Operations Of Fuzzy Numbers Research Articles

Rankings and operations for interval type-2 fuzzy numbers: a review and some new methods

Interval type-2 fuzzy numbers (IT2FNs) are a particular kind of type-2 fuzzy numbers (T2FNs). In most scientific works, since arithmetic operations required IT2FNs are simpler than those of T2FNs, mathematical calculations on IT2FNs are used more frequently rather than the T2FNs. Hence, in recent decades, the study on IT2FNs have been intensified significantly. These numbers can be explained by trapezoidal and triangular forms. In this article, first, the concept of general interval type-2 trapezoidal fuzzy numbers (GIT2TrFNs) and then arithmetic operations among them are introduced. Next, three new ranking methods are suggested for GIT2TrFN. Finally, several examples are used to illustrate and compare new ranking methods with others.

Non-linear interval-valued fuzzy numbers and their application in difference equations

This paper represents the various representation of non-linear interval-valued fuzzy number. The arithmetic operations of interval-valued fuzzy numbers are found by transmission of average method. Demonstration of non-linear interval-valued fuzzy solutions of interval-valued fuzzy difference equation is carried out. An illustrative example is also undertaken with the useful graph and table for showing usefulness for attaining the proposed concept. Finally we apply this whole concept in interval-valued fuzzy arms race model and find its solution.

A Group Decision Making Approach Based on Newly Defined Additively Consistent Interval-Valued Intuitionistic Preference Relations

Because of the complexity of interval-valued intuitionistic fuzzy set, there are few literatures investigating the additive consistency of the interval-valued intuitionistic preference relation. This paper proposes a group decision making approach with interval-valued intuitionistic preference relations. Firstly, two additive reciprocal fuzzy preference relations and two fuzzy non-preferred relations are obtained by dividing an interval-valued intuitionistic preference relation in order to avoid the operations of interval-valued intuitionistic fuzzy numbers. Here, the fuzzy non-preferred relation is interpreted as the non-preferred intensity of one alternative over another one. Moreover, the rational concept of additive consistency of interval-valued intuitionistic preference relation is defined. Secondly, optimizing models to derive priority weight vector from interval-valued intuitionistic preference relation are constructed. In particular, the priority weight vector is in terms of interval-valued intuitionistic fuzzy numbers, which is rational under the interval-valued intuitionistic fuzzy environment. Thirdly, an extended approach to address group decision making (GDM) problems is proposed. Each individual expert’s consistency index is applied to measure his/her degree of importance. Finally, an illustrative example of supplier selection is provided to illustrate the proposed approach addressing GDM problems. The feasibility to derive priority weight vector in terms of interval-valued intuitionistic fuzzy numbers using the developed approach is verified. And the proposed approach is compared with the other conventional existing ones to show its advantages.

Solving Fully Fuzzy Multi-objective Linear Programming Problem Using Nearest Interval Approximation of Fuzzy Number and Interval Programming

This paper focuses on Fully Fuzzy Multi-Objective Linear Programming (FFMOLP) problem in which all the coefficients and decision variables are LR flat fuzzy numbers, the more generalized version of fuzzy numbers and all the constraints are fuzzy inequalities. A new algorithm is proposed for solving FFMOLP problem which first converts it into the Multi-Objective Interval Linear Programming (MOILP) problem. Further, taking the help of fuzzy slack variable, fuzzy surplus variables, nearest interval approximation of fuzzy numbers and scalarization technique, MOILP is then converted into the Crisp Linear Programming (CLP) problem. It is shown that the optimal solution of CLP problem is the fuzzy Pareto optimal solution of FFMOLP problem. The main advantage of the proposed algorithm is that it transforms FFMOLP problem into Crisp Linear Programming problem. Moreover, to apply algorithm, only the knowledge of arithmetic operations of LR flat fuzzy numbers, centre and width of the closed intervals are required. At the end, to illustrate the proposed method and its effectiveness over the existing method, numerical examples are solved and compared.

