One of the most crucial problems in many decision-making **methods** is the precise evaluation of the pertinent data. Very often in real-life decision-making applications data are imprecise and **fuzzy**. A decision maker may encounter difficulty in quantifying and processing linguistic statements. Therefore it is desirable to develop decision-making **methods** which use **fuzzy** data. It is equally important to evaluate the performance of the **fuzzy** decision-making method. Hence, the development of useful **fuzzy** decision-making **methods** is really the need of the hour.

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In our daily life we often make inferences whose antecedents and consequents contain **fuzzy** concepts. Such an inference cannot be made adequately by the **methods** which are based either on classical two valued logic or on many valued logic. In order to make such an inference, Zadeh suggested an inference rule called “compositional rule of inference”. Using this inference rule, he, Mamdani, Mizumoto et al ., R. Aliev and A. Tserkovny suggested several **methods** for **fuzzy** **reasoning** in which the antecedent contain a conditional proposition with **fuzzy** concepts:

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Since the initial release in mid 2006 the grow- ing user base has voiced a number of suggestions and requests for improving KNIME’s usability and functionality. From the beginning KNIME has sup- ported open standards for exchanging data and mod- els. Early on, support for the Predictive Model Markup Language (PMML) 15 was added and most of the KNIME mining modules natively support PMML, including association analysis, clustering, regressions, neural network, and tree models. With the latest KNIME release, PMML support was en- hanced to cover PMML 4.1. See 18 for more details. Before dicussing how **fuzzy** types and learning **methods** can be integrated into KNIME, let us first discuss the KNIME architecture in more detail.

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system (KBS) or system with uncertainty. Firstly, due to the graphical description ability of FPN, **fuzzy** production rule (FPR) in KBS could be completely presented in the form of FPN. For example, Chen et al. (1990) utilized the FPNs to depict different types of FPRs, which includes ‘Simple’ rule, ‘OR’ rule, ‘AND’ rule, and multi- condition rule. Moreover, Gao et al. (2003) used a different proposition to stand for the proposition with the negation operator in the FPN. In addition,multi-output places were referred to the IF–THEN and IF–THEN–ELSE rules in KBS by Shen (2006). On the other hand, because of the parallel operation ability inherited from Petri net (PN), FPN is also broadly employed to performthe **approximate** **reasoning** for KBS (Amin and Shebl, 2014; Chen et al., 2014; Fenton et al., 2007; Gong and Wang, 2012; Hu et al., 2011; Lee et al., 2009; Liu et al., 2010; Luo and Kezunovic, 2008;Wai and Liu, 2009; Wai et al., 2010; Wu and Hsieh, 2012).According to the existing literature, inference mechanism using FPN couldbe roughly classified into three types, which areinference using reachability tree, inference usingalgebraic operation, and inferenceusing high level of FPN (HLFPN) (Ting et al., 2008; Sharma et al., 2008; Peters et al., 2009; Cheng et al., 2009; Sharma et al., 2010; Asthana et al., 2011; Abdulkareem et al., 2011; Barzegar et al., 2011; Rajpurohit et al., 2012; Liu et al., 2013a; Liu et al., 2013b; Ding et al., 2013; Wai and Lin, 2013;Bharathiet al., 2013; Chen et al., 2014; Shen et al., 2014; Chen et al., 2014).

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In fact, Gong et al. proved the basic theorem of their method (Theorem 3.1 of [6]) by considering the mathematical incorrect assumption that RL **fuzzy** numbers can be added to LR **fuzzy** num- bers (see [7]). For this reason, Kaur and Kumar [7] stated that the method proposed in [6] is valid only if either A and C are non-negative real ma- trices or L(x) = R(x) for all x ∈ [0, 1].

A cost estimate is actually the approximation of cost at the early stage of a particular project or work. Accurate cost estimation a construction project is key factor in a project’s success. But it is hard to estimate construction costs at the planning stage rapidly and precisely, when drawings, documentation etc... are still incomplete. As such, various techniques have been applied to accurately estimate construction costs at an early stage, when project information is limited. While the various techniques have their problems and consequences, there has been little effort made to determine the best technique in terms of cost estimating performance.[10] Several estimation **methods** are used in construction practice and the appropriateness of any particular method is usually reliant on the purpose it is used for, the amount of information available at the time of estimation, and the people using it.[24] The main traditional cost estimation method used for common practice is Quantity Rate Analysis In this method, the entire project is divided into small discrete work items and a unit rate is established for each item. The unit rate is then multiplied by the required quantity to find the cost for the work item. All costs are added to obtain the Estimated Total Construction Cost. Quantity rate analysis is the most accurate means of ascertaining costs based on materials and labor content. In addition to traditional cost estimating approaches, alternative cost estimation **methods** have been developed and investigated in recent years in an attempt to improve the reliability of cost forecasts in predicting the actual final costs of projects. Many

