# Fuzzy Interval Analysis

• Didier Dubois
• Etienne Kerre
Part of the The Handbooks of Fuzzy Sets Series book series (FSHS, volume 7)

## Abstract

This chapter is an overview of past and present works dealing with fuzzy intervals and their operations. A fuzzy interval is a fuzzy set in the real line whose level-cuts are intervals. Particular cases include usual real numbers and intervals. Usual operations on the real line canonically extend to operations between fuzzy quantities, thus extending the usual interval (or error) analysis to membership functions. What is obtained is a counterpart of random variable calculus, but where, contrary to the latter case, there is no compensation between variables. Many results pertaining to basic properties of fuzzy interval analysis are summed up in the chapter. Computational methods are presented, exact or approximate ones, based on parametric representations, or level-cut approximations. The generalized fuzzy variable calculus involving interactive variables is also discussed with emphasis on triangular-norm based fuzzy additions. Dual ‘optimistic’ operations on fuzzy intervals, i.e., with maximal error compensation are also presented; its interest lies in providing tools for solving fuzzy interval equations. This chapter also contains a reasoned survey of methods for comparing and ranking fuzzy intervals. The chapter includes some historical background, as well as pointers to applications in mathematics and engineering.

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### References

1. Adamo J. M. (1980). Fuzzy decision trees, Fuzzy Sets and Systems, 4, 207–219.
2. Ammar S. (1989). Determining the’ best1 decision in the presence of imprecise information, Fuzzy Sets and Systems, 29, 293–302.
3. Anile A. M., Deodato S. and Privitera G. (1995). Implementing fuzzy arithmetic, Fuzzy Sets and Systems, 72, 239–250.
4. Antonsson E. and Otto K. (1995). Imprecision in engineering design, ASME J. of Mechanical Design, 117(B), 25–32.
5. Ambrosio R. and Martini G. B. (1984). Maximum and minimum between fuzzy symbols in non-interactive and weakly non-interactive situations, Fuzzy Sets and Systems, 12, 27–35.
6. Aumann R. J. (1965). Integrals of set-valued functions, J. Math. Anal Appl., 12, 1–22.
7. Baas S. M. and Kwakernaak H. (1977). Rating and ranking of multiple aspect alternatives using fuzzy sets, Automatica, 13, 47–58.
8. Badard R. (1982). The law of large numbers for fuzzy processes and the estimation problem, Information Sciences, 28, 161–178.
9. Badard R. (1984). Fixed point theorems for fuzzy numbers, Fuzzy Sets and Systems, 13, 291–302.
10. Baekeland R. and Kerre E., (1988). Pieccwise linear fuzzy quantities: A way to implement fuzzy information into expert systems and fuzzy databases, Uncertainty and Intelligent Systems (Bouchon B., Saitta L. and Yager R. R., eds.), Lecture Notes in Computer Sciences, vol. 313, Springer-Verlag, Berlin, 119–126.Google Scholar
11. Baldwin J. F. and Guild N. C. F. (1979a). Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems, 2, 213–231.
12. Baldwin J. F. and Guild N. C. F. (1979b). Comments on the fuzzy max operator of Dubois and Prade, Int. J. Systems Science, 10, 1063–1064.
13. Ban J. (1990). Radon-Nikodym theorem and conditional expectation of fuzzy-valued measures and variables, Fuzzy Sets and Systems, 34, 383–392.
14. Ban J. (1991). Ergodic theorems for random compact sets and fuzzy variables in Banach spaces, Fuzzy Sets and Systems, 44, 71–82.
15. Baptistella L.F.B., and Ollero A. (1980). Fuzzy methodologies for interactive multicriteria optimization, IEEE Trans. Syst. Man Cybern., 10, 355–365.
16. Bellman R. (1957). Dynamic Programming, Princeton University Press, Princeton, N.J.Google Scholar
17. Bertoluzza C. and Bodini A. (1998). A new proof of Nguyen’s compatibility theorem in a more general context, Fuzzy Sets and Systems, 95, 99–102.
18. Bertoluzza C., Corral N. and Salas AA (1995). On a new class of distances between fuzzy numbers, Mathware and Soft Computing, 2, 71–84.
19. Bezdek J.C., Dubois D. and Prade H. (Eds.) (1999). Fuzzy Sets in Approximate Reasoning and Information Systems. The Handbooks of Fuzzy Sets Series (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., Boston.Google Scholar
20. Biacino L. and Lettieri A. (1989). Equations with fuzzy numbers, Information Sciences, 47, 63–76.
21. Bilgic T. and Türksen L.B. (1999). Measurement of membership functions: Theoretical and empirical work, Fundamentals of Fuzzy Sets (Dubois D. and Prade H. eds.), Kiuwer Acad. PubL, 1999. This volume.Google Scholar
22. Bloch I. and Maitre H. (1994). Fuzzy mathematical morphology, Annals of Mathematics and Artificial Intelligence, 10, 55–84.
23. Bloch L. and Maitre H. (1995). Fuzzy mathematical morphologies: A comparative study, Pattern Recognition, 28, 1341–1387.
24. Bodenhofer U. (1998). A similarity-based generalisation of fuzzy orderings, PhD Thesis, J. Kepler University, Linz, AustriaGoogle Scholar
25. Boender C., De Graan J. G and Lootsma F. (1989). Multicriteria decision analysis with fuzzy pairwise comparisons, Fuzzy Sets and Systems, 29, 133–144.
26. Bortolan G. and Degani R. (1985). A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems, 15, 1–19.
27. Bosc P., Buckles B.B., Petry F. E. and Pivert O. (1999). Fuzzy databases, Fuzzy Sets in Approximate Reasoning and Information Systems (Bezdek J.C., Dubois D. and Prade H., eds.), Kluwer Acad. Publ., New York, 403–468.
28. Bouchon-Meunier B., Kosheleva O., Kreinovich V. and Nguyen, H. T. (1997). Fuzzy numbers are the only fuzzy sets that keep invertible operations invertible, Fuzzy Sets and Systems, 91, 155–163.
29. Braae M. and Rutherford D.A. (1978). Fuzzy relations in a control setting, Kybemetes, 7, 185–188.
30. Buckles B. P. and Petry F. E. (1984). Extending the fuzzy data base with fuzzy numbers, Information Sciences, 34, 145–155.
31. Buckley J. J. (1985). Ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems, 15, 21–31.
32. Buckley J. J. (1989a). On the algebra of interactive fuzzy numbers, Fuzzy Sets and Systems, 32, 291–306.
33. Buckley J. J. (1989b). A fuzzy ranking of fuzzy numbers, Fuzzy Sets and Systems, 33, 119–122.
34. Buckley J. J., (1989c). Fuzzy complex numbers, Fuzzy Sets and Systems, 33, 333–345.
35. Buckley J. J. (1990). On the algebra of interactive fuzzy numbers: The continuous case, Fuzzy Sets and Systems, 37, 317–326.
36. Buckley J. J. (1991). Fuzzy complex analysis I: Differentiation, Fuzzy Sets and Systems, 41, 269–284.
37. Buckley J. J. (1992). Fuzzy complex analysis II: Integration, Fuzzy Sets and Systems, 49, 171–179.
38. Buckley J. J. and Ghanas S. (1989). A fast method of ranking alternatives using fuzzy numbers, Fuzzy Sets and Systems, 30, 337–338.
39. Buckley J. J. and Qu Y. (1990). Solving linear and quadratic fuzzy equations, Fuzzy Sets and Systems, 38, 43–61.
40. Buckley J. J. and Qu Y. (1991a). Solving fuzzy equations: a new solution concept, Fuzzy Sets and Systems, 39, 291–303.
41. Buckley J. J. and Qu Y. (1991b), Solving systems of linear fuzzy equations, Fuzzy Sets and Systems, 43, 33–44.
42. Burgin M. (1995). Nonclassical analysis: Fuzzy continuity and convergence, Fuzzy Sets and Systems, 75, 291–299.
43. Burgin M. and Sostak M. (1994). Fuzzyfication of the theory of continuous functions, Fuzzy Sets and Systems, 64, 71–81.
44. Campos L. and Gonzalez (1994). Further contributions to the study of average value for ranking fuzzy numbers, Int. J. Approximate Reasoning, 10, 135–163.
45. Campos L. and Verdegay J. L. (1989). Linear programming problems and ranking of fuzzy numbers, Fuzzy Sets and Systems, 32, 1–12.
