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Ant algorithm for modifying an inconsistent pairwise weighting matrix in an analytic hierarchy process

  • Advances in Intelligent Data Processing and Analysis
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Abstract

One important issue in the analytic hierarchy process (AHP) is confirming the consistency of comparison matrix to verify the logical respondent opinion. As inconsistent comparison matrix cannot be used as reference to make decisions, this paper proposes a method using an ant algorithm to modify an inconsistent pairwise weight matrix to be consistent in AHP, called ANTAHP. This method employs the matrix element as the path in an ant colony optimization to construct the tour. By laying pheromone information on their path, the ants can find the optimal matrix (or tour), which satisfies the consistency and closer to the original judgment of the decision makers. The experimental results demonstrate that the proposed algorithm is able to make consistent matrices, as well as minimize the difference index.

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References

  1. Alonso JA, Lamata MT (2006) Consistency in the analytic hierarchy process: a new approach. Int J Uncertain Fuzziness Knowl Based Syst 14(04):445–459

    Article  MATH  Google Scholar 

  2. Blum C, Sampels M (2004) An ant colony optimization algorithm for shop scheduling problems. J Math Model Algorithms 3(3):285–308

    Article  MATH  MathSciNet  Google Scholar 

  3. Cao D, Leung LC, Law J (2008) Modifying inconsistent comparison matrix in analytic hierarchy process: a heuristic approach. Decis Support Syst 44(4):944–953

    Article  Google Scholar 

  4. Chamodrakas I, Batis D, Martakos D (2010) Supplier selection in electronic marketplaces using satisfying and fuzzy AHP. Expert Syst Appl 37(1):490–498

    Article  Google Scholar 

  5. Chen SM, Lin TE, Lee LW (2014) Group decision making using incomplete fuzzy preference relations based on the additive consistency and the order consistency. Inf Sci 259:1–15

  6. Chiclana F, Herrera-Viedma E, Alonso S, Herrera F (2009) Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans Fuzzy Syst 17(1):14–23

    Article  Google Scholar 

  7. da Serra Costa JF (2011) A genetic algorithm to obtain consistency in analytic hierarchy process. Brazilian J Oper Produc Manag 8(1):55–64

    Article  Google Scholar 

  8. Dong Y, Zhang G, Hong WC, Xu Y (2010) Consensus models for AHP group decision making under row geometric mean prioritization method. Decis Support Syst 49(3):281–289

    Article  Google Scholar 

  9. 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 

  10. Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B 26(1):29–41

    Article  Google Scholar 

  11. Durán O (2011) Computer-aided maintenance management systems selection based on a fuzzy AHP approach. Adv Eng Softw 42(10):821–829

    Article  Google Scholar 

  12. 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

    Article  MATH  MathSciNet  Google Scholar 

  13. Gambardella LM, Taillard É, Agazzi G (1999) Macs-vrptw: a multiple colony system for vehicle routing problems with time windows. In: New ideas in optimization

  14. Genç S, Boran FE, Akay D, Xu Z (2010) Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations. Inf Sci 180(24):4877–4891

    Article  MATH  Google Scholar 

  15. Gomez-Ruiz JA, Karanik M, Peláez JI (2009), Improving the consistency of AHP matrices using a multi-layer perceptron-based model. In: Bio-inspired systems: computational and ambient intelligence. Springer, Berlin, pp 41–48

  16. Güngör Z, Serhadlıoğlu G, Kesen SE (2009) A fuzzy AHP approach to personnel selection problem. Appl Soft Comput 9(2):641–646

    Article  Google Scholar 

  17. Ishizaka A, Lusti M (2004) An expert module to improve the consistency of AHP matrices. Int Trans Oper Res 11(1):97–105

    Article  MATH  Google Scholar 

  18. Kwiesielewicz M, van Uden E (2002) Problem of inconsistent and contradictory judgements in pairwise comparison method in sense of AHP. In: Computational science-ICCS. Springer, Berlin, pp 468–473

