Corrections to “Aggregation Using the Linguistic Weighted Average and Interval Type-2 Fuzzy Sets”

In the previous paper, we have proposed linguistic weighted average (LWA) algorithms that can be used in distributed and hierarchical decision making. The original LWA algorithms were completely based on the representation theorem for interval type-2 fuzzy sets (IT2 FSs). In later usage, we found that when the lower membership functions (LMFs) of the inputs and weights are of different heights, the LMF of the output IT2 FS may be nonconvex and discontinuous. In this letter, a correction to the original LWA algorithms is proposed. The new LWA algorithms are simpler and easier to understand; so, it should facilitate the applications of the LWAs.      

Data-driven modeling and optimization of thermal comfort and energy consumption using type-2 fuzzy method

In the research domain of intelligent buildings and smart home, modeling and optimization of the thermal comfort and energy consumption are important issues. This paper presents a type-2 fuzzy method based data-driven strategy for the modeling and optimization of thermal comfort words and energy consumption. First, we propose a methodology to convert the interval survey data on thermal comfort words to the interval type-2 fuzzy sets (IT2 FSs) which can reflect the inter-personal and intra-personal uncertainties contained in the intervals. This data-driven strategy includes three steps: survey data collection and pre-processing, ambiguity-preserved conversion of the survey intervals to their representative type-1 fuzzy sets (T1 FSs), IT2 FS modeling. Then, using the IT2 FS models of thermal comfort words as antecedent parts, an evolving type-2 fuzzy model is constructed to reflect the online observed energy consumption data. Finally, a multiobjective optimization model is presented to recommend a reasonable temperature range that can give comfortable feeling while reducing energy consumption. The proposed method can be used to realize comfortable but energy-saving environment in smart home or intelligent buildings.     

On answering the question “Where do I start in order to solve a new problem involving interval type-2 fuzzy sets?”

This paper, which is tutorial in nature, demonstrates how the Embedded Sets Representation Theorem (RT) for a general type-2 fuzzy set (T2 FS), when specialized to an interval (I)T2 FS, can be used as the starting point to solve many diverse problems that involve IT2 FSs. The problems considered are: set theoretic operations, centroid, uncertainty measures, similarity, inference engine computations for Mamdani IT2 fuzzy logic systems, linguistic weighted average, person membership function approach to type-2 fuzzistics, and Interval Approach to type-2 fuzzistics. Each solution obtained from the RT is a structural solution but is not a practical computational solution, however, the latter are always found from the former. It is this author’s recommendation that one should use the RT as a starting point whenever solving a new problem involving IT2 FSs because it has had such great success in solving so many such problems in the past, and it answers the question “Where do I start in order to solve a new problem involving IT2 FSs?”     

Uncertainty measures for interval type-2 fuzzy sets

Fuzziness (entropy) is a commonly used measure of uncertainty for type-1 fuzzy sets. For interval type-2 fuzzy sets (IT2 FSs), centroid, cardinality, fuzziness, variance and skewness are all measures of uncertainties. The centroid of an IT2 FS has been defined by Karnik and Mendel. In this paper, the other four concepts are defined. All definitions use a Representation Theorem for IT2 FSs. Formulas for computing the cardinality, fuzziness, variance and skewness of an IT2 FS are derived. These definitions should be useful in IT2 fuzzy logic systems design using the principles of uncertainty, and in measuring the similarity between two IT2 FSs.     

Conceptual design evaluation using interval type-2 fuzzy information axiom

Concept selection is the most critical part of the design process as it determines the direction of subsequent design stages. In addition, it is a difficult task because available information for decision-making at this stage is imprecise and subjective. This necessitates the need for fuzzy decision models for selecting the best conceptual design among a set of alternatives. Although ordinary fuzzy sets cover uncertainties of linguistic words to some extent, it is recommended to use interval type-2 fuzzy sets (IT2FS) to capture potential uncertainties of words. This paper presents a new concept selection methodology that extends the fuzzy information axiom (FIA) approach to incorporate IT2FSs. The proposed methodology is called interval-type-2 fuzzy information axiom (IT2-FIA). IT2-FIA method is also enriched by using ordered weighted geometric aggregation operator to include the decision maker's attitude during the aggregation process. A case study is given to demonstrate the potential of the methodology.     

