On the sure criticality of tasks in activity networks with imprecise durations

The notion of the necessary criticality (both with respect to path and to activity) of a network with imprecisely defined (by means of intervals or fuzzy intervals) activity duration times is introduced and analyzed. It is shown, in the interval case, that both the problem of asserting whether a given path is necessarily critical and the problem of determining an arbitrary necessarily critical path (more exactly, a subnetwork covering all the necessarily critical paths) are easy. The corresponding solution algorithms are proposed. However, the problem of evaluating whether a given activity is necessarily critical does not seem to be so easy. Certain conditions are formulated which, in some situations (but not in all possible situations), allow the necessary criticality of activities to be evaluated. The results obtained for networks with interval activity duration times are generalized to the case of networks with fuzzy activity duration times. Two effective algorithms for calculating the degree of necessary criticality of a fixed path, as well as an algorithm for determining the paths that are necessarily critical to the maximum degree, are proposed.