A sim-learnheuristic algorithm for solving a capacitated dispersion problem under stochastic and non-static conditions

Elnaz Ghorbani; Juan F. Gomez; Javier Panadero; Angel A. Juan; Elnaz Ghorbani; Juan F. Gomez; Javier Panadero; Angel A. Juan

doi:10.3934/math.20241180

AIMS Mathematics

2024, Volume 9, Issue 9: 24247-24270. doi: 10.3934/math.20241180

Previous Article Next Article

Research article Special Issues

A sim-learnheuristic algorithm for solving a capacitated dispersion problem under stochastic and non-static conditions

1.
Department of Computer Science, Universitat Oberta de Catalunya, 08018 Barcelona, Spain
2.
Research Center on Production Management and Engineering, Universitat Politècnica de València, 03801 Alcoy, Spain
3.
Department of Computer Architecture and Operating Systems, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Received: 11 June 2024 Revised: 30 July 2024 Accepted: 05 August 2024 Published: 16 August 2024
MSC : 68T20, 90-08, 90-10, 90Bxx, 90B36

A fundamental assumption in addressing real-world problems is acknowledging the presence of uncertainty and dynamism. Dismissing these factors can lead to the formulation of an optimal solution for an entirely different problem. This paper presents a novel variant of the capacitated dispersion problem (CDP) referred to as the stochastic and non-static CDP. The main objective of this problem is to strategically position facilities to achieve maximum dispersion while meeting the capacity demand constraint. The proposed approach combines stochastic and non-static elements, introducing a new paradigm to address the problem. This innovation allows us to consider more realistic and flexible environments. To solve this challenging problem, a novel sim-learnheuristic algorithm is proposed. This algorithm combines a biased-randomized metaheuristic (optimization component) with a simulation component (to model the uncertainty) and a machine learning component (to model non-static behavior). The non-static part works by using black box and white box mechanisms to learn the uncertainty with some related facilities' variables. Based on an extended set of traditional benchmarks for the CDP, a series of computational experiments were carried out. The results demonstrate the effectiveness of the proposed sim-learnheuristic approach for solving the CDP under non-static and stochastic scenarios.

Keywords:

Citation: Elnaz Ghorbani, Juan F. Gomez, Javier Panadero, Angel A. Juan. A sim-learnheuristic algorithm for solving a capacitated dispersion problem under stochastic and non-static conditions[J]. AIMS Mathematics, 2024, 9(9): 24247-24270. doi: 10.3934/math.20241180

Related Papers:

[1]	Huiyang Xu . Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials. Communications in Analysis and Mechanics, 2023, 15(2): 132-161. doi: 10.3934/cam.2023008
[2]	Tingfu Feng, Yan Dong, Kelei Zhang, Yan Zhu . Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth. Communications in Analysis and Mechanics, 2025, 17(1): 263-289. doi: 10.3934/cam.2025011
[3]	Yuxuan Chen . Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity. Communications in Analysis and Mechanics, 2023, 15(4): 658-694. doi: 10.3934/cam.2023033
[4]	Yue Pang, Xiaotong Qiu, Runzhang Xu, Yanbing Yang . The Cauchy problem for general nonlinear wave equations with doubly dispersive. Communications in Analysis and Mechanics, 2024, 16(2): 416-430. doi: 10.3934/cam.2024019
[5]	Reinhard Racke . Blow-up for hyperbolized compressible Navier-Stokes equations. Communications in Analysis and Mechanics, 2025, 17(2): 550-581. doi: 10.3934/cam.2025022
[6]	Isaac Neal, Steve Shkoller, Vlad Vicol . A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy. Communications in Analysis and Mechanics, 2025, 17(1): 188-236. doi: 10.3934/cam.2025009
[7]	Mustafa Avci . On an anisotropic $\overset{\rightarrow }{p}(\cdot)$ -Laplace equation with variable singular and sublinear nonlinearities. Communications in Analysis and Mechanics, 2024, 16(3): 554-577. doi: 10.3934/cam.2024026
[8]	Fangyuan Dong . Multiple positive solutions for the logarithmic Schrödinger equation with a Coulomb potential. Communications in Analysis and Mechanics, 2024, 16(3): 487-508. doi: 10.3934/cam.2024023
[9]	Ying Chu, Bo Wen, Libo Cheng . Existence and blow up for viscoelastic hyperbolic equations with variable exponents. Communications in Analysis and Mechanics, 2024, 16(4): 717-737. doi: 10.3934/cam.2024032
[10]	Ho-Sik Lee, Youchan Kim . Boundary Riesz potential estimates for parabolic equations with measurable nonlinearities. Communications in Analysis and Mechanics, 2025, 17(1): 61-99. doi: 10.3934/cam.2025004

Abstract

1. Introduction

Fractional calculus has been concerned with integration and differentiation of fractional (non-integer) order of the function. Riemann and Liouville defined the concept of fractional order intgro-differential equations ^[1]. Fractional calculus has developed an extensive attraction in current years in applied mathematics such as physics, medical, biology and engineering ^{[2,3,4,5,6,7,8]}. Whenever dealing with the fractional integro-differential equation many authors consider the terms Caputo fractional derivative, Riemann-Liouville and Grunwald-Letnikvo ^{[9,10,11,12,13]}. The subject fractional calculus has many applications in widespread and diverse field of science and engineering such as fractional dynamics in the trajectory control of redundant manipulators, viscoelasticity, electrochemistry, fluid mechanics, optics and signals processing etc.

Fractional integro-differential equations having some uncertainties in the form of boundary conditions, initial conditions and so on ^[14,15,16]. To resolve these type of uncertainties mathematicians introduced some concepts fuzzy set theory is one of them.

Zadeh introduced the concept of fuzzy set theory ^{[17,18,19,20]}. Later on Prade and Dubois ^[21,22], Nahmias ^[23], Tanaka and Mizumoto ^[24]. All of them experienced that the fuzzy number as a location of r-cut $0 \leqslant r \leqslant 1$ .

Many authors investigated some numerical techniques related to these problem which include the existence of the solution for discontinuous ^[25], reproducing kernel algorithm ^[26], integro-differential under generalized Caputo differentiability ^[27], A domain decomposition method ^[28], fractional differential transform method ^[29], Jacobi polynomial operational matrix ^[30], global solutions for nonlinear fuzzy equations ^[31], radioactivity decay model ^[32], Caputo-Katugampola fractional derivative approach ^[33], two-dimensional legendre wavelet method ^[34], fuzzy Laplace transform ^[35], fuzzy sumudu transform ^[36]. Further we can see ^{[37,38,39,40]}

Optimal Homotopy Asymptotic Method (OHAM) is one of the powerful techniques introduced by Marinca at al. ^[41,42,43] for approximate solution of differential equations. OHAM attracted an enormous importance in solving various problems in different field of science. Iqbal et al. applied this technique to Klein-Gordon equations and singular Lane-Emden type equation ^[44]. Sheikholeslami et al. used the proposed method for investigation of the laminar viscous flow and magneto hydrodynamic flow in a permeable channel ^[45]. Hashmi et al. obtained the solution of nonlinear Fredholm integral equations using OHAM ^[46]. Nawaz at al. applied the proposed method for solution of fractional order integro-differential equations ^[47], fractional order partial differential equations ^[48] and three-dimensional integral equations ^[49].

Aim of our study is to extend OHAM for solution of system of fuzzy Volterra integro differential equation of fractional order of the following form

$D_x^\alpha u(x) = h(x) + \mathop \smallint \limits_a^x k(x, t)u(t)dt, \, \, \, \, \, 0 \leqslant \alpha \leqslant 1, \, \, x \in [0, 1],$

(1.1)

with the given initial condition

${u^k}(0), \, \, \, \, {u_0}^k(x), \, \, \, k = 1, 2, 3, ...., \eta - 1, \, \, \, \eta - 1 < \alpha < \eta , \, \, \, \, \eta \in N,$

Where $D_x^\alpha$ represents the fuzzy fractional derivative in Caputo sense for fractional order of $\alpha$ with respect to $x$ , $h:[a, b] \to {\mathbb{R}_{\cal{F}}}$ is fuzzy valued function, $k(x, t)$ is arbitrary kernel ${u_0}(x) \in {\mathbb{R}_{\cal{F}}}$ is an unknown solution. ${\mathbb{R}_{\cal{F}}}$ represent set of all fuzzy valued function on real line.

