Α-level Sets Research Articles

Enhancing group decision-making: Maximum consensus aggregation for fuzzy cross-efficiency under hesitant fuzzy linguistic information

Group decision-making (GDM) is essential as it recognizes the inherent complexity of many decision scenarios, which frequently require the collective wisdom and knowledge of multiple decision-makers (DMs) to be effectively resolved. The proposed method aims to develop fuzzy data envelopment analysis (DEA) cross-efficiency models tailored to address GDM challenges, wherein attribute values are provided by DMs using hesitant fuzzy linguistic term sets (HFLTSs). For this purpose, we initially transform HFLTSs into their corresponding fuzzy envelopes, defined as trapezoidal fuzzy numbers (TrFNs). This conversion strategy effectively minimizes the loss in assessments based on HFLTSs while retaining the inherent ambiguity of the original information. Building upon this foundation, we develop fuzzy cross-efficiency models by leveraging the α-level sets of fuzzy envelopes. These models are designed to handle fuzzy input and output variables under various α-level sets, which are capable of considering all possible attribute values for each alternative. Following this, we implement a maximum consensus model using fuzzy cross-efficiency to assign weights to DMs. These weights facilitate the aggregation of individual fuzzy cross-efficiency intervals obtained from DMs’ assessments into collective ones, which serve to rank alternatives. Finally, we showcase the effectiveness and superiority of our proposal through numerical validation and comparative analysis.

Analysis of Factors Affecting Space Teleoperation Safety Performance Based on a Hybrid Fuzzy DEMATEL Method

Teleoperation is one of the highest-risk activities in manned space activities. An appropriate set of influencing factors is essential for the safety assessment of teleoperation. The examination of the impact of a single factor or two factors on space teleoperation as done in the existing works suffers from the lack of a systematic analysis of factors. To address this gap, a hybrid α-level set incorporated in the hesitant fuzzy decision making trial and evaluation laboratory (DEMATEL) method was adopted in this study to identify and analyze teleoperation factors. The findings are as follows: (a) A system of influencing factors for teleoperation safety performance was constructed based on an extensive literature review and using the Delphi method with the aid of 30 space experts. The system covered 16 factors grouped based on individual, crew, machine, and context perspectives. (b) The α level affects the screening results of the pivotal influencing factors. For the average α level, five pivotal influencing factors, namely, team communication, basic cognitive ability, display interface information, control mode design, and task complexity, are identified from the 16 influencing factors. The present study represents a valuable effort in extracting the crucial influencing factors that affect teleoperation safety performance. It provides valuable insights from an ergonomic viewpoint and aids in determining priorities for enhancing operational safety.

New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility

The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a general aggregation function to set up the arithmetic operations of non-normal fuzzy sets. In applications, the arithmetic operations of fuzzy sets are always transferred to the arithmetic operations of their corresponding α-level sets. When the aggregation function is taken to be the minimum function, this transformation is clearly realized. Since the general aggregation function was adopted in this paper, the concept of compatibility with α-level sets is needed and is proposed, which can cover the conventional case using minimum functions as the special case.

Evaluation of Anti-Tank Guided Missiles: An integrated Fuzzy Entropy and Fuzzy CoCoSo multi criteria methodology using technical and simulation data

The Anti-Tank Guided Missiles (ATGMs) selection problem, a sub-problem of the Weapon Systems Selection Problems (WSSP), is an extremely important strategical level decision problem and is the main focus of this study. The study aims to determine a new methodology that represents accurate and reliable results from simulation model. This model decides on the anti-tank weapon system for various combat scenarios with high uncertainty. Data obtained from hundreds of simulation scenarios, with different combat conditions such as friendly and enemy forces, terrain, and environmental conditions, and technical weapon specifications is used to define solutions for the ATGM problem, such a solution proposal was not seen in any of the previous studies. For this aim, we proposed a comprehensive methodology consisting of 4 phases and 15 steps, using simulation and technical data, Fuzzy Shannon’s Entropy (F-Entropy) based on α-level sets, and Fuzzy CoCoSo with Bonferroni (F-CoCoSo’B) methods for the selection of ATGMs to be used in the close quarter combat conditions. F-Entropy method is utilized for obtaining the criteria weights and F-CoCoSo’B is used for ranking the ATGM alternatives. 3 main and 22 sub-criteria have been determined by considering not only the technical data of the weapon systems but also data obtained from JCATS simulation tool. This combination of data sources results in criteria that reflects the combat environment more precisely, when compared to previous studies where only technical data or the experience of military experts were collected. The sensitivity analysis performed to test the validity and applicability of the proposed methodology approves the stability, robustness, and practicality of the methodology. The purpose of this paper is to present the proposed methodology, findings, and insights.

