute of Basic Natural Science, Wonkwang University, Iksan, Chonbuk, Korea 570
749
Division of Mathematics and Informational
Korea
kulhur@@wonkwang.ac.kr
Su Youn
Jang
Su Youn
Jang
Division of Mathematics and Informational Statistics, and
Institute of Basic Natural Science, Wonkwang University, Iksan, Chonbuk, Korea 570
749
Division of Mathematics and Informational
Korea
suyoun123@@yahoo.co.kr
Hee won
Kang
Hee won
Kang
Dept. of Mathematics Education, Woosuk University, HujongRi
SamraeEup, Wanjukun Chonbuk, Korea 565701
Dept. of Mathematics Education, Woosuk University,
Korea
khwon@@woosuk.ac.kr
Tpure semigroup
Intuitionistic fuzzy set
intuitionistic fuzzy congruence
intuitionistic fuzzy group congruence
intuitionistic fuzzy semilattice congruence
intuitionistic fuzzy normal congruence
[[1] K. T. Atanassov, Intuitionistic fuzzy sets, in :V.Sgurev. Ed., VII ITKR0 s Session, Sofia, June##1983 Central Sci. and Techn, Library, Bul. Academy of Sciences, 1984.##[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 8796.##[3] B. Banerjee and D. Kr. Basnet, Intuitionistic fuzzy subrings and ideals, Journal of Fuzzy##Math., 11 (1) (2003), 139155.##[4] G. Birkhoff, Lattice Theory, A.M.S.Colloquium Publication, Vol XXV, 1967.##[5] R. Biswas, Intuitionistic fuzzy subgroups, Mathematical Forum, X (1989), 3746.##[6] H. Bustince and P. Burillo, Structures on intuitionistic fuzzy relations, Fuzzy Sets and Systems,##78 (78)(1996), 293303##[7] D. C¸ oker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems,##88 (1997), 8189.##[8] D. C¸ oker and A.Haydar Es, On fuzzy compactness in intuitionistic fuzzy topological spaces,##Journal of Fuzzy Math., 3 (1995), 899909.##[9] P. Das, Lattice of fuzzy congruences in inverse semigroups, Fuzzy Sets and Systems, 91 (1997),##[10] G. Deschrijver and E. E. Kerre, On the composition of intuitionistic fuzzy relations, Fuzzy##Sets and Systems, 136 (2003), 333361.##[11] H. G¨ur¸cay, D. C¸ oker and A. Haydar Es, On fuzzy continuity in intuitionistic fuzzy topological##spaces, Journal of Fuzzy Math., 5 (1997), 365378.##[12] J. M Howie, An Introduction to Semigroup Theory Academic Press, NewYork, 1976.##[13] K. Hur, S. Y. Jang and H. W. Kang, Intuitionistic fuzzy subgroupids, International Journal##of Fuzzy Logic and Intelligent Systems, 3 (1) (2003), 7277.##[14] K. Hur, H. W. Kang and H. K. Song, Intuitionistic fuzzy subgroups and subrings, Honam##Mathematical Journal, 25 (2) (2003), 1941.##[15] K. Hur, S. Y. Jang and H. W. Kang, Intuitionistic fuzzy congruences on a lattice, Journal##of Appl.Math.and Computing, 18(2005), Journal of Appl.Math. Computing, 18 (12) (2005),##[16] K. Hur, Y. B. Jun and J. H. Ryou, Intuitionistic fuzzy topological groups, Honam Math.##Journal, 26 (2) (2004), 163192.##[17] K. Hur, J. H. Kim and J. H. Ryou, Intuitionistic fuzzy topological spaces, Journal of Korea##Soc, Math. Educ. Ser. B: Pure Appl. Math., 11 (3) (2004), 243265.