convex linear combination
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Some retailers (e.g. pharmacies) are responsible for satisfying the demand for the minimum range of goods, which are generally unprofitable. With respect to such goods, the enterprise seeks to satisfy uncertain demand rather than to make profit. We assume that: (a) the vector of demand for goods of the minimum assortment in the planning period lies “between” the demand vectors of several previous periods (is a convex linear combination of these vectors); (b) the smaller the maximum unsatisfied demand (by product groups and possible vectors of demand), the greater is the reliability of meeting the demand. Under these assumptions, we address the problem of allocating a limited procurement budget among commodity groups to meet uncertain demand most reliably. The article shows that this problem is equivalent to finding an optimal strategy by Wald’s criterion in some game with nature and can be reduced to a linear programming problem. Using the problem features, we propose a fast (having quadratic complexity) algorithm for constructing an optimal procurement plan. The model can be used when planning the minimum assortment goods procurement in order to maximize the meeting demand reliability, achievable within the allocated budget. As far as we know, such a formulation of the problem has not been studied in the previous literature.

In this paper, we will investigate and discuss a new class of meromorphic univalent functions defined by multiplier transformation which is R(c, , y, ), as well as study the coefficient estimates and growth theorems, and then another line in this work, upon to get the close under the convex linear combination 

In this article, we introduce a new subclass of analytic functions, using the exponent operators of Rafid and $ q $-derivative. The coefficient estimates, extreme points, convex linear combination, radii of starlikeness, convexity, and finally integral are investigated.

The focus of this article is the introduction of a new subclass of analytic functions involving q-analogue of the Bessel function and obtained coefficient inequities, growth and distortion properties, radii of close-to-convexity, and starlikeness, as well as convex linear combination. Furthermore, we discussed partial sums, convolution, and neighborhood properties for this defined class.

In this paper, we introduce a new  class of meromorphic functions, using the exponent $ q $-derivative operator, and then look at it coefficient estimates, extreme points, convex linear combination, Radii of starlikeness, convexity and finally partial sum property are investigated.

Let function P be a probability on a finite group G, i.e. $P(g)\geq0\ $ $(g\in G),\ \sum\limits_{g}P(g)=1$ (we write $\sum\limits_{g}$ instead of $\sum\limits_{g\in G})$. Convolution of two functions $P, \; Q$ on group $G$ is \linebreak $ (P*Q)(h)=\sum\limits_{g}P(g)Q(g^{-1}h)\ \ (h\in G)$. Let $E(g)=\frac{1}{|G|}\sum\limits_{g}g$ be the uniform (trivial) probability on the group $G$, $P^{(n)}=P*...*P$ ($n$ times) an $n$-fold convolution of $P$. Under well known mild condition probability $P^{(n)}$ converges to $E(g)$ at $n\rightarrow\infty$. A lot of papers are devoted to estimation the rate of this convergence for different norms. Any probability (and, in general, any function with values in the field $R$ of real numbers) on a group can be associated with an element of the group algebra of this group over the field $R$. It can be done as follows. Let $RG$ be a group algebra of a finite group $G$ over the field $R$. A probability $P(g)$ on the group $G$ corresponds to the element $ p = \sum\limits_{g} P(g)g $ of the algebra RG. We denote a function on the group $G$ with a capital letter and the corresponding element of $RG$ with the same (but small) letter, and call the latter a probability on $RG$. For instance, the uniform probability $E(g)$ corresponds to the element $e=\frac{1}{|G|}\sum\limits_{g}g\in RG. $ The convolution of two functions $P, Q$ on $G$ corresponds to product $pq$ of corresponding elements $p,q$ in the group algebra $RG$. For a natural number $n$, the $n$-fold convolution of the probability $P$ on $G$ corresponds to the element $p^n \in RG$. In the article we study the case when a linear combination of two probabilities in algebra $RG$ equals to the probability $e\in RG$. Such a linear combination must be convex. More exactly, we correspond to a probability $p \in RG$ another probability $p_1 \in RG$ in the following way. Two probabilities $p, p_1 \in RG$ are called complementary if their convex linear combination is $e$, i.e. $ \alpha p + (1- \alpha) p_1 = e$ for some number $\alpha$, $0 <\alpha <1$. We find conditions for existence of such $\alpha$ and compare $\parallel p ^ n-e \parallel$ and $\parallel {p_1} ^ n-e \parallel$ for an arbitrary norm ǁ·ǁ.

The purpose of this paper is to derive analytic expressions for the multivariate Lorenz surface for a relevant type of models based on the class of distributions with given marginals described by Sarmanov and Lee. The expression of the bivariate Lorenz surface can be conveniently interpreted as the convex linear combination of products of classical and concentrated univariate Lorenz curves. Thus, the generalized Gini index associated with this surface is expressed as a function of marginal Gini indices and concentration indices. This measure is additively decomposable in two factors, corresponding to inequality within and between variables. We present different parametric models using several marginal distributions including the classical Beta, the GB1, the Gamma, the lognormal distributions and others. We illustrate the use of these models to measure multidimensional inequality using data on two dimensions of well-being, wealth and health, in five developing countries.

Abstract With the recent growth in data availability and complexity, and the associated outburst of elaborate modelling approaches, model selection tools have become a lifeline, providing objective criteria to deal with this increasingly challenging landscape. In fact, basing predictions and inference on a single model may be limiting if not harmful; ensemble approaches, which combine different models, have been proposed to overcome the selection step, and proven fruitful especially in the supervised learning framework. Conversely, these approaches have been scantily explored in the unsupervised setting. In this work we focus on the model-based clustering formulation, where a plethora of mixture models, with different number of components and parametrizations, is typically estimated. We propose an ensemble clustering approach that circumvents the single best model paradigm, while improving stability and robustness of the partitions. A new density estimator, being a convex linear combination of the density estimates in the ensemble, is introduced and exploited for group assignment. As opposed to the standard case, where clusters are typically associated to the components of the selected mixture model, we define partitions by borrowing the modal, or nonparametric, formulation of the clustering problem, where groups are linked with high-density regions. Staying in the density-based realm we thus show how blending together parametric and nonparametric approaches may be beneficial from a clustering perspective.

Possibility, necessity and credibility measures are used in the literature in order to deal with imprecision. Recently, Yang and Iwamura [L. Yang and K. Iwamura, Applied Mathematical Science 2(46) (2008) 2271–2288] introduced a new measure as convex linear combination of possibility and necessity measures and they determined some of its axioms. In this paper, we introduce characteristics (parameters) of a fuzzy variable based on that measure, namely, expected value, variance, semi-variance, skewness, kurtosis and semi-kurtosis. We determine some properties of these characteristics and we compute them for trapezoidal and triangular fuzzy variables. We display their application for the determination of optimal portfolios when assets returns are described by triangular or trapezoidal fuzzy variables.

     The applications of Ruscheweyh derivative are studied and discussed of class of meromorphic multivalent application. We get some interesting geometric properties, such as coefficient bound, Convex linear combination, growth and distortion bounds, radii of starlikenss ,  convexity and neighborhood property.