Different Types Of Fuzzy Numbers Research Articles

Shadowed AHP for Multi-Criteria Supplier Selection

Numerous techniques of multi-criteria decision-making (MCDM) have been proposed in a variety of business domains. One of the well-known methods is the Analytical Hierarchical Process (AHP). Various uncertain numbers are commonly used to represent preference values in AHP problems. In the case of multi-granularity linguistic information, several methods have been proposed to address this type of AHP problem. This paper introduces a novel method to solve this problem using shadowed fuzzy numbers (SFNs). These numbers are characterized by approximating different types of fuzzy numbers and preserving their uncertainty properties. The new Shadowed AHP method is proposed to handle preference values which are represented by multi-types of uncertain numbers. The new approach converts multi-granular preference values into unified model of shadowed fuzzy numbers and utilizes their properties. A new ranking approach is introduced to order the results of aggregation preferences. The new approach is applied to solve a supplier selection problem in which multi-granular information are used. The features of the new approach are significant for decision-making applications.

Ranking fuzzy numbers based on relative positions and shape characteristics

In this paper, we propose a methodology for ranking fuzzy numbers based on both relative relationships and shape characteristics of fuzzy numbers. We introduce the so-called relative fuzzy dominant degree to capture some shape characteristics instead of using an absolute value to represent fuzzy numbers. This measure addresses the dominant level that one fuzzy number is better than others in terms of its position and shape. Then, a ranking procedure is developed to compare different types of fuzzy numbers. The proposed method combines both ranking index and relative relationships of fuzzy numbers. Numerical examples show the differences of the proposed methodology compared to the previous approaches.

A New Approach for Ranking Shadowed Fuzzy Numbers and its Application

In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.

Uključivanje nesigurnosti sa različitim vrstama fazi brojeva u "DEMATEL"

Nowadays the multi-criteria decision making is very complicated due to uncertainty, vagueness, limited sources, knowledge and time. The Decision-making Trial and Evaluation Laboratory (DEMATEL) method is a widely used multi-criteria decision-making method to analyze the structure of a complex system. It is useful in analysing the cause and effect relationships between the components of the system. Fuzzy sets can be used to include uncertainty in multi-criteria decision making. Linguistic assessments of decision makers can be translated into fuzzy numbers. In this study, fuzzy numbers, intuitionistic fuzzy numbers and neutrosophic fuzzy numbers were used for the decision makers evaluations in the DEMATEL method. The aim of this study was to evaluate how different types of fuzzy numbers affect the final results. An application of risk in construction projects was selected from the literature, where seven experts used a linguistic scale to evaluate different criteria. The results showed that there are only slight differences between the weights of the criteria with regard to the type of fuzzy numbers.

WITHDRAWN: Optimization of unconstrained multi-item (EPQ) model using fuzzy geometric programming with varying fuzzification and defuzzification methods by applying python

In this paper, an unconstrained multi item inventory model for a single vendor and single buyer with constant demand is considered. Fuzzification was done by using five different types of fuzzy numbers, which are triangular, trapezoidal, pentagonal, hexagonal, and octagonal. To minimize the total cost of the (EOQ) Economic order quantity, is solved by using geometric programming, it gives maximize the optimal Shipment order quantity and optimal Shipment and The optimal policy for the developed model is determined using the geometric programming after the defuzzification of the cost function with the Robust ranking and singed distance method. We acquainted a substitute arrangement with the manual figuring’s, supplanted by python program which lessens the endeavours and blunders. Mathematical models and an affectability investigation study are given to feature the contrasts among crisp and the fuzzy cases.

A New Approach for Ranking Shadowed Fuzzy Numbers and its Application

In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.