46. Cayrol M., Farreny H. and Prade H. (1982). Fuzzy pattern matching, Kybernetes, 11, 103–116.
47. Ghanas S., De Igado M., Verdegay M., and Vila M.A. (1993). Ranking fuzzy intervals in the setting of random sets, Information Sciences, 69, 201–217.
48. Ghanas S. and Kamburovski J. (1981). The use of fuzzy variables in PERT, Fuzzy Sets and Systems, 3, 11–19.Google Scholar
49. Ghanas S. and Kolodziejczyk W. (1982). Maximum flow in a network with fuzzy arc capacities, Fuzzy Sets and Systems, 8, 165–173.
50. Ghanas S. and Kuchta D. (1998), Discrete fuzzy optimization, Fuzzy Sets in Decision Analysis Operations Research and Statistics (Slowinski R., ed.), The Handbooks of Fuzzy Sets Series, Kluwer, Boston, USA.Google Scholar
51. Ghanas S. and Nowakowski M. (1988), Single value simulation of fuzzy variable, Fuzzy Sets and Systems, 25, 43–57.
52. Chang P. T. and Lee E. S. (1994). Fuzzy arithmetics and comparison of fuzzy numbers, Fuzzy Optimization: Recent Advances (De lgado M, Kacprzyk J., Verdegay J. L., Vila M. A., eds), Physica-Verlag, Heidelberg, Germany, 69–81.Google Scholar
53. Chang W. (1981). Ranking of fuzzy utilities with triangular membership functions. Proc. Int. Conf. on Policy Analysis and Information Systems, Taipei, 263–271.Google Scholar
54. Chen S. H. (1985). Ranking fuzzy numbers with maximizing sets and minimizing sets, Fuzzy Sets and Systems, 17, 113–129.
55. Chen S. J., Hwang, C. L. and Hwang, F. P. (1992). Fuzzy Multiple Attribute Decision Making-Methods and Applications, Lecture Notes in Economics and Mathematical Systems, vol. 375, Springer Verlag, Berlin.Google Scholar
56. Chen H. K., Hsu W. K. and Chiang W. L. (1998). A comparison of the vertex method and with JHE method, Fuzzy Sets and Systems, 95, 201–214.
57. Cheng C. H. (1998). A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems, 95, 307–317.
58. Choobineh F. and Li H. (1993). An index for ordering fuzzy numbers, Fuzzy Sets and Systems, 54, 287–295.
59. Cuninghame-Green R. A. (1979). Minimax Algebra, Lecture Notes in Economical and Mathematical Systems, vol. 166, Springer-Verlag, Berlin.Google Scholar
60. Cuninghame-Green R. A. and Cechlarova V. (1995). Residuation in fuzzy algebra and some applications, Fuzzy Sets and Systems, 71, 227–239.
61. Czogala E. and Hirota K. (1986). Probabilistic Sets: Fuzzy and Stochastic Approach to Decision Control and Recognition Processes, ISR 91, Verlag TUV Rheinland, Köln.Google Scholar
62. De Baets B. (1999). Analytical solution methods for fuzzy relational equations, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., 1999. This volume.Google Scholar
63. De Baets, B., Kerre E. and Gupta M. M. (1994a). The fundamentals of fuzzy mathematical morphology — Part 1: Basic concepts, Int. J. General Systems, 23, 155–171.
64. De Baets, B., Kerre E. and Gupta M. M. (1994b). The fundamentals of fuzzy mathematical morphology — Part 2: Idempotence, convexity and decomposition, Int. J, General Systems, 23, 307–322.
65. De Baets, B., Mares, M. and Mesiar, R. (1997). T-partitions of the real line generated by idempotents shapes, Fuzzy Sets and Systems, 91, 177–184.
66. De Baets B., and Markova-Stupnanova A. (1997). Analytical expressions for the addition of fuzzy intervals, Fuzzy Sets and Systems, 91, 203–213.
67. De Campos Ibanez L.M. and Gonzalez-Munoz (1989). A subjective approach for ranking fuzzy numbers, Fuzzy Sets and Systems, 29, 145–154.
68. De lgado M., Verdegay J. L. and Vila M. A (1988). A procedure for ranking fuzzy-numbers using fuzzy relations, Fuzzy Sets and Systems, 26, 49–62.
69. De lgado M., Verdegay J. L. and Vila M. A (1994). Fuzzy numbers, definitions and properties, Mathware and Soft Computing, 1, 31–43.
70. De lgado M., Vila M. A. and Voxman W. (1998a). On a canonical representation for fuzzy numbers, Fuzzy Sets and Systems, 93, 125–135.
71. De lgado M., Vila M. A., and Voxman W. (1998b). A fuzziness measure for fuzzy numbers: Applications, Fuzzy Sets and Systems, 94, 205–216.
72. De mpster A. P. (1967). Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Stau, 38, 325–339.
74. Diamond P. and Kloeden P. (1994). Metric spaces of fuzzy sets, World Scientific, Singapore.
75. Dijkman J. G., van Haeringen H. and de Lange S. T. (1983), Fuzzy numbers, J. Math. Anal Appl., 92, 301–341.
76. Dishkant H. (1981). About membership functions estimation, Fuzzy Sets and Systems, 5, 141–147.
77. Dombi J. (1986). Properties of the fuzzy connectives in the light of general representation theory, Acta Cybernetica, 7, 313–321.
78. Dong W. and Shah H. (1987). Vertex method for computing functions of fuzzy variables, Fuzzy Sets and Systems, 24, 65–79.
79. Dong W., Shah H. and Wong F. (1985). Fuzzy computations in decision and risk analysis, Civ. Eng. Syst., 2, 201–208.
80. Dong W. and Wong F. (1987). Fuzzy weighted averages and the implementation of the extension principle, Fuzzy Sets and Systems, 21, 183–201.
81. Dong W. and Wong F. (1989). Interactive variables and fuzzy decisions, Fuzzy Sets and Systems, 29, 1–19.
82. Dubois D. (1982). A law of large numbers for nonconvex fuzzy sets of the real line, Busefal (IRIT, Université P. Sabatier, Toulouse), 9, 31–38.
83. Dubois D. (1983). A fuzzy heuristic, interactive approach to the optimal network problem, Advances in Fuzzy Sets and Possibility Theory and Applications (Wang P. P., ed.), Plenum Press, New York, 253–276.
84. Dubois D. (1987a). An application of fuzzy arithmetics to the optimization of industrial machining processes, Mathematical Modelling, 9, 461–475.
85. Dubois D. (1987b). Linear programming with fuzzy data, Analysis of Fuzzy Information (Bezdek J. C, ed.), Vol. 3, CRC Press, Boca Raton, F1., 241–264.Google Scholar
86. Dubois D., Fargier H., and Prade (1995). Fuzzy constraints in job-shop scheduling, J. Intellig. Manufacturing, 6, 215–234.
87. DuboisD., Nguyen H.T. and Prade H. (1999). Possibility theory, probability and fuzzy sets: Misunderstandings, bridges and gaps, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., 1999. This volume.Google Scholar
88. Dubois D., Ostasiewicz W. and Prade H. (1999). Fuzzy sets: History and basic notions, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., 1999. This volume.Google Scholar
89. Dubois D. and Prade H. (1978a). Operations on fuzzy numbers, Int. J. Systems Science, 9, 613–626.
90. Dubois D. and Prade H. (1978b). Comment on Tolerance analysis using fuzzy sets’ and ‘A procedure for multiple aspect decision making’, Int. J. Systems Science., 9, 357–360.
91. Dubois D. and Prade H. (1978c). Algorithmes de plus court chemin pour traiter des donnees floues, RAIRO, Serie R. O., 12, 213–227.
92. Dubois D. and Prade H. (1979a). Fuzzy real algebra: Some results, Fuzzy Sets and Systems, 2, 327–348.
93. Dubois D. and Prade H. (1979b), Operations in a fuzzy-valued logic, Inf. & Control, 43, 224–240.
94. Dubois D. and Prade H. (1980a). Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York.
95. Dubois D. and Prade H. (1980b). Systems of linear fuzzy constraints, Fuzzy Sets and Systems, 3, 37–48.
96. Dubois D. and Prade H. (1981). Additions of interactive fuzzy numbers, IEEE Trans. Automatic Control, 26, 926–936.
97. Dubois D. and Prade H. (1982a). The use of fuzzy numbers in decision analysis, Fuzzy Information and Decision Processes (Gupta M. M. and Sanchez E., eds.), North-Holland, Amsterdam, 309–321.Google Scholar
98. Dubois D., and Prade H. (1982b). Towards fuzzy differential calculus, Fuzzy Sets and Systems, 8, Part I. Integration of fuzzy mappings, 1-17; Part II. Integration on fuzzy intervals, 105-116; Part III Differentiation, 225–233.Google Scholar
99. Dubois D. and Prade H. (1982c). What does’ convergence’ mean for fuzzy numbers?, Proc. IFAC Symp. on Theory and Applications of Digital Control (Mahalonabis A. K., ed.), Pergamon Press, New York, 433–438.Google Scholar
100. Dubois D. and Prade H. (1983a). Inverse operations for fuzzy numbers, Proc. IFAC Symp. on Fuzzy Information, Knowledge Representation and Decision Processes (Sanchez E. and Gupta M. M., eds.), Pergamon Press, Oxford, 391–396.Google Scholar
101. Dubois D. and Prade H. (1983b). Ranking fuzzy numbers in the setting of possibility theory, Information Sciences, 30, 183–224.
