Solving different practical granular problems under the same system of equations
Tóm tắt
This paper contains discussion about the paper of Kreinovich (Granular Computing 1(3):171–179, 2016) in which the author suggests the thesis that in conditions of uncertainty “solving different practical problems we get different solutions of the same system of equations with the same granules”. Authors of the present paper are of the opinion that the above thesis is result of one-dimensional approach to interval analysis prevailing at present in scientific community and also used by Kreinovich. According to this approach the direct result of any arithmetic operation
$$*\in \{+,-,\times ,/\}$$
on intervals is also an interval. The main scientific contribution of the paper is showing that a granular problem analysis in an incomplete, low-dimensional space can lead to untrue conclusions because picture of the problem in this space is too poor and part of valuable, important information is lost. What is seen in the full-dimensional space of the problem space cannot be seen in an incomplete, low-dimensional space. The paper also shortly describes a multidimensional approach to interval arithmetic that is free of the above-described deficiencies.
Tài liệu tham khảo
Bader F, Nipkow T (1998) Term rewriting and all that. Cambridge University Press, Cambridge
Birkhoff G (1967) Lattice theory. Colloquium Publications—American Mathematical Society, Rhode Island
Chatterjee K, Kar S (2017) Unified granular-number-based AHP-VIKOR multi-criteria decision framework. Granul Comput 2(3):199–221
D’Aniello G, Gaeta A, Loia V, Orciuoli F (2017) A granular computing framework for approximate reasoning in situation awareness. Granul Comput 2(3):141–158
Dymova L (2011) Soft computing in economics and finance. Springer, Heidelberg
Kaucher E (1977) Uber eigenschaften und anwendungsmoglichkeiten der erweiterten intervalrechnung und des hyperbolische fastkorpers uber r. Comput Suppl 1:81–94
Kovalerchuk B, Kreinovich V (2016) Comparisons of applied tasks with intervals, fuzzy sets and probability approaches. In: Proceedings of 2016 IEEE international conference on fuzzy systems (FUZZ), Vancouver, 24–29 July 2016, pp 1478–1483
Kovalerchuk B, Kreinovich V (2017) Concepts of solutions of uncertain equations with intervals, probabilities and fuzzy sets for applied tasks. Granul Comput 2(3):121–130
Kreinovich V (2016) Solving equations (and systems of equations) under uncertainty: how different practical problems lead to different mathematical and computational formulations. Granul Comput 1(3):171–179
Livi L, Sadeghian A (2016) Granular computing, computational intelligence, and the analysis of non-geometric input spaces. Granul Comput 1(1):13–20
Lodwick W, Dubois D (2015) Interval linear systems as a necessary step in fuzzy linear systems. Fuzzy Sets Syst 281:227–251
Mazandarani M, Pariz N, Kamyad A (2017) Granular differentiability of fuzzy-number-valued functions. IEEE Trans Fuzzy Syst 1–24. doi:10.1109/TFUZZ.2017.2659731
Najariyan M, Mazandarani M, John R (2017) Type-2 fuzzy linear systems. Granul Comput 2(3):175–186
Piegat A, Landowski M (2012) Is the conventional interval arithmetic correct? J Theor Appl Comput Sci 6(2):27–44
Piegat A, Landowski M (2013) Two interpretations of multidimensional rdm interval arithmetic—multiplication and division. Int J Fuzzy Syst 15(4):488–496
Piegat A, Landowski M (2014) Correction checking of uncertain-equation solutions on example of the interval-modal method. In: Atanasov T et al (eds) Modern approaches in fuzzy sets, intuitionistic fuzzy sets, generalized nets and related topics, vol 1. Foundations, SRI PAS-IBS Pan Warsaw, Poland, pp 159–170
Piegat A, Landowski M (2015) Horizontal membership function and examples of its applications. Int J Fuzzy Syst 17(1):22–30
Piegat A, Landowski M (2017a) Fuzzy arithmetic type 1 with horizontal membership functions. In: Kreinovich V (ed) Uncertainty modeling, Springer, Cham, pp 233–250
Piegat A, Landowski M (2017b) Is an interval the right result of arithmetic operations on intervals? Int J Appl Math Comput Sci 27(3):575–590
Piegat A, Landowski M (2018) Is fuzzy number the right result of arithmetic operations on fuzzy numbers? In: Kacprzyk J, Szmidt E, Zadrony S, Atanassov K, Krawczak M (eds) Advances in fuzzy logic and technology 2017. IWIFSGN 2017, EUSFLAT 2017. Advances in intelligent systems and computing, vol 643. Springer, Cham, pp 181–194
Piegat A, Plucinski M (2015a) Computing with words with the use of inverse rdm models of membership functions. Int J Appl Math Comput Sci 25(3):675–688
Piegat A, Plucinski M (2015b) Fuzzy number addition with application of horizontal membership functions. Sci World J 2015:1–16. doi:10.1155/2015/367214
Piegat A, Plucinski M (2017) Fuzzy number division and the multi-granularity phenomenon. Bull Pol Acad Sci Tech Sci 65(4):497–511
Piegat A, Tomaszewska K (2013) Decision making under uncertainty using info-gap theory and a new multidimensional RDM interval arithmetic. Electrotech Rev 89(8):71–76
Popova E (1998) Algebraic solutions to a class of interval equations. J Univers Comput Sci 4(1):48–67
Shary S (1996) Algebraic approach to the interval linear static identification, tolerance and control problems, or one more application of kaucher arithmetic. Reliab Comput 2(1):3–33
Wang G, Yang J, Xu J (2017) Granular computing: from granularity optimization to multi-granularity joint problem solving. Granul Comput 2(3):105–120