Control and Optimization
Negar Izadi; Mohammad Taghi Dastjerdi
Abstract
In this paper, we present a new approach for achieving leader-follower consensus in a network of nonlinear dynamic agents with an undirected graph topology, using a fuzzy sliding mode controller (FSMC) for Multi-Agent Systems (MASs). Our proposed sliding mode controller ...
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In this paper, we present a new approach for achieving leader-follower consensus in a network of nonlinear dynamic agents with an undirected graph topology, using a fuzzy sliding mode controller (FSMC) for Multi-Agent Systems (MASs). Our proposed sliding mode controller is based on a separating hyperplane that effectively addresses the consensus problem in MASs. Additionally, we design a fuzzy controller to eliminate the chattering phenomenon. According to the communication graph topology and the Lyapunov stability condition, the proposed FSMC satisfies the consensus condition. One significant advantage of our approach is that the system states converge to the sliding surface quickly and remain on the surface, thereby ensuring better tracking performance. We validate the effectiveness of our proposed approach through simulation results.
Control and Optimization
Reza Akbari; Leader Navaei; Mohammad Shahriari
Abstract
This paper presents an extension of the SEIR mathematical model for infectious disease transmission to a fractional-order model. The model is formulated using the Caputo derivative of order α ∈ (0, 1]. We study the stability of equilibrium points, including ...
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This paper presents an extension of the SEIR mathematical model for infectious disease transmission to a fractional-order model. The model is formulated using the Caputo derivative of order α ∈ (0, 1]. We study the stability of equilibrium points, including the disease-free equilibrium $(E_{f})$, and the infected steady-state equilibrium $(E_{e})$ using the stability theorem of Fractional Differential Equations. The model is also analyzed under certain conditions, and it is shown that the disease-free equilibrium is locally asymptotically stable. Additionally, the extended Barbalat’s lemma is applied to the fractional-order system, and a suitable Lyapunov functional is constructed to demonstrate the global asymptotic stability of the infected steady-state equilibrium. To validate the theoretical results, a numerical simulation of the problem is conducted.
Control and Optimization
Masoomeh Ebrahimipour; Saeed Nezhadhosein; Seyed Mehdi Mirhosseini-Alizamini
Abstract
This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems. The proposed approach combines sliding mode control, a linear quadratic regulator for optimality, and gradient descent as an adaptive controller. ...
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This paper presents an optimal robust adaptive technique for controlling a certain class of uncertain nonlinear affine systems. The proposed approach combines sliding mode control, a linear quadratic regulator for optimality, and gradient descent as an adaptive controller. The convergence of the sliding mode control process is proven using two theorems based on the Lyapunov function. Simulation results for pendulum and inverted pendulum systems demonstrate that the proposed method outperforms both the linear quadratic regulator technique and the sliding mode control regarding reduced chattering and improved reaching time.
Control and Optimization
Fatemeh Babakordi; Nemat Allah Taghi-Nezhad
Abstract
This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation. Additionally, an efficient method is proposed to solve these equations by defining four solution categories: Controllable, Tolerable Solution ...
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This paper presents the introduction of two novel equation types: the partial hesitant fuzzy equation and the half hesitant fuzzy equation. Additionally, an efficient method is proposed to solve these equations by defining four solution categories: Controllable, Tolerable Solution Set (TSS), Controllable Solution Set (CSS), and Algebraic Solution Set (ASS). Furthermore, the paper establishes eight theorems that explore different types of solutions and lay out the conditions for the existence and non-existence of hesitant fuzzy solutions. The practicality of the proposed method is demonstrated through numerical examples.
Control and Optimization
Sajad Amirian; Maghsoud Amiri; Mohammad Taghi Taghavifard
Abstract
Integrating sustainability and reliability represents a synergistic approach that can be explored through the problem of a closed-loop supply chain network design (SCND). This study is conducted in three stages: mathematical modeling, model solution using exact methods, ...
