Applications of Interval Computations pp 245-290 | Cite as

# Nested Intervals and Sets: Concepts, Relations to Fuzzy Sets, and Applications

## Abstract

In data processing, we often encounter the following problem: Suppose that we have processed the measurement results \({\tilde x_1},...,{\tilde x_n}\), and, from this processing, have obtained an estimate \(\tilde y = f({\tilde x_1},...,{\tilde x_n})\) for a quantity *y* = *f*(xi,…,xn); we know the intervals x_{i} of possible values of *x* _{i}, and we want to find the interval y of possible values of *y*. Interval computations are one of the main techniques for solving this problem.

In some cases, for each i, in addition to the *guaranteed* interval x_{i} of possible values, we have a smaller interval that an expert believes to contain *x* _{i}. There may be several such *nested* intervals. In these cases, in addition to the guaranteed interval y, it is desirable to know the possible intervals of *y* that correspond to the opinions of different experts.

Techniques of such *nested interval computations* and real-life applications of these techniques are described in this paper.

## Keywords

Membership Function Fuzzy Number Linear Programming Problem Choice Function Interval Computation## Preview

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## References

- [1]O. Artbauer, “Application of interval, statistical, and fuzzy methods to the evaluation of measurements”,
*Metrologia*, 1988, Vol. 25, pp. 81–86.CrossRefGoogle Scholar - [2]K. T. Atanassov,
*Review and new results on intuitionistic fuzzy sets*, Preprint IM-MFAIS-1-88, Sofia, Bulgaria, 1988.Google Scholar - [3]K. T. Atanassov, “Operations over interval valued intuitionistic fuzzy sets”,
*Fuzzy Sets and Systems*, 1994, Vol. 64, pp. 159–174.zbMATHCrossRefMathSciNetGoogle Scholar - [4]K. T. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets”,
*Fuzzy Sets and Systems*, 1989, Vol. 31, No. 3, pp. 343–349.zbMATHCrossRefMathSciNetGoogle Scholar - [5]R. E. Bellman, L.A. Zadeh, “Decision making in a fuzzy environment”,
*Management Science*, 1970, Vol. 17, No. 4, pp. B141–B164.CrossRefMathSciNetGoogle Scholar - [6]D. Berleant, R. R. Goforth, and A. Khanna, “Comparisons of simulation with pdf’s vs. fuzzy values”,
*Proceedings of the Symposium on Computer Simulation in Industrial Engineering and in Problems of Urban Development*, Nov. 18–19, 1992, Mexico City.Google Scholar - [7]A. K. Bit, M. P. Biswal, S. S. Alam, “Fuzzy programming approach to multicriteria decision making transportation problem”,
*Fuzzy Sets and Systems*, 1992, Vol. 50, pp. 135–141.zbMATHCrossRefMathSciNetGoogle Scholar - [8]J. M. Blin, “Fuzzy relations in group decision theory”,
*J. of Cybernetics*,**4**, 17–22, 1974.CrossRefMathSciNetGoogle Scholar - [9]J. M. Blin and A. B. Whinston. “Fuzzy sets and social choice”,
*J. of Cybernetics*,**3**, 28–36, 1973.zbMATHCrossRefMathSciNetGoogle Scholar - [10]G. Bojadziev and G. Bojadzieva, “Liapunov design avoidance control for an open-end two-link robot manipulator”, In: M. Jamshidi and M. Saif (eds.),