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U n can be modelled using **fuzzy** differential certain and incompletely specified systems, equations. Differential equations which arise in real-word physical problems are often too com- plicated to solve exactly. And even if an ex- act solution is obtainable, the required calcula- tions may be too complicated to be practical, or the resulting solution may be difficult to inter- pret. When some fractional derivatives appear in a differential equation, we will have a **fuzzy** fractional differential equation. In recent years, fractional differential equations have found appli- cations in many problems in physics and engineer- ing [16, 17]. With uncertainty in initial value of these problems, **fuzzy** fractional differential equa- tions will also find applications in physics and en- gineering. Benchohra and Darwish [7] introduced an existence and uniqueness theorem for **fuzzy** integral equation of fractional order and under some assumptions gave a **fuzzy** successive itera-

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Soft set theory is a generalization of **fuzzy** set theory, which was proposed by Molodtsov [8] in 1999 to deal with uncertainty in a non-parametric manner. One of the most important steps for the theory of soft sets was to define mappings on soft sets; this was achieved in 2009 by mathematician Athar Kharal, though the results were published in 2011. Soft sets have also been applied to the problem of medical diagnosis for use in medical expert systems. **Fuzzy** soft sets have also been introduced in [10]. Mappings on **fuzzy** soft sets were defined and studied in the ground breaking work of Kharal and Ahmad.

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Abstract: Recently, many authors have been interested to introduce **fuzzy** implications over t-norms and t-conorms. In this paper, we introduce ( , ) S N and residuum **fuzzy** implication for Dubois t-norm and Hamacher's t-norm. Also, new concepts so-called ( , ) T N and residual **fuzzy** co-implication in dual Heyting Algebra are investigated. Some examples as well as application are discussed as well.

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condition of a patient in an intensive care unit. The data include heart rate, sys- tolic and diastolic blood pressures, percent of oxygen saturation in both arter- ial and venous blood, and temperature. We set up linguistic variables for each of these measurements, with five linguistic terms in each linguistic vari- able, and fuzzify the input data. The data are collected nearly continuously (1 sample every 2 s), so we can calculate rates of change for each input vari- able, and set up corresponding linguistic variables for the rates of change. Our output consists of two discrete **fuzzy** sets, one for present condition (good, fair, poor, bad) and one for changes (improving. stable, deteriorating). We write rules whose consequents are present condition (condition is good), and other rules whose consequents are rate of change (state is deteriorating). We have several rules that have the same consequent. Rule A would set the grade of membership of “deteriorating” to 0.1; rule B would set its grade of membership to 0.2; and rule C would set its grade of membership to 0.5. Since our rules are fired in parallel, all three rules are fired concurrently. To what value should we set the grade of membership of “deteriorating”? 7.5 We have five input variables with four possible values for each, and six output

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The proposed methodology for pattern recognition system is given in Figure 1. The numeral database consists binary samples fixed in a window of particular size is partitioned into overlapping and non-overlapping regions for feature extraction. A set of features are extracted to represent the sample and it is called feature vector. Feature vectors such extracted for the database samples are divided into training and testing data. The training samples feature vectors are used to construct the KB. The class label of each training and testing sample is assigned by the **fuzzy** **reasoning** technique.

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Looney’s **fuzzy** **reasoning** algorithm supported by **Fuzzy** Petri Net(FPN)is further extended for representing the **fuzzy** production rules in a knowledge based system which is provided with **fuzzy** **reasoning** ability of **fuzzy** systems . In intuitionistic **fuzzy** sets(IFS)there is an extension of Zadeh **Fuzzy** sets that possess a new additional attribute function i,e non membership and intutionistic index take into account with stronger expression ability to deal with uncertain information. In this paper a rule based **reasoning** algorithm is presented for intuitionistic **fuzzy** petrinets. intuitionistic **fuzzy** members are used to represent the confidence degree and threshold of the transition along with token values of each place. A generalized **fuzzy** petri net to intuitionistic **fuzzy** petri net and vice versa is presented in an effective way . Some new features are added to take care of the generalization of the intuitionistic nature of the **fuzzy** petri nets. Keywords - Intuitionistic **Fuzzy** Set, Intuitionistic **Fuzzy** Petri Nets, **Fuzzy** **Reasoning**, **Fuzzy** Relations,

