Extended TOPSISs for Belief Group Decision Making
Author(s)
Chao Fu*
ABSTRACT
Multiple attribute decision analysis (MADA) problems in the situation of belief group decision making (BGDM) are a special class of decision problems, where the attribute evaluations of each decision maker (DM) are represented by belief functions. In order to solve these special problems, in this paper, TOPSIS (technique for order preference by similarity to ideal solution) model is extended by three approaches, by which group preferences are aggregated in different manners. Corresponding to the three approaches, three extended TOPSIS models, the pre-model, post-model, and inter-model, are developed and their procedures are elaborated step by step. Aggregating group preferences in the three extended models respectively depends on Dempster’s rule or its modifications, some social choice functions, and some mean approaches. Furthermore, a numerical example clearly illustrates the procedures of the three extended models for BGDM.
Multiple attribute decision analysis (MADA) problems in the situation of belief group decision making (BGDM) are a special class of decision problems, where the attribute evaluations of each decision maker (DM) are represented by belief functions. In order to solve these special problems, in this paper, TOPSIS (technique for order preference by similarity to ideal solution) model is extended by three approaches, by which group preferences are aggregated in different manners. Corresponding to the three approaches, three extended TOPSIS models, the pre-model, post-model, and inter-model, are developed and their procedures are elaborated step by step. Aggregating group preferences in the three extended models respectively depends on Dempster’s rule or its modifications, some social choice functions, and some mean approaches. Furthermore, a numerical example clearly illustrates the procedures of the three extended models for BGDM.
KEYWORDS
basic belief assignment, belief group decision making, belief preferences aggregation, TOPSIS
basic belief assignment, belief group decision making, belief preferences aggregation, TOPSIS
Cite this paper
nullC. Fu, "Extended TOPSISs for Belief Group Decision Making," Journal of Service Science and Management, Vol. 1 No. 1, 2008, pp. 11-20. doi: 10.4236/jssm.2008.11002.
nullC. Fu, "Extended TOPSISs for Belief Group Decision Making," Journal of Service Science and Management, Vol. 1 No. 1, 2008, pp. 11-20. doi: 10.4236/jssm.2008.11002.
References
[1] N. Bryson, A. Mobolurin, “A Process for Generating Quantitative Belief Functions”, European Journal of Operational Research, 115(3), 1999, pp. 624-633. [2] C.T. Chen, “Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment”, Fuzzy Sets and Systems, 114(1), 2000, pp. 1-9. [3] C.T. Chen, C.T. Lin, and S.F. Huang, “A Fuzzy Approach for Supplier Evaluation and Selection in Supply Chain Management”, International Journal of Production Economics, 102(2), 2006, pp. 289-301. [4] T.C. Chu, “Facility Location Selection Using Fuzzy TOPSIS under Group Decisions”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(6), 2002, pp. 687-701. [5] A. Dempster, “Upper and Lower Probabilities Induced by Multivalued Mapping”, Annals of Mathematical Statistics, 38, 1967, pp. 325-339. [6] H. Deng, C.H. Yeh, and R. J. Willis, “Inter-Company Comparison Using Modified TOPSIS with Objective Weights”, Computers & Operations Research, 27(10), 2000, pp. 963-973. [7] Y. Deng, W. Shi, Z Zhu, and Q Liu, “Combining belief functions based on distance of evidence”, Decision Support Systems, 38(3), 2004, pp. 489-493. [8] D. Dubois, H. Prade, “Representation and combination of uncertainty with belief functions and possibility measures”, Computational Intelligence, 4, 1998, pp. 244-264. [9] Z. Elouedi, K. Mellouli, and P. Smets, “Belief Decision Trees: Theoretical Foundations”, International Journal of Approximate Reasoning, 28(2-3), 2001, pp. 91-124. [10] C. Fu, S.L. Yang, X. Ji, “A Pre-Extension of TOPSIS for Belief Group Decision Making”, International Conference on Wireless Communications, Networking and Mobile Computing, WiCOM, 2007, pp. 