The GCS data is a complex dataset consisting of different operational conditions of a gas plant. The GCS data consist of 825 data points and modeled as a time series using input generating format: [y(t − 3), y(t − **2**), y(t − 1)] with y(t) as the output. The inputs are normalised to lie between small range of [0,1], so that larger input values do not overshadow the smaller values, thereby leading to poor prediction and learning using the embedded neural network architecture. For each run of the experiments, the data are randomly sampled and split into 70% training and 30% testing set with each data point having equal chance of being chosen for training and testing. For a clear and objective discussion and **evaluation** of the three models of IT2IFLS, IT2FLS and IFLS, the Kalman filter parameters R, Q and P for both MFs and NMFs are initially set as 40, 0.01I 32

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. Other work using an alpha-planes representation has been applied, e.g. as a method for edge-detection [35] and a learning method to forecast Mackey-Glass time-series [41]. The latter showed a better performance of general **type**-**2** **fuzzy** **logic** **systems** using a simpler model known as “triangle quasi- **type**-**2** **fuzzy** **logic** system” first presented in [40]. Some other researchers used some neural network concepts or classification algorithms such as: **type** **2** Adaptive Network Based **Fuzzy** Inference System (ANFIS) [28], general **type**-**2** **fuzzy** neural network (GT2FNN) [24] and **fuzzy** C-means algorithm with a model known as “efficient tri- angular **type**-**2** **fuzzy** **logic** system” [43]. To the best of the authors’ knowledge, no attempt to employ a learning method to general **type**-**2** **fuzzy** **logic** **systems** using the vertical-slices representation has been reported. To achieve this objective, apart from using a practical **type**-reducer, some kinds of parametrization are needed for general **type**-**2** sets to allow learning or optimization techniques to deal with these parame- ters easily rather than having all the secondary grades or membership functions chosen manually. The parametrization method should preserve most of the freedom associated with GT2FLS.

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Quality of web service (QoWS) monitoring is an important component in web service as it evaluates web service delivery performance and detects problems. Our previous work proposed a **fuzzy** model for QoWS monitoring due to uncertain nature of web service environment. However, **fuzzy** models are computationally costly. In this work, we propose a parallelization implementation of the models. The objective of this paper is to compare the performance between Mamdani- and Sugeno-based **fuzzy** inference **systems** (FIS) when they are applied to the QoWS monitoring models. The results suggested that Sugeno models produced less processing time than that of Mamdani models. However, Mamdani models benefited from parallelization more than that of Sugeno models by recoding higher percentage of improvement in terms of average processing time. This work will be expanded to investigate the implementation of the models in cluster computers and using a higher **type** of **fuzzy** **logic**, namely **interval** **type**-**2** **fuzzy**.

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Recent work had proposed the use of neural based **systems** to learn the **type**-**2** FLC parameters. However, these approaches require existing data to optimize the **type**-**2** FLC. Thus, they are not suitable for applications where there is no or not sufficient data available to represent the various situations faced by the IT2FLC controller. Genetic Algorithms (GAs) do not require a priori knowledge such as a model or data but perform a search through the solution space based on natural selection, using a specified fitness function. We did not evolve the **interval** **type**-**2** FLC rule base as it will remain the same as the **interval** **type**-1 FLC rule base. However, the FLC antecedents and consequents will be represented by **interval** **type**-**2** MFs rather than **type**-1 MFs.

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GAs are general purpose search algorithms, based on natural genetics, that provide robust search capabilities in complex spaces, and thereby offer a valid approach to problem requiring efficient and effective search process. The basic idea is to maintain a population of chromosomes that evolves over time through a process of competition and controlled variation. A chromosome is representing candidate solutions to the concrete problem being solved. A GA starts with a population of randomly generated chromosomes, and advance towards better chromosomes by applying genetic operators modeled on the genetic process occurring in nature. The population undergoes evolution in a form of natural selection. During successive iterations, called generation, chromosomes in the population are rated for their adaptation as solutions, and on the basic of these **evaluation**, a new population of chromosomes is formed using a selection mechanism and specific genetic operator such as crossover and mutation. A fitness function must be devised for each problem to be solved. Given a particular chromosome, the fitness function returns a single numerical value, which is supposed to be proportional to the utility or adaptation of the solution represented by that chromosome.

