Are multidimensional RDM interval arithmetic and constrained interval arithmetic one and the same?

Document Type : Research Paper


1 Faculty of Computer Science, West Pomeranian University of Technology, Szczecin, Poland

2 Faculty of Computer Science and Telecommunications, Maritime University of Szczecin, Szczecin, Poland


This article discusses the comments made by some scientists that multidimensional interval arithmetic (MIA) is the same as constraint interval arithmetic (CIA) and multidimensional fuzzy arithmetic (MFA) is the same as constraint fuzzy arithmetic (CFA). Both types of arithmetic are briefly presented and then the difference in their dimensions, calculation methods, differences in the obtained results and the way they are used in complex calculations are shown. The answer to the question posed is presented in the conclusions.


[1] S. K. Alamanda, K. K. Boddeti, Relative distance measure arithmetic-based available transfer capability calculation with uncertainty in wind power generation, International Transactions on Electrical Energy Systems, 31(11) (2021), e-13112.
[2] R. A. Aliev, Uncertain computation-based decision theory, World Scientific, New Jersey, London, Beijing, 2018.
[3] R. Boukezzoula, L. Foulloy, D. Coquin, S. Galichet, Gradual interval arithmetic and fuzzy interval arithmetic, Granular Computing, 6 (2021), 451-471.
[4] Y. Chalco-Cano, W. A. Lodwick, B. Bede, Single level constraint interval arithmetic, Fuzzy Sets and Systems, 257 (2014), 146-168.
[5] L. Dymova, Soft computing in economics and finance, Springier, Heidelberg, New York, London, 2011.
[6] A. Ebrahimnejad, An effective computational attempt for solving fully fuzzy linear programming using MOLP problem, Journal of Industrial and Production Engineering, 39(2) (2019), 59-69.
[7] A. Ebrahimnejad, An acceptability index based approach for solving shortest path problem on a network with interval weights, Rairo Operation Research, 55 (2021), S1767-S1787.
[8] A. Ebrahimnejad, J. L. Verdegay An efficient computational approach for solving type-2 intuitionistic fuzzy numbers based transportation problems, International Journal of Computational Intelligence Systems, 9(6) (2016), 1154-1173.
[9] M. Friedman, M. Ming, A. Kandel, Fuzzy linear systems, Fuzzy Sets and Systems, 96 (1998), 201-209.
[10] G. J. Klir, B. Yuan, Fuzzy sets and fuzzy logic: Theory and applications, Prentice Hall, Upper Saddle River, NJ, 1995.
[11] J. Kolodziejczyk, A. Piegat, W. Salabun, Which alternative for solving dual fuzzy nonlinear equations is more precise?, Mathematics, 8 (2020) 1-13.
[12] M. Landowski, RDM interval method for solving quadratic interval equation, Przeglad Elektrotechniczny (Electrotechnical Review), R.93(1) (2017), 65-68.
[13] M. Landowski, Usage of RDM interval arithmetic for solving cubic interval equation, In: Kacprzyk J., et al. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT 2017, IWIFSGN 2017. Advances in Intelligent Systems and Computing, Springer, Cham, 642 (2018), 382-391.
[14] M. Landowski, Method with horizontal fuzzy numbers for solving real fuzzy linear systems, Soft Computing, 23 (2019), 3921-3933.
[15] W. A. Lodwick, Constrained interval arithmetic, CCM Report 138, February 1999.
[16] W. A. Lodwick, D. Dubois, Interval linear systems as a necessary step in fuzzy linear systems, Fuzzy Sets and Systems, 281 (2015), 227-251.
[17] W. A. Lodwick, O. A. Jenkins, Constrained intervals and interval spaces, Soft Computing, 17(8) (2013), 1393-1402.
[18] W. A. Lodwick, E. A. Untiedt, A comparison of interval analysis using constraint interval arithmetic and fuzzy interval analysis using gradual numbers, Proceedings of the NAFIPS 2008 - 2008 Annual Meeting of the North American Fuzzy Information Processing Society, New York City, NY, (2008), 1-6.
[19] M. Mazandarani, N. Pariz, Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept, ISA Transactions, 76 (2018), 1-17.
[20] M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Transactions on Fuzzy Systems, 26(1) (2018), 310-323.
[21] M. Mazandarani, Y. Zhao, Fuzzy bang-bang control problem under granular differentiability, Journal of the Franklin Institute, 355(12) (2018), 4931-4951.
