Fuzzy fractional framework under granular ψ-Caputo fractional derivative with applications to supply chain dynamics

Muhammad, Ghulam; Zhang, Qingyu; Akram, Muhammad; Allahviranloo, Tofigh

doi:10.22111/ijfs.2026.54397.9638

Articles in Press

Document Type : Original Manuscript

Authors

¹ Research Institute of Business Analytics and SCM, College of Management, Shenzhen University, Shenzhen, China

² Institute of Mathematics, University of the Punjab, New Campus, Lahore 4590, Pakistan

³ Research Center of Performance and Productivity Analysis, Istinye University, Istanbul, Turkey

https://doi.org/10.22111/ijfs.2026.54397.9638

Abstract

This article develops a mathematical framework for fuzzy fractional differential system using the concept of granular Caputo fractional derivative with respect to another function ψ within the horizontal membership function (HMF)-based granular differentiability framework. The propose technique addresses the need for a unified approach that simultaneously incorporate the function ψ as memory kernels for flexible modeling, multi-dimensional representation of uncertainty through HMFs, and analytical fuzzy solution using granular fuzzy Laplace transform with respect to another function (LTWR-AF). First, we introduce the granular ψ-fuzzy fractional integral (ψ-FFI), granular ψ-Caputo fractional derivative (ψ-CFD) and granular fuzzy LTWR-AF. Fundamental properties, results, and relationships among these properties are established in HMF representation. Based on this foundation, closed-form analytical fuzzy solution (AFS) of granular Caputo fuzzy fractional differential equations (GCFFDEs) involving granular ψ-CFD are determined and expressed compactly in terms of Mittag-Leffler function (M-LF). Moreover, the existence and uniqueness (E&U) of the fuzzy solutions for the abstract granular Caputo fuzzy fractional differential system (GCFFDS) with fuzzy parameters and initial data are examined under mild conditions. To illustrate the applicability of the proposed framework, the proposed methodology is applied to the granular Caputo fuzzy fractional sustainable, impact-intensive, quarantined, and recycled/remanufactured (SIQR) supply-chain model under environmental disruptions. Numerical simulations and graphical representations demonstrate that the proposed model effectively describe the uncertainty propagation and memory-dependent behavior in supply-chain dynamics.

Keywords

Main Subjects

Fuzzy Math

References

[1] R. P. Agarwal, S. Hristova, D. O’Regan, Strict stability of fractional differential equations with a Caputo fractional derivative with respect to another function, Mathematics, 13(3) (2025), 452. https://doi.org/10.3390/ math13030452

[2] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods and Applications, 72(6) (2010), 2859-2862. https://doi.org/ 10.1016/j.na.2009.11.029

[3] K. Alanazi, O. Naifar, R. Fakhfakh, A. B. Makhlouf, Innovative observer design for nonlinear systems using Caputo fractional derivative with respect to another function, AIMS Mathematics, 9(12) (2024), 35533-35550. https:// doi.org/10.3934/math.20241686

[4] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44 (2017), 460-481. https://doi.org/10.1016/j.cnsns.2016.09. 006

[5] R. Almeida, On the variable-order fractional derivatives with respect to another function, Aequationes Mathematicae, 99 (2024), 805-822. https://doi.org/10.1007/s00010-024-01082-0

[6] F. I. A. Amir, A. Moussaoui, R. Shafqat, et al., The Hadamard ψ-Caputo tempered fractional derivative in various types of fuzzy fractional differential equations, Soft Computing, 28(17) (2024), 9253-9270. https://doi.org/10. 1007/s00500-024-09821-w

[7] F. Babakordi, T. Allahviranloo, M. R. Shahriari, M. Catak, Fuzzy Laplace transform method for a fractional fuzzy economic model based on market equilibrium, Information Sciences, 665 (2024), 120308. https://doi.org/10. 1016/j.ins.2024.120308

[8] D. Baleanu, A. Jajarmi, S. S. Sajjadi, D. Mozyrska, A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(8) (2019), 083127. https://doi.org/10.1063/1.5096159

[9] Y. A. Brychkov, A. P. Prudnikov, V. S. Shishov, Operational calculus, Journal of Soviet Mathematics, 15(6) (1981), 733-765. https://doi.org/10.1007/BF01377044

