Rectifying the Inconsistent Fuzzy Preference Matrix in AHP Using a Multi-Objective BicriterionAnt


Analytic hierarchy process (AHP) is a decision making tool regarding the criteria analysis to obtain a priority alternative. One of the important issues in comparison matrix of AHP is the consistency. The inconsistent comparison matrix cannot be used to make decision. This paper proposes an algorithm using a modified BicriterionAnt to pursue two objectives intended to rectify the inconsistent fuzzy preference matrix, called MOBAF. The two objectives include minimizing the consistent ratio (CR) and minimizing the deviation matrix, which are in conflict with each other when rectifying the inconsistent matrix. This study uses two pheromone matrices and two heuristic distances matrices to generate the ants tour. To see the performance, MOBAF is implemented to rectify on some inconsistent fuzzy preference matrices. As a result, in addition to being able to rectify the CR, the proposed algorithm also successfully generates some non-dominated solutions that can be considered as optimal solutions.

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  1. 1.

    Bozóki S, Fülöp J, Rónyai L (2010) On optimal completion of incomplete pairwise comparison matrices. Math Comput Model 52(1):318–333

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Chaharsooghi SK, Kermani AM (2008) An intelligent multi-colony multi-objective ant colony optimization (aco) for the 0–1 knapsack problem. In: Evolutionary computation, 2008. CEC 2008. (IEEE World Congress on Computational Intelligence). IEEE Congress on IEEE, pp 1195–1202

  3. 3.

    Chiclana F, Herrera F, Herrera-Viedma E (2001) Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst 122(2):277–291

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Coello CAC, Lechuga MS (2002) MOPSO: a proposal for multiple objective particle swarm optimization. In: Proceeding of congress evolutionary computation (CEC), IEEE, vol 2, pp 1051–1056

  5. 5.

    Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197

    Article  Google Scholar 

  6. 6.

    Doerner K, Gutjahr WJ, Hartl RF, Strauss C, Stummer C (2004) Pareto ant colony optimization: a metaheuristic approach to multiobjective portfolio selection. Ann Oper Res 131(1–4):79–99

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66

    Article  Google Scholar 

  8. 8.

    Ergu D, Kou G, Peng Y, Shi Y (2011) A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP. Eur J Oper Res 213(1):246–259

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    García-Martínez C, Cordón O, Herrera F (2007) A taxonomy and an empirical analysis of multiple objective ant colony optimization algorithms for the bi-criteria tsp. Eur J Oper Res 180(1):116–148

    Article  MATH  Google Scholar 

  10. 10.

    Girsang AS, Tsai CW, Yang CS (2014) Ant colony optimization for reducing the consistency ratio in comparison matrix. In: International conference on advances in engineering and technology (ICAET), pp 577–582

  11. 11.

    Girsang AS, Tsai CW, Yang CS (2015) Ant algorithm for modifying an inconsistent pairwise weighting matrix in an analytic hierarchy process. Neural Comput Appl 26(2):313–327

    Article  Google Scholar 

  12. 12.

    Herrera-Viedma E, Herrera F, Chiclana F, Luque M (2004) Some issues on consistency of fuzzy preference relations. Eur J Oper Res 154(1):98–109

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Iredi S, Merkle D, Middendorf M (2001) Bi-criterion optimization with multi colony ant algorithms. In: Coello CAC, Aguirre AH, Zitzler E (eds) Evolutionary multi-criterion optimization. Springer, New York, pp 359–372

    Google Scholar 

  14. 14.

    Ke L, Zhang Q, Battiti R (2013) Moea/d-aco: a multiobjective evolutionary algorithm using decomposition and antcolony. IEEE T Cybern 43(6):1845–1859

    Article  Google Scholar 

  15. 15.

    Lin CC, Wang WC, Yu WD (2008) Improving AHP for construction with an adaptive AHP approach (A\(^{ 3}\)). Autom Constr 17(2):180–187

    MathSciNet  Article  Google Scholar 

  16. 16.

    Liu X, Pan Y, Xu Y, Yu S (2012) Least square completion and inconsistency repair methods for additively consistent fuzzy preference relations. Fuzzy Sets Syst 198:1–19

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    López-Ibáñez M, Stützle T (2010) Automatic configuration of multi-objective aco algorithms. In: Swarm intelligence, Springer, pp 95–106

  18. 18.

