The Analytic Hierarchy Process (AHP) enables decision-makers to prioritize alternatives. However, when an expert expresses judgments using natural language statements (e.g. words or phrases) inherent vagueness of language constructs can cause the interpretation to be imprecise. The fuzzy Analytic Hierarchy Process (FAHP) can be viewed in the context of the classical AHP expansion. While performing pairwise comparisons domain experts are accustomed to operating with verbal terms in their judgments. Most existing FAHP approaches do not consider a human’s confidence in the estimates provided. This paper presents a model that gives weight to the constraints on domains of expert assessments as they are almost always supplied with certain degrees of confidence. Interval type-2 membership functions (IT2MF) along with the probability-theoretical procedure for comparison of intervals can be applied here as suitable modeling options. Empirical comparison of FAHP that makes use of triangular fuzzy numbers and IT2MF-based FAHP is also presented.
Analytic Hierarchy Process, expert assessment, degree of confidence, fuzzy logic, linguistic label, type-1 membership function, interval type-2 membership function, Fuzzy Synthetic Extents, interval calculations, threshold values, prioritizing alternatives
Ahmed, F., & K?l?ç, K. (2015). Modification to fuzzy extent analysis method and its performance analysis. Proceedings of the 6th International Conference on Industrial Engineering and Systems Management (IESM), 435 – 438. Doi: https://doi.org/10.1109/IESM.2015.7380193
Ahmed, F., & K?l?ç, K. (2018). Fuzzy Analytic Hierarchy Process: A performance analysis of different algorithms. Fuzzy Sets and Systems, Doi: https://doi.org/10.1016/j.fss.2018.08.009.
Azadeh, A., Saberi, M., Atashbar, N.Z., Chang, E., & Pazhoheshfar, P. (2013). Z-AHP: A z-number extension of Fuzzy Analytical Hierarchy Process. Proceedings of the 7th IEEE International Conference on Digital Ecosystems and Technologies, 141–147. Doi: https://doi.org/10.1109/DEST.2013.6611344
Bellman, R., & Zadeh, L.A. (1970). Decision-making in fuzzy environment. Management Science, 17(4), B141 – B164. Doi: https://doi.org/10.1287/mnsc.17.4.B141
Buckley, J.J. (1985). Fuzzy hierarchical analysis, Fuzzy Sets and Systems, 17, 233 – 247. Doi: https://doi.org/10.1016/0165-0114(85)90090-9
Bustince, H., Fernandez, J., Hagras, H., Herrera, F., Pagola, M., & Barrenechea, E. (2015). Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wider view on their relationship. IEEE Trans. on Fuzzy Systems, 23(5), 1876 – 1882. Doi: https://doi.org/10.1109/TFUZZ.2014.2362149
Chang, D.Y. (1992). Extent analysis and synthetic decision. Optimization Techn. and Applications, 1, 352 – 355.
Chang, D.Y. (1996). Applications of the Extent Analysis Method on fuzzy AHP. European Journal of Operational Research, 95(3), 649 – 655. Doi: https://doi.org/10.1016/0377-2217(95)00300-2
Chen, S.M., & Lee, L.-W. (2010). Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. Expert Systems with Applications, 37, 824 – 833. Doi: https://doi.org/10.1016/j.eswa.2009.06.094
Chen, S.-M., Yang, M.-W., Lee, L.-W., & Yang, S.-W. (2012). Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets. Expert Systems with Applications, 39, 5295 – 5308. Doi: https://doi.org/10.1016/j.eswa.2011.11.008
Chiao, K.-P. (2016). The multi-criteria group decision making methodology using type-2 fuzzy linguistic judgments. Applied Soft Computing, 49, 189 – 211. Doi: https://doi.org/10.1016/j.asoc.2016.07.050
Fine, K. (1975). Vagueness, truth and logic. Synthese, 30(3-4), 265 – 300. Doi: https://doi.org/10.1007/BF00485047
Gimaletdinova, A.R., & Degtiarev, K.Y. (2017). Type-2 fuzzy rule-based model of urban metro positioning service. Proceedings of the Institute for System Programming (ISP RAS), 29(4), 87 – 106. Doi: 10.15514/ISPRAS-2017-29(4)-6
de Graan, J.G. (1980). Extensions of the multiple criteria analysis method of T. L. Saaty. Tech. Report 80-3. Leidschendam, the Netherlands : National Institute for Water Supply,.
Kabir, G., & Hasin, M.A.A. (2011). Comparative analysis of AHP and fuzzy AHP models for multicriteria inventory classification. International Journal of Fuzzy Logic Systems (IJFLS), 1(1), 1 – 16.
Kahneman, D. (2013). Thinking, fast and slow. New York: Farrar, Straus and Giroux.