Minimizing energy consumption and tardiness penalty for fuzzy flow shop scheduling with state-dependent setup time

This paper addresses a flow shop scheduling problem in a production system where the machine setup times depend on their prior states. State-dependent setup times exist widely in thermal facilities such as boilers and furnaces. The fuzzy set theory is introduced to describe the uncertainty of processing times and due dates in this study. The goal of the proposed fuzzy flow shop scheduling problem is to dispatch jobs to the machines and to determine the job sequence and state transition of each machine to minimize energy consumption and tardiness. To most efficiently determine the impact of uncertainty, the problem is formulated based on accurate operations of fuzzy numbers, which differ from approximate calculations in the existing literature on scheduling. To solve the problem, two common pattern matching schemes and heuristics are proposed to be combined with the classical genetic algorithm. Computational experiments show that the proposed GA performs better than the random key GA method, especially for large problems. The numerical results also provide practical implications for the proposed problem. The state-dependent setup time constraint significantly influences the scheduling results. In addition, the objective can be improved by reducing the uncertainty of processing times and due dates.

Robust and Minimum Spanning Tree in Fuzzy Environment

This paper proposes an algorithm to find the fuzzy minimum spanning tree (FMST) of an undirected weighted fuzzy graph, in which mixed fuzzy numbers, either triangular or trapezoidal, are used to represent the lengths/costs of the arcs. In the proposed algorithm, we incorporate the uncertainty in Kruskal's algorithm for MST using fuzzy number as arc length. The concept of possibility programming is used to compare between the fuzzy number (i.e., costs of arcs) and addition operation of fuzzy numbers is used to find the cost of the spanning tree. We also investigate the robust version of the FMST problem. We define two measures for robustness of an FMST: absolute robustness and relative robustness. We characterize the fuzzy worst case scenarios for a given fuzzy spanning tree for both the measures. The corresponding fuzzy robust spanning trees are respectively defined as absolute robust fuzzy spanning tree (ARFST) and relative robust fuzzy spanning tree (RRFST). We extend Kruskal's algorithm to compute the ARFST and RRFST in fuzzy environment. An example of fuzzy graph is used to illustrate the effectiveness of the proposed methods.

On fuzzy ellipsoid numbers and membership functions

In this paper, the operations of fuzzy ellipsoid numbers and the direct relationship between the joint membership function and the edge membership functions are investigated. First, we prove that the general scalar multiplication (defined by Zadeh's expansion) of fuzzy ellipsoid numbers preserves the closeness of the operation, but the general addition (defined by Zadeh's expansion) does not preserve the closeness of the operation. Then, a new addition ⊕ of fuzzy ellipsoid numbers is defined, and it is shown that the new addition ⊕ preserves the closeness of the operation, and is the best approximation of the general addition operation in all addition operations which preserves the closeness of the operation for fuzzy ellipsoid numbers. Next, a bijection is established between fuzzy n−ellipsoid number space and n dimension fuzzy vector space. Furthermore a formula is obtained to express the edge membership functions with the joint membership function, and an iterative algorithm is proposed to calculate the joint membership function value as the edge membership functions are known, and the convergence of the iterative algorithm is demonstrated.

Exponential operations for intuitionistic fuzzy numbers and interval numbers in multi-attribute decision making

Intuitionistic fuzzy numbers (IFNs) have already been applied to many fields, especially in multi-attribute decision making (MADM). Based on the basic operational laws and information aggregation methods of IFNs, MADM with intuitionistic fuzzy information has become more and more popular. In this paper, we investigate the MADM problems where the attribute values take the form of interval numbers and the weight information on the attributes are expressed as IFNs. We first propose a novel exponential operational law based on IFNs and interval numbers, and then study some of its desirable properties. Based on the exponential operational law, we put forward an intuitionistic fuzzy weighted exponential aggregation operator, and utilize it to develop a MADM method. Finally, we apply our method to solve the decision making problem under uncertainty.

