In this chapter we deal with the following nonlinear fractional programming problem: 

$$P:\mathop{{\max }}\limits_{{x \in s}} q(x) = (f(x) + \alpha )/((x) + \beta )$$

where f, g: R n → R, α, β ∈ R, S ⊆ R n . To simplify things, and without restricting the generality of the problem, it is usually assumed that, g(x) + β + 0 for all x ∈ S,S is non-empty and that the objective function has a finite optimal value.

Nonlinear Fractional Programming

Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography

This paper presents taxonomy of detailed literature reviews on bi-level programming problems (BLPPs), multi-level programming problems (MLPPs) and associated research problems, while providing detail of solution techniques at the same time. In this taxonomy of review, we classified the multi-level programming problems into two types: (i) General multi-level programming problems (MLPPs) (ii) multi-level multi-objective programming problems (ML-MOPPs) which are further sub classified based on the algorithmic and optimality studies. Bi-level programming problems (BLPPs) are considered as special cases of multi-level programming problems with two level structures. The present literature review includes approximately all prior and latest references on BLPPs, and MLPPs, related solution methodologies. The general related concepts are briefly described while associated references are included for further investigations. The aim of this paper is to provide an easy and systematic road map of currently available literature studies on BLPPs and MLPPs for future researchers.

Bi-level and Multi-Level Programming Problems: Taxonomy of Literature Review and Research Issues

Parts manufacturers use sudden death testing to reduce the testing time of experiments. The sudden death testing plan in the literature can only be applied when all observations of failure time/parameters are crisp. In practice however, it is noted that not all measurements of continuous variables are precise. Therefore, the existing sudden death test plan can be applied if failure data/or parameters are imprecise, incomplete, and fuzzy. The classical statistics have the special case of neutrosophic statistics when there are no fuzzy observations/parameters. The neutrosophic fuzzy statistics can be applied for the testing of manufacturing parts when observations are imprecise, incomplete and fuzzy. In this paper, we will design an original neutrosophic fuzzy sudden death testing plan for the inspection/testing of the electronic product or parts manufacturing. We will assume that the lifetime of the product follows the neutrosophic fuzzy Weibull distribution. The neutrosophic fuzzy operating function will be given and used to determine the neutrosophic fuzzy plan parameters through a neutrosophic fuzzy optimization problem. The results of the proposed neutrosophic fuzzy death testing plan will be implemented with the aid of an example.

Testing of Grouped Product for the Weibull Distribution Using Neutrosophic Statistics

Solving the linear fractional programming problem in a fuzzy environment: Numerical approach

In the recent years we have seen many approaches to solve fractional programming problems. In this paper, the linear fractional programming problem with interval coefficients in objective function is solved by the variable transformation. In this method a convex combination of the first and the last points of the intervals are used in place of the intervals and consequently the problem is reduced to a nonlinear programming problem. Finally, the nonlinear problem is transformed into a linear programming problem with two more constraints and one more variable compare to the initial problem. Numerical examples are illustrated to show the efficiency of the method.

/pdf/solving-linear-fractional-programming-problems-with-interval-2yfu69hjjq.pdf

Solving Linear Fractional Programming Problems with Interval Coefficients in the Objective Function. A New Approach

Los procedimientos clasicos de estimacion para los parametros de la distribucion Weibull se encuentran basados en datos precisos. Se asume usualmente que los datos observados son numeros reales precisos. Sin embargo, algunos datos recolectados podrian ser imprecisos y ser representados en la forma de numeros difusos. Por lo tanto, es necesario generalizar los metodos de estimacion estadisticos clasicos de numeros reales a numeros difusos. En este articulo, diferentes metodos de estimacion son discutidos para los  parametros de la distribucion Weibull cuando los datos disponibles estan en la forma de numeros difusos. Estos incluyen la estimacion por maxima verosimilitud, la estimacion Bayesiana y el metodo de momentos. Los procedimientos de estimacion se discuten en detalle y se comparan via simulaciones de Monte Carlo en terminos de sesgos promedios y errores cuadraticos medios.

/pdf/inference-for-the-weibull-distribution-based-on-fuzzy-data-21g6r3cvd0.pdf

Inference for the Weibull Distribution Based on Fuzzy Data

In this paper, a new method is proposed to find the fuzzy optimal solution of fully fuzzy linear Bilevel programming (FFLBLP) problems by representing all the parameters as triangular fuzzy numbers. In the proposed method, the given FFLBLP problem is decomposed into three crisp linear programming (CLP) problems with bounded variables constraints, the three CLP problems are solved separately and by using its optimal solutions, the fuzzy optimal solution to the given FFLBLP is obtained. The proposed method is easy to understand and to apply for finding the fuzzy optimal solution of FFLBLP occurring in real life situations.

A new method for solving fully fuzzy linear Bilevel programming problems

The present paper deals with the logistic equation having harvesting factor, which is studied in two cases, constant and non-constant. In fact, the nature of equilibrium points and solutions behavior has been analyzed for both of the above cases by finding the first integral, solution curve and phase diagram. Finally, a theorem which is describing the stability of a real model of single species is proved. Keywords: Equilibrium Points, Harvesting Factor, Logistic Equation, O.D.Es. MSC: 78A70, 92D25, 92D40, 35E15 .

/pdf/the-logistic-modeling-population-having-harvesting-factor-4fc4p19oo1.pdf

The logistic modeling population: Having harvesting factor

In this this paper, a parametric approach is used to address a fractional functional programming problem with interval coefficients of the type: Minimize Z(x) = |f(x)|g(x) = | ∑ k1[ai,bi]xi+[ai+1,bi+1]| ∑ k1[ci,di]xi+[ci+1,di+1] subject to Ax≤b,x≥0

Parametric approach for an absolute value linear fractional programming with interval coefficients in the objective function

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