×

zbMATH — the first resource for mathematics

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment. (English) Zbl 1140.90430
Summary: The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.

MSC:
90B50 Management decision making, including multiple objectives
90C05 Linear programming
90C70 Fuzzy and other nonstochastic uncertainty mathematical programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atanassov K.T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20, 87–96 · Zbl 0631.03040
[2] Atanassov K.T. (1999). Intuitionistic Fuzzy Sets. Springer-Verlag, Heidelberg · Zbl 0939.03057
[3] Burillo P., Bustince H. (1996). Vague sets are intuitionistic fuzzy sets. Fuzzy Sets and Systems 79, 403–405 · Zbl 0871.04006
[4] Carlsson C., Fuller R. (2000). Multiobjective linguistic optimization. Fuzzy Sets and Systems 115, 5–10 · Zbl 0955.00027
[5] Chen S.M., Tan J.M. (1994). Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems 67, 163–172 · Zbl 0845.90078
[6] Chen C.T. (2000). Extensions of the TOPSIS for group decision-making under fuzzy environment. Fuzzy Sets and Systems 114, 1–9 · Zbl 0963.91030
[7] De S.K., Biswas R., Roy A.R. (2001). An application of intuitionistic fuzzy sets in medical diagnosis. Fuzzy Sets and Systems 117, 209–213 · Zbl 0980.92013
[8] Delgado M., Verdegay J.L., Vila M.A. (1992). Linguistic decision-making models. International Journal of Intelligent System 7, 479–492 · Zbl 0756.90001
[9] Deschrijver G., Kerre E.E. (2007). On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision. Information Sciences 177, 1860–1866 · Zbl 1121.03074
[10] Erol I., Ferrell W.G. (2003). A methodology for selection problems with multiple conflicting objectives and both qualitative and quantitative criteria. International Journal of Production Economics 86, 187–199
[11] Fisher B. (2003). Fuzzy environmental decision-making: applications to air pollution. Atmospheric Environment 37, 1865–1877
[12] Gupta, J. P., Chevalier, A., & Dutta, S. (2003). Multicriteria model for risk evaluation for venture capital firms in an emerging market context. European Journal of Operational Research, available online 10 April (In Press).
[13] Herrera F., Martinez L., Sanchez P.J. (2005). Managing non-homogeneous information in group decision making. European Journal of Operational Research 166, 115–132 · Zbl 1066.90533
[14] Herrera F., Viedma E.H.-V., Verdegay J.L. (1996). A model of consensus in group decision making under Linguistic assessments. Fuzzy Sets and Systems 78, 73–87 · Zbl 0870.90007
[15] Hong D.H., Choi C.H. (2000). Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets and Systems 114, 103–113 · Zbl 0963.91031
[16] Hu Y.-C., Hu J.-S., Chen R.-S., Tzeng G.-H. (2004). Assessing weights of product attributes from fuzzy knowledge in a dynamic environment. European Journal of Operational Research 154, 125–143 · Zbl 1099.91509
[17] Hwang C.L., Yoon K. (1981). Multiple attributes decision making methods and applications. Berlin Heidelberg, Springer · Zbl 0453.90002
[18] Li D.-F. (2005a). An approach to fuzzy multiattribute decision making under uncertainty. Information Sciences 169, 97–112 · Zbl 1101.68840
[19] Li D.-F. (2005b). Multiattribute decision making models and methods using intuitionistic fuzzy sets. Journal of Computer and System Sciences 70, 73–85 · Zbl 1066.90052
[20] Li D.-F. (2004). Some measures of dissimilarity in intuitionistic fuzzy structures. Journal of Computer and System Sciences 68, 115–122 · Zbl 1052.03034
[21] Li D.-F., Cheng C.-T. (2002). New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions. Pattern Recognition Letters 23, 221–225 · Zbl 0996.68171
[22] Li, D.-F., & Yang, J.-B. (2003). A multiattribute decision making approach using intuitionistic fuzzy sets. In Proceedings of the Conference EUSFLAT 2003, Zittau, Germany, (Vol. 1, pp. 183–186).
[23] Li D.-F., Yang J.-B. (2004). Fuzzy linear programming technique for multiattribute group decision making in fuzzy environments. Information Sciences 158, 263–275 · Zbl 1064.91035
[24] Li L., Yuan X.-H., Xia Z.-Q. (2007). Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets. Journal of Computer and System Sciences 73, 84–88 · Zbl 1178.68541
[25] Liu H.-W., Wang G.-J. (2007). Multi-criteria decision-making methods based on intuitionistic fuzzy sets. European Journal of Operational Research 179, 220–233 · Zbl 1163.90558
[26] Pankowska A., Wygralak A. (2006). General IF-sets with triangular norms and their applications to group decision making. Information Sciences 176, 2713–2754 · Zbl 1149.91308
[27] Srinivasan V., Shocker A.D. (1973). Linear programming techniques for multidimensional analysis of preference. Psychometrica 38, 337–342 · Zbl 0316.92024
[28] Stanciulescu C., Fortemps Ph., Install M., Wertz V. (2003). Multiobjective fuzzy linear programming problems with fuzzy decision variables. European Journal of Operational Research 149, 654–675 · Zbl 1033.90124
[29] Szmidt E., Kacprzyk J. (1996a). Intuitionistic fuzzy sets in group decision making. NIFS 2, 15–32 · Zbl 0865.90003
[30] Szmidt E., Kacprzyk J. (1996b). Remarks on some applications of intuitionistic fuzzy sets in decision making. NIFS 2, 22–31 · Zbl 0865.90003
[31] Szmidt, E., & Kacprzyk, J. (1996c). Group decision making via intuitionistic fuzzy sets. In FUBEST’96, Sofia, Bulgaria, pp. 107–112. · Zbl 0865.90003
[32] Szmidt, E., & Kacprzyk, J. (1997). Intuitionistic fuzzy sets for more realistic group decision making. In International conference on transition to advanced market institutions and economies, Warsaw, pp. 430–433.
[33] Szmidt E., Kacprzyk J. (2001). Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems 114, 505–518 · Zbl 0961.03050
[34] Wang J., Lin Y.-I. (2003). A fuzzy multicriteria group decision making approach to select configuration items for software development. Fuzzy Sets and Systems 134, 343–363 · Zbl 1031.91019
[35] Wang R.-C., Chuu S.-J. (2004). Group decision-making using a fuzzy linguistic approach for evaluating the flexibility in a manufacturing system. European Journal of Operational Research 154, 563–572 · Zbl 1146.90398
[36] Xu Z.-Sh. (2004). A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Information Sciences 166, 19–30 · Zbl 1101.68849
[37] Xu Z.-Sh. (2007). Intuitionistic preference relations and their application in group decision making. Information Sciences 177, 2363–2379 · Zbl 1286.91043
[38] Zadeh L.A. (1965). Fuzzy sets. Information and Control 8, 338–356 · Zbl 0139.24606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.