Differential equation with interval valued fuzzy number and its applications

The paper presents an adaptation of solution of first order differential equation with initial value as interval valued triangular fuzzy number. The arithmetic operation of interval-valued triangular fuzzy number is re-established and studied with the help of fuzzy extension principle method. Demonstration of fuzzy solutions of the governing differential equation is carried out using the approaches namely, generalized Hukuhara derivative. Additionally, different illustratively examples and applications are also undertaken with the useful table and graph for usefulness for attained to the proposed approaches.

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal.

A FUZZY ALGORITHM TO EVALUATE COMPETITIVE LOCATIONS FOR INTERNATIONAL TRANSPORT LOGISTICS SYSTEM

This paper evaluates competition location decisions in a fuzzy environment and assesses international transport logistics service systems using a fuzzy model. First, the study introduces the concepts of fuzzy set theory. Second, a stepby- step fuzzy algorithm, including ten systematic procedures, is effectively represented and processed to assure convincing and effective decision making. Third, three major competitive locations in the Pacific Asia region are presented as examples to analyze the rank order and competitive scenario of international transport logistics service systems. Finally, the operations of fuzzy numbers and α-cut can be adopted as a practical tool for empirical applications in future studies.

Comparative Study of Arithmetic Nature of Heptagonal Fuzzy Numbers

The uncertainty existing in the available information of many real problems is represented by fuzzy numbers. One type of fuzzy number is Heptagonal fuzzy number and its arithmetic operations can be used to solve many decision making problems involving uncertainty. The existing arithmetic operations do not incorporate all the properties required for the implementation of Heptagonal fuzzy numbers in practical problems. The arithmetic operations of Heptagonal fuzzy numbers based on the extension principle is defined and the arithmetic behavior of Heptagonal fuzzy numbers is observed. In order to improve the efficiency of the arithmetic behavior, a modification in the arithmetic operations is done and a comparison is made with the operations based on the function principle and the extension principle. It is shown through examples, that the proposed modified arithmetic operations are more reasonable, flexible and computationally efficient.

Exponential operations of interval-valued intuitionistic fuzzy numbers

The operational laws of intuitionistic fuzzy numbers are useful in dealing with the problems under intuitionistic fuzzy circumstances. As a supplementation, the exponential operational law of intuitionistic fuzzy numbers has been defined. Consider that we may utilize interval-valued intuitionistic fuzzy numbers (IVIFNs) to express uncertain weight information comprehensively in the complex multi-attribute decision making problems, in this paper, we discuss two exponential operational laws of IVIFNs, namely, the exponential operational law of IVIFNs with crisp parameter and the exponential operational law of IVIFNs with interval-valued parameter. Several properties and aggregation methods based on the exponential operational laws are discussed. Furthermore, we propose a new method to compare the novel dual interval-valued intuitionistic fuzzy numbers. Additionally, we give two information aggregation methods over these two exponential operational laws for dealing with some special problems in multi-attribute decision making, and finally, we apply the methods to a practical problem involving the choice of the optimal powered roof support for coal extraction with a high recovery rate.

Transform Group of Monotonic Functions with the Same Monotonicity on [ $$-$$ - 1, 1] and Operations of Fuzzy Numbers

Operations of fuzzy numbers are the main content of the fuzzy mathematical analysis. This paper defines the transformation of monotonic bounded functions with same monotonicity on the symmetric interval [\(-\)1, 1], and the four fundamental operations of fuzzy numbers based on the fuzzy structured element. It not only make operations of fuzzy numbers easier, but also start a new way for studying on the theory and application of fuzzy analysis.