Parameter Optimization of Interval Type-2 Fuzzy Neural Networks Based on PSO and BBBC Methods

Interval type-2 fuzzy neural networks (IT2FNNs) can be seen as the hybridization of interval type-2 fuzzy systems (IT2FSs) and neural networks (NNs). Thus, they naturally inherit the merits of both IT2FSs and NNs. Although IT2FNNs have more advantages in processing uncertain, incomplete, or imprecise information compared to their type-1 counterparts, a large number of parameters need to be tuned in the IT2FNNs, which increases the difficulties of their design. In this paper, big bang-big crunch (BBBC) optimization and particle swarm optimization (PSO) are applied in the parameter optimization for Takagi-Sugeno-Kang (TSK) type IT2FNNs. The employment of the BBBC and PSO strategies can eliminate the need of backpropagation computation. The computing problem is converted to a simple feed-forward IT2FNNs learning. The adoption of the BBBC or the PSO will not only simplify the design of the IT2FNNs, but will also increase identification accuracy when compared with present methods. The proposed optimization based strategies are tested with three types of interval type-2 fuzzy membership functions (IT2FMFs) and deployed on three typical identification models. Simulation results certify the effectiveness of the proposed parameter optimization methods for the IT2FNNs.      

Type-2 Fuzzistics for Nonsymmetric Interval Type-2 Fuzzy Sets: Forward Problems

Interval type-2 fuzzy sets (IT2 FS) play a central role in fuzzy sets as models for words and in engineering applications of T2 FSs. These fuzzy sets are characterized by their footprints of uncertainty (FOU), which in turn are characterized by their boundaries-upper and lower membership functions (MF). The centroid of an IT2 FS, which is an IT1 FS, provides a measure of the uncertainty in the IT2 FS. The main purpose of this paper is to quantify the centroid of a non-symmetric IT2 FS with respect to geometric properties of its FOU. This is very important because interval data collected from subjects about words suggests that the FOUs of most words are non-symmetrical. Using the results in this paper, it is possible to formulate and solve forward problems, i.e., to go from parametric non-symmetric IT2 FS models to data with associated uncertainty bounds. We provide some solutions to such problems for non-symmetrical triangular, trapezoidal, Gaussian and shoulder FOUs.     

基于区间二型模糊决策粗糙集的三支决策方法

考虑到区间二型模糊数在描述高度不确定性信息方面的优势,将区间二型模糊数拓展到决策粗糙集中,提出两种区间二型模糊三支决策方法.在没有类标签的区间二型模糊信息系统中,解释损失函数与确定条件概率是需要解决的两个关键问题.首先,根据区间二型模糊数的性质,将其引入决策粗糙集中,为损失函数提供一种新的解释.其次,基于贝叶斯决策过程...     

Interval type-2 fuzzy membership function generation methods for pattern recognition

Type-2 fuzzy sets (T2 FSs) have been shown to manage uncertainty more effectively than T1 fuzzy sets (T1 FSs) in several areas of engineering [4], [6], [7], [8], [9], [10], [11], [12], [15], [16], [17], [18], [21], [22], [23], [24], [25], [26], [27] and [30]. However, computing with T2 FSs can require undesirably large amount of computations since it involves numerous embedded T2 FSs. To reduce the complexity, interval type-2 fuzzy sets (IT2 FSs) can be used, since the secondary memberships are all equal to one [21]. In this paper, three novel interval type-2 fuzzy membership function (IT2 FMF) generation methods are proposed. The methods are based on heuristics, histograms, and interval type-2 fuzzy C-means. The performance of the methods is evaluated by applying them to back-propagation neural networks (BPNNs). Experimental results for several data sets are given to show the effectiveness of the proposed membership assignments.     

A comparative study of ranking methods, similarity measures and uncertainty measures for interval type-2 fuzzy sets

Ranking methods, similarity measures and uncertainty measures are very important concepts for interval type-2 fuzzy sets (IT2 FSs). So far, there is only one ranking method for such sets, whereas there are many similarity and uncertainty measures. A new ranking method and a new similarity measure for IT2 FSs are proposed in this paper. All these ranking methods, similarity measures and uncertainty measures are compared based on real survey data and then the most suitable ranking method, similarity measure and uncertainty measure that can be used in the computing with words paradigm are suggested. The results are useful in understanding the uncertainties associated with linguistic terms and hence how to use them effectively in survey design and linguistic information processing.     

广义区间二型模糊集合的词计算

普通的模糊集合是点值为二维的一型模糊集合,二型模糊集合(Type-2 fuzzy sets, T2 FS)是点值为三维的模糊集合, T2 FS比相应的一型难以理解和计算. 为了让人们更好地理解T2 FS并推广其应用, 本文提出了广义区间二型模糊集合(Generalized interval type-2 fuzzy sets, GIT2 FS)的定义, 并将其分成三类:离散型、半离散型及连续型,分别给出相应的数学表达式与扩展原理公式,并得到了GIT2 FS在两种不同的模糊逻辑算子下的词计算.     

