The Multi-State Inclusion-Exclusion Principle: Conventional and Improved Versions

Abstract

This chapter considers an emergent variant of the classical problem of calculating the probability of the union of n events, or, more precisely, 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 under consideration herein works with multi-valued variables, with the required probability representing the dependability of a multi-state delivery network (MSDN), whose binary system success is a two-valued function defined in terms of multi-valued component successes. The chapter discusses a simple method for dealing with the aforementioned problem in terms of a standard example MSDN, where success is defined as the disjunction of prime implicants or minimal paths of the relevant network. This method employs the multi-state inclusion-exclusion (MS-IE) principle, which is associated with a multi-state generalization of the ANDing operation's idempotency property. The method is demonstrated with a detailed symbolic example of a real-world case study, and it yields a more precise version of the same numerical value obtained previously. The example demonstrates the MS-IE method's well-known flaws and extreme inefficiency, but it also reveals the path to alternative methods in which such a flaw is (partially) mitigated. The construction of a multi-state probability-ready expression(MS-PRE)  is a well-known example of these methods.  Applying the MS-IE principle to the union of fewer (factored or composite) pathways that are converted (at a low cost) to PRE form is another potential approach. An innovative technique for fusing the MS-PRE and MS-IE principles together is used as a third candidate method in this article. The use of MS-PRE is limited to 'shellable' disjointing of ORed terms, and the resulting partially orthogonalized disjunctive form is subsequently subjected to MS-IE. This new approach avoids the issues brought on by either MS-PRE or MS-IE and makes the most of both of them. The identical real-world issue that was encountered with the traditional MS-IE is effectively exploited to illustrate the method.