##[18] K. Hur, S. Y. Jang and Y. B. Jun, Intuitionistic fuzzy congruences, Far East Journal of##Math.Sci., 17 (1) (2005), 129.##[19] K. Hur, S. Y. Jang and Y. S. Ahn, Intuitionistic fuzzy equivalence relations, Honam Math.##Journal,27 (2) (2005), 159177.##[20] N. Kuroki, On a weakly idempotent semigroup in which the idempotents are central, Joetsu,##J. Math, Educ., 2 (1987), 1928.##[21] N. Kuroki,Tpure Archimedean semigroups, Comment. Math. Univ. St. Pauli, 31 (1982),##[22] N. Kuroki, Fuzzy congruence and fuzzy normal subgroups, Inform.Sci., 66 (1992), 235243.##[23] N. Kuroki, Fuzzy congruences on Tpure semigroups, Inform.Sci., 84 (1995), 239246.##[24] N. Kuroki, Fuzzy congruences on inverse semigroups, Fuzzy Sets and Systems, 87 (1997),##[25] S. J. Lee and E. P. Lee, The category of intuitionistic fuzzy topological spaces, Bull. Korean##Math. Soc., 37 (1) (2000), 6376.##[26] M. Samhan, Fuzzy congruences on semigroups, Inform.Sci., 74 (1993), 165175.##[27] T. Yijia, Fuzzy congruences on a regular semigroup, Fuzzy sets and Systems, 117 (2001),##[28] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338353.##]
GENERALIZED FUZZY POLYGROUPS
GENERALIZED FUZZY POLYGROUPS
2
2
small Polygroups are multivalued systems that satisfy grouplikeaxioms. Using the notion of “belonging ($epsilon$)” and “quasicoincidence (q)” offuzzy points with fuzzy sets, the concept of ($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroups isintroduced. The study of ($epsilon$, $epsilon$ $vee$ q)fuzzy normal subpolygroups of a polygroupare dealt with. Characterization and some of the fundamental properties ofsuch fuzzy subpolygroups are obtained. ($epsilon$, $epsilon$ $vee$ q)fuzzy cosets determined by($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroups are discussed. Finally, a fuzzy subpolygroupwith thresholds, which is a generalization of an ordinary fuzzy subpolygroupand an ($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroup, is defined and relations between twofuzzy subpolygroups are discussed.
1
small Polygroups are multivalued systems that satisfy grouplikeaxioms. Using the notion of “belonging ($epsilon$)” and “quasicoincidence (q)” offuzzy points with fuzzy sets, the concept of ($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroups isintroduced. The study of ($epsilon$, $epsilon$ $vee$ q)fuzzy normal subpolygroups of a polygroupare dealt with. Characterization and some of the fundamental properties ofsuch fuzzy subpolygroups are obtained. ($epsilon$, $epsilon$ $vee$ q)fuzzy cosets determined by($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroups are discussed. Finally, a fuzzy subpolygroupwith thresholds, which is a generalization of an ordinary fuzzy subpolygroupand an ($epsilon$, $epsilon$ $vee$ q)fuzzy subpolygroup, is defined and relations between twofuzzy subpolygroups are discussed.
59
75
B.
Davvaz
B.
Davvaz
Department of Mathematics, Yazd University, Yazd, Iran
Department of Mathematics, Yazd University,
Iran
davvaz@yazduni.ac.ir
P.
Corsini
P.
