Journal of Intelligent & Fuzzy Systems - Volume 38, issue 4

Authors: Wang, Zhihua

Article Type: Research Article

Abstract: Let k  ≥ 2 be an integer. The purpose of this paper is first to introduce the notation of Felbin’s type fuzzy normed linear spaces, and then by virtue of this notation to study some stability results concerning the more general cubic functional equation of the form f ( x + ky ) + f ( x - ky ) + f ( kx ) = k 2 f ( x + y ) + k 2 f ( x - y ) + ( k 3 - 2 k 2 + …2 ) f ( x ) in the setting of Felbin’s type fuzzy normed linear spaces by employing the direct and fixed point methods. Then some applications of our results for the stability of the cubic functional equation from a real normed space to a Banach space will be demonstrated. Furthermore, the interdisciplinary relation between the theory of Felbin’s type fuzzy spaces and the theory of functional equations are also presented in this paper. Show more

Keywords: Fixed point method, Felbin’s type fuzzy normed linear space, fuzzy real number, generalized Hyers-Ulam stability, 39B82, 39B72, 46B03, 47H10

DOI: 10.3233/JIFS-191418

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4733-4742, 2020

Authors: Khan, Zahid | Gulistan, Muhammad | Hashim, Rabiya | Yaqoob, Naveed | Chammam, Wathek

Article Type: Research Article

Abstract: The Shewart S -control chart is commonly used as one of the statistical tools for monitoring the process variability. The existing design of S -control is based on the assumption that inspected quality of the observed process is, an exact and clearly specified quantitative quantity. If however, measured data involve some vague and imprecise observations, the conventional approach of the S -control, cannot be practiced. Designing of a generalized neutrosophic S -control chart which could support the indeterminate values in the processing data is originally developed in this article. The associated properties of proposed design under neutrosophic environment have been …established in this study. The proposed chart represents a general design of the existence structure of the S-chart. Using neutrosophic average run length (ARL n ) as a performance measure, a comparative study of the proposed chart with the conventional approach of S -control under vague parameter values is evaluated. Findings both from analytical and simulation studies indicate that proposed design of S -control leads to efficient and more flexible approach over the traditional S -control. A real data example has been provided for demonstrating the implementation procedure of the proposed design. Show more

Keywords: Quality control, fuzzy charts, neutrosophic measures, S-control chart

DOI: 10.3233/JIFS-191439

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4743-4751, 2020

Authors: Ahmad, Jawad | Tahir, Ahsen | Larijani, Hadi | Ahmed, Fawad | Aziz Shah, Syed | Hall, Adam James | Buchanan, William J.

Article Type: Research Article

Abstract:  Energy uncertainty and ecological pressures have contributed to a high volatility in energy demand and consumption. The building sector accounts for 30 to 40% of the total global energy consumption. There is a high demand for novel techniques and viable energy strategies for reducing energy consumption in this domain. Energy prediction models have the potential to play a pivotal role in optimising energy consumption. The proposed work presents a new and accurate Energy Demand Prediction (EDP) model for large buildings. This approach leverages the Random Neural Network (RNN) prediction methodology. The proposed RNN-based EDP is compared with traditional Artificial Neural …Network (ANN), Support Vector Machine (SVM) and linear regression models. A large building is modelled and simulated for one year in the Integrated Environment Solutions Virtual Environment (IES-VE). Several data inputs such as air temperature, internal gain and the number of people (occupancy) are calculated from IES-VE model and provided to traditional ANN and the proposed RNN predictor. A number of test parameters such as Root Mean Square (RMSE), Normalized Root Mean Square (N-RMSE), Mean Absolute Percentage Error (MAPE) and R provide the proposed RNN model with higher accuracy over the traditional ANN, SVM and linear regression. The proposed RNN predictor provides approximately half of the error of the ANN model. The traditional ANN model gives higher error values of 2.07×, 1.83× and 2.35× for RMSE, NRMSE and MAPE, respectively as compared to the proposed RNN model. Furthermore, the error values of SVM and linear regression were also higher than the proposed EDP scheme. Show more

Keywords: Buildings, energy demand, random neural network, prediction

DOI: 10.3233/JIFS-191458

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4753-4765, 2020

Authors: Madadi, Masoumeh | Saadati, Reza

Article Type: Research Article

Abstract:  We attempt to solve some bi-additive θ -random operator inequalities and use the fixed point technique to prove the fuzzy version of Hyers-Ulam-Rassias stability of them.

Keywords: Hyers-Ulam-Rassias stability, bi-additive θ-random operator inequality, fuzzy sets, 54H20, 46L05, 39B62, 43A22

DOI: 10.3233/JIFS-191482

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4767-4777, 2020

Authors: Wang, Rui | Li, Dengfeng | Yu, Gaofeng

Article Type: Research Article

Abstract: One of the critical activities for bilateral matching decision is matching accuracy, which may be regarded as a type of bilateral matching decision problem with heterogeneous information and attribute association. This paper aims to develop a new fuzzy linear programming method to address such problems. In the proposed method, the multiple attributes are expressed as exact numbers, interval numbers, triangular fuzzy numbers, intuitionistic fuzzy numbers, linguistic terms, and neutrosophic numbers. Firstly, the distance of heterogeneous data and fuzzy measures are introduced; meanwhile, heterogeneous information attribute weights are calculated based on the Choquet integral. Then based on the psychological characteristics of …matching participants’ loss avoidance and superiority maximization, the lexicographical method is used to solve the multi-objective linear programming model to obtain the optimal bilateral transaction matching pair. Finally, an example of second-hand housing online rental-sales matching problems is analyzed to demonstrate the implementation process and applicability of the method proposed in this paper. Show more

Keywords: Bilateral matching decision, attribute association, heterogeneous information, prospect theory

