Interactive fuzzy linear programming and an application sample at a textile firm

The purpose of this study is to examine Interactive Fuzzy Linear Programming (IFLP) model by using Zimmermann, Werners, Chanas and Verdegay’s approaches that provide best decision-making under fuzzy environments. In this study, it is used the method which can model the fuzzy structure of the real world and which operates with the decision maker interactively, which aims at obtaining the best solution by continuing this interactiveness in the solution process, which includes fuzziness with more realistic approach to the system. It is showed that the importance of fuzziness concept for IFLP problems, how it is applied on real-world problems and its effects.     

On fuzzy linear programming

Fuzzy multi-objective and fuzzy Goal Programming are discussed in connection with several membership functions which are used to transform the original problem into three equivalent linear programming problems. Existence and uniqueness theorems are given. Fuzzy duality is presented, and an extension of the initial fuzzy problem arises immediately from it.     

A kind of fuzzy linear programming problems based on interval-valued fuzzy sets

Abstract. The objective of this paper is to deal with a kind of fuzzy linear programming problem based on interval-valued fuzzy sets (IVFLP) through the medium of procedure that turns IVFLP into parametric linear programming via the mathematical programming. Some useful results for the benefit of solving IVFLP are expounded and proved,developed and discussed. Furthermore,that the proposed techniques in this paper allow the decision-maker to assign a different degree of importance can provide a useful way to efficiently help the decision-maker make their decisions.     

Solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution

This paper discusses full fuzzy linear programming (FFLP) problems of which all parameters and variable are triangular fuzzy numbers. We use the concept of the symmetric triangular fuzzy number and introduce an approach to defuzzify a general fuzzy quantity. For such a problem, first, the fuzzy triangular number is approximated to its nearest symmetric triangular number, with the assumption that all decision variables are symmetric triangular. An optimal solution to the above-mentioned problem is a symmetric fuzzy solution. Every FLP models turned into two crisp complex linear problems; first a problem is designed in which the center objective value will be calculated and since the center of a fuzzy number is preferred to (its) margin. With a special ranking on fuzzy numbers, the FFLP transform to multi objective linear programming (MOLP) where all variables and parameters are crisp.     

On the fuzzy solution of LR fuzzy linear systems

In this paper, the solution of fuzzy linear system (FLS) is investigated based on a 1$1$-level expansion. To this end, 1$1$-level of FLS is solved for calculating the core of fuzzy solution and then its spreads are obtained by solving an optimization problem with a special objective function. Using our proposed method, if the computed solution satisfies the FLS, it can be as exact or approximated solution. Finally, the existence of solution of FLS is proved in details and some numerical examples are solved to illustrate the accuracy and capability of the method.     

Possibilistic linear regression analysis for fuzzy data

Fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined. Recently, it has become important to deal with fuzzy data in connection with expert knowledge. Three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data. Since our formulations can be reduced to linear programming problems, the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters. This approach can be regarded as a fuzzy interval analysis in a fuzzy environment.     

Supplier selection and order allocation problem using a two-phase fuzzy multi-objective linear programming

The aim of this paper is to solve a supplier selection problem under multi-price level and multi-product using interactive two-phase fuzzy multi-objective linear programming (FMOLP) model. The proposed model attempts to simultaneously minimize total purchasing and ordering costs, a number of defective units, and late delivered units ordered from suppliers. The piecewise linear membership functions are applied to represent the decision maker’s fuzzy goals for the supplier selection and order allocation problem, and can be resulted in more flexibility via an interactive decision-making process. To demonstrate effectiveness of the proposed model, results of applying the proposed model are shown by a numerical example. The analytical results show that the proposed approach is effective in uncertain environments and provide a reliable decision tool for integrated multi-objective supplier selection problems.     

On the algebraic solution of fuzzy linear systems based on interval theory

In this paper, fuzzy linear systems involving a crisp square matrix and a fuzzy right-hand side vector are considered. A new approach to solve such systems based on interval theory and the new concept “interval inclusion linear system” is proposed. Also, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Numerical examples are given to illustrate the efficiency of the proposed method.     

Linear programming with fuzzy parameters: An interactive method resolution

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.     

A dual approach to solve the fuzzy linear programming problem

A concept of fuzzy objective based on the Fuzzification Principle is presented. In accordance with this concept, the Fuzzy Linear Mathematical Programming problem is easily solved. A relationship of duality among fuzzy constraints and fuzzy objectives is given. The dual problem of a Fuzzy Linear Programming problem is also defined.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

A geometric approach for solving fuzzy linear programming problems

In this paper we first recall some definitions and results of fuzzy plane geometry, and then introduce some definitions in the geometry of two-dimensional fuzzy linear programming (FLP). After defining the optimal solution based on these definitions, we use the geometric approach for obtaining optimal solution(s) and show that the algebraic solutions obtained by Zimmermann method (ZM) and our geometric solutions are the same. Finally, numerical examples are solved by these two methods.     

A class of linear differential dynamical systems with fuzzy matrices

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the 2-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.     

Interactive multiobjective fuzzy random linear programming: Maximization of possibility and probability

This paper considers multiobjective linear programming problems with fuzzy random variables coefficients. A new decision making model is proposed to maximize both possibility and probability, which is based on possibilistic programming and stochastic programming. An interactive algorithm is constructed to obtain a satisficing solution satisfying at least weak Pareto optimality.     

L.P.L. a fuzzy programming language: 1. Syntactic aspects

The L.P.L. language (Linguistic oriented Programming Language) that we shall present in this paper and in a forthcoming one is a new language aiming at an implementation of the concepts of the fuzzy sets theory. In the first paper, after an introduction to the basic concepts. we shall describe the syntactic aspects of L.P.L., i.e., the data declarations structure, the statements structure, the general structure of L.P.L. models. The second paper will deal first with the basic semantic and the semantics of logical expressions, then the semantics of basic statements as well as the semantics of the control structure of programs.     

On the manpower allocation within matrix organization: A fuzzy linear programming approach

In this study, we provide a framework of fuzzy linear programming model for the function management division dealing with manpower allocation problem within matrix organization. The proposed model reveals how the function management division seeks a minimized cost and satisfies with the requirement of subordinate functional departments as far as possible under limited manpower and project budget. Numerical example illustrates that an adjustable parameter in the proposed model can be employed by the function management division to generate many different alternatives and grasp the degree of satisfaction of each subordinate functional department.     

Some duality results on linear programming problems with symmetric fuzzy numbers

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.     

Solving fuzzy queueing decision problems via a parametric mixed integer nonlinear programming method

This paper proposes a mathematical programming method to construct the membership functions of the fuzzy objective value of the cost-based queueing decision problem with the cost coefficients and the arrival rate being fuzzy numbers. On the basis of Zadeh’s extension principle, three pairs of mixed integer nonlinear programs (MINLP) parameterized by the possibility level α are formulated to calculate the lower and upper bounds of the minimal expected total cost per unit time at α, through which the membership function of the minimal expected total cost per unit time of the fuzzy objective value is constructed. To provide a suitable optimal service rate for designing queueing systems, the Yager’s ranking index method is adopted. Two numerical examples are solved successfully to demonstrate the validity of the proposed method. Since the objective value is completely expressed by a membership function rather than by a crisp value, it conserves the fuzziness of the input information, thus more information is provided for designing queueing systems. The successful extension of queueing decision models to fuzzy environments permits queueing decision models to have wider applications in practice.