102. Dubois D. and Prade H. (1983c). Twofold fuzzy sets: An approach to the representation of sets with fuzzy boundaries, based on possibility and necessity measures, Fuzzy Math. (Huazhong, China), 3(4), 53–76.
103. Dubois D. and Prade H. (1983d). On distances between fuzzy points and their use for plausible reasoning, IEEE Int. Conf. on Systems Man and Cybernetics, Bombay and New De lhi, IEEE, Pistacaway, NJ, 300–303.Google Scholar
104. Dubois D. and Prade H. (1984). Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations, Control Cybern. (Warsaw), 13, 129–146.
105. Dubois D. and Prade H. (1985), Fuzzy cardinality and the modeling of imprecise quantification, Fuzzy Sets and Systems, 16, 199–230.
106. Dubois D. and Prade H. (1986). Fuzzy sets and statistical data, Europ. J. Op. Res., 25, 345–356.
107. Dubois D. and Prade H. (1987a). Fuzzy numbers: An overview, Analysis of Fuzzy Information, Vol. I (Bezdek J., ed.), CRC Press, Boca Raton, FL, 3–39.Google Scholar
108. Dubois D. and Prade H., (1987b). The mean value of a fuzzy number. Fuzzy Sets and Systems, 24, 279–300.
109. Dubois D. and Prade H,, (eds.) (1987c). Special Issue on Fuzzy Numbers. Fuzzy Sets and Systems, 24(3).Google Scholar
110. Dubois D. and Prade H. (1988a). Possibility Theory. An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.
111. Dubois D. and Prade H. (1988b). On fuzzy syllogisms, Computational Intelligence, 4, 171–179.
112. Dubois D. and Prade H. (1989a). Processing fuzzy temporal knowledge, IEEE Trans. Syst. Man and Cybern., 19, 729–744.
113. Dubois D. and Prade H. (1989b). Order-of-magnitude reasoning with fuzzy relations, Revue d’Intelligence Artificielle (Hermes, Paris), 3(4), 69–94.Google Scholar
114. Dubois D. and Prade H. (1991a). Random sets and fuzzy interval analysis, Fuzzy Sets and Systems, 42, 87–101.
115. Dubois D. and Prade H. (1991b). On the ranking of ill-known values in possibility theory, Fuzzy Sets and Systems, 43, 311–317.
116. Dubois D. and Prade H. (1995). Possibility theory as a basis for qualitative decision theory. Proc. 24 th Int. Joint Conf. on AI, Montreal, Canada, 1924–1930.Google Scholar
117. Dubois D., Prade H. and Sandri S. (1993). On possibility/probability transformations. In: Fuzzy Logic. State of the Art, (R. Lowen, M. Roubens, eds.), Kluwer Acad. Publ., Dordrecht, 103–112.Google Scholar
118. Dubois D., Prade F.L. and Sabbadin R. (1998). Qualitative decision theory with Sugeno integrals. Proc. of the 14th Conf. on Uncertainty in Artificial Intelligence, Madison, July 24-26, 1998, (G. Cooper, S. Moral, eds.), Morgan Kaufmann, San Francisco, 121–128.Google Scholar
119. Dubois D., Prade H. and Yager R. R. (1998). Computation of intelligent fusion operations based on constrained fuzzy arithmetic, Proc. IEEE Int. Conf. on Fuzzy Systems, Anchorage, Al., 767–772.Google Scholar
120. Efstathiou J. and Bonissone P. (1979). Ranking fuzzy sets using linguistic preference relations, Proc. IEEE Int. Conf. on Systems Man and Cybern. Denver, Co, USA.Google Scholar
121. Efstathiou J. and Tong R. (1982). Ranking fuzzy sets: A decision-theoretic approach, IEEE Trans. SysL Man. Cybern., 12, 655–659.
122. Eklund P. and Gähler W. (1988). Basic notions for fuzzy topology, Fuzzy Sets and Systems, 26, 333–356.
123. Féron R. (1976). Ensembles aléatoires flous, C.R. Acad. Sci. Ser A., 282, 903–906.
124. Fishburn P. (1985). Interval Orders and Interval Graphs. Wiley, New-York.Google Scholar
125. Jenei S. and Fodor J. (1998). On continuous triangular norms, Fuzzy Sets and Systems, 100, 273–282.
126. Fodor J., Orlovski S.A., Perny P. and Roubens M. (1998). The use of fuzzy preference models in multiple criteria choice, ranking and sorting, Fuzzy Sets in Decision Analysis, Operations Research and Statistics (Slowinski R., ed.), Kluwer Acad. Publ., Boston, 69–101.
127. Fodor J. and Roubens M. (1994). Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer Acad. Publ., Dordrecht.Google Scholar
128. Fodor J. and Yager R.R. (1999). Fuzzy set-theoretic operators and quantifiers, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad, Publ., 1999. This volume.Google Scholar
129. Fortemps P. (1997). Jobshop scheduling with imprecise durations: A fuzzy approach, IEEE Trans. Fuzzy Systems, 5, 557–569.
130. Fortemps P. and Roubens M. (1996). Ranking and defuzzification methods based on area compensation, Fuzzy Sets and Systems, 82, 319–330.
131. Freeling A. N. S. (1980). Fuzzy sets and decision analysis, IEEE Trans. Syst. Man. Cybern., 10, 341–354.
132. Freeling A. N. S. (1984). Possibilities versus fuzzy probabilities-Two alternative decision-aids, Fuzzy Sets and Decision Analysis (Zimmermann H. J., Zadeh L. A. and Gaines B. R., eds.), TIMS Studies in the Management Sciences, vol. 20, North-Holland, Amsterdam, 67–71.Google Scholar
133. French S. (1984). Fuzzy decision analysis: some criticisms, Fuzzy Sets and Decision Analysis (Zimmermann H. J., Zadeh L. A. and Gaines B. R,, eds.), TIMS Studies in the Management Sciences, vol. 20, North-Holland, Amsterdam, 29–44.Google Scholar
134. Fullér R. (1990). On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers, Fuzzy Sets and Systems, 34, 347–354.
135. Fullér R. (1991). On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems, 41, 83–87.
136. Fullér R. (1992). A law of large numbers for fuzzy numbers, Fuzzy Sets and Systems, 45, 299–303.
137. Fullér R. and Keresztfalvi T. (1991). On generalization of Nguyen’s theorem, Fuzzy Sets and Systems, 41, 371–374.
138. Fullér R. and Keresztfalvi T. (1992). T-norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 51, 155–159.
139. Fullér R. and Mesiar R. (eds.) (1997). Special Issue on Fuzzy Arithmetic, Fuzzy Sets and Systems, 91(2).Google Scholar
140. Fullér R. and Triesch E. (1993). A note on the law of large numbers for fuzzy variables, Fuzzy Sets and Systems, 55, 235–236.
141. Fullér R. and Zimmermann H. J. (1992). On computation of the compositional rule of inference under triangular norms, Fuzzy Sets and Systems, 51, 267–275.
142. Fullér R. and Zimmermann H. J. (1993). Fuzzy reasoning for solving fuzzy mathematical programming problems, Fuzzy Sets and Systems, 60, 121–133.
143. Gähler S. and Gähler W. (1994). Fuzzy real numbers, Fuzzy Sets and Systems, 66, 137–158.
144. Gähler W. (1992). Fuzzy topology in Topology, Measures and Fractals, Math. Research, 67, Academia-Verlag, Berlin, 188–197.Google Scholar
145. Gebhardt A. (1995). On types of fuzzy numbers and extension principles, Fuzzy Sets and Systems, 75, 311–318.
146. Gebhardt J., Gil M. A. and Kruse R. (1998). Fuzzy set-theoretic methods in statistics, Fuzzy Sets in Decision Analysis, Operations Research and Statistics, The Handbook of Fuzzy Sets Serie, Kluwer Academic Publ., Dordrecht, The Netherlands, 311–348.Google Scholar
147. Giachetti R. E. (1996). The Mathematics of Triangular Fuzzy Numbers to support a Model of Imprecision in Design, Ph.D. Dissertation, Industrial Engineering, North-Carolina State University, US.Google Scholar
148. Giachetti R. E. and Young R. E. (1997a). Analysis of the error in standard approximation for multiplication of triangular and trapezoidal fuzzy numbers, and the development of a new approximation, Fuzzy Sets and Systems, 91, 1–15.
