Fuzzy Linear Programming Problems Research Articles

Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation

Solving fully fuzzy linear programming problems with flexible constraints based on a new order relation

Application of Multi-Choice Fuzzy Linear Programming Problem to a Garment Manufacture Company

In this paper, we consider a fuzzy multi-choice linear programming problem where some of the parameters and decision variables are trapezoidal type fuzzy numbers. In order to defuzzify a general fuzzy quantity the concept of nearest trapezoidal fuzzy number is introduced. By assuming all the decision variables as trapezoidal fuzzy number, the objective function and the left hand side of constraints are approximated to their nearest trapezoidal fuzzy number. Interpolating polynomials are formulated for all multi-choice type parameters. Multi-choice type parameters are replaced by polynomials with integer variables. Then an equivalent multi-objective non linear programming problem is established. First and second objective functions represent left and right modal values of the trapezoidal type fuzzy number, where as the third and fourth objective functions represent the left and right spreads of the trapezoidal type fuzzy number. By applying lexicographic method optimal solution is obtained. In addition, a case study on a garment manufacture company is presented to demonstrate the solution procedure.

Pareto-optimal solutions for multi-objective flexible linear programming

The aim of this paper is twofold. Firstly, to define a solution concept of Pareto-optimality for a multi-objective flexible linear programming (MOFLP) problem (or multi-objective fuzzy linear programming problem) and design a method to extract a Pareto-optimal solution of MOFLP problem from the set of optimal solutions of equivalent optimization problem formulated by Dubey and Mehra (2013). Secondly, to extend this study to multi-objective linear programming problem involving hard and flexible constraints with interval uncertainty. A flexible constraint with interval uncertainty generalizes a flexible constraint by allowing preferences to be expressed in the form of intervals. An optimistic-pessimistic approach is proposed to solve multi- objective flexible linear programming with interval uncertainty (MOFLPIU) using an interval-valued fuzzy set representation and the Hurwicz optimism-pessimism criterion.

Decision Making Approach to Fuzzy Linear Programming (FLP) Problems with Post Optimal Analysis

This paper finds solutions to the fuzzy linear program where some parameters are fuzzy numbers. In practice, there are many problems in which all decision parameters are fuzzy numbers, and such problems are usually solved by either probabilistic programming or multi objective programming methods. Unfortunately all these methods have shortcomings. In this paper, using the concept of comparison of fuzzy numbers, the author introduces a very effective method for solving these problems. This paper extends linear programming based problem in fuzzy environment. With the problem assumptions, the optimal solution can still be theoretically solved using the simplex based method. To handle the fuzzy decision variables can be initially generated and then solved and improved sequentially using the fuzzy decision approach by introducing robust ranking technique. The model is illustrated with an application and a post optimal analysis approach is obtained. The proposed procedure was programmed with MATLAB (R2009a) version software, the four dimensional slice diagram is represented to the application. Finally, numerical example is presented to illustrate the effectiveness of the theoretical results, and to gain additional managerial insights.

Branch-and-price algorithm for fuzzy integer programming problems with block angular structure

This paper deals with fuzzy integer linear programming problems with block angular structure in which the fuzzy constraints are simplified by using possibility and necessity relations. This main fuzzy problem is efficiently decomposed and is solved by a branch-and-price algorithm. In the nodes of the branch-and-price tree, the linear relaxation of the problem is solved by applying a column generation method. Also, the relationship between the optimal solutions of this problem under possibility and necessity relations is derived. To show the validation of the proposed algorithm, some results are proved. In addition, the application of this algorithm is illustrated on fuzzy multicommodity flow problem. For this case, a new branching scheme is proposed to preserve the network structure of the subproblems which are produced in the column generation method. Some examples are solved and their results are compared with the previous works. Also, the results of the proposed algorithm are reported on some large-scale benchmark instances.

