Solution of a possibilistic multiobjective linear programming problem

Solving the multiobjective possibilistic linear programming problem

The multi-objective linear fractional programming is an interesting topic with many applications in different fields. Until now, various algorithms have been proposed in order to solve the multi-objective linear fractional programming (MOLFP) problem. An important point in most of them is the use of non-linear programming with a high computational complexity or the use of linear programming with preferences of the objective functions which are assigned by the decision maker. The current paper, through combining goal programming and data envelopment analysis (DEA), proposes an iterative method to solve MOLFP problems using only linear programming. Moreover, the proposed method provides an efficient solution which fairly optimizes each objective function when the decision maker has no information about the preferences of the objective functions. In fact, along with normalization of the objective functions, their relative preferences are fairly determined using the DEA. The implementation of the proposed method is demonstrated using numerical examples.

On computational methods for solutions of multiobjective linear production programming games

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.

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.

Optimization problems with multiple criteria measuring solution quality can be modeled as multiobjective programming problems. Because the objective functions are usually in conflict, there is not a single feasible solution that can optimize all objective functions simultaneously. An optimal solution is one that is most preferred by the decision maker (DM) among all feasible solutions. An optimal solution must be nondominated but a multiobjective programming problem may have, possibly infinitely, many nondominated solutions. Therefore, tradeoffs must be made in searching for an optimal solution. Hence, the DM's preference information is elicited and used when a multiobjective programming problem is solved. The model, concepts and definitions of multiobjective programming are presented and solution methods are briefly discussed. Examples are used to demonstrate the concepts and solution methods. Graphics are used in these examples to facilitate understanding.

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, an algorithm for solving interval time-cost tradeoff transportation problemsis presented. In this problem, all the demands are defined as intervalto determine more realistic duration and cost. Mathematical methods can be used to convert the time-cost tradeoff problems to linear programming, integer programming, dynamic programming, goal programming or multi-objective linear programming problems for determining the optimum duration and cost. Using this approach, the algorithm is developed converting interval time-cost tradeoff transportation problem to the linear programming problem by taking into consideration of decision maker (DM).

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.

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.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

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.

A neural network approach to multiobjective and multilevel programming problems

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.