DOI: 10.3233/JIFS-191495

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4779-4792, 2020

Authors: Jing, Xin | Li, Shihao | Cheng, Jing | Guo, Junjun

Article Type: Research Article

Abstract: In order to solve the problem of resource waste in colleges and universities, this paper proposes a multidimensional situational information fusion method, which can be used to normalize, analyze and predict the multi-source data such as natural, humanistic and spatio-temporal data on campus so as to meet the application requirements for high-level decision-making. With this method, firstly, the event object model is used to normalize multi-source data. Then, the multidimensional situational information fusion mechanism of twice reasoning is used to obtain the real-time situation and equipment control scheme of the campus so that real-time intelligent semantic understanding is realized. In …the process of reasoning, the improved KNN prediction model is used to predict situational trends, and the prediction information is used to continue deep reasoning and mining. Finally, the real-time energy-saving regulation is carried out through control instructions. In addition, through simulation verification, experimental results show that this method can quickly identify, integrate, and predict the current real-time situation and generate reasoning results, and finally achieve the goal of intelligent control for energy saving on campus. Show more

Keywords: Smart campus, energy saving, situational information fusion, intelligent control

DOI: 10.3233/JIFS-191513

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4793-4807, 2020

Authors: Mohanta, Kartick | Dey, Arindam | Pal, Anita | Long, Hoang Viet | Son, Le Hoang

Article Type: Research Article

Abstract:  Information in many real life problems collect from multi agents, i.e., "multipolar information" exists. This multipolar information cannot be properly modeled by m - polar fuzzy graph or intutionistic fuzzy graph. An m -polar neutrosophic model is very much efficient for such real word problems which can construct more precise, flexible, and comparable system as compared to the classical, fuzzy and neutrosophic graph models. In this paper, we present the definition of m -polar neutrosophic graph model. Some new operations, such as union, join, composition and ring sum of two m -polar neutrosophic graph are defined here. We define six …new products on m -polar neutrosophic graphs namely strong product, semi strong product, complete product, direct product, cartesian product and lexicographic product. Some idea of complement, isomorphism, weak and co weak isomorphism on m -polar neutrosophic graph are introduced here. We also present several associated properties and theorems of m -polar neutrosophic graph. We introduce a model of m -polar neutrosophic graph, which is applied in evaluating the teacher’s performance of a college. The performances of the teachers are computed based on the response score (feedback) of the students of the college. We also present a numerical example to illustrate our proposed model. Show more

Keywords: Multipolar information, neutrosophic set, m-polar neutrosophic graph, union, direct product

DOI: 10.3233/JIFS-191520

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4809-4828, 2020

Authors: Fezai, Radhia | Mansouri, Majdi | Abodayeh, Kamaleldin | Nounou, Hazem | Nounou, Mohamed | Messaoud, Hassani

Article Type: Research Article

Abstract:  In this paper, a novel fault detection and isolation (FDI) framework based on kernel PCA (KPCA) and generalized likelihood ratio test (GLRT) that is capable of detecting and identifying faults is developed. Specifically, three main objectives are addressed. First, system model identification and residuals generation are addressed using KPCA model. Second, KPCA-based GLRT method is proposed to detect different types of faults in the systems. Third, partial KPCA (PKPCA)-based GLRT is developed for fault isolation. The proposed approach aims to apply a structured PKPCA -based GLRT to a set of sub-models. The fault detection and isolation performances using PKPCA-based GLRT …are illustrated through two examples: a simulated continuous stirred tank reactor (CSTR) data and an air quality monitoring network data. The obtained results demonstrate the effectiveness of the partial KPCA-based GLRT method over the partial PCA-based GLRT method. Show more

Keywords: Partial kernel principal component analysis (PKPCA), fault detection and isolation (FDI), generalized likelihood ratio test (GLRT), continuous stirred tank reactor (CSTR), air quality monitoring networks (AQMN)

DOI: 10.3233/JIFS-191525

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4829-4843, 2020

Authors: Liu, Yang | Lio, Waichon

Article Type: Research Article

Abstract: Nowadays, uncertainty theory has become a branch of axiomatic mathematics and has been studied by many researchers. In particular, uncertainty distribution is one of the most important tools to deal with indeterminate quantity in uncertainty theory. Peng and Iwamura (2010) presented a sufficient and necessary condition of a function being an uncertainty distribution. This paper gives a counterexample to illustrate this condition is not appropriate. A revision of the sufficient and necessary condition is also provided in this paper.

Keywords: Uncertainty theory, uncertain variable, uncertainty distribution

DOI: 10.3233/JIFS-191535

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4845-4854, 2020

Authors: Zhang, Peiwen | Tao, Zhifu | Liu, Jinpei | Jin, Feifei | Zhang, Junting

Article Type: Research Article

Abstract: The aim of this paper is to develop a multi-attribute group decision making (MAGDM) with picture fuzzy sets based on ELECTRE TRI method, i.e., an ELECTRE TRI based group decision making with picture fuzzy information is given. The MAGDM with picture fuzzy information based on picture fuzzy ELECTRE TRI outranking method is divided into three stages, i.e., the group decision information aggregation stage, determination of parameters and ELECTRE TRI outranking based outranking stage. A novel comparison law for picture fuzzy sets is introduced. In the group decision information aggregation stage, the concept of picture fuzzy normalized weighted Bonferroni mean (PFNWBM …) is developed. The developed decision procedure is further applied to the assessment of energy security. The numerical example shows that the developed group decision procedure is feasible and valid. Show more

Keywords: Multi-attribute group decision making, picture fuzzy sets, ELECTRE TRI, supplier selection, picture fuzzy normalized Bonferroni mean

DOI: 10.3233/JIFS-191540

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4855-4868, 2020

Authors: Liu, Yan | Wang, Xiao-Kang | Wang, Jian-Qiang | Li, Lin | Cheng, Peng-Fei