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Integrating sustainability and reliability represents a synergistic approach that can be explored through the problem of a closed-loop supply chain network design (SCND). This study is conducted in three stages: mathematical modeling, model solution using exact methods, and evaluation of the solution methods. In the first stage, a mixed-integer linear programming (MILP) model is developed in a multi-objective, multi-product, and multi-period framework. The objectives of the proposed model aim to maximize profitability, social responsibility, and reliability. In the second stage, two methods, namely Augmented $\varepsilon$-Constraint (AEC) and Normalized Normal Constraint (NNC), are implemented in the GAMS software to solve the model and identify the optimal Pareto solutions. In the third stage, the Shannon Entropy technique is employed to determine the criteria weights, and the VIKOR technique is utilized to select the superior solution method. The overall performance accuracy of the proposed model is measured using four samples from a numerical example with randomly generated data based on the objective function coefficients. The results indicate the presence of a conflict among the three objective functions. Consequently, decision-makers should consider sacrificing some profitability to enhance environmental protection and improve reliability. In terms of three criteria, run time, diversification metric, and general distance, the NNC method is given priority over the AEC method. Even when the criteria are given equal weight, the superiority of the NNC method remains unchanged. The application of the proposed model across different industries represents a significant research direction for future research.
Control and Optimization
Zeinab Barary; AllahBakhsh Yazdani Cherati; Somayeh Nemati
Abstract
This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and ...
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This paper proposes and analyzes an applicable approach for numerically computing the solution of fractional optimal control-affine problems. The fractional derivative in the problem is considered in the sense of Caputo. The approach is based on a fractional-order hybrid of block-pulse functions and Jacobi polynomials. First, the corresponding Riemann-Liouville fractional integral operator of the introduced basis functions is calculated. Then, an approximation of the fractional derivative of the unknown state function is obtained by considering an approximation in terms of these basis functions. Next, using the dynamical system and applying the fractional integral operator, an approximation of the unknown control function is obtained based on the given approximations of the state function and its derivatives. Subsequently, all the given approximations are substituted into the performance index. Finally, the optimality conditions transform the problem into a system of algebraic equations. An error upper bound of the approximation of a function based on the fractional hybrid functions is provided. The method is applied to several numerical examples, and the experimental results confirm the efficiency and capability of the method. Furthermore, they demonstrate a good agreement between the approximate and exact solutions.
Control and Optimization
Fahimeh Akhavan Ghassabzade; Mina Bagherpoorfard
Abstract
This paper aims to demonstrate the flexibility of mathematical models in analyzing carbon dioxide emissions and account for memory effects. The use of real data amplifies the importance of this study. This research focuses on developing a mathematical model utilizing fractional-order ...
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This paper aims to demonstrate the flexibility of mathematical models in analyzing carbon dioxide emissions and account for memory effects. The use of real data amplifies the importance of this study. This research focuses on developing a mathematical model utilizing fractional-order differential equations to represent carbon dioxide emissions stemming from the energy sector. By comparing simulation results with real-world data, it is determined that the fractional model exhibits superior accuracy when contrasted with the classical model. Additionally, an optimal control strategy is proposed to minimize the levels of carbon dioxide, CO2, and associated implementation costs. The fractional optimal control problem is addressed through the utilization of an iterative algorithm, and the effectiveness of the model is verified by presenting comparative results.
Control and Optimization
Abbas Ali Rezaee; Hadis Ahmadian Yazdi; Mahdi Yousefzadeh Aghdam; Sahar Ghareii
Abstract
With the advancements in science and technology, the industrial and aviation sectors have witnessed a significant increase in data. A vast amount of data is generated and utilized continuously. It is imperative to employ data mining techniques to extract and uncover knowledge ...
