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Group Assessment of Comparable Items from the Incomplete Judgments

  • Tomoe EntaniEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)

Abstract

In this study, we propose an approach to derive a group assessment of an item as its weight vector on multiple viewpoints. When there are a group of decision makers who give the judgments of the item as comparison matrices on the viewpoints, it is reasonable that the weight vector of the item is the core of those by all the decision makers. Each decision maker’s weight vector basically includes his/her given comparison matrix, which represents only a part of his/her thinking. Namely, there is an inclusion relation between a comparison and a ratio of the corresponding weights. In addition, there are items other than the target one. A decision maker gives the comparison matrices of some of the other items if s/he knows them, as well as the target one. It is natural that there is a correlation between the judgment of the target item to those of the others. The correlation is taken into consideration from the aspect of consistency of his/her judgments. We define a fuzzy degree of the consistency with all the comparison matrices s/he gives. As the consistency degree for a comparison matrix increases, it may become unable to satisfy the inclusion relation between the comparison matrix and the weight vector. Hence, we introduce a fuzzy degree of inclusion relation in order to relax it. There is a trade-off between them. Therefore, by maximizing both degrees we obtain the weight vector of the target item from the comparison matrices of the target item considering the consistency of each decision maker’s judgments. The proposed approach is applicable even in the case that a group of given comparison matrices is incomplete such that some comparisons in a comparison matrix are missing and/or the comparison matrices of some items are missing.

References

  1. 1.
    Haker, P.T.: Incomplete pairwise comparisons in the analytic hierarchy process. Math. Model. 9(11), 837–848 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bozóki, S., Fülöp, J.: Efficient weight vectors from pairwise comparison matrices. Eur. J. Oper. Res. 264(2), 419–427 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  4. 4.
    Jensen, R.E.: An alternative scaling method for priorities in hierarchical structures. J. Math. Psychol. 28(3), 317–332 (1984)CrossRefGoogle Scholar
  5. 5.
    Saaty, T.L., Vargas, L.G.: Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios. Math. Model. 5(5), 309–324 (1984)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ishizaka, A., Siraj, S.: Are multi-criteria decision-making tools useful? An experimental comparative study of three methods. Eur. J. Oper. Res. 264(2), 462–471 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Sugihara, K., Tanaka, H.: Interval evaluations in the analytic hierarchy process by possibilistic analysis. Comput. Intell. 17(3), 567–579 (2001)CrossRefGoogle Scholar
  8. 8.
    Entani, T., Sugihara, K.: Uncertainty index based interval assignment by interval AHP. Eur. J. of Oper. Res. 219(2), 379–385 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Inuiguchi, M., Innan, S.: Improving interval weight estimations in interval AHP by relaxations. J. Adv. Comput. Intell. Intell. Inform. 21, 1135–1143 (2017)CrossRefGoogle Scholar
  10. 10.
    Jalao, E.R., Wu, T., Shunk, D.: An intelligent decomposition of pairwise comparison matrices for large-scale decisions. Eur. J. Oper. Res. 238(1), 270–280 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brunelli, M., Fedrizzi, M.: Axiomatic properties of inconsistency indices for pairwise comparisons. J. Oper. Res. Soc. 66, 1–15 (2015)CrossRefGoogle Scholar
  12. 12.
    Ramík, J., Korviny, P.: Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161(11), 1604–1613 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kubler, S., Derigent, W., Voisin, A., Robert, J., Traon, Y.L., Viedma, E.H.: Measuring inconsistency and deriving priorities from fuzzy pairwise comparison matrices using the knowledge-based consistency index. Knowl.-Based Syst. 162, 147–160 (2018). Special Issue on Intelligent Decision-Making and Consensus Under Uncertainty in Inconsistent and Dynamic EnvironmentsCrossRefGoogle Scholar
  14. 14.
    Siraj, S., Mikhailov, L., Keane, J.A.: Contribution of individual judgments toward inconsistency in pairwise comparisons. Eur. J. Oper. Res. 242(2), 557–567 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Fedrizzi, M., Giove, S.: Incomplete pairwise comparison and consistency optimization. Eur. J. Oper. Res. 183(1), 303–313 (2007)CrossRefGoogle Scholar
  16. 16.
    Forman, E., Peniwati, K.: Aggregating individual judgments and priorities with the Analytic hierarchy process. Eur. J. Oper. Res. 108(1), 165–169 (1998)CrossRefGoogle Scholar
  17. 17.
    Wan, S., Wang, F., Dong, J.: Additive consistent interval-valued atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. Eur. J. Oper. Res. 263(2), 571–582 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wan, S., Wang, F., Dong, J.: A group decision making method with interval valued fuzzy preference relations based on the geometric consistency. Inf. Fusion 40, 87–100 (2018)CrossRefGoogle Scholar
  19. 19.
    Brunelli, M., Fedrizzi, M.: Boundary properties of the inconsistency of pairwise comparisons in group decisionsy. Eur. J. Oper. Res. 240, 765–773 (2015)CrossRefGoogle Scholar
  20. 20.
    Entani, T., Inuiguchi, M.: Pairwise comparison based interval analysis for group decision aiding with multiple criteria. Fuzzy Sets Syst. 274(1), 79–96 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of HyogoKobeJapan

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