Article Type: Research Article

Abstract: This paper proposes a cloud model-based Preference Ranking Organisation Method for Enrichment Evaluation (PROMETHEE) method with 2D uncertain linguistic variables (2DULVs). 2DULVs are adopted by decision makers (DMs) to evaluate each alternative under the criteria because they can provide extra evaluation information. Cloud model is adopted to depict randomness and fuzziness. The possibility degree and possibility degree index are defined to develop an improved PROMETHEE II method for sorting alternatives. Entropy weight method is used to calculate the weight of each criterion. A renewable energy performance sample is used to illustrate the applicability of the proposed method. Sensitivity analysis and …four comparative experiments demonstrate the stability and accuracy of the proposed approach. Show more

Keywords: 2D uncertain linguistic variable, cloud, possibility degree, possibility degree index, improved PROMETHEE

DOI: 10.3233/JIFS-191546

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4869-4887, 2020

Authors: Qin, Xuezhi | Lin, Xianwei | Shang, Qin

Article Type: Research Article

Abstract:  In order to introduce the long memory property of financial markets into the study of binary option pricing under fuzzy environment, the fractional Brownian motion is used to describe the dynamics of the stock price. This paper develops a new framework for pricing the binary option by using fuzzy set theory based on the long memory property of financial markets. The fuzzy price of the binary option is obtained by using a risk-neutral pricing principle and quasi-conditional expectation. To better understand the pricing model, some Greeks of this pricing model are given. In addition, the influence of the Hurst parameter …H , a measure of long memory in the financial market, on binary option pricing is analyzed. Finally, the study provides an example that study binary option by fuzzifying the maturity value of the stock price using the triangular fuzzy number. The numerical experiment demonstrates the fuzzy pricing model proposed is rational and practicable. Show more

Keywords: Binary option, fuzzy option pricing, fractional brownian motion, asset-or-nothing option

DOI: 10.3233/JIFS-191551

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4889-4900, 2020

Authors: Yin, Shizhuang | Wang, Tao

Article Type: Research Article

Abstract: In order to solve the clustering problem of unknown binary protocols, an improved k -means unknown binary protocol clustering method is proposed, which determines the initial clustering center and improves the clustering distance. Firstly, the k value is determined and the clustering center is extracted by using DCBP (Determine the initial clustering center of binary Protocol) algorithm and the change rate of error square, and then the data are clustered by improving the k -means algorithm of distance function. The unknown binary protocol bit stream is divided into different subsets of binary protocols. By improving the k -means algorithm, …the Pearson distance improves the accuracy of binary protocol clustering from 96% to 98.9%. The DCBP algorithm helps us to determine the k value accurately. The k value determined in this paper is 5, and the clustering accuracy is 98.9%. The clustering accuracy is 80% when k is 4 and 92.2% when k is 6. And the operation speed of the improved k -means algorithm is better than that of the AGNES algorithm. The algorithm is better adapted to the clustering of unknown binary protocols, and improves the accuracy of clustering and the speed of operation. Show more

Keywords: Protocol identification, unknown binary protocol, Pearson distance, determine cluster center

DOI: 10.3233/JIFS-191561

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4901-4913, 2020

Authors: Gao, Yi | Sun, Xia | Wang, Xin | Guo, Shouxi | Feng, Jun

Article Type: Research Article

Abstract:  Forum posts in Massive Open Online Courses (MOOCs) support an important way for online learners to interact with each other and with instructors. Instructors explore the sentiment from posts in MOOCs to detect learners’ trending opinions towards the course so that they can improve MOOCs. However, it is unrealistic to expect instructors to adequately track learners’ sentiment under the large number of messages exchanged on the forums. Fortunately, sentiment classification can automatically analyze learners’ emotion on the course of MOOCs from posts. Traditional classifiers based on machine learning algorithm, which often depend on human-designed features and have data sparsity problem. …In contrast to traditional approaches, we develop a novel neural network model called parallel neural network (PNNs) for sentiment classification of MOOCs discussion forum to alleviate the aforementioned problems. In our model, we design a parallel neural network structure to replace the popular serial neural network structure so that PNNs can preserve the validity of features as far as possible when neural network model training. Meanwhile, we also introduce Self-attention mechanism that automatically identifies which features play key roles in sentiment classification to obtain the important components in posts. We experiment on a public MOOCs dataset and two common sentiment classification datasets, and achieve a good performance. That means PNNs is a substantially reliable classification model for identifying the sentiment polarity of posts. The study has great potential application value on the platform of large scale courses, which can help instructors to gain the emotional tendency of learners for the course content in real time, so that timely intervention to support learning and may reduce the dropout rates. Show more

Keywords: Parallel neural network, sentiment classification, MOOCs, learners’ sentiment

DOI: 10.3233/JIFS-191572

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4915-4927, 2020

Authors: Alos, Ahmad | Dahrouj, Z.

Article Type: Research Article

Abstract: The importance of detecting faults in Unmanned Aerial Vehicles motivated researchers to work in this area over recent years. Complex relationships among UAV attributes (Sensor readings, and Commands) make the task a bit challenging. Many known algorithms consider detecting the faults by spotting data anomalies in the values of each attribute without concern for their context, which leaves an opportunity for potential improvement. The contextual faults occur when a defected sensor shows an invalid value concerning other attributes. Our contribution is a novel matrix platform for detecting the potential contextual faults. This platform consists of multiple small Decision Trees, instead …of using one huge single Decision Tree, which could be difficult and time-consuming to produce, particularly in the case of a large dataset with too many attributes. We propose to use the C4.5 decision tree algorithm to build each decision tree. The Decision Tree is a machine learning technique, which is an effective supervised method used for classification. It is computationally inexpensive and capable of dealing with noisy data. Besides, our approach uses a sliding window technique during training and testing phases, which brings into consideration the effect of the previous state of the system on the process of detecting the contextual faults. The algorithm starts by collecting the attributes of the UAV into a table of pairs, where each pair consists of two attributes; then, it defines the Decision Tree matrix by assigning one Decision Tree for each pair of attributes. The Training step includes constructing training sub-datasets using the values of sliding windows. The C4.5 algorithm uses each constructed training sub-dataset to induce one Decision Tree in the matrix. Finally, the testing step is responsible for reading the values of the sliding windows and using the concerned Decision Tree to detect the contextual faults. We evaluated our approach using Detection Rate, False Alarm Rate, Precision, and F1-score indicators. Moreover, we made a comparison with other broadly used algorithms, such as K-Means and One-Class SVM. Our approach showed superior results in detecting different types of faults (sensor-offset, sensor-stuck, sensor-drift, and sensor-cut). The DT-Matrix performance was neither affected by the small values of the outliers, nor by the number of the outliers, and this caused the DT-Matrix to work better in most of the experiments compared to the other algorithms. Show more

