On Multi-Objective Geometric Programming Problems with a Negative Degree of Difficulty

Multi-objective geometric programming (MOGP) is a powerful optimization technique widely used for solving a variety of nonlinear optimization problems and engineering problems. Generally, the parameters of a multi-objective geometric programming (MOGP) models are assumed to be deterministic and fixed. However, the values observed for the parameters in real-world MOGP problems are often imprecise and subject to fluctuations. Therefore, we use MOGP within an uncertainty based framework and propose a MOGP model whose coefficients are uncertain in nature. We assume the uncertain variables (UVs) to have linear, normal or zigzag uncertainty distributions and show that the corresponding uncertain chance-constrained multi-objective geometric programming (UCCMOGP) problems can be transformed into conventional MOGP problems to calculate the objective values. The paper develops a procedure to solve a UCCMOGP problem using an MOGP technique based on a weighted-sum method. The efficacy of this procedure is demonstrated by some numerical examples.

A multi–objective geometric programming problem contains more than one objective that needs to be achieved simultaneously. Such problems arise in many applications where two or more, sometimes conflicting objective functions have to be minimised concurrently. In this paper a new adaptive strategy called hybrid method proposed to find Pareto optimal solutions of the multi–objective geometric programming problem. Using geometric programming technique, a global best optimal solution is obtained from a set of Pareto optimal solution having a great impact on convergence of solution. The discussed hybrid method having a goal to enhance the optimisers over all performance by combining different optimisation techniques. In the proposed method we have combined e–constraint and weighted mean method and finally the result so obtained compared with the result obtained by fuzzy programming method. The solution procedure of the proposed hybrid method is illustrated by the numerical examples.

Fuzzy programming technique to solve multi-objective geometric programming problems

In this paper a general multi-objective geometric programming problem with interval parameters is proposed, and a methodology is developed to derive its solution. The model is transformed into a general geometric programming problem, and relation between the original problem and the transformed problem is established. Application of this discussion is illustrated in a waste water treatment model.

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

On computational methods for solutions of multiobjective linear production programming games

A neural network approach to multiobjective and multilevel programming problems

Convergence of exponential penalty function method for multiobjective fractional programming problems

In this paper, we focus on multiobjective two-level fuzzy random programming problems with simple recourses, in which multiple objective functions are involved in each level, shortages and excesses resulting from the violation of the constraints with fuzzy random variables are penalized, and sums of the objective functions and expectation of the amount of the penalties are minimized. To deal with such problems, the concept of estimated Pareto Stackelberg solutions of the leader is introduced under the assumption that the leader can estimate the preference of the follower as a weighting vector of the weighting problems. Employing the possibility measure for fuzzy numbers, weighting method for multiobjective programming problems, and Kuhn-Tucker approach for two-level programming problems, a nonlinear optimization problem under complementarity conditions is formulated to obtain the estimated Pareto Stackelberg solutions for the leader. A numerical example illustrates the proposed method for a multiobjective two-level fuzzy random programming problem with simple recourses. Several types of estimated Pareto Stackelberg solutions are derived corresponding to the weighting vectors and permissible possibility levels specified by the leader.

In this paper, we propose a linear programming based interactive method for multiobjective linear programming problems, in which fuzzy coefficients and random variable coefficients are involved in the objective functions simultaneously. In the proposed method, it is assumed that the decision maker has a fuzzy goal for each objective function, and such a fuzzy goal can be quantified by eliciting the membership function. Through the possibility measure and a fractile optimization model, the original problem is transformed to the well-defined multiobjective programming problem. Then, a generalized Pareto optimal concept is defined, and a linear programming based interactive algorithm is proposed to obtain a satisfactory solution from among a generalized Pareto optimal solution set.

Particle swarm optimization (PSO) was proposed by Kennedy et al. as a general approximate solution method for nonlinear programming problems. Its efficiency has been shown, but there have been left some shortcomings of the method. Thus, the authors proposed a revised PSO (rPSO) method incorporating the homomorphous mapping and the multiple stretching in order to cope with these shortcomings. In this paper, we construct an interactive fuzzy satisficing method for multiobjective nonlinear programming problems based on the rPSO. Furthermore, in order to obtain better solutions in consideration of the property of multiobjective programming problems, we incorporate the direction to nondominated solutions into the rPSO. Finally, we show the efficiency of the proposed method by applying it to numerical examples

The aim of this paper is twofold. Firstly, to define a solution concept of Pareto-optimality for a multi-objective flexible linear programming (MOFLP) problem (or multi-objective fuzzy linear programming problem) and design a method to extract a Pareto-optimal solution of MOFLP problem from the set of optimal solutions of equivalent optimization problem formulated by Dubey and Mehra (2013). Secondly, to extend this study to multi-objective linear programming problem involving hard and flexible constraints with interval uncertainty. A flexible constraint with interval uncertainty generalizes a flexible constraint by allowing preferences to be expressed in the form of intervals. An optimistic-pessimistic approach is proposed to solve multi-objective flexible linear programming with interval uncertainty (MOFLPIU) using an interval-valued fuzzy set representation and the Hurwicz optimism-pessimism criterion.

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 focus on multiobjective nonlinear programming problems with fuzzy parameters and introduce new solution concepts by assuming that the decision maker may have fuzzy goals for each of the objective functions with fuzzy parameters. In order to deal with multiobjective nonlinear programming problems with fuzzy parameters, two types of feasibility and two types of Pareto optimality are introduced by making use of several indices for ranking two fuzzy numbers proposed by Dubois and Prade. Then new solution concepts are defined by considering both two types of feasibility and Pareto optimality, and it is shown that new solutions proposed here can be obtained on the basis of nonlinear programming for appropriate satisficing levels.

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.