Decision Making with Dempster-Shafer Theory of Evidence Using Geometric Operators
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.
[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. 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.
[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.
[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. 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.
[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.
@article{"International Journal of Information, Control and Computer Sciences:61686", author = "José M. Merigó and Montserrat Casanovas", title = "Decision Making with Dempster-Shafer Theory of Evidence Using Geometric Operators", 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.
", keywords = "Decision making, aggregation operators, Dempster-
Shafer theory of evidence, Uncertainty, OWA operator, OWG
operator.", volume = "2", number = "11", pages = "3934-8", }