CARTILAGE THICKNESS, SUBCHONDRAL BONE MINERAL DENSITY, AND AMBULATORY LOADING ARE CORRELATED IN NON-OSTEOARTHRITIC FEMORAL CONDYLES

A few studies have reported correlations between structural and functional parameters in healthy knees, and sometimes alterations of these correlations with OA. These observations suggested a new concept where knee OA could be related to a maladaptation between joint features. However, studies analyzing multiple features from different domains simultaneously are missing, which makes it difficult to fully understand the importance of relationships in maintaining joint homeostasis and their role in the development of OA. To assess the correlations among cartilage thickness (CTh), subchondral bone mineral density (sBMD), and ambulatory loading in non-OA femoral condyles. As commonly done, ambulatory loading was quantified using the peak knee adduction moment during walking (pKAM), which is a proxy for the distribution of load between medial and lateral compartments. To analyze comparable data, CTh and sBMD were expressed as medial-to-lateral (M/L) ratios. Therefore, this study tested the hypothesis that CTh M/L ratios, sBMD M/L ratios, and pKAM are positively correlated in non-OA femoral condyles. This study analyzed a subgroup of 48 young (25.7 ± 4.8 years; 22 males) asymptomatic volunteers with healthy knees enrolled in the HIPPA-compliant IRB-approved Lausanne Knee Study. This study aims at characterizing the structure and function of the knee, with a particular emphasis on the relationships among knee features The study protocol included clinical evaluation, radiographic imaging (weight-bearing Schuss and lateral radiographs of both knees), CT imaging (calibrated by a solid calcium hydroxyapatite-based bone mineral reference phantom) and 3T MRI of one randomly selected knee. 3D gait recordings were performed using a 16-camera motion capture system and force plates. CTh was obtained by manually segmenting the cartilage and bone on the DESS images, building 3D mesh models of bone and cartilage, and calculating the distance between the models. sBMD was obtained by registering the bone models from the DESS segmentation to the CT images and calculating the average CT intensity in the 3 mm of bone adjacent to cartilage. Mean CTh and sBMD were then calculated for six common regions of interest (ROIs) in the medial and lateral condyles and expressed as M/L ratios. pKAM was obtained following a standard inverse dynamics procedure applied to the marker-based motion capture and force plate recordings. Data processing was done using in-house software and algorithms previously validated. The relationship among CTh M/L ratios, sBMD M/L ratios and pKAM were assessed using Pearson correlation, with an α-level set a priori at 5%. All statistically significant correlations were positive, indicating larger CTh M/L ratios in knees with larger sBMD M/L ratios and larger pKAM, and vice versa. Specifically, CTh and sBMD M/L ratios were correlated in 5 of the 6 ROIs (0.31 ≤ r ≤ 0.53; 0.001 ≤ p ≤ 0.035), CTh M/L ratios in 4 ROIs were correlated with pKAM (0.29 ≤ r ≤ 0.33; 0.021 ≤ p ≤ 0.047), and sBMD M/L ratio in one ROI was correlated with pKAM (r = 0.31; p = 0.032). The hypothesis was supported, confirming the idea that structural and mechanical features of the knee are positively correlated, or adapted, in non-OA knees. This first study to show correlations among bone, cartilage, and mechanical loading provided insight into the functioning of the joint and underlined the potential of evaluating knee health based on relationships among features rather than on individual features in isolation. The analyses in this study will be extended to OA knees in future studies in order to determine the alterations of the relationships between knee features with OA.

The Collocation Method For Fourth Order Fuzzy Boundary Value Problems

In this paper, the collocation method is considered to solve the nonhomogeneous for fourth order fuzzy boundary value problems, In which the fuzziness appeared together in the boundary conditions and in the nonhomogeneous term of the differential equation. The method of solution depends on transforming the fuzzy problem to equivalent crisp problems using the concept of α-level sets.

On solving multi-objective fully quadratic fractional optimisation model with interactive FGP approach

PurposeThe main motive behind framing this paper is to provide a compromised solution for trapezoidal fuzzy number–multi-objective fully quadratic fractional optimisation model (TrFN-MOFQFOM) by avoiding ambiguities and confusion of decision-makers (DMs). Many researchers have used Taylor's series and parametric approach to transform fractional objective function into non-fractional ones, but Taylor's series expansion is valid only up to a neighbourhood. To avoid these extra efforts, this article suggests a methodology in which numerator of objective function is optimised under the condition of optimising denominator.Design/methodology/approachThis paper suggests an efficient procedure to search for compromised solution of MOFQFOM with fuzzy coefficients using α-level set and FGP approach. Incomplete data in model is dealt with α-level set. Then after defuzzification, non-fractional models are built from fractional model to get optimal solution of every objective. Finally, the linear weighted sum of negative deviational variables is minimised to satisfy all objective functions up to maximum possible extent.FindingsOn applying suggested approach to the example given in end, the authors arrived at compromised solution having and . The applied procedure requires less computational efforts and provides the preferred compromised solution.Originality/valueThis work has not been done previously by anyone. The idea being developed here of constructing non-fractional model by dealing numerators and denominators separately is completely new. 10; In the end, an algorithm, flowchart and numerical are also given to clarify the applicability of the suggested approach.

