This study investigates and improves the operational laws of triangular intuitionistic fuzzy numbers. The triangular intuitionistic fuzzy random variable (TIFRV) is introduced on the basis of the concepts of the triangular intuitionistic fuzzy number and triangular fuzzy random variable. Related properties of a TIFRV are also proposed and verified. To solve the problem of multi-criteria decision making on aspiration levels—a situation in which criterion weights are unknown and criterion values are given in terms of TIFRVs—this study proposes a triangular intuitionistic fuzzy random decision-making method based on a combination of parametric estimation, score functions, and prospect theory. In this method, the decision maker evaluates alternatives with triangular intuitionistic fuzzy numbers in different periods of decision making and thus enables the estimation of the parameters of the triangular intuitionistic fuzzy population and the creation of an intuitionistic triangular fuzzy random matrix. An expectation–variance intuitionistic fuzzy matrix is constructed on the basis of mean–variance analysis, and a fuzzy random score function is then defined to transform a normalized expectation–variance intuitionistic fuzzy matrix into a score function matrix. Prospect theory is used to calculate the values of prospect score functions, and the information entropy method is used to determine criterion weights. This procedure generates comprehensive prospect score function values that determine the final ranking of alternatives. A practical example is presented to show the feasibility and effectiveness of the proposed approach.
AtanassovK.T., On Intuitionistic Fuzzy Sets Theory, Heidelberg, Germany: Springer, 2012.
2.
AtanassovK.T., Index Matrices: Towards an Augmented Matrix Calculus, Springer, 2014.
3.
AtanassovK.T., PasiG. and YagerR.R., Intuitionistic fuzzy interpretations of multi-criteria multi-person and multi-measurement tool decision making, International Journal of Systems Science36 (2005), 859–868.
4.
Andrés-SánchezaJ. de and PuchadesL.G.-V., Using fuzzy random variables in life annuities pricing, Fuzzy sets and systems188 (2012), 27–44.
5.
BaucellsM. and HeukampF.H., Stochastic dominance and cumulative prospect theory, Management Science52 (2006), 1409–1423.
6.
BleichrodtH., SchmidtU. and ZankH., Additive utility in prospect theory, Management Science55 (2009), 863–873.
7.
ChenZ.S. and LiY.L., An improved MULTIMOORA approach for group decision making based upon interdependent inputs of intuitionistic trapezoidal fuzzy numbers, Systems Engineering-Theory & Practice12 (2014), 3138–3156.
8.
ChenZ.S. and LiY.L., Approach for group MULTIMOORA decision making based upon prospect intuitionistic trapezoidal fuzzy number Choquet integral operator, Control & Decision29 (2014), 1053–1063.
9.
CousoI. and DuboisD., On the variability of the concept of variance for fuzzy random variables, IEEE Transactions on Fuzzy Systems17 (2009), 1070–1080.
10.
ChenX.R., Probability and statistics. Beijing: Science Press, 2000.
11.
ChenS.M. and TanJ.M., Handling multi-criteria fuzzy decision-making problems based on vague set theory, Fuzzy Sets and Systems67 (1994), 163–172.
12.
CoomanGDe, KerreE.E. and VanmassenhoveF.R.,Possibility theory: An integral theoretic approach, Fuzzy Sets and Systems46 (1992), 287–299.
13.
DuboisD. and PradeH., Possibility Theory, New York: Plenum, 1988.
14.
DeyO. and ChakrabortyD., A fuzzy random continuous review inventory system, International Journal of Production Economics132 (2011), 101–106.
15.
FanZ.P., ZhangX., ChenF.D., et al., Multiple attribute decision making considering aspiration-levels: A method based on prospect theory, Computers & Industrial Engineering65 (2013), 341–350.
16.
FanZ.P., LiuY. and ShenR.J., Risk decision analysis method for emergency response based on prospect theory, Systems Engineering-Theory & Practice32 (2012), 977–984.
17.
FengX. and LiuY.K., Measurability criteria for fuzzy random vectors, Fuzzy Optimization and Decision Making5 (2006), 245–253.
18.
González-RodríguezG., BlancoÁ., ColubiA., et al., Estimation of a simple linear regression model for fuzzy random variables, Fuzzy Sets & Systems160 (2009), 357–370.
19.
HeX.D. and ZhouX.Y., Portfolio choice under cumulative prospect theory: An analytical treatment, Management Science57 (2011), 315–331.
20.
HaoF.F., LiuY.K. and WangS., The variance formulas for triangular fuzzy random variables, In: Machine Learning and Cybernetics, International Conference on IEEE1 (2008), 612–617.
21.
KlirG.J., On fuzzy-set interpretation of possibility theory, Fuzzy Sets and Systems108 (1999), 263–273.
22.
KwakernaakH., Fuzzy random variables—I. Definitions and theorems, Information Sciences15 (1978), 1–29.
23.
KwakernaakH., Fuzzy random variables—II. Algorithms and examples for the discrete case, Information Sciences17 (1979), 253–278.
24.
KahnemanD. and TverskyA., Prospect theory: An analysis of decision under risk, Econometrica47 (1979), 263–292.
25.