Keywords: UAV, decision tree, anomaly detection, abnormal, classification, system failure, sensor faults, contextual faults, supervised algorithm

DOI: 10.3233/JIFS-191575

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4929-4939, 2020

Authors: Xue, Wei | Wang, Qi | Liu, Xiaona

Article Type: Research Article

Abstract: Although the Takagi-Sugeno-Kang (TSK) fuzzy classifier has achieved great success, how to further improve its classification performance and enhance its interpretability is still one of the most difficult challenges. Involved with the fusion of existing decision information and pre-known classification task, a newly proposed deep/hierarchical TSK fuzzy classifier (EDIPK-TSK) with interpretable fuzzy rules makes full use of the classification advantages of each base classifier to construct a multi-layer deep learning structure. This study first considers that the existing decision information of each training sub-block is sequentially projected into the subsequent sub-blocks for training. Undoubtedly, the existing decision information has played …a guiding role in the current learning process to some extent. Simultaneously, the pre-known classification task is fused into the decision information for fine-tuning of it, which can significantly improve the efficiency of guidance and accelerate the fitting speed of the model. In each layer, the use of interpretable integration input space guarantees that EDIPK-TSK is not a black box. The proposed deep classifier can realize learning by using short fuzzy rules, which ensures the satisfactory interpretability of the classifier. The final experimental results also verify that EDIPK-TSK has strong classification advantages and interpretability. Show more

Keywords: Fuzzy classifier, deep learning structure, existing decision information, interpretability, classification performance

DOI: 10.3233/JIFS-191579

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4941-4957, 2020

Authors: Chang, Wen-Jer | Chang, Chih-Ming | Lin, Yann-Hong

Article Type: Research Article

Abstract: A novel robust fuzzy controller design problem subject to multi-variance constraints and pole location constraints for nonlinear discrete-time systems with internal and external noises is studied in this paper. Based on the Takagi-Sugeno fuzzy model, the nonlinear discrete-time systems are represented by blending many linear subsystems. The control performances considered in this paper include stability requirement, pole location constraint, individual state variance constraint, and minimum output variance. Applying the Lyapunov theory, a discrete-time robust fuzzy controller is designed based on parallel distributed compensation technology and the relevant conditions are deduced in the form of linear matrix inequalities. By solving these …conditions, a discrete-time robust fuzzy controller can be obtained to satisfy the above performance constraints. At last, some simulations for controlling a nonlinear inverted pendulum system and a nonlinear ship steering system are provided to show the feasibility and applicability of the proposed robust fuzzy control method. Show more

Keywords: Robust fuzzy control, discrete-time Takagi-Sugeno fuzzy model, variance constraints and pole location constraints

DOI: 10.3233/JIFS-191600

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4959-4975, 2020

Authors: Shahzadi, Gulfam | Akram, Muhammad | Davvaz, Bijan

Article Type: Research Article

Abstract:  A Pythagorean fuzzy soft set, an extension of intuitionistic fuzzy soft set, plays an essential role to handle the vagueness in many real-life problems. We apply this concept to graph theory, and present certain new notions including, perfectly regular Pythagorean fuzzy soft graphs (PFSGs), perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum ( O ˆ NDS ) and closed neighborhood degree sum ( C ˆ NDS ) of PFSGs. …Finally, we discuss some decision-making problems of PFSGs. Show more

Keywords: Pythagorean fuzzy soft graphs, perfectly edge-regular, perfectly edge-irregular

DOI: 10.3233/JIFS-191610

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4977-4991, 2020

Authors: Taimoor, Muhammad | Aijun, Li

Article Type: Research Article

Abstract: An online fault detection, isolation, and reconstruction strategy is proposed for actuators and sensors fault detection of an aircraft. For increasing the fault detection capabilities, the Extended Kalman Filter (EKF) is used for the weight updating parameters of multi-layer perceptron (MLP) neural network. The main purpose of using the EKF is to make the weight updating parameters of MLP adaptive in order to increase the fault detection, isolation and reconstruction preciseness, efficiency and rapidness compared to the conventional MLP where the fixed learning rate due to which it has slow response to faults occurrence. Because of the online adaptation of …weighting parameters of MLP, the preciseness of the faults detection is increased. For testing and validation of the proposed strategy, the nonlinear dynamics of Boeing 747 100/200 are used. Results demonstrate that the proposed strategy has better accuracy and rapid response to fault detection compared to convention multi-layer perceptron neural network based faults detection schemes. Show more

Keywords: Actuators, sensors, fault detection and isolation, aircraft, neural networks, nonlinear systems

DOI: 10.3233/JIFS-191627

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 4993-5012, 2020

Authors: Xue, Zhan’ao | Zhao, Li-Ping | Zhang, Min | Sun, Bing-Xin

Article Type: Research Article

Abstract: Three-way decisions have become a representative of the models dealing with decision-making problems with uncertainty and fuzziness. However, most of the current models are single granular structures that cannot meet the needs of complex fuzzy environmental decision-making. Multi-granulation rough sets can better deal with fuzzy problems of multiple granularity structures. Therefore, three-way decisions will be a more reasonable decision-making model to address uncertain decision problems in the context of multiple granularity structures. In this paper, firstly we propose the four different conditional probabilities based on support intuitionistic fuzzy sets, which are referred to as support intuitionistic fuzzy probability. Then, a …multi-granulation support intuitionistic fuzzy probabilistic approximation space is defined. Secondly, we calculate the thresholds α and β by the Bayesian theory, and construct four different types of multi-granulation support intuitionistic fuzzy probabilistic rough sets models in multi-granulation support intuitionistic fuzzy probabilistic approximation space. Moreover, some properties of lower and upper approximation operators of these models are discussed. Thirdly, by combining these proposed models with three-way decision theory, the corresponding three-way decision models are constructed and three-way decision rules are derived. Finally, an example of person-job fit procedure is given to prove and compare the validity of these proposed models. Show more

