Geometric Operators in Decision Making with Minimization of Regret
We study different types of aggregation operators and
the decision making process with minimization of regret. We analyze
the original work developed by Savage and the recent work
developed by Yager that generalizes the MMR method creating a
parameterized family of minimal regret methods by using the ordered
weighted averaging (OWA) operator. We suggest a new method that
uses different types of geometric operators such as the weighted
geometric mean or the ordered weighted geometric operator (OWG)
to generalize the MMR method obtaining a new parameterized family
of minimal regret methods. The main result obtained in this method
is that it allows to aggregate negative numbers in the OWG operator.
Finally, we give an illustrative example.
[1] F. Chiclana, F. Herrera, and E. Herrera-Viedma, "The ordered weighted
geometric operator: Properties and application", in Proc. 8th Conf.
Inform. Processing and Management of Uncertainty in Knowledgebased
Systems (IPMU), Madrid, Spain, 2000, pp. 985-991.
[2] 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.
[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 business decision making, Thesis (in Spanish), Dept. Business
Administration, Univ. Barcelona, Barcelona, Spain, 2007.
[5] V. Torra, Information Fusion in Data Mining, Springer, New York,
2002.
[6] Z.S. Xu, "An Overview of Methods for Determining OWA Weights",
Int. J. Intelligent Systems, vol. 20, pp. 843-865, 2005.
[7] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating
Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[8] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[9] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[10] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[11] R.R. Yager, and D.P. Filev, "Parameterized "andlike" and "orlike"
OWA operators", Int. J. General Systems, vol. 22, pp. 297-316, 1994.
[12] R.R. Yager, "Constrained OWA Aggregation", Fuzzy Sets and Systems,
vol. 81, pp. 89-101, 1996.
[13] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[14] R.R. Yager, "E-Z OWA weights", in: Proc. 10th IFSA World Congress,
Istanbul, Turkey, 2003, pp. 39-42.
[15] R.R. Yager, "Decision making using minimization of regret", Int. J.
Approximate Reasoning, vol. 36, pp. 109-128, 2004.
[16] R.R. Yager, "Generalized OWA Aggregation Operators", Fuzzy Opt.
Decision Making, vol. 3, pp.93-107, 2004.
[17] R.R. Yager, "Centered OWA operators", Soft Computing, vol. 11, pp.
631-639, 2007.
[18] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[19] C.H. Cheng, and J.R. Chang, "MCDM aggregation model using
situational ME-OWA and ME-OWGA operators", Int. J. Uncertainty,
Fuzziness and Knowledge-Based Systems, vol. 14, pp. 421-443, 2006.
[20] F. Chiclana, F. Herrera, E. Herrera-Viedma, "Integrating multiplicative
preference relations in a multipurpose decision-making model based on
fuzzy preference relations", Fuzzy Sets and Systems, vol. 122, pp. 277-
291, 2001.
[21] F. Chiclana, F. Herrera, E. Herrera-Viedma, "Multiperson Decision
Making Based on Multiplicative Preference Relations", European J.
Operational Research, vol. 129, pp. 372-385, 2001.
[22] F. Chiclana, F. Herrera, E. Herrera-Viedma, and S. Alonso, "Induced
ordered weighted geometric operators and their use in the aggregation of
multiplicative preference relations", Int. J. Intelligent Systems, vol. 19,
pp. 233-255, 2004.
[23] F. Herrera, E. Herrera-Viedma, and F. Chiclana, "A study of the origin
and uses of the ordered weighted geometric operator in multicriteria
decision making", Int. J. Intelligent Systems, vol. 18, pp. 689-707,
2003.
[24] J.I. Peláez, J.M. Doña and A. Mesas, "Majority Multiplicative Ordered
Weighted Geometric Operators and Their Use in the Aggregation of
Multiplicative Preference Relations", Mathware & Soft Computing, vol.
12, pp. 107-120, 2005.
[25] Z.S. Xu, "EOWA and EOWG operators for aggregating linguistic labels
based on linguistic preference relations", Int. J. Uncertainty, Fuzziness
and Knowledge-Based Systems, vol. 12, pp. 791-810, 2004.
[26] Z.S. Xu, "An approach based on the uncertain LOWG and induced
uncertain LOWG operators to group decision making with uncertain
multiplicative linguistic preference relations", Decision Support
Systems, vol. 41, pp. 488-499, 2006.
[27] Z.S. Xu, and Q.L. Da, "The Ordered Weighted Geometric Averaging
Operators", Int. J. Intelligent Systems, vol. 17, pp. 709-716, 2002.
[28] Z.S. Xu, and R.R. Yager, "Some geometric aggregation operators based
on intuitionistic fuzzy sets", Int. J. General Systems, vol. 35, pp. 417-
433, 2006.
[29] R.R. Yager, and Z.S. Xu, "The continuous ordered weighted geometric
operator and its application to decision making", Fuzzy Sets and
Systems, vol. 157, pp. 1393-1402, 2006.
[30] L.J. Savage, "The theory of statistical decision", J. American Statistical
Association, vol. 46, pp. 55-67, 1951.
[31] L.J. Savage, The foundations of statistics, John Wiley & Sons, New
York, 1954.
[32] J. Azcel, and T.L. Saaty, "Procedures for synthesizing ratio
judgements", J. Mathematical Psychology, vol. 27, pp. 93-102, 1983.
[33] J. Azcel, and C. Alsina, "Synthesizing judgements: A functional
equations approach", Mathematical Modelling, vol. 9, pp. 311-320,
1987.
[1] F. Chiclana, F. Herrera, and E. Herrera-Viedma, "The ordered weighted
geometric operator: Properties and application", in Proc. 8th Conf.