The remaining paper is structured as follows: A brief overview on some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus are discussed in section 2. Analysis of the technique is presented in section 3. Proposed method is applied to solve fuzzy fractional order Volterra integro-differential equations in section 4. Result and discussion of the paper is given in section 5 and section 6 is the conclusion of the paper.

2. Preliminaries

In literature there exist various definitions of fuzzy calculus and fuzzy fractional calculus ^[50]. Some elementary concept, notations and definitions of fuzzy calculus and fuzzy fractional calculus related to this study are provided in this section.

Definition 2.1. The Riemann-Liouville fractional integral operator $I_x^\alpha$ of order $\alpha$ is ^[50]:

$I_x^\alpha u(x) = \, \left\{ \begin{array}{l} \frac{1}{{\Gamma (\alpha )}}\mathop \smallint \limits_0^x {(x - t)^{\alpha - 1}}u(t)dt = 0, \, \, \, \, \alpha > 0, \, \, \, \hfill \\ u(x), \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \alpha = 0.\, \, \hfill \\ \end{array} \right. \, \, \, \, \, \, \, \, \,$

(2.1)

Definition 2.2. Caputo partial fractional Derivative operator $D_x^\alpha$ of order $\alpha$ with respect to $x$ is defined as follow ^[50]:

$D_x^\alpha u(x) = \, \left\{ \begin{array}{l} \frac{1}{{\Gamma (\eta - \alpha )}}\mathop \smallint \limits_0^x {(x - t)^{\eta - \alpha - 1}}{u^{(n)}}(t)dt = 0, \, \, \, \, \, \, \, \, \, \, \, \eta - 1 < \alpha \leqslant \eta , \, \, \, \hfill \\ \frac{{{d^\eta }\, u(x)}}{{d{x^\eta }}}, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \alpha = \eta \in N.\, \, \hfill \\ \end{array} \right.$

(2.2)

which clearly shows that

$D_x^\alpha I_x^\alpha u(x) = u(x)$

(2.3)

Definition 2.3. A fuzzy number $\sigma$ is a mapping $\sigma :\mathbb{R} \to [0, 1]$ , satisfy the following property:

a. $\sigma$ is normal that is, $\exists \, \, {x_0} \in \mathbb{R}$ with $u({x_0}) = 1$ ^[51,52].

b. $\sigma$ is a convex fuzzy set that is, $u(\lambda x + (1 - \lambda )y) \geqslant \min \left\{ {u(x), u(y)} \right\}$ for all $x, y \in \mathbb{R}, {\text{ }}\lambda \in [0, 1].$

c. $\sigma$ is upper semi-continuous in $\mathbb{R}.$

d. $\overline {\left\{ {x \in \mathbb{R}:u(x) > 0} \right\}}$ is compact.

Definition 2.4. Parametric form of fuzzy number $\sigma$ represented by an order pair $(\underline \sigma , \bar \sigma )$ of the function $\left( {\underline \sigma (r), \bar \sigma (r)} \right)$ , satisfies the following conditions ^[52,53]:

a. $\underline \sigma (r)$ is bounded monotonic increasing left continuous $\forall \, \, r \in [0, 1].$

b. $\bar \sigma (r)$ is bounded monotonic decreasing left continuous $\forall \, \, r \in [0, 1].$

c. $\underline \sigma (r) \leqslant \bar \sigma (r)\, \, \forall \, \, r \in [0, 1].$

Definition 2.5. Addition and scalar multiplication of fuzzy number is given as:

a. $\left( {{\sigma _1} \oplus {\sigma _2}} \right) = \left( {{{\underline \sigma }_1}(r) + {{\underline \sigma }_2}(r), {{\bar \sigma }_1}(r) + {{\bar \sigma }_2}(r)} \right)$

b. $\left( {k \otimes \sigma } \right) = \left\{ \begin{array}{l} \left( {\underline \sigma (r), \bar \sigma (r)} \right), \, \, \, \, \, k \geqslant 0, \hfill \\ \left( {\underline \sigma (r), \bar \sigma (r)} \right), \, \, \, \, \, k < 0. \hfill \\ \end{array} \right.$

Definition 2.6. A fuzzy real valued function ${\sigma _1}, {\sigma _2}:[a, b] \to \mathbb{R},$ then in ^[54]:

${D_U}({\sigma _1}, {\sigma _2}) = \sup \left\{ {D({\sigma _1}(x), {\sigma _2}(x))|x \in [a, b]} \right\}.$

Definition 2.7. Assume $u:[a, b] \to {\mathbb{R}_{\cal{F}}}.$ For every partition ${\rm P} = \left\{ {{\sigma _0}, {\sigma _1}, {\sigma _2}, {\sigma _3}, ...., {\sigma _n}} \right\}$ and arbitrary ${\ell _i}:{\sigma _{i - 1}} \leqslant {\ell _i} \leqslant {\sigma _i}, {\text{ }}2 \leqslant i \leqslant n$ consider

${\mathbb{R}_p} = \mathop \Sigma \limits_{i = 2}^n u({\ell _j})({\sigma _i} - {\sigma _{i - 1}}).$ The definite integral of $u(x)$ over $[\alpha , \beta ]$ is

$\mathop \smallint \limits_\alpha ^\beta u(x)dx = \lim {\mathbb{R}_\rho },$

which show existence of limit in metric ^[55].

Definite integral exist if $u(x)$ is continuous in metric $D$ ^[51]:

$\begin{gathered} \left( {\underline {\mathop \smallint \limits_\alpha ^\beta u(x)dx} } \right) = \mathop \smallint \limits_\alpha ^\beta \underline u (x)dx, \hfill \\ \left( {\overline {\mathop \smallint \limits_\alpha ^\beta u(x)dx} } \right) = \mathop \smallint \limits_\alpha ^\beta \overline u (x)dxt. \hfill \\ \end{gathered}$

3. Application of OHAM

By considering definition 2.4. as discussed in section 2, Eq (1.1) becomes:

$\left\{ \begin{array}{l} D_x^\alpha \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \mathop \smallint \limits_a^x k(x, t)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (t, r)dt = 0, \hfill \\ D_x^\alpha \bar u(x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \mathop \smallint \limits_a^x k(x, t)\bar u(t, r)dt = 0, \hfill \\ \end{array} \right. \, \, \, \, \, 0 \leqslant \alpha \leqslant 1, \, 0 \leqslant r \leqslant 1, \, \, \, x \in [0, 1],$

(3.1)

with the given initial condition

${\left[ {{u^k}(0)} \right]^r}, \, \, \, \, \left( {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}^k(x, r), {{\bar u}_0}^k(x, r)} \right), \, \, \, k = 1, 2, 3, ...., \eta - 1, \, \, \, \eta - 1 < \alpha < \eta , \, \, \, \, \eta \in N,$

(3.2)

The homotopy of OHAM ^[41,42,43], constructed as follow:

$\left\{ \begin{array}{l} (1 - \rho )\left( {\frac{{{\partial ^\alpha }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho )}}{{\partial {t^\alpha }}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r)} \right) = H(\rho )\left( {\frac{{{\partial ^\alpha }\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho )}}{{\partial {t^\alpha }}} - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\delta } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } , r)} \right), \hfill \\ (1 - \rho )\left( {\frac{{{\partial ^\alpha }\bar \upsilon (x, r;\rho )}}{{\partial {t^\alpha }}} - \bar h(x, r)} \right) = H(\rho )\left( {\frac{{{\partial ^\alpha }\bar \upsilon (x, r;\rho )}}{{\partial {t^\alpha }}} - \bar h(x, r) - \bar \delta (\bar \upsilon , r)} \right). \hfill \\ \end{array} \right.$

(3.3)

where $\rho \in [0, 1]$ , $H(\rho ) = \sum\limits_{m\, \, \geqslant 1} {{c_m}{\rho ^m}}$ for all $\rho \ne 0$ is an auxiliary function, if $\rho = 0$ then $H(0) = 0$ where

${\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r, 0) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;1) = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} (x, r), \hfill \\ \bar \upsilon (x, r, 0) = {{\bar u}_0}(x, r)\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \bar \upsilon (x, r;1) = \bar u(x, r). \hfill \\ \end{array} \right. _{\, \, }}\,$

and ${c_m}$ represent auxiliary constants. Using Taylor's series to expand $\upsilon (x, r;\rho )$ about $\rho$ we get