149. Giachetti R. E. and Young R. E. (1997b). A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets and Systems, 91, 185–202.
150. Gil M. A. (1992), A note on the connection between fuzzy numbers and random intervals, Statistics and Probability Lett., 13, 311–319.
151. Goetschel R. H. (1997). Representations with fuzzy darts, Fuzzy Sets and Systems, 89, 77–106.
152. Goetschel R. Jr. and Voxman W. (1983). Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 10, 87–99.
153. Gonzalez A. (1990). A study of the ranking function approach through mean values, Fuzzy Sets and Systems, 35, 29–43.
154. Gonzalez A. and Vila M. A. (1991). A discrete method for studying indifference and order relations between fuzzy numbers, Information Sciences, 56, 245–258.
155. Gonzalez A. and Vila M. A. (1992). Dominance relations on fuzzy numbers, Information Sciences, 64, 1–16.
156. Gottwald S. (1984). On the existence of solutions of systems of fuzzy equations, Fuzzy Sets and Systems, 12, 301–302.
157. Gottwald S. (1993). Fuzzy Sets and Fuzzy Logic, Vieweg, Braunschweig.Google Scholar
158. Guh Y.Y., Hong C.C., Wang K.M. and Lee E.S. (1996). Fuzzy weighted average: A max-min paired elimination method, Computers Math. Applic., 32, 115–123.
159. Harman B. (1992a). On associativity of the product of modified real fuzzy numbers, Tatra Mount Math Publ., 1, 45–50.
160. Harman B. (1992b). Sum and product of the modified real fuzzy numbers, Kybernetika (Prague), Suppl. 28(1/6), 37–40.
161. Hellendoorn H. (1990). Closure properties of the compositional rule of inference, Fuzzy Sets and Systems, 35, 163–184.
162. Heilpern S. (1992). The expected value of a fuzzy number, Fuzzy Sets and Systems, 47, 81–87.
163. Heilpern S. (1997). Representation and application of fuzzy numbers, Fuzzy Sets and Systems, 91, 259–268.
164. Herencia J. A. (1997). Graded numbers and graded convergence of fuzzy numbers, Fuzzy Sets and Systems, 88, 183–194.
165. Hirota K. and Pedrycz W. (1989). Interpretation of results of ranking methods with the aid of probabilistic sets, Fuzzy Sets and Systems, 32, 263–274.
166. Höhle U. (1981). Representation theorems for L-fuzzy quantities, Fuzzy Sets and Systems, 5, 83–107.
167. Höhle U. (1987). Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic, Fuzzy Sets and Systems, 24, 263–278.
168. Hong D. H. (1995). A note on t-norm-based addition of fuzzy intervals, Fuzzy Sets and Systems, 75, 73–76.
169. Hong D. H. (1996). A note on the convergence of T-sum series of L-R fuzzy numbers, Fuzzy Sets and Systems, 77, 253–254.
170. Hong D. H. and Hwang S. Y. (1994a). On the convergence of T-sum of L-R fuzzy numbers, Fuzzy Sets and Systems, 63, 175–180.
171. Hong D. H. and Hwang S. Y. (1994b). On the compositional rule of inference under triangular norms, Fuzzy Sets and Systems, 66, 25–38.
172. Hong D. H. and Hwang S. Y. (1996). A note on the correlation of fuzzy numbers, Fuzzy Sets and Systems, 79, 401–402.
173. Hong D. H. and Hwang S. Y. (1997). A T-sum bound of LR-fuzzy numbers. Fuzzy Sets and Systems, 91, 239–252.
174. Hong D. H. and Kim H. (1996). A law of large numbers for fuzzy numbers in a Banach space, Fuzzy Sets and Systems, 77, 349–354.
175. Hong D. H. and Kim H. (1998). A note to the sum of fuzzy variables, Fuzzy Sets and Systems, 93, 121-124.Google Scholar
176. Ishibuchi H., Kwon K. and Tanaka H. (1995). A learning algorithm of fuzzy neural networks with triangular fuzzy weights, Fuzzy Sets and Systems, 71, 277–293.
177. Jacas J. and Recasens J. (1993). Fuzzy numbers and equality relations, Proc. IEEE Int. Conf. on Fuzzy Systems, San Francisco, 1298–1301.Google Scholar
178. Jain R. (1976). Tolerance analysis using fuzzy sets, Int. J. Systems Science, 7, 1393–1401.
179. Jain R. (1977). A procedure for multiple aspect decision-making, Int. J. Systems Science, 8, 1–7.
180. Jiang H. (1986). The approach to solving simultaneous linear equations that coefficients are fuzzy numbers, J. Nat. Univ. Defence Technology (Chinese), 3, 96–102.Google Scholar
181. Jimenez M. (1996). Ranking fuzzy intervals throgh the comparison of its expected intervals. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 4, 379–388.
182. Jimenez M. and Rivas J. A (1998). Fuzzy number approximation. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 6, 69–78.
183. Juang C.-H., Huang, X. H. and Elton D. J., (1991). Fuzzy information processing by Monte Carlo simulation technique, J. Civil Eng. Systems, 8, 19–25.
184. Kacprzyk J. and Fedrizzi M. (1992). Fuzzy Regression Analysis, Physica-Verlag, Heidelberg.Google Scholar
185. Kaleva O. (1985). On the convergence of fuzzy sets, Fuzzy Sets and Systems, 24, 53–65.
186. Kaleva O. (1987). Fuzzy differential equations, Fuzzy Sets and Systems, 24, 301–317.
187. Kalina M. (1997). Derivatives of fuzzy functions and fuzzy derivatives, Tatra Mount Math. Publ., 12, 27–34.
188. Kaufmann A. (1975). Introduction to the Theory of Fuzzy Subsets, Academic Press, New York.
189. Kaufmann A (1980). La simulation des ensembles flous, CNRS Round Table on Fuzzy Sets, Lyon, France (unpublished proceedings).Google Scholar
190. Kaufmann A. (1981). Hybrid Convolution, a way to combine fuzzy numbers and random numbers, Fuzzy Math. (Huazhong, China), 1(2), 1–12.
191. Kaufmann A. and Gupta M. M. (1985). Introduction to Fuzzy Arithmetic-Theory and Applications, Van Nostrand Reinhold, New York.Google Scholar
192. Kaufmann A. and Gupta M. M. (1988). Fuzzy Mathematical Models in Engineering and Management Science. North-Holland, Amsterdam.Google Scholar
193. Kawaguchi M. F., and Da-Te T. (1993). A calculation method for solving fuzzy arithmetic equations with triangular norms, Proc. 2d IEEE Int. Conf on Fuzzy Systems (FUZZ-IEEE), San Francisco, 470–476.Google Scholar
194. Kawaguchi M. F. and Da-Te T. (1994). Some algebraic properties of weakly non-interactive fuzzy numbers, Fuzzy Sets and Systems, 68, 281–291.
195. Keresztfalvi T. and Kovacs M. (1992). g, p-fuzzification of arithmetic operations, Tatra Mount Math. Publ., 1, 65–71.
196. Kerre E. (1982). The use of fuzzy numbers in electrocardiological diagnosis, Approximate Reasoning in Decision Analysis (Gupta M. M. and Sanchez E., eds.), North-Holland, Amsterdam, 277–282.Google Scholar
197. Kerre E. (1987). Fuzzy approach to ECG diagnosis, Encyclopedia of Systems and Control (Singh, M. ed.), Pergamon Press, Oxford, UK, 1405–1407.Google Scholar
198. Kerre E. (1993). Introduction to the Basic Principles of Fuzzy Set Theory and Some of its Applications. Communication & Cognition, Gent, Belgium.Google Scholar
199. Kerre E. and Van Schooten A. (1988). A deeper look on fuzzy numbers from a theoretical as well as from a practical point of view, Fuzzy Logic in Knowledge-Based Systems, Decision and Control (M. M. Gupta, T. Yamakawa, eds.), North-Holland, Amsterdam, 173–196.Google Scholar
200. Kim J. B. (1993), On product-sum of fuzzy complex numbers of an elliptic type, J. Fuzzy Math., 1(3), 611–617.
201. Kim K. and Park K. S. (1990). Ranking fuzzy numbers with index of optimum, Fuzzy Sets and Systems, 35, 143–150.
202. Kim W. J., Ko J. H. and Chung M. J. (1994). Uncertain robot environment modelling using fuzzy numbers, Fuzzy Sets and Systems, 61, 53–62.
