While generaltype-2 FLSs have and are being used and discussed more and more, there is a feeling that the additional complexity of generaltype-2fuzzy logic is not warranted by corresponding advances in performance or utility. In order to address this criticism which used to be a common criticism of type-1 fuzzy FLSs and respective publications (and is now mainly focused on type-2 FLSs), we are showing in this paper how the use of generaltype-2 FLSs has very specific advantages which are not available when employing other methods. As such, it is worth noting that “performance” is not a simple measure of a system or controller output but a measure of how satisfactory a specific system achieves its described goals – in a variety of aspects including precision, efficiency, interpretability, simplicity, maintainability, etc. At this point we would also like to clarify that this paper is not aiming to show that zSlicesbasedgeneraltype-2 FLSs are the only or even the best solution to the given problem. We firmly believe that generaltype-2fuzzysets are one tool with a specific set of properties which can be used as part of certain applications and in general – there is not the perfect tool for any application of reasonable complexity: all available tools have their specific strengths and weaknesses.
The IAA enables these capabilities while minimising the number and scope of assumptions during the model-creation process. In concrete terms, it generates non-parametric FS models of the interval-based data without preselecting a spe- cific FS type (such as triangular or Gaussian) and minimises the loss of information during the model creation process (from the original, potentially uncertain intervals). Finally, the IAA leverages the different degrees of freedom of zGT2 FSs to represent and to distinguish the different types of uncer- tainty. Specifically, agreement/variation over multiple interval samples (e.g., survey responses) from the same source (e.g., expert) is commonly captured using the primary memberships (y ∈ [0, 1]), with lower and upper bounds of primary member- ship modelling potential uncertainty about interval endpoints. Uncertainty originating from the agreement/variation between multiple sources (e.g., experts) of information is modelled through the secondary memberships of the zGT2 FSs (z ∈ [0, 1]). Note that the choice of in which dimension to model which type of uncertainty is arbitrary and either choice is valid - as long as data collection and the interpretation of the re- sulting sets is modified accordingly. To illustrate, consider the case where only a single sample/survey was administered to a large number of subjects; here, it is preferable to model inter- expert uncertainty using the primary membership, resulting in simpler and easier interpretable (type-1 FS) models.
SOFLCs with type-1 SOFLCs in controlling anesthesia delivery to maintain physiological set points for muscle relaxation and BP over the duration of a single stage operational procedure. Our results specifically showed that both the interval and zSlicesbasedgeneraltype-2 SOFLCs were able to produce a good multivariable control performance in maintaining the desired set points for muscle relaxation and BP in comparison to the type-1 SOFLCs while operating under signal and patient noise. In the case of interval type-2 SOFLC these results concur with experimental evidence that suggests that an interval type-2 FLC will give a smoother control surface in regions around the steady state over its type-1 counterpart [60, 61]. Karnik and Mendel also suggest that interval type-2 FLCs are more adaptive as they are able to realize more complex input-output relationships which cannot be achieved by a type-1 FLC . Our results also support this from the IT2 SOFLCs ability to adaptively regulate multi variable anesthesia set points based on adjusting the delivery rates of two different drugs with different pharmacodynamic and pharmacokinetic characteristics. zSlicesbasedgeneraltype-2fuzzysets slice the third dimension of a generaltype-2fuzzy set into a finite number of interval type-2fuzzysets each with a specific height that represents the secondary membership or amplitude associated with all the primary memberships over its FOU . The three dimensional MF of a zSlicesbasedgeneraltype-2fuzzy set can therefore provide greater design degrees of freedom over interval type-2fuzzysets .The simulation results we obtained for our GT2-SOFLCs were shown to support this by achieving lower steady state errors for multi variable set point control than the equivalent interval type-2 and type-1 SOFLCs.
Abstract—Facial expressions of a person representing similar emotion is not always unique. Naturally, the facial features of a subject taken from different instances of the same emotion have wide variations. In presence of two or more facial features, the variation of the attributes together makes the emotion recognition problem more complicated. This variation is the main source of uncertainty in the emotion recognition problem, which has been addressed here in two steps using type-2fuzzysets. First a type-2fuzzy face-space is constructed with the background knowledge of facial features of different subjects for different emotions. Second, the emotion of an unknown facial expression is determined based on the consensus of the measured facial features with the fuzzy face-space. Both Interval and GeneralType-2FuzzySets have been used separately to model the fuzzy face space. The interval type-2fuzzy set involves primary membership functions for m facial features obtained from n-subjects, each having l–instances of facial expressions for a given emotion. The generaltype-2fuzzy set besides employing the primary membership functions mentioned above, also involves the secondary memberships for individual primary membership curve, which has been obtained here by formulating and solving an optimization problem. The optimization problem here attempts to minimize the difference between two decoded signals: the first one being the type-1 defuzzification of the average primary membership functions obtained from the n- subjects, while the second one refers to the type-2 defuzzified signal for a given primary membership function with secondary memberships as unknown. The uncertainty management policy adopted using generaltype-2fuzzy set has resulted in a classification accuracy of 98.333% in comparison to 91.667% obtained by its interval type-2 counterpart. A small improvement (approximately 2.5%) in classification accuracy by IT2FS has been attained by pre-processing measurements using the well- known Interval Approach.
