Iranian journal of fuzzy systems
Volume:19 Issue: 1, Jan-Feb 2022

Aggregation operations play a fundamental  role in a large number of disciplines, from mathematics and natural sciences to economics and social sciences. This paper is focused on  the problem of distributivity for some special classes  of aggregation operations, and quasi-linear means. Characterization of distributivity pairs for uninorms, semi-uninorms and associative a-CAOA vs quasi-linear means is given.

Process capability indices (PCIs) have been widely used to analyze capability of the process that measures how the customer expectations have been conformed. Two of the well-known PCIs, named indices  C_{pm}  and  C_{pmk}  have been developed to consider customers' ideal value that called target value ( T ). Although, these indices have similar features of the well-known indices C_{p}  and  C_{pk} , one of the most important differences is to consider textit{T}. In real case problems, we need to add some uncertainties related with human's evaluations into process capability analysis (PCA). One of the uncertainty modelling methods called neutrosophic sets (NSs), have an important role in modeling uncertainty based on incomplete and inconsistent information. For this aim, the PCIs have been designed by using NSs to manage the uncertainties of systems and to increase sensitiveness, flexibility and to obtain more detailed results of PCA in this paper. For this aim, the indices  C_{pm} and C_{pmk} have been performed and re-designed by using single valued neutrosophic numbers for the first time in the literature. Additionally, specification limits (SLs) have been re-considered by using NSs. The neutrosophic state of the SLs provide us to have more knowledge about the process and easily applied for engineering problems that includes uncertainty. Finally, the neutrosophic process capability indices (NPCIs) widetilde{dddot{C}}_{pm} and widetilde{dddot{C}}_{pmk} have been obtained and the main formulas of them have been produced. Additionally, the proposed widetilde{dddot{C}}_{pm} and widetilde{dddot{C}}_{pmk} have been applied on real case studies from manufacturing industry. The obtained results show that the indices widetilde{dddot{C}}_{pm} and widetilde{dddot{C}}_{pmk} include more informative and flexible results to evaluate capability of process.

In this research, a Type 2 adaptive fuzzy controller approach is formulated and designed to be applied to variable speed doubly fed induction generator-based wind turbines directly connected to the grid. It brings this study to evaluate the whole operation of the system to capture the highest rate of power in the wind turbines. The controlling approach is considered to keep the stator reactive power to the ideal value. In contrast to the other researches, here the controlling technique is developed through the nonlinear systems.  By the aim of making progress in system operation, in contrast with the Type 1 adaptive fuzzy system, type two adaptive fuzzy theory is proposed to approximate a large number of uncertainties and the dynamic nonlinearities, exists in tracking errors which may limit the system performance. Feedback linearization control approach helps us to algebraically alter the system into a linearized plant. Thanks to the Lyapunov theorem, the introduced type two adaptive fuzzy approach is proved to meet the uniformly ultimately boundness (UUB) property. On the other hand, it results better tracking function. The simulation outputs represent that the proposed technique is robust enough in presence of parameter variations and unstructured uncertainties.

In this paper, a new type of  equation namely  fuzzy fractional stochastic Pantograph delay differential system  (FSPDDS) is proposed. In our previous work,  a first  extension of fuzzy stochastic differential system into   fuzzy fractional stochastic differential system by using Granular differentiability has been established. Here we study the existence and uniqueness  results for the fuzzy FSPDDS which  are obtained by using  generalized Granular  differentiability  and contraction principle with weaker conditions. This kind of equation is used in many real world problems. Finally, we provide two numerical examples for the effectiveness of the theoretical results.

Parametric time series models   typically consists of model identification, parameter estimation, model diagnostic checking, and forecasting. However compared with parametric methods, nonparametric time series models often provide  a very flexible approach to bring out the features of the observed time series. This paper suggested a novel fuzzy nonparametric method in time series models with fuzzy observations. For this purpose, a fuzzy forward fit kernel-based smoothing method was introduced to estimate fuzzy smooth functions corresponding to each observation. A simple optimization algorithm was also suggested to evaluate optimal bandwidths and autoregressive order. Several common goodness-of-fit criteria were also extended to compare the performance of the proposed fuzzy time series method compared to other fuzzy time series model based on fuzzy data. Furthermore, the effectiveness of the proposed method was illustrated through two numerical examples including a simulation study. The results indicate that the proposed model performs better than the previous ones in terms of both scatter plot criteria and goodness-of-fit evaluations.

Triangular norms and conorms on [0,1] as well as on finite chains are characterized by 4 independent properties, namely by the associativity, commutativity, monotonicity and neutral element being one of extremal points of the considered domain (top element for t-norms, bottom element for t-conorms). In the case of [0,1]domain, earlier results of Mostert and Shields on I-semigroups can be used to relax the latest three properties significantly, once the continuity of the underlying t-norm or t-conorm is considered. The aim of this short note is to show a similar result for finite chains, we significantly relax 3 basic properties of t-norms and t-conorms (up to the associativity) when the divisibility of a t-norm or of a t-conorm is considered.