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Abstract. In this paper the exact and the **approximate** solutions of **fuzzy** fractional diﬀerential equation, in the sense of Caputo Hukuhara diﬀerentiability, with a **fuzzy** condition are constructed by using the **fuzzy** Laplace transform. The obtained solutions are expressed in the form of the **fuzzy** Mittag-Leﬄer function. The presented procedure is visualized and the graphs of the obtained **approximate** solutions are drawn by using the GeoGebra package.

E ither of these **methods** would be useful for describing the early stages of the process of decay but could not be used for large values of i smce the respective series would become divergent as cr o. Rather than use the above m ethods, however, we shall obtain the solution for the flow-field in the early stages of decay by a method introduced by Meyer (1956). This method too cannot be used for large values of t but it has three advantages over either of the above **methods**. F irst, it does not employ the Riemann function at all and hence avoids complicated

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Apart from the object, a controller constitutes of the fundamental elements of control systems. However, such con- trollers, even in SISO (Single Input, Single Output) systems where a single output is controlled by a single input, may not be the most appropriate choice. Moreover, a PID controller may not be easily tuned to any setup. The paper outlines three elementary **methods** for tuning the controller. The results of the process conducted by means of Ziegler-Nichols, Cohen-Coon, and Lambda **methods** were compared in Matlab.

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There are different techniques to improve the performance of the classification systems. For ex- ample, Evolutionary **Fuzzy** Systems (See subsection 1.2.3) and Ensembles [OM99]. Ensembles of classifiers usually enhance the performance of single classifiers by inducing several classifiers and combining them so that they outperform all the models conforming it [GFB + 12]. Therefore, the basic idea of an ensemble method is to construct several classifiers from the original data and, then, aggregate their predictions when unknown instances are presented. It is observable that different generalizations of the Choquet integral achieved different perfor- mances in the same dataset. Thus, another open research line is related to the construction of an ensemble that considers different FRMs, that is, different generalizations of the Cho- quet integral. Then we could made the final decision based on the outputs of the different generalizations. The open problem is to know if this methodology offers diversity enough as it is a key factor when constructing ensembles of classifiers [Kun05]. If this combination is successful we could improve even more the behavior of FRBCSs to deal with classification problems.

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As it is obvious, our method yields trapezoidal **fuzzy** numbers with closer cores in comparison with the ones obtained from the other four **methods**. While the other four **methods** fail in approximating the m-degree polynomial form **fuzzy** numbers, our method can **approximate** all the trapezoidal, triangular and m-degree polynomial form **fuzzy** numbers.

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Seaport operations are characterised by high levels of uncertainty, as a result their risk evaluation is a very challenging task. Much of the available data associated with the system’s operations is uncertain and ambiguous, requiring a flexible yet robust approach of handling both quantitative and qualitative data as well as a means of updating existing information as new data becomes available. Conventional risk modelling approaches are considered to be inadequate due to the lack of flexibility and inappropriate structure for addressing the system’s risks. This paper proposes a novel **fuzzy** risk assessment approach to facilitating the treatment of uncertainties in seaport operations and to optimize its performance effectiveness in a systematic manner. The methodology consists of a **fuzzy** analytical hierarchy process, an evidential **reasoning** (ER) approach, **fuzzy** set theory and expected utility. The **fuzzy** analytical hierarchy process is used to analyse the complex structure of seaport operations and determine the weights of risk factors while ER is used to synthesise them. The methodology provides a robust mathematical framework for collaborative modelling of the system and allows for a step by step analysis of the system in a systematic manner. It is envisaged that the proposed approach could provide managers and infrastructure analysts a flexible tool to enhance the resilience of the system in a systematic manner.

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Abstract: In our daily life we often face some problems in which the right decision making is highly essential. But in most of the cases we become confused about the right solution. To obtain the best feasible solution of these problems we have to consider various parameters relating to the solution. For this we can use the best mathematical tool called **Fuzzy** soft set theory. In this paper we select a burning problem for the parents and successfully applied the fs-aggregation algorithm in decision making for selecting a suitable bride by the family.

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