Solving Complex Fuzzy Linear System of Equations by using QR-Decomposition Method

Many real-world engineering systems are too complex to be defined in precise terms, imprecision is often involved in any engineering design process. Fuzzy systems have an essential role in this fuzzy modeling, which can formulate uncertainty in actual environment. In many linear systems, some of the system parameters are vague or imprecise, and fuzzy mathematics is a better tool than crisp mathematics for modeling these problems, and hence solving a fuzzy linear system [11] or a fuzzy differential equation is becoming more important. The concept of fuzzy numbers and arithmetic operations with these numbers were first introduced and investigated by Zadeh [22]. A different approach to fuzzy numbers and the structure of fuzzy number spaces was given by Puri and Ralescu [16], Goetschell and Voxman [12] and Wu and Ma Ming [20]

Since Friedman et al. proposed a general model for solving an n×n fuzzy linear systems whose coefficients matrix is crisp and the right-hand side is an arbitrary fuzzy numbers vector by an embedding approach, many works have been done about how to deal with some advanced fuzzy linear systems such as dual fuzzy linear systems (DFLS), general fuzzy linear systems (GFLS), full fuzzy linear systems (FFLS), dual full fuzzy linear systems (DFFLS) and general dual fuzzy linear systems (GDFLS). These works were performed mainly by Allahviranloo et al. [2,3,4,5], Abbasbandy et al. [1,7], Zheng et al. [1,9] and Dehgham et al. [10] and so on. In traditional fuzzy linear systems, the uncertain elements were usually denoted by the parametric form of fuzzy numbers. Based on arithmetic operations of the fuzzy number, the fuzzy linear systems could be extended into crisp function linear systems. Therefore the solutions of the fuzzy linear systems can be obtained by solving the model by means of ordinary analytical and numerical methods.