We discuss ordinal sums as one of powerful tools in the aggregation theory serving, depending on the context, both as a construction method and as a representation, respectively. Up to recalling of several classical results dealing with ordinal sums, in particular dealing, e.g.,  with continuous t-norms, copulas, or recent results, e.g., concerning uninorms with continuous underlying functions, we present also several new results, such as the uniqueness of the link between t-norms or t-conorms, and related Archimedean components, problems dealing with the cardinality of the considered index sets in ordinal sums, or infinite ordinal sums of aggregation functions covering by one type of ordinal sums both t-norms and t-conorms ordinal sums.

A hesitant fuzzy number (HFN) is important as a generalization of the fuzzy number for hesitant fuzzy analysis and takes some applications that were discussed in recent literature. In this paper, we develop the hesitant fuzzy arithmetic, which is based on the extension principle for hesitant fuzzy sets. Employing this principle, standard arithmetic operations on fuzzy numbers are extended to HFNs and we show that the outcome of these operations on two HFNs are an HFN.Also we use the extension principle in HFSs for the ranking of HFNs, which may be an interesting topic.In this paper, we show that the HFNs can be ordered in a natural way. To introduce a meaningful ordering of HFNs, we use a new lattice operation on HFNs based upon extension principle and  defining the Hamming distance on them.Finally, the applications of them are explained on optimization and decision-making problems.

Testing the capability of a productive process on the basis of the flexible fuzzy quality  using Yongting's index is proposed in this paper by the Monte Carlo simulation.The theoretical approach and detailed steps of an algorithm are given to simulate the critical-value-based and also p-value-based approaches to statistical testing fuzzy quality. Also, the probability of type textit{II} error of the quality test simulated by Monte Carlo  approach. Moreover, a real-world case study is provided to show the performance of the proposed algorithm for triangular and trapezoidal fuzzy qualities.

In this paper, we propose construction methods for triangular norms (t-norms) and triangular conorms (t-conorms) on bounded lattices by using interior and closure operators, respectively. Thus, we obtain some proposed methods by Ertuu{g}rul, Karac{c}al, Mesiar cite{Ertugrul} and c{C}ayl{i} cite{Gul} as results. Also, we give some illustrative examples. Finally, we show that the introduced construction methods can not be generalized by induction to a modified ordinal sum for t-norms and t-conorms on bounded lattices. This paper has further constructed the t-norms and t-conorms on bounded lattices from a mathematical viewpoint.

We investigate the stability problems of the $n$-dimensional Cauchy-Jensen type and the n-dimensional Pexiderized Cauchy-Jensen type fuzzy number-valued functional equations in Banach spaces by using the metric defined on a fuzzy number space.Under some suitable conditions, some properties of the solutions for these equations such as existence and uniqueness are discussed. Our results can be regarded as important extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.

This paper presents an efficient and straightforward method with less computational complexities to address the linear fractional programming with fuzzy coefficients (FLFPP). To construct the approach, the concept of $alpha$-cut is used to tackle the fuzzy numbers in addition to rank them.  Accordingly, the fuzzy problem is changed into a bi-objective linear fractional programming problem (BOLFPP) by the use of interval arithmetic. Afterwards, an equivalent BOLFPP is defined in terms of the membership functions of the objectives, which is transformed into a bi-objective linear programming problem (BOLPP) applying suitable non-linear variable transformations. Max-min theory is utilized to alter the BOLPP into a linear programming problem (LPP). It is proven that the optimal solution of the LPP is an $epsilon$-optimal solution for the fuzzy problem. Four numerical examples are given to illustrate the method and comparisons are made to show the efficiency.

A new approach of fuzzy processes, the source of which are expert knowledge reflections on the states on Stationary Discrete Extremal Fuzzy Dynamic  System (SDEFDS) in extremal fuzzy time intervals, are considered.  A fuzzy-integral representation of a stationary discrete   extremal fuzzy process is given.A method and an algorithm  for identifying the transition operator of SDEFDS are developed.  The SDEFDS transition operator is restored by means of  expert knowledge reflections on the states of SDEFDS. The regularization condition for obtaining of the quasi-optimal estimator of the transition operator is represented by the theorem. The corresponding calculating algorithm is provided. The results obtained are illustrated by an example in the case of a finite set of SDEFDS states.

Uninorms and nullnorms are special 2-uninorms. In this work, we  construct 2-uninorms on bounded lattices. Let L be a bounded lattice with a nontrivial element d. Given two uninorms U_1 and U_2, defined on sublattices [0,d] and [d,1], respectively, this paper presents two methods for constructing binary operators on L which extend both U_1 and U_2.  We show that our first construction is a 2-uninorm on L if and only if U_2 is  conjunctive  and our second construction is a 2-uninorm on L if and only if U_1 is  disjunctive. Moreover, we prove that the two  2-uninorms are, respectively,  the  weakest  and the strongest 2-uninorm among all 2-uninorms, the restrictions of which on [0,d]^2 and [d,1]^2 are respectively U_1 and U_2.