[22] M. Mazandarani, Y. Zhao, Z-differential equations, IEEE Transactions on Fuzzy Systems, 28(3) (2020), 462-473.
[23] M. T. Mizukoshi, W. A. Lodwick, Interval arithmetic: WSM, CI or RDM?, In: Rayz J., et al. (eds) Explainable AI and Other Applications of Fuzzy Techniques. NAFIPS 2021. Lecture Notes in Networks and Systems, Springer, Cham, 258 (2022), 291-301.
[24] R. E. Moore, Interval analysis, Prentice-Hall, Englewood Cliffs, N.J., 1966.
[25] M. Najariyan, Y. Zhao, Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives, IEEE Transactions on Fuzzy Systems, 26(4) (2018), 2273-2288.
[26] M. Najariyan, Y. Zhao, On the stability of fuzzy linear dynamical systems, Journal of the Franklin Institute, 357(9) (2020), 5502-5522.
[27] M. Najariyan, Y. Zhao, The explicit solution of fuzzy singular differential equations using fuzzy Drazin inverse matrix, Soft Computing, 24(15) (2020), 11251-11264.
[28] A. Piegat, M. Landowski, Two interpretations of multidimensional RDM interval arithmetic - multiplication and division, International Journal of Fuzzy Systems, 15(4) (2013), 488-496.
[29] A. Piegat, M. Landowski, Horizontal membership functions and examples of its applications, International Journal of Fuzzy Systems, 17(1) (2015), 22-30.
[30] A. Piegat, M. Landowski, Multidimensional interval type 2 epistemic fuzzy arithmetic, Iranian Journal of Fuzzy Systems, 18(5) (2021), 19-36.
[31] A. Piegat, M. Landowski, Multidimensional type 2 epistemic fuzzy arithmetic based on the body definition of the type 2 fuzzy set, Applied Sciences, 11 (2021), 1-27.
[32] A. Piegat, M. Olchowy, Contextual one sector non-regular fuzzy model based on 4 knowledge points, Pomiary, Automatyka, Kontrola, 56(10) (2010), 1193-1196.
[33] A. Piegat, M. Plucinski, Fuzzy number addition with application of horizontal membership functions, The Scientific World Journal, ID 367 (2015), 1-16.
[34] A. Piegat, M. Plucinski, Computing with words with the use of inverse RDM models of membership functions, International Journal of Applied Mathematics and Computer Science, 25(3) (2015), 675-688.
[35] A. Piegat, M. Plucinski, Fuzzy number division and the multi-granularity phenomenon, Bulletin of the Polish Academy of Sciences: Technical Science, 65(4) (2017), 497-511.
[36] A. Piegat, M. Plucinski, Inclusion principle of fuzzy arithmetic results, Journal of Intelligent and Fuzzy Systems, 42(6) (2022), 4987-4998.
[37] A. Piegat, M. Plucinski, The optimal tolerance solution of the basic interval linear equation and the explanation of the Lodwick’s anomaly, Applied Sciences, 12(9) (2022), 1-21.
[38] A. Piegat, K. Tomaszewska, Decision-making under uncertainty using Info-Gap theory and a new multi-dimensional RDM interval arithmetic, Przeglad Elektrotechniczny (Electrotechnical Review), R.89(8) (2013), 71-76.
[39] M. Plucinski, Solving Zadeh’s challenge problems with the application of RDM-arithmetic, In: Rutkowski L., et al. (eds) Artificial Intelligence and Soft Computing. ICAISC 2015. LNCS, 9119. Springer, Cham, (2015), 239-248.
[40] G. Schmidt, Relational mathematics, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 132 (2011), 169-227.
[41] A. Sotoudeh-Anvari, A critical review on theoretical drawbacks and mathematical incorrect assumptions in fuzzy OR methods: Review from 2010 to 2020, Applied Soft Computing, 93 (2020), 106354.
[42] T. Sunaga, Theory of an interval algebra and its applications to numerical analysis, RAAAG Memoirs, 2 (1958), 547-564.
[43] K. Tomaszewska, A. Piegat, Application of the horizontal membership function to the uncertain displacement calculation of a composite massless rod under a tensile load, In Wilinski A., et al. (eds), Soft Computing in Computer and Information Sciences, Springer, Cham, 342 (2015), 63-72.
[44] M. Warmus, Calculus of approximations, Bulletin de L’Academie Polonaise de Sciences CI.III, 4 (1956), 253-259.
[45] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning-I, Information Sciences, 8(3) (1975), 199-249.
[46] H. J. Zimmermann, Fuzzy set theory - and its applications, Springer, Dordrecht, 1985.