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Numerous optimization problems comprise uncertain data in practical circumstances and such uncertainty can be suitably addressed using the concept of fuzzy logic. This pafa proposes a computationally efficient solution methodology to generate a set of fuzzy non-dominated solutions of a fully fuzzy multi-objective linear programming problem, which incorporates all its parameters and decision variables expressed in form of triangular fuzzy numbers. The fuzzy parameters associated with the objective functions are transformed into interval forms by utilizing the fuzzy-cuts, which subsequently generates the equivalent interval valued objective functions. The concept of centroid of triangular fuzzy numbers derives the deterministic form of the constraints. Furthermore, the scalarization process of weighting sum approach and certain concepts of interval analysis are used to generate the fuzzy non-dominated solutions from which the compromise solution can be determined based on the corresponding real valued expressions of fuzzy optimal objective values resulted due to the ranking function. Three numerical problems and one practical problem are solved for illustration and validation of the proposed approach. The computational results are also discussed as compared to some existing methods. [ABSTRACT FROM AUTHOR]

Numerous optimization problems comprise uncertain data in practical circumstances and such uncertainty can be suitably addressed using the concept of fuzzy logic. This paper proposes a computationally efficient solution methodology to generate a set of fuzzy non-dominated solutions of a fully fuzzy multi-objective linear programming problem, which incorporates all its parameters and decision variables expressed in form of triangular fuzzy numbers. The fuzzy parameters associated with the objective functions are transformed into interval forms by utilizing the fuzzy-cuts, which subsequently generates the equivalent interval valued objective functions. The concept of centroid of triangular fuzzy numbers derives the deterministic form of the constraints. Furthermore, the scalarization process of weighting sum approach and certain concepts of interval analysis are used to generate the fuzzy non-dominated solutions from which the compromise solution can be determined based on the corresponding real valued expressions of fuzzy optimal objective values resulted due to the ranking function. Three numerical problems and one practical problem are solved for illustration and validation of the proposed approach. The computational results are also discussed as compared to some existing methods.

Decision making problems with ambiguous data often arise in numerous practical fields which can be formulated as optimization models in fuzzy environment. This paper develops a new mathematical model using a proposed methodology to efficiently solve a multi-objective linear fuzzy fractional programming problem in trapezoidal fuzzy environment and generate a set of nondominated solutions. The concept of fuzzy cuts with different degrees of satisfaction is implemented which transforms the fuzzy optimization into an equivalent interval valued optimization. Subsequently, interval valued linear functions approximate the fuzzy valued fractional functions based on Taylor's series expansion. Finally, a proposed concept using weighting sum approach with varying weight vectors is utilized to design a mathematical model which generates the set of nondominated solutions. Two numerical examples including an existing problem and an additional practical problem in the field of production, are solved for the illustration of the proposed model. The results of the numerical problems are comparatively discussed with graphical analysis to justify the feasibility and applicability of the proposed model. [ABSTRACT FROM AUTHOR]

AbstractIn the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen’s inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen’s inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results.

In the paper, we study dynamic h-convexity for interval valued functions. Some generalizations of Jensen's inequality in interval valued analysis for h-convex functions on time scales are proved in the paper. In seek of applications of generalized Jensen's inequality, Hermite-Hadamard type inequalities for h-convex functions on time scales are established. Further discrete analogues of newly proved results are also presented in the paper. Some numerical examples are also provided to check the validity of the results. [ABSTRACT FROM AUTHOR]

This note introduces the concept of $ (h_1, h_2) $-convex stochastic processes using interval-valued functions. First we develop Hermite-Hadmard $ (\mathbb{H.H}) $ type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for $ (h_1, h_2) $-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples.

Suppose $X$ is a locally compact Hausdorff space and $\Omega \in \bigtriangleup$. If $ F $ is an interval valued function defined in $ \Omega $ with $F:\bar \Omega\rightarrow I_{\mathbb{R}}$. Suppose $F$ is Topological Henstock integrable, is $ F $ Sequential Henstock integrable? Therefore, the purpose of this paper is to provide a positive response to this query.

This note introduces the concept of (h1; h2)-convex stochastic processes using intervalvalued functions. First we develop Hermite-Hadmard (H:H) type inequalities, then we check the results for the product of two convex stochastic process mappings, and finally we develop Ostrowski and Jensen type inequalities for (h1; h2)-convex stochastic process. Also, we have shown that this is a more generalized and larger class of convex stochastic processes with some remark. Furthermore, we validate our main findings by providing some non-trivial examples. [ABSTRACT FROM AUTHOR]

Suppose X is a locally compact Hausdorff space and Ώ ∈ Δ. If F is an interval valued function defined in with F: Ώ → IR. Suppose F is Topological Henstock integrable, is F Sequential Henstock integrable? Therefore, the purpose of this paper is to provide a positive response to this query. [ABSTRACT FROM AUTHOR]

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical h-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial.

It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.