A Lexicographic Approach to Fuzzy Linear Assignment Problems with Different Types of Fuzzy Numbers

The fuzzy linear assignment problem (FLAP) is an extension of the classical linear assignment problem (LAP) to situations in which uncertainty in the cost coefficients is represented by fuzzy numbers. FLAP applications range from the assignment of workers to tasks to multiple-criteria decision analysis in fuzzy environments and many other engineering applications. Most FLAP formulations assume that all cost coefficients are fuzzy numbers of the same type (e.g. triangular, trapezoidal). The standard solution approach is the defuzzification of the cost coefficients, thus transforming the FLAP into a crisp LAP that can be solved by classical assignment algorithms such as the Hungarian method. It is known that defuzzification methods suffer from lack of discrimination when comparing fuzzy numbers which may lead to suboptimal assignments. The solution approach proposed in this paper is based on the theory of algebraic assignment problems and total orderings in the set of all fuzzy numbers, and it allows to solve FLAPs with different types of fuzzy numbers. More specifically, the FLAP is transformed into a lexicographic linear assignment problem (LLAP) which is solved in its place. We show, both theoretically and numerically, how this transformation overcomes the limitations present in existing approaches.

An Integrated AFS-Based SWOT Analysis Approach for Evaluation of Strategies Under MCDM Environment

This article outlines the development of a hybrid methodology aimed to help the policymakers in strategic planning. The proposed methodology integrates the axiomatic fuzzy set (AFS) theory, analytic hierarchy process (AHP), and the concept of simple additive weighting (SAW) to evaluate the strategies by strengths, weaknesses, opportunities, and threats (SWOT) analysis. The combination of AHP with SWOT yields analytically determined weights of the factors included in SWOT analysis. The SAW technique provides a flexible technique to obtain the final ranking of strategies in multi-criteria decision situations. In SAW, the strategies are described using the AFS-based AHP calculation framework for normalization and consistent ratings over the SWOT factors. The AFS theory is incorporated in the model to overcome the uncertainty and ambiguity in human decision-making processes. The proposed integrated methodology copes with the inconsistency caused by different types of fuzzy numbers and normalization methods required in solving multi-criteria decision-making (MCDM) problems. A real-world application is conducted to illustrate the utilization of the model to evaluate SWOT analysis and strategies for tourism development.

Fuzzy risk analysis in poultry farming based on a novel similarity measure of fuzzy numbers

Similarity measure between fuzzy numbers is a very important tool in fuzzy risk analysis problems where the parameters involved are linguistic terms. Such types of measure already exist in the literature. The emergence of different types of fuzzy numbers has led to generalization and modification of existing similarity measures. Moreover, newer measures are being proposed owing to the limitations and drawbacks in existing methods. It is seen that most existing similarity measures have numerous drawbacks and limitations. Namely, most of the method is based on geometric distance, hence the concept of the height is missing, which depicts a limitation on the existing method. Further, crisp-valued fuzzy numbers are not at all considered in some of the existing methods. Hence, in this paper, a new method to measure the degree of similarity between fuzzy numbers has been proposed. This measure has been discussed on generalized fuzzy numbers with different left heights and right heights. This measure is not just confined to generalized fuzzy numbers with different left heights and right heights, but also can deal with arbitrary fuzzy numbers. The proposed similarity measure is based on geometric distance, height and radius of gyration point of fuzzy number. In fact, a much generalized method has been proposed which removes all the drawbacks and limitations of the existing similarity measures. The out-performance of the proposed method is illustrated by comparing with existing methods of similarity measure. Further, the proposed method is effectively applied in risk analysis of poultry farming.

A hybrid decision support model using axiomatic fuzzy set theory in AHP and TOPSIS for multicriteria route selection

This paper presents a hybrid decision support model aimed to help the travelers in selecting best route among multiple alternatives. The proposed model consists of three parts: (i) analytic hierarchy process (AHP) to determine relative importance of route attributes, (ii) axiomatic fuzzy set (AFS) theory for description of alternative routes, and (iii) technique for order preference by similarity to ideal solution (TOPSIS)-based final selection. TOPSIS methodology is used to determine ranking order of alternative routes in multicriteria decision situations. In TOPSIS, the alternative routes are described using AFS theory to normalize the decision matrix for consistent rating of routes over attributes. The main advantage of the developed model is that it copes inconsistency caused by both, different types of fuzzy numbers and normalization methods. An illustrative example of route selection is presented to better understand the hybrid methodological process. A comparative analysis with an established multicriteria decision-making technique shows the effectiveness and validity of the hybrid model for route selection.

