The central limit theorems for fuzzy random variables

The new concept of the central limit theorem for fuzzy random variables is discussed in this paper by proposing the convergence in distribution for fuzzy random variables. We first consider the limit properties of fuzzy numbers by invoking the Hausdorff metric and then we extend it to the weak and strong convergence of fuzzy distribution functions. We provide a notion of fuzzy normal distribution. Then the central limit theorem for fuzzy random variables follows naturally.     

Methods for fuzzy complementary preference relations based on multiplicative consistency

To improve the consistency of a preference relation is a hot topic in decision making. Wang and Chen (2008) gave a simple method to construct the complete fuzzy complementary preference relation from only n − 1 pairwise comparisons. However, some values may not be in the defined scope and need to be transformed, and thus some original information may be lost in the transformation process. In this paper, we propose a new method to avoid this issue based on the multiplicative consistency of the fuzzy complementary preference relation and apply it to fuzzy Analytic Hierarchy Process (AHP). An example is further given to illustrate our method.     

Fixed point theorems on fuzzy normed linear spaces

In this paper, definitions of strongly fuzzy convergent sequence, l-fuzzy weakly convergent sequence and l-fuzzy weakly compact set are given in a fuzzy normed linear space. The concepts of fuzzy normal structure, fuzzy non-expansive mapping, uniformly convex fuzzy normed linear space are introduced and fixed point theorems for fuzzy non-expansive mappings are proved.     

Applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy AHP

The lack of consistency in decision making can lead to inconsistent conclusions. In fuzzy analytic hierarchy process (fuzzy AHP) method, it is difficult to ensure a consistent pairwise comparison. Furthermore, establishing a pairwise comparison matrix requires judgments for a level with n criteria (alternatives). The number of comparisons increases as the number of criteria increases. Therefore, the decision makers judgments will most likely be inconsistent. To alleviate inconsistencies, this study applies fuzzy linguistic preference relations (Fuzzy LinPreRa) to construct a pairwise comparison matrix with additive reciprocal property and consistency. In this study, the fuzzy AHP method is reviewed, and then the Fuzzy LinPreRa method is proposed. Finally, the presented method is applied to the example addressed by Kahraman et al. [C. Kahraman, D. Ruan, I. Do?an, Fuzzy group decision making for facility location selection, Information Sciences 157 (2003) 135-153]. This study reveals that the proposed method yields consistent decision rankings from only n − 1 pairwise comparisons, which is the same result as in Kahraman et al. research. The presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy AHP method.     

A revisited approach to linear fuzzy regression using trapezoidal fuzzy intervals

Conventional Fuzzy regression using possibilistic concepts allows the identification of models from uncertain data sets. However, some limitations still exist. This paper deals with a revisited approach for possibilistic fuzzy regression methods. Indeed, a new modified fuzzy linear model form is introduced where the identified model output can envelop all the observed data and ensure a total inclusion property. Moreover, this model output can have any kind of spread tendency. In this framework, the identification problem is reformulated according to a new criterion that assesses the model fuzziness independently from the collected data distribution. The potential of the proposed method with regard to the conventional approach is illustrated by simulation examples.     

On admissible total orders for typical hesitant fuzzy consensus measures

In this paper, we discussconsensus measures for typical hesitant fuzzy elements (THFE), which are the finite and nonempty fuzzy membership degrees under the scope of typical hesitant fuzzy sets (THFS). In our approach, we present a model that formally constructs consensus measures by means of aggregations functions, fuzzy implication-like functions and fuzzy negations, using admissible orders to compare the THFE, and also providing an analysis of consistency on them. Our theoretical results are applied into a problem of decision making with multicriteria illustrating our methodology to achieve consensus in a group of experts working with THFS.     

Combining fuzzy AHP and fuzzy Kano to optimize product varieties for smart cameras: A zero-one integer programming perspective

In an era of global customization, dominating the majority market with a single product has become increasingly difficult and almost impossible for most companies. In contrast, they must provide various product varieties that attract diverse customers, particularly when acquiring distinct market segments. In practice, however, most companies cannot effectively reduce the gap between customer requirements and design characteristics, although this impacts the profitability and future growth of companies. Meanwhile, companies often get stuck in the trade-offs between enhancing product varieties and controlling manufacturing costs. Accordingly, this paper proposes a hybrid framework that combines fuzzy analytical hierarchy process (AHP), fuzzy Kano model with zero-one integer programming (ZOIP) to incorporate customer preferences and customer perceptions into the decision-making process of product configuration. Specifically, fuzzy AHP is used to extract customer preferences for core attributes while fuzzy Kano model is utilized to elicit customer perceptions of optional attributes. Finally, by virtue of ZOIP, the optimal product varieties (smart cameras) for distinct segments are determined by maximizing overall customer utility (OCU) and taking a firm's pricing policy into account.     