Evaluation of Anti-Tank Guided Missiles (Atgms): A Novel Integrated Fuzzy Shannon's Entropy (F-Entropy) Based on Α-Level Sets and Fuzzy Cocoso with Bonferroni (F-Cocoso’B) Multi Criteria Methodology Using Technical and Simulation Data

Evaluation of Anti-Tank Guided Missiles (Atgms): A Novel Integrated Fuzzy Shannon's Entropy (F-Entropy) Based on Α-Level Sets and Fuzzy Cocoso with Bonferroni (F-Cocoso’B) Multi Criteria Methodology Using Technical and Simulation Data

Generating fuzzy intervals using gradual intervals and gradual numbers

A fuzzy interval is a fuzzy set in R such that its α-level sets are bounded closed intervals in R. The purpose of this article is to generate fuzzy intervals using gradual intervals and gradual numbers by following the mathematical expression in decomposition theorem. On the other hand, the concepts of elements and subsets of fuzzy intervals are proposed on the basis of the concept of gradual numbers. Each gradual number can induce a corresponding fuzzy set in R. We also compare the fuzziness between the generated fuzzy interval and the fuzzy set generated by the corresponding gradual number.

Discussion on Optimality Conditions of Fuzzy Optimal Control Problems

In this study, a novel methodology for detecting fuzzy optimal solutions of fuzzy optimal control problems governed by fuzzy differential equations is presented. For this purpose, first the concept of optimal solution is proposed by considering a partial ordering on fuzzy numbers space which is based on a new parametric representation for α-level set of fuzzy numbers. Using this partial ordering and parametric representation, the discussed fuzzy optimal control problems are reduced to general optimal control problems in parametric form. Then by Pontryagin’s principle, a candidate for the solution of the original problem is derived. The procedure of extracting candidate solution is illustrated for two experimental problems, which improve the recent results of Refs. 10 and 17.

Generating fuzzy sets from the families of sets

The main purpose of this paper is to establish a mechanical procedure to determine the membership functions using the data collected from the economic and engineering problems. Determining the membership functions from the collected data may depend on the subjective viewpoint of decision makers. The mechanical procedure proposed in this paper can get rid of the subjective bias of decision makers. The concept of solid families is also proposed by regarding the sets in a family to be continuously varied. The desired fuzzy sets will be generated in the sense that its α-level sets will be identical to the sets of the original family. In order to achieve this purpose, any arbitrary families will be rearranged as the nested families by applying some suitable functions to the original families that are formulated from the collected data.

Choquet-Sugeno-like operator based on relation and conditional aggregation operators

We introduce a Choquet-Sugeno-like operator generalizing many operators for bounded nonnegative functions and monotone measures from the literature, e.g., the Sugeno-like operator, the Lovász and Owen measure extensions, the F-decomposition integral with respect to a partition decomposition system, and others. The new operator is based on concepts of dependence relation and conditional aggregation operators, but it does not depend on α-level sets. We also provide conditions under which the Choquet-Sugeno-like operator coincides with some Choquet-like integrals defined on finite spaces and appeared recently in the literature, e.g., the reverse Choquet integral, the d-Choquet integral, the F-based discrete Choquet-like integral, some version of the CF1F2-integral, the CC-integrals (or Choquet-like Copula-based integral) and the discrete inclusion-exclusion integral. Some basic properties of the Choquet-Sugeno-like operator are studied.

Stabilizability And Solvability Of Fuzzy Differential Equations Using Backstepping Method

Solution and stability of systems of fuzzy ordinary differential equations are of great important and may be sometimes difficult to carry out in the theory of fuzzy differential equations. Therefore, need for efficient approaches and techniques seem to be necessary for solving and stabilizing such type of problems. In this paper the backstepping method will be modified and used for analyzing systems of fuzzy ordinary differential equations. Such type of problems that might be encountered in studying the mathematical problems resulting from modeling complex physical or engineering real life phenomena’s. The topic of this paper is considered as a part of the control theory of fuzzy differential equations, in which the adaptive backstepping method based on constructing Lyapunove function in quadratic form will be combined with the so called the α-level sets in fuzzy set theory. The reliability of the proposed approach is presented by solving three examples, linear and nonlinear, in which the orginal system may be unstable.

