Iranian Journal of Fuzzy Systems
https://ijfs.usb.ac.ir/
Iranian Journal of Fuzzy Systemsendaily1Sat, 01 Oct 2022 00:00:00 +0330Sat, 01 Oct 2022 00:00:00 +0330cover IJFS-Vol 19-No 5
https://ijfs.usb.ac.ir/article_7165.html
Possibilistic max-mean dispersion problem
https://ijfs.usb.ac.ir/article_7153.html
The Max-Mean Dispersion Problem comprises selecting a subset of elements from a set while maximising the average of a measure of dispersion. We usually compute this measure of dispersion based on some distance metric between the elements of the candidate set. However, in real-world applications, this measure of dispersion could be ill-defined or vague. For example, consider the problem of building a team to execute some kinds of project. We can see the dispersion between the members of the team as the team chemistry, so the coach is interested in maximising this chemistry. In this example, it would be very difficult to compute exactly a dispersion measure for each candidate. This is due to the lack of information about how well the candidates worked together in the past. To cope with imprecise or vague information, in this paper, we propose three mixed integer linear programming models based on possibility, necessity, and credibility measures. To the best of our knowledge, this is the first approach which explicitly considers this type of uncertainty in this optimisation problem.Are multidimensional RDM interval arithmetic and constrained interval arithmetic one and the same?
https://ijfs.usb.ac.ir/article_7154.html
This article discusses the comments made by some scientists that multidimensional interval arithmetic (MIA) is the same as constraint interval arithmetic (CIA) and multidimensional fuzzy arithmetic (MFA) is the same as constraint fuzzy arithmetic (CFA). Both types of arithmetic are briefly presented and then the difference in their dimensions, calculation methods, differences in the obtained results and the way they are used in complex calculations are shown. The answer to the question posed is presented in the conclusions.Best-possible bounds on the set of copulas with a given value of Gini's gamma
https://ijfs.usb.ac.ir/article_7155.html
In this note, pointwise best-possible (lower and upper) bounds on the set of copulas with a given value of the Gini's gamma coefficient are established. It is shown that, unlike the best-possible bounds on the set of copulas with a given value of other known measures such as Kendall's tau, Spearman's rho or Blomqvist's beta, the bounds found are not necessarily copulas, but proper quasi-copulas.Finite-time synchronization of fractional-order fuzzy Cohen-Grossberg neural networks with time delay
https://ijfs.usb.ac.ir/article_7156.html
This paper deals with the issues of the finite-time synchronization (FTS) for a class of fractional-order fuzzy Cohen-Grossberg neural networks (FOFCGNNs) with time delay. Based on the finite-time stability theory, &nbsp;fractional-order Razumikhin theorem and applying fractional-order differential inequalities and other inequality techniques, a few new and effective criteria formulated by testable algebraic inequalities are derived to ensure the FTS for the concerned models via designing a discontinuous control strategy. Finally, two numerical simulations examples are furnished to demonstrate the feasibility and effectiveness of the derived theoreticalresults.L-R representation of TA fuzzy arithmetic and its application to solving fuzzy equations
https://ijfs.usb.ac.ir/article_7157.html
Fuzzy arithmetic with standard methods such as the extension principle and$\alpha $-cut lead to restricted possibilities for solving fuzzy equations.The procedures to find a solution to a fuzzy equality with these methodsrequire strong assumptions and high computation costs. Among severalapproaches dealing with this restrictions this paper focuses on theTransmission Average (TA) fuzzy arithmetic. The shape preservation ofthe TA arithmetic operations on L-R fuzzy numbers is proven. Theseproperties together with some other algebraic properties investigatedin the paper are applied to solve fuzzy polynomial equations as well as systems of linear fuzzy equations in general form. Several examples in the paper present the advantages of TA arithmetic in solving fuzzy equations. &nbsp;It is shown that the results in this paper support the fact that TA arithmetic is an easy to implement approach in fuzzy modeling.An extended MABAC method based on prospect theory for multiple attribute group decision making under probabilistic uncertain linguistic environment
https://ijfs.usb.ac.ir/article_7158.html
