An interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters

This paper deals with multiobjective integer programming problems by considering fuzzy goals of the decision maker for objective functions. After determining the fuzzy goals of the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the augmented minimax problem which becomes an integer programming problem. For solving the problem, decoding algorithms for 0-1 programming problems are revised and ringed double strings are also introduced. Then an interactive fuzzy satisficing method is presented together with an illustrative 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.

Stability of multiobjective nonlinear programming problems with fuzzy parameters

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 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

In this paper, we propose an interactive fuzzy decomposition method for large-scale multiobjective nonlinear programming problems with the block angular structure to obtain the satisficing solution for the decision maker (DM). In the proposed method, after eliciting the membership function for each of the objective functions, the satisficing solution for the DM can be obtained on the basis of the dual decomposition method from Pareto optimal solution set by updating the reference membership values interactively. An interaction processes for a numerical example under the hypothetical DM are illustrated to indicate the feasibility of the proposed method.KeywordsMembership FunctionDecomposition MethodDual ProblemPareto Optimal SolutionNonlinear Programming ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Stability of possibilistic multiobjective nonlinear programming problems without differentiability

Here, we propose a mathematical model to analyze the multi-objective bimatrix game under entropy environment. In this new approach, the entropy function of each player is considered as an objective to the multi-objective bimatrix game. This model is named as multi-objective entropy bimatrix game model. The formulated model is a multi-objective non-linear programming problem. To solve this model, we have introduced a solution procedure, which determines the feasibility of multi-objective non-linear programming problem using fuzzy programming approach and revised Genetic Algorithm (GA). By a real coded GA we have obtained the bounds of the objectives of the said model. Then we have determined the solutions of our proposed model. Finally, numerical examples are presented to illustrate the methodology.

This chapter is devoted to several interactive fuzzy programming approaches to multiobjective stochastic programming problems. Through the use of the several stochastic models including the expectation model, the variance model, the probability model, the fractile model, and the simple recourse model together with chance constrained programming, the formulated multiobjective stochastic programming problems are transformed into deterministic ones. Assuming that the decision maker has the fuzzy goal for each of the objective functions, several interactive fuzzy satisficing methods are presented for deriving a satisficing solution for the decision maker by updating the reference membership levels.

On computational methods for solutions of multiobjective linear production programming games

Solving the multiobjective possibilistic linear programming problem

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

Multi-criteria decision making method is used in such cases having multiple targets. Multi-objective linear programming problems, which deal with uncertain measurements for both objectives and constraints, are solved by fuzzy multi-objective linear programming methods. Analytic hierarchy process (AHP), one of the multi-criteria decision-making methods based on weighting of objectives, is designed to solve complex problems. In this study, multiple criteria decision making problems and decision makers’ opinions on these problems as well as their solution processes are discussed. Firstly, a fuzzy multi objective linear programming problem is solved using the approaches of Zimmerman and Hybrid. In the literature, these methods are applied using triangular fuzzy numbers based on decision makers’ opinions only in cases where the objective function is minimization. In this study, as an alternative to the triangular fuzzy numbers, trapezoidal fuzzy numbers are defined, and both minimization and maximization of objective functions are examined. Finally, a solution algorithm for a fuzzy multi-objective linear programming problem is proposed under some given conditions. The algorithm is applied to a sample problem that involves modeling of a congress organizing which aims to place attendees to accommodations under specific objectives and constraints. In the evaluation phase of the algorithm, targets are weighted by using triangular and trapezoidal fuzzy numbers. At the end, the results of the proposed algorithm are compared with the results of other methods in the literature.

Warner (1965) introduced the randomized response model as an alternative survey technique for socially undesirable or incriminating behaviour questions in order to reduce response error, protect a respondent’s privacy, and increase response rates. In multivariate stratified surveys with multiple randomised response data the choice of optimum sample sizes from various strata may be viewed as a multi-objective nonlinear programming problem. The allocation thus obtained may be called a “compromise allocation” in sampling literature.In this paper, we have formulated two stage stratified Warner’s Randomised Response model (RRM) as a multi-objective integer non-liner optimization problem. In this problem of RRM we have minimized the square root of coefficient of variations instead of variations for different characteristics because the coefficient of variation is unit free, subject to the linear and quadratic cost constraint. The multi-objective optimization problem of RRM has been solved by lexicographic goal programming integrated with fixed priority - distance method. The solution obtained by lexicographic goal programming Integrated with fixed priority - distance have been compared with various existing approaches namely the value function approach, goal programming techniques, - constraint method and distance-based method and Khuri & Cornel distance based method. A numerical example is also been presented to illustrate the computational details.https://doi.org/10.28919/jmcs/3376