$\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\upsilon } (x, r;\rho ) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) + \sum\limits_{m\, \, \geqslant 1} {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_m}(x, r){\rho ^m}} , \hfill \\ \bar \upsilon (x, r;\rho ) = {{\bar u}_0}(x, r) + \sum\limits_{m\, \, \geqslant 1} {{{\bar u}_m}(x, r){\rho ^m}} . \hfill \\ \end{array} \right.$

(3.4)

Inserting Eq (3.4) into Eq (3.3) we get series of the problems by comparing the like power of $\rho$ given as follow:

${\rho ^0}:\, \, \, \left\{ \begin{array}{l} \, {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) = 0, \hfill \\ \, {{\bar u}_0}(x, r) - \bar h(x, r) = 0. \hfill \\ \end{array} \right.$

(3.5)

${\rho ^1}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}) + (1 + {c_1}) + {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) = 0, \hfill \\ {{\bar u}_1}(x, r) + {c_1}\delta ({{\bar u}_0}) + (1 + {c_1}) + {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.6)

${\rho ^2}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_2}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}) + {c_2}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}) + {c_2}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r)) \hfill \\ - (1 + {c_1}){{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_1}(x, r) = 0, \hfill \\ {{\bar u}_2}(x, r) + {c_1}\delta ({{\bar u}_1}) + {c_2}\delta ({{\bar u}_0}) + {c_2}(\bar h(x, r) - {{\bar u}_0}(x, r)) \hfill \\ - (1 + {c_1}){{\bar u}_1}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.7)

${\rho ^n}:\, \, \, \, \left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}(x, r) + {c_1}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}) + {c_2}\delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_{n - 1}}) + {c_3}(h + \delta ({{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}))... - {c_2}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_{n - 1}}(x, r) - (1 + {c_1}){{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_n}(x, r) = 0, \hfill \\ {{\bar u}_n}(x, r) + {c_1}\delta ({{\bar u}_n}) + {c_2}\delta ({{\bar u}_{n - 1}}) + {c_3}(h + \delta ({{\bar u}_0}))... - {c_2}{{\bar u}_{n - 1}}(x, r) - (1 + {c_1}){{\bar u}_n}(x, r) = 0. \hfill \\ \end{array} \right.$

(3.8)

For calculating the constants ${c_1}, {c_2}, {c_3}...$ , m^th order optimum solution becomes

$\left\{ \begin{array}{l} {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }^m}(x, r, {c_l}) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_0}(x, r) + \sum\limits_{k = 1}^m {{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }_k}(x, r, {c_l}), } \, \, \, \, l = 1, 2, 3, ...m, \hfill \\ {{\bar u}^m}(x, r, {c_l}) = {{\bar u}_0}(x, r) + \sum\limits_{k = 1}^m {{{\bar u}_k}(x, r, {c_l}), } \, \, \, \, l = 1, 2, 3, ...m. \hfill \\ \end{array} \right.$

(3.9)

Putting Eq (3.9) into Eq (3.1), we can found our residual given as follow:

$\left\{ \begin{array}{l} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} \left( {x, r;{c_l}} \right) = {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} }^m}(x, r;{c_l}) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{h} (x, r) - \delta (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} ), \, \, l = 1, 2, ... \hfill \\ \bar R\left( {x, r;{c_l}} \right) = {{\bar u}^m}(x, r;{c_l}) - \bar h(x, r) - \delta (\bar u), \, \, l = 1, 2, ... \hfill \\ \end{array} \right.$

(3.10)

If $R\left( {x, r;{c_l}} \right) = 0,$ then ${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} ^m}(x, r;{c_l})\, \, \& \, \, {\bar u^m}(x, r;{c_l})$ will be the exact solutions.

Optimum solution contains some auxiliary constants; the optimal values of these constants are obtained through various techniques. In the present work, we have used the least square method ^[56,57]. The method of least squares is a powerful technique for obtaining the values of auxiliary constants. By putting the optimal values of these constants in Eq (8), we obtain the OHAM solution.

4. Application and accuracy

Problem 4.1. Consider system of fuzzy fractional order Volterra integro-differential equation as ^[58]:

$\left\{ \begin{array}{l} D_x^\alpha \underline u (x, r) = \left( {r - 1} \right) + \int\limits_0^x {\underline u (t, r)dt} \hfill \\ D_x^\alpha \bar u(x, r) = \left( {1 - r} \right) + \int\limits_0^x {\bar u(t, r)dt} \hfill \\ \end{array} \right. , \, \, \, \, 0 < \alpha \leqslant 1, \, \, x \in [0, 1],$

(4.1)

subject to the fuzzy initial condition ${\left[ {u(0)} \right]^r} = \left[ {r - 1, 1 - r} \right],$ and for $\alpha = 1$ fuzzy fractional order Volterra integro-differential equations the exact solution is ${\left[ {u(x)} \right]^r} = \left[ {r - 1, 1 - r} \right]Sinh(x)$ and $0 \leqslant r \leqslant 1.$

By follow the technique as discussed in section 3, we get series of problems and their solutions as:

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _0}(x, r) + (1 - r) = 0, \hfill \\ {D_x}^\alpha {\overline u _0}(x, r) + (r - 1) = 0. \hfill \\ \end{array} \right.$

(4.2)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _1}(x, r) - 1 + r - {c_1} + r{c_1} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {\underline u _0}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) + 1 - r + {c_1} - r{c_1} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {{\bar u}_0}(x, r) - {c_1}{D_x}^\alpha \bar u(x, r) = 0. \hfill \\ \end{array} \right.$

(4.3)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_1} - {c_2} + r{c_2} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {\underline u _0}(x, r) - \hfill \\ {D_x}^\alpha {\underline u _1}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_1} + {c_2} - r{c_2} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {{\bar u}_0}(x, r) - \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.4)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _3}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_2}(t, r)dt} \right){c_1} + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_2} - {c_3} + r{c_3} + \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {\underline u _0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {\underline u _1}(x, r) - {D_x}^\alpha {\underline u _2}(x, r) - {c_1}{D_x}^\alpha {\underline u _2}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_3}(x, r) + \left( {\mathop \smallint \limits_0^x {{\bar u}_2}(t, r)dt} \right){c_1} + \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_2} + {c_3} - r{c_3} + \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {{\bar u}_0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {{\bar u}_1}(x, r) - {D_x}^\alpha {{\bar u}_2}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_2}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.5)

Their solutions are

$\left\{ \begin{array}{l} {\underline u _0}(x, r) = \frac{{( - 1 + r){x^\alpha }}}{{\alpha \, \Gamma (\alpha )}} \hfill \\ {{\bar u}_0}(x, r) = - \frac{{( - 1 + r){x^\alpha }}}{{\alpha \, \Gamma (\alpha )}} \hfill \\ \end{array} \right. ,$

(4.6)

$\left\{ \begin{array}{l} {\underline u _1}(x, r) = - \frac{{( - 1 + r){x^{1 + 2\alpha }}{c_1}}}{{\Gamma (2 + 2\alpha )}}, \hfill \\ {{\bar u}_1}(x, r) = \frac{{( - 1 + r){x^{1 + 2\alpha }}{c_1}}}{{\Gamma (2 + 2\alpha )}}. \hfill \\ \end{array} \right.$

(4.7)

$\left\{ \begin{array}{l} {\underline u _2}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( {\frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 3\alpha )}} - \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + 2\alpha )}}} \right), \hfill \\ {{\bar u}_2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 3\alpha )}} + \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + 2\alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.8)

$\left\{ \begin{array}{l} {\underline u _3}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( { - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} + \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}} - \frac{{{c_1} + 2c_1^2 + c_1^3 + {c_2} + 2{c_1}{c_2} + {c_3}}}{{\Gamma (2 + 2\alpha )}}} \right), \hfill \\ {{\bar u}_3}(x, r) = ( - 1 + r){x^{1 + 2\alpha }}\left( {\frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} - \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}} + \frac{{{c_1} + 2c_1^2 + c_1^3 + {c_2} + 2{c_1}{c_2} + {c_3}}}{{\Gamma (2 + 2\alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.9)

Adding (4.6), (4.7), (4.8) and (4.9), one can construct $\underline u (x, r)$ & $\bar u(x, r)$ :

