Fuzzy Reliability of Weighted-((f / (r, s)), k)/ (m, n) G System

Introduction

Every real life engineering system consists of elements connected in different configurations such as series, parallel, series-parallel, k-out-of-n etc. Of course they often have to deal with the reliability evaluation of these systems. Needless to say, recently especial attention has been paid to k-out-of-n systems due to its wide applications in engineering systems. A number of researchers like (Chung; 1988 and Bueno & Carmo; 2007) and lot of others carried out research on reliability assessment of k-out-of-n: G (F) system. It needs to be pointed out here that in this system if each element is having its own weight within the system, then it is renamed as weighted-k-out-of-n: G (F) system. Quite a few authors like Wu and Chen (1994), Li and Zuo (2008) and Li et al. (2008), Wang et al. (2012) and Negi and Singh (2015) have done incredible works in the field of reliability assessment of weighted-k-out-of-n: G (F) system. There are different extensions of k-out-of-n: G (F) system, consecutive k-out-of-n: G (F) system is one of them. Lam and Ng (2001) and Bollinger (1982) contributed significantly in this extension. The two-dimensional version of consecutive- k-out-of-n: F system is known as a consecutive-(R, S)-out-of-(M, N): F system denoted by (R, S)/ (M, N): F. This version is generated by Salvia and Lasher (1990) and used in a sensor system, a pattern system, an x-ray diagnostic system etc. This system consists of M x N components arranged like an (M, N) matrix and fails iff the system has an (R, S) sub matrix that contains all failed components. Habib et al. (2010) evaluated the reliability of this system by putting conditions on the number of failed components. Noguchi et al. (1996), Yamamoto et al. (1995) and Yuge et al. (2000) discussed the reliability of connected-(R, S)-out-of-(M, N): F system. The reliability bounds of connected -(R, S)-out-of-(M, N): F system is derived by Malinowski and preuss (1996). Another system termed as x-within-consecutive-(R, S)-out-of-(M, N): F system is introduced by Makri et al. (1996) and obtained the lower and upper bounds for its reliability. This system fails iff x components in an arbitrary (R, S) sub lattice fail. The system can be designated as x-(R, S)/ (M, N): F system. Clearly if x = RS, then x-(R, S)/ (M, N): F system becomes (R, S)/ (M, N): F system.

Zuo et al. (2000) proposed a combined model of a k-out-of-M x N: F and an (R,S)/(M,N): F system which is described as an ((R, S), k)/(M,N): F system. This system fails if at least k components fail in the system, or there is at least one cluster (size RS > k) of failed components. If k = MN, it clearly looks like the (R, S)/(M,N): F system. Though they used recursive equation method to assess the reliability of the system but it cannot be applied to all realistic situations. Furthermore, there are some engineering systems in which each component has its own importance i.e., weight in the system. Correspondingly as stated earlier this system is renamed as weighted-((f / (r, s)), k)/ (m, n): F system. Further, sometimes, it is very hard to get precise data of system parameters which leads to the uncertainty in the system parameters. It is a hard fact that reliability of the system depends upon failure data. Therefore if uncertainties are present in the system parameters, it becomes difficult to find out exact reliability characteristics of the system and so we move towards the fuzzy reliability theory.