Fuzzy Multi-objective Linear Programming Problem Research Articles

Modified new operations for triangular intuitionistic fuzzy numbers

Intuitionistic fuzzy sets (IFS) are a generalization of the concept of fuzzy set. In standard intuitionistic fuzzy arithmetic operations, we have some grievances in subtraction and division operations. In this paper, modified new operations for subtraction and division on triangular intuitionistic fuzzy numbers (TIFNS) are defined. Finally an illustrative example for solving Intuitionistic fuzzy multi-objective linear programming problem (IFMOLPP) using these modified operators is provided.

Interactive Decision Making for Fuzzy Random Multiobjective Linear Programming Problems with Variance Covariance Matrices Through Fractile Optimization

In this paper, we focus on fuzzy random multiobjective linear programming problems with variance covariance matrices through fractile optimization, and propose an interactive decision making method to obtain a satisfactory solution. In the proposed method, it is assumed that the decision maker has fuzzy goals for not only permissible probability levels but also the corresponding objective functions. Such fuzzy goals are quantified by eliciting the corresponding membership functions. Using the fuzzy decision, such two kinds of membership functions are integrated, andDf-Pareto optimal solution concept is defined in the integrated membership space. By using the bisection method and the convex programming technique, a satisfactory solution is obtained from among aDf-Pareto optimal solution set through the interaction with the decision maker.

Ranking of Hexagonal Fuzzy Numbers for Solving Multiobjective Fuzzy Linear Programming Problem

Ranking of Hexagonal Fuzzy Numbers for Solving Multiobjective Fuzzy Linear Programming Problem

Algorithm for Solving Fuzzy Multiobjective Linear Fractional Programming Problem by Additive Weighted Method

In this paper attention has been paid to the study of multiobjective linear fractional programming problem (MOLFPP) by using fuzzy set theoretic approach. In this approach, MOLFPP is transformed into multiobjective linear programming problem (MOLPP) by suitable transformation. In algorithm-I, MOLFPP is transformed into MOLPP by using fuzzy set theory and the pareto optimal solution of the transformed MOLPP is obtained by applying Zimmermann’s min-operator model and simplex method. Further we have used additive weighted method to modify the above approach. Algorithm-II has been presented to find the pareto optimal solution of MOLFPP by applying additive weighted method. To demonstrate the applicability of the proposed approach, one numerical example is solved to find the pareto optimal solution by applying this two algorithms.

An interactive fuzzy satisficing method for random fuzzy multiobjective linear programming problems through fractile criteria optimization with possibility

This paper considers multiobjective linear programming problems where each coefficient of the objective functions isexpressed by a random fuzzy variable. A new decision making model is proposed by incorporating the concept of fractilecriteria optimization into a possibilistic programming model. An interactive fuzzy satisficing method is presented forderiving a satisficing solution for a decision maker efficiently by updating the reference membership levels. In theproposed method, it is shown that the transformed deterministic problems for obtaining Pareto optimal solutions can besolved by using some convex programming techniques. An illustrative numerical example is provided to clarify theproposed method.

A bipolar approach in fuzzy multi-objective linear programming

The traditional frameworks for fuzzy linear optimization problems are inspired by the max–min model proposed by Zimmermann using the Bellman–Zadeh extension principle to aggregate all the fuzzy sets representing flexible (fuzzy) constraints and objective functions together. In this paper, we propose an alternative approach to model fuzzy multi-objective linear programming problems (FMOLPPs) from a perspective of bipolar view in preference modeling. Bipolarity allows us to distinguish between the negative and the positive preferences. Negative preferences denote what is unacceptable while positive preferences are less restrictive and express what is desirable. This framework facilitate a natural fusion of bipolarity in FMOLPPs. The flexible constraints in a fuzzy multi-objective linear programming problem (FMOLPP) are viewed as negative preferences for describing what is somewhat tolerable while the objective functions of the problem are viewed as positive preferences for depicting satisfaction to what is desirable. This approach enables us to handle fuzzy sets representing constraints and objective functions separately and combine them in distinct ways. After aggregating these fuzzy sets separately, coherence (or consistency) condition is used to define the fuzzy decision set.

