5.2 Activity Time Analysis

Activity Time Analysis

Why this topic matters for the PE Construction exam: The foundation of a reliable schedule is accurate activity durations. The exam frequently requires you to calculate how many days a task will take based on crew sizes, productivity rates, and total quantities. You must also be prepared to evaluate probabilistic scheduling using PERT formulas.

Once the sequence of activities is established, the next critical step is determining the duration of each activity. A schedule is only as good as its time estimates. Underestimating durations leads to unachievable schedules and liquidated damages, while overestimating leads to uncompetitive bids and resource inefficiencies.

Determining Activity Durations

The duration of an activity is a function of the scope of work (quantity) and the resources applied to it (productivity rate). The fundamental formula for calculating duration is:

$\text{Duration} = \frac{\text{Total Quantity}}{\text{Productivity Rate}}$

Units and Conversion

When using this formula, ensure your units align. If the quantity is in cubic yards (CY) and the productivity rate is in cubic yards per hour (CY/hr), the resulting duration will be in hours. Schedules are typically built in standard working days.

Conversion to Days: $\text{Duration (Days)} = \frac{\text{Duration (Hours)}}{\text{Working Hours per Day}}$

Adjusting for Crew Size

Productivity rates are often given per worker or per standard crew. If you increase the crew size, the overall productivity rate increases, reducing the duration.

$\text{Overall Productivity Rate} = (\text{Production per Crew}) \times (\text{Number of Crews})$

[!WARNING] Exam Trap: Be careful with the assumption of perfect linear scaling. Doubling the crew size does not always exactly halve the duration due to space constraints, diminishing returns, and the "stacking of trades." However, unless otherwise stated in the exam problem, assume linear scaling applies.

Worked Calculation: Productivity Duration

Scenario: A contractor needs to excavate 4,500 cubic yards (CY) of soil. A standard excavation crew can remove 150 CY per 8-hour workday. The contractor assigns two crews to the task. How many workdays will the excavation take?

Step 1: Determine the total productivity rate. Since there are 2 crews, the total daily productivity is: $150 \text{ CY/crew/day} \times 2 \text{ crews} = 300 \text{ CY/day}$

Step 2: Calculate the duration. $\text{Duration} = \frac{4,500 \text{ CY}}{300 \text{ CY/day}} = 15 \text{ days}$

Three-Point Estimating (PERT)

In construction, durations are rarely deterministic. Weather, supply chain issues, and unforeseen site conditions introduce variability. The Program Evaluation and Review Technique (PERT) is a statistical tool used to manage this uncertainty.

Instead of a single point estimate, PERT uses three time estimates for each activity:

1. Optimistic Time ($O$ or $t_o$): The minimum possible duration if everything goes perfectly.
2. Most Likely Time ($M$ or $t_m$): The best estimate of the duration under normal conditions.
3. Pessimistic Time ($P$ or $t_p$): The maximum duration if significant, but reasonable, negative issues occur.

Expected Duration Formula

PERT calculates a weighted average, known as the Expected Duration ($t_e$), giving the Most Likely estimate four times the weight of the extremes.

$t_e = \frac{O + 4M + P}{6}$

Standard Deviation and Variance

PERT also allows us to quantify the uncertainty of the estimate. The Standard Deviation ($\sigma$) represents the spread of the data, and the Variance ($V$) is the square of the standard deviation.

$\text{Standard Deviation } (\sigma) = \frac{P - O}{6}$ $\text{Variance } (V) = \left( \frac{P - O}{6} \right)^2$

These statistical measures are highly testable on the PE exam, especially when determining the probability of a project finishing on time by summing the variances of activities on the critical path.

Worked Calculation: PERT

Scenario: A bridge decking activity has the following estimated durations:

• Optimistic ($O$): 12 days
• Most Likely ($M$): 18 days
• Pessimistic ($P$): 30 days

Calculate the Expected Duration ($t_e$) and Standard Deviation ($\sigma$).

Expected Duration: $t_e = \frac{12 + 4(18) + 30}{6} = \frac{12 + 72 + 30}{6} = \frac{114}{6} = 19 \text{ days}$

Standard Deviation: $\sigma = \frac{30 - 12}{6} = \frac{18}{6} = 3 \text{ days}$

Note that the Expected Duration (19 days) is pulled slightly higher than the Most Likely duration (18 days) because the Pessimistic estimate (30 days) is further away from the Most Likely estimate than the Optimistic estimate (12 days) is. This is known as a right-skewed distribution.