# American Institute of Mathematical Sciences

December  2015, 8(6): 1139-1154. doi: 10.3934/dcdss.2015.8.1139

## Dynamic systems based on preference graph and distance

 1 Business School of Sichuan University, Chengdu 610065, China, China 2 School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China 3 Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, 1068 Xueyuan Avenue, Shenzhen University Town, Shenzhen 518055, China

Received  May 2015 Revised  August 2015 Published  December 2015

A group decision-making approach fusing preference conflicts and compatibility measure is proposed , focused on dynamic group decision making with preference information of policymakers at each time describing with dynamic preference Hasse diagram with the identification framework of relation between alternative pairs is H=$\{\succ,\parallel,\succeq,\preceq,\approx,\prec,\phi\}$, and the preference graph may contain incomplete decision making alternatives. First, the relationship between preference sequences is be defined on the basis of concepts about preference, preference sequence and preference graph; and defining the decision function that can reflect dynamic preference, such as conflict ,comply support and preference distance measure. Finally, through the perspective of conflict and compatible aggregating the comprehensive preference of each decision makers in each period, and by establishing the optimization model based on lattice preference distance measure to assemble group preference, gives the specific steps of the decision making. The feasibility and effectiveness of the approach proposed in this paper are illustrated with a numerical example.
Citation: Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139
##### References:
 [1] S. B. Amor and J. M. Martel, A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations, European Journal of Operational Research, 237 (2014), 1165-1169. doi: 10.1016/j.ejor.2014.03.036. [2] M. Z. Angiz, A. Tajaddini, A. Mustafa and M. J. Kamali, Ranking alternatives in a preferential voting system using fuzzy concepts and data envelopment analysis, Comput. Ind. Eng., 63 (2012), 784-790. [3] J. R. Busemeyera and T. J. Pleskac, Theoretical tools for understanding and aiding dynamic decision making, Journal of Mathematical Psychology, 53 (2009), 126-138. doi: 10.1016/j.jmp.2008.12.007. [4] G. Campanella and R. A. Ribeiro, A framework for dynamic multiple-criteria decision making, Decision Support Systems, 52 (2011), 52-60. doi: 10.1016/j.dss.2011.05.003. [5] S. Y. Chen and G. T. Fu, Combining fuzzy iteration model with dynamic programming to solve multiobjective multistage decision making problems, Fuzzy Sets and Systems, 152 (2005), 499-512. doi: 10.1016/j.fss.2004.10.006. [6] Y. L. Chen and L. C. Cheng, An approach to group ranking decisions in a dynamic environment, Decision Support Systems, 48 (2010), 622-634. doi: 10.1016/j.dss.2009.12.003. [7] Y. L. Chen and L. C. Cheng, Mining maximum consensus sequences from group ranking data, European Journal of Operational Research, 198 (2009), 241-253. doi: 10.1016/j.ejor.2008.09.004. [8] Y. L. Chen, L. C. Cheng and P. H. Huang, Mining consensus preference graphs from users' ranking data, Decision Support Systems, 54 (2013), 1055-1064. doi: 10.1016/j.dss.2012.10.031. [9] F. Chiclana, J. M. T. García, M. J. Moral and E. Herrera-viedma, A statistical comparative study of different similarity measures of consensus in group decision making, Information Sciences, 221 (2013), 110-123. doi: 10.1016/j.ins.2012.09.014. [10] S. J. Chuu, Selecting the advanced manufacturing technology using fuzzy multiple attributes group decision making with multiple fuzzy information, Computers & Industrial Engineering, 57 (2009), 1033-1042. doi: 10.1016/j.cie.2009.04.011. [11] D. Engelage, Optimal stopping with dynamic variational preferences, Journal of Economic Theory, 146 (2011), 2042-2074. doi: 10.1016/j.jet.2011.06.014. [12] Z. P. Fan, Q. Yue, B. Feng and Y. Liu, An approach to group decision-making with uncertain