An analysis of fully fuzzy linear programming with fuzzy decision variables through logistics network design problem

Recently, there is a growing attention by the researchers to solve and interpret the analysis of fully fuzzy linear programming problems in which all of the parameters as well as the decision variables are considered as fuzzy numbers. Under a fully uncertain environment where all of the data are stated as fuzzy, presenting the reasonable range of values for the decision variables may be comparatively better than the currently available crisp solutions so as to provide ranges of flexibility to decision makers. However, there is still a scarcity of solution methodologies on fuzzy mathematical programs with fuzzy decision variables. Based on this motivation, a new parametric method which is mainly based on α-cut representation of fuzzy intervals is proposed in this paper by incorporating the decision maker's attitude toward risk. In order to illustrate validity and practicality of the proposed method, it is applied to a generic reverse logistics network design model including fuzzy decision variables. To the best of our knowledge, this is the first study in the literature which presents fuzzy efficient solutions and analysis for a fully fuzzy reverse logistics network design problem with fuzzy decision variables. The provided solutions by the proposed method are also compared to the available solution methodologies from the literature in terms of computational efficiency, solution quality and ease of use. By using the proposed method, the decision makers can be supported by yielding fuzzy efficient solutions under different uncertainty levels and risk attitudes. The computational results have also shown that more reliable and necessarily precise solutions can be generated by the proposed method for a risk-averse decision maker.

ON THE RATIO OF FUZZY NUMBERS – EXACT MEMBERSHIP FUNCTION COMPUTATION AND APPLICATIONS TO DECISION MAKING

In the present paper, we propose a new approach to solving the full fuzzy linear fractional programming problem. By this approach, we provide a tool for making good decisions in certain problems in which the goals may be modelled by linear fractional functions under linear constraints; and when only vague data are available. In order to evaluate the membership function of the fractional objective, we use the α-cut interval of a special class of fuzzy numbers, namely the fuzzy numbers obtained as sums of products of triangular fuzzy numbers with positive support. We derive the α-cut interval of the ratio of such fuzzy numbers, compute the exact membership function of the ratio, and introduce a way to evaluate the error that arises when the result is approximated by a triangular fuzzy number. We analyse the effect of this approximation on solving a full fuzzy linear fractional programming problem. We illustrate our approach by solving a special example – a decision-making problem in production planning.

Progress on Fuzzy Mathematical Programming: A personal perspective

Fuzzy Linear Programming is among the best and most studied topics in the Fuzzy Sets and Systems area. In this paper, the author describes the main developments, results and solution methods achieved in this topic throughout the last 35 years, from a personal point of view. Short descriptions on different Fuzzy Linear Programming problems are provided and future research lines are briefly pointed out.

A revised simplex method of solving degenerate fuzzy number linear programming problems

A revised simplex method of solving degenerate fuzzy number linear programming problems

Evolving a linear programming technique for MAGDM problems with interval valued intuitionistic fuzzy information

It is believed that multi attribute decision making problem is an ill-defined and unstructured problem. This difficulty intensifies while considering the uncertainty of decision makers’ information about the problem. In recent years, interval valued intuitionistic fuzzy sets (hereafter IVIFs) as a generalization of ordinal fuzzy sets, became a well-known and widely applied framework for dealing with uncertainty of decision making problems. However, the mathematical programming aspects of interval valued intuitionistic fuzzy sets besides their applications in decision making problems are neglected. To reinforce the mathematical programming approach in IVIF environment, an IVIF multi attribute group decision making problem is formulated as a linear programming model. Using a variable transformation and the notion of aggregation operators, the proposed model is transformed into an equivalent linear programming model solvable by common approaches. Application of the proposed method is represented in a group decision making problem and the results are compared with similar methods, proving the compatibility of the proposed method with previous ones. The solid and understandable logic with computational easiness are the main advantages of the proposed method. Solving interval valued intuitionistic fuzzy linear programming problems can be applied lucratively in other problems being formulated in this context.