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With the advancements in science and technology, the industrial and aviation sectors have witnessed a significant increase in data. A vast amount of data is generated and utilized continuously. It is imperative to employ data mining techniques to extract and uncover knowledge from this data. Data mining is a method that enables the extraction of valuable information and hidden relationships from datasets. However, the current aviation data presents challenges in effectively extracting knowledge due to its large volume and diverse structures. Air Traffic Management (ATM) involves handling Big data, which exceeds the capacity of conventional acquisition, matching, management, and processing within a reasonable timeframe. Aviation Big data exists in batch forms and streaming formats, necessitating the utilization of parallel hardware and software, as well as stream processing, to extract meaningful insights. Currently, the map-reduce method is the prevailing model for processing Big data in the aviation industry. This paper aims to analyze the evolving trends in aviation Big data processing methods, followed by a comprehensive investigation and discussion of data analysis techniques. We implement the map-reduce optimization of the K-Means algorithm in the Hadoop and Spark environments. The K-Means map-reduce is a crucial and widely applied clustering method. Finally, we conduct a case study to analyze and compare aviation Big data related to air traffic management in the USA using the K-Means map-reduce approach in the Hadoop and Spark environments. The analyzed dataset includes flight records. The results demonstrate the suitability of this platform for aviation Big data, considering the characteristics of the aviation dataset. Furthermore, this study presents the first application of the designed program for air traffic management.
Control and Optimization
Ali Asghar Hojatifard; Nader Kanzi; Shahriar Farahmand Rad
Abstract
This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems. These problems involve inequality constraints whose index set depends on the decision vector, and all emerging functions are assumed ...
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This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems. These problems involve inequality constraints whose index set depends on the decision vector, and all emerging functions are assumed to be locally Lipschitz. We introduce a new constraint qualification for these problems. Building upon this qualification, we derive an upper estimate for the Clarke sub-differential of the value function of the problem. Furthermore, we demonstrate the necessary optimality conditions for properly efficient solutions to the problem.
Control and Optimization
Farid Pourofoghi; Davood Darvishi Salokolaei
Abstract
Fractional programming is a significant nonlinear planning tool within operation research. It finds applications in diverse domains such as resource allocation, transportation, production programming, performance evaluation, and finance. In ...
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Fractional programming is a significant nonlinear planning tool within operation research. It finds applications in diverse domains such as resource allocation, transportation, production programming, performance evaluation, and finance. In practical scenarios, uncertainties often make it challenging to determine precise coefficients for mathematical models. Consequently, utilizing indefinite coefficients instead of definite ones is recommended in such cases. Grey systems theory, along with probability theory, randomness, fuzzy logic, and rough sets, is an approach that addresses uncertainty. In this study, we address the problem of linear fractional programming with grey coefficients in the objective function. To tackle this problem, a novel approach based on the variable change technique proposed by Charnes and Cooper, along with the convex combination of intervals, is employed. The article presents an algorithm that determines the solution to the grey fractional programming problem using grey numbers, thus capturing the uncertainty inherent in the objective function. To demonstrate the effectiveness of the proposed method, an example is solved using the suggested approach. The result is compared with solutions obtained using the whitening method, employing Hu and Wong's technique and the Center and Greyness Degree Ranking method. The comparison confirms the superiority of the proposed method over the whitening method, thus suggesting adopting the grey system approach in such situations.
Control and Optimization
Hanifa Mosawi; Mostafa Tavakolli; Khatere Ghorbani-Moghadam
Abstract
Graph coloring is a crucial area of research in graph theory, with numerous algorithms proposed for various types of graph coloring, particularly graph p-distance coloring. In this study, we employ a recently introduced graph coloring algorithm to develop a hybrid algorithm approximating the chromatic ...
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Graph coloring is a crucial area of research in graph theory, with numerous algorithms proposed for various types of graph coloring, particularly graph p-distance coloring. In this study, we employ a recently introduced graph coloring algorithm to develop a hybrid algorithm approximating the chromatic number p-distance, where $p$ represents a positive integer number. We apply our algorithm to molecular graphs as practical applications of our findings.
Control and Optimization
Mohamad Reza Ramezani-al; Samira Sabeti
Abstract
Guaranteed cost control (GCC) is an impressive method of controlling nonlinear systems, incredibly uncertain switched systems. Most of the recent studies of GCC on uncertain switched linear systems have been concerned with asymptotic stability analysis. In this paper, ...
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Guaranteed cost control (GCC) is an impressive method of controlling nonlinear systems, incredibly uncertain switched systems. Most of the recent studies of GCC on uncertain switched linear systems have been concerned with asymptotic stability analysis. In this paper, a new robust switching law for time-delay uncertain switched linear systems is designed. First, the switching law is designed, and second, a state-feedback controller based on Lyapunov-Krasovskii Functional (LKF) is designed. Also, using Linear Matrix Inequality (LMI) particular condition for the existence of a solution of obtained switching law and controller is achieved. Consequently, in the presented theorems, the exponential stability of the overall system under switching law and controller is analyzed. Finally, theoretical results are verified via presenting an example.
