Critical Path Analysis (Edexcel A-Level Further Mathematics): Revision Notes

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11.1.3 Critical Path Analysis

Critical Path Analysis (CPA) is a method for analysing an activity network to determine the minimum project duration, identify critical activities, and calculate key timings. It also helps determine the minimum number of workers required to complete the project efficiently.

Key Concepts

Critical Path

The critical path is the longest path through the network. It determines the minimum time required to complete the project. Activities on the critical path have zero float.

Timings

  1. Earliest Event Time (EET): The earliest time an event can occur, considering all preceding activities.
  • Calculated through a forwards pass from start to finish.
  1. Latest Event Time (LET): The latest time an event can occur without delaying the project.
  • Calculated through a backwards pass from finish to start.
  1. Earliest Start Time (EST) and Earliest Finish Time (EFT):
  • EST: Earliest time an activity can start (equal to the EET of its preceding event).
  • EFT: Earliest time an activity can finish:
EFT=EST+Duration\text{EFT} = \text{EST} + \text{Duration}
  1. Latest Start Time (LST) and Latest Finish Time (LFT):
  • LFT: Latest time an activity can finish without delaying the project (equal to the LET of its succeeding event).
  • LST: Latest time an activity can start:
LST=LFT−Duration\text{LST} = \text{LFT} - \text{Duration}
  1. Float: The amount of time an activity can be delayed without affecting the project duration.
Float=LFT−EFT\text{Float} = \text{LFT} - \text{EFT}

Steps in Critical Path Analysis

  1. Draw the Activity Network
  2. Forward Pass
  3. Backward Pass
  4. Identify Critical Activities
  5. Calculate Lower Bound for Workers
  1. Draw the Activity Network:
  • Ensure all dependencies are included, and activities are labelled with their durations.
  1. Forward Pass:
  • Start at the start node.
  • Calculate the EET for each event:
EET of current event=max⁡(EFT of all preceding activities)\text{EET of current event} = \max(\text{EFT of all preceding activities})
  1. Backward Pass:
  • Start at the end node.
  • Calculate the LET for each event:
LET of current event=min⁡(LST of all succeeding activities)\text{LET of current event} = \min(\text{LST of all succeeding activities})
  1. Identify Critical Activities:
  • Activities with zero float are critical.
  • The sequence of critical activities forms the critical path.
  1. Calculate Lower Bound for Workers:
  • The lower bound for workers is given by:
Lower Bound=⌈Total Activity TimeProject Duration⌉\text{Lower Bound} = \lceil \frac{\text{Total Activity Time}}{\text{Project Duration}} \rceil

Worked Example

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Question Consider the precedence table below:

ActivityDurationImmediate Predecessors
A3-
B2A
C4A
D5B, C
E3C
F6D, E

Step 1: Draw the Activity Network

Construct the network with a single start node, a single end node, and all dependencies shown.

Step 2: Forward Pass

  1. Start at AA: EET(A)=0+3=3\text{EET}(A) = 0 + 3 = 3
  2. BB: EET(B)=EET(A)+2=5\text{EET}(B) = \text{EET}(A) + 2 = 5
  3. CC: EET(C)=EET(A)+4=7\text{EET}(C) = \text{EET}(A) + 4 = 7
  4. DD: EET(D)=max⁡(EET(B),EET(C))+5=12\text{EET}(D) = \max(\text{EET}(B), \text{EET}(C)) + 5 = 12
  5. EE: EET(E)=EET(C)+3=10\text{EET}(E) = \text{EET}(C) + 3 = 10
  6. FF: EET(F)=max⁡(EET(D),EET(E))+6=18\text{EET}(F) = \max(\text{EET}(D), \text{EET}(E)) + 6 = 18

Step 3: Backward Pass

  1. Start at FF: LET(F)=EET(F)=18\text{LET}(F) = \text{EET}(F) = 18
  2. DD: LET(D)=LET(F)−6=12\text{LET}(D) = \text{LET}(F) - 6 = 12
  3. EE: LET(E)=LET(F)−6=12\text{LET}(E) = \text{LET}(F) - 6 = 12
  4. BB: LET(B)=LET(D)−5=7\text{LET}(B) = \text{LET}(D) - 5 = 7
  5. CC: LET(C)=min⁡(LET(D),LET(E))−4=8\text{LET}(C) = \min(\text{LET}(D), \text{LET}(E)) - 4 = 8
  6. AA: LET(A)=min⁡(LET(B),LET(C))−3=3\text{LET}(A) = \min(\text{LET}(B), \text{LET}(C)) - 3 = 3

Step 4: Identify Critical Activities

  • Activities with : EET=LET\text{EET} = \text{LET}
  • A,C,D,FA, C, D, F Critical Path: A → C → D → F

Project Duration: 18 units.

Step 5: Calculate Lower Bound for Workers

  1. Total Activity Time:
3(A)+2(B)+4(C)+5(D)+3(E)+6(F)=233 (A) + 2 (B) + 4 (C) + 5 (D) + 3 (E) + 6 (F) = 23
  1. Lower Bound for Workers:
Lower Bound=⌈2318⌉=2\text{Lower Bound} = \lceil \frac{23}{18} \rceil = 2

At least 2 workers are required to complete the project in the shortest possible time.

Note Summary

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Common Mistakes

  1. Incorrect Timings in Forwards/Backwards Pass Forgetting to take the maximum for EET or minimum for LET leads to incorrect critical paths.

  2. Confusion Between Float and Timings Misinterpreting float as the difference between EET and LET instead of LFT and EFT.

  3. Missing Dependencies Failing to include all immediate predecessors or successors can cause errors in the network.

  4. Incorrect Lower Bound Calculation Using project duration instead of total activity time or vice versa.

  5. Overlooking Critical Activities Not correctly identifying zero float activities as critical.

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Key Formulas/Theorems

  1. Earliest Finish Time (EFT):
EFT=EST+Duration\text{EFT} = \text{EST} + \text{Duration}
  1. Latest Start Time (LST):
LST=LFT−Duration\text{LST} = \text{LFT} - \text{Duration}
  1. Float:
Float=LFT−EFT\text{Float} = \text{LFT} - \text{EFT}
  1. Lower Bound for Workers:
Lower Bound=⌈Total Activity TimeProject Duration⌉\text{Lower Bound} = \lceil \frac{\text{Total Activity Time}}{\text{Project Duration}} \rceil

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