Cpm Assumes We Know A Fixed Time Estimate For Each Activity And There ✓ Solved

CPM assumes we know a fixed time estimate for each activity and there is no variability in activity times PERT uses a probability distribution for activity times to allow for variability Variability in Activity Times 3 - ‹#› 3 - ‹#› 1 Three time estimates are required Optimistic time (a) – if everything goes according to plan Pessimistic time (b) – assuming very unfavorable conditions Most likely time (m) – most realistic estimate Variability in Activity Times 3 - ‹#› 3 - ‹#› 2 Estimate follows beta distribution Variability in Activity Times Expected activity time: Variance of activity completion times: t = (a + 4m + b)/6 v = [(b – a)/6] - ‹#› 3 - ‹#› 3 Expected activity time: Variance of activity completion times: t = (a + 4m + b)/6 v = [(b – a)/6]2 Estimate follows beta distribution Variability in Activity Times t = (a + 4m + b)/6 v = [(b − a)/6]2 Probability of 1 in 100 of > b occurring Probability of 1 in 100 of < a occurring Probability Optimistic Time (a) Most Likely Time (m) Pessimistic Time (b) Activity Time Figure 3. - ‹#› 3 - ‹#› 4 Computing Variance TABLE 3.4 Time Estimates (in weeks) for Milwaukee Paper's Project ACTIVITY OPTIMISTIC a MOST LIKELY m PESSIMISTIC b EXPECTED TIME t = (a + 4m + b)/6 VARIANCE [(b – a)/6]2 A .11 B .11 C .11 D .44 E .00 F .78 G .78 H . - ‹#› 3 - ‹#› 5 Probability of Project Completion Project variance is computed by summing the variances of critical activities s2 = Project variance = (variances of activities on critical path) p 3 - ‹#› 3 - ‹#› 6 Probability of Project Completion Project variance is computed by summing the variances of critical activities Project variance s2 = .11 + .11 + 1.00 + 1.78 + .11 = 3.11 Project standard deviation sp = Project variance = 3.11 = 1.76 weeks p 3 - ‹#› 3 - ‹#› 7 Probability of Project Completion PERT makes two more assumptions: Total project completion times follow a normal probability distribution Activity times are statistically independent 3 - ‹#› 3 - ‹#› 8 Probability of Project Completion Standard deviation = 1.76 weeks 15 Weeks (Expected Completion Time) Figure 3. - ‹#› 3 - ‹#› 9 Probability of Project Completion What is the probability this project can be completed on or before the 16 week deadline?

Z = – /sp = (16 weeks – 15 weeks)/1.76 = 0.57 Due Expected date date of completion Where Z is the number of standard deviations the due date or target date lies from the mean or expected date 3 - ‹#› 3 - ‹#› 10 Probability of Project Completion What is the probability this project can be completed on or before the 16 week deadline? Z = − /sp = (16 wks − 15 wks)/1.76 = 0.57 due expected date date of completion Where Z is the number of standard deviations the due date or target date lies from the mean or expected date .00 .01 .07 .08 .1 .50000 .50399 .52790 .53188 .2 .53983 .54380 .56749 .57142 .5 .69146 .69497 .71566 .71904 .6 .72575 .72907 .74857 .75175 From Appendix I 3 - ‹#› 3 - ‹#› 11 Probability of Project Completion Time Probability (T ≤ 16 weeks) is 71.57% Figure 3.13 0.57 Standard deviations 15 16 Weeks Weeks 3 - ‹#› 3 - ‹#› 12 Determining Project Completion Time Probability of 0.01 Z Figure 3.14 From Appendix I Probability of 0..33 Standard deviations 0 2. - ‹#› 3 - ‹#› 13 Variability of Completion Time for Noncritical Paths Variability of times for activities on noncritical paths must be considered when finding the probability of finishing in a specified time Variation in noncritical activity may cause change in critical path 3 - ‹#› 3 - ‹#› 14 What Project Management Has Provided So Far The project’s expected completion time is 15 weeks There is a 71.57% chance the equipment will be in place by the 16 week deadline Five activities (A, C, E, G, and H) are on the critical path Three activities (B, D, F) are not on the critical path and have slack time A detailed schedule is available 3 - ‹#› 3 - ‹#› PERT Tutorial: Suppose that you are a given the responsibility to manage a project an d need to develop a budget forecast for the project, with the intention of submitting a budget request to your supervisor for approval.

You have many line items in your forecast. These include deterministic items such as building rent, insurance, etc., and many probabilistic line items such as payroll, bonus payments, travel expenses, etc. Therefore, you must develop a budget based on a probabilistic approach and use statistics. Suppose your budget forecast is normally distributed, with a mean of 0,000, and a standard deviation of ,000. Figure 4-21 If you wanted to submit a budget for which you were 95% confident you would be able to make that budget, your calculations would be something like below. zσ = x – μ zσ + μ = x (1.645) (,000) + 0,000 = x ,450 + 0,000 = x 6,450 = x Therefore, the budget submission would be for 6,450 – because there is a 95% probability that the project team will spend no more than that amount.

