In some multi-attribute decision making (MADM) problems, various relationships among the decision attributes should be considered. This paper investigates the prioritization relationship of attributes in MADM with trapezoidal intuitionistic fuzzy numbers (TrIFNs). TrIFNs are a special intuitionistic fuzzy set on a real number set and have the better capability to model ill-known quantities. Firstly, the weighted possibility means of membership and non-membership functions for TrIFNs are defined. Hereby, a new lexicographic ranking method for TrIFNs is presented. Then, a series of trapezoidal intuitionistic fuzzy prioritized aggregation operators are developed, including the trapezoidal intuitionistic fuzzy prioritized score (TrIFPS) operator, trapezoidal intuitionistic fuzzy prioritized weighted average (TrIFPWA) operator, trapezoidal intuitionistic fuzzy prioritized “and” (TrIFP-AND) operator and trapezoidal intuitionistic fuzzy prioritized “or” (TrIFP-OR) operator. Some desirable properties of these operators are also discussed. By utilizing the TrIFPWA operator, the attribute values of alternatives are integrated into the overall ones, which are used to rank the alternatives. Thus, a new method is proposed for solving the prioritized MADM problems with TrIFNs. Finally, the applicability of the proposed method is illustrated with a supply chain collaboration example.
 K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1) (1986), 87-96.  K. Atanassov and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, 31(3) (1989), 343-349.  C. Carlsson and R. Fullr, On possibilistic mean value and deviation of fuzzy numbers, Fuzzy Sets and Systems, 122 (2001), 315-326.  R. Fullr and P. Majlender, On weighted possibilistic mean and variance of fuzzy numbers, Fuzzy Sets and Systems, 136 (2003), 363-374.  D. F. Li, A note on using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly, Microelectronics Reliability, 48(10) (2008), 17-41.  D. F. Li, A ratio ranking method of triangular intuitionistic fuzzy numbers and its appli- cation to MADM problems, Computers and Mathematics with Applications, 60 (2010), 1557-1570.  D. F. Li, J. X. Nan and M. J. Zhang, A ranking method of triangular intuitionistic fuzzy numbers and application to decision making, International Journal of Computational In- telligence Systems, 3(5) (2010), 522-530.  B. Q. Li and W. He, Intuitionistic fuzzy PRI-AND and PRI-OR aggregation operators, Information Fusion, 14 (2013), 450-459.  R. Mansini, M. W. P. Savelsbergh and B. Tocchella, The supplier selection problem with quantity discounts and truckload shipping, Omega, 40(4) (2012), 445-455.  J. X. Nan, D. F. Li and M. J. Zhang, A lexicographic method for matrix games with payos of triangular intuitionistic fuzzy numbers, International Journal of Computational Intelligence Systems, 3(3) (2010), 280-289.  M. H. Shu, C. H. Cheng and J. R. Chang, Using intuitionistic fuzzy sets for fault tree analysis on printed circuit board assembly, Microelectronics Reliability, 46(12) (2006), 2139-2148.  S. P. Wan, D. F. Li and Z. F. Rui, Possibility mean, variance and covariance of triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24(4) (2013), 847- 858.  S. P. Wan and D. F. Li, Possibility mean and variance based method for multi-attribute decision making with triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems, 24 (2013), 743-754.  S. P. Wan, Q. Y. Wang and J. Y. Dong, The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers, Knowledge-Based Sys- tems, 52 (2013), 65-77.  S. P. Wan and J. Y. Dong, Possibility method for triangular intuitionistic fuzzy multi- attribute group decision making with incomplete weight information, International Journal of Computational Intelligence Systems, 7(1) (2014), 65-79.  S. P. Wan and D. F. Li. Fuzzy LINMAP approach to heterogeneous MADM considering comparisons of alternatives with hesitation degrees, Omega, 41(6)(2013), 925-940.  S. P. Wan and D. F. Li. Atanassovs intuitionistic fuzzy programming method for hetero- geneous multiattribute group decision making with Atanassovs intuitionistic fuzzy truth degrees, IEEE Transaction on Fuzzy Systems, 22(2)(2014), 300-312.  S. P. Wan and J. Y. Dong, Interval-valued intuitionistic fuzzy mathematical programming method for hybrid multi-criteria group decision making with interval-valued intuitionistic fuzzy truth degree, Information Fusion, 26 (2015), 49-65.
