Network analysis pert and cpm
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PERT CPM network analysis operations research explained
Here's a comprehensive overview of Network Analysis: PERT and CPM.
Network Analysis: PERT & CPM
Network analysis is a project planning and scheduling technique that represents a project as a network of activities and events to determine the optimal sequence, timing, and resource allocation.
Key Definitions
| Term | Meaning |
|---|---|
| Activity | A task that consumes time and/or resources |
| Event (Node) | A point in time marking start/end of activities |
| Network/Arrow Diagram | Graphical representation of activities and dependencies |
| Critical Path | Longest path through the network; determines project duration |
| Float/Slack | Leeway an activity has without delaying the project |
PERT vs CPM — Comparison
| Feature | PERT | CPM |
|---|---|---|
| Full form | Program Evaluation and Review Technique | Critical Path Method |
| Developed by | US Navy (Polaris missile, 1958) | DuPont/Remington Rand (1957) |
| Activity duration | Probabilistic (uncertain) | Deterministic (known) |
| Focus | Time | Time + Cost |
| Best for | R&D, new projects | Construction, repetitive projects |
| Time estimates | 3 estimates (optimistic, most likely, pessimistic) | 1 estimate |
CPM — Critical Path Method
Step-by-Step Process
1. List all activities and their predecessor relationships.
2. Draw the Network Diagram
- Activities on Arrow (AOA) or Activities on Node (AON)
- Dummy activities used in AOA to show dependencies without consuming time
3. Assign durations to each activity.
4. Forward Pass — Calculate Earliest Start (ES) and Earliest Finish (EF):
EF = ES + Duration
ES of successor = Maximum EF of all predecessors
5. Backward Pass — Calculate Latest Start (LS) and Latest Finish (LF):
LF of last event = EF of last event
LS = LF − Duration
LF of predecessor = Minimum LS of all successors
6. Calculate Float/Slack:
Total Float (TF) = LS − ES = LF − EF
Free Float (FF) = ES of successor − EF of activity
7. Identify Critical Path: All activities where TF = 0
Example
| Activity | Predecessor | Duration (days) |
|---|---|---|
| A | — | 3 |
| B | — | 5 |
| C | A | 4 |
| D | B | 2 |
| E | C, D | 6 |
- Path A→C→E = 3+4+6 = 13 days
- Path B→D→E = 5+2+6 = 13 days
- Both are critical paths (TF = 0 for all activities)
PERT — Program Evaluation and Review Technique
Three Time Estimates
For each activity, three durations are estimated:
| Symbol | Name | Meaning |
|---|---|---|
| a (t_o) | Optimistic time | Best-case scenario |
| m (t_m) | Most likely time | Most probable duration |
| b (t_p) | Pessimistic time | Worst-case scenario |
Expected Time (te) — Beta Distribution
$$t_e = \frac{a + 4m + b}{6}$$
Variance (σ²) of an Activity
$$\sigma^2 = \left(\frac{b - a}{6}\right)^2$$
Project Duration & Probability
- Expected project duration = Sum of t_e along the critical path
- Variance of project = Sum of σ² along the critical path
- Standard deviation σ = √(project variance)
To find the probability of completing by a deadline T:
$$Z = \frac{T - T_e}{\sigma}$$
Use the Z-value with a standard normal distribution table.
Network Diagram Rules (AOA)
- Each activity is represented by an arrow (→)
- Each node (circle) is an event
- Network must have a single start and single end node
- No loops allowed
- Dummy activities (dashed arrows, zero duration) show logical dependencies without consuming time
- No two activities can share the same start and end nodes (use dummies to resolve this)
Float/Slack — Types
| Type | Formula | Meaning |
|---|---|---|
| Total Float | LF − EF (or LS − ES) | Max delay without affecting project completion |
| Free Float | ES(next) − EF(current) | Delay without affecting successor's ES |
| Independent Float | ES(next) − LF(prev) − Duration | Delay without affecting any activity |
Critical activities have Total Float = 0
CPM — Cost-Time Trade-off (Crashing)
CPM also models cost:
- Normal time / Normal cost: Planned duration and cost
- Crash time / Crash cost: Minimum time at maximum cost
- Cost slope = (Crash cost − Normal cost) / (Normal time − Crash time)
Crashing strategy: Crash activities on the critical path with the lowest cost slope first.
PERT — Worked Example
| Activity | a | m | b | t_e | σ² |
|---|---|---|---|---|---|
| A | 1 | 3 | 5 | 3 | 4/9 |
| B | 2 | 4 | 12 | 4.67 | 2.78 |
| C | 3 | 5 | 7 | 5 | 4/9 |
Critical path A→C: T_e = 3+5 = 8 weeks, σ² = 4/9 + 4/9 = 8/9, σ = 0.943
Probability of completing in ≤ 9 weeks:
Z = (9−8)/0.943 = 1.06 → P ≈ 85.5%
Key Differences: CPM vs PERT Summary
| CPM | PERT | |
|---|---|---|
| Time | Deterministic | Probabilistic |
| Cost analysis | Yes | No |
| Best use | Civil, construction | Research, defense |
| Uncertainty handled | No | Yes |
| Output | Critical path + cost | Critical path + probability |
Applications
- CPM: Construction projects, plant maintenance, software development
- PERT: Aerospace, defense, R&D projects, new product development
- Both are used for: project scheduling, resource allocation, delay analysis, progress monitoring
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