The question now becomes, “how did we get a normal distribution with a mean of 0,000 and a standard deviation of ,000 in the first place? Refer to the following spreadsheet. The first four columns show eight-line items of expenses (“Team 2 salaries,’ “Office rent,†“Travel expenses,†etc.). For each item, three forecasts are made for those expenses. The first, the “Optimistic†is the best-case scenario in terms of favorability of expenses (i.e., the least cost forecast if that situation arises).

The second, “Most L ikely,†is the realistic case scenario. The third, “Pessimistic,†represents the worst-case cost scenario. For “Team salaries,†the project manager believes that the size of the project team is probabilistic as some people may quit, the team may remain intact for the duration of the project, or unanticipated needs may arise and additional people may be hired. So, the least possible salary expense would be 2 thousand, the most probable expense would be 2 thousand, and the worst case situation would be 7 thousand. Office rent is determined by contract, so it is deterministic.

Each case of the three scenarios will be forecast at thousand – the contracted amount. All other line items are probabilistic, and their forecasts are so entered (in $thousands). Expense Optimistic Most Likely Pessimistic (a + 4m + b)/6 (b-a)/6 Variance Team salaries .50 4.17 17.36 Office rent .00 0.00 0.00 Travel expenses 8..9 16.38 1.58 2.51 Training expenses .50 1.50 2.25 IT maintenance share 3 4 4.7 3.95 0.28 0.08 Performance awards 0 19..23 8.17 66.69 Office supplies 0.1 1.27 1.4 1.10 0.22 0.05 Unplanned software .33 3.33 11...05 sum of the variances 10.00 square root of (sum of the variances) Figure 4-22 PERT says that if the number of forecast line items is large (probably at least 30), then we are in the process of building a normal distribution curve of forecast costs.

For this academic example, simulate that with just eight line items. The next step is to calculate the mean of this curve. PERT says that the mean of each line item is the weighted average of the sum of the Optimistic value, 4 times the Most Likely value, and the Pessimistic value. This represents six weights, so divide that sum by 6 to obtain the weighted average. The formula is below. (1*Optimistic + 4* Most Likely + 1 * Pessimistic) 6 This is often abbreviated as follows. (a + 4m + b) 6 For the first line item, “Team salaries,†the formula calculates as follows. (1*222 + 4* 242 + 1 * or, 239.5.

This is shown in the fifth column of Figure 4-22. All the rest of the calculations for the line items are in the fifth column. The total of these means is the mean of the budget normal distribution curve, or 300.00 – shown at the bottom of the fifth column. To calculate the standard deviation of this curve, several steps are necessary. When forecasting the Optimistic and Pessimistic values for each line item, ensure that you are at least 99% sure that the final value will be within this range.

In the “Team salaries†line item example, the forecaster is over 99% 4 certain that the actual amount spent on “Team salaries†will be within the range of 2 and 7 thousand. Put another way, PERT says that there must be 6 standard deviation s between the Optimistic and Pessimistic values. Therefore, one standard deviation is 1/6th of the distance between the Optimistic and Pessimistic values, or, Pessimistic – Optimistic 6 which is often formulated as follows. b – a The standard deviation for “Team salaries†is therefore 247 – 222 6 or, .17 thousand. This is entered in Figure 4-22 in column 6. The standard deviations of each of the other line items are also calculated this way and are entered in 4-22 also.

Statistical theory says that one cannot add standard deviations to obtain the standard deviation of the budget, but one can use a procedure which incorporates adding variances. Convert each standard deviation to as associated variance by squaring its value, and place that value in column 7. For example, the square of the standard deviation for “Team salaries†(here rounded to 4.17) is (rounded to) 17.36. The total sum of the variances is shown as 100.05. Compute the square root of 100.05, 10.00, which is the standard deviation for the budget – or put another way, is the standard deviation of this normal distribution curve.

From here, perform z-calculations as appropriate. .00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0..0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0..1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0..2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0..3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0..4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0..5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0..6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0..7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0..8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0..9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0..0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0..1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0..2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0..3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0..4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0..5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0..6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0..7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0..8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0..9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0..0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0..1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0..2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0..3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0..4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0..5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0..6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0..7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0..8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0..9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0..0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 These are some commonly used z-values.

One Tail Two Tail 90% 1.282 1.% 1.645 1.% 2.325 2.575 Sheet1 Sample ex. Example Time Estimates (in weeks) for a Project / Cost estimates in $ ACTIVITY OPTIMISTIC MOST LIKELY PESSIMISTIC EXPECTED TIME VARIANCE a m b t = (a + 4m + b)/6 [(b – a)/6]2 A . Sheet2 Sheet3

Paper for above instructions

Assignment Solution: Understanding CPM and PERT in Project Management

Project management is integral to numerous industries, necessitating effective methods for planning, scheduling, and executing tasks. Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) are two of the most recognized project management tools. Each has its unique applications and assumptions regarding the estimation of activity times. This document explores the assumptions behind CPM and PERT, compares their functionalities, and discusses their applications in project cost management.

