Abstract

A project schedule contains a network of activities, the activity durations, the early and late finish dates for each activity, and the associated total float or slack times, the difference between the late and early dates. Here I show that the distribution of activity durations and total floats of construction project schedules exhibit a power law scaling. The power law scaling of the activity durations is explained by a historical process of specialization fragmenting old activities into new activities with shorter duration. In contrast, the power law scaling of the total floats distribution across activities is determined by the activity network. I demonstrate that the power law scaling of the activity duration distribution is essential to obtain a good estimate of the project delay distribution, while the actual total float distribution is less relevant. Finally, using extreme value theory and scaling arguments, I provide a mathematical proof for reference class forecasting for the project delay distribution. The project delay cumulative distribution function is \(G(z) = \exp ( - (z_c/z)^{1/s})\), where \(s>0\) and \(z_c>0\) are shape and scale parameters. Furthermore, if activity delays follow a lognormal distribution, as the empirical data suggests, then \(s=1\) and \(z_c \sim N^{0.20}d_{\max }^{1+0.20(1-\gamma _d)}\), where N is the number of activities, \(d_{\max }\), the maximum activity duration in units of days and \(\gamma _d\), the power law exponent of the activity duration distribution. These results offer new insights about project schedules, reference class forecasting and delay risk analysis.

Graphic abstract

A process of activities duration fragmentation explains the emergence of scaling in the activities duration distribution.

中文翻译：

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项目进度中出现规模化问题

摘要

项目进度表包含活动网络、活动持续时间、每项活动的最早和最晚完成日期以及相关的总浮动时间或松弛时间、最晚日期和最早日期之间的差异。在这里，我展示了建设项目进度表的活动持续时间和总浮动时间的分布呈现幂律缩放。活动持续时间的幂律缩放可以通过专业化将旧活动分割成持续时间较短的新活动的历史过程来解释。相反，活动中总浮动量分布的幂律缩放由活动网络决定。我证明了活动持续时间分布的幂律缩放对于获得项目延迟分布的良好估计至关重要，而实际的总浮动分布则不太相关。最后，使用极值理论和标度参数，我为项目延迟分布的参考类预测提供了数学证明。项目延迟累积分布函数为\(G(z) = \exp ( - (z_c/z)^{1/s})\)，其中\(s>0\)和\(z_c>0\)为形状和尺度参数。此外，如果活动延迟遵循对数正态分布，如经验数据所示，则\(s=1\)和\(z_c \sim N^{0.20}d_{\max }^{1+0.20(1-\gamma _d)}\)，其中N是活动数量，\(d_{\max }\)是最大活动持续时间（以天为单位），\(\gamma _d\)是活动持续时间分布的幂律指数。这些结果提供了有关项目进度、参考类预测和延误风险分析的新见解。

图文摘要

活动持续时间碎片化的过程解释了活动持续时间分布中缩放的出现。