Representation method of integrated importance measure in gradient

Aim. The introduction of the full paper reviews a number of papers in the open literature and then proposes the representation method in the title, which is explained in sections 1 and 2. Section 1 briefs the system assumptions and the equation of integrated importance measure (IIM). The core of section 2 consists of: (1) we use the gradient method, which is given by eq. (4) to describe the IIM as in eq. (5); (2) we analyze the physical meaning of the geometry of IIM and the relationships between IIM and gradient as indicated in Theorem 1; (3) we discuss the characteristics of IIM in gradient for typical systems in Theorems 2 and 3 and their respective Corollaries 1 and 2; (4) we get that IIM can be determined by the inner product of gradient and vector. Section 3 presents the numerical examples of series and parallel systems. Computer simulation results, presented in Figs. 1 through 6, and their analysis verify the physical meaning of the geometry of IIM in two dimensional space and three dimensional space.