Utilization of Symmetric Switching Functions in the Symbolic Reliability Analysis of Multi-State k-out-of-n Systems

Abstract

Symmetric switching functions (SSFs) play a prominent role in the reliability analysis of a binary k-out-of-n: G system, which is a dichotomous system that is successful if and only if at least k out of its n components are successful. The aim of this paper is to extend the utility of SSFs to the reliability analysis of a multi-state k-out-of-n: G system, which is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of non-repairable multi-state k-out-of-n: G systems with independent non-identical components. The paper utilizes algebraic techniques of multiple-valued logic (together with known properties of SSFs) to evaluate each of the multiple levels of the system output as an individual binary or propositional function of the system multi-valued inputs. The formula of each of these levels is then written as a probability–ready expression, thereby allowing its immediate conversion, on a one-to-one basis, into a probability or expected value. The symbolic reliability analysis of a commodity-supply system (which serves as a standard gold example of a multi-state k-out-of-n: G system) is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.

Reliability Analysis of a Commodity-Supply Multi-State System Using the Map Method

“…A binary k-out-of-n: G system is uniquely defined as a dichotomous system that is successful if and only if at least k out of its n components are successful , By contrast, a multi-state k-out-of-n: G system does not possess a unique definition [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. The definition adopted herein is that this system is a multi-state system (MSS) whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level)) whenever at least k m components are in state m or above for all m such that 1 ≤ m ≤ j [34,[40][41][42][43].…”

“…In this paper, we a study a standard multi-state system, which was proposed and studied by Tian et al [34], and further studied by Fadhel et al [44], Mo et al [40], Rushdi [41], Rushdi & Al-Amoudi [42,43]. The system (shown in Fig.…”

“…We have recently reported several solutions of the aforementioned problem, and our present paper offers yet another solution of this problem. In our earlier solutions, we employed purely-algebraic methods of multivalued logic, in which we handled multi-valued variables either directly [41] or through some binary encoding [42,43], with various map versions used occasionally for verification. In this paper, however, we deliberately avoid the mathematically-demanding algebraic manipulations in [41][42][43] by employing the Karnaugh map [45][46][47][48][49][50] as the sole vehicle for our manipulations.…”

“…In our earlier solutions, we employed purely-algebraic methods of multivalued logic, in which we handled multi-valued variables either directly [41] or through some binary encoding [42,43], with various map versions used occasionally for verification. In this paper, however, we deliberately avoid the mathematically-demanding algebraic manipulations in [41][42][43] by employing the Karnaugh map [45][46][47][48][49][50] as the sole vehicle for our manipulations. There is a long history of utilization of the Karnaugh map as a probability map (or reliability map) in the binary case [51][52][53][54][55][56][57][58][59].…”

“…The organization of the remainder of this paper is as follows. Section 2 retrieves from Rushdi [41] a mathematical description of the example multi-state k-out-of-n system. Section 3 implements a purely-map analysis of the system.…”