Cyber security is a hot topic in recent years and one of the main performances in the decision-making is the choice of cyber security service providers (CSSPs). For enterprises and government units, the selection of appropriate NSSPs needs to be measured and evaluated from multiple perspectives, which is obviously a multiple attribute decision making (MADM) or multiple attribute group decision making (MAGDM). In this paper, the traditional MABAC method is improved by using the prospect theory, and the evaluation information is collected and sorted by using PULTS. The improved PUL-PT-MABAC method can not only deal with the uncertainty well, but also fully consider the influence of the psychological state of decision maker (DM) on the decision result. More importantly, we apply the improved PUL-PT-MABAC model to the selection of NSSPs and prove the reliability of this proposed method by taking advantage of comparative analysis with three existed methods. These results show that the model can deal with practical problems, and has good practicability and science.Characterizations of $L$-order $L$-convex spaces
https://ijfs.usb.ac.ir/article_7159.html
In this paper, the concepts of $L$-enclosed $L$-order space, $L$-order $L$-concave space, $L$-internal $L$-order space and $L$-order $L$-convex filter are introduced. The main results are:(1) the categories of $L$-order $L$-convex spaces, $L$-enclosed $L$-order spaces, $L$-order $L$-concave spaces and $L$-internal $L$-order spaces are isomorphic; (2) the category of $L$-order convergence spaces based on $L$-order $L$-convex filters is topological; (3) there is a Galois correspondence between the category of $L$-order convergence spaces and that of $L$-order $L$-convex spaces.Supervisory adaptive interval type-2 fuzzy sliding mode control for planar cable-driven parallel robots using Grasshopper optimization
https://ijfs.usb.ac.ir/article_7160.html
Design of an adaptive supervisory fuzzy sliding mode control for a planar cable-driven parallel robot is aided in this paper. The fuzzy logic controller is proposed to generate the switching control signal without occurring the chattering problem. For this purpose, an adaptive mechanism is suggested for online tuning of the output gain of the fuzzy sliding mode controller. Moreover, for better tracking, a supervisory control system is considered for online tuning of the PID sliding surface gains. The Grasshopper Optimization Algorithm is suggested for optimization of the membership functions selected for the fuzzy sliding surface. The stability proof of the closed-loop system is derived by using the Lyapunov stability theorem. Simulation results are reported to show the merits of the proposed controller on reduced chatter, and system robustness against parameter uncertainty, load disturbance, and nonlinearities.Diagonal conditions and uniformly continuous extension in $\top$-uniform limit spaces
https://ijfs.usb.ac.ir/article_7161.html
We study suitable diagonal conditions for $\top$-uniform limit spaces. A dual diagonal condition is shown to be a suitable axiom for uniform regularity. We characterize this regularity concept by closures of $L$-sets. We apply all these diagonal axioms and prove an extension theorem for uniformly continuous mappings defined on a dense subspace.An improvement in integrating clustering method and neural network to extract rules and application in diagnosis support
https://ijfs.usb.ac.ir/article_7162.html
Most of chronic liver diseases without suitable treatment will lead to cirrhosis of the liver, eventually progressing to liver cancer. Thus, early diagnosis is very important in detecting the liver diseases and suggesting the treatment at the right time. A useful model that effectively predicts the patient's liver fibrosis has great importance in reducing the load on doctors, especially in lower-level hospitals. In this paper, a new model combining semi-supervised learning method and fuzzy min max neural network with selective fuzzy rule set rendering is proposed. Cirrhosis level is evaluated by APRI and FIB-4. The proposed method is experimented on data sets from machine learning databases, including UCI and CS. Apart from that, our method is also implemented on the liver data set collected from the hospitals of Thai Nguyen province. The comparison among our proposed method and other related ones is also given. The obtained results show that our proposed model has better performance than compared methods in terms of execution time and the number of rules.HFC: Data clustering based on hesitant fuzzy decision making
https://ijfs.usb.ac.ir/article_7163.html
