PERT and CPM Study Notes

Project Management: Network-Based Prescriptive Analytics

• Project management involves planning, organizing, motivating, and controlling resources to achieve specific goals within specific time-frames and budgets. To effectively manage complex projects, two complementary network techniques are widely used:

• PERT (Program Evaluation & Review Technique)

• CPM (Critical Path Method)

These methods provide a visual representation of project tasks and their dependencies, enabling managers to identify critical activities, estimate completion times, and optimize resource allocation.

PERT – Program Evaluation & Review Technique

• Short history

• Developed in 1957 by the U.S. Navy Special Projects Office in conjunction with Booz, Allen & Hamilton. This development was a direct response to the urgent need for a method to manage the highly complex and uncertain development of the Polaris missile program during the Cold War. The scale of this project was immense, involving approximately 250 prime contractors and over 9,000 subcontractors.

• Core purpose

• PERT is designed to model a project as a network of interconnected activities and significant events (milestones). Its primary goal is to help managers navigate project uncertainty, particularly regarding activity durations. It aims to answer five classic managerial questions:

When will the entire project finish? (Providing a probabilistic estimate of completion time).
When should every activity realistically start and finish? (Calculating earliest and latest possible times).
Which activities MUST finish on time to avoid delaying the entire project? These are known as critical activities, as they lie on the critical path.
Can resources be strategically shifted from non-critical activities to critical activities without negatively impacting the overall project schedule? (Leveraging slack).
Where should top management concentrate their attention and oversight efforts to ensure project success?

• Practical language

• In a PERT network, Nodes represent events or milestones, marking the beginning or end of activities, or significant points in the project. They generally do not consume time or resources.

• Arcs (or arrows) represent activities, which are tasks that consume time (and, in the case of CPM, cost) and resources. Each activity has an estimated duration.

PERT Activity-Time Estimation

• Crucially, the time required for each activity in PERT is considered uncertain. Therefore, instead of a single deterministic estimate, a probability distribution (commonly the Beta distribution) is employed. The Beta distribution is often preferred because it can model various shapes, is bounded (activities have a finite minimum and maximum time), and is skewed, which often reflects real-world activity completion times.

• Three time estimates are gathered from experienced engineers or subject matter experts for each activity to capture this uncertainty:

a = optimistic time: The shortest possible time an activity could take, assuming everything goes exceptionally well (approximately a 1% chance the project could be completed within this time).
m = most likely time: The most probable time an activity will take under normal circumstances (the mode of the distribution).
b = pessimistic time: The longest possible time an activity might take, assuming unfavorable conditions or delays (approximately a 1% chance the project could be completed in this much time).

• Formulas (derived from the Beta distribution using the method of moments):

• Expected duration ( $t$ ): This formula provides a weighted average, giving more emphasis to the most likely estimate:
$t = \frac{a + 4m + b}{6}$

• Standard deviation ( $\sigma$ ): This measures the variability or spread of the activity's duration. A larger standard deviation indicates greater uncertainty.
$\sigma = \frac{b - a}{6}$

Pioneer Audio Case Study – Response 1000 Speaker Launch

• Ken Vaughn (President) seeks a flawless product promotion and launch for the new Response 1000 Speaker. This requires precise coordination of numerous tasks.

• Rollie Tillman (Director, Advertising Promotion) is responsible for planning these intertwined tasks, which include literature creation, sales-force training, media releases, in-store demonstrations, and more. A critical aspect of network diagrams is ensuring correct logical sequencing; thus, a dummy activity is often inserted where a dependency exists but no actual work or time is consumed. These are represented by dashed lines and have zero duration.

Step 1 – Activity List

• The project involves 12 real, time-consuming activities, plus 1 dummy activity (referred to as F in this case study).

• Precedence relationships (extracts illustrating dependencies):

• Activity D and Activity E can only begin after Activity A is completed.

