A method for solving linear programming with interval-valued trapezoidal fuzzy variables

2018 ◽  
Vol 52 (3) ◽  
pp. 955-979 ◽  
Author(s):  
Ali Ebrahimnejad
Keyword(s):  
Linear Programming ◽  
Fuzzy Numbers ◽  
Auxiliary Problem ◽  
Coefficient Matrix ◽  
Uncertain Parameters ◽  
Fuzzy Variables ◽  
Right Hand ◽  
Fuzzy Cost ◽  
The Right ◽  
Interval Valued

An efficient method to handle the uncertain parameters of a linear programming (LP) problem is to express the uncertain parameters by fuzzy numbers which are more realistic, and create a conceptual and theoretical framework for dealing with imprecision and vagueness. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand side, and/or the elements of the coefficient matrix. The aim of this article is to introduce a formulation of FLP problems involving interval-valued trapezoidal fuzzy numbers for the decision variables and the right-hand-side of the constraints. We propose a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To do this, we first define an auxiliary problem, having only interval-valued trapezoidal fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. It is demonstrated that study of LP problems with interval-valued trapezoidal fuzzy variables gives rise to the same expected results as those obtained for LP with trapezoidal fuzzy variables.

Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients

2013 ◽  
pp. 40-63
Author(s):  
Ali Ebrahimnejad ◽  
Seyed Hadi Nasseri ◽  
Sayyed Mehdi Mansourzadeh
Keyword(s):  
Linear Programming ◽  
Auxiliary Problem ◽  
Simplex Algorithm ◽  
Upper Bounds ◽  
Bounded Linear ◽  
Fuzzy Variables ◽  
Decision Variables ◽  
Bounded Variables ◽  
Linear Programming Problems ◽  
Fuzzy Cost

In most practical problems of linear programming problems with fuzzy cost coefficients, some or all variables are restricted to lie within lower and upper bounds. In this paper, the authors propose a new method for solving such problems called the bounded fuzzy primal simplex algorithm. Some researchers used the linear programming problem with fuzzy cost coefficients as an auxiliary problem for solving linear programming with fuzzy variables, but their method is not efficient when the decision variables are bounded variables in the auxiliary problem. In this paper the authors introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus.

Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients

2011 ◽  
Vol 2 (1) ◽  
pp. 96-120 ◽  
Author(s):  
Ali Ebrahimnejad ◽  
Seyed Hadi Nasseri ◽  
Sayyed Mehdi Mansourzadeh
Keyword(s):  
Linear Programming ◽  
Auxiliary Problem ◽  
Simplex Algorithm ◽  
Lower And Upper Bounds ◽  
Bounded Linear ◽  
Fuzzy Variables ◽  
Decision Variables ◽  
Bounded Variables ◽  
Linear Programming Problems ◽  
Fuzzy Cost

In most practical problems of linear programming problems with fuzzy cost coefficients, some or all variables are restricted to lie within lower and upper bounds. In this paper, the authors propose a new method for solving such problems called the bounded fuzzy primal simplex algorithm. Some researchers used the linear programming problem with fuzzy cost coefficients as an auxiliary problem for solving linear programming with fuzzy variables, but their method is not efficient when the decision variables are bounded variables in the auxiliary problem. In this paper the authors introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus.

scholarly journals A comparative study on interval arithmetic operations with intuitionistic fuzzy numbers for solving an intuitionistic fuzzy multi–objective linear programming problem

2017 ◽  
Vol 27 (3) ◽  
pp. 563-573 ◽  
Author(s):  
Rajendran Vidhya ◽  
Rajkumar Irene Hepzibah
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Method ◽  
Arithmetic Operations ◽  
Intuitionistic Fuzzy Numbers ◽  
Multi Objective ◽  
Intuitionistic Fuzzy ◽  
Interval Valued

AbstractIn a real world situation, whenever ambiguity exists in the modeling of intuitionistic fuzzy numbers (IFNs), interval valued intuitionistic fuzzy numbers (IVIFNs) are often used in order to represent a range of IFNs unstable from the most pessimistic evaluation to the most optimistic one. IVIFNs are a construction which helps us to avoid such a prohibitive complexity. This paper is focused on two types of arithmetic operations on interval valued intuitionistic fuzzy numbers (IVIFNs) to solve the interval valued intuitionistic fuzzy multi-objective linear programming problem with pentagonal intuitionistic fuzzy numbers (PIFNs) by assuming differentαandβcut values in a comparative manner. The objective functions involved in the problem are ranked by the ratio ranking method and the problem is solved by the preemptive optimization method. An illustrative example with MATLAB outputs is presented in order to clarify the potential approach.

scholarly journals Postoptimization of the model of water supply for urban and industrial agglomeration