5725-5728. [11] C. Fu, S.L. Yang, “Solutions to Belief Group Decision Making Using Extended TOPSIS”, International Conference on Management Science and Engineering, 2007, pp. 458-463. [12] C. Fu, S.L. Yang, W.X. Lu, “An Extended TOPSIS for Belief Group Decision Making”, International Conference on Fuzzy Systems and Knowledge Discovery, 2007, pp. 551-555. [13] R. Haenni, “Are alternatives to Dempster’s rule of combination real alternatives? Comments on “About the belief function combination and the conflict management problem””, Information Fusion, 3(4), 2002, pp. 237-239. [14] C.L. Hwang, M.J. Lin, Group Decision Making under Multiple Criteria, Berlin: Springer-Verlag, Berlin, 1987. [15] C.L. Hwang, and K. Yoon, Multiple Attribute Decision Making, Berlin: Springer-Verlag, Berlin, 1981. [16] G.R. Jahanshahloo, F.H. Lotfi, and M. Izadikhah, “Extension of the TOPSIS Method for Decision-Making Problems with Fuzzy Data”, Applied Mathematics and Computation, 181(2), 2006, pp. 1544-1551. [17] G.R. Jahanshahloo, F.H. Lotfi and M. Izadikhah, “An Algorithmic Method to Extend TOPSIS for Decision-Making Problems with Interval Data”, Applied Mathematics and Computation, 175(2), 2006, pp. 1375-1384. [18] M.S. Kuo, G.H. Tzeng, and W.C. Huang, “Group Decision-Making Based on Concepts of Ideal and Anti-ideal Points in a Fuzzy Environment”, Mathematics and Computer Modelling, vol. 45, no. 3-4, 324-339, Feb. 2007. [19] E. Lefevre, O. Colot, and P. Vannoorenberghe, “Belief function combination and conflict management”, Information Fusion, 3, 2002, pp. 149-162. [20] D.F. Li, “Compromise Ratio Method for Fuzzy Multi-attribute Group Decision Making”, Applied Soft Computing, 7(3), 2007, pp. 807-817. [21] C.K. Murphy, “Combining belief functions when evidence conflicts”, Decision Support Systems, 29(1), 2000, pp. 1-9. [22] T.L. Saaty, The Analytic Hierarchy Process (edition 2), RWS publication, Pittsburgh, PA, 1990. [23] G. Shafer, A Mathematical Theory of Evidence, Princeton: Princeton University Press, Princeton, 1976. [24] A. Shanian, O. Savadogo, “TOPSIS Multiple-criteria Decision Support Analysis for Material Selection of Metallic Bipolar Plates for Polymer Electrolyte Fuel Cell”, Journal of Power Sources, 159(2), 2006, pp. 1095-1104. [25] H.S. Shih, H.J. Shyur, and E.S. Lee, “An Extension of TOPSIS for Group Decision Making”, Mathematical and Computer Modelling, 45(7-8), 2007, pp. 801-813. [26] P. Smets, “The combination of evidence in the transferable belief model”, IEEE Transaction on Pattern Analysis and Machine Intelligence, 12(5), 1990, pp. 447-458. [27] P. Smets, “Analyzing the combination of conflicting belief functions”, Information Fusion, 8(4), 2007, pp. 387-412. [28] P. Smets, K. Kennes, “The Transferable Belief Model”, Artificial Intelligence, 66 (2), 1994, pp. 191-234. [29] T.C. Wang, T.H. Chang, “Application of TOPSIS in Evaluating Initial Training Aircraft under a Fuzzy Environment”, Expert Systems with Applications, 33(4), 2007, pp. 870-880. [30] Y.M. Wang, Y. Luo, and Z.S. Hua, “A Note on Group Decision-Making Based on Concepts of Ideal and Anti-ideal Points in a Fuzzy Environment”, Mathematical and Computer Modelling, 46(9-10), 2007, pp. 1256-1264 [31] S.K.M. Wong, P. Lingras, “Representation of Qualitative User Preference by Quantitative Belief Functions”, IEEE Transactions on Knowledge and Data Engineering, 6(1), 1994, pp. 72-78. [32] R.R. Yager, “On the Dempster-Shafer Framework and New Combination Rules”, Information Sciences, 41(2), 1987, pp. 93-137. [33] R.R. Yager, “Quasi-associative operations in the combination of evidence”, Kybernetes, 16, 1987, 37-41. [34] A.Ben Yaghlane, T. Denoeux, and K. Mellouli, “Constructing Belief Functions from Qualitative Expert Opinions”, Information and Communication Technologies, ICTTA’06, 1, 2006, pp. 1363-1368. [35] J.B. Yang, “Rule and Utility Based Evidential Reasoning Approach for Multiattribute Decision Analysis under Uncertainties”, European Journal of Operational Research, 131, 2001, pp. 31-61. [36] M. Zeleny, “A Concept of Compromise Solutions and the Method of the Displaced Ideal”, Computers and Operations Research, 1(3-4), 1974, pp. 479-496.