$alpha$-Plane Representation for Type-2 Fuzzy Sets: Theory and Applications

This paper 1) reviews the alpha-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the alpha-cut representation of a type-1 FS (T1 FS) and is useful for both theoretical and computational studies of and for T2 FSs; 2) proves that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using alpha-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each alpha-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 alpha -planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and 7) compares T1, IT2, and triangle Q-T2 FLSs to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.     

A perceptual computing-based method to prioritize failure modes in failure mode and effect analysis and its application to edible bird nest farming

A failure mode and effect analysis (FMEA) procedure that incorporates a novel Perceptual Computing (Per-C)–based Risk Priority Number (RPN) model is proposed in this paper. The proposed model considers linguistic uncertainties and vagueness of words, because it is more natural to use words, instead of numerals, for an FMEA user to express his/her knowledge when he/she provides an assessment. Therefore, it is important to consider the inherited uncertainties in words used by humans for assessment as an additional risk factor in the entire FMEA reasoning process. As such, we propose to use Per-C to analyze the uncertainties in words provided by different FMEA users. There are three potential sources of risks. Firstly, the risk factors of Severity (S), Occurrence (O), and Detection (D) are graded using words by each FMEA user, and indicated as interval type-2 fuzzy sets (IT2FSs). Secondly, the relative importance of S, O, and D are reflected by the weights given by each FMEA user in words, which are indicated as IT2FSs. Thirdly, the expertise level of each FMEA user is reflected by words, which are expressed as IT2FSs too. The proposed Per-C-RPN model allows these three sources of risks from each FMEA user to be considered and combined in terms of IT2FSs. A case study related to edible bird nest farming in Borneo Island is reported. The results indicate the effectiveness of the proposed model. In summary, this paper contributes to a new Per-C-RPN model that utilizes imprecise assessment grades pertaining to group decision making in FMEA.     

Industrial applications of type-2 fuzzy sets and systems: A concise review

Data, as being the vital input of system modelling, contain dissimilar level of imprecision that necessitates different modelling approaches for proper analysis of the systems. Numbers, words and perceptions are the forms of data that has varying levels of imprecision. Existing approaches in the literature indicate that, computation of different data forms are closely linked with the level of imprecision, which the data already have. Traditional mathematical modelling techniques have been used to compute the numbers that have the least imprecision. Type-1 fuzzy sets have been used for words and type-2 fuzzy sets have been employed for perceptions where the level of imprecision is relatively high. However, in many cases it has not been easy to decide whether a solution requires a traditional approach, i.e., type-1 fuzzy approach or type-2 fuzzy approach. It has been a difficult matter to decide what types of problems really require modelling and solution either with type-1 or type-2 fuzzy approach. It is certain that, without properly distinguishing differences between the two approaches, application of type-1 and type-2 fuzzy sets and systems would probably fail to develop robust and reliable solutions for the problems of industry. In this respect, a review of the industrial applications of type-2 fuzzy sets, which are relatively novel to model imprecision has been considered in this work. The fundamental focus of the work has been based on the basic reasons of the need for type-2 fuzzy sets for the existing studies. With this purpose in mind, type-2 fuzzy sets articles have been selected from the literature using the online databases of ISI-Web of Science, ScienceDirect, SpringerLink, Informaworld, Engineering Village, Emerald and IEEE Xplore. Both the terms “type-2 fuzzy” and “application” have been searched as the main keywords in the topics of the studies to retrieve the relevant works. The analysis on the industrial applications of type-2 fuzzy sets/systems (FSs) in different topics allowed us to summarize the existing research areas and therefore it is expected be useful to prioritize future research topics. This review shows that there are still many opportunities for application of type-2 FSs for several different problem domains. Shortcomings of type-1 FSs can also be considered as an opportunity for the application of type-2 FSs in order to provide a better solution approach for industrial problems.     