Keywords: Support intuitionistic fuzzy sets, rough sets, support intuitionistic fuzzy probabilistic, multi-granulation rough sets, three-way decisions

DOI: 10.3233/JIFS-191657

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5013-5031, 2020

Authors: Ghasemzadeh, Mehdi | Hadidi, Khayrollah

Article Type: Research Article

Abstract: In this paper, the design and simulation results of a general-purpose fuzzy logic controller (FLC) with mixed-signal (analog and digital) inputs and digital outputs are presented. Based on a new strategy, it provides simplicity and high speed from the analog prospective and a total digital system advantages with unchanged digital system properties. A novel and reliable structure with respect to other topologies for the fuzzifier section is designed which enhances the accuracy and the velocity. In order to detect minimum and maximum of the input currents at the same time, an inference engine consisting of a min & max circuit …is an addition. The benchmark for the defuzzifier in the proposed design is simplicity and through a simple approach, the center of area (COA) is attributed to the defuzzifier. The proposed controller circuit consists of two inputs, sixteen rules and one output designed in 0.35μ m CMOS standard technology and simulated with MATLAB systematically. The total controller circuit is simulated with HSPICE simulator (BSIM3v3 parameters) and the layouts were extracted with Cadence Virtuoso v 5.1. The inference speed of the controller is about 41.3 MFLIPS (fuzzy logic inference per second) and power consumption is 3.2 mW. Show more

Keywords: A/D converter, CMOS fuzzy controller, current mode circuits, defuzzifier, fuzzifier, fuzzy logic

DOI: 10.3233/JIFS-191672

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5033-5044, 2020

Authors: Rajendra Thilahar, C. | Sivaramakrishnan, R.

Article Type: Research Article

Abstract: A tele-operated robot stereo vision system is used for stretching out the operator’s eye-hand motion and its distance based co-ordination with experts. The major challenge is the reduction of communication delay by using effective decisions to avoid tele-operation instability. This problem can be handled effectively by using the principles of Augmented Reality which provides facilities for superimposing virtual objects onto the real video images of the workspace to create a simulation plan in the client system. In this paper, we propose a new feature selection algorithm called Fuzzy Rules and Information Gain Ratio based Feature Selection Algorithm for selecting the …optimal number of features from the full set of available features. Also, a new Fuzzy Rule based Neuro-Genetic Classification Algorithm is proposed in this paper for classifying the augmented images more accurately. The main advantages of the proposed model are reduction in classification and communication time and increase in decision accuracy. Show more

Keywords: Image classification, augmented reality, virtual reality, fuzzy systems, robot motion

DOI: 10.3233/JIFS-191674

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5045-5054, 2020

Authors: Qiao, Sha | Zhu, Ping

Article Type: Research Article

Abstract:  A large number of studies have investigated the systems of fuzzy relation equations and inequalities, which have a much wider field of application. In this paper, we study several types of systems of fuzzy relation equations and inequalities consisting of a given family of k -ary fuzzy relations, where natural number k  ≥ 2, and three unknown fuzzy relations over complete residuated lattices and meet-continuous lattices. Their solutions are triples of fuzzy relations. For the systems of fuzzy relation inequalities, we give the greatest solutions contained in a given triple of fuzzy relations, the least solutions containing a given triple of …fuzzy relations, or give maximal solutions contained in or containing a given triple of fuzzy relations, or belonging to a given interval of triples of fuzzy relations over complete residuated lattices and complete meet-continuous lattices. For the systems of fuzzy relation equations, we present a method of computing maximal solutions contained in a given triple of fuzzy relations and a method of computing minimal solutions containing a given triple of fuzzy relations. Furthermore, we provide some conditions under which there exist the greatest solutions contained in and the least solutions containing a given triple of fuzzy relations. Show more

Keywords: Fuzzy relation, fuzzy relation equation, fuzzy relation inequality, complete residuated lattice, complete meet-continuous lattice

DOI: 10.3233/JIFS-191695

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5055-5076, 2020

Authors: Lin, Chih-Hong

Article Type: Research Article

Abstract: A six phase copper rotor induction motor (SPCRIM) drive system still exists in lots of nonlinear characteristics such as the added load torque, the Stribeck effect torque, the the cogging torque, the coulomb friction torque and the parameters variations. Due to some uncertainties effects, the using linear controller can not achieve better control performance for the SPCRIM drive system. To obtain better performance, a clever backstepping control system using two adaptive laws and a hitting function is proposed for controlling the SPCRIM drive system. To improve larger chattering phenomenon under uncertainties affects for aforementioned control system, the clever backstepping control …system using two adaptive laws, a revised recurrent fuzzy neural network (RRFNN) and a compensated controller is proposed to estimate the required lumped uncertainty and to compensate the minimum reconstructed error of the estimated law. Furthermore, the corrected particle swarm optimization (CPSO) algorithm by using variable dynamic inertia weight and variable dynamic constriction factor with segment regulation mechanics that is the innovativeness for using the CPSO algorithm is adopted to regulate four variable learning rates of the weights in the RRFNN to speed-up parameter’s convergence. Finally, comparative performances through some experimental results are verified that the clever backstepping control system using two adaptive laws, a RRFNN and a compensated controller has better control performances than those of the proposed methods for the SPCRIM drive system. Show more

Keywords: Backstepping control, copper rotor induction motor, Lyapunov stability, particle swarm optimization, recurrent fuzzy neural network

