Decision Making using Maximization of Negret

We analyze the problem of decision making under ignorance with regrets. Recently, Yager has developed a new method for decision making where instead of using regrets he uses another type of transformation called negrets. Basically, the negret is considered as the dual of the regret. We study this problem in detail and we suggest the use of geometric aggregation operators in this method. For doing this, we develop a different method for constructing the negret matrix where all the values are positive. The main result obtained is that now the model is able to deal with negative numbers because of the transformation done in the negret matrix. We further extent these results to another model developed also by Yager about mixing valuations and negrets. Unfortunately, in this case we are not able to deal with negative numbers because the valuations can be either positive or negative.

Authors:

• José M. Merigó
• Montserrat Casanovas

Keywords:

• Decision Making
• Aggregation operators
• Negret
• OWA operator
• OWG operator.

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] 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.
[3] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New
Trends and applications, Physica-Verlag, New York, 2002.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 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.
[10] J.M. Merig├│, and M. Casanovas, "Geometric Operators in Decision
Making with Minimization of Regret", Int. J. Computer Systems
Science and Engineering, submitted for publication.
[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, "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.
[13] Z.S. Xu, and Q.L. Da, "The Ordered Weighted Geometric Averaging
Operators", Int. J. Intelligent Systems, vol. 17, pp. 709-716, 2002.
[14] R.R. Yager, "On generalized measures of realization in uncertain
environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[15] R.R. Yager, Families of OWA operators, Fuzzy Sets and Systems, vol.
59, pp. 125-148, 1993.
[16] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty
Fuzziness Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[17] R.R. Yager, and D.P. Filev, "Parameterized "andlike" and "orlike"
OWA operators", Int. J. General Systems, vol. 22, pp. 297-316, 1994.
[18] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators",
Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[19] R.R. Yager, "E-Z OWA weights", in: Proc. 10th IFSA World Congress,
Istanbul, Turkey, 2003, pp. 39-42.
[20] R.R. Yager, "Decision making using minimization of regret", Int. J.
Approximate Reasoning, vol. 36, pp. 109-128, 2004.
[21] R.R. Yager, "Centered OWA operators", Soft Computing, vol. 11, pp.
631-639, 2007.
[22] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging
Operators: Theory and Applications, Kluwer Academic Publishers,
Norwell, MA, 1997.
[23] 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.
[24] 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.
[25] L.J. Savage, "The theory of statistical decision", J. American Statistical
Association, vol. 46, pp. 55-67, 1951.
[26] L.J. Savage, The foundations of statistics, John Wiley & Sons, New
York, 1954.
[27] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York,
1980.
[28] J. Azcel, and T.L. Saaty, "Procedures for synthesizing ratio
judgements", J. Mathematical Psychology, vol. 27, pp. 93-102, 1983.
[29] J. Azcel, and C. Alsina, "Synthesizing judgements: A functional
equations approach", Mathematical Modelling, vol. 9, pp. 311-320,
1987.