Degree Formula Research Articles

Priority degrees for hesitant fuzzy sets: Application to multiple attribute decision making

In this paper, some drawbacks to existing ordering relations for hesitant fuzzy sets are examined using examples. To overcome these flaws, a priority degree formula for comparing two hesitant fuzzy sets is presented and the desirable priority degree properties studied. Then, based on an introduced priority degree formula, a new hesitant fuzzy multiple attribute decision making methodology is proposed. Finally, a numerical example together with a comparison analysis is given to illustrate the effectiveness and feasibility of the new approach to decision making applications.

Decoupling relationship between industrial growth and carbon emissions in three northeastern provinces of China : Analysis based on stability and deviation

The low-carbon industrial processes of all provinces were different in China, and the decoupling quality between industrial economic development and carbon emissions also existed differences in different provinces. In order to develop more efficient carbon reduction policy, this paper used decoupling model, decoupling value stability degree, decoupling quality degree and decoupling quality deviation degree formula to analyze horizontally and vertically the decoupling quality of five industries in three northeastern provinces of China, including primary industry; industry; construction; transportation, storage and postal industry; wholesale, retail, accommodation and catering industry. The paper concludes: firstly, the decoupling stabilities between five industries development and carbon emissions are different in three northeastern provinces of China. Liaoning and Jilin province is relatively stable compared with Heilongjiang province. Secondly, the decoupling quality among the five industries development and carbon emissions are different in three northeastern provinces of China. Jilin and Liaoning province has better decoupling quality than Heilongjiang province. Thirdly, the low carbonization process has gap in three northeastern provinces of China. It is better in low carbonization process of Jilin province and Liaoning province, whereas there is relatively big gap for low carbonization process among the different places in Liaoning province. Finally, this dissertation puts forward countermeasures and suggestions to provide references for the development of low-carbon industry in Jilin province.

Degree of Global Competitiveness: A Case on Jute, Kenaf & Allied Fibres Industry

Global competition is the services or products provided by competing companies that serve international customers. Competition on a local scale where people are competing against one another for one common goal - the best in their core competency. Global competition has allowed companies to buy and sell their services internationally, which opens the door to increased profits and flattens the playing field in business. Degree of competitiveness helps to examine the present status and future prospects of an industry. This paper is an attempt to measure the extent of global competitiveness in world Jute, Kenaf and Allied Fibres (JAF) industry. Roy (2006) (1-√∑ 𝑀𝑖 2 ) the formula for degree of competition is applied where Mi is the market share of each individual nation in global JAF production, i ranging from 1 to k. For each year we can calculate one such measure, generating there by a time series data. Based on time series analysis one can indicate the past trend and future direction. The position of India has also been indicted along with top ten nations.

Comparisons of probabilistic linguistic term sets for multi-criteria decision making

The theory of probabilistic linguistic term sets (PLTSs) is very useful in dealing with the multi-criteria decision making (MCDM) problems in which there is hesitancy in providing linguistic assessments; and PLTSs allow experts to express their preferences on one linguistic term over another. The existing approaches associated with PLTSs are limited or highly complex in real applications. Hence, the main purpose of this paper is to establish more appropriate comparison method and develop a more efficient way to handle with MCDM problems. We first put forward a diagram method to analyze the structures of PLTSs and develop the visualized way for readers to comprehend. Then a possibility degree formula is given for ranking PLTSs. Based on the new comparison method and the theory of the fuzzy preference relation, an efficient decision-making framework is proposed to solve real-life problems under linguistic environment. Conventional TOPSIS methods combined with PLTSs are also included for comparison. All results demonstrate the practicality of the new framework. Finally, we also seek out relationship between PLTSs and hesitant fuzzy linguistic term sets (HFLTSs), and compare the new formula with the similar approaches to HFLTSs’ rating.

A variant of the method of quadratures for solving integral equations with fractional integral of Weyl in the main part

In this paper the method of mechanical quadrature solutions fractional integral equation. Computational scheme quadrature method is based on the quadrature formula of rectangles with equidistant nodes, which is the formula of the highest trigonometric degree of accuracy, using a regularizing parameter. This decision is taken for the approximate trigonometric interpolation polynomial constructed from the values that make up the solution of the quadrature method. The substantiation of the method in Holder spaces.

A formula for the algebraic degree in semidefinite programming

In this paper we use the Bott residue formula in equivariant cohomology to show a formula for the algebraic degree in semidefinite programming.