• Activity I depends on both Activity D and the completion of the dummy activity F. This often signifies that I cannot start until D is done AND some other logical condition represented by F (which itself might depend on other activities) is met.

• The critical final launch activity L depends on the completion of activities I, G, and K, meaning all three must be finished before L can begin.

Step 2 – Draw the Network

• The project network is constructed with nodes numbered sequentially from 1 to 9. A dummy arc is explicitly drawn between nodes 5 and 6 (representing activity F) to correctly maintain the logical flow and precedence, ensuring that the network accurately reflects all dependencies even where no direct work is involved.

Step 3 – Compute Individual Activity Times (wk)

(Only key results are shown here; the formulas for expected time (t) and standard deviation (σ) were listed earlier.)

• Activities A, B, and C all have an expected duration of $t=2$ weeks and a standard deviation of $\sigma=0.33$ weeks, indicating relatively short and consistent durations.

• The tasks with the longest variance (and thus greatest uncertainty) are G ( $t=7, \sigma=2.00$ ) and H ( $t=6, \sigma=2.00$ ). These activities carry the most risk in terms of time variability and should be closely monitored.

• Activity L is a deterministic task, meaning its duration is fixed and certain ( $t=4, \sigma=0$ ). This implies no uncertainty in its completion time.

Step 4 – Forward & Backward Passes

• These two passes are fundamental algorithms used to determine earliest and latest possible start and finish times for all activities within the network.

• Forward pass formula: To compute earliest times, starting from the project's beginning. The Earliest Finish (EF) time for an activity is its Earliest Start (ES) time plus its expected duration (t): $EF = ES + t$ . This pass determines the earliest possible completion time for the entire project.

• Backward pass formula: To compute latest times, starting from the project's determined earliest finish time and working backward. The Latest Start (LS) time for an activity is its Latest Finish (LF) time minus its expected duration (t): $LS = LF - t$ . This pass identifies the latest an activity can start or finish without delaying the overall project.

• The final earliest project finish time calculated at the terminal node (node 9 in this case) is $19$ weeks. This is the shortest possible time the project can be completed, given the activity durations and dependencies.

Step 5 – Slack & Critical Path

• Slack (or Float) is the amount of time an activity can be delayed without delaying the entire project. It is calculated as the difference between the Latest Start and Earliest Start times, or Latest Finish and Earliest Finish times: $Slack = LS-ES = LF-EF$ .

• Activities with zero slack are on the unique critical path. These are the activities that MUST finish on time; any delay in a critical activity will directly delay the entire project. For the Pioneer Audio Case Study, the critical path is: $C \rightarrow J \rightarrow K \rightarrow L$ . This path represents the longest sequence of activities from start to finish.

• Non-critical activities and their respective slack allowances:

• A (5 wk), B (4 wk), D (6 wk), E (5 wk), G (6 wk), H (4 wk), I (5 wk). These activities have flexibility and can be delayed by the indicated weeks without affecting the 19-week project completion target.

Step 6 – Managerial Conclusions

The expected completion time for the entire project is $19$ weeks. This provides a baseline target for the launch.
Managers must closely monitor activities C, J, K, and L as they form the critical path. Any unexpected slip or delay in these activities will directly postpone the full product launch.
To accelerate the project if needed, resources (e.g., personnel, equipment) can be borrowed or reallocated from non-critical activities (A, B, D, E, G, H, I) to critical ones. This rebalancing allows for potential project compression without immediately increasing costs or risks.
Non-critical activities possess inherent slack. They may slip by an amount up to their allocated slack without jeopardizing the target project completion date. This flexibility can be useful for managing resource leveling or addressing minor unforeseen issues.
The detailed ES (Earliest Start) and EF (Earliest Finish) schedules for all activities are fully defined. This comprehensive scheduling enables precise planning for workforce deployment, material procurement, and overall logistical management.