$\left\{ \begin{array}{l} \underline u (x, r) = ( - 1 + r){x^\alpha }\left( \begin{gathered} \frac{1}{{\Gamma (1 + \alpha )}} - \frac{{{x^{3 + 3\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} + \frac{{{x^{2 + 2\alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 3\alpha )}} - \hfill \\ \frac{{{x^{1 + \alpha }}(2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3})}}{{\Gamma (2 + 2\alpha )}} \hfill \\ \end{gathered} \right), \hfill \\ \bar u(x, r) = ( - 1 + r){x^\alpha }\left( \begin{gathered} - \frac{1}{{\Gamma (1 + \alpha )}} + \frac{{{x^{3 + 3\alpha }}c_1^3}}{{\Gamma (4 + 4\alpha )}} - \frac{{{x^{2 + 2\alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 3\alpha )}} + \hfill \\ \frac{{{x^{1 + \alpha }}(2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3})}}{{\Gamma (2 + 2\alpha )}} \hfill \\ \end{gathered} \right). \hfill \\ \end{array} \right.$

(4.10)

Values of ${c_1}, \, \, {c_2}$ and ${c_3}$ contain is in Eq (4.10)

Substituting the values from into Eq (4.10), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.75$ respectively is as follow

$\alpha = 0.7$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2751369{x}^{0.7}-0.08602097{x}^{2.4}-0.25{x}^{2.4}(-0.00904995+0.0376716{x}^{1.7})\\ -0.25{x}^{2.4}(0.000425859-0.00198165{x}^{1.7}+0.0021734872{x}^{3.4}), \\ \overline{u}(x, r)\approx 0.2751369{x}^{0.7}+0.08602097{x}^{2.4}-0.25{x}^{2.4}(0.00904995-0.0376716{x}^{1.7})-\\ 0.25{x}^{2.4}(-0.000425859+0.00198165{x}^{1.7}-0.0021734872{x}^{3.4}).\end{array} \right.$

(4.11)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2684178{x}^{0.8}-0.0685589{x}^{2.6}-0.25{x}^{2.6}(-0.005391327+0.023297809{x}^{1.8})\\ -0.25{x}^{2.6}(0.000197513-0.0009160448{x}^{1.8}+0.001008410504{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.2684178{x}^{0.8}+0.0685589{x}^{2.6}-0.25{x}^{2.6}(0.0053913-0.023297809{x}^{1.8})\\ -0.25{x}^{2.6}(-0.000197513+0.0009160448{x}^{1.8}-0.001008410504{x}^{3.6}).\end{array} \right.$

(4.12)

$\alpha = 0.9$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.2599385{x}^{0.9}-0.0540269{x}^{2.8}-0.25{x}^{2.8}(-0.0031639+0.01418901{x}^{1.9})\\ -0.25{x}^{2.8}(0.00008989-0.0004154745{x}^{1.9}+0.000458881{x}^{3.8}), \\ \overline{u}(x, r)\approx 0.2599385{x}^{0.9}+0.0540269{x}^{2.8}-0.25{x}^{2.8}(0.0031639-0.014189097{x}^{1.9})\\ -0.25{x}^{2.8}(-0.00008989+0.0004154745{x}^{1.9}-0.000458881{x}^{3.8}).\end{array} \right.$

(4.13)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.25x-0.042114377{x}^{3}-0.25{x}^{3}(-0.001829736+0.0085133796{x}^{2})\\ -0.25{x}^{3}(0.0000401535-0.0001849397{x}^{2}+0.00020487753{x}^{4}), \\ \overline{u}(x, r)\approx 0.25x+0.042114377{x}^{3}-0.25{x}^{3}(0.001829736-0.0085133796{x}^{2})\\ -0.25{x}^{3}(-0.0000401535+0.0001849397{x}^{2}-0.0002048775{x}^{4}).\end{array} \right.$

(4.14)

Table 1. at r = 0.75.

$\alpha$	${\underline c _1}$ & ${\overline c _1}$	${\underline c _2}$ & ${\overline c _2}$	${\underline c _3}$ & ${\overline c _3}$
0.7	−1.0257850714449026	5.298291106236844×10⁻⁴	−3.040859671410477×10⁻⁵
0.8	−1.0193406988378892	3.249294721058776×10⁻⁴	−1.5364294422415488×10⁻⁵
0.9	−1.014446487354385	1.9694362983845834×10⁻⁴	−7.63055570435551×10⁻⁶
1	−1.0107450504316333	1.1791102776455743×10⁻⁴	−3.779171763451589×10⁻⁶

| Show Table

DownLoad: CSV

Substituting the values from into Eq (4.10), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.5$ respectively is as follow

$\alpha = 0.7$

$\left\{ \begin{array}{l} \underline u (x, r) \approx - 0.5502737{x^{0.7}} - 0.172041940{x^{2.4}} - 0.5{x^{2.4}}( - 0.00904995 + 0.0376716{x^{1.7}}) \hfill \\ - 0.5{x^{2.4}}(0.0004258594 - 0.0019816476{x^{1.7}} + 0.0021734872{x^{3.4}}), \hfill \\ \bar u(x, r) \approx 0.5502737{x^{0.7}} + 0.172041940{x^{2.4}} - 0.5{x^{2.4}}(0.00904995 - 0.03767164{x^{1.7}}) \hfill \\ - 0.5{x^{2.4}}( - 0.0004258594 + 0.0019816476{x^{1.7}} - 0.0021734872{x^{3.4}}). \hfill \\ \end{array} \right.$

(4.15)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.53683564{x}^{0.8}-0.137117858{x}^{2.6}-0.5{x}^{2.6}(-0.00539133+0.02329781{x}^{1.8})\\ -0.5{x}^{2.6}(0.00019751-0.0009160448{x}^{1.8}+0.0010084105{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.53683564{x}^{0.8}+0.137117858{x}^{2.6}-0.5{x}^{2.6}(0.00539133-0.02329781{x}^{1.8})\\ -0.5{x}^{2.6}(-0.00019751+0.0009160448{x}^{1.8}-0.0010084105{x}^{3.6}).\end{array} \right.$

(4.16)

$\alpha = 0.9$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5198771{x}^{0.9}-0.10805378{x}^{2.8}-0.5{x}^{2.8}(-0.0031640+0.0141891{x}^{1.9})\\ -0.5{x}^{2.8}(0.0000898945-0.0004154745{x}^{1.9}+0.000458881{x}^{3.8}), \\ \overline{u}(x, r)\approx 0.5198771{x}^{0.9}+0.10805378{x}^{2.8}-0.5{x}^{2.8}(0.0031640-0.0141891{x}^{1.9})\\ -0.5{x}^{2.8}(-0.0000898945+0.0004154745{x}^{1.9}-0.000458881{x}^{3.8}).\end{array} \right.$

(4.17)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5x-0.0842288{x}^{3}-0.5{x}^{3}(-0.0018298+0.008513{x}^{2})-0.5{x}^{3}\\ (0.0000401612-0.00018494622{x}^{2}+0.0002048777{x}^{4}), \\ \overline{u}(x, r)\approx 0.5x+0.0842288{x}^{3}-0.5{x}^{3}(0.0018298-0.008513{x}^{2})-0.5{x}^{3}\\ (-0.000040162+0.00018494622{x}^{2}-0.0002048777{x}^{4}).\end{array} \right.$

(4.18)

Table 2. at r = 0.5.

