Efficiency and duality in nonsmooth multiobjective fractional programming involving η-pseudolinear functions

Optimality and Duality for Nonsmooth Multiobjective Fractional Programming with Generalized Invexity

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.

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.

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

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.

Convergence of exponential penalty function method for multiobjective fractional programming problems

We describe an improvement of Chergui and Moulaï’s method (2008) that generates the whole efficient set of a multiobjective integer linear fractional program based on the branch and cut concept. The general step of this method consists in optimizing (maximizing without loss of generality) one of the fractional objective functions over a subset of the original continuous feasible set; then if necessary, a branching process is carried out until obtaining an integer feasible solution. At this stage, an efficient cut is built from the criteria’s growth directions in order to discard a part of the feasible domain containing only nonefficient solutions. Our contribution concerns firstly the optimization process where a linear program that we define later will be solved at each step rather than a fractional linear program. Secondly, local ideal and nadir points will be used as bounds to prune some branches leading to nonefficient solutions. The computational experiments show that the new method outperforms the old one in all the treated instances.

A new class of (not necessarily differentiable) multiobjective fractional programming problems with E-differentiable functions is considered. The so-called parametric E-Karush–Kuhn–Tucker necessary optimality conditions and, under E-convexity hypotheses, sufficient E-optimality conditions are established for such nonsmooth vector optimization problems. Further, various duality models are formulated for the considered E-differentiable multiobjective fractional programming problems and several E-duality results are derived also under appropriate E-convexity hypotheses.

A class of multiobjective fractional programming problems is considered and duality results are established through a parametric approach. These results give a multiobjective analog of some of the well known results of a single ratio programming problem.

The purpose of the paper is to present some approaches for multiobjective fractional programming with set coefficients in the objective functions. Among other cases, we discuss for linear fractional multiobjective programming some conservative and nonconservative approaches related to Soyster’s mathematical programming with set-inclusive constraints and Dantzig’s generalized linear programming, respectively. The vectorial max-min optimization will be also used as a conservative approach for inexact multiobjective programming. In particular, the conservative approach will be applied to mathematical programming with nondecreasing objectives with respect to the inexact coefficients. Linear-fractional objective functions will be studied from both points of view (conservative and nonconservative). Also, some applications to the fuzzy linear-fractional multiobjective programming will be suggested.

A classification scheme for fractional programs is introduced. Following a brief review of results in single-ratio fractional programming we focus on three classes of multi-ratio fractional programs: (a) minimization of the largest of several ratios, (b) minimization of the sum of ratios, and (c) multiobjective fractional programs.

The authors deal with multiobjective linear programming problems with fuzzy random variable coefficients. Since the problem is ill-defined due to both fuzziness and randomness, we propose a decision making model based on E-model, which is a useful model in stochastic programming, and a possibility measure. First, we show that the formulated problem is reduced to a multiobjective linear fractional programming problem. After defining a Pareto optimal solution based on the expected value of possibility measure, we construct a solution algorithm for solving a minimax problem. Further, we consider interactive decision making using reference points and give numerical examples.

On computational methods for solutions of multiobjective linear production programming games

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.