The formula for the completion time of project networks

TL;DR

This work reframes project completion time as a linear transformation $\tau = ||\mathbf{R}\mathbf{t}||_{\infty}$ where $\mathbf{R}$ encodes simple paths and $\mathbf{t}$ contains activity durations, enabling algebraic and spectral analyses of network structure. By examining the nullspace of $\mathbf{R}$, performing SVD, and introducing spectral networks, the paper reveals how topology, durations, and uncertainty shape path and activity relevance, and introduces the project stress metric and Moore-Penrose inverse to analyze and invert the path-duration mapping. A systematic mapping review establishes novelty for the algebraic implications of the proposed formulation, while the discussion connects these ideas to incidence-based LP constraints and classical critical-path concepts. Collectively, the framework offers new insights for path/activity prioritization, schedule optimization, and stress assessment in project management, with potential for more compact network representations via spectral decompositions.

Abstract

This paper formulates the completion time $τ$ of a project network as $ τ=\|\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.

Figures (2)

• Figure 1: Illustrative toy example of a project network
• Figure 2: Mapping for theorical frame.

Theorems & Definitions (8)

• Example 1.1
• Example 1.2
• Example 1.3
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Example 2.5