DOI: 10.3233/JIFS-191712

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5077-5093, 2020

Authors: Imran, Muhammad | Siddiqui, Muhammad Kamran | Baig, Abdul Qudair | Shaker, Hani

Article Type: Research Article

Abstract:  The topological descriptor are numerical parameters of a graph which characterize its topology and are usually graph invariants. Nowadays, Biological science is an energizing and quickly creating branch of knowledge together with the topological descriptor. In recent years, the investigation of living things has experienced huge extension. All the living things are composed of a fundamental unit of life called cells. The microbiology is a science that deals with the living creatures that can not be seen by naked eyes like bacteria, viruses. In recent years, eccentricity based topological indices gain a lot of importance in many disciplines like chemistry, …computer science, integrated circuits, electric circuits, communication networks, biological networks. In a connected graph G , the vertex set V  (G ) shows the bacteria and the edge set E  (G ) shows the relationship between two bacterium. Mostly, the reproduction of bacteria and other microorganisms occur by binary fission process. The topological indices play a vital and useful role in indicating and analyzing physical, chemical and biological properties of any molecular graph. In this paper, we have computed eccentric polynomial and eccentric atom bond connectivity index of hyper binary trees networks (k -level) and relate these networks to biological networks. Also discuss how biological activities of these networks work in daily life. Show more

Keywords: Biological networks, molecular structure descriptor, eccentric atom bond connectivity, eccentric connectivity polynomial, binary tree, hyper binary trees networks

DOI: 10.3233/JIFS-191714

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5095-5105, 2020

Authors: Akram, Muhammad | Dudek, Wieslaw A. | Habib, Amna | Al-Kenani, Ahmad N.

Article Type: Research Article

Abstract: The imperfect competition models are equipped by fuzzy set theory with direct assessments of uncertainty. An appropriate point of departure for origination of a system with potentially broader coverage can be provided in view of fuzzy sets. In this way, several extensions of fuzzy set have been introduced to deal with uncertain and ambiguous information including relationships between objects. The q-rung picture fuzzy (q-RPF) model, which inherits the virtues of q-rung orthopair fuzzy set and picture fuzzy set, is one of the convenient way to represent such information. In order to exhibit interactions in various economic structures the conception of …q-RPF economic competition graphs can be employed. Thus the intention of present study is to deal with q-rung picture fuzzy competition graphs (q-RPFCGs) and in particular, q-rung picture fuzzy economic competition graphs (q-RPFECGs) with its generalizations: q-RPF k -economic competition graphs; p -economic competition q-RPFGs; and m -step q-RPFECGs through several important results. Furthermore, this paper offers a brief review for perfect and imperfect competition in competitive market structures and sketch q-RPFECGs to represent duopoly, oligopoly, and monopolistic competitions in graph theoretic approach. Also, it designs an algorithm to calculate the strength of economic competition among buyers and sellers in imperfect competitive markets with q-RPF information. Show more

Keywords: q-Rung picture fuzzy competition graphs, q-rung picture fuzzy economic competition graphs, imperfect competition models, duopoly, oligopoly, monopolistic competition

DOI: 10.3233/JIFS-191726

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5107-5126, 2020

Authors: Vinh An, Truong | Van Hoa, Ngo

Article Type: Research Article

Abstract: In this work, a new class of generalized fractional integral equations involving the kernel ψ -function in the fuzzy setting is introduced. With this problem, we can recover a wide class of fractional fuzzy integral equations by choosing the kernel ψ -function. In this sense, we provide sufficient conditions for the existence, uniqueness of solutions and δ -Ulam-Hyers-Rassias stability of the given problems. Some examples are given to illustrate our main results.

Keywords: δ-Ulam-Hyers-Rassias, kernel ψ-functions, Fuzzy fractional integral equations

DOI: 10.3233/JIFS-191743

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5127-5141, 2020

Authors: Vinh An, Truong | Van Hoa, Ngo

Article Type: Research Article

Abstract:  In this work, we consider a new form of fuzzy fractional Volterra integral equations (FFVIEs) involving the generalized kernel functions. By using the monotone iterative technique (MIT) combined with the method of lower and upper solutions, the existence of extremal solutions of FFVIEs is established. Some examples are given to illustrate our main results.

Keywords: Fuzzy fractional integral equations, generalized kernel functions, extremal solutions, monotone iterative technique

DOI: 10.3233/JIFS-191746

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5143-5155, 2020

Authors: Akbari, Reza | Dabbagh, Rahim | Ghoushchi, Saeid Jafarzadeh

Article Type: Research Article

Abstract: One of the most crucial components in risk management in an organization is detection of risk modes in a system, prioritization of them and making plans in order to enact corrective actions. And one of the common methods for prioritization of risks is the conventional Failure Mode Effects Analysis (FMEA). Although this approach is widely used in different industries, it suffers from some shortcomings, which can lead to failures in reaching reality-based results. This research study, therefore, proposed an approach in three phases for the compensation of the shortcomings of the FMEA method. In the first phase, the FMEA method …was used to detect different risk modes and then assign values to the Risk Priority Number (RPN) determinant factors. In the second phase, the weights of the triple factors were calculated by means of Fuzzy Best-Worst Method (FBWM) and experts’ opinions. And finally, with respect to the outputs of previous phases, the risks were ranked by means of the proposed Z-WASPAS method. In addition to the assignment of different weights to the triple factors and considering the feature of uncertainty in these factors, the proposed approach paid attention to reliability in the risk modes via the Z-Numbers theory. The proposed approach was applied in the operation processes of Mes-e Sarcheshmeh molybdenum factory in Iran and the results indicated a full ranking of risks compared to other conventional methods such as FMEA and fuzzy WASPAS. Show more

