Symbolic Reliability Analysis of a Multi-State Network

This paper deals with the reliability of a multi-state delivery network with multiple suppliers, transfer stations and markets (depicted as vertices), connected by branches of multi-state capacities, delivering a certain commodity or service between their end vertices. We utilize a symbolic logic expression of the network success to satisfy the market demand within budget and production capacity limitations even when subject to deterioration. This system success is a two-valued function expressed in terms of multi-valued component successes, and it has been obtained in the literature in minimal form as the disjunction of particular prime implicants or minimal paths of the pertinent network. This paper provides a systematic procedure for converting this minimal expression into a probability-ready expression (PRE). We successfully extrapolate the PRE concept from the two-valued logical domain to the multi-valued logical domain. This concept is important since it allows a direct transformation of a random logical expression, on a one-to-one one, 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 statistical expectation of the expression is its probability of being equal to 1, and is simply called reliability of the multi-state delivery network (MSDN).