Predictions on dynamic evolution of compositional mixing degree in two-component aggregation

The compositional distribution in two-component aggregative mixing of initially bidisperse particle populations can be described by a Guassian-type function, which is determined by the mixing degree χ (assessed quantitatively by the mass-normalized power density of excess component A), and the overall mass fraction ϕ (a known value from the initial feeding condition) of component A. It is known that χ will reach a steady-state value χ∞ over time (factually, after attaining the self-preserving size distribution), and χ∞ is only relevant to ϕ, namely the feeding condition. However, the dynamic evolution of χ before the attainment of a steady-state value is not exactly known. In this paper, the fast differentially-weighted Monte Carlo method for population balance modeling was used to predict the dependence of time-varied χ on initial feeding conditions through hundreds of systematically varied simulations. It is found that χ is subject to an exponential decay, largely depending on the ratio of steady-state mixing degree and its initial value (χ∞/χ0). With the explored exponential formulas for the dynamic mixing degree, it is possible to attain an optimum control on the compositional distributions during two-component aggregation processes through selecting the initial feeding parameters, and the time needed for reaching a steady-state is investigated.

Connecting the linguistic hierarchy and the numerical scale for the 2-tuple linguistic model and its use to deal with hesitant unbalanced linguistic information

The 2-tuple linguistic representation model is widely used as a basis for computing with words (CW) in linguistic decision making problems. Two different models based on linguistic 2-tuples (i.e., the model of the use of a linguistic hierarchy and the numerical scale model) have been developed to address term sets that are not uniformly and symmetrically distributed, i.e., unbalanced linguistic term sets (ULTSs). In this study, we provide a connection between these two different models and prove the equivalence of the linguistic computational models to handle ULTSs. Further, we propose a novel CW methodology where the hesitant fuzzy linguistic term sets (HFLTSs) can be constructed based on ULTSs using a numerical scale. In the proposed CW methodology, we present several novel possibility degree formulas for comparing HFLTSs, and define novel operators based on the mixed 0–1 linear programming model to aggregate the hesitant unbalanced linguistic information.

Maximum likelihood degree of Fermat hypersurfaces via Euler characteristics

Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the maximum likelihood degree of Fermat hypersurfaces. We give a formula of the maximum likelihood degree in terms of the constants β μ , ν \beta _{\mu , \nu } , which is defined to be the number of complex solutions to the system of equations z 1 ν = z 2 ν = ⋯ = z μ ν = 1 z_1^\nu =z_2^\nu =\cdots =z_\mu ^\nu =1 and z 1 + ⋯ + z μ + 1 = 0 z_1+\cdots +z_\mu +1=0 .

On degrees and birationality of the maps $$X_0(N)\rightarrow \mathbb P^2$$ X 0 ( N ) → P 2 constructed via modular forms

In this paper we prove a formula which relates the degree of a curve which is the image of a mapping \(z\longmapsto (f(z): g(z): h(z))\) constructed out of three linearly independent modular forms of the same integral or half-integral weight into \(\mathbb P^2\) and the degree of that map. Based on the formula, we present a test for birationality of the map. As an example, we compute the formula for the total degree i.e., the degree considered as a polynomial of two (independent) variables of the classical modular polynomial (or the degree of the canonical model of \(X_0(N)\)). We give an interesting example to the test for birationality which leads us to make a question on existence of specific explicit and simple model of \(X_0(N)\). We prove our claim when \(N=p\) is a prime.

Involutions of varieties and Rost’s degree formula

Abstract To an algebraic variety equipped with an involution, we associate a cycle class in the modulo two Chow group of its fixed locus. This association is functorial with respect to proper morphisms having a degree and preserving the involutions. Specialising to the exchange involution of the square of a complete variety, we obtain Rost’s degree formula in arbitrary characteristic (this formula was proved by Rost and Merkurjev in characteristic not two).

Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making

The theory of hesitant fuzzy linguistic term sets (HFLTSs) is a powerful technique used to describe hesitant situations, which are typically assessed by experts using several possible linguistic values or rich expressions instead of a single term. The union of HFLTSs with respect to each expert, that is, an extended HFLTS (EHFLTS), further facilitates the elicitation of linguistic assessments for addressing group decision-making problems because EHFLTSs can deal with generalized (either consecutive or non-consecutive) linguistic terms. In this study, we propose proportional HFLTSs (PHFLTSs), which include the proportional information of each generalized linguistic term. The mathematical form for a PHFLTS is consistent with that for a linguistic distribution assessment. However, the underlying meanings of the proportions associated with generalized linguistic terms are different. PHFLTSs can be viewed as a special method for performing linguistic distribution assessments. PHFLTSs are recognized as a useful extension of HFLTSs and a possibility distribution for HFLTSs under different assumptions. We define the basic operations with closed properties among PHFLTSs on the basis of t-norms and t-conorms. We then propose a probability theory-based outranking method for PHFLTSs by providing possibility degree formulas. We also study two fundamental aggregation operators for PHFLTSs, namely, the proportional hesitant fuzzy linguistic weighted averaging operator and the proportional hesitant fuzzy linguistic ordered weighted averaging operator. Several important properties of these aggregation operators are investigated. Finally, we use the proposed multiple criteria group decision-making model in practical applications.

