文档介绍：Redundancy Optimization ProblemKai YaoSchool of Management, University of Chinese Academy of Sciences, Beijing 100190, ******@ Problem DescriptionConsider a redundant system consisting ponents. For ponenti, there is only one type ofelements available,i= 1,2,· · ·, n. We usexito indicate the numbers of theith types of elements selectedas redundant elements,i= 1,2,· · ·, n, respectively. The redundant elements are arranged in one of twoways: parallel or standby (for example, see Figs. 1 and 2, respectively). Letξijindicate the lifetimes ofthejth redundant elements ponentsi,Ti(x,ξ) indicate the lifetimes ponentsi, andT(x,ξ)indicate the system lifetime, whereξ= (ξ11, ξ12,· · ·, ξ1x;ξ21, ξ22,· · ·, ξ2x;· · ·;ξn1, ξn2,· · ·, ξnx), i=1,2,· · ·, n, j= 1,2,· · ·, 1: Parallel Redundancy SystemFigure 2: Parallel Redundancy SystemThe redundancy optimization problem is to ?nd the optimal value ofx= (x1, x2,· · ·, xn) such thatsome system performance are optimized under some cost AssumptionWe make the following assumptions about the redundant . Lifetimes of elementsξijare random fuzzy variables,i= 1,2,· · ·, n, j= 1,2,· · ·, . There is no ele