Model Answer
0 min readIntroduction
In modern project management, adhering to deadlines is paramount, and often, contracts include penalty clauses for delays. These penalties are a significant cost factor, and companies employ various statistical and operational techniques to minimize the risk of incurring them. This question assesses the ability to apply probabilistic reasoning to a construction project scenario, specifically calculating the likelihood of avoiding penalties and estimating the potential financial impact if penalties are triggered. The core concept revolves around understanding the distribution of project completion times and its relation to the contractual deadline.
Understanding the Problem
The question lacks specific details regarding the project completion time distribution and the penalty structure. To proceed, we need to make some reasonable assumptions. Let's assume:
- The project completion time follows a normal distribution.
- The mean completion time is 100 days.
- The standard deviation of completion time is 10 days.
- The deadline for completion is 110 days.
- The penalty is a fixed amount of $100 per day for each day beyond the deadline.
These assumptions allow us to apply statistical methods to estimate the probability of avoiding penalties and the expected penalty amount.
Calculating the Probability of Avoiding Penalties
To find the probability of avoiding penalties, we need to calculate the probability that the project completion time is less than or equal to 110 days. This can be done by calculating the Z-score and using a standard normal distribution table or a statistical software.
Z-score Calculation
The Z-score is calculated as:
Z = (X - μ) / σ
Where:
- X = Deadline (110 days)
- μ = Mean completion time (100 days)
- σ = Standard deviation (10 days)
Z = (110 - 100) / 10 = 1
Probability Calculation
Using a standard normal distribution table, the probability of Z ≤ 1 is approximately 0.8413. This means there is an 84.13% chance that the project will be completed within the deadline and the company will not have to pay penalty fees.
Calculating the Expected Penalty Fee
To calculate the expected penalty fee, we need to consider the probability of exceeding the deadline and the expected number of days beyond the deadline. First, we find the probability of exceeding the deadline:
P(Completion Time > 110) = 1 - P(Completion Time ≤ 110) = 1 - 0.8413 = 0.1587
Next, we need to calculate the expected number of days beyond the deadline, given that the deadline is exceeded. This requires understanding the properties of the normal distribution. The expected value of the excess time is given by:
E[Excess Time] = σ * (φ(z) / (1 - Φ(z)))
Where:
- σ = Standard deviation (10 days)
- z = Z-score (1)
- φ(z) = Probability density function of the standard normal distribution at z = 1 (approximately 0.2420)
- Φ(z) = Cumulative distribution function of the standard normal distribution at z = 1 (0.8413)
E[Excess Time] = 10 * (0.2420 / (1 - 0.8413)) = 10 * (0.2420 / 0.1587) ≈ 15.26 days
Finally, we calculate the expected penalty fee:
Expected Penalty Fee = P(Completion Time > 110) * E[Excess Time] * Penalty per day
Expected Penalty Fee = 0.1587 * 15.26 * $100 ≈ $242.48
Penalty Fee for 50 Triplexes
If the total number of triplexes to be built is 50, the total expected penalty fee would be:
Total Expected Penalty Fee = $242.48 * 50 = $12,124
Conclusion
Based on the assumed parameters, the probability that the company will not have to pay penalty fees is approximately 84.13%. If penalties are incurred, the expected penalty fee per triplex is approximately $242.48, resulting in a total expected penalty fee of $12,124 for 50 triplexes. It's crucial to remember that these calculations are based on the assumed normal distribution and penalty structure. A more accurate assessment would require detailed project data and a thorough risk analysis.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.