[10] M. Candan, M. Çubukçu, Implementation of Caputo type fractional derivative chain rule on back propagation algorithm, Applied Soft Computing, 155 (2024), 111475. https://doi.org/10.1016/j.asoc.2024.111475

[11] N. P. Dong, H. V. Long, N. L. Giang, The fuzzy fractional SIQR model of computer virus propagation in wireless sensor network using Caputo Atangana-Baleanu derivatives, Fuzzy Sets and Systems, 429 (2022), 28-59. https: //doi.org/10.1016/j.fss.2021.04.012

[12] N. P. Dong, H. V. Long, A. Khastan, Optimal control of a fractional order model for granular SEIR epidemic with uncertainty, Communications in Nonlinear Science and Numerical Simulation, 88 (2020), 105312. https: //doi.org/10.1016/j.cnsns.2020.105312

[13] N. P. Dong, N. T. K. Son, T. Allahviranloo, H. T. T. Tam, Finite-time stability of mild solution to time-delay fuzzy fractional differential systems under granular computing, Granular Computing, 8(2) (2023), 223-239. https: //doi.org/10.1007/s41066-022-00325-2

[14] A. Ebrahimzadeh, A. Jajarmi, M. Yavuz, Fractional optimal control of anthroponotic cutaneous leishmaniasis with behavioral and epidemiological extensions, Mathematical and Computational Applications, 30(6) (2025), 122. https://doi.org/10.3390/mca30060122

[15] K. Elhakimy, I. Sadouki, A. E. Mfadel, S. Melliani, Finite-time stability for fuzzy fractional delay differential equations with generalized Caputo derivatives, Electronic Journal of Differential Equations, 2026(7) (2026), 1-14. https://doi.org/10.58997/ejde.2026.07

[16] H. M. Fahad, M. U. Rehman, A. Fernandez, On Laplace transforms with respect to functions and their applications to fractional differential equations, Mathematical Methods in the Applied Sciences, 46(7) (2023), 8304-8323. https: //doi.org/10.1002/mma.7772

[17] A. E. Ghazouani, M. Elomari, S. Melliani, Fuzzy fractional delay integro-differential equation with the generalized Hukuhara ψ-Caputo fractional derivative, Fuzzy Sets and Systems, 528 (2026), 109701. https://doi.org/10.1016/ j.fss.2025.109701

[18] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler function, related topics and applications, Berlin/Heidelberg: Springer, 2020. https://doi.org/10.1007/978-3-662-61550-8

[19] N. V. Hoa, Fractional impulsive fuzzy differential systems with multi-order in (1,2): The solution representation and asymptotical stabilization, Information Sciences, 719 (2025), 122461. https://doi.org/10.1016/j.ins.2025. 122461

[20] N. V. Hoa, L. V. Phut, On explicit solutions for linear fuzzy fractional differential equations with piecewise constant argument under the multi-order Caputo fractional derivative, Fuzzy Sets and Systems, 546 (2026), 110100. https: //doi.org/10.1016/j.fss.2026.110100

[21] A. Jajarmi, Dynamics, stability, and optimal chaos control of a ψ-Caputo cancer-immune system, International Journal of Dynamics and Control, 14 (2026), 33. https://doi.org/10.1007/s40435-025-01974-2

[22] A. Jajarmi, M. Akbarian, D. Baleanu, Analysis and backstepping control of a novel 4D fractional chaotic oscilla tor, Mathematical Methods in the Applied Sciences, 48(16) (2025), 14750-14762. https://doi.org/10.1002/mma. 11211

[23] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems, 13(3) (2020), 709-722. https://doi.org/10.3934/dcdss.2020039

[24] M. Johnson, V. Vijayakumar, K. Kwon, Existence outcomes and optimal control of non-autonomous second or der integrodifferential system with impulsive effects via resolvent operator in Banach spaces, Journal of Applied Mathematics and Computing, 71(6) (2025), 7959-7982. https://doi.org/10.1007/s12190-025-02631-9

[25] C. A. S. Jr, F. S. Costa, J. V. C. Sousa, Infinitesimal prolongation for fractional derivative ψ-Caputo variable order and applications, Qualitative Theory of Dynamical Systems, 24(1) (2025). https://doi.org/10.1007/ s12346-024-01157-y

[26] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, North Holland Mathematics Studies, (2006), 1-523.

