A Short Review of Fuzzy Linear Equation Systems

A Short Review of Fuzzy Linear Equation Systems

Hale Gonce Kocken (Yildiz Technical University, Turkey) and Inci Albayrak (Yildiz Technical University, Turkey)
DOI: 10.4018/978-1-4666-5888-2.ch687
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Background

Wide range of real world applications in many areas including financial engineering, scientific management and engineering technology are using linear equation systems for modeling and solving their respective problems. In many situations, the estimation of the system parameters is imprecise because of the lack of exact information, environmental and changeable economic conditions, etc. The uncertainty of parameters involved in the process of actual mathematical modeling is often represented by fuzzy numbers, so it is important to develop mathematical models that would appropriately treat fuzzy linear equation systems.

One of the most important applications of linear equation systems to electrical engineering is to analyze electronic circuits that cannot be described using the rules for resistors in series or parallel. The goals are to calculate the current flowing in each branch of the circuit and to calculate the voltage at each node of the circuit, which are known Loop Current and Nodal Voltage Analysis, respectively. Owing to environmental conditions, tolerance in the elements and noise, these analyses can be modeled in the form of fuzzy linear equation systems.

Another important application of linear equation systems is input-output analysis developed by V. Leontief. Input-output analysis is an extremely effective tool used in more than 70 countries over the world for manufacturing processes optimization, economy condition improvement and intersectors costs allocation analysis. Considering dynamic nature of economics, it can be easily stated that extending this analysis to fuzzy version of linear equation system is significant.

There is a vast literature on the investigation of solutions for fuzzy linear equation systems. Early works in the literature are on to linear equation systems whose coefficient matrix is crisp and the right hand vector is fuzzy, that is known as Fuzzy Linear Equation System (FLS). The crispness of the coefficient matrix makes the modeling of real life problems restricted. Linear systems, whose all the parameters i.e. both coefficient matrix and right hand vector are fuzzy, are named Fully Fuzzy Linear Equation System (FFLS). The main intend of FFLS is to widen the scope of FLS in scientific applications by removing the crispness assumption on the entries of coefficient matrix. Besides FLS and FFLS, there exist the dual forms of both systems in the literature.

Generally, both FLS and FFLS are handled under two main headings: square (n x n) and nonsquare (m x n) forms. Most of the works in the literature deal with square form. Fuzzy elements of these systems can be taken as triangular, trapezoidal or generalized fuzzy numbers in general or parametric form. While triangular fuzzy numbers are widely used in earlier works, trapezoidal fuzzy numbers are neglected for a long time. Besides, there exist lots of works using the parametric and level cut representation of fuzzy numbers. Another classification for FFLS can be made also depending on whether FFLS has sign restrictions on its parameters. Having sign restrictions for FFLS means that all parameters of FFLS are assumed as positive. Since the parameters are assumed as positive in the most of the papers, further work is needed for FFLS with arbitrary (no restrictions on sign) fuzzy numbers.

Key Terms in this Chapter

Fully Fuzzy Linear Equation System: A linear equation system whose all parameters are fuzzy.

Linear Equation System: A set of linear equations involving the same set of variables.

Iterative Methods: A method to solve a problem by generating a sequence of approximate solutions progressively.

Fuzzy Linear Equation System: A linear equation system involving some fuzzy parameters and variables.

Solution of a System: An assignment of numbers to the variables such that all equations are satisfied.

Fuzziness: A way of handling the uncertainty.

Direct Methods: A method to solve a problem by a finite sequence of operations.

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