$\alpha$	${\underline c _1}$ & ${\overline c _1}$	${\underline c _2}$ & ${\overline c _2}$	${\underline c _3}$ & ${\overline c _3}$
0.7	−1.0257850714449026	5.298291106236844×10⁻⁴	−3.040859671410477×10⁻⁵
0.8	−1.0193406988378892	3.249294721058776×10⁻⁴	−1.5364294422415488×10⁻⁵
0.9 1	−1.014446487354385 −1.0107453381292266	1.9694362983845834×10⁻⁴ 1.1800167363027721×10⁻⁴	−7.630555570435551×10⁻⁶ −3.726389827252244×10⁻⁶

| Show Table

DownLoad: CSV

Problem 4.2. Consider system of fuzzy fractional order Volterra integro-differential equation as ^[59]:

$D_x^\alpha u(x, r) + \int\limits_0^t {u(t, r)dt = 0} , \, \, \, \, 0 < \alpha \leqslant 1, \, \, x \in [0, 1],$

(4.19)

subject to the fuzzy initial condition ${\left[ {u(0)} \right]^r} = \left[ {r - 1, 1 - r} \right],$ and the exact solution is $\underline u (x, r) = \left( {r - 1} \right){E_{\alpha + 1}}\left( { - {t^{\alpha + 1}}} \right), \bar u(x, r) = \left( {1 - r} \right){E_{\alpha + 1}}\left( { - {t^{\alpha + 1}}} \right),$

where ${E_{\alpha + 1}}$ is a Mittag-Leffler function and $0 \leqslant r \leqslant 1.$

By follow the technique as discussed in section 3, we get series of problems and their solutions as:

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.20)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _1}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {\underline u _0}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_1}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_1} - {D_x}^\alpha {{\bar u}_0}(x, r) - {c_1}{D_x}^\alpha \bar u(x, r) = 0. \hfill \\ \end{array} \right.$

(4.21)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _2}(x, r) + \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {\underline u _0}(x, r) - {D_x}^\alpha {\underline u _1}(x, r) - {c_1}{D_x}^\alpha {\underline u _0}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_2}(x, r) + \left( { - \mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_2} - {c_2}{D_x}^\alpha {{\bar u}_0}(x, r) - {D_x}^\alpha {{\bar u}_1}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_0}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.22)

$\left\{ \begin{array}{l} {D_x}^\alpha {\underline u _3}(x, r) - \left( {\mathop \smallint \limits_0^x {{\underline u }_2}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\underline u }_1}(t, r)dt} \right){c_2} - \left( {\mathop \smallint \limits_0^x {{\underline u }_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {\underline u _0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {\underline u _1}(x, r) - {D_x}^\alpha {\underline u _2}(x, r) - {c_1}{D_x}^\alpha {\underline u _2}(x, r) = 0, \hfill \\ {D_x}^\alpha {{\bar u}_3}(x, r) - \left( {\mathop \smallint \limits_0^x {{\bar u}_2}(t, r)dt} \right){c_1} - \left( {\mathop \smallint \limits_0^x {{\bar u}_1}(t, r)dt} \right){c_2} - \left( {\mathop \smallint \limits_0^x {{\bar u}_0}(t, r)dt} \right){c_3} - {c_3}{D_x}^\alpha {{\bar u}_0}(x, r) \hfill \\ - {c_2}{D_x}^\alpha {{\bar u}_1}(x, r) - {D_x}^\alpha {{\bar u}_2}(x, r) - {c_1}{D_x}^\alpha {{\bar u}_2}(x, r) = 0. \hfill \\ \end{array} \right.$

(4.23)

And their solutions are

$\left\{ \begin{array}{l} {\underline u _0}(x, r) = r - 1, \hfill \\ {{\bar u}_0}(x, r) = 1 - r. \hfill \\ \end{array} \right.$

(4.24)

$\left\{ \begin{array}{l} {\underline u _1}(x, \underline r ) = \frac{{( - 1 + r){x^{1 + \alpha }}{c_1}}}{{(\alpha + {\alpha ^2})\Gamma (\alpha )}}, \hfill \\ {\overline u _1}(x, \overline r ) = - \frac{{( - 1 + r){x^{1 + \alpha }}{c_1}}}{{\alpha (1 + \alpha )\Gamma (\alpha )}}, \hfill \\ \end{array} \right.$

(4.25)

$\left\{ \begin{array}{l} {\underline u _2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( {\frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} + \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + \alpha )}}} \right), \hfill \\ {{\bar u}_2}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} - \frac{{{c_1} + c_1^2 + {c_2}}}{{\Gamma (2 + \alpha )}}} \right). \hfill \\ \end{array} \right.$

(4.26)

$\left\{ \begin{array}{l} {\underline u _3}(x, r) = \frac{{( - 1 + r){x^{1 + \alpha }}\left( {{c_2} + {c_1}\left( {1 + \frac{{{x^{2 + 2\alpha }}c_1^2}}{{\Gamma (4 + 2\alpha )}} + {c_1}(2 + {c_1}) + 2{c_2} + \frac{{2{x^{1 + \alpha }}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 3\alpha )}}} \right) + {c_3}} \right)}}{{\Gamma (1 + \alpha )}}, \hfill \\ {{\bar u}_3}(x, r) = ( - 1 + r){x^{1 + \alpha }}\left( { - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 3\alpha )}} - \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (3 + 2\alpha )}} - \frac{{{c_2} + {c_1}({{(1 + {c_1})}^2} + 2{c_2}) + {c_3}}}{{\Gamma (2 + \alpha )}}} \right), \hfill \\ \end{array} \right.$

(4.27)

Adding (4.24), (4.25), (4.26) and (4.27), one can construct $\underline u (x, r)$ & $\bar u(x, r)$ :

$\left\{ \begin{array}{l} \underline u (x, r) = - 1 + r + ( - 1 + r){x^{1 + \alpha }}\left( \begin{gathered} \frac{{{x^{1 + \alpha }}c_1^2}}{{\Gamma (3 + 2\alpha )}} + \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (1 + \alpha )\Gamma (4 + 2\alpha )}} + \frac{{2{x^{1 + \alpha }}{c_1}({c_1} + c_1^2 + {c_2})}}{{\Gamma (1 + \alpha )\Gamma (3 + \alpha )}} \hfill \\ + \frac{{{c_1}(2 + {c_1}) + {c_2}}}{{\Gamma (2 + \alpha )}} + \frac{{{c_2} + {c_1}({{(1 + {c_1})}^2} + 2{c_2}) + {c_3}}}{{\Gamma (1 + \alpha )}} \hfill \\ \end{gathered} \right), \hfill \\ \bar u(x, r) = 1 - r + ( - 1 + r){x^{1 + \alpha }}\left( \begin{gathered} - \frac{{{x^{2 + 2\alpha }}c_1^3}}{{\Gamma (4 + 3\alpha )}} - \frac{{{x^{1 + \alpha }}{c_1}({c_1}(3 + 2{c_1}) + 2{c_2})}}{{\Gamma (3 + 2\alpha )}} - \hfill \\ \frac{{2{c_2} + {c_1}(3 + {c_1}(3 + {c_1}) + 2{c_2}) + {c_3}}}{{\Gamma (2 + \alpha )}} \hfill \\ \end{gathered} \right). \hfill \\ \end{array} \right.$

(4.28)

Values of ${c_1}, \, \, {c_2}$ and ${c_3}$ contain in Eq (4.28)

Substituting the values from and into Eq (4.28), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $\alpha$ taking $r = 0.5$ is as follow

$\alpha = 0.2$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.2}(-0.905948+0.325139{x}^{1.2}-0.055807{x}^{2.4}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.2}(0.905948-0.325139{x}^{1.2}+0.055807{x}^{2.4}).\end{array} \right.$

(4.29)

$\alpha = 0.4$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.4}(-0.804545+0.210093{x}^{1.4}-0.025570{x}^{2.8}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.4}(0.80454545-0.210093{x}^{1.4}+0.025570{x}^{2.8}).\end{array} \right.$

(4.30)

$\alpha = 0.6$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.6}(-0.699349+0.128177{x}^{1.6}-0.010450{x}^{3.2}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.6}(0.699349-0.128177{x}^{1.6}+0.010450{x}^{3.2}).\end{array} \right.$

(4.31)

$\alpha = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{1.8}(-0.596450+0.074561{x}^{1.8}-0.0038863{x}^{3.6}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{1.8}(0.5964503-0.074561{x}^{1.8}+0.0038863{x}^{3.6}).\end{array} \right.$

(4.32)

$\alpha = 1$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.5-0.5{x}^{2}(-0.499992+0.04163089{x}^{2}-0.001336253{x}^{4}), \\ \overline{u}(x, r)\approx 0.5-0.5{x}^{2}(0.4999914-0.04163076{x}^{2}+0.001336247{x}^{4}).\end{array} \right.$

(4.33)

Table 3. at r = 0.5.

$\alpha$	${\underline c _1}$	${\underline c _2}$	${\underline c _3}$
0.2	−0.8038238618267683	7.6178003377104005×10⁻³	−1.334733828882206×10⁻³
0.4	−0.739676946061329	0.02530379950927192	−3.78307542584476×10⁻³
0.6	−0.6725325561865596	0.04771922184372877	−8.140310215705777×10⁻³
0.8	−0.6062340661192892	0.06889015391501904	−0.015746150088743013
1.0	−0.5432795308983783	0.08615024033359142	−0.026582381449582644

| Show Table

DownLoad: CSV

Table 4. at r = 0.5.

