15.4 Project Management: CPM, PERT, Network Analysis & Crashing

Project Management: CPM, PERT, Network Analysis & Crashing

Infrastructure expansion projects in Coal India Limited—such as commissioning a 10 Mtpa coal washery, erecting a cross-country overland belt conveyor, or sinking a new vertical mine shaft—involve thousands of interdependent activities, stringent safety deadlines, and substantial capital outlays. Network analysis techniques (CPM and PERT) provide the rigorous mathematical backbone for project scheduling, risk assessment, and time-cost trade-off optimization.

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1. Network Construction Rules & Representation

Network Notations

1. Activity-on-Arrow (AOA): Arrows represent activities consuming time and resources; nodes (circles) represent instantaneous events (milestones).
2. Activity-on-Node (AON / Precedence Diagramming): Nodes represent activities; arrows represent precedence relationships. Eliminates the need for dummy activities.

  AOA Model:          [Event 1] ---- Activity A (t) ----> [Event 2]

  AON Model:          +---------------+          +---------------+
                      |  Activity A   | -------> |  Activity B   |
                      +---------------+          +---------------+

Network Construction Rules

1. Each activity is uniquely identified by its tail (start) event $i$ and head (end) event $j$ ($i < j$).
2. No two activities can share both identical starting and identical ending events.
3. Dummy Activities ($-!-!-\to$): Fictitious activities with zero duration and zero resource consumption used to:
   • Maintain unique numerical identity between parallel activities.
   • Establish correct logical precedence without introducing unintended false dependencies.
4. Common Network Errors:
   • Looping / Cycling: Circular dependency path ($A \to B \to C \to A$), preventing project completion.
   • Dangling: An intermediate event having incoming arrows but no outgoing arrows (disconnect from the final project milestone).
5. Fulkerson's Rule for Node Numbering:
   • The initial project event has no incoming arrows; label it node $1$.
   • Delete all outgoing arrows from node $1$. Identify new nodes with no remaining incoming arrows and number them $2, 3, \dots$ consecutively.
   • Repeat until the final event is reached. For every activity $(i, j)$, $i < j$.

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2. Critical Path Method (CPM - Deterministic)

CPM assumes deterministic (single, well-known) activity durations derived from standard work study data.

                         FORWARD PASS (Earliest Times)
               ES_j = max { EF_i } = max { ES_i + t_ij }
                                   |
                                   v
                         BACKWARD PASS (Latest Times)
               LF_i = min { LS_j } = min { LF_j - t_ij }

Forward Pass Computations (Earliest Times)

• Earliest Start Time ($ES_i$): Earliest time an activity can begin: $ES_1 = 0, \quad ES_j = \max_{i} \{ ES_i + t_{ij} \} = \max_i \{ EF_{ij} \}$
• Earliest Finish Time ($EF_{ij}$): $EF_{ij} = ES_i + t_{ij}$

Backward Pass Computations (Latest Times)

• Latest Finish Time ($LF_j$): Latest time an activity can finish without delaying overall project completion: $LF_{\text{terminal}} = ES_{\text{terminal}} = T_E, \quad LF_i = \min_j \{ LF_j - t_{ij} \} = \min_j \{ LS_{ij} \}$
• Latest Start Time ($LS_{ij}$): $LS_{ij} = LF_j - t_{ij}$

Detailed Float Formulations

Float (or slack) represents schedule flexibility.

  ES_i                           EF_ij                     LF_j
   |===============================|------------------------|
   |<------ Duration t_ij -------->|<---- TOTAL FLOAT ----->|
   
   |===============================|------------|-----------|
                                   |<-- FREE -->|<- HEAD ->|
                                   |    FLOAT   |   SLACK  |

1. Total Float ($TF_{ij}$): Maximum time by which an activity can be delayed without delaying the overall project completion time: $\mathbf{TF_{ij} = LF_j - EF_{ij} = LS_{ij} - ES_i = LF_j - ES_i - t_{ij}}$
2. Free Float ($FF_{ij}$): Maximum time by which an activity can be delayed without delaying the earliest start of any immediately succeeding activity: $\mathbf{FF_{ij} = ES_j - EF_{ij} = ES_j - ES_i - t_{ij} = TF_{ij} - S_j}$ (where $S_j = LF_j - ES_j$ is the head event slack).
3. Independent Float ($IF_{ij}$): Time by which an activity can be delayed if preceding activities finish at their latest times and succeeding activities start at their earliest times. It affects no other activity in the network: $\mathbf{IF_{ij} = \max(0, \; ES_j - LF_i - t_{ij}) = \max(0, \; FF_{ij} - S_i)}$ (where $S_i = LF_i - ES_i$ is the tail event slack).
4. Interfering Float ($ITF_{ij}$): The portion of total float that consumes head event slack: $ITF_{ij} = TF_{ij} - FF_{ij} = S_j$
5. Fundamental Float Hierarchy: $\mathbf{TF_{ij} \ge FF_{ij} \ge IF_{ij} \ge 0}$
6. Critical Path Identification: The critical path is the continuous chain of activities connecting the start event to the end event where Total Float is minimum (zero) ($TF = 0, FF = 0, IF = 0$). It is the longest duration path through the network.