Abstract —In this paper a new approach is presented to model interval-based data using FuzzySets (FSs). Specifically, we show how both crisp and uncertain intervals (where there is uncertainty about the endpoints of intervals) collected from individual or multiple survey participants over single or repeated surveys can be modelled using type-1, interval type-2, or generaltype-2 FSs based on zSlices. The proposed approach is designed to minimise any loss of information when transferring the interval- based data into FS models, and to avoid, as much as possible assumptions about the distribution of the data. Furthermore, our approach does not rely on data pre-processing or outlier removal which can lead to the elimination of important information. Different types of uncertainty contained within the data, namely intra- and inter-source uncertainty, are identified and modelled using the different degrees of freedom of type-2 FSs, thus provid- ing a clear representation and separation of these individual types of uncertainty present in the data. We provide full details of the proposed approach, as well as a series of detailed examples based on both real-world and synthetic data. We perform comparisons with analogue techniques to derive fuzzysets from intervals, namely the Interval Approach (IA) and the Enhanced Interval Approach (EIA) and highlight the practical applicability of the proposed approach.
RBFNN, it was determined the optimal value to provide a high level of generalisation is with 4 hidden units, where each has 3 fuzzy rules. Table VI shows the average generalisation performance of 20 trials, the number of parameters per each model as well as the Average Training Time ATT of each GT2 RBFN model with respect to an FWSIRM , SANFIS , RBFNN, IT2 RBFNN , E-RBFNN  and the ANFIS system According to Table VI, the highest trade-off between accuracy and model simplicity is obtained by the RBFNN of GT2 using a NT algorithm. From Table VI, it it is clear for most of GT2 RBFNN models the training time is comparable to that of some models such as the BPNN and RBFNN. It is worth noting, the generalisation performance of an E- RBFNN is higher than an IT2 RBFNN and similar to a GT2 RBFNN. Both, GT2 RBFNN and E-RBFNN treat uncertainty as measure for ambiguity. However, a GT2 RBFNN quantifies uncertainty as a deficiency that results not only from imprecise boundaries in the fuzzysets (vagueness or fuzziness), but also as nonspecificity that refers to information-based imprecision, whereas an E-RBFNN defines ambiguity as a variation of the output of the ensemble members over unlabeled data. That means, uncertainty quantification is useful in an ensemble only if there is a disagreement among on some inputs . TABLE VI: COMPARISON OF THE AVERAGE PERFORMANCE OF 20 TRIALS OF DIFFERENT MODELS IN EXAMPLE 2.
Generalised Type-2FuzzySets The technique as described in this paper applies to interval type-2fuzzysets; we plan to extend it to generalised sets. The PRES as a Substitute for an RES It is far easier to calculate the PRES membership function than that of an RES, but of course the defuzzified value of the PRES is only an approximation. It would be useful to investigate (both mathematically and experimentally) the extent to which accuracy is lost in replacing an RES by the PRES.
Data in the real world is gathered from many sources. Considering that none of the processes in the real world is ‘ideal’, the data collected generally possess some kind of uncertainty, and information about some features or characteristics of the data are either incorrect or incomplete. Uncertainty is, therefore, a major challenge in real world applications, and there was a need felt, to have a theoretical framework within the discipline of Mathematics and Computer Science for handling such unknown, incomplete or uncertain information 3 . A solution was provided by Zadeh  in 1965 when he proposed the fuzzy set theory to bridge the gap between certain and uncertain environments. Although Zadeh’s work did not attract much attention initially, from the mid-70’s a number of scientists began examining this area. Several professional associations were started, and the first conference was held in USA in 1982. European and Asian researchers had also started to show increased attention to the field since the mid-70’s. In 1985, the International Fuzzy Systems Association (IFSA) was created which held its first conference in Palma de Mallorca, in Spain. Today, fuzzy set theory has emerged as a powerful way of quantitatively representing and manipulating the imprecision in a variety of problems, both theoretical and applied. It is now an established field with thousands of researchers studying different theoretical or practical aspects, as can be verified by bibliometrics.