Control and Optimization
Jahangir Alizadeh; Hamid Khaloozadeh
Abstract
In the present study, a novel methodology is developed to enlarge the Region of Attraction (ROA) at the point of equilibrium of an input-affine nonlinear control system. Enlarging the ROA for non-polynomial dynamical systems is developed by designing a nonlinear state feedback controller ...
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In the present study, a novel methodology is developed to enlarge the Region of Attraction (ROA) at the point of equilibrium of an input-affine nonlinear control system. Enlarging the ROA for non-polynomial dynamical systems is developed by designing a nonlinear state feedback controller through the State-Dependent Riccati Equation (SDRE). Consequently, its process is defined in the form of Sum-of-Squares (SOS) optimization problem with control and non-control constraints. Of note, the proposed technique is effective in estimating the ROA for a nonlinear system functioning on polynomial or non-polynomial dynamics. In the present study, the application of the proposed scheme are shown by numerical simulations.
Control and Optimization
Zahra Abbasi; Nasser Akhoundi
Abstract
Product reviews in E-commerce websites such as restaurants, movies, E-commerce products, etc., are essential resources for consumers to make purchasing decisions on various items. In this paper, we model discovering groups with maximum descriptively ...
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Product reviews in E-commerce websites such as restaurants, movies, E-commerce products, etc., are essential resources for consumers to make purchasing decisions on various items. In this paper, we model discovering groups with maximum descriptively from E-commerce website of the form $<i,u,s>$, where $i\in \mathcal{I}$ (the set of items or products), $u\in \mathcal{U}$ (the set of users) and $s$ is the integer rating that user $u$ has assigned to the item $i$. Labeled groups from user's attributes are found by solving an optimization problem. The performance of the approach is examined by some experiments on real data-sets.
Control and Optimization
Mohammad Darvisahzadeh; Ahmad Shahvarani Semnani; Hassan Alamolhodaei,; Hassan Behzadi
Abstract
Experiences of teaching Integral have indicated that the vast majorities of Iranian university students commit numerous errors while solving integral problems and have weak skills in this field; we might even say that they hide away from integral and consider it the nightmare of mathematics. On ...
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Experiences of teaching Integral have indicated that the vast majorities of Iranian university students commit numerous errors while solving integral problems and have weak skills in this field; we might even say that they hide away from integral and consider it the nightmare of mathematics. On the other hand, Integral is the base of pure and applied mathematics for all students of science, especially engineering, which some of their lessons are dependent on it directly or indirectly, so it is important to pay attention to it. Through descriptive method-exposed factor, an exam has been conducted in the form of three questions, the first of which is consisted of 4 sections on fifty students from different fields, and then interviews were conducted with a few of those students about their answers in order to study the students' behaviors when solving integral problems and to determine the type of their errors. By analyzing the performance of students in this test, we can see that students often struggle with integral and mostly have a feeble performance in solving trigonometric integrals. They want to learn computational integral instead of how to conceptualize integral in their minds correctly. The error most committed by university students was procedural errors, which arise from using derivative instead of integral. Besides most of the mistakes happen in solving definite integrals, and calculating finite areas between two curves. This is due to a lack of understanding of integrals and a lack of information in other areas of mathematics.
Control and Optimization
Mostafa Nouri Jouybari; Yahya Talebi Rostami; Siyamak Firouzian
Abstract
In this study, $R$ and $M$ are assumed to be a commutative ring with non-zero identity $M$ and an $R$-module, respectively. Scalar Product Graph of $M$, denoted by $G_R(M)$, is a graph with the vertex-set $M$ and two different vertices $a$ and $b$ in ...