  19. Lakoff G (1975) Hedges: a study in meaning criteria and the logic of fuzzy concepts. Springer, Berlin

    Google Scholar 

  20. Lin C, Kou G, Ergu D (2013) An improved statistical approach for consistency test in AHP. Ann Oper Res 211(1):289–299

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. 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

    Article  MATH  MathSciNet  Google Scholar 

  23. Parpinelli RS, Lopes HS, Freitas AA (2002) Data mining with an ant colony optimization algorithm. IEEE Trans Evol Comput 6(4):321–332

    Article  Google Scholar 

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

    Google Scholar 

  25. Saaty TL (1982) Decision Making for leaders: The analytical hierarchy process for decisions in a complex work. Lifetime Learning Publications

  26. Saaty TL, Vargas LG (2001) Models, methods, concepts, and applications of the analytic hierarchy process. Kluwer, Dordrecht

    Book  Google Scholar 

  27. Shelokar PS, Jayaraman VK, Kulkarni BD (2004) An ant colony approach for clustering. Analytica Chimica Acta 509(2):187–195

    Article  Google Scholar 

  28. Siraj S, Mikhailov L, Keane J (2012) A heuristic method to rectify intransitive judgments in pairwise comparison matrices. Eur J Oper Res 216(2):420–428

    Article  MATH  MathSciNet  Google Scholar 

  29. Triantaphyllou E, Mann SH (1990) An evaluation of the eigenvalue approach for determining the membership values in fuzzy sets. Fuzzy Sets Syst 35(3):295–301

    Article  MathSciNet  Google Scholar 

  30. Vargas LG (1982) Reciprocal matrices with random coefficients. Math Model 3(1):69–81

    Article  MATH  MathSciNet  Google Scholar 

  31. 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 

  32. Xia M, Xu Z, Chen J (2013) Algorithms for improving consistency or consensus of reciprocal [0, 1]-valued preference relations. Fuzzy Sets Syst 216:108–133

    Article  MATH  MathSciNet  Google Scholar 

  33. 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 

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

  35. Xu Y, Li KW, Wang H (2013) Consistency test and weight generation for additive interval fuzzy preference relations. Soft Comput. doi:10.1007/s00500-013-1156-x

  36. 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

    Article  MathSciNet  Google Scholar 

  37. Xu Z (2011) Consistency of interval fuzzy preference relations in group decision making. Appl Soft Comput 11(5):3898–3909

    Article  Google Scholar 

  38. Xu Z, Chen J (2008) Group decision-making procedure based on incomplete reciprocal relations. Soft Comput 12(6):515–521

    Article  MATH  Google Scholar 

  39. 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 

  40. Zeshui X, Cuiping W (1999) A consistency improving method in the analytic hierarchy process. Eur J Oper Res 116(2):443–449

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the editors and anonymous reviewers for their valuable comments and suggestions on the paper that greatly improve the quality of the paper This work was supported in part by the Ministry of Science and Technology of Taiwan, R.O.C., under Contracts NSC102-2221-E-041-006 and NSC 102-2219-E-006-001.

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Authors and Affiliations

  1. Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, ROC

    Abba Suganda Girsang & Chu-Sing Yang

  2. Department of Applied Informatics and Multimedia, Chia Nan University of Pharmacy and Science, Tainan, Taiwan, ROC

    Chun-Wei Tsai

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  1. Abba Suganda Girsang
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  2. Chun-Wei Tsai
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Correspondence to Chun-Wei Tsai.

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Girsang, A.S., Tsai, CW. & Yang, CS. Ant algorithm for modifying an inconsistent pairwise weighting matrix in an analytic hierarchy process. Neural Comput & Applic 26, 313–327 (2015). https://doi.org/10.1007/s00521-014-1630-0

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  • DOI: https://doi.org/10.1007/s00521-014-1630-0

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