9.2 Project Scheduling, Critical Path Method (CPM) & PERT

Project Scheduling, Critical Path Method (CPM) & PERT

Building robust project schedules is essential for coordinating complex supply chain operations, such as facility relocations, new supplier onboarding, and global technology rollouts. Project scheduling translates scope deliverables into sequenced activities with estimated durations, logical dependencies, and assigned resources. Supply chain managers must master quantitative scheduling tools—specifically the Critical Path Method (CPM) and the Program Evaluation and Review Technique (PERT)—to manage schedule risk and make data-driven trade-offs.

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1. Network Diagrams & Activity Logic Dependencies

Project schedules are modeled graphically using Activity-on-Node (AON) network diagrams, where nodes represent activities and arrows depict precedence relationships. The logical dependencies between activities dictate execution sequence:

• Finish-to-Start (FS): The successor activity cannot start until the predecessor activity finishes (e.g., Supplier Audit must finish before Contract Award can start). This is the most common dependency.
• Start-to-Start (SS): The successor activity cannot start until the predecessor activity starts (e.g., System User Training can start once System Software Testing starts).
• Finish-to-Finish (FF): The successor activity cannot finish until the predecessor activity finishes (e.g., Equipment Documentation cannot finish until Equipment Installation finishes).
• Start-to-Finish (SF): The successor activity cannot finish until the predecessor activity starts (rare in supply chain projects).
• Leads and Lags: A Lead allows an acceleration of the successor activity (overlapping work), while a Lag imposes a mandatory delay between activities (e.g., 5-day lag for concrete curing before building warehouse racks).

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2. Critical Path Method (CPM) Mechanics

The Critical Path Method (CPM) is a deterministic technique used to determine the longest path of dependent activities through a network diagram and identify the minimum total duration required to complete the project.

Forward Pass & Backward Pass Step-by-Step Rules

1. Forward Pass (Early Dates): Moves from project start to finish to calculate Early Start (ES) and Early Finish (EF).
   • $EF = ES + \text{Duration}$
   • For activities with multiple predecessors, $ES = \max(EF_{\text{predecessors}})$.
2. Backward Pass (Late Dates): Moves from project finish to start to calculate Late Finish (LF) and Late Start (LS).
   • $LS = LF - \text{Duration}$
   • For activities with multiple successors, $LF = \min(LS_{\text{successors}})$.

Step-by-Step Worked CPM Numerical Example

Consider a supply chain network setup project with 6 activities (Durations in business days):

Step 1: Forward Pass Calculations

• Activity A: $ES = 0$, $EF = 0 + 3 = 3$.
• Activity B: Predecessor A. $ES = 3$, $EF = 3 + 5 = 8$.
• Activity C: Predecessor A. $ES = 3$, $EF = 3 + 4 = 7$.
• Activity D: Predecessor B. $ES = 8$, $EF = 8 + 6 = 14$.
• Activity E: Predecessor C. $ES = 7$, $EF = 7 + 3 = 10$.
• Activity F: Predecessors D ($EF = 14$) and E ($EF = 10$). Merge node rule: $ES = \max(14, 10) = 14$. $EF = 14 + 2 = 16$.
• Total Project Duration = 16 Days.

Step 2: Backward Pass Calculations

• Set $LF$ of final Activity F equal to total project duration: $LF_F = 16$.
• Activity F: $LF = 16$, $LS = 16 - 2 = 14$.
• Activity D: Successor F. $LF = 14$, $LS = 14 - 6 = 8$.
• Activity E: Successor F. $LF = 14$, $LS = 14 - 3 = 11$.
• Activity B: Successor D. $LF = 8$, $LS = 8 - 5 = 3$.
• Activity C: Successor E. $LF = 11$, $LS = 11 - 4 = 7$.
• Activity A: Successors B ($LS = 3$) and C ($LS = 7$). Burst node rule: $LF = \min(3, 7) = 3$. $LS = 3 - 3 = 0$.

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3. Float (Slack) Analysis: Total Float vs. Free Float

Float represents schedule flexibility. Supply chain managers analyze float to allocate resources effectively and identify non-critical activities that can absorb delays without disrupting final milestones.

• Total Float (TF): The amount of time an activity can be delayed from its early start date without delaying the overall project finish date. $\text{Total Float (TF)} = LS - ES = LF - EF$
• Free Float (FF): The amount of time an activity can be delayed without delaying the early start date of any immediate successor activity. $\text{Free Float (FF)} = ES_{\text{immediate successor}} - EF$

Comprehensive CPM & Float Analysis Table

Critical Path Analysis: The path A → B → D → F has zero Total Float ($TF = 0$) and a total duration of 16 days. Any delay in A, B, D, or F directly delays project completion. Conversely, Activity C has 4 days of Total Float; delaying Activity C by up to 4 days will consume float but will not delay project completion.

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4. Program Evaluation and Review Technique (PERT)

When activity durations are uncertain due to external supply chain variability (e.g., international ocean freight transit times), PERT 3-Point Estimation provides a probabilistic estimate based on beta probability distributions.

PERT 3-Point Estimates & Formulas

• Optimistic Duration ($O$): The minimum activity duration assuming best-case conditions.
• Most Likely Duration ($M$): The most realistic activity duration under normal conditions.
• Pessimistic Duration ($P$): The maximum activity duration assuming worst-case conditions.

$\text{Expected Duration } (E) = \frac{O + 4M + P}{6}$ $\text{Activity Standard Deviation } (\sigma) = \frac{P - O}{6}$ $\text{Activity Variance } (\sigma^2) = \left(\frac{P - O}{6}\right)^2$

Step-by-Step Worked PERT Numerical Example

Scenario: A procurement team estimates the lead time to import specialized automated sorting machinery:

• Optimistic ($O$) = 8 business days
• Most Likely ($M$) = 11 business days
• Pessimistic ($P$) = 20 business days

$\text{Expected Duration } (E) = \frac{8 + 4(11) + 20}{6} = \frac{8 + 44 + 20}{6} = \frac{72}{6} = \mathbf{12.0 \text{ days}}$ $\text{Standard Deviation } (\sigma) = \frac{20 - 8}{6} = \frac{12}{6} = \mathbf{2.0 \text{ days}}$ $\text{Variance } (\sigma^2) = (2.0)^2 = \mathbf{4.0 \text{ days}^2}$

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5. Schedule Compression Techniques

When a project falls behind schedule or executive leadership demands an earlier go-live date, project managers apply two schedule compression techniques:

• Crashing: Adding extra resources (e.g., authorizing contractor overtime, expediting air freight) to critical path activities at the lowest incremental cost. Trade-off: Increases total project cost.
• Fast-Tracking: Executing sequential critical path activities in parallel or overlapping them. Trade-off: Increases project risk and rework potential.