The Induced Generalized Hybrid Averaging Operator and its Application in Financial Decision Making

We present the induced generalized hybrid averaging (IGHA) operator. It is a new aggregation operator that generalizes the hybrid averaging (HA) by using generalized means and order inducing variables. With this formulation, we get a wide range of mean operators such as the induced HA (IHA), the induced hybrid quadratic averaging (IHQA), the HA, etc. The ordered weighted averaging (OWA) operator and the weighted average (WA) are included as special cases of the HA operator. Therefore, with this generalization we can obtain a wide range of aggregation operators such as the induced generalized OWA (IGOWA), the generalized OWA (GOWA), etc. We further generalize the IGHA operator by using quasi-arithmetic means. Then, we get the Quasi-IHA operator. Finally, we also develop an illustrative example of the new approach in a financial decision making problem. The main advantage of the IGHA is that it gives a more complete view of the decision problem to the decision maker because it considers a wide range of situations depending on the operator used.

Authors:

• José M. Merigó
• Montserrat Casanovas

Keywords:

• Decision making
• Aggregation operators
• OWA operator
• Generalized means
• Selection of investments.

References:

[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] G. Beliakov, A. Pradera, and T. Calvo, Aggregation Functions: A guide
for practitioners, Springer-Verlag, Berlin, 2007.
[3] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New Trends
and applications, Physica-Verlag, New York, 2002.
[4] J.M. Merig├│, New Extensions to the OWA Operators and its application
in decision making, PhD Thesis (in Spanish), Dept. Business
Administration, Univ. Barcelona, Barcelona, Spain, 2008
[5] J.M. Merig├│, and M. Casanovas, "Induced aggregation operators in
decision making with Dempster-Shafer belief structure", Int. J.
Intelligent Systems (to be published).
[6] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[7] R.R. Yager, "Families of OWA operators", Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[8] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[9] N. Karayiannis, "Soft Learning Vector Quantization and Clustering
Algorithms Based on Ordered Weighted Aggregation Operators", IEEE
Trans. Neural Networks, vol. 11, 1093-1105, 2000.
[10] R.R. Yager, "Generalized OWA Aggregation Operators", Fuzzy Opt.
Decision Making, vol. 3, pp.93-107, 2004.
[11] J. Dujmovic, "Weighted conjunctive and disjunctive means and their
application in system evaluation", Publikacije Elektrotechnickog
Faculteta Beograd, Serija Matematika i Fizika, No. 483, pp. 147-158,
1974.
[12] H. Dyckhoff, and W. Pedrycz, "Generalized means as model of
compensative connectives", Fuzzy Sets and Systems, vol. 14, pp. 143-
154, 1984.
[13] G. Beliakov, "Learning Weights in the Generalized OWA Operators",
Fuzzy Opt. Decision Making, vol. 4, pp. 119-130, 2005.
[14] G.H. Hardy, J.E. Littlewood, and G. P├│lya, Inequalities, Cambridge
University Press, Cambridge, 1934.
[15] A.N. Kolmogoroff, "Sur la notion de la moyenne", Atti Della Accademia
Nazionale Dei Lincei, vol. 12, pp. 388-391, 1930.
[16] M. Nagumo, "Uber eine klasse der mittelwerte", Japanese J. of
Mathematics, vol. 6, pp. 71-79, 1930.
[17] J. Fodor, J.L. Marichal, and M. Roubens, "Characterization of the
ordered weighted averaging operators", IEEE Trans. Fuzzy Systems, vol.
3, pp. 236-240, 1995.
[18] J.M. Merig├│, and A.M. Gil-Lafuente, "The induced generalized OWA
operator", Information Sciences, vol. 179, pp. 729-741, 2009.
[19] R.R. Yager, "Induced aggregation operators", Fuzzy Sets and Systems,
vol. 137, pp. 59-69, 2003.
[20] R.R. Yager, and D.P. Filev, "Induced ordered weighted averaging
operators", IEEE Trans. Syst. Man Cybern., vol. 29, pp. 141-150, 1999.
[21] J.M. Merig├│, and M. Casanovas, "The fuzzy generalized OWA
operator", In Proceedings of the Conference SIGEF 2007, pp. 504-517,
Poiana-Brasov, Romania, 2007.
[22] J.M. Merig├│, M. Casanovas, "The uncertain generalized OWA operator
and its application in the selection of financial strategies", In
Proceedings of the International Conference AEDEM 2007, pp. 547-
556, Krakow, Poland, 2007.
[23] J.H. Wang, and J. Hao, "A new version of 2-tuple fuzzy linguistic
representation model for computing with words", IEEE Trans. Fuzzy
Systems, vol. 14, pp. 435-445, 2006.
[24] P.A. Schaefer, and H.B. Mitchell, "A generalized OWA operator", Int. J.
Intelligent Systems, vol. 14, pp. 123-143, 1999.
[25] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating
Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[26] Z.S. Xu, "A method based on linguistic aggregation operators for group
decision making with linguistic preference relations", Information
Sciences, vol. 166, pp. 19-30, 2004.
[27] Z.S. Xu, "A Note on Linguistic Hybrid Arithmetic Averaging Operator
in Multiple Attribute Group Decision Making with Linguistic
Information", Group Decision and Negotiation, vol. 15, pp. 593-604,
2006.
[28] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness and Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[29] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[30] R.R. Yager, E-Z OWA weights, In Proceedings of the 10th International
Fuzzy Systems Association (IFSA) World Congress, Istanbul, Turkey, pp.
39-42, 2003.
[31] R.R. Yager, "Centered OWA operators", Soft Computing, vol. 11, pp.
631-639, 2007.
[32] R.R. Yager, and D.P. Filev, "Parameterized "andlike" and "orlike"
OWA operators", Int. J. General Systems, vol. 22, pp. 297-316, 1994.