A Note on Solving a Fully Intuitionistic Fuzzy Linear Programming Problem based on Sign Distance

A Note on Solving a Fully Intuitionistic Fuzzy Linear Programming Problem based on Sign Distance

A Fully Fuzzy DEA Approach for Cost and Revenue Efficiency Measurements in the Presence of Undesirable Outputs and Its Application to the Banking Sector in India

This paper extends the conventional cost efficiency (CE) and revenue efficiency (RE) models to fully fuzzy environments to account for real situations where input–output data and their corresponding prices are not known precisely. Owing to the importance of the presence of undesirable outputs in the production process, these are also incorporated into the production technologies of the proposed models. This paper endeavours to propose fully fuzzy CE (FFCE) and fully fuzzy RE (FFRE) models where input–output data and prices include uncertainty of fuzzy forms, in particular, of triangular membership forms. Further, the concepts of fully fuzzy linear programming problems (FFLPPs) and linear ranking functions are used to transform FFCE and FFRE models into the crisp linear programming problems (LPPs), and to evaluate fuzzy CE (FCE) and fuzzy RE (FRE) measures of the decision-making units as triangular fuzzy numbers. Moreover, the proposed models are compared with some existing approaches and are also illustrated with an application to the banking sector in India for proving their acceptability and effectiveness in real-world systems.

Solving a Fuzzy Linear Programming Problem with Triangular Membership Function

This paper deals with the linear problem under the equality constraints whose right hand side parameters are asymmetric triangular fuzzy numbers. The right hand side values are defuzzified by using the triangular membership functions. Each of these triangular functions is algebraically replaced with two piecewise linear functions. And the desired objective value is also a fuzzy number. The membership function of the objective is assumed to be piecewise linear. Then, according to Zimmermann’s approach, by using these membership functions and the max-min rule, the fuzzy linear problem is transformed into the crisp linear programming problem. Consequently a fuzzy solution is obtained by solving this linear programming problem.

A New Method For Solving Fuzzy Linear Programming Problems Based on The Fuzzy Linear Complementary Problem (FLCP)

Linear programming (LP) is one of the most widely used methods in the area of optimization. Dealing with the formulation of LP problems, the parameters of objective function and constraints should be assigned by experts. In most cases, precise data have been used, but in most of the real-life situations, these parameters are imprecise and ambiguous. In order to deal with the problem of ambiguity and imprecision, fuzzy numbers can be appropriate. By replacing precise numbers with fuzzy numbers, LP problems change to fuzzy linear programming (FLP) problems. So FLPs can be considered as a broader category in comparison to LPs. Considering the above-mentioned points, FLP problems play an important rule in operational researches hence there is a need to investigate these problems. In this paper, a new method for solving the FLP problems is presented in which the coefficients of the objective function and the values of the right-hand side are represented by fuzzy numbers, while the elements of the coefficient matrix are represented by real numbers. To this end, we develop the Karush–Kuhn–Tucker (KKT) optimality conditions for FLP problems. Then, every FLP problem is converted to a fuzzy linear complementary problem (FLCP) by considering KKT conditions. In order to solve the FLCP problems, ranking functions and Lemke’s algorithm are used. Consequently, the solution to primal and dual problems of FLP is obtained. In addition to simplicity in calculations and feasibility, this method solves the primal and dual problems of FLP simultaneously. In order to illustrate the proposed method, some numerical examples are considered.

A hybrid method for solving fuzzy semi-infinite linear programming problems

Abilities of fuzzy models in more adaptations with the real world phenomena causes that in this paper, we remodel semi-infinite linear programming problems as fuzzy problems which contained the crisp objective function and the infinite number of fuzzy constraints. Then, we introduce a new hybrid solution method for these fuzzy semi-infinite linear programming problems. This solution technique is a kind of cutting plane algorithm in which its sub-problems were solved by using the Zimmermann-method. Convergence of the presented algorithm is proved, and some numerical test examples are given and also the obtained results are compared.