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In this work, different versions of the training and the testing data are generated, i.e. the data is corrupted with a zero-mean uniform noise for different SNRs. We use 12 noise levels in training and testing data. Specifically, we use discretized levels from 0dB to 20dB with increments of **2**, as well as the original NF data set (noise-free). Fig. 4 shows examples of the training and the testing data of the the MG time series at NF data and two different SNR level (10 and 0 dBs). Table I shows the delta (δ) values for different noise levels from MG time series training data corrupted by different levels of noise. These values are used to design the T1 non-singleton inputs and will be detailed in the next subsection.

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office equipment, spacecraft power **systems**, laptop computers, and telecommunications equipment, as well as DC motor drives [4]. Several control techniques for DC–DC converters have been reported in the literature, such as linear based control techniques, sliding mode control technique, and **fuzzy** **logic** control technique. Although the structure and design of linear based control techniques are simple, their performance usually depends on the working conditions of the controlled system. Sliding mode control technique needs a system model to be designed. One of the most important problems in design of this controller is control chattering [5]. Traditional **fuzzy** techniques provide for the output voltage regulation against input voltage However, the performance of this controller depends on the experience and knowledge of human experts. In general, trial-and-error tuning procedure is used to adjust parameters of the rule base and membership sets [6]. This means that these parameters will be change from one expert to another expert. The controlled system performance may be undesirably affected from these uncertainty conditions. Thus, a **type**-**2** **fuzzy** controller will be highly suitable to tackle the uncertainty which occurs in traditional **fuzzy** **logic** controllers. Karnik and Mendel [7], [8] established a complete **Type**-**2** FLS theory to handle uncertainties in FLS parameters.

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ecently, the **fuzzy** **systems** and control are regarded as the most widely used application of **fuzzy** **logic** system [8-11, 20, 23, 26, 31, 32]. In traditional **fuzzy** system models, the structure is characterized by using **type** 1 **fuzzy** sets, which are defined on a universe of discourse, map an element of the universe of discourse onto a precise number in the unit **interval** [0, 1]. The concept of **type**-**2** **fuzzy** sets was initially proposed by Zadeh as an extension of typical **fuzzy** sets (called **type**-1) [34]. Mendel and Karnik developed a complete theory of **type**-**2** **fuzzy** **logic** **systems** (T2FLSs) [12, 21, 26]. Recently, T2FLSs have attracted more attention in many literatures and special issues [5, 9, 15, 21, 26]).

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In a GPU each SGC trip implies in at least two hours of shutdown until the SGC system can be restarted, which represents a cost of thousands of dollars for the company. This work shows the potentiality, simplicity and viability of a **fuzzy** inference system using only **interval** **type**-**2** **fuzzy** sets to instrument fault detection and diagnosis. The IFDD system developed can be applicable and useful for a variety of real-world **systems**.

According to The the **fuzzy** control theory, which was presented by Zadeh on the theory of **fuzzy** **systems**, has attracted the attention of many researchers in controlling structures [10]. The remarkable features of this method have been greatly appreciated. This method solves the need for precise mathematical modeling of the structure by applying a series of innovational rules. Other features of this control algorithm can be its robustness against the uncertainties and errors in the various parts of the control system such as data, loads, structure model, measurements, etc. Another important feature of this method is the ability to use it in non-linear **systems**. Due to the nature of non-linear behavior of structures, this method can be used to control structures. Using human knowledge and experience in controller design and the possibility of adapting the control system can be considered as the other advantages of this method than in comparison with other control methods. In this paper, the **type**-**2** **fuzzy** **systems**, which are in fact a development of **type**-1 **fuzzy** **systems**, are applied. In the following, the equations and components of the **type**-**2** **fuzzy** system are briefly described. The **type**-**2** **fuzzy** set is represented by Eq. (**2**) and Eq. (3)uations **2** and 3.

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Abstract: An overview and a derivation of **interval** **type**-**2** fussy **logic** system (IT2 FLS), which can handle rule’s uncertainties on continuous domain, having good number of applications in real world. This work focused on the performance of an IT2 FLS that involves the operations of fuzzification, inference, and output processing. The out- put processing consists of **Type**-Reduction (TR) and defuzzification. This work made IT2 FLS much more accessible to FLS modelers, because it provides mathematical formulation for calculating the derivatives. Presenting extend to representation of T2 FSs on continuous domain and using it to derive formulas for operations, we developed and extended the derivation of the union of two IT2 FSs to the derivation of the intersection and union of N-IT2 FSs that is based on various concepts. The derivation of all the formulas that are related with an IT2 and these formulas de- pend on continuous domain with multiple rules. Each rule has multiple antecedents that are activated by a crisp number with T2 singleton fuzzification (SF). Then, we have shown how those results can be extended to T2 non- singleton fuzzification (NSF). We are derived the relationship between the consequent and the domain of uncertainty (DOU) of the T2 fired output FS. As well as, provide the derivation of the general form at continuous domain to cal- culate the different kinds of **type**-reduced. We have also applied an IT2 FLS to medical application of Heart Diseases (HDs) and an IT2 provide rather modest performance improvements over the T1 predictor. Finally, we made a com- parison of HDs result between IT2 FLS using the IT2FLS in MATLAB and the IT2 FLS in Visual C# models with T1 FISs (Mamdani, and Takagi-Sugeno).