203. Klawonn F. and Kruse R. (1993). Equality relations as a basis for fuzzy control, Fuzzy Sets and Systems, 54, 147–156.
204. Klement E. P. (1981). Operations on fuzzy sets and fuzzy numbers related to triangular norms, Proc. Ilth IEEE Int. Symp. on Multiple-Valued Logic, Oklahoma City, 218–225.Google Scholar
205. Klement E. P. (1985). Integration of fuzzy valued functions, Revue Roumaine Math. Pures Appl., 30, 375–384.
206. Klement E. P. (1987). Strong law of large numbers for random variables with values in the fuzzy real line, IFSA Commun, Math. Chapt., 7–11.Google Scholar
207. Klement E. P., Mesiar R. and Pap E. (1999). Triangular Norms, a book to appear.Google Scholar
208. Klir G. J. (1997). Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems, 91, 165–175.
209. Klir G. J. (1999). Measures of uncertainty and information, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., 1999. This volume.Google Scholar
210. Kolesarova A. (1995). Additions preserving the linearity of fuzzy intervals, Tatra Mount Math. Publ., 6, 75–81.
211. Kolesarova A. (1997). Similarity preserving t-norm based additions of fuzzy numbers, Fuzzy Sets and Systems, 91, 215–229.
212. Kolesarova A. (1998). Triangular norm-based addition preserving linearity of T-sums of linear fuzzy intervals, Mathware and Soft Computing, 5, 91–98.
213. Kolesarova A. and Riecan B. (1993). T-fuzzy observables, Int. J. Theor. Physics, 32, 1691–1707.
214. Kolodziejczyk W. (1986). Orlovsky’s concept of decision-making with a fuzzy relation — Further results. Fuzzy Sets and Systems, 20, 11–20.
215. Kosheleva O., Cabrera S.D., Gibson G.A. and Koshelev M. (1997). Fast implementations of fuzzy arithmetic operation using fast Fourier transform, Fuzzy Sets and Systems, 91, 269–277.
216. Kruse R. (1987). On a software tool for statistics with linguistic data, Fuzzy Sets and Systems, 24, 377–383.
217. Kruse R. and Meyer K, D (1987). Statistics with Vague Data, Kluwer Academic, Dordrecht, Netherlands.Google Scholar
218. Kwakernaak H. (1978). Fuzzy random variables — Vol. I: Definitions and theorems, Information Sciences, 15, 1–29.
219. Kwakernaak H. (1979), Fuzzy random variables —-Part II: Algorithms and examples in the discrete case, Information Sciences, 17, 253–278.
220. Kwiesielewicz M. (1998). A note on the fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems, 95, 161–173.
221. Lai Y. J. and Hwang C. L. (1992). Fuzzy Mathematical Programming-Methods and Applications, Lecture Notes in Economics and Mathematical Systems, vol. 394, Springer-Verlag, Berlin.Google Scholar
222. Lee D. H. and Park D. (1997). An efficient algorithm for fuzzy weighted average, Fuzzy Sets and Systems, 87, 39–45.
223. Lee K. M., Cho C. H. and Lee-Hwang H. (1994). Ranking fuzzy numbers with satisfaction function, Fuzzy Sets and Systems, 64, 295–311.
224. Li RJ. and Lee E. S. (1987). Ranking fuzzy numbers: A comparison, Proc. North-American Fuzzy Inf. Processing Soc. Workshop (NAFIPS’87), Purdue University, West Lafayette, IN, 169–204.Google Scholar
225. Li E. S. and Lee R. J. (1988). Comparison of fuzzy numbers based on the probability measure of fuzzy events, Comp. & Math. with Appl., 15, 887–896.
226. Ling C. H. (1965). Representation of associative functions, Publ. Math. Debrecen, 12, 189–212.
227. Liou T. S. and Wang M. J. J. (1992a). Ranking fuzzy numbers with integral value, Fuzzy Sets and Systems, 50, 247–255.
228. Liou T. S. and Wang M. J. J. (1992b). Fuzzy weighted average: An improved algorithm. Fuzzy Sets and Systems, 307–317.Google Scholar
229. Liu X. (1994). On the continuity of fuzzy number valued function, Fuzzy Sets and Systems, 68, 245–247.
230. Lootsma F. A. (1985). Performance evaluation of nonlinear optimization methods via pairwise comparison and fuzzy numbers, Math. Prog., 33, 93 et seq.Google Scholar
231. Löwen R. (1980). Convex fuzzy sets, Fuzzy Sets and Systems, 3, 291–310.
232. Löwen R. (1983a). On (R(L),⊕), Fuzzy Sets and Systems, 10, 203–209.
233. Lowen R. (1983b). Hyper-spaces of fuzzy sets, Fuzzy Sets and Systems, 9, 287–311.
234. Lowen R. (1985). The order aspect of the fuzzy real line, Manuscripta Math., 39, 293–309.
235. Lowen R. (1996). Fuzzy Set Theory, Kluwer, Dordrecht.Google Scholar
236. Mabuchi S (1988). An approach to the comparison of fuzzy subsets with an α-cut dependent index, IEEE Trans. Syst. Man. Cybern., 18, 264–272.
237. Mandic N. J. and Mamdani E. H. (1980). A linguistic fuzzy calculator. Res. Rep. No. 9, Queen Mary College, University of London, UK.Google Scholar
238. Mares M. (1977a). How to handle fuzzy quantities, Kybernetika (Prague), 13, 23–40.
239. Mares M. (1977b). On fuzzy quantities with real and integer values, Kyhernetika (Prague), 13, 41–56.
240. Mares M. (1993a). Algebraic equivalences over fuzzy quantities, Kyhernetika (Prague), 29(2), 121–132.
241. Mares M. (1993b). Remarks on fuzzy quantities with finite support, Kybernetika (Prague), 29(2), 133–143.
242. Mares M. (1994), Computation Over Fuzzy Quantities, CRC Press, Boca RatonGoogle Scholar
243. Mares M. and Mesiar R, (1996). Processing of sources of fuzzy quantities, Proc. IPMU’96, Granada, 359–363.Google Scholar
244. Markov S. M. (1995). On directed interval arithmetics and its applications, J. Universal Computer Science, 1(7), 510–521.Google Scholar
245. Markov S. M. (1996). On the foundations of interval mathematics, Scientific Computing and Validated Numerics (Proc. SCAN-95) (Alefeld G. and Frommer, A., eds.), Akademie-Verlag, Berlin, 307 et seq.Google Scholar
246. Markova A. (1995). Additions of L-R fuzzy numbers, Busefal (IRIT, Université P. Sabatier, Toulouse), 63, 25–29.Google Scholar
247. Markova A. (1997). Idempotents of the T-addition of fuzzy numbers, Tatra Mount Math Publ., 12.Google Scholar
248. Markova A. (1998). T-sum of L-R fuzzy numbers, Fuzzy Sets and Systems, 85, 379–384.
249. Markova-Stupnanova A. (1997a). A note to the addition of fuzzy intervals based on the continuous Archimedean t-norm, Fuzzy Sets and Systems, 91, 253–258.
250. Markova-Stupnanova A. (1997b). Pseudo-convolutions and their idempotents. Proc. 7th Inter. Fuzzy Syst. Assoc. World Cong. (IFSA’97), Prague, June 25–29, 1997, Vol. 1, Academia, 484–487.Google Scholar
251. Markova-Stupnanova A. (1998). A note on recent results on the law of large numbers for fuzzy numbers, BUSEFAL (IRIT, Univ. P. Sabatier, Toulouse), 76, 12–18.Google Scholar
252. Matheron G. (1975). Random Sets and Integral Geometry, John Wiley & Sons, New York.
253. McCahon C. and Lee E. S. (1990). Comparing fuzzy numbers: the proportion of the optimum method, Int. J. Approximate Reasoning, 4, 159–181.
