In the present chapter, we give an overview of computational iterative schemes for fuzzy system of linear equations. We also consider fully fuzzy linear systems (FFLS) and demonstrate a class of the existing iterative methods using the splitting approach for calculating the solution. Furthermore, the main aim in this work is to design a numerical procedure for improving this algorithm. Some numerical experiments are illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm, we report the numerical results of large-scaled fuzzy problems.
TopIntroduction
Unfailing real world problems in economics, finance, mechanics etc. can lead to solving a system of linear equations. There are many methods for solving linear systems, see(Barrett et al., 1994; Edalatpanah, 2008; Eisenstat, Elman, & Schultz, 1983; Greenbaum, 1997; Martins, Trigo, & Evans, 2007; Saad, 2003; Saberi Najafi & Edalatpanad, 2013a; Saberi Najafi & Edalatpanah, 2013b, 2013c, 2014b; Saberi Najafi & Edalatpanah, 2011; Varga, 2009; Young, 2014; Zhang, Huang, Cheng, & Wang, 2012) and the references therein. Let us consider the following linear systems
Ax=b,
(1)However, when the estimation of the system coefficients is imprecise and only some vague knowledge about the actual values of the parameters is available, it may be convenient to represent some or all of them with fuzzy numbers. Fuzzy data is being used as a natural way to describe uncertain data. Fuzzy concept was introduced by Zadeh (Zadeh, 1965, 1972) and following his work, many papers and books were published in fuzzy system theory; see (Bellman & Zadeh, 1970; Bezdek, 2013; Chen, 2000; Cordón, 2001; Driankov, Hellendoorn, & Reinfrank, 2013; Hájek, 1998; Höppner, 1999; Jang & Sun, 1995; Kusko, 1993; Pham, Xu, & Prince, 2000; Ragin, 2000; Sheridan, 1992; Wasserman, 1993; Yager & Filev, 1994; Zadeh, 1997; Zimmermann, 2001).We refer the reader to (Kaufmann & Gupta, 1991) for more information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to study a variety of problems including fuzzy metric spaces (Alaca, Turkoglu, & Yildiz, 2006; Gregori, Romaguera, & Veeramani, 2006; J. H. Park, 2004), fuzzy differential equations (Bede & Gal, 2005; Buckley, Eslami, & Feuring, 2002; Kaleva, 1987; Khastan, Nieto, & Rodríguez-López, 2011; Khastan & Rodríguez-López, 2015; Malinowski, 2012; J. Y. Park & Han, 2000), particle physics (El Naschie, 2004a, 2004b), Game theory (Kacher & Larbani, 2008; Larbani, 2009; Maeda, 2000; Oliveira & Petraglia, 2014; Saberi Najafi & Edalatpanah, 2012a; Yang & Gao, 2014), optimization (Amid, Ghodsypour, & O’Brien, 2006; Edalatpanah & Shahabi, 2012; Guua & Wu, 1999; Huang, Baetz, & Patry, 1995; Inuiguchi, Ichihashi, & Kume, 1990; Lee & Li, 1993; Najafi & Edalatpanah, 2013d; Rommelfanger, 2007; Shamooshaki, Hosseinzadeh, & Edalatpanah, 2014; Słowiński, 1986; Yu, 2002), fuzzy linear systems(Allahviranloo, 2004; Asady, Abbasbandy, & Alavi, 2005; Dehghan & Hashemi, 2006b; Dehghan, Hashemi, & Ghatee, 2006, 2007; Dubois & Prade, 1980; Friedman, Ming, & Kandel, 1998; Ma, Friedman, & Kandel, 2000; Nasseri, Sohrabi, & Ardil, 2008; Saberi Najafi & Edalatpanah, 2012b), and so on.