Quality-based supplier selection and evaluation using fuzzy data

Since fuzzy quality data are ubiquitous in the real world, under this fuzzy environment, the supplier selection and evaluation on the basis of the quality criterion is proposed in this paper. The C_pk index has been the most popular one used to evaluate the quality of supplier’s products. Using fuzzy data collected from q2 possible suppliers’ products, fuzzy estimates of q suppliers’ capability indices are obtained according to the form of resolution identity that is a well-known theorem in fuzzy sets theory. Certain optimization problems are formulated and solved to obtain α-level sets for the purpose of constructing the membership functions of fuzzy estimates of C_pki. These membership functions are sorted by using a fuzzy ranking method to choose the preferable suppliers. Finally, a numerical example is illustrated to present the possible application by incorporating fuzzy data into the quality-based supplier selection and evaluation.     

A web-based decision support system for multi-criteria inventory classification using fuzzy AHP methodology

We propose an inventory classification system based on the fuzzy analytic hierarchy process (AHP), a commonly used tool for multi-criteria decision making problems. We integrate fuzzy concepts with real inventory data and design a decision support system assisting a sensible multi-criteria inventory classification. We report on a study conducted in a small electrical appliances company and validate the design of the proposed multi-criteria inventory classification system and its underlying fuzzy AHP model.     

On the order of the preference intensities in fuzzy AHP

We show that a recently discovered fundamental problem with the Analytic Hierarchy Process (AHP) concerning the meaning of the resultant preference intensities is also evident for the fuzzy AHP. We prove that if there is a judgmental inconsistency in the fuzzy pair-wise comparisons, it is impossible to ensure the preservation of the order regarding to preference intensities in the resultant priority vector. Further, it is shown with an example from the published literature that the order of the preference intensities may not be preserved even there is no inconsistency in the judgment set, albeit it is possible to comply with this order via using fuzzy preference programming (FPP) methodology. Finally, it is proved that if the interval judgments regarding to the decompositions of original judgments to - level sets are consistent, FPP guarantees the preservation of the order of the preference intensities at those levels.     

偏好关系的表示和收缩

针对偏好经常会改变的问题,提出偏好的舍弃操作——偏好收缩,而对于偏好收缩来说,偏好收缩的最小化和保持严格偏序性是很重要的。给出可用于表示有限和无限偏好关系的框架,并针对偏好的收缩问题提出算法,计算偏好关系对基本收缩的全收缩,使收缩后的偏好满足严格偏序,对全收缩进行最小化,得到修正后的偏好全收缩。仿真结果表明,修正后的全收缩小于修正前的全收缩。     

A joint economic-lot-size model for purchaser and vendor in fuzzy sense

This paper investigates a group of computing schemas for joint economic lot size as fuzzy values of the economic lot size model for purchaser and vendor. We express the fuzzy order quantity/production lot size for the purchaser/vendor as the normal triangular fuzzy number (q₁, q₀, q₂) and then we solve the aforementioned optimization problem under the condition 0 < q₁ < q₀ < q₂. We find that, after defuzzification, the joint total relevant cost is slightly higher than in the crisp model.     

Group decision making based on incomplete multiplicative and fuzzy preference relations

The main aim of this paper is to investigate the group decision making on incomplete multiplicative and fuzzy preference relations without the requirement of satisfying reciprocity property. This paper introduces a new characterization of the multiplicative consistency condition, based on which a method to estimate unknown preference values in an incomplete multiplicative preference relation is proposed. Apart from the multiplicative consistency property among three known preference values, the method proposed also takes the multiplicative consistency property among more than three values into account. In addition, two models for group decision making with incomplete multiplicative preference relations and incomplete fuzzy preference relations are presented, respectively. Some properties of the collective preference relation are further discussed. Numerical examples are provided to make a discussion and comparison with other similar methods.     

Data-driven fuzzy modeling for Takagi-Sugeno-Kang fuzzy system

This paper presents a systematic approach to design first order Tagaki-Sugeno-Kang (TSK) fuzzy systems. This approach attempts to obtain the fuzzy rules without any assumption about the structure of the data. The structure identification and parameter optimization steps in this approach are carried out automatically, and are capable of finding the optimal number of the rules with an acceptable accuracy. Starting with an initial structure, the system first tries to improve the structure and, then, as soon as an improved structure is found, it fine tunes its rules’ parameters. Then, it goes back to improve the structure again to find a better structure and re-fine tune the rules’ parameters. This loop continues until a satisfactory solution (TSK model) is found. The proposed approach has successfully been applied to well-known benchmark datasets and real-world problems. The obtained results are compared with those obtained with other methods from the literature. Experimental studies demonstrate that the predicted properties have a good agreement with the measured data by using the elicited fuzzy model with a small number of rules. Finally, as a case study, the proposed approach is applied to the desulfurization process of a real steel industry. Comparing the proposed approach with some other fuzzy systems and neural networks, it is shown that the developed TSK fuzzy system exhibits better results with higher accuracy and smaller size of architecture.     