CPM – Critical Path Method

• Developed concurrently with PERT in 1957, but by Remington Rand and Du Pont. CPM was specifically created for industrial projects, particularly for chemical-plant maintenance and construction, where activity durations were more predictable and cost considerations were paramount.

• CPM goes beyond time estimates by adding an explicit cost dimension to the PERT time network. This allows managers to analyze the trade-offs between project duration and total project cost.

• It requires reasonably accurate direct-cost data for each activity at both normal and crashed levels. This precision makes it highly suitable for projects where historical data or precise cost estimations are available.

• CPM utilizes the same precedence logic and node/arc representation as PERT, making the transition between the two methodologies straightforward if cost data is available.

Normal vs. Crash Concepts

• Normal time/cost: This represents the typical duration and associated cost of an activity under routine and efficient operating conditions. It mirrors PERT’s “most likely” estimate but is treated as a deterministic value in CPM.

• Crash time/cost: This refers to the minimum feasible duration for an activity if additional money is spent to expedite it. Crashing involves incurring extra direct costs (e.g., through expedited crews, overtime pay, premium freight, using more resources, or specialized equipment) to shorten the activity's duration. It is assumed that there is a practical limit to how much an activity can be shortened.

• The relationship between time and cost in crashing is generally convex: each additional week saved in an activity’s duration costs progressively more money. This is because initial reductions are usually achieved by the most efficient means, while further reductions require increasingly expensive measures.

CPM Worked Example – Chapel Hill Expressway (Concrete Catch Basin)

• This example involves activities A–E, each with a defined normal duration and a shorter crash duration, along with their associated costs.

• Normal plan:

• The critical path for the normal scenario is A–C–E. The project finishes in $16$ weeks, with a total direct cost of $35,000$ .

• Fully crashed plan:

• Even when fully crashed, the critical path remains A–C–E. The project duration is reduced to $10$ weeks, but the total direct cost increases significantly to $47,000$ . This demonstrates the upper bound of cost for the minimum duration.

Time–Cost Trade-off Table (Normal vs Crash)

• This table (not fully displayed here, but conceptually central) would detail each activity's normal time/cost, crash time/cost, max reduction (normal time - crash time), and the associated cost per week of reduction. This cost per week of reduction is crucial for intelligent crashing.

• Based on such a table, the cheapest per-week reduction is achieved by activity E ( $333$ /wk), indicating it provides the most cost-effective way to shorten the project duration. The highest cost per week of reduction is activity B ( $3,000$ /wk), meaning it's very expensive to accelerate.

Intelligent Crashing Process (Table 2)

This process outlines how to systematically reduce project duration while minimizing additional costs, by making incremental reductions strategically:

Start at the normal project duration and cost: Begin with the baseline of $16$ weeks and a direct cost of $35,000$ .
Identify and crash the cheapest activity on the current critical path: The critical path is initially A–C–E. Activity E is the cheapest to crash ( $333$ /wk). Crash E by its maximum of $3$ weeks. New duration: $13$ weeks ( $16 - 3$ ). New cost: $36,000$ (added $333 \times 3 = 1,000$ to the original $35,000$ ).
Re-evaluate the critical path and crash the next cheapest: The network’s critical path is still A–C–E. The next cheapest activity on this path is activity C ( $1,000$ /wk). Crash C by $1$ week. New duration: $12$ weeks ( $13 - 1$ ). New cost: $37,000$ (added $1,000$ ).
A new critical path emerges – address it: At this point, a new critical path emerges: B–D, which also has a duration of $12$ weeks. To reduce the overall project time further, both critical paths must be shortened simultaneously. Identify the cheapest activity on the new critical path (B–D): Activity D ( $1,500$ /wk). Crash D by $2$ weeks. New duration: $10$ weeks ( $12 - 2$ ). New cost: $40,000$ (added $1,500 \times 2 = 3,000$ ).
Address multiple critical paths simultaneously: Now, multiple critical paths exist (A–C–E and B–D), both with a duration of $10$ weeks. To achieve further reduction, activities on both paths must be crashed. Crash activity A ( $2,000$ /wk) on the A–C–E path by its remaining $2$ weeks, while simultaneously accounting for the need to maintain or reduce the B-D path. The duration remains $10$ weeks (as further reduction would require crashing activities on both paths simultaneously for a higher combined cost, or if no more activities can be crashed). New cost: $44,000$ (added $2,000 \times 2 = 4,000$ ).