A fuzzy robust linear programming problem with hybrid variables

This paper proposes a fuzzy robust linear programming (FRLP) problem where the coefficients of objective function and constraints contain both fuzziness and randomness. It is shown that the original FRLP problem is transformed into deterministic linear programming problems by new methods based on the chance–constrained programming. In the presented new methods, we use an extended chance–constrained programming based on possibility and necessity measures for fuzzy stochastic constraints in FRLP problem. Finally, a numerical example is also given to show the applicability of the methods discussed in this paper.

FFLP problem with symmetric trapezoidal fuzzy numbers

Article history: Received December 10, 2014 Received in revised format: January 4, 2015 Accepted January 5, 2015 Available online January 5 2015 The most popular approach for solving fully fuzzy linear programming (FFLP) problems is to convert them into the corresponding deterministic linear programs. Khan et al. (2013) [Khan, I. U., Ahmad, T., & Maan, N. (2013). A simplified novel technique for solving fully fuzzy linear programming problems. Journal of Optimization Theory and Applications, 159(2), 536546.] claimed that there had been no method in the literature to find the fuzzy optimal solution of a FFLP problem without converting it into crisp linear programming problem, and proposed a technique for the same. Others showed that the fuzzy arithmetic operation used by Khan et al. (2013) had some problems in subtraction and division operations, which could lead to misleading results. Recently, Ezzati et al. (2014) [Ezzati, R., Khorram, E., & Enayati, R. (2014). A particular simplex algorithm to solve fuzzy lexicographic multi-objective linear programming problems and their sensitivity analysis on the priority of the fuzzy objective functions. Journal of Intelligent and Fuzzy Systems, 26(5), 2333-2358.] defined a new operation on symmetric trapezoidal fuzzy numbers and proposed a new algorithm to find directly a lexicographic/preemptive fuzzy optimal solution of a fuzzy lexicographic multiobjective linear programming problem by using new fuzzy arithmetic operations, but their model was not fully fuzzy optimization. In this paper, a new method, by using Ezzati et al. (2014)’s fuzzy arithmetic operation and a fuzzy version of simplex algorithm, is proposed for solving FFLP problem whose parameters are represented by symmetric trapezoidal fuzzy number without converting the given problem into crisp equivalent problem. By using the proposed method, the fuzzy optimal solution of FFLP problem can be easily obtained. A numerical example is provided to illustrate the proposed method. Growing Science Ltd. All rights reserved. 5 © 201

Intuitionistic fuzzy non linear programming problem: Modeling and optimization in manufacturing systems

This paper develops an approach for solving non-linear programming problem in intuitionistic fuzzy environment. Since, fuzziness may occur in some or all parameters of a system, we discuss both situations in this study. We transform the intuitionistic fuzzy model into a fuzzy model using the concept of component wise optimization. We propose a non-linear membership function and utilize it to convert the fuzzy model into a crisp model. For application, we consider some manufacturing systems having some or all intuitionistic fuzzy parameters. After modeling these systems, we apply our methodology for justification.

The Knowledge of Expert Opinion in Intuitionistic Fuzzy Linear Programming Problem

In real life, information available for certain situations is vague and such uncertainty is unavoidable. One possible solution is to consider the knowledge of experts on the parameters involved as intuitionistic fuzzy data. We examine a linear programming problem in which all the coefficients are intuitionistic in nature. An approach is presented to solve an intuitionistic fuzzy linear programming problem. In this proposed approach, a procedure for allocating limited resources effectively among competing demands is developed. An example is given to highlight the illustrated study.

A Parametric Approach to Solve Multi Objective Fuzzy Linear Programming Problem

A new method has been proposed by Senthilkumar and Rajendran (1) for solving fuzzy linear programming problem in parametric form. This paper extends this method for solving multi-objective fuzzy linear programming problem with fuzzy variables in parametric form. To obtain optimal solution, problem is converted into two auxiliary crisp linear programming problems corresponding to each objective functions. A numerical example is given to check the feasibility of the proposed method.