2018 ◽  
Vol 23 ◽  
pp. 00035
Author(s):  
Jacek Wawrzosek ◽  
Szymon Ignaciuk
Keyword(s):  
Linear Programming ◽  
Water Supply ◽  
Programming Models ◽  
Industrial Agglomeration ◽  
Infinitely Many Solutions ◽  
Nodal Solutions ◽  
Parametric Linear Programming ◽  
Right Hand ◽  
The Right

A case study of the tools used by an analyst of the economic aspects of the operation of the water supply network has been undertaken in this paper. All issues discussed here are formulated by using degenerated linear programming models ( PL ). Below, it is noted that the linear dependence of binding constraints ( CO ) distorts standard postoptimization procedures in PL. This observed fact makes postoptimization analysis mostly unhelpful for an average analyst due to problems with the int erpretation of ambiguous sensitivity reports which are obtained from popular computer packages. In standard postoptimization methods, changes to single parameters of the right-hand vector CO are analyzed or referred to parametric linear programming that unfortunately requires prior knowledge of mathematically and economically justified vectors of changes of right-hand sides CO. Therefore, it is suggested that modifications are introduced to some of the postoptimization procedures in this work. For issues in the field of hydrology, the following were presented: interpretation and methods of generating justified vectors of changes of right-hand sides of limiting conditions. And so, the procedure of generating infinitely many solutions of the dual issue based on certain vectors orthogonal to the vector of right-hand sides of constraint conditions was demonstrated. Furthermore, the same orthogonal vectors were used to obtain nodal solutions of the dua0l model and the corresponding vectors of changes of the entire right-hand sides of the constraint conditions. Then, managerial interpretation was applied to this way of proceeding. The methods presented in the work serve to improve the functioning of the system of water supply.

scholarly journals Linear Programming and Fuzzy Optimization to Substantiate Investment Decisions in Tangible Assets

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 121
Author(s):  
Marcel-Ioan Boloș ◽  
Ioana-Alexandra Bradea ◽  
Camelia Delcea
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Hybrid Model ◽  
Fuzzy Numbers ◽  
Investment Decision ◽  
Fuzzy Optimization ◽  
Simplex Algorithm ◽  
Fuzzy Variables ◽  
Decision Variables ◽  
Tangible Assets

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

scholarly journals Type-2 Duality Triangular Fuzzy Fractional Transportation Problem using Goal Programming Technique

2020 ◽  
Vol 8 (6) ◽  
pp. 1295-1302
Keyword(s):  
Goal Programming ◽  
Transportation Problem ◽  
Fuzzy Numbers ◽  
Programming Technique ◽  
Intuitionistic Fuzzy Numbers ◽  
Fuzzy Variables ◽  
Proposed Model ◽  
Fuzzy Parameters ◽  
Interval Valued

The authors focused goal programming technique for solving type-2 duality fuzzy fractional transportation problem by using interval-valued triangular intuitionistic fuzzy numbers. The proposed method solves three types of models in which variables are taken as type-2 fuzzy variables which have spring up to the fuzzy fractional transportation problem. In these models, duality is applied and a particular type of non-linear membership functions are used to resolve duality fractional transportation problem including fuzzy parameters. A numerical example for examining the performance of the proposed model is envisaged here.

scholarly journals Comparing different ranking functions for solving fuzzy linear programming problems with fuzzy cost coefficients

2021 ◽  
Vol 4 (2) ◽  
pp. 3-17
Author(s):  
Betsabé Pérez Garrido ◽  
Szabolcs Szilárd Sebrek ◽  
Viktoriia Semenova
Keyword(s):  
Linear Programming ◽  
Objective Function ◽  
Fuzzy Numbers ◽  
Fuzzy Linear Programming ◽  
Ranking Functions ◽  
Statistical Software ◽  
Linear Programming Problems ◽  
Fuzzy Cost ◽  
The Cost ◽  
Cost Vector

In many applications of linear programming, the lack of exact information results in various problems. Nevertheless, these types of problems can be handled using fuzzy linear programming. This study aims to compare different ranking functions for solving fuzzy linear programming problems in which the coefficients of the objective function (the cost vector) are fuzzy numbers. A numerical example is introduced from the field of tourism and then solved using five ranking functions. Computations were carried out using the FuzzyLP package implemented in the statistical software R.

scholarly journals A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1302
Author(s):  
Hong-Xiu Zhong ◽  
Xian-Ming Gu ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Coefficient Matrix ◽  
Complex Symmetric Linear Systems ◽  
Right Hand ◽  
Orthogonality Conditions ◽  
Free Block ◽  
Complex Symmetric ◽  
The Right ◽  
Conjugate Residual

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

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