254. McCain R. A. (1983). Fuzzy confidence intervals, Fuzzy Sets and Systems, 10, 281–290.
255. Mesiar R. (1993). Fuzzy measurable functions, Fuzzy Sets and Systems, 59, 35–42.
256. Mesiar R. (1995). Computation over LR-fuzzy numbers, Proc. CIFT95, Trento, 165–176.Google Scholar
257. Mesiar R. (1996a). LR-fuzzy numbers, Proc. 6th Int. Conf. On Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU’96), Granada, Spain, 337–342.Google Scholar
258. Mesiar R. (1996b). A note on the T-sum of L-R fuzzy numbers, Fuzzy Sets and Systems, 79, 259–261.
259. Mesiar R. (1997a). Shape preserving additions of fuzzy intervals, Fuzzy Sets and Systems, 86, 73–78
260. Mesiar R. (1997b). Triangular norm-based additions of fuzzy intervals, Fuzzy Sets and Systems, 91, 231–237.
261. Mesiar R. (1998). Approximations of continuous t-norms by strict t-norms with smooth generators, BUSEFAL (IRIT, Université Paul Sabatier, Toulouse), n° 75, 72–79.Google Scholar
262. Miyakawa M., Nakamura K., Ramik J. and Rosenberg I. G. (1993). Joint canonical fuzzy numbers, Fuzzy Sets and Systems, 53, 39–49.
263. Miyakoshi M. and Shimbo M. (1984a). A strong law of large numbers for fuzzy-random variables, Fuzzy Sets and Systems, 12, 133–142.
264. Miyakoshi M. and Shimbo M. (1984b), An individual ergodic theorem for fuzzy random variables, Fuzzy Sets and Systems, 13, 285–290.
265. Mizumoto M. and Tanaka K. (1976a). The four operations of arithmetic on fuzzy numbers, Syst. Comput. Controls, 7(5), 73–81.
266. Mizumoto M. and Tanaka K. (1976b). Algebraic properties of fuzzy numbers, Proc. Int. Conf, On Cybernetics and Society, Washington, DC, 559–563.Google Scholar
267. Mizumoto M. and Tanaka K. (1976c). Some properties of fuzzy sets of type 2, Inf. Control, 31, 312–340.
268. Mizumoto M. and Tanaka K. (1979). Some properties of fuzzy numbers, Advances in Fuzzy Set Theory and Applications (Gupta M. M., Ragade R. K. and Yager R. R., eds.), North-Holland, Amsterdam, 153–165.Google Scholar
269. Moore R. (1966). Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
270. Moore R. (1979). Methods and Applications of Interval Analysis, SIAM Studies on Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
271. Murakami S., Maeda H. and Imamura S. (1989). Fuzzy decison analysis on the development of centralized regional energy control systems. Proc. IFAC Symp. on Fuzzy Information, Knowledge Representation and Decision Processes (Sanchez E. and Gupta M. M, eds.), Pergamon Press, Oxford, 363–368.Google Scholar
272. Murthy C. A, Pal S. K. and Dutta Majumder D. (1985). Correlation bet ween two membership functions, Fuzzy Sets and Systems, 17, 23–38.
273. Nahmias S. (1978). Fuzzy variables, Fuzzy Sets and Systems, 1, 97–110.
274. Nahmias S. (1979). Fuzzy variables in a random environment, Advances in Fuzzy Set Theory (Gupta M.M., Ragade R. and Yager R. R., eds.), North-Holland, Amsterdam, 165–180.Google Scholar
275. Nakahara Y. (1998). User oriented ranking criteria and application to fuzzy mathematical programming problems, Fuzzy Sets and Systems, 94, 275–286.
276. Nakamura K. (1986). Preference relations on a set of fuzzy utilities as a basis for decision-making, Fuzzy Sets and Systems, 20, 147–162.
277. Nakamura K. (1990). Canonical fuzzy number of dimension two and fuzzy utility difference for understanding preferential judgements, Information Sciences, 50, 1–22.
278. Nanda S. (1989). On sequences of fuzzy numbers, Fuzzy Sets and Systems, 33, 123–126.
279. Negoita C. V. (1978). Management Applications of Systems Theory, Birkhauser Verlag, BaselGoogle Scholar
280. Nguyen H. T. (1978). A note on the extension principle for fuzzy sets, J. Math. Anal Appl., 64, 369–380.
281. Nguyen H. T. and Kreinovitch V., Nesterov V. and Nakamura M. (1997). On hardware support for interval computations and for soft computing: theorems, IEEE Trans. on Fuzzy Systems, 5, 108–127.
282. Nguyen H. T., Kreinovitch V. and Wojciechowski P. (1998). Strict Archimedean t-norms and t-co-norms as universal approximators, Int. J. Approximate Reasoning, 18, 239–249.
283. Oftedal H. (1981). Imprecision of specification of information systems parameters: A study of decision point probabilities, Inform. Syst., 6, 101–109.
284. Orlovsky S.A. (1978). Decision-making with a fuzzy preference relation, Fuzzy Sets and Systems, 1, 155–168.
285. Otto K. and Antonsson E. (1991). Trade-off strategies in engineering design, Research in Engineering Design (Springer Verlag), 3, 87–104.
286. Otto K. and Antonsson E. (1994). Design parameter selection in the presence of noise, Research in Engineering Design (Springer Verlag), 6, 234–246.
287. Otto K., Lewis A. D. and Antonsson E. (1993). Approximating α-cuts with the vertex method, Fuzzy Sets and Systems, 55, 43–50.
288. Ovchinnikov S. and Migdal M. (1987). On ranking fuzzy sets, Fuzzy Sets and Systems, 24, 113–117.
289. Pal N. and Bezdek J.C. (1999). Quantifying different facets of fuzzy uncertainty, Fundamentals of Fuzzy Sets (Dubois D. and Prade H., eds.), Kluwer Acad. Publ., 1999. This volume.Google Scholar
290. Pan Y. and Yuan B. (1997). Bayesian inference of fuzzy probabilities, Int. J. of General Systems, 26(1-2), 73–90.
291. Papoulis A. (1965). Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York.Google Scholar
292. Parratt L, G. (1971). Probability and Experimental Errors in Science, Dover, NewYork.Google Scholar
293. Pedrycz W. (1985). On generalized fuzzy relation equations and their applications, J. Math. Anal Appl., 107, 520–536.
294. Pedrycz W. (1986). Ranking multiple aspect alternatives — Fuzzy relational approach, Automatica, 22, 251–253.
295. Pedrycz W. (1994). Why triangular membership functions?, Fuzzy Sets and Systems, 64, 21–30.
296. Prade H. (1979). Using fuzzy set theory in a scheduling problem, Fuzzy Sets and Systems, 2, 153–165.
297. Prade H. (1980). Operations research with fuzzy data, Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems (Wang P. P. and Chang S. K., eds.), Plenum Press, New York, 155–170.Google Scholar
298. Prade H. (1982). Modal semantics and fuzzy set theory. In: Fuzzy Set and Possibility Theory. Recent De velopments. (R.R. Yager, ed.), Pergamon Press, New York, 232–246.Google Scholar
299. Prade H. (1984). Lipski’s approach to incomplete information data bases restated and generalized in the setting of Zadeh’s possibility theory, Inf. Syst., 9, 27–42.
300. Prade H. and Testemaie C. (1984). Generalizing database relational algebra for the treatment of incomplete/uncertain information and vague queries, Information Sciences, 34, 115–143.
301. Procyk T. S. and Marndani M. M. (1979). A linguistic self organizing process controller, Automatica, 15, 15–30.
302. Puncochar J., Drahos M. and Vrba J. (1996). Fuzzy number as a product of geometrical construction, Fuzzy Sets and Systems, 83, 43–50.
303. Puri M. and Ralescu D. (1981), Différentielle d’une fonction floue, C. R. Acad. Sci. Paris, Ser. I, 293, 237–239.
304. Puri M. and Ralescu D. (1983). Differentials of fuzzy functions, J. Math. Anal. Appl., 91, 552–558.
305. Puri M. and Ralescu D. (1986), Fuzzy random variables, J. Math. Anal. Appl., 114, 409–422.
306. Puri M. L. and Ralescu D. (1991). Convergence theorems for fuzzy martingales, J. Math. Anal Appl., 160, 107–122.
307. Quadrat J. P. (1990). Théorèmes asymptotiques en programmation dynamique, C. R. Acad. Sci Paris, Série I, 311, 745–748.
308. Ramik J. (1986). Extension principle in fuzzy optimization, Fuzzy Sets and Systems, 19, 29–37.
309. Ramik J. and Nakamura K. (1993). Canonical fuzzy numbers of dimension two, Fuzzy Sets and Systems, 54(2), 167–181.
310. Ramik J. and Rimanek J. (1985). Inequality relation between fuzzy numbers and its use in fuzzy optimization. Fuzzy Sets and Systems, 16, 123–138.