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3. Program Evaluation and Review Technique (PERT - Probabilistic)

PERT is designed for R&D and non-repetitive projects where activity durations are uncertain. It models activity duration $t$ as a Beta distribution defined by three time estimates.

  Probability ^
    Density   |          Most Likely (tm)
              |                 /\
              |                /  \
              |   Optimistic  /    \    Pessimistic
              |      (to)    /      \      (tp)
              |       |     /        \      |
            0 +-------+----+----------+-----+-----> Time (t)
                           |<- te ->|

Three Time Estimates

1. Optimistic Time ($t_o$): Minimum possible duration if everything proceeds exceptionally well (probability $< 1%$ of occurring faster).
2. Most Likely Time ($t_m$): Modal duration under normal working conditions.
3. Pessimistic Time ($t_p$): Maximum duration if adverse conditions and bottlenecks occur (probability $< 1%$ of exceeding).

Statistical Parameters of PERT

• Expected Activity Duration ($t_e$): Weighted mean of the Beta distribution: $\mathbf{t_e = \frac{t_o + 4 t_m + t_p}{6}}$
• Activity Variance ($\sigma^2$) and Standard Deviation ($\sigma$): $\mathbf{\sigma^2 = \left(\frac{t_p - t_o}{6}\right)^2, \quad \sigma = \frac{t_p - t_o}{6}}$

Project Duration Statistics & Central Limit Theorem

By the Central Limit Theorem, the sum of independent random activity durations along the critical path converges to a Normal Distribution $\mathcal{N}(T_e, \sigma_{\text{proj}}^2)$:

• Expected Project Completion Time ($T_e$): $T_e = \sum_{k \in \text{Critical Path}} t_{e, k}$
• Project Variance ($\sigma_{\text{proj}}^2$): $\sigma_{\text{proj}}^2 = \sum_{k \in \text{Critical Path}} \sigma_k^2$
• Project Standard Deviation ($\sigma_{\text{proj}}$): $\sigma_{\text{proj}} = \sqrt{\sum_{k \in \text{Critical Path}} \sigma_k^2}$ (Crucial Rule: Variances add along the critical path; standard deviations do NOT add algebraically).

Probability of Meeting a Target Schedule Date ($T_s$)

The standard normal deviate $Z$ is computed as:

$\mathbf{Z = \frac{T_s - T_e}{\sigma_{\text{proj}}}}$

                 STANDARD NORMAL DISTRIBUTION (Z-VALUES)
  -------------------------------------------------------------------------
  Z = -2.0  --> P =  2.28%     |  Z = +0.0  --> P = 50.00%
  Z = -1.0  --> P = 15.87%     |  Z = +1.0  --> P = 84.13%
  Z = -0.5  --> P = 30.85%     |  Z = +1.28 --> P = 90.00%
  Z = +0.67 --> P = 75.00%     |  Z = +2.0  --> P = 97.72%

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4. Project Crashing and Time-Cost Optimization

Project crashing compresses total project duration by allocating additional direct resources (overtime, extra shifts, special equipment) to critical activities.

  Cost ^
       |   Crash Point (CT, CC)
    CC |      *
       |       \    Cost Slope = (CC - NC) / (NT - CT)
       |        \    
    NC |---------\---------* Normal Point (NT, NC)
       |          \        |
       |           \       |
     0 +------------+------+------------------> Time
                   CT      NT

Key Parameters

• Normal Time ($NT$) & Normal Cost ($NC$): Standard baseline duration and direct cost.
• Crash Time ($CT$) & Crash Cost ($CC$): Minimum achievable duration and resulting inflated direct cost.
• Cost Slope ($S$): Marginal direct cost per unit time saved: $\mathbf{S = \frac{\text{Crash Cost} - \text{Normal Cost}}{\text{Normal Time} - \text{Crash Time}} = \frac{\Delta C}{\Delta t}}$

Total Project Cost Components

1. Direct Costs: Increase monotonically as duration is shortened.
2. Indirect (Overhead) Costs: Decrease linearly with shortened duration ($C_{\text{ind}} = \text{daily overhead} \times T$).
3. Optimal Project Duration: Occurs at the minimum point of the Total Cost Curve ($TC = \text{Direct} + \text{Indirect}$), where marginal crashing cost slope equals the daily indirect cost rate.

  Cost ^
       |       \           /  Total Cost Curve
       |        \         / 
       | Direct  \  Opt. /   Indirect Cost
       | Cost     \  *  /     (Overhead)
       |           \/  /     / 
       |___________/\_/_____/_____________
       0             |                    --> Duration
                Optimal Time

Systematic Project Crashing Algorithm

1. Calculate the critical path, initial project duration $T_E$, and baseline total cost (Sum of all Normal Direct Costs + Indirect Costs for $T_E$).
2. Calculate the Cost Slope $S$ for every network activity.
3. Identify the active critical path(s).
4. Among critical activities that have remaining crash capability ($NT > CT$), select the activity with the lowest Cost Slope ($S$).
5. If parallel critical paths exist, crash activities simultaneously on all parallel paths such that $\sum S$ is minimized, or crash a single common critical activity.
6. Crash the selected activity by the maximum permissible amount until:
   • The activity reaches its crash limit ($CT$), or
   • A new parallel path becomes critical.
7. Recalculate critical paths and total costs. Repeat steps 4–6 until no further crashing is possible or the cost slope exceeds the daily indirect cost rate.