Fuzzification of vector-valued functions

Fuzzification of real-valued functions using the extension principle is frequently used to study engineering and economic problems when fuzzy data are taken into account. In this article, we generate a fuzzy function by fuzzifying a vector-valued function using the extension principle. On the other hand, it is also well known that the membership function of each fuzzy set can be expressed in terms of its α-level sets. Roughly speaking, each fuzzy set can be decomposed into its α-level sets. Therefore, we generate another fuzzy function by fuzzifying a vector-valued function on the basis of the decomposition form. The main focus is to establish their equivalence under some suitable conditions. Without extra conditions, we have difficulty to establish their equivalence. However, their relationship can be realized on the basis of the degree of fuzzy uncertainty. In these settings, we suggest the use of the decomposition form rather than the extension principle to fuzzify vector-valued functions.

Dual extension principle and dual fuzzification using lower level sets: representation theorems

ABSTRACT The extension principle is always used to fuzzify the crisp functions to obtain the fuzzy functions in which the α-level sets of fuzzy sets are involved in the calculation. In this paper, we shall consider the concepts of upper and lower α-level sets of fuzzy sets. On the other hand, we shall establish the so-called dual extension principle using the concept of dual fuzzy set, which can also be used to fuzzify the crisp function into fuzzy functions. This kind of technique can also be called the dual fuzzification. The benefit of this paper is to provide another technique called dual fuzzification to fuzzify the crisp function into fuzzy functions. The relationship between the fuzzified function using the extension principle and the dual-fuzzified function using the dual extension principle will be established in this paper.

RP-T-Fuzzy Soft Subrings and Ideals of Soft Rings

In this paper we introduce a concept which is called RP-T-fuzzysoft subring and examine some properties of the restricted intersection, therestricted union, the ∧-intersection and the product of their families. A con-dition to make the restricted union of RP-T-fuzzy soft subrings of a soft ringto be RP-T-fuzzy soft subring of this soft ring is determined. A correlationbetween the RP-T-fuzzy soft subring of a soft ring and α-level sets of this softring is demonstrated. The RP-T-fuzzy soft subrings under some binary opera-tions are investigated. Moreover, the image and pre-image of RP-T-fuzzy softsubrings under fuzzy soft homomorphisms is examined. Finally, we present theconcept of RP-T-fuzzy soft ideal and we investigate the analogue propertiesfor them.

AN ASSESSMENT OF AWARENESS, ATTITUDE, AND PRACTICE REGARDING RELATIONSHIP BETWEEN DIABETES MELLITUS & PERIODONTITIS AMONG GENERAL DENTAL & MEDICAL PRACTITIONERS IN 3 MAJOR CITIES OF MADHYA PRADESH, INDIA- A SURVEY.

Background: Diabetes mellitus and periodontal diseases evidently have a bidirectional relationship. To apply this into daily practise, an awareness among dentists and physicians of the same is imperative. Thus, a KAP (knowledge, attitude and practice) survey was conducted among dental and medical practitioners about the association between diabetes mellitus and periodontitis. Methodology: A survey was conducted among 563 dental & medical practitioners (263 physicians and 300 dentists) in 3 cities of Madhya Pradesh. Information on knowledge about complications related to periodontal conditions, attitude regarding diabetes and periodontal disease and the relation between the two was collected. Association between categorical was computed using Chi square test, with the α-level set at 0.05. Results: Dentists were more aware about impact of diabetes complications on periodontal health as compared to physicians (p<0.05). Majority of dentists referred their patients to physicians for medical evaluation and were more interested in learning about bidirectional relationship between diabetes mellitus & periodontal health, while fewer physicians referred their patients to dentists. The preferred mode of learning for dental practitioners was through conferences or CDEs, while medical practitioners preferred professional journals. Conclusion: The knowledge, awareness and practice regarding bidirectional association between diabetes and periodontitis of dentists fared better than the physicians.

Arithmetics of Vectors of Fuzzy Sets

The arithmetic operations of fuzzy sets are completely different from the arithmetic operations of vectors of fuzzy sets. In this paper, the arithmetic operations of vectors of fuzzy intervals are studied by using the extension principle and a form of decomposition theorem. These two different methodologies lead to the different types of membership functions. We establish their equivalences under some mild conditions. On the other hand, the α-level sets of addition, difference and scalar products of vectors of fuzzy intervals are also studied, which will be useful for the different usage in applications.

α-level Fuzzy Soft Sets

In this paper, based on soft lattices, with the help of fuzzy level (cut) set, -level fuzzy soft sets and -level fuzzy soft lattices are defined, and the structure and characteristics of our definitions are explained with examples, at the same time, their di erences and relations are compared with classic soft set.

On a Fuzzy Problem with Variable Coefficient by Fuzzy Laplace Transform

This study is on solutions of a fuzzy problem with variable coefficient. Solutions are found by fuzzy Laplace transform. Generalized differentiability is used. It is searched whether the solutions are valid 𝛼-level sets or not. Examples are solved on studied problem. Conclusions are given.