preference ordinals, Computers & Industrial Engineering, 58 (2010), 51-57. doi: 10.1016/j.cie.2009.08.001. [13] C. X. Guo, A method for aggregating group preference based on pair-wise comparison with random binary relations under interval belief structures, Mathematics & Information Sciences, 6 (2012), 869-880. [14] C. X. Guo, H. Gong and Y. H. Guo, Approach for random lattice order ranking based on preference entropy under interval belief degree circumstance, Operations Research and Management Science, 22 (2013), 21-29. [15] Y. Guo, Lattice Order Making[M], Shanghai Science and Technology Publishing House, Shanghai, 2003. [16] T. M. Gureckis and B. C. Love, Learning in noise: Dynamic decision-making in a variable environment, Journal of Mathematical Psychology, 53 (2009), 180-193. doi: 10.1016/j.jmp.2009.02.004. [17] C. X. Guo and Y. Peng, Lattice order group decision making with interval probability based on prospect theory, Group Decis. Negot., 24 (2015), 753-775. [18] E. D. Hahn, Judgmental consistency and consensus in stochastic multicriteria decision making, Expert Systems with Applications, 37 (2010), 3784-3791. doi: 10.1016/j.eswa.2009.11.042. [19] C. H. Han, J. K. Kim and S. H. Choi, Prioritizing engineering characteristics in quality function deployment with incomplete information: A linear partial ordering approach, Int. J. Production Economics, 91 (2004), 235-249. doi: 10.1016/j.ijpe.2003.09.001. [20] J. D. Hey and J. A. Knoll, Strategies in dynamic decision making - An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology, 32 (2011), 399-409. doi: 10.1016/j.joep.2011.02.011. [21] B. Huang and C. X. Guo, Intuitionistic fuzzy multigranulation rough sets, Information Sciences, 277 (2014), 299-320. doi: 10.1016/j.ins.2014.02.064. [22] Y. Huang and J. W. Hutchinson, The roles of planning, learning, and mental models in repeated dynamic decision making, Organizational Behavior and Human Decision Processes, 122 (2013), 163-176. doi: 10.1016/j.obhdp.2013.07.001. [23] K. Jabeur and J. M. Martel, A collective choicemethod based on individual preferences relational systems(prs), Eur. J. Oper. Res., 177 (2007), 1549-1565. doi: 10.1016/j.ejor.2005.10.028. [24] K. Jabeur and J. M. Martel, An agreement index with respect to a consensus preorder, Group Decis. Negot., 19 (2010), 571-590. doi: 10.1007/s10726-009-9160-3. [25] K. Jabeur and J. M. Martel, An ordinal sorting method for group decision-making, Eur. J. Oper. Res., 180 (2007), 1272-1289. doi: 10.1016/j.ejor.2006.05.032. [26] K. Jabeur, J. M. Martel and A. Guitouni, Deriving a minimum distance-based collective preorder: A binary mathematical programming approach, OR Spectr., 34 (2012), 23-42. doi: 10.1007/s00291-009-0192-5. [27] K. Jabeur, J. M. Martel and S. B. Khélif, A distance-based collective preorder integrating the relative importance of the group's members, Group Decis. Negotiat., 13 (2004), 327-349. doi: 10.1023/B:GRUP.0000042894.00775.75. [28] S. Jullien-Ramasso, G. Mauris, L. Valet and P. Bolon, A decision support system for animated film selection based on a multi-criteria aggregation of referees' ordinal preferences, Expert Syst. Appl., 39 (2012), 4250-4257. doi: 10.1016/j.eswa.2011.09.109. [29] D. Lerche, S. Y. Matsuzaki, P. B. Sørensen, L. Carlsen and O. J. Nielsen, Ranking of chemical substances based on the Japanese Pollutant Release and Transfer Register using partial order theory and random linear extensions, Chemosphere, 55 (2004), 1005-1025. doi: 10.1016/j.chemosphere.2004.01.023. [30] D. Lerche and P. B. Sørensen, Evaluation of the ranking probabilities for partial orders based on random linear extensions, Chemosphere, 53 (2003), 981-992. doi: 10.1016/S0045-6535(03)00558-7. [31] Y. H. Lin , P. C. Lee and H. I. Ting, Dynamic multi-attribute decision making model with grey number evaluations, Expert Systems with Applications, 35 (2008), 1638-1644. doi: 10.1016/j.eswa.2007.08.064. [32] L. C. Ma, Visualizing preferences on spheres for group decisions based on multiplicative preference relations, European