Simulation and Limiting Behavior of Random Fuzzy Intervals

Romaniuk, M.; Parchami, Abbas; Grzegorzewski, P.

doi:10.22111/ijfs.2026.53290.9431

Document Type : Research Paper

Authors

¹ Systems Research Institute Polish Academy of Sciences, Newelska 6, 01-447 Warszawa

² عضو هییت علمی

³ Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa

https://doi.org/10.22111/ijfs.2026.53290.9431

Abstract

Fuzzy random variables combine the modeling of imprecision (fuzzy component) and unpredictability (caused by random effects) into a single entity.
Statistical samples of such units are widely used; therefore, their direct, numerically efficient generation is necessary. Typically, these samples consist of triangular or trapezoidal fuzzy numbers.
This paper describes some theoretical results and simulation algorithms for another useful family of fuzzy numbers, called the LR fuzzy numbers with interval cores. Starting from a simulation perspective on the piecewise linear LR fuzzy numbers with the interval cores, we consider their limiting behavior, which leads to some of their interesting properties and provides a numerically efficient algorithm for simulating a sample consisting of such fuzzy values.
As a result, we obtain a new perspective on a way of introducing the random fuzzy intervals.

Keywords

Main Subjects

Decision Making

References

[1] A. B´aez-S´anchez, A. C. Moretti, M. Rojas-Medar, On polygonal fuzzy sets and numbers, Fuzzy Sets and Systems,
209 (2012), 54-65. https://doi.org/10.1016/j.fss.2012.04.003

[2] A. Blanco-Fernandez, M. R. Casals, A. Colubi, N. Corral, M. Garca-Barzana, M. A. Gil, G. Gonzalez-Rodriguez,
M. Lopez, M. Montenegro, M. A. Lubiano, A. B. Ramos-Guajardo, S. de la Rosa de Saa, B. Sinova, Random fuzzy
sets: A mathematical tool to develop statistical fuzzy data analysis, Iranian Journal of Fuzzy Systems, 10(2) (2013),
1-28. https://doi.org/10.22111/ijfs.2013.609

[3] A. Calcagn`ı, P. Grzegorzewski, M. Romaniuk, Bayesianize fuzziness in the statistical analysis of fuzzy data, International
Journal of Approximate Reasoning, 186 (2025), 109495. https://doi.org/10.1016/j.ijar.2025.109495

[4] F. P. Cantelli, Sulla determinazione empirica della leggi di probabilita, Giornale dell’Istituto Italiano degli Attuari,
4 (1933), 421-224.

[5] A. Colubi, J. S. Dom´ınguez-Menchero, M. L´opez-D´ıaz, D. A. Ralescu, On the formalization of fuzzy random
variables, Information Sciences, 133(1-2) (2001), 3-6. https://doi.org/10.1016/S0020-0255(01)00073-1

[6] A. Colubi, C. Fern´andez-Garc´ıa, M. Gil, Simulation of random fuzzy variables: An empirical approach to statistical/
probabilistic studies with fuzzy experimental data, IEEE Transactions on Fuzzy Systems, 10(3) (2002), 384-390.
https://doi.org/10.1109/TFUZZ.2002.1006441

[7] L. Coroianu, M. Gagolewski, P. Grzegorzewski, Piecewise linear approximation of fuzzy numbers: Algorithms,
arithmetic operations and stability of characteristics, Soft Computing, 23(19) (2019), 9491-9505. https://doi.
org/10.1007/s00500-019-03800-2

[8] I. Couso, D. Dubois, Statistical reasoning with set-valued information: Ontic vs. epistemic views, International
Journal of Approximate Reasoning, 55(7) (2014), 1502-1518. https://doi.org/10.1016/j.ijar.2013.07.002

[9] D. Dubois, H. Prade, Fuzzy sets and systems: Theory and applications, Academic Press, Boston, (1980).

[10] R. A. Fisher, The use of multiple measurements in taxonomic problems, Annals of Eugenics, 7(Part II) (1936),
179-188. https://doi.org/10.1111/j.1469-1809.1936.tb02137.x

[11] V. Glivenko, Sulla determinazione empirica della leggi di probabilita, Giornale dell’Istituto Italiano degli Attuari,
4 (1933), 92-99.

