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 nonrisky activities in order to better focus on those activities that might have an impact on the overall project objective.
In this article, the Kendall's tau rank correlation cruciality index (CRI(τ) (tau)) 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).
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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 blackandwhite view on the critical parts of a project, and should be refined to capture more detailed sensitivity information.
Cruciality index CRI(τ)
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 the
CRI(r) is the Pearson's productmoment which is a traditional measure of the degree of linear relationship between two variables (see “Measuring time sensitivity in a project: the cruciality index (Pearson’s productmoment)”). However, when it is conjectured that the relation between the variables is nonlinear, the Kendall's tau rank correlation CRI(τ) might be a good alternative. The CRI(τ) assumes that the values for the variables (i.e. activity durations and project durations) are converted to ranks, measures the degree of correspondence between two rankings and assesses the significance of this correspondence. The measure is a socalled nonparametric measure to deal with situations where the strict statistical assumptions of the parametric CRI(r) measure are not met.
The Kendall's tau rank correlation cruciality index CRI(τ) can be calculated as follows:
CRI(τ) = (4 * PairwiseComparisonValue) / {nrs * (nrs  1)}  1
with nrs the number of simulation runs.
The value for the variable PairwiseComparisonValue is calculated by performing ((nrs  1) + (nrs  2) + ... + 1) comparisons of activity and project duration values between the all simulation runs. In each comparison between simulation runs x and y, the following value VAL is calculated:
if {(AD^{x}  AD^{y}) * (SD^{x}  SD^{y})} > 0 then VAL = 1 else VAL = 0
with
AD^{x}: The activity duration of simulation run x
SD^{x}: The simulated project duration of simulation run x
The value for the variable PairwiseComparisonValue in the CRI(τ) formula is then the sum of all VAL values over all comparisons.
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(τ) can be easily calculated as illustrated for activity 2 of figure 1. The value for the PairwiseComparisonValue is equal to the sum of all VAL values as calculated below:
Simulation runs 1 and 2: (12  23) * (29  102) > 0 so VAL = 1
Simulation runs 1 and 3: (12  38) * (29  109) > 0 so VAL = 1
...
Simulation runs 9 and 10: (35  44) * (119  129) > 0 so VAL = 1
In total, there are 45 combinations. The number of times the VAL variable is equal to 1 is given for all simulations runs. As an example, simulation run 1 will be compared with all 9 others, and 9 of the comparisons give a VAL = 1. Simulation run 2 will only be compared with 8 other simulation runs (with run 3 to run 10) and only 6 comparisons give a VAL = 1. The complete list of 45 comparisons results in the following number of VAL = 1 observations:
Simulation run 1: 9 (= all)
Simulation run 2: 6 (of 8)
Simulation run 3: 5 (of 7)
Simulation run 4: 5 (of 6)
Simulation run 5: 3 (of 5)
Simulation run 6: 2 (of 4)
Simulation run 7: 2 (of 3)
Simulation run 8: 1 (of 2)
Simulation run 9: 1 (= all)
The sum of all VAL values is equal to PairwiseComparisonValue = 34, and hence, the CRI(τ) can be easily calculated as follows:
CRI(τ) = (4 * 34) / (10 * 9)  1 = 0.56.
Although activity 2 lies on the critical path in the baseline schedule, the cruciality index (based on Kendall’s tau rank correlation) is lower than 1. Figure 2 shows the CRI(τ) values for all project activities, ranging between 7% and 56%. More precisely, the values are equal to 0.47, 0.51, 0.56, 0.11, 0.47, 0.11, 0.07 and 0.33 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.
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Figure 2: The cruciality index (Kendall’s tau rank correlation) values for the 8 activities after 10 simulation runs