Solving fuzzy multi-objective linear programming problems using deviation degree measures and weighted max–min method

This paper proposes a method for solving fuzzy multi-objective linear programming (FMOLP) problems where all the coefficients are triangular fuzzy numbers and all the constraints are fuzzy equality or inequality. Using the deviation degree measures and weighted max–min method, the FMOLP problem is transformed into crisp linear programming (CLP) problem. If decision makers fix the values of deviation degrees of two side fuzzy numbers in each constraint, then the δ-pareto-optimal solution of the FMOLP problems can be obtained by solving the CLP problem. The bigger the values of the deviation degrees are, the better the objectives function values will be. So we also propose an algorithm to find a balance-pareto-optimal solution between two goals in conflict: to improve the objectives function values and to decrease the values of the deviation degrees. Finally, to illustrate our method, we solve a numerical example.

Solution for Fuzzy Multi-Objective Linear Programming Problem

Solution for Fuzzy Multi-Objective Linear Programming Problem

Optimizing Human Diet Problem Based on Price and Taste Using Multi-Objective Fuzzy Linear Programming Approach

Low price and good taste of foods are regarded as two major factors for optimal human nutrition. Due to price fluctuations and taste diversity, these two factors cannot be certainly and determinately evaluated. This problem must be viewed from another perspective because of the uncertainty about the amount of nutrients per unit of foods and also diversity of people’s daily needs to receive them.This paper discusses human diet problem in fuzzy environment. The approach deals with multi-objective fuzzy linear programming problem using a fuzzy programming technique for its solution. By prescribing a diet merely based on crisp data, some ofthe realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Results indicated uncertainty about factors of nutrition diet -including taste and price, amount of nutrients and their intake- would affect diet quality, making the proposed diet more realistic.

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.

Taylor series approach to fuzzy multiobjective linear fractional programming

This paper presents the use of a Taylor series for fuzzy multiobjective linear fractional programming problems (FMOLFP). The Taylor series is a series expansion that a representation of a function. In the proposed approach, membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a Taylor series are unified. Thus, the problem is reduced to a single objective. Practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach.

A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

For modelling imprecise data the literature offers two different methods: either the use of probability distributions or the use of fuzzy sets. In our opinion these two concepts should be used in parallel or in combination, dependent on the real situation. Moreover, in many economic problems, the well-known probabilistic or fuzzy solution procedures are not suitable because the stochastic variables do not have a simple classical distribution and the fuzzy values are not fuzzy intervals. For example, in case of investment problems the coefficients may often be described by means of more complex distributions or more general fuzzy sets. In this case we propose to distinguish several scenarios and to describe the parameters of the different scenarios by fuzzy intervals. For solving such stochastic linear programs with fuzzy parameters we propose a new method, which retains the original objective functions dependent on the different states of nature. It is based on the integrated chance constrained program introduced by Klein Haneveld [On integrated chance constraints, in: Gargnano (Ed.), Stochastic Programming, Springer, Berlin, 1986, pp. 194–209] and the interactive solution process FULPAL, see Rommelfanger [Fuzzy Decision Support-Systeme - Entscheiden bei Unschärfe, second ed., Springer, Berlin, Heidelberg, 1994; FULPAL: an interactive method for solving multiobjective fuzzy linear programming problems, in: R. Slowinski, J. Teghem (Eds.), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Reidel Publishing Company, Dordrecht, 1990, pp. 279–299; FULPAL 2.0—an interactive algorithm for solving multicriteria fuzzy linear programs controlled by aspiration levels, in: D. Scheigert (Ed.), Methods of Multicriteria Decision Theory, Pfalzakademie Lamprecht, 1995, pp. 21–34; The advantages of fuzzy optimization models in practical use, Fuzzy Optim. Decision Making 3 (2004) 295–310] and Rommelfanger and Slowinski [Fuzzy linear programming with single or multiple objective functions, in: R. Slowinski (Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Norwell, MA, 1998, pp. 179–213]. An extensive numerical example illustrates the efficiency and the generality of the proposed new method.