$\alpha$	${\overline c _1}$	${\overline c _2}$	${\overline c _3}$
0.2	−0.9072542694138958	3.5727393445527333×10⁻³	3.637119200533134×10⁻⁴
0.4	−0.9409187563211361	1.9803045412863643×10⁻³	1.7807647588483674×10⁻⁴
0.6	−0.9636043521097131	9.808463578935658×10⁻⁴	7.369503633679291×10⁻⁵
0.8	−0.9781782948007163	4.4492367269725435×10⁻⁴	2.6729929232143615×10⁻⁵
1.0	−0.9872029432879605	1.896941835601795×10⁻⁴	1.021455544474314×10⁻⁵

| Show Table

DownLoad: CSV

Substituting the values from and into Eq (4.28), the approximate solutions for $\underline u (x, r)$ & $\bar u(x, r)$ at different values of $r$ taking $\alpha = 0.5$ is as follow

$r = 0$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -1-{x}^{1.5}(-0.75199032254+0.165168588631{x}^{1.5}-0.016559344247{x}^{3.}), \\ \overline{u}(x, r)\approx 1-{x}^{1.5}(0.75199032256-0.165168588653{x}^{1.5}+0.016559344262{x}^{3.}).\end{array} \right.$

(4.34)

$r = 0.2$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.8-0.8{x}^{1.5}(-0.751990322550+0.165168588{x}^{1.5}-0.01655934425{x}^{3.}), \\ \overline{u}(x, r)\approx 0.8-0.8{x}^{1.5}(0.751990322554-0.165168588{x}^{1.5}+0.01655934426{x}^{3.}). \end{array} \right.$

(4.35)

$r = 0.4$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.6-0.6{x}^{1.5}(-0.7519903225+0.1651685886{x}^{1.5}-0.016559344250{x}^{3.}), \\ \overline{u}(x, r)\approx 0.6-0.6{x}^{1.5}(0.7519903225-0.1651685886{x}^{1.5}+0.0165593442496{x}^{3.}).\end{array} \right.$

(4.36)

$r = 0.6$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.4-0.4{x}^{1.5}(-0.751990322550+0.16516858863{x}^{1.5}-0.01655934425{x}^{3.}), \\ \overline{u}(x, r)\approx 0.4-0.4{x}^{1.5}(0.751990322554-0.16516858865{x}^{1.5}+0.01655934426{x}^{3.}).\end{array} \right.$

(4.37)

$r = 0.8$

$\left\{ \begin{array}{l} \underset{\_}{u}(x, r)\approx -0.1999910-0.1999910{x}^{1.5}(-0.7519903+0.16516859{x}^{1.5}-0.0165593{x}^{3.}), \\ \overline{u}(x, r)\approx 0.19999910-0.19999910{x}^{1.5}(0.7519903-0.16516859{x}^{1.5}+0.0165593{x}^{3.}).\end{array} \right.$

(4.38)

Table 5. at

$\alpha$ = 0.5.

$r$	${\underline c _1}$	${\underline c _2}$	${\underline c _3}$
$0$	$- 0.7062087686601037$	$0.03638991875243272$	$- 5.6065011933001474 \times {10^{ - 3}}$
$0.2$	$- 0.7062087687083373$	$0.03638991875755982$	$- 5.606501189519195 \times {10^{ - 3}}$
$0.4$	$- 0.7062087686911187$	$0.03638991875566347$	$- 5.606501190876118 \times {10^{ - 3}}$
$0.6$	$- 0.7062087687083373$	$0.03638991875755982$	$- 5.606501189519195 \times {10^{ - 3}}$
$0.8$	$- 0.7062087686312865$	$0.036389918749298394$	$- 5.606501195566148 \times {10^{ - 3}}$

| Show Table

DownLoad: CSV

Table 6. at

$\alpha$ = 0.5.

$r$	${\overline c _1}$	${\overline c _2}$	${\overline c _3}$
$0$	$- 0.9534544876637709$	$1.4110239362094313 \times {10^{ - 3}}$	$1.1669890178958486 \times {10^{ - 4}}$
$0.2$	$- 0.9534544876175205$	$1.4110239407705756 \times {10^{ - 3}}$	$1.1669890245071123 \times {10^{ - 4}}$
$0.4$	$- 0.9534544874104965$	$1.4110239614371703 \times {10^{ - 3}}$	$1.1669890547775563 \times {10^{ - 4}}$
$0.6$	$- 0.9534544876175205$	$1.4110239407705756 \times {10^{ - 3}}$	$1.1669890245071123 \times {10^{ - 4}}$
$0.8$	$- 0.9534544876289686$	$1.4110239398903034 \times {10^{ - 3}}$	$1.1669890235330204 \times {10^{ - 4}}$

| Show Table

DownLoad: CSV

5. Result and discussion

– show the values of auxiliary constant at different values of $r$ & $\alpha$ for both lower and upper solution of OHAM for the solved problems. and show the comparison of absolute error of 3^rd order OHAM with Fractional Residual Power Series (FRPS) Method for 5-approximated solution and $k = 5$ for both lower and upper solutions of OHAM at different value of $\alpha$ for problem 1. Comparison of absolute error of 3^rd orders OHAM for both lower and upper solution of OHAM are shown in and . Numerical result show that OHAM provide more accuracy as compared to the other method and as $\alpha \to 1$ the approximate solution become very close to the exact solution. Graphical representation confirmed the convergence of fractional order solution towards the integer order solution. In graphical representation of OHAM at $\alpha = 0.7, \, \, 0.8, \, \, 0.9\, , \, \, 1, \, \, r = 0.75$ and $\alpha = 0.7, \, \, 0.8, \, \, 0.9\, , \, \, 1, \, \, r = 0.50$ are discussed for both $\underline u (x, r)$ & $\bar u(x, r)$ for problem 1. Figures 2 and 3 show the comparison of OHAM with the exact solution at different values of and taking r = 0.75 & r = 0.5 respectively for problem 1. represent the comparison of OHAM at $\alpha = 0.2, \, 0.4, \, \, 0.6, \, \, 0.8, \, \, 1, \, \, r = 0.5$ and $r = 0, \, \, 0.2, \, 0.4, \, \, 0.6, \, \, 0.8, \, \, \alpha \, = 0.5$ for both $\underline u (x, r)$ and $\bar u(x, r)$ for problem 2. Figure 5 shows the comparison of OHAM with the exact solution at different values of and r = 0.5 while Figure 6 shows the comparison of OHAM with the exact solution at different values of r and = 0.5 for problem 2.

Table 7. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, r)$ and Fractional Residual Power Series (FRPS) ^[54] Method for 5-approximated solution and

$k = 5$ for problem 1.

$r$	$x$	FRPS ^[58] $\alpha = 0.7$	OHAM	FRPS ^[58] $\alpha = 0.8$	OHAM	FRPS ^[58] $\alpha = 0.9$	OHAM	FRPS ^[58] $\alpha = 1$	OHAM
0.75	0.2	0.042797	0.040621	0.025514	0.024764	0.011512	0.011321	6.35273×10⁻¹⁰	2.14676×10⁻⁹
	0.4	0.059664	0.051698	0.035840	0.032584	0.016392	0.015405	8.14507×10⁻⁸	1.00832×10⁻⁸
	0.6	0.075769	0.058997	0.045171	0.037635	0.020545	0.018031	1.39554×10⁻⁶	1.11336×10⁻⁸
	0.8	0.094364	0.066136	0.055863	0.04232	0.025182	0.020362	1.04955×10⁻⁵	6.21567×10⁻⁹
0.50	0.2	0.085595	0.081241	0.051027	0.049528	0.011321	0.022643	1.27055×10⁻⁹	4.25946×10⁻⁹
	0.4	0.119328	0.103396	0.071680	0.065167	0.015405	0.030811	1.62901×10⁻⁷	1.98877×10⁻⁸
	0.6	0.151537	0.117994	0.090342	0.075269	0.018031	0.036062	2.79107×10⁻⁶	2.12923×10⁻⁸
	0.8	0.188728	0.132271	0.111723	0.084640	0.020362	0.040725	2.09911×10⁻⁵	1.00120×10⁻⁸

| Show Table

DownLoad: CSV

Table 8. Comparison of Absolute Error (Abs Err.) of 3rd order OHAM for

$\bar u(x, r)$ and Fractional Residual Power Series (FRPS) ^[58] Method for 5-approximated solution and

$k = 5$ for problem 1.