*Robotics and Manufacturing*, ASME Press, N.Y., 1990, Vol. 3, pp. 307–314.Google Scholar - [11]G. Bojadziev and G. Bojadzieva, “Avoidance control of an open-end robot manipulator with two links subject to uncertainties of fuzzy nature”, In: Mohammad Jamshidi, Charles Nguyen, Ronald Lumia, and Junku Yuh (Editors),
*Intelligent Automation and Soft Computing. Trends in Research, Development, and Applications*.*Proceedings of the First World Automation Congress (WAC’94), August 14–17, 1994, Maui, Hawaii*, TSI Press, Albuquerque, NM, 1994, Vol. 1, pp. 439–444.Google Scholar - [12]E. L. Deporter, K. P. Ellis, “Optimization of project networks with goal programming and fuzzy linear programming”,
*Computers in Industrial Engineering*, 1990, Vol. 19, pp. 500–504.CrossRefGoogle Scholar - [13]W. M. Dong, W. L. Chiang, H. C. Shah, “Fuzzy information processing in seismic hazard analysis and decision making”,
*International Journal of Soil Dynamics and Earthquake Engineering*, 1987, Vol. 6, No. 4., pp. 220–226.CrossRefGoogle Scholar - [14]W. Dong and F. Wong, “Fuzzy weighted averages and implementation of the extension principle”,
*Fuzzy Sets and Systems*, 1987, Vol. 21, pp. 183–199.zbMATHCrossRefMathSciNetGoogle Scholar - [15]D. Dubois, M. Grabisch, and H. Prade, “Gradual rules and the approximation of control laws”, In: H. T. Nguyen, M. Sugeno, R. Tong, and R. Yager (eds.),
*Theoretical aspects of fuzzy control*, J. Wiley, N.Y., 1995, pp. 117–146.Google Scholar - [16]D. Dubois and H. Prade.
*Fuzzy sets and systems: theory and applications*, Academic Press, N.Y., London, 1980.zbMATHGoogle Scholar - [17]D. Dubois, H. Prade, “Random sets and fuzzy interval analysis”,
*Fuzzy Sets and Systems*, 1991, Vol. 42, pp. 87–101.zbMATHCrossRefMathSciNetGoogle Scholar - [18]R. Fuller, T. Keresztfalvi, “On generalization of Nguyen’s theorem”,
*Fuzzy Sets and Systems*, 1990, Vol. 4, pp. 371–374.MathSciNetGoogle Scholar - [19]E. Gardeñes, A. Trepat, J. M. Janer, “Approaches to simulation and to the linear problem in SIGLA system”,
*Freiburger Intervall-Berichte*, 1981, No. 81/8.Google Scholar - [20]Y. Gentilhomme, “Les ensembles flou en linsguistique”,
*Cahiers de Ling. Theor. et Appl.*, 1968, Vol. 5, pp. 47–65.Google Scholar - [21]V. A. Gerasimov, M. Yu. Shustrov, “Numerical operations with fuzzy objects”, In: S. P. Shary and Yu. I. Shokin (editors),
*Interval Analysis*, Krasnoyarsk, Academy of Sciences Computing Center, Technical Report No. 17, 1990, pp. 11–15 (in Russian).Google Scholar - [22]J. A. Goguen, “
*L*-fuzzy sets”,*Journal of Mathematical Analysis and Applications*, 1967, Vol. 18, pp. 145–174.zbMATHCrossRefMathSciNetGoogle Scholar - [23]J. A. Goguen, “The logic of inexact reasoning”,
*Synthese*, 1969,**19**, 325–373, 1969 (reprinted in D. Dubois, H. Prade, R. Yager (eds.),*Reading in Fuzzy Sets for Intelligent Systems*, Morgan Kaufmann, San Mateo, CA, 1994, pp. 417–441 ).zbMATHCrossRefGoogle Scholar - [23]J. A. Goguen, “The logic of inexact reasoning”,
*Synthese*, 1969,**19**, 325–373, 1969 (reprinted in D. Dubois, H. Prade, R. Yager (eds.),*Reading in Fuzzy Sets for Intelligent Systems*, Morgan Kaufmann, San Mateo, CA, 1994, pp. 417–441 ).Google Scholar - [24]I. R. Goodman, “Algebraic and probabilistic bases for fuzzy sets and the development of fuzzy conditioning”, In: I. R. Goodman, M. M. Gupta, H. T. Nguyen, and G. S. Rogers (eds.),
