Optimality and duality results for E-differentiable multiobjective fractional programming problems under E-convexity

Convergence of exponential penalty function method for multiobjective fractional programming problems

In this paper, we present a multi-objective fractional probabilistic programming problem where the parameters in the objective functions and right-hand side constraints are multi-choice in nature. Multi-choice multi-objective objective fractional probabilistic programming problem is formulated. The multi-choice parameters in the right-hand side of constraints are random variables following Cauchy distribution. Multi-choice multi-objective fractional probabilistic programming problem is transformed to deterministic equivalent multi-choice multi-objective fractional programming problem using chance-constrained programming problem. Lagrange interpolating polynomial approach is proposed to deal with multi-choice parameters. The multi-choice multi-objective fractional programming problem is transformed to non-linear mixed integer multi-objective programming problem. An \(\epsilon -\) constrained method is applied to find the compromise solutions of non-linear mixed integer multi-objective programming problem using different values of \(\epsilon \). LINGO software is applied to find the optimal solution of single objective non-linear mixed integer programming problem. Finally, a numerical example is given to illustrate the proposed programming problem.KeywordsMulti-choice programming problemMulti-objective programming problemProbabilistic programming problemFractional programming problem\(\epsilon \)-constraint method

The present work defines interval and interval valued function in terms of parameters. Two types of multi-objective programming problems are considered: one is a general multi-objective interval fractional programming problem and the other is a parametric form of the first problem, where the objective and constraint functions are interval valued. Relationships between the solutions of both problems are developed, and some of the results are illustrated through numerical example.

We establish necessary and sufficient conditions for efficiency of multiobjective\nfractional programming problems involving r-invex functions. Using the optimality\nconditions, we investigate the parametric type dual, Wolfe type dual and Mond-Weir\ntype dual for multiobjective fractional programming problems concerning r-invexity.\nSome duality theorems are also proved for such problem in the framework of\nr-invexity.

Necessary and sufficient saddle-point- and stationary-point-type proper efficiency conditions are established for a class of continuous-time multiobjective fractional programming problems with convex operator inequality and affine operator equality constraints. Subsequently, the form and contents of these proper efficiency results are utilized for constracting a Lagrangian-type, two parametric, and four semiparametric duality models, and for proving appropriate duality theorems. These proper efficiency and duality results contain, as special cases, similar results for continuous-time programming problems with nonfractional multiple, single fractional, and conventional objective functions, which are patricular cases of the principal problem studied in this paper.

This paper proposes an interactive fuzzy programming method for seeking a satisfactory solution for multiobjective two‐level linear fractional programming problems in which the decision makers in the upper and lower levels have several objectives, by first setting up the fuzzy goals for several objectives of each decision maker and seeking a satisfactory solution for the degree of satisfaction of two decision makers in a cooperative manner. In the proposed method, the decision maker at the upper level sets the minimal satisfactory levels for each fuzzy goal and the decision maker at the lower level determines the aspiration levels. The minimal satisfactory levels are treated as a constraint and the solution closest to the aspiration levels of the decision maker at the lower level is computed. The ratio of the aggregative degrees of satisfaction of the decision makers in the upper and lower levels with the obtained solution is evaluated by using partial information on the preference of the decision makers. If the ratio of the degrees of satisfaction satisfies the given condition and the decision maker at the upper level is satisfied with this solution, then the interaction is complete. Otherwise, a satisfactory solution is sought by updating the minimal satisfactory and aspiration levels. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(9): 54–62, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10047

In this paper, we propose an alternate modified solution procedure based on fuzzy goal programming (FGP) approach for solving bi-level multiobjective linear fractional programming problems (BL-MOLFPPs) with some major modifications in the algorithm suggested by Abo-Sinna and Baky (2010). The proposed modified technique is simpler and requires less computational works than that of earlier technique. In the formulation of FGP models, each objective function (at both the levels - leader's level and follower's level) as well as control variables (of leader's level) are transformed into fuzzy goals with modified membership functions. Then FGP approach is used for achieving the highest degree of each of membership goals. To avoid decision deadlock situations and difficulties associated with bi-level programming problems, solution preferences by decision makers (at leader's and follower's levels) are not taken into account in proposed algorithm. A comparative analysis is also carried out with numerical example in order to show efficiency of proposed algorithm over earlier technique.

The purpose of this paper is to consider a class of nondifferentiable multiobjective fractional programming problems in which every component of the objective function contains a term involving the support function of a compact convex set. Based on the (C,α,ρ,d)-convexity, sufficient optimality conditions and duality results for weakly efficient solutions of the nondifferentiable multiobjective fractional programming problem are established. The results extend and improve the corresponding results in the literature.

On computational methods for solutions of multiobjective linear production programming games

A generalized nondifferentiable fractional optimization problem (GFP), which consists of a maximum objective function defined by finite fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions, is considered. Recently, Kim et al. [Journal of Optimization Theory and Applications 129 (2006), no. 1, 131-146] proved optimality theorems and duality theorems for a nondifferentiable multiobjective fractional programming problem (MFP), which consists of a vector-valued function whose components are fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions. In fact if <TEX>$\overline{x}$</TEX> is a solution of (GFP), then <TEX>$\overline{x}$</TEX> is a weakly efficient solution of (MFP), but the converse may not be true. So, it seems to be not trivial that we apply the approach of Kim et al. to (GFP). However, modifying their approach, we obtain optimality conditions and duality results for (GFP).

In this paper, new classes of nondifferentiable generalized invex functions are introduced. Further, nonsmooth vector optimization problems with functions belonging to the introduced classes of (generalized) (Φ,ρ)-type I functions are considered. Sufficient optimality conditions and duality results for such classes of nonsmooth vector optimization problems are established. It turns out that the presented results are proved also for such nonconvex vector optimization problems in which not all functions constituting them possess the fundamental property of invexity.

Optimality and Duality for Nonsmooth Multiobjective Fractional Programming with Generalized Invexity

In this paper, a new class of nonconvex nonsmooth multiobjective programming problems with both inequality and equality constraints is considered. Namely, the sufficient optimality conditions and Mond-Weir duality results are established for such nondifferentiable multicriteria optimization problems in which the involved functions are nondifferentiable (b,?,?,?)w-univex functions (not necessarly with respect to the same functions b and ? and ?). Then the aforesaid results developed here under (b,?,?, ?)w-uinvexity are applicable for a larger class of nonsmooth vector optimization problems than under other generalized convexity notions existing in the literature.

A neural network approach to multiobjective and multilevel programming problems

In this paper, a new generalised class of higher order (F, α, ρ, d)-V-type I function is introduced for a non-smooth multi-objective fractional programming problem involving support functions. The newly defined class extends several known classes in the literature has been justified through a non-trivial example. In the framework of new concept, we determine conditions under which a fractional function becomes higher order (F, α, ρ, d)-V-type I function and do some computational work to substantiate the analysis. Further, we establish Karush-Kuhn-Tucker type sufficient optimality conditions and derive various duality results for higher order Mond-Weir type and Schaible type dual programs.