A General Method for Consistency Improving in Decision-Making Under Uncertainty

  • Virgilio López-Morales
  • Joel Suárez-Cansino
  • Ruslan Gabbasov
  • Anilu Franco Arcega
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11288)


In order to ascertain and solve a particular Multiple Criteria Decision Making (MCDM) problem, frequently a diverse group of experts must share their knowledge and expertise, and thus uncertainty arises from several sources. In those cases, the Multiplicative Preference Relation (MPR) approach can be a useful technique. An MPR is composed of judgements between any two criteria components which are declared within a crisp rank and to express decision maker(s) (DM) preferences. Consistency of an MPR is obtained when each expert has her/his information and, consequently, her/his judgments free of contradictions. Since inconsistencies may lead to incoherent results, individual Consistency should be sought after in order to make rational choices. In this paper, based on the Hadamard’s dissimilarity operator, a methodology to derive intervals for MPRs satisfying a consistency index is introduced. Our method is proposed through a combination of a numerical and a nonlinear optimization algorithms. As soon as the synthesis of an interval MPR is achieved, the DM can use these acceptably consistent intervals to express flexibility in the manner of her/his preferences, while accomplishing some a priori decision targets, rules and advice given by her/his current framework. Thus, the proposed methodology provides reliable and acceptably consistent Interval MPR, which can be quantified in terms of Row Geometric Mean Method (RGMM) or the Eigenvalue Method (EM). Finally, some examples are solved through the proposed method in order to illustrate our results and compare them with other methodologies.


Decision-making support systems Multiple criteria decision-making Analytic hierarchy process Consistency Multiplicative preference relations Uncertain decision-making 


  1. 1.
    Beale, E.M.L.: Numerical methods. In: Abadie, J. (ed.) Nonlinear Programming. North-Holland, Amsterdam (1967)Google Scholar
  2. 2.
    Bronshtein, I.N., Semendyayev, K.A., Musiol, G., Mühlig, H.: Handbook of Mathematics. Springer, Heidelberg (2015). Scholar
  3. 3.
    Cabrerizo, F.J., Herrera-Viedma, E., Pedrycz, W.: A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts. Eur. J. Oper. Res. 230(3), 624–633 (2013). Scholar
  4. 4.
    Campanella, G., Ribeiro, R.A.: A framework for dynamic multiple-criteria decision making. Decis. Support Syst. 52, 52–60 (2011)CrossRefGoogle Scholar
  5. 5.
    Chiclana, F., Mata, F., Martínez, L., Herrera-Viedma, E., Alonso, S.: Integration of a consistency control module within a consensus decision making model. Int. J. Uncertainty Fuzziness Knowl.-Based Syst. 16(01), 35–53 (2008)CrossRefGoogle Scholar
  6. 6.
    Dong, Y., Zhang, G., Hong, W.C., Xu, Y.: Consensus models for AHP group decision making under row geometric mean prioritization method. Decis. Support Syst. 49(3), 281–289 (2010)CrossRefGoogle Scholar
  7. 7.
    Herrera-Viedma, E., Cabrerizo, F.J., Kacprzyk, J., Pedrycz, W.: A review of soft consensus models in a fuzzy environment. Inf. Fusion 17, 4–13 (2014). Scholar
  8. 8.
    Huang, C.C., Lin, S.H.: Sharing knowledge in a supply chain using the semantic web. Expert Syst. Appl. 37(4), 3145–3161 (2010)CrossRefGoogle Scholar
  9. 9.
    IEOM: International conference in Dhaka, Bangladesh. In: Proceedings of the 2010 International Conference on Industrial Engineering and Operations Management, pp. 48–83, January 2010Google Scholar
  10. 10.
    Ma, L., Li, H.: Using Gower plots and decision balls to rank alternatives involving inconsistent preferences. Decis. Support Syst. 51, 712–719 (2011)CrossRefGoogle Scholar
  11. 11.
    Ribeiro, R., Moreira, A., van den Broek, P., Pimentel, A.: Hybrid assessment method for software engineering decisions. Decis. Support Syst. 51, 208–219 (2011)CrossRefGoogle Scholar
  12. 12.
    Saaty, T.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  13. 13.
    Saaty, T.: A ratio scale metric and the compatibility of ratio scales: the possibility of arrow’s impossibility theorem. Appl. Math. Lett. 7(6), 45–49 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Srdjevic, B.: Linking analytic hierarchy process and social choice methods to support group decision-making in water management. Decis. Support Syst. 42, 2261–2273 (2007)CrossRefGoogle Scholar
  15. 15.
    Urena, R., Chiclana, F., Morente-Molinera, J.A., Herrera-Viedma, E.: Managing incomplete preference relations in decision making: a review and future trends. Inf. Sci. 302, 14–32 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, L.: Compatibility and group decision making. Syst. Eng. Theory Pract. 20, 92–96 (2002)Google Scholar
  17. 17.
    Wu, Z., Xu, J.: A consistency and consensus based decision support model for group decision making with multiplicative preference relations. Decis. Support Syst. 52(3), 757–767 (2012)CrossRefGoogle Scholar
  18. 18.
    Yu, L., Lai, K.: A distance-based group decision-making methodology for multiperson multi-criteria emergency decision support. Decis. Support Syst. 51, 307–315 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Virgilio López-Morales
    • 1
  • Joel Suárez-Cansino
    • 1
  • Ruslan Gabbasov
    • 1
  • Anilu Franco Arcega
    • 1
  1. 1.Centro de Investigación en Tecnologías de Información y SistemasUniv. Aut. del Edo. de HidalgoPachucaMexico

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