Deduction of the Bromilow's time-cost model from the fractal nature of activity networks

TL;DR

This paper derives Bromilow's time-cost model from the fractal properties of activity networks, linking the model's exponent to network scaling, and provides empirical evidence relating project network structure to duration and cost scaling.

Contribution

It offers a theoretical derivation of Bromilow's model based on activity network fractality, connecting the exponent to network scaling properties.

Findings

Bromilow's exponent B equals 1 minus the network scaling exponent alpha.

Projects with lower serial/parallel ratios have lower B values.

Forecasting duration from cost is more reliable for projects with high serial/parallel ratios.

Abstract

In 1969 Bromilow observed that the time $T$ to execute a construction project follows a power law scaling with the project cost $C$, $T\sim C^B$ [Bromilow 1969]. While the Bromilow's time-cost model has been extensively tested using data for different countries and project types, there is no theoretical explanation for the algebraic scaling. Here I mathematically deduce the Bromilow's time-cost model from the fractal nature of activity networks. The Bromislow's exponent is $B=1-\alpha$, where $1-\alpha$ is the scaling exponent between the number of activities in the critical path $L$ and the number of activities $N$, $L\sim N^{1-\alpha}$ with $0\leq\alpha<1$ [Vazquez et al 2023]. I provide empirical data showing that projects with low serial/parallel (SP)% have lower $B$ values than those with higher SP%. I conclude that the Bromilow's time-cost model is a law of activity networks, the Bromilow's exponent is a network property and forecasting project duration from cost should be limited to projects with high SP%.