Multiobjective fuzzy linear programming problems with fuzzy decision variables

In this paper, a multiobjective decision-making process is modeled by a multiobjective fuzzy linear programming problem with fuzzy coefficients for the objectives and the constraints. Moreover, the decision variables are linked together because they have to sum up to a constant. Most of the time, the solutions of a multiobjective fuzzy linear programming problem are compelled to be crisp values. Thus the fuzzy aspect of the decision is partly lost and the decision-making process is constrained to crisp decisions. We propose a method that uses fuzzy decision variables with a joint membership function instead of crisp decision variables. First, we consider lower-bounded fuzzy decision variables that set up the lower bounds of the decision variables. Then, the method is generalized to lower–upper-bounded fuzzy decision variables that set up also the upper bounds of the decision variables. The results are closely related to the special type of problem we are coping with, since we embed a sum constraint in the joint membership function of the fuzzy decision variables. Numerical examples are presented in order to illustrate our method.

A goal programming procedure for fuzzy multiobjective linear fractional programming problem

This paper presents a goal programming (GP) procedure for fuzzy multiobjective linear fractional programming (FMOLFP) problems. In the proposed approach, which is motivated by Mohamed (Fuzzy Sets and Systems 89 (1997) 215), GP model for achievement of the highest membership value of each of fuzzy goals defined for the fractional objectives is formulated. In the solution process, the method of variable change on the under- and over- deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using linear goal programming (LGP) methodology. The approach is illustrated by two numerical examples.

Sensitivity analysis in fuzzy multiobjective linear fractional programming problem

In this paper, we study measurement of sensitivity for changes of violations in the aspiration level for the fuzzy multiobjective linear fractional programming problem.

A fuzzy multiobjective program with quasiconcave membership functions and fuzzy coefficients

A method is proposed for solving a fuzzy multi-objective linear programming problem (FMP) with quasiconcave membership functions and fuzzy coefficients. The proposed method first expresses a piecewise function as the summation of absolute terms. Then we search for the interval where the optimal solution is allocated by finding the corresponding points with same value of membership functions. After that, the problem is solved by goal programming techniques. Comparing with other FMP methods, the proposed method does not need to add extra zero–one variables, to divide the original problem into several sub-problems, or transforming all original quasiconcave functions into concave functions. In addition, the proposed method could solve a FMP problem with fuzzy coefficients to obtain a solution closing to a global optimum.

Evolutionary algorithm solution to fuzzy problems: Fuzzy linear programming

In this paper we wish to find solutions to the fully fuzzified linear program where all the parameters and variables are fuzzy numbers. We first change the problem of maximizing a fuzzy number, the value of the objective function, into a multi-objective fuzzy linear programming problem. We prove that fuzzy flexible programming can be used to explore the whole undominated set to the multi-objective fuzzy linear program. An evolutionary algorithm is designed to solve the fuzzy flexible program and we apply this program to two applications to generate good solutions.

非対称型ファジィ多目的線形計画問題に対する二段階アプローチ

非対称型ファジィ多目的線形計画問題に対する二段階アプローチ

Neural network technique for fuzzy multiobjective linear programming

Neural Network(NN) is well-known as one of powerful computing tools to solve optimization problems. Due to the massive computing unit-neurons and parallel mechanism of neural network approach we can solve the large-scale problem efficiently and optimal solution can be gotten. In this paper, we intoroduce improvement of the two-phase approach for solving fuzzy multiobjectve linear programming problem with both fuzzy objectives and constraints and we propose a new neural network technique for solving fuzzy multiobjective linear programming problems. The procedure and efficiency of this approach are shown with numerical simulations.

角型構造の大規模多目的ファジィ線形計画問題に対する分解手法の有効性

In this paper, we focus on large-scale multiobjective fuzzy linear programming problems with the block angular structure and examine the efficiency of the Dantzig-Wolfe decomposition method in the interactive fuzzy satisficing method proposed by M. Sakawa et al. After overviewing the Dantzig-Wolfe decomposition method (1961) and the interactive fuzzy satisficing method, three-objective linear programming problems with 15 coupling constraints are considered in order to show the efficiency of the Dantzig-Wolfe decomposition method over the revised simplex method. Through a lot of computational experiments on workstation for numerical examples with both 50 and 200 variables, the advantages of the Dantzig-Wolfe decomposition method are discussed with respect to processing time and required memory storage.