The structure and realization of a polygonal fuzzy neural network

As a generalization of ladder fuzzy numbers, the polygonal fuzzy numbers can be approximately represented by a group of ordered real numbers, it not only keeps the closeness of arithmetic operations, and also inherits some excellent properties of ladder fuzzy numbers. To modify defects of the definition of the original polygonal fuzzy numbers, the method of an equidistant subdivision is firstly introduced, and the specific algorithm of general fuzzy numbers into the polygonal fuzzy numbers by two examples is explained in detail. Secondly, using the extension operations of polygonal fuzzy numbers, a polygonal fuzzy neural network is constructed by mathematical methods, and the coefficients transform formulas of the new network are obtained. In addition, it is also proved that the new network still possesses approximation with respect to a continuous polygonal fuzzy valued function. Finally, the approximation algorithm and realization process of the polygonal fuzzy neural networks to a two-polygonal fuzzy valued function are given by means of an example.

Derivative and Differential Operations of Intuitionistic Fuzzy Numbers

The innovative idea of Atanassov's intuitionistic fuzzy sets (A-IFSs) is to get a more comprehensive and detailed description of the ambiguity and uncertainty by introducing a membership function and a nonmembership function. Each element in an A-IFS is expressed by an ordered pair, which is called an intuitionistic fuzzy number (IFN). In this paper, we first describe the change values of IFNs when considering them as variables and classify these change values based on the basic operations for IFNs. Second, we depict the convergences of sequences of IFNs by the subtraction and division operations. Moreover, we develop some intuitionistic fuzzy functions (IFFs) and study in detail their continuities, derivatives, and differentials.

Modified new operations for triangular intuitionistic fuzzy numbers

Intuitionistic fuzzy sets (IFS) are a generalization of the concept of fuzzy set. In standard intuitionistic fuzzy arithmetic operations, we have some grievances in subtraction and division operations. In this paper, modified new operations for subtraction and division on triangular intuitionistic fuzzy numbers (TIFNS) are defined. Finally an illustrative example for solving Intuitionistic fuzzy multi-objective linear programming problem (IFMOLPP) using these modified operators is provided.

A SYSTEMATIC APPROACH FOR SOLVING MIXED INTUITIONISTIC FUZZY TRANSPORTATION PROBLEMS

In this paper, we formulate a transportation problem in which supplies, demands and costs are different types of real, fuzzy and intuitionistic fuzzy numbers. The new multiplication operations of triangular intuitionistic fuzzy numbers are employed to get the optimal solution. Mixed Intuitionistic fuzzy zero point method is proposed to find the optimal solution in terms of triangular intuitionistic fuzzy numbers. The proposed method is easy to understand and to apply for finding the intuitionistic fuzzy optimal solution in a mixed intuitionistic fuzzy transportation problem. The method is illustrated by a numerical example. AMS Subject Classification: 03E72, 03F55, 90B06

Parameters optimization of polygonal fuzzy neural networks based on GA-BP hybrid algorithm

Based on the extensive operations of polygonal fuzzy numbers, a GA-BP hybrid algorithm for polygonal fuzzy neural network is designed. Firstly, an optimal solution is obtained by the global searching ability of GA algorithm for the untrained polygonal fuzzy neural network. Secondly, some parameters for connection weights and threshold values are appropriately optimized by using an improved BP algorithm. Finally, through a simulation example, we demonstrate that the GA-BP hybrid algorithm based on the polygonal fuzzy neural network can not only avoid the initial values’ dependence and local convergence of the original BP algorithm, but also overcome a blindness problem of the traditional GA algorithm.

Fuzzy risk analysis based on the novel fuzzy ranking with new arithmetic operations of linguistic fuzzy numbers

The project of managing risk analysis problems plays an important task in present world. The arithmetic operations of some fuzzy risk analysis methods do not satisfy some properties of linguistic fuzzy numbers and the comparison of the trapezoidal fuzzy numbers does not coincide with the perception of the individual. In this paper, we present a new ranking on generalized fuzzy numbers and new arithmetic operations such as fuzzy addition, fuzzy multiplication, and fuzzy division on linguistic fuzzy numbers. Also, we present an efficient approach for fuzzy risk analysis based on new arithmetic operators of the linguistic trapezoidal fuzzy numbers and new fuzzy ranking.