In a clustering task, choosing a proper clustering algorithm and obtaining qualified clusters are crucial issues. Sometimes, a clustering algorithm is chosen based on the data distribution, but data distributions are not known beforehand in real world problems. In this case, we hesitate which clustering algorithm to choose. In this paper, this hesitation is modeled by a hesitant fuzzy multi criteria decision making problem {\small (HFMCDM)} in which some clustering algorithms play the role of experts. Here, we consider fuzzy {\footnotesize C}-means {\small (FCM)} and agglomerative clustering algorithms as representative of two popular categories of clustering algorithms partitioning and hierarchical clustering methods, respectively.Then, we propose a new clustering procedure based on hesitant fuzzy decision making approaches {\small (HFC)} to decide which of the {\small FCM} family or hierarchical clustering algorithms is suitable for our data. This procedure ascertains a good clustering algorithm using neutrosophic {\small FCM} ({\small NFCM}) through a two phases process. The {\small HFC} procedure not only makes a true decision about applying partitioning clustering algorithms, but also improves the performance of {\small FCM} and evolutionary kernel intuitionistic fuzzy c-means clustering algorithm ({\small EKIFCM}) with construction hesitant fuzzy partition {\small (HFP)} conveniently. Experimental results show that the clustering procedure is applicable and practical. According to {\small HFC} procedure, it should be mentioned that it is possible to replace the other clustering algorithms that belong to any partitioning and hierarchical clustering methods. Also, we can consider other categories of clustering algorithms.On some categories of triangular norms on the real unit interval
https://ijfs.usb.ac.ir/article_7164.html
In this work, we introduce some categories of triangular norms in which truth values belong to the real unit interval, where arrows are a generalization of automorphisms. We investigate the existence of products, coproducts, equalizers and &nbsp;coequalizers in these categories. Moreover, we show that Theorems 2.29, 2.30 in \cite{Yousefi_Mashinchi_Mesiar_2021} are false by providing counterexamples.(2205-7438) Connections between commutative rings and some algebras of logic
https://ijfs.usb.ac.ir/article_7139.html
In this paper using the connections between some subvarieties of residuated lattices, we investigated some properties of the lattice of ideals in commutative and unitary rings. We give new characterizations for commutative rings A in which Id(A) is an MV-algebra, a Heyting algebra or a Boolean algebra and we establish connections between these types of rings. We are very interested in the finite case and we present summarizing statistics. We show that the lattice of ideals in a finite commutative ring is a Boolean algebra or an MV-algebra (which is not Boolean). Using this result we generate the binary block codes associated to the lattice of ideals in finite commutative rings and we present a new way to generate all (up to an isomorphism) finite MV-algebras using rings.
&nbsp;translate IJFS-Vol 19-No 5
https://ijfs.usb.ac.ir/article_7166.html
(2110-6985) A new model to protect an important node against two threatening agents
https://ijfs.usb.ac.ir/article_7081.html
One of the main goals of network planners is the protection of important nodes in a network against natural disasters,security threats, attacks, and so on. Given the importance of this issue, a new model is presented in this paper forprotecting an important node in a typical network based on a defensive location problem where the two agents threatenthis node. The protecting facilities location problem with two agents is formulated as a three-level programmingproblem. The decision maker in the upper level is a network planner agent. The planner agent wants to find the bestpossible location of protecting facilities to protect the important node against threatening agents. The second andthird levels problems are stated as the shortest path problems in the network in which the edges are weighted withpositive values. In this work, the genetic, variable neighborhood search, simulated annealing algorithms are used tosolve the problem. The performance of the used metaheuristic algorithms on this class of problems is investigated by atest problem that is generated randomly. Then, t-test are used to compare the performance of these algorithms. Thebest results are obtained by the variable neighborhood search algorithm.(2201-7201) Numerical solution for interval initial value problems based on interactive arithmetic
https://ijfs.usb.ac.ir/article_7109.html
This work studies Interval Initial Value Problems (IIVPs), where the derivative is given by the generalized Hukuhara derivative ($gH$-derivative) and the initial condition is given by an interval. The focus of the paper is to provide the numerical approximations for the solutions associated with the $gH$-derivative of IIVPs. This article considers the Euler numerical method, where the classical arithmetic operation is adapted for intervals. The arithmetic considered here is obtained using sup-$J$ extension principle, where $J$ is a particular family of joint possibility distributions. This family gives raise to different types of interactivity and this work shows what kind of interactivity is necessary in the numerical method, in order to approximate the solution via $gH$-derivative. To illustrate the results, the paper focuses in the decay Malthusian model.(2201-7171) The analysis of a fractional network-based epidemic model with saturated treatment function and fuzzy transmission