Keywords: Failure mode effects analysis, HSE, Z-Numbers, fuzzy BWM, WASPAS

DOI: 10.3233/JIFS-191749

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5157-5173, 2020

Authors: Shukla, Shilpi | Jain, Madhu

Article Type: Research Article

Abstract: Deep learning is far and wide considered to be the most powerful method in computer vision fields, which has a lot of applications such as image recognition, robot navigation systems, and self-driving cars. Recent developments in neural networks have led to an efficient end-to-end architecture to human activity representation and classification. In light of these recent events in deep learning, there is now much considerable concern about developing less expensive computation and memory-wise methods. This paper presents an optimized end-to-end approach named stochastic deep conviction network (SDCN) formulated using the deep learning method. It comprises of deep learning method namely …deep belief network (DBN), two supervised machine learning algorithm support vector machine (SVM) and decision tree (DT) with optimization capability for speech emotion identification. In the beginning, pre-processing is performed and the features are automatically extracted from the input speech signal by the DBN. Since speech signal features loses most of the information and the performance cannot be guaranteed because dynamic interactions can generate uncountable emotion-specific experiences that have the same core feeling state but different perceptual inclinations so DBN provides more robust features. The next step is to classify the emotions in the training phase; here the SVM classifier is chosen which performs dual classification. In order to enhance this classification process, defects must be reduced and the best discrimination of the extracted features should be obtained hence particle swarm optimization (PSO) technique is being added along with SVM classifier in the training phase. To reduce the over fitting problem and risks of a single classifier a DT is being used in the testing phase for the exact identification of emotions (anger, disgust, fear, happiness, neutral and sadness) and therefore it obtains better performance than a single classifier. The complication of the decision tool is that it can increase the computation time. Thus to eliminate this defect whale optimization (WO) technique is being added to the decision tree to reduce the complexity of the system, which in turn lessens the time taken for recognizing the emotion of the speech signal. This formulated proposed SDCN system improves the recognition rate accurately. In this work, theMATLAB environment is being preferred to perform speech emotion recognition. Using the proposed technique the achieved accuracy of emotion detection is above 95% and the identification of various emotions exceeds 98% recognition rate with a computation time of 23 seconds, which has not been achieved so far by any other existing techniques. Show more

Keywords: Stochastic deep conviction network, restricted Boltzmann machine, particle swarm optimization, support vector machine, whale optimization

DOI: 10.3233/JIFS-191753

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5175-5190, 2020

Authors: Liu, Jinpei | Zheng, Yun | Jin, Feifei | Li, Hongyan | Chen, Huayou

Article Type: Research Article

Abstract: Considering the influence of undesirable outputs in environmental assessments of the practical production process, this paper proposes a DEA cross-efficiency method and fuzzy preference relation based on semi-disposability of undesirable outputs for environmental assessments, which combines self-evaluation with peer-evaluation and avoids potentially unrealistic weighting scheme. We first develop a new input-oriented DEA model based on semi-disposability of undesirable outputs. Then, the corresponding DEA cross-efficiency method is proposed to evaluate the homogeneous decision-making units (DMUs) and the additive fuzzy preference relation is constructed using the DEA cross-efficiency scores. An output-oriented DEA model is adopted to derive the priority vector from the …constructed additive fuzzy preference relation. Furthermore, we summarize the specific procedures for environmental assessments based on DEA cross-efficiency and fuzzy preference relation. Finally, the practical example of industrial environment assessment of the Yangtze River Economic Belt is provided to show the applicability and effectiveness of the proposed method. Show more

Keywords: DEA cross-efficiency, semi-disposability, performance evaluation, preference relations, undesirable outputs

DOI: 10.3233/JIFS-191777

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5191-5201, 2020

Authors: Duong, Truong Thi Thuy | Phong, Le Thai | Hoi, Le Quoc | Thao, Nguyen Xuan

Article Type: Research Article

Abstract: The Quality Function Deployment (QFD) problem is an effective tool for translating the customer’s voice into product characteristics, which reduces production costs and improves customer satisfaction. The purpose of this paper is to put forward a new similarity measure based on the concept of interval neutrosophic sets and propose a novel decision making model that combines QFD and similarity measure to evaluate and select market segments. The interval neutrosophic set is used to express the importance of criteria WHATs, technical characteristics HOWs, correlations of WHATs and HOWs and the aggregated rating of each market segment.

Keywords: Interval neutrosophic set, decision making, QFD, market segment, similaity measure

DOI: 10.3233/JIFS-191790

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5203-5214, 2020

Authors: Zhao, Mengke | Wu, Jian | Cao, Mingshuo | Yu, Zhaoyan

Article Type: Research Article

Abstract:  This article proposes a novel multi-criteria group decision making (MCGDM) approach with multi-granularity hesitant fuzzy linguistic term set (HFLTS). It consists three aspects: (1) The processing algorithms for multi-granularity HFLTS; (2) G-DEMATEL model based on HFLTS; (3) A consensus model with the feedback mechanism. To do that, the relative projection model for multi-granular hesitant fuzzy language information is presented and the similarity degree between individual decision matrices based on relative projection is defined. On the basis, the similarity degree is used to determine the expert’s weight vectors, and the SD-MGHIOWA operator is defined to aggregate the experts opinions. The traditional …G-DEMATEL model is improved by the multi-granular hesitant fuzzy language and a new model is built to analyze the correlation between criterion and weight vector. Furthermore, consensus degree is defined from three levels to identify the inconsistent experts, and a feedback mechanism is activated to generate recommendation advice for the inconsistent experts to increase consensus degree. After that, a comprehensive score mechanism of alternatives is designed to select the most appropriate alternative after the consensus is reached. The main characteristics of the proposed MCGDM is that it not only considers the correlation between criterions but also provides the consensus model with the feedback mechanism in the context of hesitant fuzzy language. Finally, an example is provided to illustrate the feasibility and effectiveness of the developed method, which are then compared to the existing methods. Show more

Keywords: Consensus, MCGDM,G-DEMATEL, HFLTS, relative projection, similarity degree

DOI: 10.3233/JIFS-191805

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5215-5229, 2020

Authors: Nguyen, Luong V. | Thu, Nguyen T.