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In this study, $R$ and $M$ are assumed to be a commutative ring with non-zero identity $M$ and an $R$-module, respectively. Scalar Product Graph of $M$, denoted by $G_R(M)$, is a graph with the vertex-set $M$ and two different vertices $a$ and $b$ in $M$ are connected if and only if there exists $r$ belong to $R$ such that $a=rb$ or $b=ra$. This paper studies some properties of such weakly perfect graphs.
Control and Optimization
Sara Mansourinasab; Mahdi Sojoodi; Seyed Reza Moghadasi
Abstract
Conventional model predictive control (MPC) methods are usually implemented to systems with discrete-time dynamics laying on smooth vector space $ \mathbf{R}^n$. In contrast, the configuration space of the majority of mechanical systems is not expressed as Euclidean space. Therefore, ...
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Conventional model predictive control (MPC) methods are usually implemented to systems with discrete-time dynamics laying on smooth vector space $ \mathbf{R}^n$. In contrast, the configuration space of the majority of mechanical systems is not expressed as Euclidean space. Therefore, the MPC method in this paper has developed on a smooth manifold as the configuration space of the attitude control of a 3D pendulum. The Lie Group Variational Integrator (LGVI) equations of motion of the 3D pendulum have been considered as the discrete-time update equations since the LGVI equations preserve the group structure and conserve quantities of motion. The MPC algorithm is applied to the linearized dynamics of the 3D pendulum according to its LGVI equations around the equilibrium using diffeomorphism. Also, as in standard MPC algorithms, convex optimization is solved at each iteration to compute the control law. In this paper, the linear matrix inequality (LMI) is used to solve the convex optimization problem under constraints. A numerical example illustrates the design procedure.
Control and Optimization
seyed mehdy shafiof; Javad Askari; Maryam Shams Solary
Abstract
In this paper, a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly. First, the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs). Then, ...
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In this paper, a modern method is presented to solve a class of fractional optimal control problems (FOCPs) indirectly. First, the necessary optimality conditions for the FOCP are obtained in the form of two fractional differential equations (FDEs). Then, the unknown functions are approximated by the hybrid functions, including Bernoulli polynomials and Block-pulse functions based on the spectral Ritz method. Also, two new methods are proposed for calculating the left Caputo fractional derivative and right Riemann-Liouville fractional derivative operators of the hybrid functions that are proportional to the Ritz method. The FOCP is converted into a system of the algebraic equations by applying the fractional derivative operators and collocation method, which determines the solution of the problem. Error estimates for the hybrid function approximation, fractional operators and, the proposed method are provided. Finally, the efficiency of the proposed method and its accuracy in obtaining optimal solutions are shown by some test problems.
Control and Optimization
Farzad Rahpeymaii; majid rostami
Abstract
In this paper, two extended three-term conjugate gradient methods based on the Liu-Storey ({\tt LS}) conjugate gradient method are presented to solve unconstrained optimization problems. A remarkable property of the proposed methods is that the search direction always satisfies ...
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In this paper, two extended three-term conjugate gradient methods based on the Liu-Storey ({\tt LS}) conjugate gradient method are presented to solve unconstrained optimization problems. A remarkable property of the proposed methods is that the search direction always satisfies the sufficient descent condition independent of line search method, based on eigenvalue analysis. The global convergence of proposed algorithms is established under suitable conditions. Preliminary numerical results show that the proposed methods are efficient and robust to solve the unconstrained optimization problems.
Control and Optimization
Alireza Ahangarani Farahani; Abbas Dideban
Abstract
The existing modeling methods using Petri Nets, have been successfully applied to model and analyze dynamic systems. However, these methods are not capable of modeling all dynamic systems such as systems with the current sample time signals, systems including various ...
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The existing modeling methods using Petri Nets, have been successfully applied to model and analyze dynamic systems. However, these methods are not capable of modeling all dynamic systems such as systems with the current sample time signals, systems including various subsystems and multi-mode systems. This paper proposes Hybrid Time Delay Petri Nets (HTDPN) to solve the problem. In this approach, discrete and continuous Petri Nets are combined so that the continuous PNs part and the discrete PNs are responsible for past time samples and current sample time, respectively. To evaluate the performance of the proposed tool, it is employed to model a legless piezoelectric capsubot robot as a multi modes system and a $PID$ controller, in which the gains tuned by the Genetic Algorithm are designed for the resulting model by HTDPN. Results show that the proposed method is faster in terms of mathematical calculations which can reduce the simulation time and complexity of complicated systems. It would be observed that the proposed approach makes the $PID$ controller design simpler as well. In addition, a comparative study of capsubot has been performed. Simulation results show that the presented method is encouraging compared to the predictive control, which is used in the literature.