Abstract—In this paper we will present two theorems for the join and meet operations for generaltype-2fuzzysets with arbitrary secondary memberships, which can be non-convex and/or non-normal type-1 fuzzysets. These results will be used to derive the join and meet operations of the more general descriptions of interval type-2fuzzysets presented in , where the secondary grades can be non-convex. Hence, this work will help to explore the potential of type-2fuzzy logic systems which use the general forms of interval type-2fuzzysets which are not equivalent to interval valued fuzzysets. Several examples for both generaltype-2 and the more general forms of interval type-2fuzzysets are presented.
ABSTRACT: A low power programmable frequency divider is proposed in this paper which is appropriate for WLAN applications. Multi- modulus architecture in dynamic logic with the minimum number of transistors is designed in 0.18µm CMOS technology. By using mixer, bandpass filter and switches, the divide ratios improved to 18. A technique is implemented in the dynamic 2-3 programmable divider cell for decreasing the glitches which leads to low power consumption. Based on simulation results it works up to 5GHz, with the average power about 37nW. Under a supply voltage of 1.8V, the total chip area of the multi- modulus programmable divider is 3100µm 2 .
During propagation, when several paths to the trustee exist via various trusted neighbors of trustor, then the evaluated indirect trust from various paths need to be joined and this process is termed as trust aggregation. Generally, users would like to employ various consensus operators for aggregation so that they could get utmost information about the trustee from several paths 3 . As Fig. 2 shows, a user
Incorporation of sustainability into supply chain management has attracted the greatest interest from practitioners and researchers. Although in the past decade, sustainability has become an important goal for companies, non-profit-making organizations, and governments, it is difficult to measure the extent to which an institution is sustainable or a sustainable growth line. Analysing sustainability is always subjective, and thus the decision-making models are crucial in this environment. In the practice of sustainability, TBL principles expand the traditional accounting by considering environmental and social impacts. This research presents an integrated fuzzymulti-criteria evaluation fremework SSS in the context of TBL. Firstly, criteria weights are evaluated by using FAHP method. Based on the weights of sustainability criteria, we observed that economic criteria are the most effective factor the considered factors. Then, fuzzy TOPSIS method is performed to rank potential suppliers both type-1 fuzzysets and type-2fuzzysets considering economic, environmental and social main criteria and their sub-criteria. Performance of the fuzzysets was compared for the integrated method. Interval type-2fuzzysets are most noteworthy because the mathematics required for such clusters is much simpler than the mathematics required for a generaltype-2fuzzysets. Additionally, interval type-2fuzzysets allow to come from the linguistic uncertainties in human thinking style and capture the vagueness of this style rather than type-1 fuzzysets. This methodology may be used together with another multi-criteria methods such as AHP-ELECTRE, or AHP-VIKOR. Also, different fuzzy MCDM methods such as interval hesitant or intuitionistic fuzzysets can be used in the future.
Feature subset selection is a data preprocessing step for pattern recognition, machine learning and data mining. In real world applications an excess amount of features present in the training data may result in significantly slowing down of the learning process and may increase the risk of the learning classifier to over fit redundant features. Fuzzy rough set plays a prominent role in dealing with imprecision and uncertainty. Some problem domains have motivated the hybridization of fuzzy rough sets with kernel methods. In this paper, the Exponential kernel is integrated with the fuzzy rough sets approach and an Exponential kernel approximation basedfuzzy rough set method is presented for feature subset selection. Algorithms for feature ranking and reduction based on fuzzy dependency and exponential kernel functions are presented. The performance of the Exponential kernel approximation basedfuzzy rough set is compared with the Gaussian kernel approximation and the neighborhood rough sets for feature subset selection. Experimental results demonstrate the effectiveness of the Exponential kernel basedfuzzy rough sets approach for feature selection in improving the classification accuracy in comparison to Gaussian kernel approximation and neighborhood rough sets approach.