KumarR.S. and GoswamiA., A continuous review production-inventory system in fuzzy random environment: Minmax distribution free procedure, Computers & Industrial Engineering79 (2015), 65–75.
26.
KumarR.S. and GoswamiA., A fuzzy random EPQ model for imperfect quality items with possibility and necessity constraints, Applied Soft Computing34 (2015), 838–850.
27.
LiuY.K. and LiuB.D., On minimum-risk problems in fuzzy random decision systems, Computers & Operations Research32 (2005), 257–283.
28.
LiuY.K., The approximation method for two-stage fuzzy random programming with recourse, IEEE Transactions on Fuzzy Systems15 (2007), 1197–1208.
29.
LiuY.K. and LiuB.D., Fuzzy random variables: A scalar expected value operator, Fuzzy Optimization and Decision Making2 (2003), 143–160.
30.
LiuY.K. and LiuB.D., A class of fuzzy random optimization: Expected value models, Information Sciences115 (2003), 89–102.
31.
LiuY.K. and LiuB.D., Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems11 (2003), 195–215.
32.
LiuY.K. and GaoJ., The independence of fuzzy variables with applications to fuzzy random optimization, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems15 (2007), 1–20.
33.
LiD.F., A note on “using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly”, Microelectronics Reliability48 (2008), 1741.
34.
LiuB.D., Uncertainty theory: An introduction to its axiomatic foundations, Springer, 2004.
35.
LiuH.W. and WangG.J., Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research179 (2007), 220–233.
36.
LiX. and LiuB.D., A sufficient and necessary condition for credibility measures, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems14 (2006), 527–535.
37.
LiuY., TangW.S. and ZhaoR.Q., Reliability and mean time to failure of unrepairable systems with fuzzy random lifetimes, IEEE Transactions on Fuzzy Systems15 (2007), 1009–1026.
38.
MousaviS.M., JolaiF. and Tavakkoli-MoghaddamR., A fuzzy stochastic multi-attribute group decision-making approach for selection problems, Group Decision and Negotiation22 (2013), 207–233.
39.
PasiG. and YagerR.R., Modeling the concept of majority opinion in group decision making, Information Sciences176 (2006), 390–414.
40.
PasiG., YagerR.R. and AtanassovK.T., Intuitionistic fuzzy graph interpretations of multi-person multi-criteria decision making: Generalized net approach, Intelligent Systems, 2004 Proceedings 2004 2nd International IEEE Conference Vol 2 IEEE2004.
41.
PasiG., Fuzzy sets in information retrieval: State of the art and research trends, Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, SpringerBerlin Heidelberg, (2008), pp. 517–535.
42.
RaoR.V., Decision Making in Manufacturing Environment Using Graph Theory and Fuzzy Multiple Attribute Decision Making Methods, Springer2012.
43.
SakawaM. and MatsuiT., Interactive fuzzy random cooperative two-level linear programming through level sets based probability maximization, Expert Systems with Applications40 (2013), 1400–1406.
44.
SchmidtU. and ZankH., Risk aversion in cumulative prospect theory, Management Science54 (2008), 208–216.
45.
ShuM.H., ChengC.H. and ChangJ.R., Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly, Microelectronics Reliability46 (2006), 2139–2148.
46.
TverskyA. and KahnemanD., Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty5 (1992), 297–323.
47.
WangJ.Q. and GongL., Interval probability fuzzy random multi-criteria decision-making approach based on expectation-hybrid entropy, Control and Decision24 (2009), 1065–1069.
48.
WangJ.Q. and NieR.R., Multi-criteria group decision-making method based on intuitionistic trapezoidal fuzzy information, Systems Engineering-Theory & Practice32 (2012), 1747–1753.
49.
WanS.P. and LiD.F., Possibility mean and variance based method for multi-attribute decision making with triangular intuitionistic fuzzy numbers, Journal of Intelligent and Fuzzy Systems24 (2013), 743–754.
50.
WangM., Fuzzy random multi-attribute decision making method, The Fifth IEEE International Conference on Fuzzy Systems and Knowledge Discovery, 2008.
51.
WatadaJ., LinL.C., QiangM., et al., A Fuzzy Random Variable Approach to Restructuring of Rough Sets through Statistical Test, Rough Sets, Fuzzy Sets, Data Mining and Granular Computing, Springer , Berlin Heidelberg, 2009, pp. 269–277.
52.
XuJ.P. and SongX., Multi-objective dynamic layout problem for temporary construction facilities with unequal-area departments under fuzzy random environment, Knowledge-Based Systems81 (2015), 30–45.
53.
YagerR.R., Determining equivalent values for possibilistic variables, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics31 (2001), 19–31.
54.
ZarghamiM., SzidarovszkyF. and ArdakanianR., A fuzzy-stochastic OWA model for robust multi-criteria decision making, Fuzzy Optimization and Decision Making7 (2008), 1–15.
55.
ZhangZ. and XuJ.P., A Mean-semivariance model for stock portfolio selection in fuzzy random environment, IEEE International Conference on Industrial Engineering and Engineering Management, 2008.
56.
ZadehL.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems100 (1999), 9–34.