311. Ramik J. and Rommelfanger H. (1993). A single-and multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients, Fuzzy Sets and Systems, 57, 203–288.
312. Ramik J. and Rommelfanger H. (1996). Fuzzy mathematical progamming based on some new inequality relations, Fuzzy Sets and Systems, 81, 77–87.
313. Rao M. B. and Rashed A. (1981). Some comments on fuzzy variables, Fuzzy Sets and Systems, 6, 285–292.
314. Requena L, De lgado M. and Verdegay J. L. (1994). Automatic ranking of fuzzy numbers with the criterion of a decision-maker learnt by an artificial neural network, Fuzzy Sets and Systems, 64, 1–21.
315. Requena L, Blanco A., De lgado M. and Verdegay J, L. (1995). A decision personnal index of fuzzy numbers based on neural network, Fuzzy Sets and Systems, 73, 185–201.
316. Rockafellar R. T. (1970). Convex Analysis, Princeton Univ. Press, Princeton, NJ.Google Scholar
317. Rodabaugh S. E. (1982). Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems, 8, 39–51.
318. Roubens M. (1990). Inequality constraints between fuzzy numbers and their use in mathematical programming, Stochastic vs. Fuzzy Approaches to Multiobjective Mathematical Programming (Slowinski R. and Teghem J., eds.), Kluwer Academic, Dordrecht, Netherlands, 321–330.Google Scholar
319. Roubens M. and Vincke P. (1985). Preference Modelling, Springer Verlag, Berlin.Google Scholar
320. Roubens M. and Vincke P. (1988). Fuzzy possibility graphs and their application to ranking fuzzy numbers, Non-Conventional Preference Relations in Decision Making (Kacprzyk J. and. Roubens M., eds.), 119–128.Google Scholar
321. Saade J. J. and Schwarzlander H. (1992). Ordering fuzzy sets over the real line: An approach besed on decision making under uncertainty, Fuzzy Sets and Systems, 50, 237–246.
322. Saaty T. L. (1980). The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, McGraw-Hill, New York.Google Scholar
323. Sage A. P. (1981). Sensitivity analysis in systems for planning and decision support, J. Franklin Inst., 312, 265–291.
324. Sakawa M. (1993). Fuzzy Sets and Interactive Multiobjective Optimization, Plenum Press, New York.
325. Salakhutdinov R.R., Salakhutdinov R.Z. (1998), The law of large numbers and T-stability fuzzy numbers, BUSEFAL (IRIT, Univ. P. Sabatier), 74, 64–67.Google Scholar
326. Sanchez E. (1984). Solution of fuzzy equations with extended operations, Fuzzy Sets and Systems, 12, 237–248.
327. Sanchez E. (1996). Truth-qualification and fuzzy relations in natural languages, application to medical diagnosis, Fuzzy Sets and Systems, 84, 155–167.
328. Sarna M. (1989). Fuzzy relations on fuzzy and non fuzzy numbers — Fast computation formulas, Fuzzy Sets and Systems, 29, 155–163.
329. Sarna M. (1998). Fuzzy relations on fuzzy and non fuzzy numbers; fast computation formulas: II. Fuzzy Sets and Systems, 93, 63–74.
330. Schweizer B. (1975). Multiplications on the space of probability distribution functions, Aeq. Math., 12, 156–183.
331. Schweizer B. and Sklar A. (1963). Associative functions and abstract semi-groups, Publ. Math. Debrecen, 10, 69–81.
332. Schweizer B. and Sklar A. (1983), Probabilistic Metric Spaces, North Holland, New York.Google Scholar
333. Scott M. and Antonsson E. (1995). Aggregation functions for engineering design trade-offs, Proc. of the 9th Int. Conf on Design Theory and Methodology, Vol. 2, 389–396.Google Scholar
334. Sebastian H.-J. and Antonsson E. (eds.) (1996). Fuzzy Sets in Engineering Design and Configuration, Kluwer Academic, Dordrecht, The Netherlands.Google Scholar
335. Shafer G. (1976). A Mathematical Theory of Evidence, Princeton University Press, Princeton.Google Scholar
336. Sinha D. (1990). A general theory of fuzzy arithmetic, Fuzzy Sets and Systems, 36, 339–364.
337. Slowinski R (1998). Fuzzy sets in decision analysis operations research and statistics, The Handbooks of Fuzzy Sets Series, Kluwer, Boston, USA.Google Scholar
338. Schmeidler, D. (1989)Subjective probability and expected utility without additivity. Econometrica, 57, 571–587.
339. Schmucker K. J. (1984). Fuzzy Sets, Natural Language Conputations and Risk Analysis, Computer Science Press, Rockville, MD.Google Scholar
340. Sinha D., Dougherty, E.R. (1995). A general axiomatic theory of intrinsically fuzzy mathematical morphologies, IEEE Trans, on Fuzzy Systems, 3, 389–403.
341. Slowinski R. (Ed.) (1998). Fuzzy Sets in Decision Analysis, Operations Research and Statistics. The Handbooks of Fuzzy Sets Series (D. Dubois, H. Prade, eds,), Kluwer Acad. Publ., Boston.
342. Song Q. and Chissom B, S. (1993a). Fuzzy time series and its models, Fuzzy Sets and Systems, 54, 269–277.
343. Song Q. and Chissom R. S. (1993b). Forecasting enrollments with fuzzy time series: Part I, Fuzzy Sets and Systems, 54, 1–9.
344. Song Q, and Chissom B. S. (1994). Forecasting enrollments with fuzzy time series: Part II, Fuzzy Sets and Systems, 62, 1–8.
345. Song Q., Leland R. L. and Chissom B. S. (1995). A new fuzzy time-series model of fuzzy number observations, Fuzzy Sets and Systems, 73, 341–348.
346. Stanford R. E. (1982). The set of limiting distributions for a Markov chain with fuzzy transition probabilities, Fuzzy Sets and Systems, 7, 71–78.
347. Stein W. E. (1983). A note on convexity and fuzzy random variables, BUSEFAL (IRIT, Université P. Sabatier, Toulouse), 14, 43–46.
348. Stein W. E. and Talati K. (1981). Convex fuzzy random variables, Fuzzy Sets and Systems, 6, 271–283.
349. Steyaert H., Van Parys F., Baekeland R. and Kerre E. (1995). Implementation of piecewise linear fuzzy quantities, Int. J. Intelligent Systems, 10, 1049–1059.
350. Stojakovic M. (1994). Fuzzy valued measure. Fuzzy Sets and Systems, 65, 95–104.
351. Stojakovic M. and Stojakovic Z. (1996). Additions and series of fuzzy sets, Fuzzy Sets and Systems, 83, 341–346.
352. Subrahmanyam P. V. and Sudarsanam S. K. (1994). On some fuzzy functional equations, Fuzzy Sets and Systems, 64, 333–338.
353. Tamura N. and Horiuchi K. (1993). VSOP fuzzy numbers and fuzzy comparison relations, Proc. FUZZ-IEEE, San Francisco, 1287–1292.Google Scholar
354. Tanaka H. and Asai K. (1984a). Fuzzy linear programming with fuzzy numbers, Fuzzy Sets and Systems, 13, 1–10.
355. Tanaka H. and Asai K. (1984b). Fuzzy solution in fuzzy linear programming problems, IEEE Trans. Syst. Man. Cybern., 14, 325–328.
356. Tanaka H. and Diamond P. (1998). Fuzzy regression analysis, Fuzzy Sets in Decision Analysis, Operations Research and Statistics (Slowinski, R., ed.), The Handbook of Fuzzy Sets Series, Kluwer Academic Publ., Dordrecht, 349–390.Google Scholar
357. Tong R. M. and Bonissone P. P. (1980). A linguistic approach to decision making with fuzzy sets, IEEE Trans. Syst. Man. Cybern., 10, 716–723.
358. Triesch E. (1993). On the convergence of product-sum series of L-R fuzzy numbers, Fuzzy Sets and Systems, 53, 189–192.
359. Tseng T. Y. and Klein C. (1988). A survey and comparative study of ranking procedures in fuzzy decision making. Working Paper n° 8812101, Dept. of Industrial Engineering, University of Missouri-Columbia, USA.Google Scholar
360. Tseng T. Y. and Klein C. (1989). New algorithm for the ranking procedure in fuzzy decision-making, IEEE Trans. Syst. Man. Cybern., 19, 1289–1296.
