Using Fuzzy Numbers in Heavy Aggregation Operators
We consider different types of aggregation operators
such as the heavy ordered weighted averaging (HOWA) operator and
the fuzzy ordered weighted averaging (FOWA) operator. We
introduce a new extension of the OWA operator called the fuzzy
heavy ordered weighted averaging (FHOWA) operator. The main
characteristic of this aggregation operator is that it deals with
uncertain information represented in the form of fuzzy numbers (FN)
in the HOWA operator. We develop the basic concepts of this
operator and study some of its properties. We also develop a wide
range of families of FHOWA operators such as the fuzzy push up
allocation, the fuzzy push down allocation, the fuzzy median
allocation and the fuzzy uniform allocation.
[1] R.R. Yager, "On Ordered Weighted Averaging Aggregation Operators
in Multi-Criteria Decision Making", IEEE Trans. Systems, Man and
Cybernetics, vol. 18, pp. 183-190, 1988.
[2] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New
Trends and applications, Physica-Verlag, New York, 2002.
[3] G. Canfora, and L. Troiano, "An Extensive Comparison between OWA
and OFNWA Aggregation", in: Proc. of the 8th SIGEF Congress,
Napoli, Italy, 2001.
[4] S.J. Chen, and S.M. Chen, "A new method for handling multi-criteria
fuzzy decision making problems using FN-IOWA operators",
Cybernetics and Systems, vol. 34, pp. 109-137, 2003.
[5] J.M. Merig├│, New Extensions to the OWA Operators and its application
in business decision making, Thesis (in Spanish), Dept. Business
Administration, Univ. Barcelona, Barcelona, Spain, 2007.
[6] J.M. Merig├│, and M. Casanovas, "Geometric Operators in Decision
Making with Minimization of Regret", Int. J. Computer Systems
Science and Engineering, vol. 1, pp. 111-118, 2007.
[7] J.M. Merig├│, and M. Casanovas, "Induced and uncertain heavy ordered
weighted averaging operators", Fuzzy Sets and Systems, to be published.
[8] H.B. Mitchell, and D.D. Estrakh, "An OWA operator with fuzzy ranks",
Int. J. Intelligent Systems, vol. 13, pp. 69-81, 1998.
[9] V. Torra, Information Fusion in Data Mining, Springer, New York,
2002.
[10] Z.S. Xu, "A fuzzy ordered weighted geometric operator and its
application in fuzzy AHP", Systems, Engineering and Electronics, vol.
24, pp. 31-33, 2002.
[11] Z.S. Xu, "An Overview of Methods for Determining OWA Weights",
Int. J. Intelligent Systems, vol. 20, pp. 843-865, 2005.
[12] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating
Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[13] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[14] R.R. Yager, "Families of OWA operators", Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[15] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[16] R.R. Yager, "Constrained OWA Aggregation", Fuzzy Sets and Systems,
vol. 81, pp. 89-101, 1996.
[17] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[18] R.R. Yager, "Heavy OWA Operators", Fuzzy Optim. Decision Making,
vol. 1, pp. 379-397, 2002.
[19] R.R. Yager, "Monitored heavy fuzzy measures and their role in decision
making under uncertainty", Fuzzy Sets and Systems, vol. 139, pp. 491-
513, 2003.
[20] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[21] S.S.L. Chang, and L.A. Zadeh, "On fuzzy mapping and control", IEEE
Trans. Systems, Man and Cybernetics, vol. 2, pp. 30-34, 1972.
[22] L.A. Zadeh, "The Concept of a Linguistic Variable and its application to
Approximate Reasoning. Part 1", Information Sciences, vol. 8, pp. 199-
249, "Part 2", Information Sciences, vol. 8, pp. 301-357, "Part 3",
Information Sciences, vol. 9, pp. 43-80, 1975.
[23] D. Dubois, and H. Prade, "Operations on fuzzy numbers", Int. J.
Systems Science, vol. 9, pp. 613-626, 1978.
[24] D. Dubois, and H. Prade, Fuzzy Sets and Systems: Theory and
Applications, Academic Press, New York, 1980.
[25] A. Kaufmann, and M.M. Gupta, Introduction to fuzzy arithmetic,
Publications Van Nostrand, Rheinhold, 1985.
[26] A. Kaufmann, and J. Gil Aluja, Management techniques for the
treatment of uncertainty, (in Spanish), Ed. Hispano-europea, Spain,
1987.
[27] A. Kaufmann, J. Gil-Aluja, and A. Terce├▒o, Mathematics for economic
and business management, (in Spanish), Ed. Foro Científico, Barcelona,
Spain, 1994.
[28] M. Mizumoto, and K. Tanaka, "The four operations of arithmetic on
fuzzy numbers", Systems, Computing and Control, vol. 7, pp. 73-81,
1976.
[29] S. Nahmias, "Fuzzy variables", Fuzzy Sets and Systems, vol. 1, pp. 97-
110, 1978.
[30] K.J. Arrow, L. Hurwicz, "An optimality criterion for decision making
under ignorance", in: C.F. Carter, J.L. Ford (Eds.), Uncertainty and
Expectations in Economics, Kelley, New Jersey, 1972.
[1] R.R. Yager, "On Ordered Weighted Averaging Aggregation Operators
in Multi-Criteria Decision Making", IEEE Trans. Systems, Man and
Cybernetics, vol. 18, pp. 183-190, 1988.