In this paper we focus on real situations where certain perfectly divisible estate has to be divided among claimants who can merely indicate the range of their claims, and the available amount is smaller than the aggregated claim. Funds' allocation of a firm among its divisions, taxation problems, priority problems, distribution of costs of a joint project among the agents involved, various disputes including those generated by inheritance, or by cooperation in joint projects based on restricted willingness to pay, fit into this framework. The corresponding claim of each claimant can vary within a closed interval or fuzzy interval. For claims, fuzzy intervals are applied whenever the claimants can distinguish a possibility of attaining the amount of estate, and/or its membership degree of a possibility of attainment. When claims of claimants have fuzzy interval uncertainty, we settle such type of division problems by transforming it into bankruptcy problems under interval uncertainty by interval valued mappings. A similar approach is applied to deal with uncertainty of estate to be divided. Here, a probability interpretation can also be considered e.g. in taxation problems. We consider the division problems under uncertainty of claims and/or estate and present bankruptcy rule, which are consistent with the classical bankruptcy proportional rule. Several examples are presented to illustrate particular problems and solution concepts. [ABSTRACT FROM AUTHOR]

It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings. [ABSTRACT FROM AUTHOR]

The role of integral inequalities can be seen in both applied and theoretical mathematics fields. According to the definition of convexity, it is possible to relate both concepts of convexity and integral inequality. Furthermore, convexity plays a key role in the topic of inclusions as a result of its definitional behavior. The importance and superior applications of convex functions are well known, particularly in the areas of integration, variational inequality, and optimization. In this paper, various types of inequalities are introduced using inclusion relations. The inclusion relation enables us firstly to derive some Hermite-Hadamard inequalities (H.H-inequalities) and then to present Jensen inequality for harmonical h-Godunova-Levin interval-valued functions (GL-IVFS) via Riemann integral operator. Moreover, the findings presented in this study have been verified with the use of useful examples that are not trivial. [ABSTRACT FROM AUTHOR]

Given a nonconvex minimization problem where the objective function is nonlinear and twice differentiable. To gain more information about the objective function, it is essential to obtain all its stationary points and study the behaviour of these points. Since many nonlinear functions are expressible as polynomials via interpolation, there is a need to devise fast and accurate algorithms in finding root(s) of the interpolating polynomial. Through interval computation, the Weierstrass-like parallel iterative methods are known for their efficiency in finding polynomial zeros. However, these schemes are highly dependent on the midpoints of each interval in generating successive intervals. In this study, we propose a scaling function on some Weierstrass-like parallel iterative methods such that the procedures are less dependent on the generated midpoints, hence allowing a more efficient search for the zeros while reducing the width of the intervals. The proposed procedures with the shifted centres of the enclosing intervals are tested on 120 problems and we compare their efficiency with the existing Weierstrass-like methods in terms of the number of iterations and largest final interval width. The results indicate that the proposed procedures outperform the original procedures, giving more reduction on the final interval width with a lesser number of iterations. [ABSTRACT FROM AUTHOR]

In this paper a non-smooth optimization problem is studied in an uncertain environment. The objective function of this problem is interval valued function. We introduce the class of LU−(p,r)−[ρL,ρU]−(η,θ)LU−(p,r)−[ρL,ρU]−(η,θ)-invex interval valued functions about the Clarke generalized gradient. Then, through non trivial examples, we illustrate that the class of functions introduced exists. Based upon the proposed invexity assumptions, the sufficient optimality conditions are established. Further, we derive weak, strong and strict converse duality theorems for Mond-Weir type and Wolfe type dual programs. Some examples are also given in order to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper we deal with well known Optimal Allocation Problem (OAP) and its solution under uncertainty. We define a special interval uncertainty by extending the well known concept of fuzzy intervals (or, fuzzy numbers), defining a new concept of extended fuzzy interval and its subspace: extended linear fuzzy interval. Then we present some examples and derive basic properties. We formulate the corresponding optimization problem with the extended fuzzy interval coefficients and, in particular, extended fuzzy interval variables and derive its optimal solution in the form of extended fuzzy intervals. Some numerical examples are presented in order to illustrate particular problems and solution concepts. [ABSTRACT FROM AUTHOR]

In this paper, we introduce twice weakly differentiable and twice H- differentiable interval valued functions. The existence of twice H-differentiable interval-valued function and its relation with twice weakly differentiable func- tions are presented. Interval valued bonvex and generalized bonvex func- tions involving twice H-differentiability are proposed. Under the proposed settings, necessary conditions are elicited naturally in order to achieve LU- efficient solution. Mangasarian type dual is discussed for a nondifferentiable multiobjective programming problem and appropriate duality results are also derived. The theoretical developments are illustrated through non-trivial nu- merical examples. [ABSTRACT FROM AUTHOR]

We devote this paper to study of multiobjective programming problems with interval valued objective functions. For this, we consider two order relations LU and LS on the set of all closed intervals and propose several concepts of Pareto optimal solutions and generalized convexity. Based on generalized convexity (viz. LU and LS-pseudoconvexity) and generalized differentiability (viz. GH-differentiablity) of interval valued functions, the KKT optimality conditions for aforesaid problems are obtained. The theoretical development is illustrated by suitable examples. [ABSTRACT FROM AUTHOR]

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