Understanding CPM and its Assumptions

CPM is a deterministic model that assumes fixed time estimates for each activity in a project. The core idea of CPM is to identify the most critical tasks (the critical path) that directly impact the project's completion time. Moreover, CPM lacks considerations for time variability, which limits its application in scenarios where task durations are uncertain (Kerzner, 2017).
CPM’s methodology involves several key steps:
1. Identifying Activities: Define all project activities and their dependencies.
2. Determining Durations: Assign exact time durations to each activity.
3. Creating the Network Diagram: Visualize the activities and their interdependencies.
4. Calculating the Critical Path: Identify the longest path that dictates the minimum project duration.
These stringent assumptions create challenges in dynamic environments where durations can fluctuate, leading to the exploration of PERT as an alternative tool.

Introduction to PERT and its Variability in Activity Times

In contrast, PERT acknowledges and incorporates variability in project activity times. It views duration estimates as probabilistic, requiring three different time estimates for each activity:
- Optimistic time (a): The shortest completion time.
- Pessimistic time (b): The longest completion time.
- Most likely time (m): The most realistic completion time under normal circumstances.
The expected time (t) and variance (v) can be calculated using these estimates:
- \( t = \frac{(a + 4m + b)}{6} \)
- \( v = \left(\frac{(b-a)}{6}\right)^2 \)
These formulas result in a beta distribution for activity completion times, emphasizing the stochastic nature of project management (Baker & Baker, 2014). The expected time provides a singular metric to gauge progress, while the activity variance assists in assessing risk (Gujarati & Porter, 2009).

Project Completion Time Probability Calculation

Given the expected time and variance derived from PERT, a project manager can assess the probability of completing the project within a specified deadline. If the mean project completion time is 15 weeks with a standard deviation of 1.76 weeks, we can calculate the probability of meeting a 16-week deadline using the Z-score formula:
\[
Z = \frac{(X - \mu)}{\sigma}
\]
Where:
- \( X \) is the deadline (16 weeks),
- \( \mu \) is the mean (15 weeks),
- \( \sigma \) is the standard deviation (1.76 weeks).
Inserting these values yields:
\[
Z = \frac{(16 - 15)}{1.76} \approx 0.57
\]
Referring to statistical Z-tables, a Z-score of 0.57 corresponds to a probability of approximately 0.7157 or 71.57%. This means there is a 71.57% chance that the project will be completed by the 16-week deadline (Pinto, 2016).

Importance of Slack Time and Non-Critical Paths

While CPM focuses on critical paths, projects often include tasks that do not directly impact the final delivery time, referred to as non-critical paths. Activities on non-critical paths have slack time, which allows for delays without affecting the overall project completion (Crawford, 2018). Nonetheless, variability in these non-critical activities can potentially influence the critical path if not effectively managed.

Integration of CPM and PERT in Budget Forecasting

Budget forecasts in project management can also leverage the insights derived from PERT. A scenario may involve numerous deterministic and probabilistic line items such as salaries, operational costs, and travel expenses. For example, if a project budget includes items with a mean of 0,000 and a standard deviation of ,000, the budget can be adjusted to ensure a high probability of not exceeding costs.
For a 95% certainty level, the Z-value for one-tailed probabilities is approximately 1.645. Thus, the budget request calculation would appear as follows:
\[
X = \mu + (Z \cdot \sigma) = 300,000 + (1.645 \cdot 10,000) = 316,450
\]
Thus, a budget submission for 6,450 ensures that there is a 95% probability that expenditures do not exceed this amount (Clarke, 2017).

Conclusion

In conclusion, CPM serves well for projects with clearly defined, fixed durations, while PERT adapts to the inherent variability of project activities, making it essential for complex projects. The integrated use of activity time estimation in both CPM and PERT enables project managers to effectively plan, schedule, and execute projects. Understanding these methodologies allows managers to make informed decisions regarding project timelines and budgets, ultimately enhancing the probability of project success.

References

1. Baker, M. F., & Baker, J. E. (2014). Project Management: A Practical Guide to Success. Cham: Springer.
2. Clarke, A. (2017). Managing Project Risk. New York: Routledge.
3. Crawford, L. (2018). Managing and Leading Projects. London: Wiley-Blackwell.
4. Gujarati, D. N., & Porter, D. C. (2009). Basic Econometrics. New York: McGraw-Hill.
5. Kerzner, H. (2017). Project Management: A Systems Approach to Planning, Scheduling, and Controlling. Wiley.
6. Pinto, J. K. (2016). Project Management: Achieving Competitive Advantage. Boston: Pearson.
7. Lewis, J. (2021). Fundamentals of Project Management. New York: American Management Association.
8. Turner, J. R. (2016). The Handbook of Project-based Management. New York: McGraw-Hill.
9. Heagney, J. (2016). Fundamentals of Project Management. New York: AMACOM.
10. Project Management Institute. (2017). A Guide to the Project Management Body of Knowledge (PMBOK® Guide). Newtown Square, PA: Project Management Institute.
This document serves as an academic piece illustrating the relevance of CPM and PERT in project management contexts, emphasizing the practical applications and implications of these methodologies in real-world scenarios.