$r$	$x$	FRPS ^[58] $\alpha = 0.7$	OHAM	FRPS ^[58] $\alpha = 0.8$	OHAM	FRPS ^[58] $\alpha = 0.9$	OHAM	FRPS ^[58] $\alpha = 1$	OHAM
0.75	0.2	0.085595	0.081242	0.025514	0.024764	0.011512	0.011321	6.35273×10⁻¹⁰	2.14676×10⁻⁹
	0.4	0.119328	0.103396	0.035840	0.032584	0.016392	0.015405	8.14507×10⁻⁸	1.00832×10⁻⁸
	0.6	0.151537	0.117994	0.045171	0.037635	0.020545	0.018031	1.39554×10⁻⁶	1.11336×10⁻⁸
	0.8	0.188728	0.132271	0.055862	0.04232	0.025182	0.020362	1.04955×10⁻⁵	6.21567×10⁻⁹
0.50	0.2	0.042797	0.040621	0.051027	0.049528	0.023025	0.022643	1.27055×10⁻⁹	4.25946×10⁻⁹
	0.4	0.059664	0.051698	0.071680	0.065167	0.032784	0.030811	1.62901×10⁻⁷	1.98877×10⁻⁸
	0.6	0.075769	0.058997	0.090342	0.075269	0.041090	0.036062	2.79107×10⁻⁶	2.12923×10⁻⁸
	0.8	0.094364	0.066136	0.111723	0.084640	0.050364	0.040725	2.09911×10⁻⁵	1.0012×10⁻⁸

| Show Table

DownLoad: CSV

Table 9. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, \underline r )$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.5$ for problem 2.

$x$	$\underline u (x, r)$ $\alpha = 0.4$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 0.6$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 0.8$	$\bar u(x, r)$	$\underline u (x, r)$ $\alpha = 1$	$\bar u(x, r)$
0.2	1.2736×10⁻⁵	1.2736×10⁻⁵	3.2527×10⁻⁶	3.2527×10⁻⁶	6.93075×10⁻⁷	6.93076×10⁻⁷	1.29476×10⁻⁷	1.44585×10⁻⁷
0.4	2.3337×10⁻⁶	2.3337×10⁻⁶	2.4426×10⁻⁶	2.4426×10⁻⁶	9.50573×10⁻⁷	9.50573×10⁻⁷	2.67538×10⁻⁷	3.26771×10⁻⁷
0.6	9.9993×10^-6	9.9993×10⁻⁶	1.8451×10⁻⁶	1.8451×10⁻⁶	3.8544×10⁻⁸	3.85438×10⁻⁸	1.10198×10⁻⁷	2.39031×10⁻⁷
0.8	4.1251×10⁻⁶	4.1251×10⁻⁶	1.3416×10⁻⁷	1.3416×10⁻⁷	6.55017×10⁻⁸	6.55017×10⁻⁸	9.64628×10⁻⁹	2.27883×10⁻⁷
1.0	1.312×10⁻⁵	1.312×10⁻⁵	1.8373×10⁻⁶	1.8373×10⁻⁶	5.49341×10⁻⁸	5.4934×10⁻⁸	7.7356×10⁻⁸	3.9735×10⁻⁷

| Show Table

DownLoad: CSV

Table 10. Comparison of Absolute Error (Abs Err.) of 3^rd order OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ taking

$\alpha = 0.5$ for problem 2.

$x$	$\underline u (x, r)$ $r = 0.4$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 0.6$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 0.8$	$\bar u(x, r)$	$\underline u (x, r)$ $r = 1$	$\bar u(x, r)$
0.	0.	0.	0.	0.	0.	0.	0.	0.
0.2	1.0578×10⁻⁵	1.0578×10⁻⁵	7.9338×10⁻⁶	7.9338×10⁻⁶	5.28921×10⁻⁶	5.28921×10⁻⁶	2.64461×10⁻⁶	2.64461×10⁻⁶
0.4	4.8940×10⁻⁶	4.8940×10⁻⁶	3.6705×10⁻⁶	3.6705×10⁻⁶	2.44698×10⁻⁶	2.44698×10⁻⁶	1.22349×10⁻⁶	1.22349×10⁻⁶
0.6	7.4825×10⁻⁶	7.4825×10⁻⁶	5.6119×10⁻⁶	5.6119×10⁻⁶	3.74125×10⁻⁶	3.74125×10⁻⁶	1.87062×10⁻⁶	1.87062×10⁻⁶
1.8	1.8038×10⁻⁶	1.8038×10⁻⁶	1.35291×10⁻⁶	1.35291×10⁻⁶	9.01939×10⁻⁷	9.01939×10⁻⁷	4.5097×10⁻⁷	4.5097×10⁻⁷

| Show Table

DownLoad: CSV

Figure 1. Solution plot of OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ &

$\alpha$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 2. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.75$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 3. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.50$ for problem 1.

DownLoad: Full-Size Img PowerPoint

Figure 4. Solution plot of OHAM for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ &

$\alpha$ for problem 2.

DownLoad: Full-Size Img PowerPoint

Figure 5. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$\alpha$ taking

$r = 0.5$ for problem 2.

DownLoad: Full-Size Img PowerPoint

Figure 6. Solution plot of OHAM and Exact for

$\underline u (x, r)$ &

$\bar u(x, r)$ at different values of

$r$ taking

$\alpha = 0.5$ for problem 2.