Kahraman, C., Öztay?i, B., Sar?, I.U., & Turano?lu, E. (2014). Fuzzy Analytic Hierarchy Process with interval type-2 fuzzy sets. Knowledge-Based Systems, 59, 48 – 57. Doi: https://doi.org/10.1016/j.knosys.2014.02.001
Kang, B., Deng Y., & Sadiq R. (2018). Total utility of z-number. Applied Intelligence, 48(3), 703 – 729. Doi: https://doi.org/10.1007/s10489-017-1001-5
Kang, B., Wei, D., Li, Y., & Deng, Y. (2012). A method of converting z-number to classical fuzzy number. Journal of Information and Computational Science, 9(3), 703 – 709.
Keefe, R. (2007). Theories of vagueness. Cambridge, UK: Cambridge University Press.
Klir, G.J., & Yuan, B. (1995). Fuzzy sets and fuzzy logic: Theory and applications. Upper Saddle River, New Jersey: Prentice Hall. Doi: https://doi.org/10.1080/03081079708945184
Krohling, R., Pacheco A.G.C., & Dos Santos, G.A. (2017). TODIM and TOPSIS with z-numbers. In Pan, Y. Lu, X. (Eds). Frontiers of Information Technology & Electronic Engineering. Zhejiang University Press & Springer.
van Laarhoven, P.J.M., & Pedrycz, W. (1983). A fuzzy extension of Saaty’s priority theory. Fuzzy Sets and Systems, 11(1-3), 229 – 241. Doi: https://doi.org/10.1016/S0165-0114(83)80082-7
Lootsma, F.A. (1980). Saaty’s priority theory and the nomination of a senior professor in operations research. European Journal of Operational Research, 4(6), 380 – 388. Doi: https://doi.org/10.1016/0377-2217(80)90189-7
Mendel, J.M. (2017). Uncertain rule-based fuzzy systems: Introduction and new directions, 2nd ed. Springer International Publishing. Doi: https://doi.org/10.1007/978-3-319-51370-6_1
Mendel, J.M., & John, R.I. (2002). Type-2 fuzzy sets made simple, IEEE Transactions on Fuzzy Systems, 10(2), 117 – 127. Doi: https://doi.org/10.1109/91.995115
Mendel, J.M., John, R.I. & Liu, F. (2006). Interval type-2 fuzzy logic systems made simple. IEEE Transactions on Fuzzy Systems, 14(6), 808 – 821. Doi: https://doi.org/10.1109/TFUZZ.2006.879986
Mendel, J.M., Hagras, H., Bustince, H., & Herrera, F. (2016). Comments on “Interval type-2 fuzzy sets are generalization of interval-valued fuzzy sets: Towards a wide view on their relationship”. IEEE Transactions on Fuzzy Systems, 24(1), 249 – 250. Doi: https://doi.org/10.1109/TFUZZ.2015.2446508
Mendel, J.M., Hagras, H., & John, R.I. (n.d.). Standard background material about interval type-2 fuzzy logic systems that can be used by all authors. Retrieved from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.125.4195&rep=rep1&type=pdf
Mendel, J.M., & Wu, H. (2006). Type-2 fuzzistics for symmetric interval
type-2 fuzzy sets: Part 1, forward problems. IEEE Transactions on Fuzzy Systems,
14(6), 781 – 792. Doi: https://doi.org/10.1109/TFUZZ.2006.881441
Nie, M., & Tan, W.W. (2008). Towards an efficient type-reduction method for interval type-2 fuzzy logic systems. Proceedings of IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence), 1425 – 1432. Doi: https://doi.org/10.1109/FUZZY.2008.4630559
Runkler, T., Chen, C., & John, R. (2018). Type reduction operators for interval type-2 defuzzification. Information Sciences, 467, 464 – 476. Doi: https://doi.org/10.1016/j.ins.2018.08.023
Runkler, T., Coupland, S., & John, R. (2017). Interval type-2 fuzzy decision making. International Journal of Approximate Reasoning, 80, 217 – 224. Doi: https://doi.org/10.1016/j.ijar.2016.09.007
Saaty, R.W. (1987). The Analytic Hierarchy Process – What it is and how it is used. Mathematical Modelling, 9(3-5), 161 – 176. Doi: https://doi.org/10.1016/0270-0255(87)90473-8
Saaty, T.L. (1980). The Analytic Hierarchy Process: Planning, priority setting, resource allocation (Decision-Making Series). McGraw-Hill Publ. (Russian translation: ????? ?. (1993). ???????? ???????. ????? ??????? ????????. ?.: ????? ? ?????).