https://ijfs.usb.ac.ir/article_7111.html
For understanding the influence of malware attacking on complex heterogeneous networks, this work studies a fractional network-based SIRS epidemic model with fuzzy transmission and saturated treatment function. Firstly, we apply the next-generation method to obtain the basic reproductive ratio $\mathcal{R}_0$, that is an important threshold value in the investigation of asymptotic behavior of the proposed epidemic model. The obtained theoretical results indicates that the value $\mathcal{R}_0$ significantly depends on the topology structure of the underlying network and the malware load. In addition, we give a threshold value $\tilde{\mathcal{R}}_0&gt;\mathcal{R}_0$ that not only determines the existence of endemic equilibrium &nbsp;$\mathbf{E}_\ast$ but also ensures the clean of malware programs on the network. At last, the sensitivity analysis of the threshold value $\mathcal{R}_0$ and some graphical simulations are presented to illustrate for the theoretical results.(2112-7107) Distributivity between 2-uninorms and Mayorâ€™s aggregation operators
https://ijfs.usb.ac.ir/article_7112.html
The issue of distributivity of aggregation operators is crucial for many different areas such as the utility theory andthe integral theory. The topic of this paper is distributivity between 2-uninorms and Mayor&rsquo;s aggregation operators.The presented research is an extension and upgrade of the previously obtained results. The full characterization ofdistributive pairs of operators is given.&nbsp;(2202-7227) A quadratic optimization problem with bipolar fuzzy relation equation constraints
https://ijfs.usb.ac.ir/article_7116.html
This paper studies the quadratic programming problem subject to a system of bipolar fuzzy relation equations withthe max-product composition. A characterization of structure of its feasible domain is presented using the lower andupper bound vector on its solution set. A sufficient condition is proposed which under the condition, a component ofone of its optimal solutions is the corresponding component of either the lower or upper bound vector. Some sufficientconditions are suggested to reveal one of its optimal solutions without resolution of the problem. Furthermore, somesufficient conditions are then given to determine some components from one of its optimal solutions. Based on theseconditions, we can simplify the problem and reduce its dimensions. The simplified problem can be reformulated to an0-1 mixed integer programming problem. Other unknown variables can be found by solving the current problem.(2106-6710) A new method to solve linear programming problems in the environment of picture fuzzy sets
https://ijfs.usb.ac.ir/article_7117.html
Picture fuzzy set is characterized by neutral membership function along with the membership and non-membership functions, and is, therefore, more general than the intuitionistic fuzzy set which is only characterized by membership and non-membership functions. In this paper, first, we are going to point out a drawback and try to fix it by the existing trapezoidal picture fuzzy number. Furthermore, we define an $LR$ flat picture fuzzy number, which is a generalization of trapezoidal picture fuzzy numbers. We also discuss a linear programming model with $LR$ flat picture fuzzy numbers as parameters and variables and present a method to solve these type of problems using a generalized ranking function.(2203-7306) Picture m-polar fuzzy soft sets and their application in decision-making problems
https://ijfs.usb.ac.ir/article_7126.html
The aim of this paper is to introduce a new multiple attribute decision-making model named picture m-polar fuzzy soft sets, which is a combination of the soft sets and picture m-polar fuzzy set. Some operations and properties of the new model, including subset, equal, union, intersection, and complement are discussed.&nbsp;Further, the basic six definitions, three theorems, and six examples on picture m-polar fuzzy soft sets are explained. Lastly, we construct a new methodology to extend the TOPSIS to picture m-polar fuzzy soft sets (i.e., an application of &nbsp;picture m-polar fuzzy soft sets) in which capable of different objects recognizing belonging to the same family is constructed and illustrated its applicability via a numerical example.(2112-7123) Generalization of rough fuzzy sets based on a fuzzy ideal
https://ijfs.usb.ac.ir/article_7127.html
Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on %arbitrary fuzzy relation $R$ and a fuzzy ideal $\ell$ on a universal finite set $X$. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and &nbsp;fuzzy closure of a rough fuzzy set are discussed. These definitions, if $\ell$ is restricted to $\ell^{\circ} = \{\overline{0}\}$, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.(2204-7314) A note on uniform continuity of super-additive transformations of aggregation functions
https://ijfs.usb.ac.ir/article_7128.html