The concept of fuzzy setwas introduced by Lotfi.A.Zadeh.It is an extension of the classical sets. In  and , Atanassov introduced the concept of intuitionistic fuzzy set(IFS), using a degree of membership and a degree of non-membership, under the constraint that the sum of the two degrees does not exceed one. Modal operators, topological operators, level operators, negation operators and aggregation operators are different groups of operators over the IFS due to Atanassov. Atanassov defined level operators 𝑃 𝛼,𝛽 and 𝑄 𝛼,𝛽 over IFS. In 2008, Atanassov studied some relations between intuitionistic fuzzy negations and intuitionistic fuzzylevel operators 𝑃 𝛼,𝛽 and 𝑄 𝛼,𝛽 .Parvathi and Geetha defined some level operators, max-min implication operators and 𝑃 𝛼,𝛽 and 𝑄 𝛼,𝛽 operators on temporal intuitionistic fuzzy sets.In , T.K.Mondal and S.K.Samanta
WhereV ∗ is the voltage reference value andstate(i)the current state of the genericith HB. In other words, dvcorresponds to the normalized voltage that the selectedkth HB has to produce in the next sampling period on the basis of its current voltage level and the subsequent one. Under steady state operation usually|dv| <1; however it is possible, especially during fast transients of the voltage reference, that the absolute value ofdv becomes larger than 1. Before performing any commutation, the modulator checks if the selected kth HB is able to switch, considering its current state, and how the subsequent commutation will affect the DC-Link voltage balancing. The following three cases, valid fordv >0and referred to the selectedkth HB state, are possible.
The DG granted this EU Work Climate project to the Institute of Labor Studies (IEL) at ESADE Business School. The fundamental aim of this project is to conduct a comparative analysis amongst 14 member countries based on standardized data pertaining to employment conditions supplied by Great Place to Work® Institute Europe. The latter has developed over the years a meth- odology (standardized instruments and sampling proce- dures) for data collection enabling them to choose the “best company to work for” in each country. The data bank includes information provided by employees and managers in over 2,500 companies in 14 EU member states in three consecutive years (2003-2005). The IEL gathered an international team of experts specialized in secondary data analyses of the Great Place to Work® data bank in order to identify trends and benchmark cases. Please kindly refer to the references for further informa- tion on this research and background on the Great Place to Work® model.
FLSs have been widely developed and utilized in the field of fuzzy logic control . Various types of ordinary (type-1) fuzzy PID (including PI and PD) controllers have been proposed in the literature     . A fuzzy controller based on the decomposition of the multivariable rule base into simple rule bases has been studied and compared to several PID-type FLCs and hybrid-type FLCs in . In , a fuzzy rule base shifting scheme is proposed for systems with time delay to improve system performance. Analytical structure of the fuzzy PID controller and conditions for bounded-input bounded-output stability of fuzzy PID control systems are obtained in . The performance of conventional PID controllers has been compared with T1-FLC through different simulations in . In , it has been shown that PID controllers can be realized by fuzzy control and simplified fuzzy reasoning methods. The main difficulty in T1-FLC design is to determine the parameters of the fuzzy logic controllers (e.g. membership functions, rules, scaling factors) for inputs and outputs of a fuzzy system. To ease the FLC design process, the researchers proposed a general methodology to systematically construct a fuzzy logic controller based on the existence of a linear controller in . This methodology guarantees identical performance to an existing linear controller. Since the performances of controllers are identical, it has been advised to use expert knowledge to improve the performance of fuzzy controller by appropriately changing the rule base.
To analyze incomplete information, Pawlak  pro- posed the theory of rough sets, which has a widespread application in artificial intelligence, data analysis and pattern recognition. Originally, Pawlak designed his model using an equivalence relation to describe the in- discernibility between objects. This way, a set can be approximated by a lower and upper approximation. The former consists of equivalence classes which are in- cluded in the set, while the latter consists of equivalence classes which have a non-empty intersection with the set. However, the use of an equivalence relation may be too strict for the applications. Therefore, many general- izations of Pawlak’s model which use coverings instead of a partition have been presented. The first authors proposing covering based rough sets were ˙ Zakowski  and Pomykała . Yao and Yao presented an extensive survey on covering based rough sets in 2012 .
Abstract. Soft Set theory is proposed to deal with uncertainties embedded in decision making. In our daily life we often face some problems in which the correct decision making is essential. In this paper, to overcome this problem, the concept based on complement of interval-valued fuzzy soft sets have been discussed. Finally the algorithm based on complement of Interval-Valued fuzzy soft sets is proposed with an example to illustrate the new approach.
In this section, we introduce the notions of anti-hesitant fuzzy UP-subalgebras, anti- hesitant fuzzy UP-filters, anti-hesitant fuzzy UP-ideals and anti-hesitant fuzzy strongly UP-ideals of UP-algebras, provide the necessary examples and prove its generalizations. Definition 6. A hesitant fuzzy set H on a A is called an anti-hesitant fuzzy UP-subalgebra of A if it satisfies the following property: for any x, y ∈ A,