[12] G. Gonz´alez-Rodr´ıguez, A. Colubi, W. Trutschnig, Simulation of fuzzy random variables, Information Sciences,
179(5) (2009), 642-653. https://doi.org/10.1016/j.ins.2008.10.018

[13] G. Gonz´alez-Rodr´ıguez, M. Montenegro, A. Colubi, M. ´A. Gil, Bootstrap techniques and fuzzy random variables:
Synergy in hypothesis testing with fuzzy data, Fuzzy Sets and Systems, 157(19) (2006), 2608-2613. https://doi.
org/10.1016/j.fss.2003.11.021

[14] P. Grzegorzewski, Statistics with vague data and the robustness to data representation, in: D. Dubois, M. A.
Lubiano, H. Prade, M. ´A. Gil, P. Grzegorzewski, O. Hryniewicz (eds.), Soft Methods for Handling Variability and
Imprecision, 100-107, Springer, Berlin, Heidelberg, (2008). https://doi.org/10.1007/978-3-540-85027-4_13

[15] P. Grzegorzewski, O. Hryniewicz, M. Romaniuk, Flexible bootstrap for fuzzy data based on the canonical representation, International Journal of Computational Intelligence Systems, 13 (2020), 1650-1662. https://doi.org/10.
2991/ijcis.d.201012.003

[16] P. Grzegorzewski, O. Hryniewicz, M. Romaniuk, Flexible resampling for fuzzy data, International Journal of Applied
Mathematics and Computer Science, 30(2) (2020), 281-297. https://doi.org/10.34768/amcs-2020-0022

[17] P. Grzegorzewski, M. Romaniuk, Bootstrap methods for epistemic data, International Journal of Applied Mathematics
and Computer Science, 32(2) (2022), 288-297. https://doi.org/10.34768/amcs-2022-0021

[18] P. Grzegorzewski, M. Romaniuk, Bootstrapped tests for epistemic fuzzy data, International Journal of Applied
Mathematics and Computer Science, 34(2) (2024), 277-289. https://doi.org/10.61822/amcs-2024-0020

[19] M. Hanss, Applied fuzzy arithmetic. An introduction with engineering applications, Springer, 2005. https://doi.
org/10.1007/b138914

[20] G. Hesamian, M. G. Akbari, J. Zendehdel, Location and scale fuzzy random variables, International Journal of
Systems Science, 51(2) (2020), 229-241. https://doi.org/10.1080/00207721.2019.1701131

[21] E. Kayacan, M. A. Khanesar, Fuzzy neural networks for real time control applications, Butterworth-Heinemann,
2016. https://doi.org/10.1016/C2014-0-02444-6

[22] V. Kr¨atschmer, A unified approach to fuzzy random variables, Fuzzy Sets and Systems, 123(1) (2001), 1-9. https:
//doi.org/10.1016/S0165-0114(00)00038-5

[23] R. Kruse, The strong law of large numbers for fuzzy random variables, Information Sciences, 28(3) (1982), 233-241.
https://doi.org/10.1016/0020-0255(82)90049-4

[24] H. Kwakernaak, Fuzzy random variables, part I: Definitions and theorems, Information Sciences, 15(1) (1978),
1-29. https://doi.org/10.1016/0020-0255(78)90019-1

[25] M. A. Lubiano, A. Salas, C. Carleos, S. de la Rosa de S´aa, M. ´A. Gil, Hypothesis testing-based comparative analysis
between rating scales for intrinsically imprecise data, International Journal of Approximate Reasoning, 88 (2017),
128-147. https://doi.org/10.1016/j.ijar.2017.05.007

[26] M. A. Lubiano, A. Salas, M. ´A. Gil, A hypothesis testing-based discussion on the sensitivity of means of fuzzy data
with respect to data shape, Fuzzy Sets and Systems, 328 (2017), 54-69. https://doi.org/10.1016/j.fss.2016.
10.015

[27] O. Mersmann, Microbenchmark: Accurate timing functions, R package version 1.5.0, (2024). https://CRAN.
R-project.org/package=microbenchmark

[28] S. P. Millard, EnvStats: An R package for environmental statistics, Springer, New York, 2013. https://doi.org/
10.1007/978-1-4614-8456-1