Journal of Operational Research, 203 (2010), 176-184. doi: 10.1016/j.ejor.2009.07.008. [33] Z. X. Ma, Applying theory of partially ordered sets to study data envelopment analysis, Journal of Systems Engineering, 3 (2002), 3-8. [34] Z. X. MA, Method of data envelopment analysis based on the theory of partially ordered sets, Systems Engineering-Theory & Practice, 4 (2003), 12-17. [35] G. Martinelli, J. Eidsvik and R. Hauge, Dynamic decision making for graphical models applied to oil exploration, European Journal of Operational Research, 230 (2013), 688-702. doi: 10.1016/j.ejor.2013.04.057. [36] R. Mu, Z. X. Ma and W. Cui, Data envelopment analysis method based on poset theory, Systems Engineering and Electronics, 35 (2013), 350-356. [37] G. Munda, Intensity of preference and related uncertainty in non-compensatory aggregation rules, Theory Dec., 73 (2012), 649-669. doi: 10.1007/s11238-012-9317-4. [38] T. D. Nielsen and J. Y. Jaffray, Dynamic decision making without expected utility: An operational approach, European Journal of Operational Research, 169 (2006), 226-246. doi: 10.1016/j.ejor.2004.05.029. [39] J. H. Park, H. J. Cho and Y. C. Kwun, Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making, Computers & Mathematics with Applications, 65 (2013), 731-744. doi: 10.1016/j.camwa.2012.12.008. [40] R. R. Yager, On the fusion of imprecise uncertainty measures using belief structures, Information Science, 181 (2011), 3199-3209. doi: 10.1016/j.ins.2011.02.010. [41] B. Roy and R. Slowinski, Criterion of distance between technical programming and socio-economic priority, RAIRO Recherche Opérationnelle, 27 (1993), 45-60. [42] A. M. Saks and B. E. Ashforth, Change in job search behaviors and employment outcomes, Journal of Vocational Behavior, 56 (2000), 277-287. doi: 10.1006/jvbe.1999.1714. [43] Z. X. Su, M. Y. Chen, G. P. Xia and L. Wang, An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications, 38 (2011), 15286-15295. doi: 10.1016/j.eswa.2011.06.022. [44] J. Thenie and J. P. Vial, Step decision rules for multistage stochastic programming: A heuristic approach, Automatica, 44 (2008), 1569-1584. doi: 10.1016/j.automatica.2008.02.001. [45] J. M. Wang, Robust optimization analysis for multiple attribute decision making problems with imprecise information, Ann Oper Res, 197 (2012), 109-122. doi: 10.1007/s10479-010-0734-x. [46] X. Y. Wang and H. Y. Meng, A multi-stage dynamic decision-making model of mine resources exploitation with many running units-theoretical analysis, Procedia Earth and Planetary Science, 1 (2009), 1654-1660. doi: 10.1016/j.proeps.2009.09.254. [47] Z. S. Xu and R. R. Yager, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning, 48 (2008), 246-262. doi: 10.1016/j.ijar.2007.08.008.

show all references

##### References:
 [1] S. B. Amor and J. M. Martel, A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations, European Journal of Operational Research, 237 (2014), 1165-1169. doi: 10.1016/j.ejor.2014.03.036. [2] M. Z. Angiz, A. Tajaddini, A. Mustafa and M. J. Kamali, Ranking alternatives in a preferential voting system using fuzzy concepts and data envelopment analysis, Comput. Ind. Eng., 63 (2012), 784-790. [3] J. R. Busemeyera and T. J. Pleskac, Theoretical tools for understanding and aiding dynamic decision making, Journal of Mathematical Psychology, 53 (2009), 126-138. doi: 10.1016/j.jmp.2008.12.007. [4] G. Campanella and R. A. Ribeiro, A framework for dynamic multiple-criteria decision making, Decision Support Systems, 52 (2011), 52-60. doi: 10.1016/j.dss.2011.05.003. [5] S. Y. Chen and G. T. Fu, Combining fuzzy iteration model with dynamic programming to solve multiobjective multistage decision making problems, Fuzzy Sets and Systems, 152 (2005), 499-512. doi: 10.1016/j.fss.2004.10.006. [6] Y. L. Chen and L. C. Cheng, An approach to group ranking decisions in a dynamic environment, Decision Support Systems, 48 (2010), 622-634. doi: 10.1016/j.dss.2009.12.003. [7] Y. L. Chen and L. C. Cheng, Mining maximum consensus sequences from group