The multiobjective stochastic CRITIC–TOPSIS approach for solving the shipboard crane selection problem

The duty of shipboard cranes is to lift and lower loads, as well as to handle floating facilities to lower or higher positions by means of fixed wire ropes, pulleys, and hook, and so forth. Hence, they play an important role in the productivity of servicing or manufacturing systems. Since each crane has distinguished properties than the others with respect to criteria and decision-makers (DMs) may express the different standpoints regarding them, the crane selection problem (CSP) can be considered as a group multicriteria decision-making (MCDM) problem. In this paper, interval type-2 fuzzy sets (IT2FSs) are first used to evaluate cranes with respect to criteria. The synthetic value method of IT2FSs is then handled to integrate the ratings expressed as IT2FSs of each crane with respect to criteria into the single fuzzy rating. Finally, the multiobjective criteria importance through inter-criteria correlation (CRITIC)–technique for order of preference by similarity to ideal solution (TOPSIS) approach is applied to solve the CSP in which CRITIC and TOPSIS are used to determine the objective weights and score of cranes, respectively. In addition, the limit distance mean (LDM) is introduced for ranking interval type-2 fuzzy ratings in the above two techniques. In contrast, to demonstrate the potential application, the proposed methodology is implemented in a real case study and the ranking results are compared with those published in the literature.     

区间二型模糊相似度与包含度

相似度与包含度是模糊集合理论中的两个重要概念,但对于二型模糊集合的研究还较为少见．鉴于此,提出了新的区间二型模糊相似度与包含度．首先选择了二者的公理化定义;然后基于公理化定义提出了新的计算公式,并讨论了二者的相互转换关系;最后通过实例来验证二者的性能,并将区间二型模糊相似度与Yang-Shih聚类方法相结合,用于高斯区间二型模糊集合的聚类分析,得到了合理的层次聚类树．仿真实例表明新测度具有一定的实用价值．     

Interval Type-2 Fuzzy Logic Systems Made Simple

To date, because of the computational complexity of using a general type-2 fuzzy set (T2 FS) in a T2 fuzzy logic system (FLS), most people only use an interval T2 FS, the result being an interval T2 FLS (IT2 FLS). Unfortunately, there is a heavy educational burden even to using an IT2 FLS. This burden has to do with first having to learn general T2 FS mathematics, and then specializing it to an IT2 FSs. In retrospect, we believe that requiring a person to use T2 FS mathematics represents a barrier to the use of an IT2 FLS. In this paper, we demonstrate that it is unnecessary to take the route from general T2 FS to IT2 FS, and that all of the results that are needed to implement an IT2 FLS can be obtained using T1 FS mathematics. As such, this paper is a novel tutorial that makes an IT2 FLS much more accessible to all readers of this journal. We can now develop an IT2 FLS in a much more straightforward way     

A vector similarity measure for linguistic approximation: Interval type-2 and type-1 fuzzy sets

Fuzzy logic is frequently used in computing with words (CWW). When input words to a CWW engine are modeled by interval type-2 fuzzy sets (IT2 FSs), the CWW engine’s output can also be an IT2 FS, , which needs to be mapped to a linguistic label so that it can be understood. Because each linguistic label is represented by an IT2 FS , there is a need to compare the similarity of and to find the most similar to . In this paper, a vector similarity measure (VSM) is proposed for IT2 FSs, whose two elements measure the similarity in shape and proximity, respectively. A comparative study shows that the VSM gives more reasonable results than all other existing similarity measures for IT2 FSs for the linguistic approximation problem. Additionally, the VSM can also be used for type-1 FSs, which are special cases of IT2 FSs when all uncertainty disappears.     

Encoding Words Into Interval Type-2 Fuzzy Sets Using an Interval Approach

This paper presents a very practical type-2-fuzzistics methodology for obtaining interval type-2 fuzzy set (IT2 FS) models for words, one that is called an interval approach (IA). The basic idea of the IA is to collect interval endpoint data for a word from a group of subjects, map each subject's data interval into a prespecified type-1 (T1) person membership function, interpret the latter as an embedded T1 FS of an IT2 FS, and obtain a mathematical model for the footprint of uncertainty (FOU) for the word from these T1 FSs. The IA consists of two parts: the data part and the FS part. In the data part, the interval endpoint data are preprocessed, after which data statistics are computed for the surviving data intervals. In the FS part, the data are used to decide whether the word should be modeled as an interior, left-shoulder, or right-shoulder FOU. Then, the parameters of the respective embedded T1 MFs are determined using the data statistics and uncertainty measures for the T1 FS models. The derived T1 MFs are aggregated using union leading to an FOU for a word, and finally, a mathematical model is obtained for the FOU. In order that all researchers can either duplicate our results or use them in their research, the raw data used for our codebook examples, as well as a MATLAB M-file for the IA, have been put on the Internet at: http://sipi.usc.edu/$sim$mendel.