Inform. Processing and Management of Uncertainty in Knowledgebased
Systems (IPMU), Madrid, Spain, 2000, pp. 985-991.
[2] 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.
[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 business decision making, Thesis (in Spanish), Dept. Business
Administration, Univ. Barcelona, Barcelona, Spain, 2007.
[5] V. Torra, Information Fusion in Data Mining, Springer, New York,
2002.
[6] Z.S. Xu, "An Overview of Methods for Determining OWA Weights",
Int. J. Intelligent Systems, vol. 20, pp. 843-865, 2005.
[7] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating
Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[8] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[9] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[10] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[11] R.R. Yager, and D.P. Filev, "Parameterized "andlike" and "orlike"
OWA operators", Int. J. General Systems, vol. 22, pp. 297-316, 1994.
[12] R.R. Yager, "Constrained OWA Aggregation", Fuzzy Sets and Systems,
vol. 81, pp. 89-101, 1996.
[13] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[14] R.R. Yager, "E-Z OWA weights", in: Proc. 10th IFSA World Congress,
Istanbul, Turkey, 2003, pp. 39-42.
[15] R.R. Yager, "Decision making using minimization of regret", Int. J.
Approximate Reasoning, vol. 36, pp. 109-128, 2004.
[16] R.R. Yager, "Generalized OWA Aggregation Operators", Fuzzy Opt.
Decision Making, vol. 3, pp.93-107, 2004.
[17] R.R. Yager, "Centered OWA operators", Soft Computing, vol. 11, pp.
631-639, 2007.
[18] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[19] C.H. Cheng, and J.R. Chang, "MCDM aggregation model using
situational ME-OWA and ME-OWGA operators", Int. J. Uncertainty,
Fuzziness and Knowledge-Based Systems, vol. 14, pp. 421-443, 2006.
[20] F. Chiclana, F. Herrera, E. Herrera-Viedma, "Integrating multiplicative
preference relations in a multipurpose decision-making model based on
fuzzy preference relations", Fuzzy Sets and Systems, vol. 122, pp. 277-
291, 2001.
[21] F. Chiclana, F. Herrera, E. Herrera-Viedma, "Multiperson Decision
Making Based on Multiplicative Preference Relations", European J.
Operational Research, vol. 129, pp. 372-385, 2001.
[22] F. Chiclana, F. Herrera, E. Herrera-Viedma, and S. Alonso, "Induced
ordered weighted geometric operators and their use in the aggregation of
multiplicative preference relations", Int. J. Intelligent Systems, vol. 19,
pp. 233-255, 2004.
[23] F. Herrera, E. Herrera-Viedma, and F. Chiclana, "A study of the origin
and uses of the ordered weighted geometric operator in multicriteria
decision making", Int. J. Intelligent Systems, vol. 18, pp. 689-707,
2003.
[24] J.I. Peláez, J.M. Doña and A. Mesas, "Majority Multiplicative Ordered
Weighted Geometric Operators and Their Use in the Aggregation of
Multiplicative Preference Relations", Mathware & Soft Computing, vol.
12, pp. 107-120, 2005.
[25] Z.S. Xu, "EOWA and EOWG operators for aggregating linguistic labels
based on linguistic preference relations", Int. J. Uncertainty, Fuzziness
and Knowledge-Based Systems, vol. 12, pp. 791-810, 2004.
[26] Z.S. Xu, "An approach based on the uncertain LOWG and induced
uncertain LOWG operators to group decision making with uncertain
multiplicative linguistic preference relations", Decision Support
Systems, vol. 41, pp. 488-499, 2006.
[27] Z.S. Xu, and Q.L. Da, "The Ordered Weighted Geometric Averaging
Operators", Int. J. Intelligent Systems, vol. 17, pp. 709-716, 2002.
[28] Z.S. Xu, and R.R. Yager, "Some geometric aggregation operators based
on intuitionistic fuzzy sets", Int. J. General Systems, vol. 35, pp. 417-
433, 2006.
[29] R.R. Yager, and Z.S. Xu, "The continuous ordered weighted geometric
operator and its application to decision making", Fuzzy Sets and
Systems, vol. 157, pp. 1393-1402, 2006.
[30] L.J. Savage, "The theory of statistical decision", J. American Statistical
Association, vol. 46, pp. 55-67, 1951.
[31] L.J. Savage, The foundations of statistics, John Wiley & Sons, New
York, 1954.
[32] J. Azcel, and T.L. Saaty, "Procedures for synthesizing ratio
judgements", J. Mathematical Psychology, vol. 27, pp. 93-102, 1983.
[33] J. Azcel, and C. Alsina, "Synthesizing judgements: A functional
equations approach", Mathematical Modelling, vol. 9, pp. 311-320,
1987.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60846", author = "José M. Merigó and Montserrat Casanovas", title = "Geometric Operators in Decision Making with Minimization of Regret", abstract = "We study different types of aggregation operators and
the decision making process with minimization of regret. We analyze
the original work developed by Savage and the recent work
developed by Yager that generalizes the MMR method creating a
parameterized family of minimal regret methods by using the ordered
weighted averaging (OWA) operator. We suggest a new method that
uses different types of geometric operators such as the weighted
geometric mean or the ordered weighted geometric operator (OWG)
to generalize the MMR method obtaining a new parameterized family
of minimal regret methods. The main result obtained in this method
is that it allows to aggregate negative numbers in the OWG operator.
Finally, we give an illustrative example.", keywords = "Decision making, Regret, Aggregation operators,OWA operator, OWG operator.", volume = "4", number = "3", pages = "398-8", }