ranking data, European Journal of Operational Research, 198 (2009), 241-253. doi: 10.1016/j.ejor.2008.09.004. [8] Y. L. Chen, L. C. Cheng and P. H. Huang, Mining consensus preference graphs from users' ranking data, Decision Support Systems, 54 (2013), 1055-1064. doi: 10.1016/j.dss.2012.10.031. [9] F. Chiclana, J. M. T. García, M. J. Moral and E. Herrera-viedma, A statistical comparative study of different similarity measures of consensus in group decision making, Information Sciences, 221 (2013), 110-123. doi: 10.1016/j.ins.2012.09.014. [10] S. J. Chuu, Selecting the advanced manufacturing technology using fuzzy multiple attributes group decision making with multiple fuzzy information, Computers & Industrial Engineering, 57 (2009), 1033-1042. doi: 10.1016/j.cie.2009.04.011. [11] D. Engelage, Optimal stopping with dynamic variational preferences, Journal of Economic Theory, 146 (2011), 2042-2074. doi: 10.1016/j.jet.2011.06.014. [12] Z. P. Fan, Q. Yue, B. Feng and Y. Liu, An approach to group decision-making with uncertain preference ordinals, Computers & Industrial Engineering, 58 (2010), 51-57. doi: 10.1016/j.cie.2009.08.001. [13] C. X. Guo, A method for aggregating group preference based on pair-wise comparison with random binary relations under interval belief structures, Mathematics & Information Sciences, 6 (2012), 869-880. [14] C. X. Guo, H. Gong and Y. H. Guo, Approach for random lattice order ranking based on preference entropy under interval belief degree circumstance, Operations Research and Management Science, 22 (2013), 21-29. [15] Y. Guo, Lattice Order Making[M], Shanghai Science and Technology Publishing House, Shanghai, 2003. [16] T. M. Gureckis and B. C. Love, Learning in noise: Dynamic decision-making in a variable environment, Journal of Mathematical Psychology, 53 (2009), 180-193. doi: 10.1016/j.jmp.2009.02.004. [17] C. X. Guo and Y. Peng, Lattice order group decision making with interval probability based on prospect theory, Group Decis. Negot., 24 (2015), 753-775. [18] E. D. Hahn, Judgmental consistency and consensus in stochastic multicriteria decision making, Expert Systems with Applications, 37 (2010), 3784-3791. doi: 10.1016/j.eswa.2009.11.042. [19] C. H. Han, J. K. Kim and S. H. Choi, Prioritizing engineering characteristics in quality function deployment with incomplete information: A linear partial ordering approach, Int. J. Production Economics, 91 (2004), 235-249. doi: 10.1016/j.ijpe.2003.09.001. [20] J. D. Hey and J. A. Knoll, Strategies in dynamic decision making - An experimental investigation of the rationality of decision behaviour, Journal of Economic Psychology, 32 (2011), 399-409. doi: 10.1016/j.joep.2011.02.011. [21] B. Huang and C. X. Guo, Intuitionistic fuzzy multigranulation rough sets, Information Sciences, 277 (2014), 299-320. doi: 10.1016/j.ins.2014.02.064. [22] Y. Huang and J. W. Hutchinson, The roles of planning, learning, and mental models in repeated dynamic decision making, Organizational Behavior and Human Decision Processes, 122 (2013), 163-176. doi: 10.1016/j.obhdp.2013.07.001. [23] K. Jabeur and J. M. Martel, A collective choicemethod based on individual preferences relational systems(prs), Eur. J. Oper. Res., 177 (2007), 1549-1565. doi: 10.1016/j.ejor.2005.10.028. [24] K. Jabeur and J. M. Martel, An agreement index with respect to a consensus preorder, Group Decis. Negot., 19 (2010), 571-590. doi: 10.1007/s10726-009-9160-3. [25] K. Jabeur and J. M. Martel, An ordinal sorting method for group decision-making, Eur. J. Oper. Res., 180 (2007), 1272-1289. doi: 10.1016/j.ejor.2006.05.032. [26] K. Jabeur, J. M. Martel and A. Guitouni, Deriving a minimum distance-based collective preorder: A binary mathematical programming approach, OR Spectr., 34 (2012), 23-42. doi: 10.1007/s00291-009-0192-5. [27] K. Jabeur, J. M. Martel and S. B. Khélif, A distance-based collective preorder integrating the relative importance of the group's members, Group Decis. Negotiat., 13 (2004), 327-349. doi: 10.1023/B:GRUP.0000042894.00775.75. [28] S. Jullien-Ramasso, G. Mauris, L. Valet