Stability of multiobjective nonlinear programming problems with fuzzy parameters

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.

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 fuzziness induced in the decision making in the actual system in general, comprises fuzziness in the formulation of the problem and that in the judgments of the decision maker (DM) as a human being. This paper proposes an interactive fuzzy satisficing method, which takes into consideration the fuzziness in problem formulation as well as that in the judgments of DM. The multiobjective nonlinear programming problem containing fuzzy parameters is transformed into the α ‐multiobjective nonlinear programming problem. After determining the fuzzy goal of DM for each objective function by eliciting the membership function, the following reference values are set: the fuzziness index α of DM for the fuzzy parameter; and the reference membership values, which are reference for the membership functions. For the specified α and the reference membership values, the corresponding augmented minimax problem is solved, and DM is supplied with the α‐Pareto optimal, together with the trade‐off ratios among membership functions and the trade‐off ratios between α and the membership functions. Obtaining the result, the DM takes the information concerning current α‐Pareto optimal solution and the trade‐off ratios into consideration, and respond by updating the reference membership values and the value of α if necessary. In this way the satisficing solution finally is derived for DM, thereby ensuring the α‐Pareto optimality. This is the proposed interactive fuzzy satisficing method. The interactive process is demonstrated for a numerical example by the interactive computer program.

The objective of this paper is to develop matrix games with pay-offs of triangular hesitant fuzzy elements (THFEs). To solve such games, a new methodology has been derived based on the notion of weighted average operator and score function of THFEs. Firstly, we formulate two non-linear programming problems with THFEs. Then applying the score function of THFEs, we transform these two problems into two non-linear multi-objective programming problems with triangular fuzzy numbers (TFNs). Finally, the Lexicographic method is used to solve these two multi-objective programming problems. A market share problem is considered to show the validity and applicability of the proposed methodology.

Stability of possibilistic multiobjective nonlinear programming problems without differentiability

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

Solving the multiobjective possibilistic linear programming problem

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.

In the real-world optimization problems, coefficients of the objective function are not known precisely and can be interpreted as fuzzy numbers. In this paper we define the concepts of optimality for linear programming problems with fuzzy parameters based on those for multiobjective linear programming problems. Then by using the concept of comparison of fuzzy numbers, we transform a linear programming problem with fuzzy parameters to a multiobjective linear programming problem. To this end, we propose several theorems which are used to obtain optimal solutions of linear programming problems with fuzzy parameters. Finally some examples are given for illustrating the proposed method of solving linear programming problem with fuzzy parameters.

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.

The sufficient optimality conditions and duality results have recently been given for the following generalised convex programming problem:where the funtion f and g satisfyfor some η: X0 × X0 → ℝnIt is shown here that a relaxation defining the above generalised convexity leads to a new class of multi-objective problems which preserves the sufficient optimality and duality results in the scalar case, and avoids the major difficulty of verifying that the inequality holds for the same function η(. , .). Further, this relaxation allows one to treat certain nonlinear multi-objective fractional programming problems and some other classes of nonlinear (composite) problems as special cases.

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

In this paper, a method is developed to derive the acceptable ranges of objective values for a multi-objective linear fractional programming problem(MOLFPP) with fuzzy parameters both in objectives and constraints. \(\alpha \)- and \(\beta \)-cuts are respectively used in the objectives and constraints to specify the degrees of satisfaction and transform the fuzzy parameters into closed intervals. Using variable transformation and Taylor series expansion, the interval-valued fractional objectives are approximated by intervals of linear functions. The objective functions are assigned proper weights using analytic hierarchy process. Weighting sum method is used to transform the interval-valued multiple objectives into single objective. MOLFPP in interval-valued form is equivalently formulated as two linear problems which derive the acceptable ranges of objective values. Two numerical examples are illustrated to demonstrate the proposed 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.

A method for optimal design of automotive body assembly using multi-material construction