A New Algorithm for Generating All Minimal Vectors for the <formula formulatype="inline"> <tex Notation="TeX">$q$</tex> </formula> SMPs Reliability Problem With Time and Budget Constraints

Abstract

The quickest path reliability problem evaluates the probability of transmitting some given units of flow from a source node to a sink node through a single minimal path in a stochastic-flow network within some specified units of time. This problem has been recently extended to the case of transmitting flow through $q$ separate minimal paths (SMPs) simultaneously within a budget constraint. Here, we propose a new algorithm to solve such problems. The algorithm finds all the minimal vectors for which a given demand level $d$ units of flow can be transmitted through $q$ SMPs from a source node to a sink node satisfying some given time and budget limits. In our proposed algorithm, a special Diophantine system is needed to be solved, for which we provide a particularly efficient procedure with linear time complexity in terms of the number of its solutions. The complexity results are derived to demonstrate the efficiency of our proposed algorithm in comparison with others. To illustrate the practical efficiency of the algorithm, a network example is considered to compare our algorithm with a recently proposed one. Moreover, a comparative computational experiment is conducted on an extensive collection of random test problems using the performance profile of Dolan and More.