The formula for the completion time of project networks

TL;DR

This paper introduces a novel mathematical formulation for project network completion time using linear algebra and spectral analysis, providing new insights into network structure and project stress.

Contribution

It formulates project completion time as a matrix norm, introduces spectral networks, and defines project stress, offering a new analytical framework for project network analysis.

Findings

Spectral networks condense project structure.

The formulation links network topology with linear algebra.

A method to compute activity durations for desired path times.

Abstract

This paper formulates the completion time $\tau$ of a project network as $ \tau =\|\mathbf{R} \mathbf{t} \|_\infty $ where the rows of $\mathbf{R}$ are simple paths of the network and $\mathbf{t}$ is a column vector representing the duration of the activities. Considering this product as a linear transformation leads to interesting findings on the topological relevance of both paths and activities using singular value decomposition. The notion of spectral networks is introduced to condense the fundamental structure of the project network. A definition of project stress is introduced to establish a comparison index between two alternatives in terms of slack. Additionally, the Moore-Penrose inverse of $\mathbf{R}$ is presented to find the configuration of the durations of the activities resulting in a given simple path duration vector. Then, the systematic mapping review process carried out to assess our claims' novelty is reported. Finally, we reflect on the notion of relevance for paths and activities and the relationship of the incidence matrix with the proposed approach.