scholarly journals Solution of a bi-criteria problem of rating alternatives using tropical optimization

2019 ◽  
pp. 15-32
Author(s):  
Nikolai K. Krivulin ◽  
◽  
Margarita A. Tsobenko ◽  
Keyword(s):  
Pareto Front ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Third Order ◽  
Tropical Mathematics ◽  
All Solutions ◽  
Computational Procedures ◽  
Necessary And Sufficient

A problem is considered to evaluate scores (priorities, weights) of alternatives through the results of pairwise comparisons according to two criteria. A formal derivation and computational procedures of the solution to the problem are described, using methods of tropical mathematics, which studies algebraic systems with specially defined operations of addition and multiplication. The problem is reduced to simultaneous approximation of two matrices of pairwise comparisons by a common consistent matrix, in the Chebyshev metric in logarithmic scale. First, auxiliary variables are introduced to represent the minima of the objective functions, and a parameterized inequality is derived, which determines the set of solutions to the original optimization problem. The necessary and sufficient conditions for the existence of solutions of the inequality are used to evaluate the values of parameters, which correspond to the Pareto front of the problem. All solutions of the inequality under the obtained values are taken as a Pareto-optimal solution for the problem. To illustrate the computational procedures used, numerical examples of evaluating scores of alternatives are given for problems with matrices of the third order.

scholarly journals Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 303
Author(s):  
Nikolai Krivulin
Keyword(s):  
Decision Problem ◽  
Optimization Problem ◽  
Pareto Frontier ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Optimal Solution ◽  
Algebraic Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Idempotent Algebra

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

scholarly journals Oscillation Criteria of Solutions of Third Order Neutral Integro-Differential Equations

2021 ◽  
pp. 3642-3647
Author(s):  
Taghreed Hussein Abed ◽  
Sattar Naser Ketab ◽  
Hussain Ali Mohamad
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Third Order ◽  
Oscillation Criteria ◽  
All Solutions ◽  
Necessary And Sufficient

      Some necessary and sufficient conditions are obtained that guarantee the oscillation of all solutions of two types of neutral integro-differential equations of third order. The integral is used in the sense of Riemann-Stieltjes. Some examples were included to illustrate the obtained results

scholarly journals Oscillations of Third Order Half Linear Neutral Differential Equations

2015 ◽  
Vol 12 (3) ◽  
pp. 625-631
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Oscillation Criterion ◽  
Necessary And Sufficient Conditions ◽  
Third Order ◽  
Neutral Differential Equations ◽  
The Third ◽  
All Solutions ◽  
Necessary And Sufficient

In this paper the oscillation criterion was investigated for all solutions of the third-order half linear neutral differential equations. Some necessary and sufficient conditions are established for every solution of (a(t)[(x(t)±p(t)x(?(t) ) )^'' ]^? )^'+q(t) x^? (?(t) )=0, t?t_0, to be oscillatory. Examples are given to illustrate our main results.

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

Characterization of Extreme Points of Multi-Stochastic Tensors

2016 ◽  
Vol 16 (3) ◽  
pp. 459-474 ◽  
Author(s):  
Rihuan Ke ◽  
Wen Li ◽  
Mingqing Xiao
Keyword(s):  
Operation Research ◽  
Convex Combination ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Stochastic Matrices ◽  
Third Order ◽  
New Approach ◽  
3 Dimensional ◽  
Necessary And Sufficient

AbstractStochastic matrices play an important role in the study of probability theory and statistics, and are often used in a variety of modeling problems in economics, biology and operation research. Recently, the study of tensors and their applications became a hot topic in numerical analysis and optimization. In this paper, we focus on studying stochastic tensors and, in particular, we study the extreme points of a set of multi-stochastic tensors. Two necessary and sufficient conditions for a multi-stochastic tensor to be an extreme point are established. These conditions characterize the “generators” of multi-stochastic tensors. An algorithm to search the convex combination of extreme points for an arbitrary given multi-stochastic tensor is developed. Based on our obtained results, some expression properties for third-order and n-dimensional multi-stochastic tensors (${n=3}$ and 4) are derived, and all extreme points of 3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple way. As an application, a new approach for the partially filled square problem under the framework of multi-stochastic tensors is given.

scholarly journals On the asymptotic behavior of solutions of certain third-order nonlinear differential equations

2005 ◽  
Vol 2005 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third ◽  
All Solutions

We establish sufficient conditions under which all solutions of the third-order nonlinear differential equation x ⃛+ψ(x,x˙,x¨)x¨+f(x,x˙)=p(t,x,x˙,x¨) are bounded and converge to zero as t→∞.

On bounded nonoscillatory solutions of third-order nonlinear differential equations

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Ivan Mojsej ◽  
Alena Tartaľová
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Banach Fixed Point Theorem ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Topological Approach ◽  
The Third ◽  
Necessary And Sufficient

AbstractThis paper is concerned with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. We give the necessary and sufficient conditions guaranteeing the existence of bounded nonoscillatory solutions. Sufficient conditions are proved via a topological approach based on the Banach fixed point theorem.

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