Saaty, T.L. (1994a). Fundamentals of decision making and priority theory with the Analytic Hierarchy Process, 1st ed. Pittsburgh, PA: RWS Publications.
Saaty, T.L. (1994b). Highlights and critical points in the theory and application of the Analytic Hierarchy Process. European Journal of Operational Research, 74(3), 426 – 447. Doi: https://doi.org/10.1016/0377-2217(94)90222-4
Saaty, T.L., & Tran, L.T. (2007). On the invalidity of fuzzifying numerical judgments in the Analytic Hierarchy Process. Mathematical and Computer Modeling, 46(7-8), 962 – 975. Doi: https://doi.org/10.1016/j.mcm.2007.03.022
Sevastianov, P. (2007). Numerical methods for interval and fuzzy number comparison based on probabilistic approach and Dempster-Shafer theory. Information Sciences, 177, 4645 – 4661. Doi: https://doi.org/10.1016/j.ins.2007.05.001
Sevastyanov, P.V., Rog, P., & Venberg A.V. (2002). A constructive numerical method for the comparison of intervals. Proceedings of the 4th International Conference on Parallel Processing and Applied Mathematics (LNCS, 2328, eds, Wyrzykowski, R., et al.), 756 – 761. Doi: https://doi.org/10.1007/3-540-48086-2_84
Shapiro, A.F., & Koissi, M.-C. (2017). Fuzzy logic modifications of the Analytic Hierarchy Process. Insurance: Mathematics and Economics, 75, 189 – 202. Doi: https://doi.org/10.1016/j.insmatheco.2017.05.003
Wang, Y.-M., Luo, Y., & Hua, Z. (2008). On the extent analysis method for fuzzy AHP and its applications. European Journal of Operational Research, 186, 735 – 747. Doi: https://doi.org/10.1016/j.ejor.2007.01.050
Wierman, M.J. (1997). Central values of fuzzy numbers – defuzzification. Information Sciences, 100 (1-4), 207 – 215. Doi: https://doi.org/10.1016/S0020-0255(96)00278-2
Wu, D., & Mendel, J.M. (2007a). Uncertainty measures for interval type-2 fzzy sets. Information Sciences, 177, 5378 – 5393. Doi: https://doi.org/10.1016/j.ins.2007.07.012
Yaakob, A.M., & Gegov, A. (2016). Interactive TOPSIS based group decision making methodology using Z-Numbers. International Journal of Computational Intelligence Systems, 9(2), 311 – 324. Doi: http://dx.doi.org/10.1080/18756891.2016.1150003
Zadeh, L.A. (1965). Fuzzy sets. Information and Control, 8(3), 338 – 353. Doi: https://doi.org/10.1016/S0019-9958(65)90241-X
Zadeh, L.A. (2011). A note on z-numbers. Information Sciences, 181(14), 2923 – 2932. Doi: https://doi.org/10.1016/j.ins.2011.02.022
Zhü, K. (2014). Fuzzy Analytic Hierarchy Process: Fallacy of the popular methods. European Journal of Operational Research, 236, 209 – 217. Doi: https://doi.org/10.1016/j.ejor.2013.10.034
Copyright of all articles published in IJAHP is transferred to Creative Decisions Foundation (CDF). However, the author(s) reserve the following:
- All proprietary rights other than copyright, such as patent rights.
- The right to grant or refuse permission to third parties to republish all or part of the article or translations thereof. In case of whole articles, such third parties must obtain permission from CDF as well. However, CDF may grant rights with respect to journal issues as a whole.
- The right to use all or parts of this article in future works of their own, such as lectures, press releases, reviews, textbooks, or reprint books.
- The authors affirm that the article has been neither copyrighted nor published, that it is not being submitted for publication elsewhere, and that if the work is officially sponsored, it has been released for open publication.
The only exception to the statements in the paragraph above is the following: If an article published in IJAHP contains copyrighted material, such as a teaching case, as an appendix, then the copyright (and all commercial rights) of such material remains with the original copyright holder.
CDF will receive permission for publication of copyrighted material in IJAHP. This permission is not transferable to third parties. Permission to make electronic and paper copies of part or all of the articles, including all computer files that are linked to the articles, for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage.
This permission does not apply to previously copyrighted material, such as teaching cases. In paper copies of the article, the copyright notice and the title of the publication and its date should be visible. To copy otherwise is permitted provided that a per-copy fee is paid.
To republish, to post on servers, or redistribute to lists requires that you post a link to the IJAHP article, which is available in open access delivery mode. Do not upload the article itself.
Authors are permitted to present a talk, based on a paper submitted to or accepted by IJAHP, at a conference where the paper would not be published in a copyrighted publication either before or after the conference and where the author did not assign copyright to the conference or related publisher.