*Conditional logic in expert systems*, North Holland, Amsterdam, 1991.Google Scholar - [25]I. R. Goodman, M. M. Gupta, H. T. Nguyen, and G. S. Rogers (eds.),
*Conditional logic in expert systems*, North Holland, Amsterdam, 1991.Google Scholar - [26]M. Grabisch, H. T. Nguyen, and E. A. Walker,
*Fundamentals of uncertainty calculi with applications to fuzzy inference*, Kluwer, Dordrecht, Netherlands, 1995.Google Scholar - [27]Z. X. Gu and H. Q. Yang,
*Basic theory of fuzzy information process*, Chengdu Institute of Radio Engineering Publishing House, P.R. China, 1989 (in Chinese).Google Scholar - [28]A. Kandel, G. Langholtz (editors),
*Fuzzy Control Systems*, CRC Press, Boca Raton, FL, 1994.zbMATHGoogle Scholar - [29]A. Kaufmann and M. M. Gupta,
*Introduction to fuzzy arithmetic*, Van Nostrand Reinhold Co., N.Y., 1985.zbMATHGoogle Scholar - [30]R. B. Kearfott, “A Review of Techniques in the Verified Solution of Constrained Global Optimization Problems,” This Volume.Google Scholar
- [31]E. P. Klement, B. Kovalerchuk, “Interval mode of fuzzy control”, In:
*Proceedings of the International Conference on Interval and Stochastic Methods in Science and Engineering INTERVAL’92*, Moscow, 1992, Vol. 2, pp. 45–46.Google Scholar - [32]G. J. Klir and T. A. Folger,
*Fuzzy sets, uncertainty, and information*, Prentice-Hall, U.K., 1988.zbMATHGoogle Scholar - [33]G. J. Klir and Bo Yuan,
*Fuzzy sets and fuzzy logic. Theory and applications*, Prentice Hall, NJ, 1995.zbMATHGoogle Scholar - [34]L. J. Kohout, “Fuzzy interval-valued inference system with para-consistent and grey set extensions”,
*Reliable Computing*, 1995, Supplement (Extended Abstracts of APIC’95: International Workshop on Applications of Interval Computations, El Paso, TX, Feb. 23–25, 1995), pp. 107–110.Google Scholar - [35]R. Kruse, J. Gebhardt, and F. Klawonn,
*Foundations of fuzzy systems*, J. Wiley & Sons, Chichester, England, 1994.Google Scholar - [36]Y.-J. Lai, C.-L. Hwang,
*Fuzzy mathematical programming. Method and application*, Springer-Verlag, Berlin, 1992.Google Scholar - [37]W. A. Lodwick, “Analysis of structure in fuzzy linear programs”,
*Fuzzy Sets and Systems*, 1990, Vol. 38.Google Scholar - [38]S. G. Loo,
*Cybernetica*, 1977, Vol. 20, pp. 201–210.zbMATHGoogle Scholar - [39]R. P. Loui, “Interval-based decisions for reasoning systems”, In: L. N. Kanal and J. F. Lemmer,
*Uncertainty in Artificial Intelligence*, Elsevier, North Holland, 1986, pp. 459–472.Google Scholar - [40]E. H. Mamdani, “Application of fuzzy algorithms for control of simple dynamic plant”,
*Proceedings of the IEE*, 1974, Vol. 121, No. 12, pp. 1585–1588.Google Scholar - [41]E. H. Mamdani, “Application of fuzzy logic to approximate reasoning using linguistic systems”,
*IEEE Transactions on Computing*, 1977, Vol. 26, pp. 1182–1191.zbMATHCrossRefGoogle Scholar - [42]R. Lopez De Mantaras, “From intervals to possibility distributions: adding flexibility to reasoning under uncertainty”,