Article Type: Research Article

Abstract: In this paper, we prove a fixed point theorem for intuitionistic fuzzy mappings concerning F -contractions without using the Pompeiu – Hausdorff distance between cut sets of intuitionistic fuzzy mappings. This result and its consequences extend and generalize several results in the literature. An application to the existence for solutions of a delay integral equation of Volterra type and an example are given to illustrate the usability of our results.

Keywords: Intuitionistic fuzzy mappings, intuitionistic fuzzy fixed point, nonlinear F-contractions, delay integral equations

DOI: 10.3233/JIFS-191806

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5231-5240, 2020

Authors: Ashraf, Shahzaib | Abdullah, Saleem | Aslam, Muhammad

Article Type: Research Article

Abstract: The spherical fuzzy set (SFS) is one of the most important concepts to accommodate more uncertainties than the intuitionistic fuzzy set, Pythagorean fuzzy set, picture fuzzy set and hence its applications are more extensive. Keeping the feature and the importance of the SFS, the objective of this paper is to present some robust symmetric operational laws for SFSs. Associated with these laws, we define some series of new aggregation operators named as spherical fuzzy (SF) symmetric weighted averaging, SF ordered weighted averaging and SF hybrid weighted averaging operators to aggregate the SF information. Afterwards, we present a group decision making …technique to solve the multi attribute group decision making (MAGDM) based on proposed symmetric aggregation operators and illustrate with a numerical example of renewable energy source selection as a real-life practical example to validate it. A comparative analysis is also conducted to show the superiorities of the proposed method. Show more

Keywords: Spherical fuzzy sets, symmetric sum, symmetric aggregation operators, decision making technique

DOI: 10.3233/JIFS-191819

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5241-5255, 2020

Authors: Cheng, Yan | You, Cuilian

Article Type: Research Article

Abstract: In this paper, semi-implicit and implicit Euler schemes for homogeneous fuzzy differential equations with perturbation term reflected by Liu process are introduced. As to the application with implicit term in numerical scheme, we must make the result gradually explicit by iterative method. In order to obtain numerical method with higher accuracy than fuzzy Euler method, fuzzy trapezoidal scheme is derived. Fuzzy trapezoidal scheme is an implicit formation, which is complicated and cumbersome in computational processing. For the sake of this problem, fuzzy Euler-trapezoidal method is proposed to simplify the algorithm. Furthermore, the convergence properties are investigated for numerical methods. At …last, local convergence is proved better than global convergence. Show more

Keywords: Credibility, fuzzy differential equations, Liu process, numerical methods, convergence

DOI: 10.3233/JIFS-191856

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5257-5266, 2020

Authors: Shi, Gang | Li, Xiaohua | Jia, Lifen

Article Type: Research Article

Abstract: Uncertain differential equation plays an important role in dealing with dynamical systems with uncertainty. Multi-dimensional uncertain differential equation is a type of differential equation driven by multi-dimensional Liu processes. Stability analysis of a multi-dimensional means insensitivity of the state of a system to small changes in the initial state. This paper focuses on the stability in p -th moment for multi-dimensional uncertain differential equation. The concept of stability in p -th moment for multi-dimensional uncertain differential equation is presented. Some stability theorems for the solution of multi-dimensional uncertain differential equation are given, in which some sufficient conditions for a multi-dimensional …uncertain differential equation being stable in p -th moment and a sufficient and necessary condition for a linear multi-dimensional uncertain differential equation being stable in p -th moment are provided. In addition, this paper discusses the relationships among stability in p -th moment, stability in measure and stability in mean. Show more

Keywords: Uncertainty theory, multi-dimensional uncertain differential equation, moment stability

DOI: 10.3233/JIFS-191880

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5267-5277, 2020

Authors: Seman, Laio Oriel | Rodrigues Machado, Victor Hugo | Koehler, Luiz Alberto | Camponogara, Eduardo

Article Type: Research Article

Abstract:  This paper presents adjustments and validation of a model for the process of passenger boarding and alighting of BRT (Bus Rapid Transport) and express bus systems. Such models are of fundamental importance for the implementation of strategies to control the operation of those systems, which allow for quality and efficiency gains in the service. First, a generalized disjunctive programming model is presented in order to adapt a bus headway control formulation according to the number of onboard passengers. Later, as a way to improve the model, real data were collected from the boarding/alighting times of passengers in the Trunk line …10 system in the city of Blumenau, Brazil. These data were used in a two-stage fuzzy regression to take into account the uncertainties associated with the dwell time of the passengers. The actual data collected were used in simulations of the enhanced bus headway control system, showing that even in the presence of disturbances, the system can still operate properly without bunching. Show more

Keywords: Fuzzy regression, BRT systems, headway control, dwell time, GDP

DOI: 10.3233/JIFS-191904

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5279-5293, 2020

Authors: Xiao, Zhiyong | Gong, Zengtai | Liu, Kun

Article Type: Research Article

Abstract:  As a key theoretical basis of fuzzy analysis, a new representation of the two-dimension fuzzy numbers is proposed in this paper. We also present a new method for solving the fuzzy system of linear equations with the two-dimension fuzzy data based on the new representation of the two-dimension fuzzy numbers, which may translate a fuzzy system of linear equations into two real systems of linear equations, one is an n  × n real system of linear equations and the other is a (2n ) × (2n ) real system of linear equations, where the coefficients, the right hand term and the unknown …term of the (2n ) × (2n ) real systems are all positive real numbers. Accordingly, we compare the classical methods with the technique on the basis of the study of the two-dimensional fuzzy linear system proposed in this paper for an one-dimensional fuzzy linear system, and the results shows that they have the same solutions. Finally, the conditions of the existence of the solutions for a fuzzy system of linear equations are discussed, and the examples are given to show the efficiency and effectiveness of the method investigated in this article. Show more

Keywords: Fuzzy numbers, two-dimension fuzzy numbers, fuzzy linear system, fuzzy system of linear equations

DOI: 10.3233/JIFS-191927

Citation: Journal of Intelligent & Fuzzy Systems, vol. 38, no. 4, pp. 5295-5315, 2020