Error Analysis Methods for Group Decision Making Based on Hesitant Fuzzy Preference Relation

Hesitant fuzzy preference relation (HFPR) is an effective way to depict the decision makers’ preferences over the objects (alternatives or attributes) in the process of group decision making. Each component of the HFPR is characterized by several possible values and can express the decision makers’ hesitant information comprehensively. To make a decision with the HFPR, it is very necessary to find a proper technique for deriving the priority weights from the HFPR. In this paper, we use the error analysis as a tool to develop several straightforward methods for the priorities of the HFPR. We first define the expected value and the average value of each hesitant fuzzy element in the HFPR. Then based on the error analysis, we come up with the interval midpoint method, the average value method, and the difference method to derive the priority weights from the HFPR. After that, we discuss the relations among these methods, and utilize them and the possibility degree formula to develop an approach to decision making with the HFPR. Finally, we demonstrate the effectiveness and practicality of our approach through a case study concerning the investment problem in liquor enterprise.

Minimal cubature formulas of degree 3 for a torus in ℝ3

We construct minimal cubature formulas of degree 3 for a torus in R3. The cases of a degenerate torus with radius r = 1 and a torus with arbitrary radius r > 1 are considered separately.

An Integrated Diagnostic Framework to Manage Organization Sustainable Growth: An Empirical Case

This research aims to develop a quantitative diagnostic framework by combining the Weisbord six-box model with the growth management model to focus on an organization’s internally driven sustainable management system. The research adopted an instrument developed by Preziosi and an extended Weisbord six-box model. The research employed a survey to collect 180 samples in a Chinese petrol company and applied the comparative method: (a) the average score method; and (b) the entropy method to confirm the growth level of the company. The survey also attempted to identify corresponding top growth influence factors using the obstacle degree formula. The results showed that the integrated diagnostic framework worked well to diagnose a regional but large Chinese petroleum company. In other words, the research successfully quantified the growth position and top influence factors and helped put forward specific suggestions to drive the organization of sustainable development. The method confirmed this organization during the fourth phase of five phases. In addition, top influence factors hindering the internal growth were (a) the lack of task engagement with energy and time; (b) the lack of personal work units; and (c) a poor division of labor for reaching sustainable growth rates. The research provides a generic theoretical framework support to incorporate growth management models into an organizational diagnosis to obtain sustainable growth. It further highlights and practices guidelines in examining actual growth management levels in companies and discusses top influence factors to design efficient management systems to pursue organizational growth in a multitude of industrial contexts.

An approach to group decision making with hesitant information and its application in credit risk evaluation of enterprises

Hesitant multiplicative preference relation (HMPR) contains much more comprehensive information than the traditional multiplicative preference relations. The HMPR is a useful tool to help the decision makers express their preferences in group decision making under uncertainty. The key of group decision making with the HMPR is to derive the priority weights from the HMPR. Thus, an efficient and practical priority method should be put forward so as to ensure the reasonability of the final decision result. In order to do that, in this paper, we first introduce the expected value and the geometric average value of hesitant multiplicative element (HME) which is the component of the HMPR. Then from different perspectives, we utilize the error-analysis technique to put forward three novel methods for the priorities of the HMPR, i.e., the expectation value method, the geometric average value method, and the multiplicative deviation method. We also investigate the relationships among these methods, and develop an approach to group decision making with the HMPR by using the methods and the possibility degree formula. Finally, by constructing the indicator system for credit risk evaluation of supply chain enterprises, we make a detailed case study concerning Lu-Zhou-Lao-Jiao (the well-known liquor enterprise in China) to demonstrate our approach.

A New Definition of Intuitionistic Fuzzy Similarity Degree

As far as the problem of intuitionistic fuzzy cluster analysis is concerned, this paper proposes a new formula of similarity degree with attribute weight of each index. We conduct a fuzzy cluster analysis based on the new intuitionistic fuzzy similarity matrix, which is constructed via this new weighted similarity degree method and can be transformed into a fuzzy similarity matrix. Moreover, an example is given to demonstrate the feasibility and validity of this method.

An approach to multiplicative linguistic group decision making based on possibility degrees

AbstractIn order to compare two uncertain multiplicative linguistic variables, a possibility degree formula has been developed and its properties are also studied. Then a possibility degree matrix (also considered as a reciprocal preference relation) is constructed to compare a collection of uncertain multiplicative linguistic variables. Two models are further provided to derive the priority vector from the possibility degree matrix based on the additive consistency and multiplicative consistency. Especially, if the parameters are assigned specific values, then the models reduce to the existing ones. A group decision making method has been developed to deal with the situations where the preferences on alternatives are expressed by uncertain multiplicative linguistic variables. In this method, the possibility degree matrix is first constructed, from which the priority of alternatives is obtained using the developed models. At the end, an example is given to illustrate the proposed method.

16,051 formulas for Ottaviani's invariant of cubic threefolds

We provide explicit combinatorial formulas for Ottaviani's degree 15 invariant which detects cubics in 5 variables that are sums of 7 cubes. Our approach is based on the chromatic properties of certain graphs and relies on computer searches and calculations.

The maximum likelihood degree of Fermat hypersurfaces

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.