*Reliable Computing*, 1995, Supplement (Extended Abstracts of APIC’95: International Workshop on Applications of Interval Computations, El Paso, TX, Feb. 23–25, 1995 ), pp. 151–152.Google Scholar - [43]M. Mareŝ,
*Computation over fuzzy quantities*, CRC Pres, Boca Raton, FL, 1994.zbMATHGoogle Scholar - [44]D. Mitra, M. L. Gerard, P. Srinivasan, and A. E. Hands, “A possibilistics interval constraint problem: fuzzy temporal reasoning”,
*FUZZ-IEEE’94*, IEEE Press, 1994, Vol. 2, pp. 1434–1439.Google Scholar - [45]E. A. Musaev, “The support of interval computations in high-level languages”, In:
*Proceedings of the 1st Soviet-Bulgarian Seminar on Numerical Processing, Pereslavl-Zalessky, October 19-24, 1987*, Pereslavl-Zalessky, VINITI, 1988, Publ. No. 2634-B89, pp. 110–121 (in Russian).Google Scholar - [46]E. A. Musaev, “Unexpected aspect of interval approach in a task of optimal currency exchange rates”, In:
*International Conference on Interval and Computer-Algebraic Methods in Science and Engineering (Interval’94), St Petersburg, Russia, March 7–10, 1994*, Abstracts, pp. 179–180.Google Scholar - [47]A. S. Narin’yani,
*Subdefinite sets - a new datatype for knowledge representation*, Academy of Sciences, Siberian Branch, Computing Center, Novosibirsk, Technical Report No. 232, 1980 (in Russian).Google Scholar - [48]A. S. Narin’yani, “Tools that simulate data incompleteness, and their usage in knowledge representation”, In:
*Knowledge representation and simulation of the understanding process*, Academy of Sciences, Siberian Branch, Computing Center, Novosibirsk, 1980 (in Russian).Google Scholar - [49]A. S. Narin’yani, “Subdefiniteness in knowledge representation and processing systems”,
*Transactions of USSR Acad. of Sciences, Technical Cybernetics*, 1986, No. 5, pp. 3–28 (in Russian).Google Scholar - [50]H. Nazaraki and I. B. Türksen, “An integrated approach for syllogistic reasoning and knowledge consistency level maintenance”,
*IEEE Transactions on Systems, Man, and Cybernetics*, 1994, Vol. 24, No. 4, pp. 548–563.CrossRefGoogle Scholar - [51]C. V. Negoita and D. A. Ralescu, “Representation theorems for fuzzy concepts”,
*Kybernetes*, 1975, Vol. 4, pp. 169–174.zbMATHCrossRefGoogle Scholar - [52]C. V. Negoita and D. A. Ralescu,
*Applications of fuzzy sets to systems analysis*, John Wiley and Sons, N.Y., Toronto, 1975.zbMATHGoogle Scholar - [53]H. T. Nguyen, “A note on the extension principle for fuzzy sets”,
*J. Math. Anal. and Appl.*1978, Vol. 64, pp. 359–380.CrossRefGoogle Scholar - [54]H. T. Nguyen and V. Kreinovich, “Interval Sessions at NAFIPS/IFIS/NASA’94,”
*Reliable Computing*, 1995, Vol. 1, No. 1, pp. 93–98.CrossRefGoogle Scholar - [55]H. T. Nguyen, V. Kreinovich, and Qiang Zuo, “Interval-Valued Degrees of Belief: Applications of Interval Computations to Expert Systems and Intelligent Control” (submitted to
*Reliable Computing*).Google Scholar - [56]S. A. Orlovsky, “On programming with fuzzy constraint sets”,
*Kybernetes*, 1977, Vol. 6, No. 3, pp. 197–201.zbMATHCrossRefGoogle Scholar - [57]S. A. Orlovsky, “Problems of decision-making in case of uncertainty of initial information”, Moscow, Nauka Publ., 1981 (in Russian).Google Scholar
