Remarks and corrections to the triangular approximations of fuzzy numbers using ��-weighted valuations

A recent paper was dedicated to find the nearest fuzzy triangular approximations of a fuzzy number by using ??-weighted valuations. We prove, by simple examples, that the results of approximations are not always triangular fuzzy numbers and that in fact they are not fuzzy sets. We give a correct solution of the problem of approximation in a more general case, and we study the properties of identity, additivity, translation invariance, scale invariance, and monotonicity of the new approximation operator.     

On the class of λ-statistically convergent difference sequences of fuzzy numbers

In this study, we introduce the sets $\left[ V,\lambda ,p\right] _{\Updelta }^{{\mathcal{F}}},\left[ C,1,p\right] _{\Updelta }^{{\mathcal{F}}}$ and examine their relations with the classes of $ S_{\lambda }\left( \Updelta ,{\mathcal{F}}\right)$ and $ S_{\mu }\left( \Updelta ,{\mathcal{F}}\right)$ of sequences for the sequences $\left( \lambda _{n}\right)$ and $\left( \mu _{n}\right) , 0<p<\infty $ and difference sequences of fuzzy numbers.     

On λ-statistical convergence of difference sequences of fuzzy numbers

In this paper the concept of strongly -Cesàro summability of a sequence of fuzzy numbers is introduced. Also some inclusion relations between the set of strongly -Cesàro convergent and -statistically convergent sequences of fuzzy numbers are given.     

RM approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Syst Appl 36:6833–6842, 2009) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L–R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets Syst 118:375–385, 2001).     

A new approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. Cheng (Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95, 307–317) pointed out that the proof of the statement “Ranking of generalized fuzzy numbers does not depend upon the height of fuzzy numbers” stated by Liou and Wang (Liou, T. S., & Wang, M. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, 247–255) is incorrect. In this paper, by giving an alternative proof it is proved that the above statement is correct. Also with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36, 6833–6842) is incorrect. The main aim of this paper is to modify the Liou and Wang approach for the ranking of L–R type generalized fuzzy numbers. The main advantage of the proposed approach is that the proposed approach provide the correct ordering of generalized and normal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfy all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems, 118, 375–385).     

Selecting the optimal shift numbers using fuzzy control model: a paint factory’s facility application

Number of shifts is one of the most important criteria for the production planners to minimize the production costs. The first measurement of the facility, where we executed the application, is optimizing the manpower rotation in B12 production, a special kind of chemical, which is used for paint binders. Seasonal variations in demand of B12 add more importance to the number of shifts in the facility. The purpose of this article is to optimize the shift periods subject to raw material, shipping date, inventory and demand constraints. The model is developed for a paint factory’s reactor facility. The production system has a raw material inventory, a reactor, and a finished goods inventory. We use fuzzy control to optimize the number of shifts under the constraints, given before.     

Posture control of robot manipulators with fuzzy voice commands using a fuzzy coach–player system

This paper presents a method of controlling robot manipulators with fuzzy voice commands. Recently, there has been some research on controlling robots using information-rich fuzzy voice commands such as 'go little slowly' and learning from such commands. However, the scope of all those works was limited to basic fuzzy voice motion commands. In this paper, we introduce a method of controlling the posture of a manipulator using complex fuzzy voice commands. A complex fuzzy voice command is composed of a set of fuzzy voice joint commands. Complex fuzzy voice commands can be used for complicated maneuvering of a manipulator, while fuzzy voice joint commands affect only a single joint. Once joint commands are learned, any complex command can be learned as a combination of some or all of them, so that, using the learned complex commands, a human user can control the manipulator in a complicated manner with natural language commands. Learning of complex commands is discussed in the framework of fuzzy coach–player model. The proposed idea is demonstrated with a PA-10 redundant manipulator.     

A new metric for L–R fuzzy numbers and its application in fuzzy linear systems

In this paper, a metric based on modified Euclidean metric on interval numbers, for L–R fuzzy numbers with fixed $L(\cdot)$ and $R(\cdot)$ is introduced. Then, it is applied for solving L–R fuzzy linear system (L–R-FLS) with fuzzy right-hand side, so that L–R-FLS is transformed to the minimization problem. The solution of the mentioned non-linear programming problem is our favorite fuzzy number vector solution. Two constructive Algorithms are proposed in detail and the method is illustrated by solving several numerical examples.     