[2] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New
Trends and applications, Physica-Verlag, New York, 2002.
[3] G. Canfora, and L. Troiano, "An Extensive Comparison between OWA
and OFNWA Aggregation", in: Proc. of the 8th SIGEF Congress,
Napoli, Italy, 2001.
[4] S.J. Chen, and S.M. Chen, "A new method for handling multi-criteria
fuzzy decision making problems using FN-IOWA operators",
Cybernetics and Systems, vol. 34, pp. 109-137, 2003.
[5] J.M. Merig├│, New Extensions to the OWA Operators and its application
in business decision making, Thesis (in Spanish), Dept. Business
Administration, Univ. Barcelona, Barcelona, Spain, 2007.
[6] J.M. Merig├│, and M. Casanovas, "Geometric Operators in Decision
Making with Minimization of Regret", Int. J. Computer Systems
Science and Engineering, vol. 1, pp. 111-118, 2007.
[7] J.M. Merig├│, and M. Casanovas, "Induced and uncertain heavy ordered
weighted averaging operators", Fuzzy Sets and Systems, to be published.
[8] H.B. Mitchell, and D.D. Estrakh, "An OWA operator with fuzzy ranks",
Int. J. Intelligent Systems, vol. 13, pp. 69-81, 1998.
[9] V. Torra, Information Fusion in Data Mining, Springer, New York,
2002.
[10] Z.S. Xu, "A fuzzy ordered weighted geometric operator and its
application in fuzzy AHP", Systems, Engineering and Electronics, vol.
24, pp. 31-33, 2002.
[11] Z.S. Xu, "An Overview of Methods for Determining OWA Weights",
Int. J. Intelligent Systems, vol. 20, pp. 843-865, 2005.
[12] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating
Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[13] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[14] R.R. Yager, "Families of OWA operators", Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[15] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[16] R.R. Yager, "Constrained OWA Aggregation", Fuzzy Sets and Systems,
vol. 81, pp. 89-101, 1996.
[17] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[18] R.R. Yager, "Heavy OWA Operators", Fuzzy Optim. Decision Making,
vol. 1, pp. 379-397, 2002.
[19] R.R. Yager, "Monitored heavy fuzzy measures and their role in decision
making under uncertainty", Fuzzy Sets and Systems, vol. 139, pp. 491-
513, 2003.
[20] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[21] S.S.L. Chang, and L.A. Zadeh, "On fuzzy mapping and control", IEEE
Trans. Systems, Man and Cybernetics, vol. 2, pp. 30-34, 1972.
[22] L.A. Zadeh, "The Concept of a Linguistic Variable and its application to
Approximate Reasoning. Part 1", Information Sciences, vol. 8, pp. 199-
249, "Part 2", Information Sciences, vol. 8, pp. 301-357, "Part 3",
Information Sciences, vol. 9, pp. 43-80, 1975.
[23] D. Dubois, and H. Prade, "Operations on fuzzy numbers", Int. J.
Systems Science, vol. 9, pp. 613-626, 1978.
[24] D. Dubois, and H. Prade, Fuzzy Sets and Systems: Theory and
Applications, Academic Press, New York, 1980.
[25] A. Kaufmann, and M.M. Gupta, Introduction to fuzzy arithmetic,
Publications Van Nostrand, Rheinhold, 1985.
[26] A. Kaufmann, and J. Gil Aluja, Management techniques for the
treatment of uncertainty, (in Spanish), Ed. Hispano-europea, Spain,
1987.
[27] A. Kaufmann, J. Gil-Aluja, and A. Terce├▒o, Mathematics for economic
and business management, (in Spanish), Ed. Foro Científico, Barcelona,
Spain, 1994.
[28] M. Mizumoto, and K. Tanaka, "The four operations of arithmetic on
fuzzy numbers", Systems, Computing and Control, vol. 7, pp. 73-81,
1976.
[29] S. Nahmias, "Fuzzy variables", Fuzzy Sets and Systems, vol. 1, pp. 97-
110, 1978.
[30] K.J. Arrow, L. Hurwicz, "An optimality criterion for decision making
under ignorance", in: C.F. Carter, J.L. Ford (Eds.), Uncertainty and
Expectations in Economics, Kelley, New Jersey, 1972.
@article{"International Journal of Information, Control and Computer Sciences:53309", author = "José M. Merigó and Montserrat Casanovas", title = "Using Fuzzy Numbers in Heavy Aggregation Operators", abstract = "We consider different types of aggregation operators
such as the heavy ordered weighted averaging (HOWA) operator and
the fuzzy ordered weighted averaging (FOWA) operator. We
introduce a new extension of the OWA operator called the fuzzy
heavy ordered weighted averaging (FHOWA) operator. The main
characteristic of this aggregation operator is that it deals with
uncertain information represented in the form of fuzzy numbers (FN)
in the HOWA operator. We develop the basic concepts of this
operator and study some of its properties. We also develop a wide
range of families of FHOWA operators such as the fuzzy push up
allocation, the fuzzy push down allocation, the fuzzy median
allocation and the fuzzy uniform allocation.", keywords = "Aggregation operators, Fuzzy numbers, Fuzzy OWAoperator, Heavy OWA operator.", volume = "2", number = "12", pages = "4076-6", }