    López-Ibáñez M, Stützle T (2010) The impact of design choices of multiobjective antcolony optimization algorithms on performance: an experimental study on the biobjective tsp. In: Proceedings of the 12th annual conference on genetic and evolutionary computation, ACM pp 71–78

  19. 19.

    Ma J, Fan ZP, Jiang YP, Mao JY, Ma L (2006) A method for repairing the inconsistency of fuzzy preference relations. Fuzzy Sets Syst 157(1):20–33

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ma W (1994) A practical approach to modify pair wise comparison matrices and two criteria of modificatory effectiveness. J Sci Syst Eng 4:37–58

    Google Scholar 

  21. 21.

    Mahfouf M, Chen MY, Linkens DA (2004) Adaptive weighted particle swarm optimisation for multi-objective optimal design of alloy steels. In: Parallel problem solving from nature-PPSN VIII, Springer, pp 762–771

  22. 22.

    Mytakidis T, Vlachos A (2008) Maintenance scheduling by using the bi-criterion algorithm of preferential anti-pheromone. Leonardo J Sci 12(16):143–164

    Google Scholar 

  23. 23.

    Orlovsky S (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1(3):155–167

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Pinto D, Barán B (2005) Solving multiobjective multicast routing problem with a new ant colony optimization approach. In: Proceedings of the 3rd international IFIP/ACM Latin American conference on networking, ACM, pp 11–19

  25. 25.

    Saaty TL (1980) The analytic hierarchy process: planning, priority setting, resources allocation. McGraw-Hill, New York

    MATH  Google Scholar 

  26. 26.

    Srinivas N, Deb K (1994) Multiobjective optimization using nondominated sorting in genetic algorithms. Evol Comput 2(3):221–248

    Article  Google Scholar 

  27. 27.

    T’kindt V, Monmarché N, Tercinet F, Laügt D (2002) An ant colony optimization algorithm to solve a 2-machine bicriteria flowshop scheduling problem. Eur J Oper Res 142(2):250–257

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Wu Z, Xu J (2012) A consistency and consensus based decision support model for group decision making with multiplicative preference relations. Decis Support Syst 52(3):757–767

    Article  Google Scholar 

  29. 29.

    Xu Y, Da Q, Wang H (2011) A note on group decision-making procedure based on incomplete reciprocal relations. Soft Comput 15(7):1289–1300

    Article  MATH  Google Scholar 

  30. 30.

    Xu Y, Gupta JN, Wang H (2013) The ordinal consistency of an incomplete reciprocal preference relation. Fuzzy Sets Syst 246:62–77

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Xu Y, Patnayakuni R, Wang H (2013) Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations. Appl Math Model 37(4):2139–2152

    MathSciNet  Article  Google Scholar 

  32. 32.

    Xu Y, Wang H (2013) Eigenvector method, consistency test and inconsistency repairing for an incomplete fuzzy preference relation. Appl Math Model 37(7):5171–5183

    MathSciNet  Article  Google Scholar 

  33. 33.

    Xu Z (2000) On consistency of the weighted geometric mean complex judgement matrix in ahp. Eur J Oper Res 126(3):683–687

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Xu Z, Da Q (2003) An approach to improving consistency of fuzzy preference matrix. Fuzzy Optim Decis Mak 2(1):3–12

    MathSciNet  Article  Google Scholar 

  35. 35.

    Yager RR (1982) Fuzzy set possibility theory recent developments, vol 1. Pergamon, New York

    MATH  Google Scholar 

  36. 36.

    Yang I, Wang WC, Yang TI (2012) Automatic repair of inconsistent pairwise weighting matrices in analytic hierarchy process. Autom Constr 22:290–297

    Article  Google Scholar 

  37. 37.

    Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195

    Article  Google Scholar 

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The authors would like to thank the anonymous reviewers for their valuable comments and suggestions on the paper. This work was supported in part by the Ministry of Science and Technology of Taiwan, R.O.C., under Contracts MOST103-2221-E-197-034 and MOST104-2221-E-197-005.

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Correspondence to Chun-Wei Tsai.

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Girsang, A.S., Tsai, CW. & Yang, CS. Rectifying the Inconsistent Fuzzy Preference Matrix in AHP Using a Multi-Objective BicriterionAnt. Neural Process Lett 44, 519–538 (2016).

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  • Analytic hierarchy process
  • Inconsistent
  • Fuzzy preference matrix
  • Bicriterionant