5.2 CPM, Float, Updating & Schedule Compression

5.2 CPM, Float, Updating & Schedule Compression

Critical Path Method (CPM) Mathematical Network Analysis

The Critical Path Method is a deterministic scheduling technique that calculates the earliest and latest possible start and finish dates for each project activity, determines total and free float, and identifies the sequence of critical activities that dictates the overall project duration.

Node Architecture

In standard Activity-on-Node (AON) representation, each node contains six fundamental data fields:

┌───────────────────────────────────────────────┐
│  Early Start (ES)    │   Duration (D)   │   Early Finish (EF)   │
├───────────────────────────────────────────────┤
│                 ACTIVITY DESCRIPTION          │
├───────────────────────────────────────────────┤
│  Late Start (LS)     │   Total Float    │   Late Finish (LF)    │
└───────────────────────────────────────────────┘

Forward Pass Calculations (Early Dates)

The forward pass proceeds from left to right through the network to determine the earliest possible dates each activity can begin and complete based on dependency logic:

1. Initial Activity: For the starting activity ($A$), $\text{Early Start } (ES_A) = 0$ (or Day 1 in calendar-day conventions; Day 0 convention is standard for mathematical calculations).
2. Early Finish (EF): $\text{EF} = \text{ES} + \text{Duration}$
3. Successor Early Start: For any successor activity with multiple predecessors, the Early Start is the maximum of the Early Finishes of all immediate predecessors: $\text{ES}_{\text{successor}} = \max(\text{EF}_{\text{predecessors}})$

Backward Pass Calculations (Late Dates)

The backward pass proceeds from right to left through the network, starting at the target project completion date, to determine the latest dates an activity can begin and finish without extending the project duration:

1. Terminal Activity: For the final activity ($Z$), $\text{Late Finish } (LF_Z) = \text{Early Finish } (EF_Z)$ (unless a fixed contractual milestone is imposed).
2. Late Start (LS): $\text{LS} = \text{LF} - \text{Duration}$
3. Predecessor Late Finish: For any predecessor activity followed by multiple successors, the Late Finish is the minimum of the Late Starts of all immediate successors: $\text{LF}_{\text{predecessor}} = \min(\text{LS}_{\text{successors}})$

Float (Slack) Calculations

Float represents schedule flexibility. There are two primary types of float evaluated in commercial contracting:

• Total Float (TF): The amount of time an activity can be delayed from its Early Start without delaying the overall project completion date (or violating a mandatory completion milestone): $\text{Total Float (TF)} = \text{LS} - \text{ES} = \text{LF} - \text{EF}$
• Free Float (FF): The amount of time an activity can be delayed without delaying the Early Start of any immediate successor activity: $\text{Free Float (FF)} = \min(\text{ES}_{\text{successors}}) - \text{EF}_{\text{current activity}}$
• Interfering Float: The portion of Total Float that, if utilized by an activity, will delay the Early Start of subsequent activities without delaying the project completion date ($\text{Interfering Float} = \text{TF} - \text{FF}$).

Worked Example: 6-Activity Network

Consider a commercial substation footing project with the following activities:

• Step 1 (Forward Pass):
  • Activity A: $ES=0, EF=4$.
  • Activity B: $ES=4, EF=10$.
  • Activity C: $ES=0, EF=7$.
  • Activity D (Predecessors B [$EF=10$] and C [$EF=7$]): $ES = \max(10, 7) = 10; EF = 10 + 5 = 15$.
  • Activity E (Predecessor B [$EF=10$]): $ES=10; EF = 10 + 3 = 13$.
  • Activity F (Predecessors D [$EF=15$] and E [$EF=13$]): $ES = \max(15, 13) = 15; EF = 15 + 4 = 19$.
• Step 2 (Backward Pass): Total Project Duration = 19 Days. $LF_F = 19, LS_F = 19 - 4 = 15$.
  • Activity D: $LF = LS_F = 15; LS = 15 - 5 = 10$.
  • Activity E: $LF = LS_F = 15; LS = 15 - 3 = 12$.
  • Activity B (Successors D [$LS=10$] and E [$LS=12$]): $LF = \min(10, 12) = 10; LS = 10 - 6 = 4$.
  • Activity C: $LF = LS_D = 10; LS = 10 - 7 = 3$.
  • Activity A: $LF = LS_B = 4; LS = 4 - 4 = 0$.
• Step 3 (Float & Critical Path): Activities A, B, D, and F have $\text{Total Float} = 0$. The Critical Path is A → B → D → F (19 Days). Activity C has $TF = 3$ days and $FF = 10 - 7 = 3$ days. Activity E has $TF = 2$ days and $FF = 15 - 13 = 2$ days.

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The Critical Path Definition & Project Impact

The Critical Path is defined as the longest continuous sequence of activities through the network diagram that establishes the minimum total project duration. Activities on the critical path have zero (or minimum) Total Float.

Key Principles of the Critical Path:

1. Zero Flexibility: Any delay in a critical path activity causes a day-for-day delay in the final project completion date unless schedule compression is applied.
2. Near-Critical Paths: Paths with very low total float (e.g., 1 to 3 days) must be monitored closely, as minor delays can consume their float and cause them to become critical.
3. Float Ownership: Standard construction contracts (including AIA A201 general conditions) generally treat project float as a shared, expiring resource that belongs to the project rather than exclusively to the owner or general contractor, available on a first-come, first-served basis.

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Schedule Compression Techniques: Crashing vs. Fast-Tracking

When a project falls behind schedule or the owner demands an accelerated completion date, the general contractor must evaluate schedule compression methods.

Calculating Cost Slope for Crashing

To compress a schedule at the lowest possible financial cost, the contractor calculates the Crash Cost Slope for each critical activity: $\text{Cost Slope} = \frac{\text{Crash Cost} - \text{Normal Cost}}{\text{Normal Duration} - \text{Crash Duration}} = \frac{\Delta \text{Cost}}{\Delta \text{Time}}$

Decision Rule: Always crash the activity on the critical path that has the lowest cost slope first, up to its maximum crash limit, before crashing higher-cost activities.