[29] A. Parchami, P. Grzegorzewski, M. Romaniuk, Statistical simulations with LR random fuzzy numbers, Statistical
Papers, 65(6) (2024), 3583-3600. https://doi.org/10.1007/s00362-024-01533-5

[30] A. Parchami, P. Grzegorzewski, M. Romaniuk, Calculating probabilities with LR fuzzy random variables, Soft
Computing, 29 (2025), 5129-5141. https://doi.org/10.1007/s00500-025-10877-5

[31] M. L. Puri, D. A. Ralescu, Fuzzy random variables, Journal of Mathematical Analysis and Applications, 114(2)
(1986), 409-422. https://doi.org/10.1016/0022-247X(86)90093-4

[32] A. Ramos-Guajardo, A. Blanco-Fern´andez, G. Gonz´alez-Rodr´ıguez, Applying statistical methods with imprecise
data to quality control in cheese manufacturing, in: Soft Modeling in Industrial Manufacturing, P. Grzegorzewski,
A. Kochanski, and J. Kacprzyk (eds.), 127-147, Springer, (2019). https://doi.org/10.1007/
978-3-030-03201-2_8

[33] C. P. Robert, G. Casella, Monte Carlo statistical methods, Springer-Verlag, Berlin, Heidelberg, (2005). https:
//doi.org/10.1007/978-1-4757-4145-2

[34] M. Romaniuk, Imprecise approaches to analysis of insurance portfolio with catastrophe bond, in: M. J. Lesot, et al.
(eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 20220, 3-16,
Springer, (2020). https://doi.org/10.1007/978-3-030-50153-2_1

[35] M. Romaniuk, P. Grzegorzewski, Resampling fuzzy numbers with statistical applications: FuzzyResampling package,
The R Journal, 15(1) (2023), 271-283. https://doi.org/10.32614/RJ-2023-036

[36] M. Romaniuk, P. Grzegorzewski, A. Parchami, FuzzySimRes: Epistemic bootstrap – an efficient tool for statistical
inference based on imprecise data, The R Journal, 16(2) (2024), 175-190. https://doi.org/10.32614/
RJ-2024-016

[37] M. Romaniuk, O. Hryniewicz, Estimation of maintenance costs of a pipeline for a U-shaped hazard rate function
in the imprecise setting, Eksploatacja i Niezawodno´s´c–Maintenance and Reliability, 22(2) (2020), 352-362. https:
//doi.org/10.17531/ein.2020.2.18

[38] M. Romaniuk, O. Hryniewicz, Discrete and smoothed resampling methods for interval-valued fuzzy numbers, IEEE
Transactions on Fuzzy Systems, 29(3) (2021), 599-611. https://doi.org/10.1109/TFUZZ.2019.2957253

[39] V. V. Sahakyan, An improved algorithm for generation of truncated normal distributed random numbers, Mathematical Problems of Computer Science, 42 (2014), 73-80.

[40] J. Shen, J. Zhou, Calculation formulas and simulation algorithms for entropy of function of LR fuzzy intervals,
Entropy, 21(3) (2019), 289. https://doi.org/10.3390/e21030289

[41] W. Stute, On a generalization of the Glivenko-Cantelli theorem, Z. Wahrscheinlichkeitstheorie verw Gebiete, 35
(1976), 167-175. https://doi.org/10.1007/BF00533322

[42] S. Sundaramoorthy, G. Karunanidhi, A systematic analysis on performance and computational complexity of sorting
algorithms, Discover Computing, 28 (2025), 250. https://doi.org/10.1007/s10791-025-09724-w

[43] H. G. Tucker, A generalization of the Glivenko-Cantelli theorem, The Annals of Mathematical Statistics, 30(3)
(1959), 828-830. https://doi.org/10.1214/aoms/1177706212

[44] A. W. Vaart, J. A. Wellner, Glivenko-Cantelli theorems, in: A. W. Vaart, J. A. Wellner, Weak Convergence and
Empirical Processes, 122-126, Springer, New York, (1996). https://doi.org/10.1007/978-1-4757-2545-2_16

Simulation and Limiting Behavior of Random Fuzzy Intervals

References

References

Volume 23, Issue 2March and April 2026Pages 1-16

Volume 23, Issue 2
March and April 2026
Pages 1-16