Interval linear systems as a necessary step in fuzzy linear systems

Abstract

International audienceThis article clarifies what it means to solve a system of fuzzy linear equations, relying on the fact that they are a direct extension of interval linear systems of equations, already studied in a specific interval mathematics literature. We highlight four distinct definitions of a systems of linear equations where coefficients are replaced by intervals, each of which based on a generalization of scalar equality to intervals. Each of the four extensions of interval linear systems has a corresponding solution set whose calculation can be carried out by a general unified method based on a relatively new concept of constraint intervals. We also consider the smallest multidimensional intervals containing the solution sets. We propose several extensions of the interval setting to systems of linear equations where coefficients are fuzzy intervals. This unified setting clarifies many of the anomalous or inconsistent published results in various fuzzy interval linear systems studies

Solving different practical granular problems under the same system of equations

“…The present result (32) has been achieved without using improper intervals used by Kaucher arithmetic, Kaucher (1977) in which b b. Improper intervals are not realizable physically and because of this reason they are not accepted by authors of the present paper and by other scientists such as Lodwick and Dubois (2015). Figure 10 shows the visualization of the fully identified results set C active AB .…”

“…It also has been applied in fuzzy arithmetic (RDM-FA) based on l-cuts, Plucinski (2015a, b, 2017), Landowski (2015, 2017a) and Mazandarani et al (2017). RDM-IA uses epistemic approach to interval calculations presented in Lodwick and Dubois (2015). The subject of arithmetic operations in RDM-IA is not the full set of values contained in interval ½x ¼ ½x; x but one single, true, precise value x pr contained in the interval, which has really occurred.…”

“…But in practical problems we also are interested in the set Z poss containing all states (x, y, z) which can occur in a system when our knowledge about inputs x, y is only approximate and of interval character. The set Z poss contains an infinite number of elements and can be called ontic set (Lodwick and Dubois 2015) The set Z poss is visualized in Fig. 3.…”

Is an interval the right result of arithmetic operations on intervals?

“…Let * be one of the operations {+, −, ×, ÷}. In terms of SI arithmetic, each operation * for independent variables can be defined as follows (Lodwick and Dubois, 2015):…”

A Novel Weak Fuzzy Solution for Fuzzy Linear System

“…Due to the uncertain parameters appearing in substantive problems, people often use fuzzy number in a real or complex form to describe fuzzy factors of the complex or real linear system, thus discovering new and easy-implemented schemes that would suitably deal with the fuzzy linear systems (FLS) and solve them, and enabling an expansion to a hotspot exploration [6][7][8][9][10][11][12][13][14][15][16].…”