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Atanassov [7] extended the concept of Zadeh’s **fuzzy** sets to **intuitionistic** **fuzzy** sets, hereafter referred to as AIFSs, which handle uncertainty by taking into account both the membership and non-membership degrees of an element x to a **fuzzy** set A together with extra degree of indeterminacy (hesitation). With AIFS, the **fuzzy** characteristic of “neither this or that” (neutral state) can be described, thus providing IFS the flexibility and the ability to capture more information than FS [8]. AIFSs are found to be useful for dealing with vagueness [9], [10]. Szmidt and Kacprzyk [11] state that AIFSs are useful in problem domains where the use of linguistic variable to describe the problem in terms of membership functions only seems too restrictive. According to Olej and Hajek [12], the representation of attributes by means of membership and non- membership functions provides a better way to express uncer- tainty. Castillo et al. [13] pointed out that the non-membership degrees or **intuitionistic** **fuzzy** indices enable the representation of imperfect knowledge and also allow adequate description of many real world problems. According to [14], when dealing with the problem of vagueness where there is insufficient information leading to an inability to satisfactorily specify the membership function, the AIFS theory becomes more suitable than **fuzzy** sets to deal with such problems. It is argued that AIFS is a tool for a more human consistent reasoning under imperfectly defined facts and imprecise knowledge [15].

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are tuned using GDA. The results in Table III show that the IT2IFLS outperforms both forms of T2FLSs. Our approach is also compared with three evolving T2FLSs namely, self evolving **interval** **type**-**2** **fuzzy** neural network (SEIT2FNN) utilising IT2FS in the antecedents and TSK **interval** **type**-1 set in the consequent, TSK-**type**-based self-evolving compen- satory IT2FNN (TSCIT2FNN) which utilises IT2FS in the antecedent and a crisp linear model in the consequent and evolving **type**-**2** neural **fuzzy** inference system (eT2FIS) with antecedent T2FS and Mamdani-**type** consequent. As shown in Table III, IT2IFLS exhibits a low level of RMSE over these evolving T2FLSs. In particular, the performance of IT2IFLS is compared with **Type**-**2** TSK **Fuzzy** Neural System (**Type**-**2** TSK FNS) [28], TSCIT2FNN [26] and SIT2FNN [29], which also utilised the parameter β to adjust the contribution of upper and lower membership values in their final outputs. The results show a clear performance improvement of IT2IFLS over **Type**- **2** TSK FNS, TSCIT2FNN and SIT2FNN. We also constructed an IFLS in order to compare the performance of the IT2IFLS with its T1 model on system identification. From Table III, there is a significant performance improvement of IT2IFLS over IFLS on system identification.

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Abstract—**Fuzzy** **logic** **systems** have been extensively applied for solving many real world application problems because they are found to be universal approximators and many methods, particularly, gradient descent (GD) methods have been widely adopted for the optimization of **fuzzy** membership functions. Despite its popularity, GD still suffers some drawbacks in terms of its slow learning and convergence. In this study, the use of decoupled extended Kalman filter (DEKF) to optimize the parameters of an **interval** **type**-**2** **intuitionistic** **fuzzy** **logic** system of Tagagi-Sugeno-Kang (IT2IFLS-TSK) **fuzzy** inference is proposed and results compared with IT2IFLS gradient descent learning. The resulting **systems** are evaluated on a real world dataset from Australia’s electricity market. The IT2IFLS-DEKF is also compared with its **type**-1 variant and **interval** **type**-**2** **fuzzy** **logic** system (IT2FLS). Analysis of results reveal performance superiority of IT2IFLS trained with DEKF (IT2IFLS-DEKF) over IT2IFLS trained with gradient descent (IT2IFLS-GD). The proposed IT2IFLS-DEKF also outperforms its **type**-1 variant and IT2FLS on the same learning platform.

