Project Management: PERT/CPM, Critical Path, Network Diagrams

TAKEAWAYS:

• PERT/CPM are quantitative techniques to model project activities as networks and identify the longest path (critical path) that determines project duration.
• A network diagram visually represents dependencies between activities using nodes (events) and arrows (activities).
• Critical path is the sequence of activities with zero slack, directly impacting project completion time.
• Time estimates (optimistic, most likely, pessimistic) are used in PERT to calculate expected time and variance for probabilistic analysis.
• CPM uses deterministic time estimates (fixed durations) to find the critical path and optimize resource allocation.
• Slack (or float) measures how much an activity can delay without affecting the project’s deadline.

1. Introduction to Project Management

Projects are temporary endeavors with defined start and end dates, aiming to deliver unique outputs (e.g., building a bridge, launching a software product, or organizing an event). Operations Research (OR) provides tools like PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) to:

• Break projects into activities and events.
• Identify dependencies between tasks.
• Calculate project duration and critical path.
• Optimize resource allocation and costs.

Key Definitions

2. PERT vs. CPM: Key Differences

3. Network Diagrams

A network diagram visually maps project activities and their dependencies. It consists of:

• Nodes (Circles): Represent events (start/end of activities).
• Arrows: Represent activities (directed from start to end node).
• Dumy Activities: Used to maintain logical sequence when activities are independent but must follow a specific order.

How to Draw a Network Diagram

1. Identify all activities and their predecessors.
2. Draw nodes for start and end events.
3. Draw arrows for activities, labeling them with:
   • Activity name.
   • Duration (in weeks/days).
4. Add dummy activities if needed (e.g., Activity A → Activity B → Activity C, but A and C are independent).

Example: Network Diagram for a Simple Project

Consider a project with the following activities and dependencies:

graph TD
    A["Start"] --> B["Activity A (5)"]
    A --> C["Activity C (4)"]
    B --> D["Activity B (3)"]
    C --> D
    D --> E["Activity D (2)"]
    E --> F["End"]

4. Calculating Expected Time and Variance (PERT)

PERT uses three time estimates for each activity:

• Optimistic (a): Best-case scenario time.
• Most Likely (m): Most probable time.
• Pessimistic (b): Worst-case scenario time.

Expected Time (TE): $$TE=\frac{a+4m+b}{6}$$

Variance (σ²): $${\sigma }^2={\left(\frac{b-a}{6}\right)}^2$$

Standard Deviation (σ): $$\sigma =\sqrt{{\sigma }^2}$$

Worked Example: PERT Calculation

Given the following activity data for a project:

Step 1: Calculate TE and σ for each activity.

• Activity A: $$T{E}_A=\frac{2+4(4)+6}{6}=\frac{2+16+6}{6}=\frac{24}{6}=4\text{ weeks}$$ $${\sigma }_A^2={\left(\frac{6-2}{6}\right)}^2={\left(\frac{4}{6}\right)}^2={\left(\frac{2}{3}\right)}^2=\frac{4}{9}$$

• Activity B: $$T{E}_B=\frac{3+4(5)+7}{6}=\frac{3+20+7}{6}=\frac{30}{6}=5\text{ weeks}$$ $${\sigma }_B^2={\left(\frac{7-3}{6}\right)}^2={\left(\frac{4}{6}\right)}^2=\frac{4}{9}$$

• Activity C: $$T{E}_C=\frac{1+4(3)+5}{6}=\frac{1+12+5}{6}=\frac{18}{6}=3\text{ weeks}$$ $${\sigma }_C^2={\left(\frac{5-1}{6}\right)}^2={\left(\frac{4}{6}\right)}^2=\frac{4}{9}$$

Step 2: Draw the network diagram (assume A → B → C).

Step 3: Identify the critical path.

• Path 1: A → B → C = 4 + 5 + 3 = 12 weeks.
• This is the critical path (no slack).

5. CPM: Critical Path and Slack Calculation

CPM uses fixed durations to find the critical path and calculate slack for non-critical activities.

Key Terms:

• Early Start (ES): Earliest time an activity can start.
• Early Finish (EF): ES + Duration.
• Late Start (LS): Latest time an activity can start without delaying the project.
• Late Finish (LF): LS + Duration.
• Slack (Float): LS - ES (or LF - EF).

Worked Example: CPM Calculation

Consider the following project with fixed durations:

Step 1: Calculate ES and EF for all activities.

• Activity A:
  • ES = 0 (start of project).
  • EF = ES + Duration = 0 + 5 = 5 weeks.
• Activity B (depends on A):
  • ES = EF of A = 5 weeks.
  • EF = 5 + 3 = 8 weeks.
• Activity C (depends on A):
  • ES = EF of A = 5 weeks.
  • EF = 5 + 4 = 9 weeks.
• Activity D (depends on B and C):
  • ES = max(EF of B, EF of C) = max(8, 9) = 9 weeks.
  • EF = 9 + 2 = 11 weeks.

Step 2: Calculate LF and LS (working backward).