[1] N. Bryson, A. Mobolurin, “A Process for Generating Quantitative Belief Functions”, European Journal of Operational Research, 115(3), 1999, pp. 624-633. [2] C.T. Chen, “Extensions of the TOPSIS for Group Decision-Making under Fuzzy Environment”, Fuzzy Sets and Systems, 114(1), 2000, pp. 1-9. [3] C.T. Chen, C.T. Lin, and S.F. Huang, “A Fuzzy Approach for Supplier Evaluation and Selection in Supply Chain Management”, International Journal of Production Economics, 102(2), 2006, pp. 289-301. [4] T.C. Chu, “Facility Location Selection Using Fuzzy TOPSIS under Group Decisions”, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(6), 2002, pp. 687-701. [5] A. Dempster, “Upper and Lower Probabilities Induced by Multivalued Mapping”, Annals of Mathematical Statistics, 38, 1967, pp. 325-339. [6] H. Deng, C.H. Yeh, and R. J. Willis, “Inter-Company Comparison Using Modified TOPSIS with Objective Weights”, Computers & Operations Research, 27(10), 2000, pp. 963-973. [7] Y. Deng, W. Shi, Z Zhu, and Q Liu, “Combining belief functions based on distance of evidence”, Decision Support Systems, 38(3), 2004, pp. 489-493. [8] D. Dubois, H. Prade, “Representation and combination of uncertainty with belief functions and possibility measures”, Computational Intelligence, 4, 1998, pp. 244-264. [9] Z. Elouedi, K. Mellouli, and P. Smets, “Belief Decision Trees: Theoretical Foundations”, International Journal of Approximate Reasoning, 28(2-3), 2001, pp. 91-124. [10] C. Fu, S.L. Yang, X. Ji, “A Pre-Extension of TOPSIS for Belief Group Decision Making”, International Conference on Wireless Communications, Networking and Mobile Computing, WiCOM, 2007, pp. 5725-5728. [11] C. Fu, S.L. Yang, “Solutions to Belief Group Decision Making Using Extended TOPSIS”, International Conference on Management Science and Engineering, 2007, pp. 458-463. [12] C. Fu, S.L. Yang, W.X. Lu, “An Extended TOPSIS for Belief Group Decision Making”, International Conference on Fuzzy Systems and Knowledge Discovery, 2007, pp. 551-555. [13] R. Haenni, “Are alternatives to Dempster’s rule of combination real alternatives? Comments on “About the belief function combination and the conflict management problem””, Information Fusion, 3(4), 2002, pp. 237-239. [14] C.L. Hwang, M.J. Lin, Group Decision Making under Multiple Criteria, Berlin: Springer-Verlag, Berlin, 1987. [15] C.L. Hwang, and K. Yoon, Multiple Attribute Decision Making, Berlin: Springer-Verlag, Berlin, 1981. [16] G.R. Jahanshahloo, F.H. Lotfi, and M. Izadikhah, “Extension of the TOPSIS Method for Decision-Making Problems with Fuzzy Data”, Applied Mathematics and Computation, 181(2), 2006, pp. 1544-1551. [17] G.R. Jahanshahloo, F.H. Lotfi and M. Izadikhah, “An Algorithmic Method to Extend TOPSIS for Decision-Making Problems with Interval Data”, Applied Mathematics and Computation, 175(2), 2006, pp. 1375-1384. [18] M.S. Kuo, G.H. Tzeng, and W.C. Huang, “Group