Corsini
Dipartimento Di Matematica E Informatica, Via Delle Scienze 206, 33100
Udin, Italy
Dipartimento Di Matematica E Informatica,
Iran
corsini@dimi.uniud.it
Polygroups fuzzy set
($epsilon$
$epsilon$ $vee$ q)fuzzy subpolygroup
Fuzzy logic
Implication operator
[[1] N. Ajmal and K. V. Thomas, A complete study of the lattice of fuzzy congruence and fuzzy##normal subgroups, Information Sciences, 82 (1995), 197218.##[2] R. Ameri and M. M. Zahedi, Hyperalgebraic systems, Italian J. Pure Appl. Math., 6 (1999),##[3] J. M. Anthony and H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl., 69 (1979),##[4] S. K. Bhakat, (2, 2 _q)fuzzy normal, quasinormal and maximal subgroups, Fuzzy Sets and##Systems, 112 (2000), 299312.##[5] S. K. Bhakat, (2 _q)level subsets, Fuzzy Sets and Systems, 103 (1999), 529533.##[6] S. K. Bhakat and P. Das, On the definition of a fuzzy subgroup, Fuzzy Sets and Systems, 51##(1992), 235241.##[7] S. K. Bhakat and P. Das, (2, 2 _q)fuzzy subgroup, Fuzzy Sets and Systems, 80 (1996),##[8] S. K. Bhakat and P. Das, Fuzzy subrings and ideals redefined, Fuzzy Sets and Systems, 81##(1996), 383393.##[9] P. Bhattacharya, Fuzzy subgroups: some characterizations. II, Information Sciences, 38##(1986), 293297.##[10] P. Bonansinga and P. Corsini, Sugli omomorfismi di semiipergruppi e di ipergruppi, B.U.M.I.,##1B, 1982.##[11] S. D. Comer, Polygroups derived from cogroups, J. Algebra, 89 (1984), 397405.##[12] S. D. Comer, Combinatorial aspects of relations, Algebra Universalis, 18 (1984), 7794.##[13] S. D. Comer, Extension of polygroups by polygroups and their representations using colour##schemes, Lecture notes in Meth., No 1004, Universal Algebra and Lattice Theory, (1982),##[14] S. D. Comer, A new fundation for theory of relations, Notre Dame J. Formal Logic, 24##(1983), 8187.##[15] S. D. Comer, Hyperstructures associated with character algebra and color schemes, New##Frontiers in Hyperstructures, Hadronic Press, (1996), 4966.##[16] P. Corsini, Prolegomena of hypergroup theory, Second Edition, Aviani Editor, 1993.##[17] P. Corsini, A new connection between hypergroups and fuzzy sets, Southeast Asian Bull.##Math., 27 (2)(2003), 221229.##[18] P. Corsini, Join spaces, power sets, fuzzy sets, Proceedings of the fifth Int. Congress on Algebraic##Hyperstructures, Jasi, Romania, July 410, 1993, Hadronic Press, Inc., Palm Harbor,##[19] P. Corsini and V. Leoreanu, Applications of hyperstructures theory, Advanced in Mathematics,##Kluwer Academic Publishers, 2003.##[20] P. Corsini and V. Leoreanu, Fuzzy sets and join spaces associated with rough sets, Rend.##Circ. Mat., Palarmo 51 (2002), 527536.##[21] P. Corsini and I. Tofan, On fuzzy hypergroups, Pure Math. Appl., 8 (1997), 2937.##[22] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl., 85 (1981), 264269.##[23] B. Davvaz, Fuzzy and anti fuzzy subhypergroups, Proc. International Conference on Intelligent##and Cognitive Systems, (1996), 140144.##[24] B. Davvaz, Fuzzy Hvgroups, Fuzzy Sets and Systems, 101 (1999), 191195.##[25] B. Davvaz, Fuzzy Hvsubmodules, Fuzzy Sets and Systems, 117 (2001), 477484.##[26] B. Davvaz and N. S. Poursalavati,On polygroup hyperrings and representations of polygroups,##J. Korean Math. Soc., 36 (6)(1999), 10211031.##[27] B. Davvaz, On polygroups and weak polygroups, Southeast Asian Bull. Math., 25 (2001),##[28] B. Davvaz, Elementary topics on weak polygroups, Bull. Korean Math. Soc., 40 (1) (2003),##[29] B. Davvaz, Fuzzy weak polygroups, Proc. 8th International Congress on Algebraic Hyperstructures##and Applications, 19 Sep., 2002, Samothraki, Greece, Spanidis Press, (2003),##[30] B. Davvaz, Polygroups with hyperoperators, J. Fuzzy Math., 9 (4) (2001), 815823.##[31] B. Davvaz, TLsubpolygroups of a polygroup, Pure Math. Appl., 12 (2) (2001), 137145.##[32] B. Davvaz, On polygroups and permutation polygroups, Math. Balkanica (N.S.) 14 (12)##(2000), 4158.##[33] A. Hasankhani and M. M. Zahedi, Fpolygroups and fuzzy subFpolygroups, J. Fuzzy Math.##6 (1) (1998), 97110.