- [58]M. Sasaki, M. Gen, K. Ida, “A method for solving reliability optimization problems by fuzzy multiobjective 0-1-linear programming”,
*Electronics and Communications*, 1991, Part 3, Vol. 74, pp. 106–116 (in Japanese).Google Scholar - [59]K. J. Schmucker,
*Fuzzy sets, natural language computations, and risk analysis*, Computer Science Press, Rockville, MD, 1984.zbMATHGoogle Scholar - [60]Tong Shaocheng, “Interval number and fuzzy number linear programmings”,
*Fuzzy Sets and Systems*, 1994, Vol. 66, pp. 301–306.CrossRefMathSciNetGoogle Scholar - [61]G. Sommer, M. A. Pollatschek, “A fuzzy programming approach to an air pollution regulation problem”,
*European Journal of Operations Research*, 1978, Vol. 10, pp. 303–313.Google Scholar - [62]N. Tamura and K. Horiuchi, “VSOP fuzzy numbers and fuzzy comparison relations”,
*Proceedings of the Second IEEE International Conference on Fuzzy Systems*, San Francisco, CA, March 28–April 1, 1993, Vol. II, pp. 1287–1292.CrossRefGoogle Scholar - [63]H. Tanaka, T. Okuda, K. Asai, “On fuzzy mathematical programming”,
*J. of Cybernetics*, 1973, Vol. 3, No. 4, pp. 37–46.zbMATHCrossRefMathSciNetGoogle Scholar - [64]H. Tanaka, K. Asai, “Fuzzy linear programming problems with fuzzy numbers”,
*Fuzzy Sets and Systems*, 1984, Vol. 13, pp. 1–10.zbMATHCrossRefMathSciNetGoogle Scholar - [65]C.-C. Tsai, C.-H. Chu, T. A. Barta, “Fuzzy linear programming approach to manufacturing cell formation”,
*FUZZ-IEEE’94*, IEEE Press, 1994, Vol. 2, pp. 1406–1411.Google Scholar - [66]K. Uehara, M. Fujise. “Fuzzy inference based on families of α-level sets”,
*IEEE Transactions on Fuzzy Systems*, 1993, Vol. 1, No. 2, pp. 111–124.CrossRefGoogle Scholar - [67]G. Wiedy, H. J. Zimmermann. “Media selection and fuzzy linear programming”,
*J. Oper. Res. Soc.*, 1978, Vol. 31, pp. 342–249.Google Scholar - [68]R. R. Yager, “A characterization of the extension principle”,
*Fuzzy Sets and Systems*, 1986, Vol. 18, No. 3, pp. 205–217.zbMATHCrossRefMathSciNetGoogle Scholar - [69]H. Q. Yang, H. Yao, and J. D. Jones, “Calculating functions of fuzzy numbers”,
*Fuzzy Sets and Systems*, 1993, Vol. 55, pp. 273–283.CrossRefMathSciNetGoogle Scholar - [70]A. V. Yazenin, “Fuzzy variables and fuzzy mathematical programming”, In:
*Models of alternative selection in a fuzzy medium*, Riga Polytechnical Institute, Riga, Latvia, 1984 (in Russian).Google Scholar - [71]A. V. Yazenin,
*Fuzzy mathematical programming*, Kalinin University, Kalinin, Russia, 1986 (in Russian).Google Scholar - [72]A. V. Yazenin, “Fuzzy and stochastic programming”,
*Fuzzy Sets and Systems*, 1987, Vol. 22.Google Scholar - [73]A. V. Yazenin, “Possibility programming models in fuzzy optimization”,
*Tekhnischeskaya Kibernetika*, 1991, Vol. 29, No. 5 (in Russian); English translation in*Journal of Computer and Systems Sciences International*, 1992, Vol. 31, No. 2.Google Scholar - [74]A. V. Yazenin, “Possibilistic and interval linear programming”,
*Tekhnischeskaya Kibernetika*, 1993, No. 5, pp. 149–155 (in Russian); English translation in*Journal of Computer and Systems Sciences International*, 1994, Vol. 32, No. 6, pp. 154–160.Google Scholar - [74]A. V. Yazenin, “Possibilistic and interval linear programming”,
