Precedence Pattern Permutations Creating Criticality Constellations: Exploring a Conjecture on Non-linear Activities with Continuous Links

The inaugural challenge of the 2016 Creative Construction Conference has posed two related questions on how many possible criticality constellations with different behavior for delays and acceleration exist and how said constellations can occur for non-linearly and monotonously progressing activities that have continuous relations. This paper systematically solves these questions by performing a thorough literature review, assembling theoretical foundations for link constellations, performing a computer simulation of all possible permutations, and providing a mathematical proof by contradiction. It is found that (for the initially assumed self-contained activities in a network schedule that exhibit only a linearly growing production) the three newly hypothesized criticality constellations cannot exist. Non-linear activity constellations with diverging or converging relative productivities are examined next. Lags in networks become buffers in linear schedules. It is found that a non-linear curvature of the progress may induce middle-to-middle relations besides those between starts and finishes. If multiple curvatures are allowed, then partial segments can form relations, which increases the number of criticality constellations.