• Activity D:
  • LF = Project end time = 11 weeks.
  • LS = LF - Duration = 11 - 2 = 9 weeks.
• Activity C:
  • LF = min(LF of D) = 11 weeks (since D depends on C).
  • LS = LF - Duration = 11 - 4 = 7 weeks.
• Activity B:
  • LF = min(LF of D) = 11 weeks.
  • LS = LF - Duration = 11 - 3 = 8 weeks.
• Activity A:
  • LF = min(LF of B, LF of C) = min(11, 11) = 11 weeks.
  • LS = LF - Duration = 11 - 5 = 6 weeks.

Step 3: Calculate Slack for each activity.

• Activity A: LS - ES = 6 - 0 = 6 weeks slack.
• Activity B: LS - ES = 8 - 5 = 3 weeks slack.
• Activity C: LS - ES = 7 - 5 = 2 weeks slack.
• Activity D: LS - ES = 9 - 9 = 0 weeks slack (critical activity).

Critical Path: A → C → D (total duration = 5 + 4 + 2 = 11 weeks).

6. Real-World Applications

In the Real World

1. Daraz (E-commerce Platform)

   • Idea Used: CPM for Order Fulfillment
   • How: Daraz uses CPM to schedule warehouse activities (packing, shipping, inventory updates) to ensure orders are delivered on time. The critical path might include:
     • Receiving order (Activity A).
     • Picking items from inventory (Activity B, depends on A).
     • Packing the order (Activity C, depends on B).
     • Shipping to customer (Activity D, depends on C).
   • Worked Example: If Activity C (packing) takes 2 hours and is on the critical path, delaying it by 1 hour delays the entire order delivery.

2. Pathao (Ride-Hailing App)

   • Idea Used: PERT for Ride Dispatch
   • How: Pathao uses PERT to estimate ride dispatch times, accounting for:
     • Driver availability (optimistic: 2 mins, most likely: 5 mins, pessimistic: 10 mins).
     • Traffic conditions (optimistic: 3 mins, most likely: 8 mins, pessimistic: 15 mins).
   • Worked Example: For a ride from Thapathali to Gyaneshwor, the critical path might be:
     • Driver assigned (TE = 4 mins).
     • Driver reaches pickup point (TE = 7 mins).
     • Total expected time = 4 + 7 = 11 mins.

3. Nepal Electricity Authority (NEPSE) – Dam Construction

   • Idea Used: CPM for Dam Construction Phases
   • How: NEPSE uses CPM to schedule activities like:
     • Excavation (Activity A).
     • Foundation laying (Activity B, depends on A).
     • Concrete pouring (Activity C, depends on B).
     • Dam completion (Activity D, depends on C).
   • Worked Example: If Activity C (concrete pouring) takes 6 months and is on the critical path, delaying it by 1 month delays the dam’s completion by 1 month.

Visual: Critical Path in Daraz Order Fulfillment

graph TD
    Start --> A["Receive Order (0)"]
    A --> B["Pick Items (2)"]
    A --> C["Check Inventory (1)"]
    B --> D["Pack Order (3)"]
    C --> D
    D --> E["Ship Order (1)"]
    E --> End
    legend["Critical Path: A → D → E (Total: 6 hours)"]

7. Advantages and Disadvantages

8. Exam Tips

1. Understand the Difference Between PERT and CPM:

   • PERT uses probabilistic estimates (3-time estimates).
   • CPM uses fixed durations.
   • Always check the question to determine which method to apply.

2. Draw Network Diagrams Clearly:

   • Label all activities, durations, and dependencies.
   • Use dummy activities where necessary to maintain logical flow.

3. Calculate Expected Time and Variance (PERT):

   • Memorize the formulas: $$TE=\frac{a+4m+b}{6},{\sigma }^2={\left(\frac{b-a}{6}\right)}^2$$
   • Practice calculating TE and variance for multiple activities.

4. Identify the Critical Path:

   • For CPM, calculate ES, EF, LS, LF, and slack for all activities.
   • The critical path is the sequence with zero slack.

5. Worked Examples Are Key:

   • Past exam questions often provide activity tables with predecessors and durations.
   • Always draw the network diagram and label all paths to identify the critical path.

6. Real-World Context:

   • Relate PERT/CPM to construction, IT projects, or logistics (e.g., Daraz, Pathao).
   • Explain how delays in critical activities impact project timelines.

Sample Exam Question (Solved)

Question: A project consists of the following activities with given durations and dependencies. Draw the network diagram and determine the critical path and project duration.

Solution:

1. Network Diagram:

2. CPM Calculations:

   • Activity A: ES = 0, EF = 5.
   • Activity B: ES = 5, EF = 8.
   • Activity C: ES = 5, EF = 9.
   • Activity D: ES = max(8, 9) = 9, EF = 11.
   • Slack:
     • A: LS = 6, ES = 0 → Slack = 6.
     • B: LS = 8, ES = 5 → Slack = 3.
     • C: LS = 7, ES = 5 → Slack = 2.
     • D: LS = 9, ES = 9 → Slack = 0 (critical).

3. Critical Path: A → C → D (Total Duration = 5 + 4 + 2 = 11 weeks).

Common Mistakes to Avoid

• Ignoring dummy activities: If two activities are independent but must follow a specific order, add a dummy activity.
• Incorrectly calculating ES/EF: Always start from the start node and move forward.
• Misidentifying the critical path: The critical path is the one with zero slack.
• Confusing PERT and CPM: PERT uses probabilistic estimates; CPM uses fixed durations.