361. Tsukamoto Y., Nikiforuk P. N. and Gupta M. M. (1981). On the comparison of fuzzy sets using fuzzy chopping, Proc. 8th Triennial World Congress IFAC, Vol. 5, Pergamon Press, Oxford, 46–52.Google Scholar
362. Tsang E. (1993). Foundations of Constraint Satisfaction, Academic Press, New York.Google Scholar
363. Umano M. (1982). Fredom-0: A fuzzy data base system, Fuzzy Information and Decision Processes (Gupta M. M. and Sanchez E., eds.), North-Holland, 339–347.Google Scholar
364. Van Laatroven P. J. M. and Pedrycz W. (1983). A fuzzy extension of Saaty’s priority theory, Fuzzy Sets and Systems, 11, 229–241.
365. Van Leekwijk W. and Kerre E. (1998). Defuzzification: criteria and classification. University of Gent, Belgium. To appear in Fuzzy Sets and Systems. Google Scholar
366. Viertl R. (1996). Statistical Methods for Non Precise Data.CRC Press, Boca Raton, FL.Google Scholar
367. Vrba J. (1992). A note on inverses in arithmetics with fuzzy numbers, Fuzzy Sets and Systems, 50, 267–278.
368. Wagenknecht M. and Hartmann K. (1983). On fuzzy rank-ordering in poly-optimization, Fuzzy Sets and Systems, 11, 253–264.
369. Wang P. Z. (1983). From the fuzzy statistics to the falling random subsets, Advances in Fuzzy Sets, Possibility Theory and Applications (Wang P. P., ed.), Plenum Press, New York, 81–96.
370. Wang X. (1997). A comparative study of the ranking methods for fuzzy quantities, PhD. Thesis, University of Gent, Belgium.Google Scholar
371. Wang X. and Ha M. (1994), Solving a system of fuzzy linear equations, Fuzzy Optimization: Recent Advances (Delgado M., Kacprzyk J., Verdegay J. L. and Vila M. A., eds.), Physica-Verlag, Heidelberg, Germany, 102–108.Google Scholar
372. Wang X. and Kerre E. (1996). On the classification and the dependencies of the ordering methods, Fuzzy Logic Foundations and Industrial Applications (Rua D., ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 73–88.
373. Wang X. and Kerre E. (1998). Reasonable properties for the ordering of fuzzy quantities, Fuzzy Sets and Systems, to appear.Google Scholar
374. Wang X., Kerre E. and Da Ruan. (1995a). Consistency and weights of judgment matrix in fuzzy AHP, Int. J. of Uncertainty, Fuzziness and Knowledge-based Systems, 3, 35–46.
375. Wang X., Kerre E. and Da Ruan. (1995b). Transitivity of fuzzy orderings based on pairwise comparisons, Int. J. Fuzzy Math., 3, 455–463.
376. Wang X. and Da Ruan. (1995). On the transitivity of fuzzy preference relations in ranking fuzzy numbers, Fuzzy Set Theory and Advanced Mathematical Applications (Rua D., ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 155–173.
377. Wang Z. and Klir G. J. (1992), Fuzzy Measure Theory, Plenum Press, New York.
378. Wasowski J. (1997). On solutions to fuzzy equations, Control and Cybern., 26, 653–658.
379. Watson S. R., Weiss J. J. and Donnell M. (1979). Fuzzy decision analysis, IEEE Trans, Syst. Man. Cybern., 9, 1–9.
380. Williamson R. C. (1991). The law of large numbers for fuzzy variables under a general triangular norm extension principle, Fuzzy Sets and Systems, 41, 55–81.
381. Willmott R. (1981). Mean measures of containment and equality between fuzzy sets, Proc. 11th Int. Symp. on Multiple Valued Logic, Oklahoma City, 183–190.Google Scholar
382. Woods K., Otto K. and Antonsson E. (1992). Engineering design calculations under uncertainty, Fuzzy Sets and Systems, 52, 1-20Google Scholar
383. Wu C. and Qiu J. (1998). Some remarks for fuzzy complex analysis, Fuzzy Sets and Systems, to appear.Google Scholar
384. Yager R. R. (1979a). A note on probabilities of fuzzy events, Information Sciences, 18, 113–129.
385. Yager R. R. (1979b). On solving fuzzy mathematical relationships, Inf Control, 41, 29–55.
386. Yager R. R. (1979c). Ranking fuzzy subsets over the unit interval, Proc. 17th IEEE Int. Confon Decision and Control, San Diego, CA, 1535–1437.Google Scholar
387. Yager R. R. (1980a). On the lack of inverses in fuzzy arithmetic, Fuzzy Sets and Systems, 4, 73–82.
388. Yager R. R. (1980b). On choosing between fuzzy subsets, Kybernetes, 9, 151–154.
389. Yager R. R. (1981). A procedure for ordering fuzzy subsets of the unit interval, Information Sciences, 24, 143–161.
390. Yager R. R. (1982). Measuring tranquility and anxiety in decision-making: An application of fuzzy sets. Int. J. General Systems, 9, 249–260.
391. Yager R. R. (1986). A characterization of the extension principle, Fuzzy Sets and Systems, 18, 205–219.
392. Yager R. R. and Filev D. (1993). On the issue of defuzzification and selection based on a fuzzy set, Fuzzy Sets and Systems, 55, 255–271.
393. Yager R. R. and Filev D. (1994). Essentials of Fuzzy Modeling and Control, Wiley, New York.Google Scholar
394. Yager R. R. and Kelman (1996). A. fusion of fuzzy information with consideration of compatibility, partial aggregation and reinforcement, Int. J. Approximate Reasoning, 15, 93–122.
395. Yang HQ. Hua Y. and Jones J. D. (1993). Calculating functions of fuzzy numbers, Fuzzy Sets and Systems, 55, 273–283.
396. Yoon K. P. (1996). A probabilistic approach to rank complex fuzzy numbers, Fuzzy Sets and Systems, 80, 167–176.
397. Young R. C. (1931). The algebra of many-valued quantities, Math. Ann., 104, 260–290.
398. Yu C. (1993). Correlation of fuzzy numbers, Fuzzy Sets and Systems, 55, 303–307.
399. Yuan Y. (1991). Criteria for evaluating fuzzy ranking methods, Fuzzy Sets and Systems, 44, 139–157.
400. Zadeh L. A. (1965). Fuzzy sets, Inf. Control, 8, 338–353.
401. Zadeh L. A. (1975a). The concept of a linguistic variable and its application to approximate reasoning, Information Sciences, Part I: 8, 199-249; Part II: 8, 301-357; Part III: 9, 43–80.
402. Zadeh L. A. (1975b). Calculus of fuzzy restrictions, Fuzzy Sets and Their Applications to Cognitive and De cision Processes (Zadeh L. A., Fu K. S., Shimura M. and Tanuka K., eds.), Academic Press, New York, 1–39.Google Scholar
403. Zadeh L. A. (1978). Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, 3–28.
404. Zadeh L. A. (1979). A theory of approximate reasoning, Machine Intelligence, Vol. 9 (Hayes J. E., Michie D. and Mikulich L. I., eds.), John Wiley & Sons, New York, 149–194.Google Scholar
405. Zadeh L. A. (1983), A computational approach to linguistic quantifiers in natural language, Comp. Math. Appl., 9, 149–184.Google Scholar
406. Zadeh L. A. (1996). Fuzzy logic-computing with words, IEEE Trans. Fuzzy Systems, 4, 103–111.
407. Zhang G. Q. (1993). The convergence of a sequence of fuzzy integrals of fuzzy number-valued functions on the fuzzy set, Fuzzy Sets and Systems, 59, 43–57.
408. Zhang D. and Wang Z. (1993). Fuzzy integrals of fuzzy-valued functions, Fuzzy Sets and Systems, 54, 63–67.
409. Zhao R, and Govind R. (1991a). Algebraic characteristics of extended fuzzy numbers, Information Sciences, 54, 103–130.
410. Zhao R. and Govind R. (1991b). Solutions of algebraic equations involving generalized fuzzy numbers, Information Sciences, 56, 199–243.
411. Zhao R. and Govind R. (1991c). Defuzzification of fuzzy intervals, Fuzzy Sets and Systems, 43, 45–56.
412. Zhu Q. and Lee E. S. (1992). Comparison and ranking of fuzzy numbers, Fuzzy Regression Analysis (Kacprzyk J. and Fedrizzi M., eds.), Omnitech Press, Warsaw, Poland, 21–44.Google Scholar
413. Zimmermann H. J. (Ed.) (1999), Practical Applications of Fuzzy Technologies, The Handbooks of Fuzzy Sets Series (D. Dubois, Prade H., eds.), Kluwer Academic Publ., New York.Google Scholar