*Tekhnischeskaya Kibernetika*, 1993, No. 5, pp. 149–155 (in Russian); English translation in*Journal of Computer and Systems Sciences International*, 1994, Vol. 32, No. 6, pp. 154–160.MathSciNetGoogle Scholar - [75]J. Yen and N. Pfluger, “Path planning and execution using fuzzy logic”, In:
*AIAA Guidance, Navigation and Control Conference*, New Orleans, LA, 1991, Vol. 3, pp. 1691–1698.Google Scholar - [76]J. Yen and N. Pfluger, “Designing an adaptive path execution system”, In:
*IEEE International Conference on Systems, Man and Cybernetics*, Charlottesville, VA, 1991.Google Scholar - [77]J. Yen, N. Pfluger, and R. Langari, “A defuzziflcation strategy for a fuzzy logic controller employing prohibitive information in command formulation”,
*Proceedings of IEEE International Conference on Fuzzy Systems, San Diego, CA, March 1992*.Google Scholar - [78]L. Zadeh, “Fuzzy sets”,
*Information and control*, 1965, Vol. 8, pp. 338–353.zbMATHCrossRefMathSciNetGoogle Scholar - [79]L. A. Zadeh, “Outline of a new approach to the analysis of complex systems and decision processes”,
*IEEE Transactions on Systems, Man and Cybernetics*, 1973, Vol. 3, pp. 28–44.zbMATHCrossRefMathSciNetGoogle Scholar - [80]L. A. Zadeh, “The concept of linguistic variable and its application to approximate reasoning”,
*Information Sciences*, 1975, Vol. 8, pp. 199–249.CrossRefMathSciNetGoogle Scholar - [81]L. A. Zadeh, “A theory of common sense knowledge”, In: H. J. Scala, S. Termini, and E. Trillas (eds.),
*Issues of Vagueness*, Dordrecht, Reidel, 1984, pp. 257–296.Google Scholar - [82]L. A. Zadeh, “Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions”,
*IEEE Trans. SMC*,, 1985, Vol. SMC-15, pp. 754–763.MathSciNetGoogle Scholar - [83]H. J. Zimmermann, “Description and optimization of fuzzy systems”,
*Intl. J. of General Systems*, 1976, Vol. 2, No. 4, pp. 209–215.zbMATHCrossRefGoogle Scholar - [84]H. J. Zimmermann, “Fuzzy programming and linear programming with several objective functions”,
*Fuzzy Sets and Systems*, 1978, Vol. 3, pp. 45–55.CrossRefGoogle Scholar - [85]H. J. Zimmermann, “Fuzzy mathematical programming”,
*Computers and Operations Research*, 1983, Vol. 10, pp. 291–298.CrossRefMathSciNetGoogle Scholar - [86]H. J. Zimmermann,
*Fuzzy set theory and its applications*, Kluwer, Dordrecht, 1985.Google Scholar - [87]H. J. Zimmermann, “Application of fuzzy set theory to mathematical programming”,
*Information Sciences*, 1985, Vol. 36, pp. 29–58.zbMATHCrossRefMathSciNetGoogle Scholar - [88]H. J. Zimmermann,
*Fuzzy set theory and its applications*, Kluwer, 1991.zbMATHGoogle Scholar - [89]V. S. Zyuzin, “An iterative method for solving a system of segment algebraic equations”, In:
*Differential equations and functions theory*, Saratov University Publ., Saratov, 1987, pp. 72–82 (in Russian).Google Scholar - [90]V. S. Zyuzin, “Twins and a method for solving systems of twin equations”, In:
*Interval Analysis*, Krasnoyarsk, Academy of Sciences Computing Center, Technical Report No. 6, 1988, pp. 19–21 (in Russian).Google Scholar