 # A Short Review on Fuzzy System of Linear Equations Applications

Hale Gonce Kocken (Yildiz Technical University, Turkey) and Inci Albayrak (Yildiz Technical University, Turkey)
DOI: 10.4018/978-1-5225-9531-1.ch006

## Abstract

Fuzzy system of linear equations (FSLE) plays a major role in various areas such as operational research, physics, statistics, economics, engineering, and social sciences since the parameters of FSLE are not always exactly known and stable in real-life problems. This effect may follow the lack of exact information, changeable economic conditions, etc. Although there exist many review papers on the solution methods for FSLE, they are not based on the applications. This chapter has attempted to provide a short review on real-life applications of FSLE. In addition, for the common application areas, the fundamental models and the solution methods are presented considering the most cited and leading papers in the literature.
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## Introduction

The fuzzy sets has been investigated handling the relationship between fuzzy logic and probability for the perception-based final decision after it was introduced by Zadeh in 1960s. After development of fuzzy set theory, financial, engineering, electrical problems, etc. are reduced to linear equation systems using intervals or fuzzy values. It is also known that differential equations represent many problems in mathematical models and some of these problems can be rewritten as a system of linear equations for solving numerically.

FSLE has important role by using fuzzy modeling considering uncertainty in actual environment. System of linear equations has wide applications that are characterized by the lacking of the sufficient parameter values. FSLE arises in various areas such as social sciences, engineering, image processing etc. Fuzzy mathematics is a better tool than the crisp mathematics for modelling the real life problems. The fuzzy sets/numbers can represent data with indeterminate information. In a fuzzy set, fuzzy numbers are proven to be very suitable for expressing vague, imprecise, uncertain values (Kaufmann & Gupta, 1985; Hanss, 2005). Thus, to develop the mathematical models in many applications, the information obtained from decision makers may have partial determinacy due to a lack of data, measurement errors or limited knowledge. To overcome these ambiguities, many researchers used fuzzy numbers instead of crisp numbers.

In many FSLEs, some or all of the system parameters are vague or imprecise. If all system parameters (coefficient matrix and right hand side vector) are fuzzy numbers, then the corresponding system of linear equations are named Fully Fuzzy System of Linear Equations (FFSLE). The main intent of FFSLE is to widen the scope of FSLE in scientific applications by removing the crispness assumption on parameters. The most general structure of a FSLE is called Dual Fuzzy System of Linear Equations (DFSLE). Dual forms of FSLE and FFSLE provide modelling real life problems in a more flexible and realistic way. Thus, in this review paper, we focus on dual forms. This advantage of dual systems in the modeling increases the usability and importance of the scope in interdisciplinary research areas.

There is a limited number of works for DFSLE and Dual Fully Fuzzy System of Linear Equations (DFFSLE), while FSLE has a vast literature. The studies involving real life applications of dual systems are encountered rarely (Kocken & Albayrak, 2015). After Friedman et al. (1998) introduced a model to solve a FSLE whose coefficient matrix is crisp and the right-hand side column is an arbitrary fuzzy number vector, in Ma et al. (2000), the authors studied duality in fuzzy linear systems where and are real matrices. Ma et al. (2000) which can be considered as the first study in the area investigates the existence of the solutions. They claimed that this type of system cannot be replaced by a fuzzy linear system , since there is no element such that , for an arbitrary fuzzy number .

Muzzioli and Reynaerts (2006) studied the FFSLE of the dual form . A solution of this system is presented by converting the original DFFSLE into two crisp linear systems. Abbasbandy et al. (2008) proposed a numerical method based on pseudo-inverse calculation for finding minimal solution of DFSLE. Two necessary and sufficient conditions for the minimal solution existence are given.

## Key Terms in this Chapter

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

Fuzziness: A way of handling the uncertainty.

Supply Demand Analysis: Determining the equilibrium point between supply and demand which indicates the consumer's tendency, desire, or need to goods according to the price.

Input-Output Analysis: An analytical technique to analyze the interdependence of various industries or sectors in an economy.

Circuit Analysis: An analysis to find all the currents and voltages in an electrical network using Kirchhoff’s laws.

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

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

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

Computer Tomography: A computerized X-ray imaging procedure that constructs a cross-sectional view of a human body.

Dual Fuzzy Linear Equation System: A linear equation system having fuzzy linear equation systems in both sides.

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