A liaison among inclusion-exclusion, probability-ready expressions and Boole-Shannon expansion for multi-state reliability

Abstract EN

This paper deals with an emergent variant of the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, i.e., the probability of this disjunction being equal to one.

The variant considered herein deals with multi-valued variables, in which the required probability stands for the reliability of a multi-state delivery network (MSDN), whose system success is a two-valued function expressed in terms of multi-valued component successes.

The paper discusses four approaches for handling the afore-mentioned problem in terms of a standard MSDN, whose success is known in minimal form as the disjunction of its prime implicants, which are the minimal paths of the pertinent network.

The paper briefly outlines and discusses two standard solutions via the utilization of the multi-state inclusion-exclusion (MS-IE) principle, and via the construction of a multi-state probability-ready expression (MS-PRE).

We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain, and employ it for a direct transformation of a random logical expression, on a one-to-one basis, to its statistical expectation form, simply by replacing all logic variables by their statistical expectations, and also substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing).

The main contribution of the paper is to provide two systematic and more efficient procedures for handling the required problem.

The first procedure uses the multi-state Boole-Shannon expansion, while the second procedure applies the MS-IE principle to fewer (factored or composite) paths that are set (at minimal cost) to PRE form.

The four approaches discussed are illustrated with a detailed symbolic example of a real-case study, and each of them produces a more precise version of the same numerical value that was obtained earlier by the method of recursive sum of disjoint products (RSDP).

The paper is a part of an ongoing activity that strives to provide a pedagogical treatment of multi-state reliability problems, and to establish a clear and insightful interrelationship between the two-state modeling and the multi-state one by stressing that multi-valued concepts are natural and simple extensions of two-valued ones.

American Psychological Association (APA)

Rushdi, Ali Muhammad Ali& Amashah, Mutazz Husayn. 2021. A liaison among inclusion-exclusion, probability-ready expressions and Boole-Shannon expansion for multi-state reliability. Journal of King Abdulaziz University : Computing and Information Technology Sciences،Vol. 10, no. 2, pp.1-17.
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