Robust spatial flood vulnerability assessment for Han River using fuzzy TOPSIS with α-cut level set

This study aims to improve the general flood vulnerability approach using fuzzy TOPSIS based on α-cut level sets which can reduce the uncertainty inherent in even fuzzy multi-criteria decision making process. Since fuzzy TOPSIS leads to a crisp closeness for each alternative, it is frequently argued that fuzzy weights and fuzzy ratings should be in fuzzy relative closeness. Therefore, this study used a modified α-cut level set based fuzzy TOPSIS to develop a spatial flood vulnerability approach for Han River in Korea, considering various uncertainties in weights derivation and crisp data aggregation. Two results from fuzzy TOPSIS and modified fuzzy TOPSIS were compared. Some regions which showed no or small ranking changes have their centro-symmetric distributions, while other regions whose rankings varied dynamically, have biased (anti-symmetric) distributions. It can be concluded that α-cut level set based fuzzy TOPSIS produce more robust prioritization since more uncertainties can be considered. This method can be applied to robust spatial vulnerability or decision making in water resources management.     

Development of optimal water supply plan using integrated fuzzy Delphi and fuzzy ELECTRE III methods—Case study of the Gamasiab basin

This paper presents a novel method for the development of an optimal water supply plan showcased using data from the Gamasiab basin, located in Kermanshah province, Iran, concerning new dams that are being constructed in this semi-arid region. In this paper, a new group multi-criteria decision-making (MCDM) plan is proposed by combining two MCDM methods based on the fuzzy Delphi and fuzzy ELECTRE III methods that convert the experts' opinions to triangular fuzzy numbers based on the level of uncertainty associated with various quantitative and qualitative criteria. Considering the opinions of four non-stakeholder experts and data analysis using the fuzzy Delphi method, the criteria were evaluated. Then, by analysing the results using the fuzzy ELECTRE III method, the final ranking of scenarios is obtained. A sensitivity analysis was conducted to assess the effect of uncertainty on the performance of the decision-making system in scenarios ranking. The total expense, flood control, reservoir capacity and diversion and water transfer played a significant role in selecting the optimal scenario. Additionally, a hydrologic model was developed to evaluate the performance of the optimal scenario in terms of qualitative criteria. The data indicated that there was a good agreement between the results obtained from the hydrological model and the scenario ranking by the employed method. Altogether, a comparison of the proposed method with other MCDM methods, including fuzzy analytic hierarchy process and fuzzy technique for order preference by simulation of ideal solution, indicated that the results of the employed method matched more closely to the local experts' opinion.     

Stabilization of Rössler chaotic dynamical system using fuzzy logic control algorithm

This paper proposes a fuzzy logic control algorithm (FLCA) to stabilize the Rössler chaotic dynamical system. The fuzzy logic control system is based on a Takagi-Sugeno-Kang inference engine and the stability analysis in the sense of Lyapunov is carried out using Lyapunov’s direct method. The new FLCA is formulated to offer sufficient inequality stability conditions. The asymptotic complexity of our algorithm is analyzed and proved to be lower in comparison with that of linear matrix inequality-based FLCAs. A set of simulation results illustrates the effectiveness of the proposed FLCA.     

Hierarchical optimization of boiler–turbine unit using fuzzy stable model predictive control

This paper develops a hierarchical control system structure based on the Takagi–Sugeno fuzzy model to achieve an optimal control of a boiler–turbine unit. In the upper layer of the hierarchy, an optimal reference governor is designed to find the optimal operating point. A disturbance term is introduced to the fuzzy model to lump the modeling mismatch and unknown disturbance. Thus, the effect of plant behavior variation can be removed and the operating point found can be feasible to control. In the lower layer, a stable model predictive controller is developed to track the optimal set-points while guaranteeing the input-to-state stability of the system. Fuzzy Lyapunov function and appropriate slack and collection matrices are used to reduce the conservatism of stability design and improve the performance. Through the estimation of the disturbance term using an observer, the two layers in the hierarchy are coupled and the integrated system can realize a dynamic optimal control of the boiler–turbine unit, even in the case of severe plant behavior variations.     

Optimization of multiple input–output fuzzy membership functions using clonal selection algorithm

A clonal selection algorithm (CLONALG) inspires from clonal selection principle used to explain the basic features of an adaptive immune response to an antigenic stimulus. It takes place in various scientific applications and it can be also used to determine the membership functions in a fuzzy system. The aim of the study is to adjust the shape of membership functions and a novice aspect of the study is to determine the membership functions. Proposed method has been implemented using a developed CLONALG program for a multiple input–output (MI–O) fuzzy system. In this study, GA and binary particle swarm optimization (BPSO) are used for implementing the proposed method as well and they are compared. It has been shown that using clonal selection algorithm is advantageous for finding optimum values of fuzzy membership functions     

A discussion on “Calculating fuzzy inverse matrix using fuzzy linear equation system”

In the present note it is shown that the examples presented in a recent paper by Basaran are incorrect. The correct exact solutions are also presented here. In this way, a new direction for approximating fuzzy inverse matrix is proposed. In this paper, the first step we solve the center part of fully fuzzy linear system, then we solve the spread part of the equation system.     