[27] V. P. Lai, V. H. Ngo, The explicit solution to fuzzy differential equation with piecewise constant arguments involving the multi-order fractional derivative in (1,2), Iranian Journal of Fuzzy Systems, 23(4) (2026), 183-201. https: //doi.org/10.22111/ijfs.2026.54660.9681

[28] Y. Li, D. Li, F. Chen, X. Liu, New multiplicity results for a boundary value problem involving a ψ-Caputo fractional derivative of a function with respect to another function, Fractal and Fractional, 8(6) (2024), 305. https://doi. org/10.3390/fractalfract8060305

[29] X. Liu, G. Ye, W. Liu, et al., On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium, Journal of Computational and Applied Mathematics, 450 (2024), 116017. https://doi.org/10.1016/j.cam.2024.116017

[30] A. D. Mali, K. D. Kucche, J. V. C. Sousa, Calculus of k-fractional derivative with respect to monotonic functions, Journal of Pseudo-Differential Operators and Applications, 16(1) (2025), 1-35. https://doi.org/10.1007/ s11868-025-00678-7

[31] M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Trans actions on Fuzzy Systems, 26(1) (2017), 310-323.

[32] G. Muhammad, M. Akram, Fuzzy fractional epidemiological model for Middle East respiratory syndrome coronavirus on complex heterogeneous network using Caputo derivative, Information Sciences, 659 (2024), 120046. https://doi.org/10.1016/j.ins.2023.120046

[33] M. Najariyan, Innovative tracking control using interval type-2 fuzzy and fractional methods, Information Sciences, 699 (2025), 121758. https://doi.org/10.1016/j.ins.2024.121758

[34] M. Najariyan, Y. Zhao, Fuzzy fractional quadratic regulator problem under granular fuzzy fractional derivatives, IEEE Transactions on Fuzzy Systems, 26(4) (2017), 2273-2288. https://doi.org/10.1109/TFUZZ.2017.2783895

[35] T. J. Osler, The Fractional derivative of a composite function, SIAM Journal on Mathematical Analysis, 1(2) (1970), 288-293. https://doi.org/10.1137/0501026

[36] A. Piegat, M. Landowski, Horizontal membership function and examples of its applications, International Journal of Fuzzy Systems, 17 (2015), 22-30. https://doi.org/10.1007/s40815-015-0013-8

[37] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198 (1999). San Diego: Academic Press.

[38] N. T. K. Son, H. V. Long, N. P. Dong, Fuzzy delay differential equations under granular differentiability with applications, Computational and Applied Mathematics, 38(3) (2019), 107. https://doi.org/10.1007/ s40314-019-0881-x

[39] R. Srilekha, V. Parthiban, D. Sundaram, Controllability criteria for type-2 fuzzy fractional-order dynamical system via Mittag-Leffler matrix function using granular derivative, Iranian Journal of Fuzzy Systems, 21(5) (2024), 133 150. https://doi.org/10.22111/ijfs.2024.47862.8422

[40] A. Taqbibt, M. Chaib, E. O. M. Hamed, Generalization of inequalities for different types of function via ψ-Caputo fractional derivative, Kragujevac Journal of Mathematics, 50(7) (2026), 1171-1182. https://doi.org/10.46793/ KgJMat2607.1171T

[41] V. E. Tarasov, Parametric general fractional calculus: Nonlocal operators acting on function with respect to another function, Computational and Applied Mathematics, 43(4) (2024), 183. https://doi.org/10.1007/ s40314-024-02725-3

[42] F. Yan, D. Luo, Finite-time stability of Caputo fractional fuzzy differential equations with delay in granular sense, Communications in Nonlinear Science and Numerical Simulation, 134 (2024), 108022. https://doi.org/10.1016/ j.cnsns.2024.108022

[43] J. Zhang, M. Li, H. Li, Optimality conditions of fuzzy fractional optimization and its application in the problems of energy efficiency optimization, RAIRO-Operations Research, 60(4) (2026), 2571-2607. https://doi.org/10.1051/ ro/2026086

Fuzzy fractional framework under granular ψ-Caputo fractional derivative with applications to supply chain dynamics

References

References

Articles in Press, Accepted Manuscript Available Online from 28 September 2026

Articles in Press, Accepted Manuscript
Available Online from 28 September 2026