 S. P. Wan and J. Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making, Journal of Computer and System Sciences, 80(1) (2014), 237-256.  S. P. Wan and J. Y. Dong, Method of trapezoidal intuitionistic fuzzy number for multi- attribute group decision, Control and Decision, 25(5) (2010), 773-776.  S. P. Wan, Power average operators of trapezoidal intuitionistic fuzzy numbers and appli- cation to multi-attribute group decision making, Applied Mathematical Modelling, 37(6) (2013), 4112-4126.  S. P. Wan and J. Y. Dong, Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making, Applied Soft Comput- ing, 29 (2015), 153-168.  S. P.Wan, Method based on fractional programming for interval-valued intuitionistic trape- zoidal fuzzy number multi-attribute decision making, Control and Decision, 27 (3) (2012), 455-458.  S. P. Wan, Multi-attribute decision making method based on interval-valued trapezoidal intuitionistic fuzzy number, Control and Decision, 6 (2011), 857-866.  J. Q. Wang, Overview on fuzzy multi-criteria decision-making approach, Control and De- cision, 23(6) (2008), 601-607.  J. Q. Wang and Z. Zhang, Aggregation operators on intuitionistic trapezoidal fuzzy num- ber and its application to multi-criteria decision making problems, Journal of Systems Engineering and Electronics, 20(2) (2009), 321-326.  J. Q.Wang, R. R. Nie, H. Y. Zhang and X. H. Chen, New operators on triangular intuition- istic fuzzy numbers and their applications in system fault analysis, Information Sciences, 251 (2013), 79-95.  H. M.Wang, Y. J. Xu and J. M. Merig, Prioritized aggregation for non-homogeneous group decision making in water resource management, Economic Computation and Economic Cybernetics Studies and Research, 48(1) (2014), 247-258.  G. W. Wei, Some arithmetic aggregation operators with trapezoidal intuitionistic fuzzy numbers and their application to group decision making, Journal of Computers, 3(2010), 345-351.  G. W. Wei, Hesitant fuzzy prioritized operators and their application to multiple attribute decision making, Knowledge-Based Systems, 31(7) (2012), 176-182.  G. W. Wei and J. M. Merig, Methods for strategic decision making problems with immedi- ate probabilities in intuitionistic fuzzy setting, Scientia Iranica, 19(6) (2012), 1936-1946.  J. Wu and Q. W. Cao, Some families of geometric aggregation operators with intuitionistic trapezoidal fuzzy numbers, Applied mathematical modeling, 37 (1/2) (2013), 318-327.  J. Wu and Y. J. Liu, An approach for multiple attribute group decision making problems with interval-valued intuitionistic trapezoidal fuzzy numbers, Computers and Industrial Engineering, 66 (2013), 311-324.  Y. J. Xu, T. Sun and D. F. Li, Intuitionistic fuzzy Prioritized OWA aggregation operator and its application to multi-criteria decision making, Control and Decision, 26(1) (2011), 129-132.  R. R. Yager, Modeling prioritized multi-criteria decision making, IEEE Transactions on Systems, Man and Cybernetics, Part B. Cybernetics, 34 (2004), 2396-2404.  R. R. Yager, Prioritized aggregation operators, International Journal of Approximate Rea- soning, 48 (2008), 263-274.  R. R. Yager, Prioritized OWA aggregation, Fuzzy Optimization and Decision Making, 8 (2009), 245-262.  H. B. Yan, V. N. Huynh, Y. Nakamori and T. Murai, On prioritized weighted aggregation in multi-criteria decision making, Expert Systems with Applications, 38 (2011), 812-823.  X. H. Yu and Z. S. Xu, Prioritized intuitionistic fuzzy aggregation operators, Information Fusion, 14(1) (2013), 108-116.  D. Yu, J. M. Merig and L. G. Zhou, Interval-valued multiplicative intuitionistic fuzzy preference relations, International Journal of Fuzzy Systems, 15(4) (2013), 412-422.
 D. J. Yu, Intuitionistic fuzzy prioritized operators and their application in multi-criteria group decision making, Technological and Economic Development of Economy, 19(1) (2013), 1-21.  D. J. Yu, Y. Y.Wu and T. Lu, Interval-valued intuitionistic fuzzy prioritized operators and their application in group decision making, Knowledge-Based Systems, 30 (2012), 57-66.  L. A. Zadeh, Fuzzy sets., Information and Control, 18 (1965), 338-353.  X. Zhang, F. Jian and P. D. Liu, A grey relational projection method for multi-attribute decision making based on intuitionistic trapezoidal fuzzy number, Applied Mathematical Modelling, 37(5) (2013), 3467-3477.