• The sponsor’s target of a $10$ -week duration was achieved at a total cost of $44,000$ . This is significantly less than the original fully crashed cost of $47,000$ and well within an incremental budget of $12,000$ over normal (only $9,000$ over normal was actually spent). This highlights the efficiency of intelligent crashing over arbitrary crashing.

Comparative Insights & Managerial Takeaways

• PERT primarily focuses on managing project time uncertainty by using probabilistic duration estimates, providing a range of possible completion times and the likelihood of meeting deadlines.

• CPM excels at integrating time-cost trade-offs, allowing managers to find the most cost-effective way to compress a project or to determine the minimum cost for a given deadline.

• Both techniques fundamentally rely on identical precedence networks, meaning the visual representation of tasks and their dependencies. Conversion from a PERT network to a CPM network (and vice-versa) is conceptually trivial if the necessary cost or probabilistic data is acquired.

• Critical path identification is a core output of both methods and provides crucial insights:

• Schedule risk: Any delay in an activity located on the critical path will directly translate into a delay for the entire project, making these activities high-priority for monitoring and management.

• Budget priorities: When considering crashing, budget allocation for acceleration only yields benefits if the crashing is applied to activities that are on (or become part of) the critical path. Crashing non-critical activities simply consumes resources without accelerating the project.

• Slack times are invaluable buffers. They indicate how much flexibility exists for non-critical activities. This flexibility can be leveraged for resource reallocation (moving resources from activities with high slack to critical ones) and are excellent indicators of overall scheduling flexibility within the project.

• Ethical & practical considerations are vital in applying these techniques:

• Over-crashing (reducing durations excessively) can have severe negative consequences, including endangering worker safety due to rushed conditions, compromising product quality due to shortcuts, and damaging team morale.

• Transparent communication of slack within the project team is essential. Withholding information about slack can lead to hidden schedule abuse (e.g., team members intentionally delaying tasks knowing they have buffer time), which can undermine overall project efficiency and trust.

• Probabilistic estimates (in PERT) serve as a constant reminder to decision-makers of the inherent uncertainty in project durations. It is critical never to promise deterministic completion dates without also providing confidence intervals, acknowledging the statistical probability of meeting that deadline.

Key Equations & Reminders

• Expected activity time: $t = \frac{a+4m+b}{6}$

• Activity standard deviation: $\sigma = \frac{b-a}{6}$

• Earliest Finish: $EF = ES + t$

• Latest Start: $LS = LF - t$

• Slack: $SL = LS-ES = LF-EF$

• Variance of an activity: $\sigma^2$

• Project duration variance (sum of variances for independent activities on the critical path): $\sigma<em>{proj}^2 = \sum</em>{i\in CP} \sigma_i^2$

• This project duration variance is useful for calculating the probability of meeting a deadline, typically using a normal approximation (Z-score calculation).

Study Tips

• Always sketch the network diagram visually before performing any computations or filling in numbers. Visual clarity significantly helps in understanding dependencies and avoids algebraic errors.

• It is crucial to recompute the critical path after each crashing step in CPM, as the critical path can and often will shift as activity durations are altered.

• Use color-coding within your network diagrams for better visual distinction: for instance, red for critical activities, green for non-critical activities, and dashed lines for dummy activities.

• In exams or assignments, clearly state the formulas you are using before substituting numbers. Graders often award partial credit for demonstrating a correct understanding of the method, even if a numerical error occurs.