Algorithms of discrete optimization and their application to problems with fuzzy coefficients

An approach to solving optimization problems with fuzzy coefficients in objective functions and constraints is described. It consists in formulating and solving one and the same problem within the framework of mutually related models with constructing equivalent analogs with fuzzy coefficients in objective functions alone. It enables one to maximally cut off dominated alternatives “from below” as well as “from above”. Since the approach is applied within the context of fuzzy discrete optimization problems, several modified algorithms of discrete optimization are discussed. These algorithms are associated with the method of normalized functions, are based on a combination of formal and heuristic procedures, and allow one to obtain quasi-optimal solutions after a small number of steps, thus overcoming the computational complexity posed the NP-completeness of discrete optimization problems. The subsequent contraction of the decision uncertainty regions is associated with reduction of the problem to multiobjective decision making in a fuzzy environment with using techniques based on fuzzy preference relations. The techniques are also directly applicable to situations in which the decision maker is required to choose alternatives from a set of explicitly available alternatives. The results of the paper are of a universal character and can be applied to the design and control of systems and processes of different purposes as well as the enhancement of corresponding CAD/CAM systems and intelligent decision making systems. The results of the paper are already being used to solve problems of power engineering.     

On fuzzy equality and approximation in fuzzy logic

The paper is a contribution to the theory of fuzzy logic in narrow sense with evaluated syntax (FLn). We show that the concepts of fuzzy equality and the provability degree enable to generalize the concept of fuzzy approximation. In the second part of the paper we return to the Mamdani-Assilian formula, which is formed on the basis of the so called totally bounded fuzzy equality and using which we can approximate any function with the prescribed accuracy.This paper has been supported by Grant A1187901/99 of the GA AV R and the project VS96037 of MMT of the Czech Republic.     

Generating fuzzy rules from training instances for fuzzy classification systems

In recent years, many methods have been proposed to generate fuzzy rules from training instances for handling the Iris data classification problem. In this paper, we present a new method to generate fuzzy rules from training instances for dealing with the Iris data classification problem based on the attribute threshold value α, the classification threshold value β and the level threshold value γ, where α  [0, 1], β  [0, 1] and γ  [0, 1]. The proposed method gets a higher average classification accuracy rate than the existing methods.     

Integration of fuzzy logic, genetic algorithms and neural networks in collaborative parametric design

The solution to a design problem is extracted through the exploitation of the design knowledge in the context of a space of solution alternatives. The design process incorporates a series of decision making and knowledge management issues, which should be often addressed through collaboration among diverse stakeholders. The alternative solutions must usually be shaped under different formalisms and evaluated against commonly accepted objective criteria.The current paper presents an approach that integrates soft-computing techniques in order to facilitate the computer-aided collaboration among designers. CopDeSC (Collaborative parametric Design with Soft-Computing) is the name of the system developed in order to implement this approach whose key features are: (a) the collaborative structuring of design parameter hierarchies, (b) the modeling of the design objectives through fuzzy preferences stated by the designers on certain design parameters, (c) the deployment of genetic algorithms for locating the optimum solution and (d) the utilization of records of elite solutions that are submitted in a neuro-fuzzy approximation in order to produce a simplified problem formulation, suitable for addressing redesign tasks in significantly less computational time.The efficiency of CopDeSC is evaluated in an example case of the parametric design of oscillating conveyor that has been conducted by a group of designers.     

Distance and similarity measures for fuzzy operators

We introduce and study a new family of normalized distance measures between binary fuzzy operators, along with its dual family of similarity measures. Both are based on matrix norms and arise from the study of the aggregate plausibility of set-operations. We also suggest a new family of normalized distance measures between fuzzy sets, based on binary operators and matrix norms, and discuss its qualitative and quantitative features. All measures proposed are intended for applications and may be customized according to the needs and intuition of the user.     

A new fuzzy peer assessment methodology for cooperative learning of students

In this paper, a new fuzzy peer assessment methodology that considers vagueness and imprecision of words used throughout the evaluation process in a cooperative learning environment is proposed. Instead of numerals, words are used in the evaluation process, in order to provide greater flexibility. The proposed methodology is a synthesis of perceptual computing (Per-C) and a fuzzy ranking algorithm. Per-C is adopted because it allows uncertainties of words to be considered in the evaluation process. Meanwhile, the fuzzy ranking algorithm is deployed to obtain appropriate performance indices that reflect a student's contribution in a group, and subsequently rank the student accordingly. A case study to demonstrate the effectiveness of the proposed methodology is described. Implications of the results are analyzed and discussed. The outcomes clearly demonstrate that the proposed fuzzy peer assessment methodology can be deployed as an effective evaluation tool for cooperative learning of students.