Analysis of an industrial system under uncertain environment by using different types of fuzzy numbers

Uncertainties play a dominant role in the performance analysis of the system. For managing it, fuzzy set theory and its corresponding triangular fuzzy numbers have been utilized by most of the researchers for quantifying the data. However, in this manuscript, this hypothesis has been calmed by defining the different types of numbers, namely gamma, normal, Cauchy and triangular for uncertainties. Based on it, behavior, performance and sensitivity analysis of the system have been investigated at different levels of confidence and the preferences as provided by the decision makers towards the data. Based on it, various expressions of the system such as failure rate, repair time, reliability, availability etc., are obtained corresponding to these different types of the numbers. From the computed results, it is concluded that these indices are reduced range of prediction as compared to the existing approaches. A numerical example has been taken for demonstrating the approach.

Transportation Problem with Triangular Mixed Intutionistic Fuzzy Numbers Solved By BCM

In this paper, we formulate a transportation problem in which sources, destinations and costs are different types of fuzzy numbers. We used real, fuzzy and intuitionistic fuzzy numbers are employed to get the optimal solution. Mixed intuitionistic fuzzy BCM is used to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The method is illustrated by a numerical examples.

Analyzing the Ranking Method for Fuzzy Numbers in Fuzzy Decision Making Based on the Magnitude Concepts

Ranking fuzzy numbers is an important component in the decision-making process with the last few decades having seen a large number of ranking methods. Ezzati et al. (Expert Syst Appl 39:690–695, 2012) proposed a revised approach for ranking symmetric fuzzy numbers based on the magnitude concepts to overcome the shortcoming of Abbasbandy and Hajjari’s method. Despite its merits, some shortcomings associated with Ezzati et al.’s approach include: (1) it cannot consistently rank the fuzzy numbers and their images; (2) it cannot effectively rank symmetric fuzzy numbers; and (3) it cannot rank non-normal fuzzy numbers. This paper thus proposes a revised method to rank generalized and/or symmetric fuzzy numbers in parametric forms that can surmount these issues. In the proposed ranking method, a novel magnitude of fuzzy numbers is proposed. To differentiate the symmetric fuzzy numbers, the proposed ranking method takes into account the decision maker’s optimistic attitude of fuzzy numbers. We employ several comparative examples and an application to demonstrate the usages and advantages of the proposed ranking method. The results conclude that the proposed ranking method effectively resolves the issues with Ezzati et al.’s ranking method. Moreover, the proposed ranking method can differentiate different types of fuzzy numbers.

Revenue Efficiency of Fuzzy Sample Decision Making Unit

Revenue efficiency measurement is one of the most important issues in data envelopment analysis (DEA). Most of the proposed models calculate the revenue efficiency of decision making units (DMUs) which play a main role in the formation of the production possibility set by implementing exact or fuzzy data. The revenue efficiency value of a sample decision making unit with exact and fuzzy data has not been investigated by these models yet. There exist different types of fuzzy numbers, however, only a special type of them has been used in revenue efficiency models with fuzzy data. The concept of vector has not been employed to calculate the measure of the revenue efficiency in any of the studies conducted thus far. However, in this article the authors propose a model for evaluating the revenue efficiency measure of a fuzzy sample DMU without the limitations of previous models with regards to the formation of the production possibility set. In the proposed model, data can be selected from different types of fuzzy numbers and there is no limitation on the type of the used fuzzy data. In addition, the current article employs the concept of vector for revenue efficiency assessment for the first time.

The optimal solution for several different degrees of feasibility for fuzzy linear and non-linear programming problems

This paper proposes a novel meta-heuristic based algorithm which provides the optimal solution for several different degrees of feasibility for fuzzy linear and nonlinear programming problems. The proposed method has got the ability for solving those problems in which all coefficients of the objective function and constraints are represented by LR fuzzy numbers with linear and/or non-linear membership function. To solve the fuzzy problems, this paper provides a new hybrid genetic algorithm accompanied by a new proposed method of simulating fuzzy coefficients which 1) eliminates the need of applying defuzzification methods and/or expected interval methods, and 2) allows dealing with different types of fuzzy numbers, properly. In order to show the performance of the proposed method, it is compared with “M. Jimenez, M. Arenas, A. Bilbao and M.V. Rodriguez, Linear programming with fuzzy parameters: An interactive method resolution, European Journal of Operational Research 177 (2007), 1599–1609.”. Computational results reveal that the proposed method is superior to the Jimenez et al. [13] method from the viewpoint of feasibility and optimality.

