Fuzzy Multi-objective Linear Programming Problem using DM's Perspective

In this paper, a two-stage method has been proposed for solving Fuzzy Multi-objective Linear Programming Problem (FMOLPP) with Interval Type-2 Triangular Fuzzy Numbers (IT2TFNs) as its coefficients. In the first stage of problem solving, the imprecise nature of the problem has been handled. All technological coefficients given by IT2TFNs are first converted to a closed interval and then the objectives are made crisp by reducing a closed interval into a crisp number and constraints are made crisp by using the concept of acceptability index. The amount by which a specific constraint can be relaxed is decided by the decision maker and thus the problem reduces to a crisp multi-objective linear programming problem (MOLPP). In the second stage of problem solving, the multi-objective nature of the problem is handled by using fuzzy mathematical programming approach. In order to explain the methodology, two numerical examples of the proposed methodology in Production planning and Diet planning problems have also been worked out in this paper.

Degree estimates of one class cut-offs for improper integrals

The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

Some strong convergence theorems for eigenvalues of general sample covariance matrices

The aim of this paper is to investigate the spectral properties of sample covariance matrices under a more general population. We consider a class of matrices of the form [Formula: see text], where [Formula: see text] is a [Formula: see text] nonrandom matrix and [Formula: see text] is an [Formula: see text] matrix consisting of i.i.d standard complex entries. [Formula: see text] as [Formula: see text] while [Formula: see text] can be arbitrary but no smaller than [Formula: see text]. We first prove that under some mild assumptions, with probability 1, for all large [Formula: see text], there will be no eigenvalues in any closed interval contained in an open interval which is outside the supports of the limiting distributions for all sufficiently large [Formula: see text]. Then we get the strong convergence result for the extreme eigenvalues as an extension of Bai-Yin law.

Encoding a Categorical Independent Variable for Input to TerrSet’s Multi-Layer Perceptron

The profession debates how to encode a categorical variable for input to machine learning algorithms, such as neural networks. A conventional approach is to convert a categorical variable into a collection of binary variables, which causes a burdensome number of correlated variables. TerrSet’s Land Change Modeler proposes encoding a categorical variable onto the continuous closed interval from 0 to 1 based on each category’s Population Evidence Likelihood (PEL) for input to the Multi-Layer Perceptron, which is a type of neural network. We designed examples to test the wisdom of these encodings. The results show that encoding a categorical variable based on each category’s Sample Empirical Probability (SEP) produces results similar to binary encoding and superior to PEL encoding. The Multi-Layer Perceptron’s sigmoidal smoothing function can cause PEL encoding to produce nonsensical results, while SEP encoding produces straightforward results. We reveal the encoding methods by illustrating how a dependent variable gains across an independent variable that has four categories. The results show that PEL can differ substantially from SEP in ways that have important implications for practical extrapolations. If users must encode a categorical variable for input to a neural network, then we recommend SEP encoding, because SEP efficiently produces outputs that make sense.

On Symmetric Riemann-Derivatives

The basic properties like monotoni city, Darboux property, mean value property of symmetric Riemann-derivatives of order n of a real valued function f at a point x of its domain (a closed interval) is studied. In some cases, function is considered to be continuous or semi-continuous.

On Symmetric Riemann-Derivatives

The basic properties like monotoni city, Darboux property, mean value property of symmetric Riemann-derivatives of order n of a real valued function f at a point x of its domain (a closed interval) is studied. In some cases, function is considered to be continuous or semi-continuous.

One-dimensional dynamical systems

The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.

Multiple criteria decision making based on Hamy mean operators under the environment of spherical fuzzy sets

Aggregation operators are widely applied to accumulate the vague and uncertain information in these days. Hamy mean (HM) operators play a vital role to accumulate the information. HM operators give us a more general and stretchy approach to develop the connections between the arguments. Spherical fuzzy sets (SpFSs), the further extension of picture fuzzy sets (PcFSs) that handle the data in which square sum of membership degree (MD), non-membership degree (NMD) and neutral degree (ND) always lie between closed interval [0, 1]. In the present article, we modify the HM operators like spherical fuzzy HM (SpFHM) operator and weighted spherical fuzzy HM (WSpFHM) operator to accumulate the spherical fuzzy (SpF) information. Moreover, various properties and some particular cases of SpFHM and the WSpFHM operators are discussed in details. Also, to compare the results obtained from the HM operators a score function is developed. Based on WSpFHM operator and score function, a model for multiple criteria decision-making (MCDM) is established to resolve the MCDM problem. To check the significance and robustness of the result, a comparative analysis and sensitivity analysis is also performed.

Integral inequalities in probability theory revisited

In [1], the following conjecture was proposed concerning the distribution of ages in a closed interval [0, A]

Numerical Integration Implementation Using Trapezoidal Rule Method To Calculate Aproximation Area Of West Java Province

An area can be shaped into a regular shape or an irregular shape. There is an area of irregular shape which is restricted by an unknown function, to determine that area must use a numerical integration. One of numerical integration methods is Trapezoidal Rule by replacing (????) with an integral approach function which can be evaluated, then let the (????) approximated by a linear polynomial in the certain interval, denoted as closed interval . This study is going to calculate the area of West Java Province by using this method with several different number of partitions in each quadrant such as, 9 partitions, 11 partitions, and 36 partitions in for different quadrants. This study provides the final result of the approximate area which will be compared with the actual area based in the error of result. The main finding is the approximate total area will be closer to the actual area followed by the increasing number of partitions.