Control and Optimization
Kamal Fallahi
Abstract
In this paper, we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph under two different conditions. We then support our main theorem by a non-trivial ...
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In this paper, we introduce a new type of graph contraction using a special class of functions and give a best proximity point theorem for this contraction in complete metric spaces endowed with a graph under two different conditions. We then support our main theorem by a non-trivial example and give some consequences of best proximity point of it for usual graphs.
Control and Optimization
Mohammad Hadi Noori Skandari
Abstract
In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space ...
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In this paper, first the space of hyperbolic tangent functions is introduced and then the universal approximator property of this space is proved. In fact, by using this space, any nonlinear continuous function can be uniformly approximated with any degree of accuracy. Also, as an application, this space of functions is utilized to design feedback control for a nonlinear dynamical system.
Control and Optimization
Ghasem Ahmadi; Mohammad Teshnehlab; Fahimeh Soltanian
Abstract
o enhance the performances of rough-neural networks (R-NNs) in the system identification, on the base of emotional learning, a new stable learning algorithm is developed for them. This algorithm facilitates the error convergence by increasing the memory depth of R-NNs. ...
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o enhance the performances of rough-neural networks (R-NNs) in the system identification, on the base of emotional learning, a new stable learning algorithm is developed for them. This algorithm facilitates the error convergence by increasing the memory depth of R-NNs. To this end, an emotional signal as a linear combination of identification error and its differences is used to achieve the learning laws. In addition, the error convergence and the boundedness of predictions and parameters of the model are proved. To illustrate the efficiency of proposed algorithm, some nonlinear systems including the cement rotary kiln are identified using this method and the results are compared with some other models.
Control and Optimization
Saeed Nezhadhosein
Volume 2, Issue 2 , December 2017, , Pages 1-14
Abstract
In this paper, Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems. Firstly, using necessary conditions for optimality, the problem is changed into a two-boundary value problem (TBVP). Next, ...
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In this paper, Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems. Firstly, using necessary conditions for optimality, the problem is changed into a two-boundary value problem (TBVP). Next, Haar wavelets are applied for converting the TBVP, as a system of differential equations, in to a system of matrix algebraic equations, as Haar matrix equations using Kronecker product. Then the error analysis of the proposed method is presented. Some numerical examples are given to demonstrate the efficiency of the method. The solutions converge as the number of approximate terms increase.
Control and Optimization
Davood Darvishi Salookolaei; Sifeng Liu; Parvin Babaei
Volume 2, Issue 2 , December 2017, , Pages 15-32
Abstract
Considering the fact that Iran is situated in an arid and semi-arid region, rainfall prediction for the management of water resources is very important and necessary. Researchers have proposed various prediction methods that have been utilized in such areas as water and meteorology, especially water ...
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Considering the fact that Iran is situated in an arid and semi-arid region, rainfall prediction for the management of water resources is very important and necessary. Researchers have proposed various prediction methods that have been utilized in such areas as water and meteorology, especially water resources management. The present study aimed at predicting rainfall amounts using Grey Prediction Method. It is a novel approach in confrontation with uncertainties in the aquiferous region of Babolrud to serve for the water resources management purposes. Therefore, expressing the concepts of Grey Prediction Methods using the collected data, at a 12-year timeframe of 2006 and 2017, rainfall prediction in 2018 and 2022 were also implemented with three methods GM(1,1), DGM(2,1) and Verhulest models. According to the calculated error and the predictive power, GM(1,1) method is better than other models and was placed within the set of good predictions. Also, we predict that in 2027, there might be a drought. According to the small samples and calculations required in this approach, the method is suggested for rainfall prediction in inexact environments. The authors can use fuzzy grey systems to predict the amount of rainfall in uncertaint environments.