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dataset is a time series that shows the number of lynx trapped in the Mckenzie river district per year in northern Canada and corresponds to the period 1821-1934. Similar to previous studies such as [26]–[28], the logarithms to the base 10 of the data are used in the analysis. Figures 4 and 5 show the original and the logarithmic transformed data of the Canadian lynx series respectively, with a periodicity of approximately 10 years. The series consists of 114 observations of which 100 samples are used for training and the remaining 14 are used for testing in order to validate the effectiveness of the model proposed in this study. Similar to [28], the maximum training epoch adopted is 2000. As shown in Table III, IT2IFLS outperforms the listed non-**fuzzy** approaches on the Canadian lynx dataset.

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order derivative based) methods have been widely used as an optimisation strategy for the parameters of **fuzzy** **systems** [21], [37]. The difficulties associated with GD methods however, are slow convergence and the possibility of getting stuck in local minima, leading to poor solutions [43]. This can be compen- sated for by combining the first-order GD with a higher-order derivative-based method such as the Kalman filter (KF)-based algorithms which have a smaller possibility of getting stuck in local minima [44]. In a different application domain, the hybrid learning utilising KF-based and GD techniques has shown good performance. For instance, Mendez et al. [38] proposed a hybrid learning approach for IT2 FLS of TSK- **type** otherwise known as **interval** **type**-1 non-singleton **type**-**2** TSK FLS ANFIS (IT2 NSFLS1 ANFIS) utilising recursive Kalman-**type** filter (REFIL) to tune the consequent parameters and the steepest descent back propagation method to tune the antecedent parameters. The developed model was applied to the prediction of transfer bar surface temperature. Experimen- tal **evaluation** revealed that the IT2 NSFLS1 ANFIS trained with hybrid REFIL-BP had the lowest prediction error on test data compared to other learning approaches investigated in their study. However, the basic KF works well for linear dynamic **systems** with white process and measurement noise but real world problems are non-linear. Hence, for nonlinear **systems**, we have extended the linear KF used in [38] through a process of linearisation where the nonlinear function is linearised around the current parameter estimates.

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This section describes the complete algorithm using rough set based on **intuitionistic** **type**-**2** **fuzzy** c- means clustering for robust and fast segmentation, which is a bottleneck to restrict the application of magnetic resonance imaging in clinic, and the segmentation of brain MRI now is confronted with presence of uncertainty and noise, many various kinds of algorithms have been proposed to handle this problem. In this paper, a hybrid clustering algorithm combined with a new **intuitionistic** **fuzzy** factor and local spatial information is proposed, where randomness is handled by **type**-**2** **fuzzy** **logic**, vagueness could be dealt with the rough set, and the **intuitionistic** **fuzzy** **logic** can address the external noises. The proposed algorithm is listed in the following three subsections:

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Definition 2.1[10]A **fuzzy** graph is a pair of functions where is a **fuzzy** subset of a non-empty set and is a symmetric **fuzzy** relation on . The underlying crisp graph of is denoted by where .

Abstract – In this paper, we first introduce some operations on the **interval**-valued **intuitionistic** **fuzzy** sets, such as Hamacher sum, Hamacher product, etc., and further develop the induced **interval**-valued **intuitionistic** **fuzzy** Hamacher correlated averaging (I-IVIFHCA) operator. The prominent characteristic of the operators is that they can not only consider the importance of the elements or their ordered positions, but also reflect the correlation among the elements or their ordered positions. We have applied the I-IVIFHCA operators to multiple attribute decision making with **interval**- valued **intuitionistic** **fuzzy** information.

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With an increase in the number of input variables, the possible set of **fuzzy** rules increases rapidly. For instance, if each variable (both input and output) has p **fuzzy** subsets, then for a FLS with q inputs and one output, the total number of the possible rules is p q - 1. It is difficult to determine a small subset of rules from such a large “rule space” that would be suitable for controlling the process. In principle, there is no a general method for the **fuzzy** **logic** setup, although a heuristic and iterative procedure for altering the membership functions to improve performance has been proposed [8], even this is not optimal. Recently, many researchers have considered a number of intelligent schemes for the task of optimizing the **fuzzy** rules and membership functions. There have been several attempts both under supervised and self-organized paradigms for obtaining a good rule base. Some of these methods use neural networks [9] and others use genetic algorithms (GA) [10]. The rule base tuning has been attempted primarily in two ways: through tuning of membership functions of a given rule set or through selection of an “optimal” subset of rules from all possible rules.

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