Extending and completing earlier results on lifting certain continuity properties of aggregation functions by super- and sub-additive transformations (J. Mahani Math. Res. Center 8 (2019) 37--51, and Iranian J. Fuzzy Sets 17 (2020) 2, 165--168), we prove that uniform continuity of multi-dimensional aggregation functions is preserved under super-additive transformations.(2111- 7076) A parametric similarity measure for extended picture fuzzy sets and its application in pattern recognition
https://ijfs.usb.ac.ir/article_7129.html
This article advances the idea of extended picture fuzzy set (E-PFS), which is especially an augmentation of generalised spherical fuzzy set (GSFS) by releasing the restricted selection of $p$ in the description of GSFSs.% restricted release of $p$ selection in {GSFS} theory.Moreover, by the use of triangular conorm term in the description of E-PFS, it indeed widens the scope of E-PFS not only compared to picture fuzzy set (PFS) and spherical fuzzy set (SFS), but also to GSFS. In the sequel, a given fundamental theorem concerning E-PFS depicts its more ability in comparison with the special types to deal with the ambiguity and uncertainty.%involved in other types of PFS, SFS and GSFS.Further, we propose a parametric E-PFS similarity measure which plays a critical role in information theory.In order for revealing the advantages and authenticity of E-PFS similarity measure, we exhibit its applicability in multiple criteria decision making entitling the recognition of building material, the recognition of patterns, and the selection process of mega project(s) in developing countries.Furthermore, through the experimental studies, we demonstrate that E-PFS is able to handle uncertain information in real-life decision procedures with no extra parameter, and it has a prominent role in decision making whenever the concepts of &nbsp;PFS, SFS and GSFS do not make sense.(2201-7187) A picture fuzzy distance measure and its application to pattern recognition problems
https://ijfs.usb.ac.ir/article_7133.html
The picture fuzzy sets are very useful in those uncertain problems which could not be solved by fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, fermatean fuzzy sets, and q-rung orthopair fuzzy sets. For example, medical diagnosis, personnel selection, human voting, etc. All of these problems require answers of the type no, yes, abstain, and refusal. To compare two picture fuzzy sets, the distance measures play an important role. There are a lot of studies about the distance measures of picture fuzzy sets available in the literature. However, all of these distance measures lead to unreasonable results in most of the problems. So, we in this paper suggest a new distance measure for picture fuzzy sets that is more effective than all of the available distance measures. We also demonstrate its utility in classification and diagnostic problems and contrast its performance with the available ones.(2109-6901) The Craig interpolation property for rational Godel logic
https://ijfs.usb.ac.ir/article_7134.html
In this article, the Craig interpolation property for rational Godel logic is studied. Despite classical Godel logic, this property can be proved in this new extension of Godel logic. This new predicate version of Godel logic is similar to continuous logic and also, its semantics is extended similar to metric model theory with some differences.(2201-7215) On the global stabilization of perturbed nonlinear fuzzy control systems
https://ijfs.usb.ac.ir/article_7137.html
In this paper, we deal with the global practical exponential stabilization of a class of perturbed Takagi-Sugeno fuzzy control systems. The terms of perturbations are supposed uniformly &nbsp;bounded by some known &nbsp;functions and in certain cases not necessarily smooth. We prove that the solution of the closed-loop system with a linear fuzzy controller convergeto &nbsp;a neighborhood of the origin. We use common quadratic Lyapunov function &nbsp;and parallel distributed compensation controller techniques to study the asymptotic behavior of the solutions of fuzzy system. Numerical simulations are given to validate the proposed approach.(2201-7178) A novel hesitant fuzzy linguistic term sets approach and its application on acceptance sampling plans
https://ijfs.usb.ac.ir/article_7145.html