DownLoad: Full-Size Img PowerPoint

6. Conclusions

In the research paper, a powerful technique known as Optimal Homotopy Asymptotic Method (OHAM) has been extended to the solution of system of fuzzy integro differential equations of fractional order. The obtained results are quite interesting and are in good agreement with the exact solution. Two numerical equations are taken as test examples which show the behavior and reliability of the proposed method. The extension of OHAM to system of fuzzy integro differential equations of fractional order is more accurate and as a result this technique will more appealing for the researchers for finding out optimum solutions of system of fuzzy integro differential equations of fractional order.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	A. Ala, M. Deveci, E. A. Bani, A. H. Sadeghi, Dynamic capacitated facility location problem in mobile renewable energy charging stations under sustainability consideration, Sustain. Comput. Infor., 41 (2024), 100954. https://doi.org/10.1016/j.suscom.2023.100954 doi: 10.1016/j.suscom.2023.100954
[2]	N. Aydin, A. Murat, B. S. Mordukhovich, Sample intelligence-based progressive hedging algorithms for the stochastic capacitated reliable facility location problem, Artif. Intell. Rev., 57 (2024), 135.
[3]	M. Bieniek, A note on the facility location problem with stochastic demands, Omega, 55 (2015), 53–60. https://doi.org/10.1016/j.omega.2015.02.006 doi: 10.1016/j.omega.2015.02.006
[4]	M. Chica, A. A. Juan, C. Bayliss, O. Cordón, W. D. Kelton, Why simheuristics? benefits, limitations, and best practices when combining metaheuristics with simulation, SORT-Stat. Oper. Res. T., 44 (2020), 311–334.
[5]	B. R. Cobb, R. Rumí, A. Salmerón, Inventory management with log-normal demand per unit time, Comput. Oper. Res., 40 (2013), 1842–1851. https://doi.org/10.1016/j.cor.2013.01.017 doi: 10.1016/j.cor.2013.01.017
[6]	P. B. de Oliveira, R. S. de Camargo, G. de Miranda Júnior, A. X. Martins, A computational study of a decomposition approach for the dynamic two-level uncapacitated facility location problem with single and multiple allocation, Comput. Ind. Eng., 151 (2021), 106964.
[7]	A. Duarte, R. Marti, Tabu search and grasp for the maximum diversity problem, Eur. J. Oper. Res., 178 (2007), 71–84. https://doi.org/10.1016/j.ejor.2006.01.021 doi: 10.1016/j.ejor.2006.01.021
[8]	E. Ghorbani, T. Fluechter, L. Calvet, M. Ammouriova, J. Panadero, A. A. Juan, Optimizing energy consumption in smart cities' mobility: Electric vehicles, algorithms, and collaborative economy, Energies, 16 (2023).
[9]	F. Glover, K. Ching-Chung, K. S. Dhir, A discrete optimization model for preserving biological diversity, Appl. Math. Model., 19 (1995), 696–701. https://doi.org/10.1016/0307-904X(95)00083-V doi: 10.1016/0307-904X(95)00083-V
[10]	F. Glover, C. C. Kuo, K. Dhir, Heuristic algorithms for the maximum diversity problem, J. Inform. Optim. Sci., 1998, 19. https://doi.org/10.1080/02522667.1998.10699366
[11]	J. F. Gomez, A. Martínez-Gavara, J. Panadero, A. A. Juan, R. Martí, A forward-backward simheuristic for the stochastic capacitated dispersion problem, Mathematics, 12 (2024), 909. https://doi.org/10.3390/math12060909 doi: 10.3390/math12060909
[12]	J. F. Gomez, J. Panadero, R. D. Tordecilla, J. Castaneda, A. A. Juan, A multi-start biased-randomized algorithm for the capacitated dispersion problem, Mathematics, 10 (2022).
[13]	J. F. Gomez, A. R. Uguina, J. Panadero, A. A. Juan, A learnheuristic algorithm for the capacitated dispersion problem under dynamic conditions, Algorithms, 16 (2023), 532. https://doi.org/10.3390/a16120532 doi: 10.3390/a16120532
[14]	A. Grasas, A. A. Juan, J. Faulin, J. de Armas, H. Ramalhinho, Biased randomization of heuristics using skewed probability distributions: A survey and some applications, Comput. Ind. Eng., 110 (2017), 216–228. https://doi.org/10.1016/j.cie.2017.06.019 doi: 10.1016/j.cie.2017.06.019
[15]	S. D. Jena, J. F. Cordeau, B. Gendron, Solving a dynamic facility location problem with partial closing and reopening, Comput. Oper. Res., 67 (2016), 143–154. https://doi.org/10.1016/j.cor.2015.10.011 doi: 10.1016/j.cor.2015.10.011
[16]	M. Landete, J. Peiró, H. Yaman, Formulations and valid inequalities for the capacitated dispersion problem, Networks, 81 (2023), 294–315. https://doi.org/10.1002/net.22132 doi: 10.1002/net.22132
[17]	Z. Lu, A. Martínez-Gavara, J. K. Hao, X. Lai, Solution-based tabu search for the capacitated dispersion problem, Expert Syst. Appl., 223 (2023), 119856.
[18]	J. Maisano, A. Radchik, T. Ling, A log-normal model for demand forecasting in the national electricity market, ANZIAM J., 57 (2016), 369–383. https://doi.org/10.21914/anziamj.v57i0.8910 doi: 10.21914/anziamj.v57i0.8910
[19]	R. Marti, A. Martinez-Gavara, J. Sanchez-Oro, The capacitated dispersion problem: An optimization model and a memetic algorithm, Memetic Comp., 13 (2021), 131–146. https://doi.org/10.1007/s12293-020-00318-1 doi: 10.1007/s12293-020-00318-1
[20]	A. Martinez-Gavara, T. Corberan, R. Marti, Grasp and tabu search for the generalized dispersion problem, Expert Syst. Appl., 173 (2021), 114703. https://doi.org/10.1016/j.eswa.2021.114703 doi: 10.1016/j.eswa.2021.114703
[21]	R. Martí, M. Gallego, A. Duarte, A branch and bound algorithm for the maximum diversity problem, Eur. J. Oper. Res., 200 (2010), 36–44. https://doi.org/10.1016/j.ejor.2008.12.023 doi: 10.1016/j.ejor.2008.12.023
[22]	M. Marufuzzaman, R. Gedik, M. S. Roni, A benders based rolling horizon algorithm for a dynamic facility location problem, Comput. Ind. Eng., 98 (2016), 462–469. https://doi.org/10.1016/j.cie.2016.06.029 doi: 10.1016/j.cie.2016.06.029
[23]	S. Mišković, Z. Stanimirović, I. Grujičić, An efficient variable neighborhood search for solving a robust dynamic facility location problem in emergency service network, Electron. Notes Discrete Math., 47 (2015), 261–268. https://doi.org/10.1016/j.endm.2014.11.034 doi: 10.1016/j.endm.2014.11.034
[24]	N. Mladenović, R. Todosijević, D. Urošević, M. Ratli, Solving the capacitated dispersion problem with variable neighborhood search approaches: From basic to skewed vns, Comput. Oper. Res., 139 (2022), 105622. https://doi.org/10.1016/j.cor.2021.105622 doi: 10.1016/j.cor.2021.105622
[25]	R. H. Pearce, M. Forbes, Disaggregated benders decomposition and branch-and-cut for solving the budget-constrained dynamic uncapacitated facility location and network design problem, Eur. J. Oper. Res., 270 (2018), 78–88.
[26]	J. Peiró, I. Jiménez, J. Laguardia, R. Martí, Heuristics for the capacitated dispersion problem, Int. T. Oper. Res., 28 (2021), 119–141. https://doi.org/10.1111/itor.12799 doi: 10.1111/itor.12799
[27]	B. S. Pimentel, G. R. Mateus, F. A. Almeida, Stochastic capacity planning and dynamic network design, Int. J. Prod. Econ., 145 (2013), 139–149. https://doi.org/10.1016/j.ijpe.2013.01.019 doi: 10.1016/j.ijpe.2013.01.019
[28]	R. M. Rosati, A. Schaerf, Multi-neighborhood simulated annealing for the capacitated dispersion problem, Expert Syst. Appl., 255 (2024), 124484. https://doi.org/10.1016/j.eswa.2024.124484 doi: 10.1016/j.eswa.2024.124484
[29]	D. J. Rosenkrantz, G. K. Tayi, S. Ravi, Facility dispersion problems under capacity and cost constraints, J. Comb. Optim., {\bf4 } (2000), 7–33.
[30]	A. Silva, D. Aloise, L. C. Coelho, C. Rocha, Heuristics for the dynamic facility location problem with modular capacities, Eur. J. Oper. Res., 290 (2021), 435–452. https://doi.org/10.1016/j.ejor.2020.08.018 doi: 10.1016/j.ejor.2020.08.018
[31]	L. Wang, Z. Zhang, C. Wu, D. Xu, X. Zhang, Approximation algorithms for the dynamic k-level facility location problems, Theor. Comput. Sci., 853 (2021), 43–56.
[32]	J. Xu, S. Sen, Ensemble variance reduction methods for stochastic mixed-integer programming and their application to the stochastic facility location problem, INFORMS J. Comput., 36 (2024), 587–599. https://doi.org/10.1287/ijoc.2021.0324 doi: 10.1287/ijoc.2021.0324

This article has been cited by:

1.	Tareq Manzoor, S. Iqbal, Mohd Asif Shah, A note on the slip effects of an Oldroyd 6-constant fluid: Optimal homotopy asymptotic method, 2022, 10, 2296-424X, 10.3389/fphy.2022.1003000
2.	Laiq Zada, Rashid Nawaz, Wasim Jamshed, Rabha W. Ibrahim, El Sayed M. Tag El Din, Zehba Raizah, Ayesha Amjad, New optimum solutions of nonlinear fractional acoustic wave equations via optimal homotopy asymptotic method-2 (OHAM-2), 2022, 12, 2045-2322, 10.1038/s41598-022-23644-5
3.	HIMAYAT ULLAH JAN, HAKEEM ULLAH, MEHREEN FIZA, ILYAS KHAN, ABDULLAH MOHAMED, ABD ALLAH A. MOUSA, MODIFICATION OF OPTIMAL HOMOTOPY ASYMPTOTIC METHOD FOR MULTI-DIMENSIONAL TIME-FRACTIONAL MODEL OF NAVIER–STOKES EQUATION, 2023, 31, 0218-348X, 10.1142/S0218348X23400212
4.	RI ZHANG, NEHAD ALI SHAH, ESSAM R. EL-ZAHAR, ALI AKGÜL, JAE DONG CHUNG, NUMERICAL ANALYSIS OF FRACTIONAL-ORDER EMDEN–FOWLER EQUATIONS USING MODIFIED VARIATIONAL ITERATION METHOD, 2023, 31, 0218-348X, 10.1142/S0218348X23400285
5.	Nagwa Saeed, Deepak Pachpatte, Fuzzy Solutions of Fuzzy Fractional Parabolic Integro Differential Equations, 2025, 8, 2619-9653, 81, 10.32323/ujma.1631793

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)