Study of the potential of alternative crops by integration of multisource data using a neuro‐fuzzy technique

This work proposes a neuro‐fuzzy method for suggesting alternative crop production over a region using integrated data obtained from land‐survey maps as well as satellite imagery. The methodology proposed here uses an artificial neural network (multilayer perceptron, MLP) to predict alternative crop production. For each pixel, the MLP takes vector input comprising elevation, rainfall and goodness values of different existing crops. The first two components of the aforementioned input, that is, elevation and rainfall, are determined from contour information of land‐survey maps. The other components, such as goodness values of different existing crops, are based on the productivity estimates of soil determined by fuzzyfication and expert opinion (on soil) along with production quality by the Normalized Difference Vegetation Index (NDVI) obtained from satellite imagery. The methodology attempts to ensure that the suggested crop will also be a high productivity crop for that region.     

A SET-THEORETIC VIEW OF BELIEF FUNCTIONS Logical operations and approximations by fuzzy sets†

A body of evidence in the sense of Shafer can be viewed as an extension of a probability measure, but as a generalized set as well. In this paper we adopt the second point of view and study the algebraic structure of bodies of evidence on a set, based on extended set union, intersection and complementation. Several notions of inclusion are exhibited and compared to each other. Inclusion is used to compare a body of evidence to the product of its projections. Lastly, approximations of a body of evidence under the form of fuzzy sets are derived, in order to squeeze plausibility values between two grades of possibility. Through all the paper, it is pointed out that a body of evidence can account for conjunctive as well as a disjunctive information, i.e. the focal elements can be viewed either as sets of actual values or as restrictions on the (unique) value of a variable.     

Detection of phishing attacks in Iranian e-banking using a fuzzy–rough hybrid system

Phishing is a method of stealing electronic identity in which social engineering and website forging methods are used in order to mislead users and reveal confidential information having economic value. Destroying the trust between users in business network, phishing has a negative effect on the budding area of e-commerce. Developing countries such as Iran have been recently facing Internet threats like phishing, whose methods, regarding the social differences, may be different from other experiences. Thus, it is necessary to design a suitable detection method for these deceits. The aim of current paper is to provide a phishing detection system to be used in e-banking system in Iran. Identifying the outstanding features of phishing is one of the important prerequisites in design of an accurate system; therefore, in first step, to identify the influential features of phishing that best fit the Iranian bank sites, a list of 28 phishing indicators was prepared. Using feature selection algorithm based on rough sets theory, six main indicators were identified as the most effective factors. The fuzzy expert system was designed using these indicators, afterwards. The results show that the proposed system is able to determine the Iranian phishing sites with a reasonable speed and precision, having an accuracy of 88%.     

Image segmentation using Atanassov’s intuitionistic fuzzy sets

The problem of segmentation in spite of all the work over the last decades, is still an important research field and also a critical preprocessing step for image processing, mostly due to the fact that finding a global optimal threshold that works well for all kind of images is indeed a very difficult task that, probably, will never be accomplished.During the past years, fuzzy logic theory has been successfully applied to image thresholding. In this paper we describe a thresholding technique using Atanassov’s intuitionistic fuzzy sets (A-IFSs). This approach uses Atanassov’s intuitionistic index values for representing the hesitance of the expert in determining whether the pixel belongs to the background or that it belongs to the object. First, we describe the general framework of this approach to bi-level thresholding. Then we present its natural extension to multilevel thresholding. This multilevel threshold methodology segments the image into several distinct regions which correspond to a background and several objects.Segmentation experimental results and comparison with Otsu’s multilevel thresholding algorithm for the calculation of two and three thresholds are presented.     

WEEE treatment strategies’ evaluation using fuzzy LINMAP method

Electrical and electronic equipments (EEE) have already begun to be accumulated at the garbage dumps. This garbage accumulation brings big danger to the environment and human health. That’s why one should look for exploring the ways to dispose of these wastes and emphasize the waste treatment strategies. Waste treatment strategies also contribute either to local or global economies by creating a new sector and employment opportunities, and by reducing use of scarce resources. In this paper, a linear programming technique for multidimensional analysis of preference (LINMAP) method for solving multi-attribute group decision making (MAGDM) problems with preference information on alternatives in fuzzy environment is developed. The aim is to develop a fuzzy LINMAP model for evaluation and select of a waste treatment strategy for EEE. Thus, three treatment strategy alternatives and eight criteria are determined. The best strategy is selected and the key criterion is found accordingly. The best alternative is found to be treating waste electrical and electronic equipments (WEEE) by reuse and recycling methods.