Project Management techniques like CPM and PERT help plan complex projects with multiple activities. They break down projects, determine sequences, and identify the critical path - the longest sequence of dependent activities that sets the project's minimum completion time.

These methods calculate float times, allowing for flexible scheduling of non-critical activities. PERT adds probability calculations, estimating project completion dates based on optimistic, most likely, and pessimistic time estimates for each activity.

Critical Path Method and PERT

Fundamentals of CPM and PERT

• CPM and PERT project management techniques plan, schedule, and control complex projects with multiple interdependent activities
• CPM uses deterministic method with fixed activity durations
• PERT employs probabilistic method considering uncertainty in activity durations
• Both methods break down projects into individual activities, determine sequences and dependencies, and identify critical path
• Critical path represents longest sequence of dependent activities determining minimum project completion time
• Float (slack time) indicates amount of time an activity can be delayed without affecting overall project completion
• PERT calculates expected activity durations and project completion probabilities using three time estimates (optimistic, most likely, pessimistic)
• These techniques facilitate resource allocation, schedule compression, and risk assessment in project management

PERT Calculations and Probability

• PERT uses three time estimates to calculate expected activity duration: $Expected Duration = \frac{a + 4m + b}{6}$ Where a = optimistic, m = most likely, b = pessimistic
• Variance for each activity calculated as: $Variance = (\frac{b - a}{6})^2$
• Central limit theorem applied in PERT assumes normal distribution for project completion time
• Probability calculations enable estimation of project completion by specific dates
• Example: Activity A has estimates a=3, m=5, b=7 days Expected duration = (3 + 45 + 7) / 6 = 5 days Variance = ((7 - 3) / 6)^2 = 0.44

Project Network Diagrams

Activity-on-Node (AON) Representation

• AON diagrams represent activities as nodes (boxes) and dependencies as connecting arrows
• Forward pass calculations determine early start (ES) and early finish (EF) times
• Backward pass calculations determine late start (LS) and late finish (LF) times
• Precedence relationships between activities include:
  • Finish-to-Start (FS) most common type
  • Start-to-Start (SS) successor can start once predecessor starts
  • Finish-to-Finish (FF) successor can finish once predecessor finishes
  • Start-to-Finish (SF) rarely used, successor finishes after predecessor starts
• Example AON node format:

  |  Activity Name   |
  | ES |      | EF |
  |    | Float |    |
  | LS |      | LF |
  

Activity-on-Arrow (AOA) Representation

• AOA diagrams represent activities as arrows and events as nodes (circles)
• Dummy activities may be required to represent complex logical relationships
• Events represent the start and finish of activities
• AOA diagrams use similar forward and backward pass calculations as AON
• Example AOA format:

  O------->O------->O
  (Event) (Activity) (Event)
  

• Network diagrams constructed to minimize crossed lines and clearly show project's logical flow
• Example: A software development project with activities like requirements gathering, design, coding, testing, and deployment

Critical Path and Project Duration

Critical Path Calculation

• Critical path identified as longest path through network diagram from start to finish
• Project duration determined by total duration of activities on critical path
• Critical activities have zero float time
• Steps to calculate critical path:
  1. Construct network diagram
  2. Perform forward pass to calculate early start and early finish times
  3. Perform backward pass to calculate late start and late finish times
  4. Identify activities with zero float (critical activities)
  5. Connect critical activities to form critical path
• Example: A construction project with critical path activities foundation laying, framing, roofing, and interior finishing

Float Time Calculation

• Float times calculated as difference between activity's latest allowable time and earliest expected time
• Total float represents amount of time an activity can be delayed without delaying project completion date
• Free float indicates amount of time an activity can be delayed without delaying early start of any succeeding activity
• Calculation formulas: $Total Float = LS - ES = LF - EF$ $Free Float = ES(successor) - EF(activity)$
• Example: Activity B with ES=5, EF=8, LS=7, LF=10 Total float = 7 - 5 = 10 - 8 = 2 days If successor activity C has ES=9, free float of B = 9 - 8 = 1 day

Activity Duration Impact

Critical Path Changes

• Changes in activity durations can alter critical path and overall project duration
• Increasing duration of critical activity directly increases project completion time
• Decreasing duration of critical activity may:
  • Shorten project duration
  • Shift critical path to another sequence of activities
• Changes in non-critical activities' durations may consume float time without affecting project completion date, unless float exceeded
• Example: Reducing duration of a critical software testing phase from 10 to 8 days shortens project by 2 days if no other critical path emerges

Schedule Compression Techniques

• Crashing involves reducing project duration by adding resources to critical activities, often at increased cost
  • Example: Adding extra workers to complete a construction task faster
• Fast-tracking involves performing activities in parallel that were originally planned to be sequential
  • Potentially introduces new risks
  • Example: Starting software development before design phase is fully complete
• Sensitivity analysis assesses how changes in individual activity durations affect overall project schedule and critical path
• Resource leveling required when changes in activity durations lead to resource conflicts or overallocation
  • Example: Adjusting schedule to prevent overallocation of specialized equipment across multiple activities