Decision-Making Based on Concepts of Ideal and Anti-ideal Points in a Fuzzy Environment”, Mathematics and Computer Modelling, vol. 45, no. 3-4, 324-339, Feb. 2007. [19] E. Lefevre, O. Colot, and P. Vannoorenberghe, “Belief function combination and conflict management”, Information Fusion, 3, 2002, pp. 149-162. [20] D.F. Li, “Compromise Ratio Method for Fuzzy Multi-attribute Group Decision Making”, Applied Soft Computing, 7(3), 2007, pp. 807-817. [21] C.K. Murphy, “Combining belief functions when evidence conflicts”, Decision Support Systems, 29(1), 2000, pp. 1-9. [22] T.L. Saaty, The Analytic Hierarchy Process (edition 2), RWS publication, Pittsburgh, PA, 1990. [23] G. Shafer, A Mathematical Theory of Evidence, Princeton: Princeton University Press, Princeton, 1976. [24] A. Shanian, O. Savadogo, “TOPSIS Multiple-criteria Decision Support Analysis for Material Selection of Metallic Bipolar Plates for Polymer Electrolyte Fuel Cell”, Journal of Power Sources, 159(2), 2006, pp. 1095-1104. [25] H.S. Shih, H.J. Shyur, and E.S. Lee, “An Extension of TOPSIS for Group Decision Making”, Mathematical and Computer Modelling, 45(7-8), 2007, pp. 801-813. [26] P. Smets, “The combination of evidence in the transferable belief model”, IEEE Transaction on Pattern Analysis and Machine Intelligence, 12(5), 1990, pp. 447-458. [27] P. Smets, “Analyzing the combination of conflicting belief functions”, Information Fusion, 8(4), 2007, pp. 387-412. [28] P. Smets, K. Kennes, “The Transferable Belief Model”, Artificial Intelligence, 66 (2), 1994, pp. 191-234. [29] T.C. Wang, T.H. Chang, “Application of TOPSIS in Evaluating Initial Training Aircraft under a Fuzzy Environment”, Expert Systems with Applications, 33(4), 2007, pp. 870-880. [30] Y.M. Wang, Y. Luo, and Z.S. Hua, “A Note on Group Decision-Making Based on Concepts of Ideal and Anti-ideal Points in a Fuzzy Environment”, Mathematical and Computer Modelling, 46(9-10), 2007, pp. 1256-1264 [31] S.K.M. Wong, P. Lingras, “Representation of Qualitative User Preference by Quantitative Belief Functions”, IEEE Transactions on Knowledge and Data Engineering, 6(1), 1994, pp. 72-78. [32] R.R. Yager, “On the Dempster-Shafer Framework and New Combination Rules”, Information Sciences, 41(2), 1987, pp. 93-137. [33] R.R. Yager, “Quasi-associative operations in the combination of evidence”, Kybernetes, 16, 1987, 37-41. [34] A.Ben Yaghlane, T. Denoeux, and K. Mellouli, “Constructing Belief Functions from Qualitative Expert Opinions”, Information and Communication Technologies, ICTTA’06, 1, 2006, pp. 1363-1368. [35] J.B. Yang, “Rule and Utility Based Evidential Reasoning Approach for Multiattribute Decision Analysis under Uncertainties”, European Journal of Operational Research, 131, 2001, pp. 31-61. [36] M. Zeleny, “A Concept of Compromise Solutions and the Method of the Displaced Ideal”, Computers and Operations Research, 1(3-4), 1974, pp. 479-496.