and P. Bolon, A decision support system for animated film selection based on a multi-criteria aggregation of referees' ordinal preferences, Expert Syst. Appl., 39 (2012), 4250-4257. doi: 10.1016/j.eswa.2011.09.109. [29] D. Lerche, S. Y. Matsuzaki, P. B. Sørensen, L. Carlsen and O. J. Nielsen, Ranking of chemical substances based on the Japanese Pollutant Release and Transfer Register using partial order theory and random linear extensions, Chemosphere, 55 (2004), 1005-1025. doi: 10.1016/j.chemosphere.2004.01.023. [30] D. Lerche and P. B. Sørensen, Evaluation of the ranking probabilities for partial orders based on random linear extensions, Chemosphere, 53 (2003), 981-992. doi: 10.1016/S0045-6535(03)00558-7. [31] Y. H. Lin , P. C. Lee and H. I. Ting, Dynamic multi-attribute decision making model with grey number evaluations, Expert Systems with Applications, 35 (2008), 1638-1644. doi: 10.1016/j.eswa.2007.08.064. [32] L. C. Ma, Visualizing preferences on spheres for group decisions based on multiplicative preference relations, European Journal of Operational Research, 203 (2010), 176-184. doi: 10.1016/j.ejor.2009.07.008. [33] Z. X. Ma, Applying theory of partially ordered sets to study data envelopment analysis, Journal of Systems Engineering, 3 (2002), 3-8. [34] Z. X. MA, Method of data envelopment analysis based on the theory of partially ordered sets, Systems Engineering-Theory & Practice, 4 (2003), 12-17. [35] G. Martinelli, J. Eidsvik and R. Hauge, Dynamic decision making for graphical models applied to oil exploration, European Journal of Operational Research, 230 (2013), 688-702. doi: 10.1016/j.ejor.2013.04.057. [36] R. Mu, Z. X. Ma and W. Cui, Data envelopment analysis method based on poset theory, Systems Engineering and Electronics, 35 (2013), 350-356. [37] G. Munda, Intensity of preference and related uncertainty in non-compensatory aggregation rules, Theory Dec., 73 (2012), 649-669. doi: 10.1007/s11238-012-9317-4. [38] T. D. Nielsen and J. Y. Jaffray, Dynamic decision making without expected utility: An operational approach, European Journal of Operational Research, 169 (2006), 226-246. doi: 10.1016/j.ejor.2004.05.029. [39] J. H. Park, H. J. Cho and Y. C. Kwun, Extension of the VIKOR method to dynamic intuitionistic fuzzy multiple attribute decision making, Computers & Mathematics with Applications, 65 (2013), 731-744. doi: 10.1016/j.camwa.2012.12.008. [40] R. R. Yager, On the fusion of imprecise uncertainty measures using belief structures, Information Science, 181 (2011), 3199-3209. doi: 10.1016/j.ins.2011.02.010. [41] B. Roy and R. Slowinski, Criterion of distance between technical programming and socio-economic priority, RAIRO Recherche Opérationnelle, 27 (1993), 45-60. [42] A. M. Saks and B. E. Ashforth, Change in job search behaviors and employment outcomes, Journal of Vocational Behavior, 56 (2000), 277-287. doi: 10.1006/jvbe.1999.1714. [43] Z. X. Su, M. Y. Chen, G. P. Xia and L. Wang, An interactive method for dynamic intuitionistic fuzzy multi-attribute group decision making, Expert Systems with Applications, 38 (2011), 15286-15295. doi: 10.1016/j.eswa.2011.06.022. [44] J. Thenie and J. P. Vial, Step decision rules for multistage stochastic programming: A heuristic approach, Automatica, 44 (2008), 1569-1584. doi: 10.1016/j.automatica.2008.02.001. [45] J. M. Wang, Robust optimization analysis for multiple attribute decision making problems with imprecise information, Ann Oper Res, 197 (2012), 109-122. doi: 10.1007/s10479-010-0734-x. [46] X. Y. Wang and H. Y. Meng, A multi-stage dynamic decision-making model of mine resources exploitation with many running units-theoretical analysis, Procedia Earth and Planetary Science, 1 (2009), 1654-1660. doi: 10.1016/j.proeps.2009.09.254. [47] Z. S. Xu and R. R. Yager, Dynamic intuitionistic fuzzy multi-attribute decision making, International Journal of Approximate Reasoning, 48 (2008), 246-262. doi: 10.1016/j.ijar.2007.08.008.
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