Hesitant fuzzy linguistic term set (HFLTS) is an approach giving ability to obtain more flexible decision-making (DM) process by integrating linguistic fuzzy modeling (LFM) with hesitative expert judgments. Although HFLTS is widely-studied in the literature and many enhancements are made on HFLTS procedure, none of these enhancements gives ability to continue with precise fuzzy modeling (PFM) in decision process. LFM has a big drawback about accuracy because of the dependency between the size of term set and the comprehensiveness of fuzzy sets (FSs). This issue creates a very critical difficulty in modeling of DM problems that need sensitive evaluations by using HFLTS. This paper aims to solve this problem by proposing a novel HFLTS methodology that is usable for DM problems that need sensitive calculations in the decision stage. The proposed methodology integrates 2-tuple LFM and linguistic fuzzy modifiers with HFLTS to overcome the accuracy problem and obtain more sensitive and flexible decision procedure. This paper also presents an envelopment transformation technique to aggregate expert assessments as a fuzzy membership function instead of membership grades. It becomes possible to keep interpretability in a certain level and achieve sensitive results at the same time with the help of these modifications. The proposed HFLTS approach is analyzed on a real case example from manufacturing industry for acceptance sampling plans (ASP) that is a DM problem requiring sensitive calculations. As another originality of the paper, the main formulations of ASP are derived based on hesitant fuzzy defectiveness information. The obtained results are also compared with some existing enhancements of the HFLTS and the success of the proposed methodology is proved in terms of sensitive calculation.(2201-7200) Pseudo L-algebras
https://ijfs.usb.ac.ir/article_7148.html
We introduce generalized structures of L-algebras, called pseudo L-algebras, which are the multiplication reduct of pseudo hoops and are structures combining two L-algebras with one compatible order. We prove that every pseudo hoop gives rise to a pseudo L-algebra and every pseudo effect algebra gives rise to a pseudo L-algebra. The self-similarity is the most important property of an L-algebra $L$, which guarantees to induce a multiplication on $L$. We introduce a notion of self-similar pseudo L-algebras and prove that a self-similar pseudo L-algebra becomes an L-algebra if and only if the multiplication $\odot$ is commutative. We get some interesting results for self-similar pseudo L-algebras: (1) The negative cone $G^-$ of an $\ell$-group $G$ can be seen as a self-similar pseudo L-algebra. (2) Every self-similar pseudo L-algebra is a pseudo hoop. Next, we introduce the notion of self-similar closures of pseudo L-algebras and obtain a self-similar closure by a recursive method. Given a pseudo L-algebra $(L, \rightarrow, \rightsquigarrow ,1)$, we can generate a free semigroup $(A, \ast)$ by the set $L\setminus \{1\}$. Furthermore, we let $S(L)=A\cup\{1\}$ and define a binary operation $\odot$ on $S(L)$. Then we extend the operations $\rightarrow$ and $\rightsquigarrow$ from $L$ to $S(L)$, and prove that $(S(L), \rightarrow, 1)$ and $(S(L), \rightsquigarrow, 1)$ are two cycloids, respectively. Furthermore, under some conditions, $(S(L), \rightarrow, \rightsquigarrow, 1)$ becomes a self-similar pseudo L-algebra. &nbsp;Finally, we introduce the notion of the structure group of pseudo L-algebras, and give an interesting example to show how to extend a pseudo L-algebra $L$ into the pseudo self-similar closure $S(L)$, and furthermore, derive it's structure group $G(L)$.(2111-7030) VIKOR-based group decision-making method for software quality assessment
https://ijfs.usb.ac.ir/article_7149.html
Software quality is an important research direction in software engineering domain, which is a multi-dimen-sional evaluation problem.The VIsekriterijumska optimizacija i KOmpromisno Resenje (VIKOR) technique is a comprehensivemethod for handing the multi-dimensional evaluation problems. The projection is commonly used to measure the closeness&nbsp;between two decision objects in decision science.However, two research questions are found in this work: (1) There is no specific concrete regretmatrix in current VIKOR technique; &nbsp;(2) The existing projection measures are not always reasonable in interval-valued intuitionistic fuzzy setting. Two research gaps are filled systematically in this paper: (1) A specific concrete regret matrix is provided in this extended VIKOR-based group decision-making method; (2) A new normalized projection measure is provided in order to measure the closeness between two decision matrices.The decision procedures are also provided in this paper. A practical application to the software quality assessment is introduced.Some experimental comparisons are provided in order to illustrate the feasibility and practicability&nbsp;of introduced method.(2102-6507) Fuzzy product rule with applications
https://ijfs.usb.ac.ir/article_7150.html
In this paper, we establish the GH-derivative of multiplication of fuzzy functions, the so-called product rule. For the first time, a product rule is constructed while both of the multiplied functions are assumed to be fuzzy without any restriction on the signs of multiplied functions. The rule is extracted based on the MCE-product and its property of distributivity. Then, we propose two important applications of the fuzzy product rule: An integration by parts formula for fuzzy functions and solving a nonlinear fuzzy differential equation. Some illustrative examples are given to verify the theoretical results.