##[34] S. Ioulidis, Polygroups et certains de leurs properietes, Bull. Greek Math. Soc., 22 (1981),##[35] Ath. Kehagias, Lfuzzy join and meet hyperoperations and the associated Lfuzzy hyperalgebras,##Rend. Circ. Mat., Palermo, 51 (2002), 503526.##[36] Ath. Kehagias, An example of Lfuzzy join space, Rend. Circ. Mat., Palermo, 52 (2003),##[37] Leoreanu, Direct limit and inverse limit of join spaces associated with fuzzy sets, Pure Math.##Appl., 11 (2002), 509512.##[38] F. Marty, Sur une generalization de la notion de group, 8th Congress Math. Scandenaves,##Stockholm, (1934), 4549.##[39] J. Mittas, Hypergroupes canoniques, Math. Balkanica, Beograd, 2 (1972), 165179.##[40] W. Prenowitz, Projective geometries as multigroups, Amer. J. Math., 65 (1943), 235256.##[41] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512517.##[42] R. Roth, Character and conjugacy class hypergroups of a finite group, Annali di Matematica##Pura ed Appl., 105 (1975), 295311.##[43] I. Tofan and A. 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NEW CRITERIA FOR RULE SELECTION IN FUZZY LEARNING CLASSIFIER SYSTEMS
NEW CRITERIA FOR RULE SELECTION IN FUZZY LEARNING CLASSIFIER SYSTEMS
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Designing an effective criterion for selecting the best rule is a major problem in theprocess of implementing Fuzzy Learning Classifier (FLC) systems. Conventionally confidenceand support or combined measures of these are used as criteria for fuzzy rule evaluation. In thispaper new entities namely precision and recall from the field of Information Retrieval (IR)systems is adapted as alternative criteria for fuzzy rule evaluation. Several differentcombinations of precision and recall are redesigned to produce a metric measure. These newlyintroduced criteria are utilized as a rule selection mechanism in the method of Iterative RuleLearning (IRL) of FLC. In several experiments, three standard datasets are used to compare andcontrast the novel IR based criteria with other previously developed measures. Experimentalresults illustrate the effectiveness of the proposed techniques in terms of classificationperformance and computational efficiency.
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Designing an effective criterion for selecting the best rule is a major problem in theprocess of implementing Fuzzy Learning Classifier (FLC) systems. Conventionally confidenceand support or combined measures of these are used as criteria for fuzzy rule evaluation. In thispaper new entities namely precision and recall from the field of Information Retrieval (IR)systems is adapted as alternative criteria for fuzzy rule evaluation. Several differentcombinations of precision and recall are redesigned to produce a metric measure. These newlyintroduced criteria are utilized as a rule selection mechanism in the method of Iterative RuleLearning (IRL) of FLC. In several experiments, three standard datasets are used to compare andcontrast the novel IR based criteria with other previously developed measures. Experimentalresults illustrate the effectiveness of the proposed techniques in terms of classificationperformance and computational efficiency.
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MEHDI
EFTEKHARI
MEHDI
EFTEKHARI
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING, SHIRAZ UNIVERSITY,
SHIRAZ, IRAN
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING,
Iran
eftekhar@shirazu.ac.ir
MANSOUR
ZOLGHADRI JAHROMI
MANSOUR
ZOLGHADRI JAHROMI
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING, SHIRAZ
UNIVERSITY, SHIRAZ, IRAN
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING,
Iran
zjahromi@shirazu.ac.ir
SERAJEDDIN
KATEBI
SERAJEDDIN
KATEBI
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING, SHIRAZ UNIVERSITY,
SHIRAZ, IRAN
DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING,
Iran
katebi@shirazu.ac.ir
Fuzzy classification
Rule evaluation criteria
Information retrieval
Iterative rule learning
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