# Measuring time sensitivity in a project: The cruciality index (Pearson’s product-moment)

Schedule Risk Analysis (SRA) is a simple yet effective technique to connect the risk information of project activities to the baseline schedule, in order to provide sensitivity information of individual project activities to assess the potential impact of uncertainty on the final project duration. A traditional schedule risk analysis requires four steps, as described in “Schedule Risk Analysis: How to measure your baseline schedule’s sensitivity?”, to report activity sensitivity measures that evaluate each activity’s time estimate on a scale of risk. These sensitivity measures can be used by the project manager to distinguish between risky and non-risky activities in order to better focus on those activities that might have an impact on the overall project objective.

In this article, the Pearson’s product-moment cruciality index (CRI(r)) calculations are illustrated on a fictitious example project network with 8 activities, as displayed in figure 1. The number above each node is the baseline duration estimate (in days). ?
Figure 1: A fictitious example project network

The baseline schedule has a planned duration PD = 90 days and the critical path is equal to the activity sequence 1 - 2 - 5 - 8. However, the baseline schedule is only an estimate of the real project duration and hence, the real project duration and the critical path might differ from the baseline schedule estimates. Consequently, the critical path is a black-and-white view on the critical parts of a project, and should be refined to capture more detailed sensitivity information.

Cruciality index CRI(r)

The cruciality index (CRI) is such a measure that calculates the correlation between the activity duration and the total project duration, as follows:

CRI = |correlation(AvgAD, AvgSD)|

with
|x|: The absolute value of x
AvgAD: Average activity duration
AvgSD: Average simulated project duration

This measure reflects the relative importance of an activity and calculates the portion of total project duration uncertainty that can be explained by the uncertainty of an activity, and can be measured in different ways.

One way to measure CRI(r) is the Pearson's product-moment which is a traditional measure of the degree of linear relationship between two variables. The correlation is 1 in the case of a clear positive linear relationship, −1 in the case of a clear negative linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the activity duration and the total project duration. The closer the coefficient is to either −1 or 1, the stronger the correlation between these two variables. The absolute value of this correlation measure obviously lies between 0 (no relation) and 1 (perfect relation). The Pearson's product-moment cruciality index CRI(r) can be calculated as follows:

CRI(r) = |sum{(AD - AvgAD) * (SD - AvgSD)} / SQRT{sum(AD - AvgAD)2 * sum(SD - AvgSD)2}|

with sum{x} the sum of all x-values over all simulation runs.

Table 1 displays 10 simulation runs where each activity has a certain duration which might differ from the original baseline duration estimate given in figure 1. The simulated project duration (SD) for activities 1 to 8 per simulation run is also given. Each simulation run reflects a possible real project progress scenario where unexpected changes in the activity estimates might occur.

Table 1: 10 simulation runs with random activity durations
 1 2 3 4 5 6 7 8 SD Run 1 4 12 1 4 5 4 7 8 29 Run 2 22 23 14 5 28 26 5 29 102 Run 3 25 38 12 15 24 20 13 22 109 Run 4 25 25 15 13 25 25 13 10 88 Run 5 21 42 12 7 30 21 14 23 116 Run 6 28 44 7 9 15 20 10 22 109 Run 7 19 21 14 13 23 24 14 28 99 Run 8 12 36 14 9 19 17 7 24 91 Run 9 28 35 7 12 28 27 13 28 119 Run 10 27 44 6 5 30 10 11 28 129

Based on the simulation runs, the cruciality index CRI(r) can be calculated with the intermediate calculations illustrated in table 2 for activity 2 of figure 1. In the example, the values for (AD - AvgAD)2, (SD - AvgSD)2 and (AD - AvgAD) * (SD - AvgSD) are calculated for activity 2 with AvgAD = E(AD) = 32 and AvgSD = E(SD) = 99.1 as the averages or expected values of the AD and SD columns of table 1.

Based on the values in table 2, the various components of the CRI(r) formula can be easily calculated as follows:

• sum{(AD - AvgAD)2} = 1,100 and is equal to the sum of the second column of table 2
• sum{(SD - AvgSD)2} = 6,882.90 and is equal to the sum of the third column of table 2
• sum{(AD - AvgAD) * (SD - AvgSD)} = 2,188.0 and is equal to the sum of the last column of table 2
and consequently, the cruciality index is equal to

CRI(r) = 2,188.0/ SQRT(1,100 * 6,882.90) = 0.80.

Table 2: Intermediate calculations for the CRI(r) of activity 2
 (AD - AvgAD)2 (SD - AvgSD)2 (AD - AvgAD) * (SD - AvgSD) Run 1 400 4,914.01 1,402.0 Run 2 81 8.41 -26.1 Run 3 36 98.01 59.4 Run 4 49 123.21 77.7 Run 5 100 285.61 169.0 Run 6 144 98.01 118.8 Run 7 121 0.01 1.1 Run 8 16 65.61 -32.4 Run 9 9 396.01 59.7 Run 10 144 894.01 358.8 SUM 1,100 6,882.90 2,188.0

Although activity 2 lies on the critical path of the baseline schedule, the cruciality index (based on Pearson’s product-moment) is lower than 1. Figure 2 shows the CRI(r) values for all project activities, ranging between 27% and 86%. More precisely, the values are equal to 0.86, 0.80, 0.34, 0.27, 0.84, 0.53, 0.45 and 0.77 for activities 1 to 8. Obviously, a schedule risk analysis should be done with care for the following two reasons: (1) the input distributions of the activity duration should reflect reality, and (2) the number of simulation runs should exceed 10 to provide more reliable results.

? Figure 2: The cruciality index (Pearson’s product-moment) values for the 8 activities after 10 simulation runs

The graph of figure 2 can be used to set action thresholds that make a distinction between sensitive and insensitive activities to better steer the project control process during project progress. More information can be found in “Bottom-up project control: Setting action thresholds using schedule risk analysis”. Apart from the cruciality index, other time sensitivity measures can also be calculated for the same simulation runs, as discussed in “Schedule Risk Analysis: Measuring the time sensitivity of an activity”.

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