Decision Making with Dempster-Shafer Theory of Evidence Using Geometric Operators
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 32799
##### Decision Making with Dempster-Shafer Theory of Evidence Using Geometric Operators

Authors: José M. Merigó, Montserrat Casanovas

Abstract:

We study the problem of decision making with Dempster-Shafer belief structure. We analyze the previous work developed by Yager about using the ordered weighted averaging (OWA) operator in the aggregation of the Dempster-Shafer decision process. We discuss the possibility of aggregating with an ascending order in the OWA operator for the cases where the smallest value is the best result. We suggest the introduction of the ordered weighted geometric (OWG) operator in the Dempster-Shafer framework. In this case, we also discuss the possibility of aggregating with an ascending order and we find that it is completely necessary as the OWG operator cannot aggregate negative numbers. Finally, we give an illustrative example where we can see the different results obtained by using the OWA, the Ascending OWA (AOWA), the OWG and the Ascending OWG (AOWG) operator.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1079618

References:

[1] A.P. Dempster, "Upper and lower probabilities induced by a multivalued mapping", Annals of Mathematical Statistics, vol. 38, pp. 325- 339, 1967.
[2] A.P. Dempster, "A generalization of Bayesian inference", J. Royal Statistical Society B, vol. 30, pp. 205-247, 1968.
[3] G. Shafer, Mathematical Theory of Evidence, Princeton University Press, Princeton, NJ, 1976.
[4] R.P. Srivastava, T. Mock, Belief Functions in Business Decisions, Physica-Verlag, Heidelberg, 2002.
[5] R.R. Yager, L. Liu, Classic Works of the Dempster-Shafer Theory of Belief Functions, Springer-Verlag, Berlin, 2008.
[6] 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.
[7] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A guide for practitioners, Springer-Verlag, Berlin, 2007.
[8] T. Calvo, G. Mayor, and R. Mesiar, Aggregation Operators: New Trends and applications, Physica-Verlag, New York, 2002.
[9] X. Liu, "The solution equivalence of minimax disparity and minimum variance problems for OWA operators", Int. J. Approximate Reasoning, vol. 45, pp. 68-81, 2007.
[10] 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.
[11] V. Torra, Information Fusion in Data Mining, Springer, New York, 2002.
[12] Y.M. Wang, C. Parkan, "A preemptive goal programming method for aggregating OWA operator weights in group decision making", Information Sciences, vol. 177, pp. 1867-1877, 2007.
[13] Z.S. Xu, "An Overview of Methods for Determining OWA Weights", Int. J. Intelligent Systems, vol. 20, pp. 843-865, 2005.
[14] Z.S. Xu, and Q.L. Da, "An Overview of Operators for Aggregating Information", Int. J. Intelligent Systems, vol. 18, pp. 953-969, 2003.
[15] R.R. Yager, "On generalized measures of realization in uncertain environments", Theory and Decision, vol. 33, pp. 41-69, 1992.
[16] R.R. Yager, "Families of OWA operators", Fuzzy Sets and Systems, vol. 59, pp. 125-148, 1993.
[17] R.R. Yager, "On weighted median aggregation", Int. J. Uncertainty Fuzziness and Knowledge-Based Systems, vol. 2, pp. 101-113, 1994.
[18] R.R. Yager, and D.P. Filev, "Parameterized "andlike" and "orlike" OWA operators", Int. J. General Systems, vol. 22, pp. 297-316, 1994.
[19] R.R. Yager, "Constrained OWA Aggregation", Fuzzy Sets and Systems, vol. 81, pp. 89-101, 1996.
[20] R.R. Yager, "Quantifier Guided Aggregation Using OWA operators", Int. J. Intelligent Systems, vol. 11, pp. 49-73, 1996.
[21] R.R. Yager, "Decision making using minimization of regret", Int. J. Approximate Reasoning, vol. 36, pp. 109-128, 2004.
[22] R.R. Yager, "Generalized OWA Aggregation Operators", Fuzzy Opt. Decision Making, vol. 3, pp.93-107, 2004.
[23] R.R. Yager, "Centered OWA operators", Soft Computing, vol. 11, pp. 631-639, 2007.
[24] R.R. Yager, and J. Kacprzyck, The Ordered Weighted Averaging Operators: Theory and Applications, Kluwer Academic Publishers, Norwell, MA, 1997.
[25] 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.
[26] 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.
[27] F. Chiclana, F. Herrera and 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.
[28] F. Chiclana, F. Herrera and E. Herrera-Viedma, "Multiperson Decision Making Based on Multiplicative Preference Relations", European J. Operational Research, vol. 129, pp. 372-385, 2001.
[29] F. Chiclana, F. Herrera and 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.
[30] 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.
[31] J.I. Pel├íez, J.M. Do├▒a 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.
[32] 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.
[33] 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.
[34] Z.S. Xu, and Q.L. Da, "The Ordered Weighted Geometric Averaging Operators", Int. J. Intelligent Systems, vol. 17, pp. 709-716, 2002.
[35] 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.
[36] 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.
[37] R.R. Yager, "Decision Making Under Dempster-Shafer Uncertainties", Int. J. General Systems, vol. 20, pp. 233-245, 1992.
[38] K.J. Engemann, H.E. Miller and R.R. Yager, "Decision making with belief structures: an application in risk management", Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 4, pp. 1-26, 1996.
[39] R.R. Yager, "Uncertainty modeling and decision support", Reliability Engineering and System Safety, vol. 85, pp. 341-354, 2004.