08 Project Management
These are network-based methods, designed to assist in planning, scheduling, and control of projects.
Objective¶
- Determine the minimum possible completion time for the project.
- Determine a range of start and end time for each activity, so that the project can be completed in minimum time.
Notations¶
| \(\text{ES}_j = \square_j\) | Earliest occurance time of event \(j\) |
| \(\text{LC}_j = \triangle_j\) | Latest completion time of event \(j\) |
| \(D_{ij}\) | Duration of activity between \(i\) and \(j\) |
Activities¶
| Critical | Non-Critical | |
|---|---|---|
| Leeway in determining start time | ā | ā , within some limit |
| Leeway in determining end time | ā | ā |
| Conditions | 1. \(\square_i = \triangle_i\) 2. \(\square_j = \triangle_j\) 3. \(\square_j - \square_i = \triangle_j - \triangle_i = D_{ij}\) |
where \(D_{ij}\) = given distance between 2 nodes
Methods¶
| CPM | PERT | |
|---|---|---|
| Full Form | Critical Path Method | Project Evaluation Review Technique |
| Assumption for activity | deterministic durations | probabilistic durations |
| Duration of activity | Fixed | Determined based on - most optimistic timeĀ \(a\) - most likely timeĀ \(m\) - pessimistic timeĀ \(b\) |
| Procedure | 2 passes 1. Forward pass determines earliest occurance times; take path with max duration if \(\exists\) multiple paths 2. Backward pass determines latest completion times; take path with min duration if \(\exists\) multiple paths 3. Find critical paths 4. Find the float for non-critical activities | 1. Calculate distance and variance 2. Solve like CPM 3. Calculate cumulative E(D_i) and Var 4. Calculate required probabilities using \(z\)Ā distribution In case of ties, take the max variance path, thereby reflecting more uncertainty |
| Average durationĀ \(\bar D = \frac{a+4m+b}{6}\) VarianceĀ \(= \left(\frac{b-a}{6}\right)^2\) |
Float¶
| Free Float | Total Float |
|---|---|
| \(\square_j - \square_i - D_{ij}\) | \(\triangle_j - \square_i - D_{ij}\) |
| Case | |
|---|---|
| FF = 0 | Any delay will cause delay in starting successive activities |
| FF < TF | We have leeway in starting the project as FF units For any excess delay (FF < d < T), starting successive activities will be delayed |
| FFĀ = TF | Activities may be scheduled anywhere between the earliest start time & the latest completion time without delaying the project |