Fuzzy system of linear equations with crisp coefficients

This paper proposed a new and simple solution approach to solve a general fuzzy system of linear equations. In the original system, the coefficient matrix is considered as real crisp whereas the unknown variable vector and right hand side vector are considered as fuzzy. Here, three different types of fuzzy numbers viz. triangular, trapezoidal and Gaussian fuzzy numbers are taken into consideration. First the system is solved in term of fuzzy centre then this solution is used with width to get the final solution of the original system. Solution theorems are stated and proved related to the proposed method. The method is also used to solve some known example problems found in the references and the results are compared to show the efficacy of the proposed method.

A revised method for ranking fuzzy numbers using maximizing set and minimizing set

A large number of methods have been proposed for ranking fuzzy numbers in the last few decades. Nevertheless, none of these methods can always guarantee a consistent result for every situation. Some of them are even non-intuitive and not discriminating. Chen proposed a ranking method in 1985 to overcome these limitations and simplify the computational procedure based on the criteria of total utility through maximizing set and minimizing set. However, there were some shortcomings associated with Chen’s ranking method. Therefore, we propose a revised ranking method that can overcome these shortcomings. Instead of considering just a single left and a single right utility in the total utility, the proposed method considers two left and two right utilities. In addition, the proposed method also takes into account the decision maker’s optimistic attitude of fuzzy numbers. Several comparative examples and an application demonstrating the usage, advantages, and applicability of the revised ranking method are presented. It can be concluded that the revised ranking method can effectively resolve the issues with Chen’s ranking method. Moreover, the revised ranking method can be used to differentiate different types of fuzzy numbers.

A Fuzzy Pay-Off Method for Real Option Valuation

Real option analysis offers interesting insights on the value of assets and on the profitability of investments, which has made real options a growing field of academic research and practical application. Real option valuation is, however, often found to be difficult to understand and to implement due to the quite complex mathematics involved. Recent advances in modeling and analysis methods have made real option valuation easier to understand and to implement. This paper presents a new method (fuzzy pay-off method) for real option valuation using fuzzy numbers that is based on findings from earlier real option valuation methods and from fuzzy real option valuation. The method is intuitive to understand and far less complicated than any previous real option valuation model to date. The paper also presents the use of number of different types of fuzzy numbers with the method and an application of the new method in an industry setting.

An entropy based solid transportation problem for general fuzzy costs and time with fuzzy equality

In this paper, a multi-objective, solid transportation problem (STP) with imprecise unit costs and route-wise travel-times (general fuzzy numbers) is considered. The sources’ availabilities, destinations’ demands and capacities of conveyances are also represented by different types of fuzzy numbers like general, trapezoidal and triangular numbers. The transportation problem (T. P.) has been formulated with and without entropy function defined by Shannon’s measure of entropy. Multi-objective problems are formed with different criteria and reduced to single objective optimization problems using Zimmermann’s approach and possibility measure of the fuzzy equality. For the entropy based model, the multi-objective problem is converted to a single objective problem using weighted average of the objectives. Generalized Reduced Gradient (GRG) method is used to find the optimal solutions for a set of given numerical data. The models under two types of formulation are numerically solved and compared. The results of the models with and without entropy are also obtained and compared.

Multi-item fuzzy inventory model for deteriorating items with finite time-horizon and time-dependent demand

This paper develops a finite time-horizon fuzzy multi-deteriorating inventory model with/without shortage, where the demand increases linearly with time. Here, the total profit is to be maximized under the limitation on investment. In this problem, total profit, total investment cost and the time-horizon are fuzzy in nature. The impreciseness in the above objective and constraint goals have been expressed by fuzzy linear/nonlinear membership functions and vagueness in time-horizon by different types of fuzzy numbers. Results are illustrated with numerical examples.