Project Scheduling

Scheduling

Scheduling is fixing the start and finish times of each activity on a calendar, respecting logic, durations and resource limits.

Purposes

• Know when each activity will be done; determine the completion date.
• Plan procurement of materials, labour and equipment.
• Coordinate subcontractors and agencies.
• Provide a baseline for monitoring progress and cash flow.

Bar chart (Gantt chart)

Developed by Henry L. Gantt (early 20th century). Activities are listed vertically and each is shown as a horizontal bar along a time scale; bar length represents duration.

Construction of a bar chart

1. List the activities (from the WBS).
2. Estimate durations.
3. Decide the sequence and start dates.
4. Draw bars against the time scale; show actual progress by a second bar or shading.

Advantages

• Simple to prepare and understand — even by non-technical people.
• Clear visual picture of time-scale and overlapping activities.
• Useful for small projects and for presenting summaries.
• Can show planned vs actual progress and resource loading.

Limitations

1. Interdependencies between activities are not shown — the effect of delay in one activity on others cannot be seen.
2. Does not identify critical activities or floats.
3. Uncertainty in durations cannot be represented.
4. Difficult to update and to use for large, complex projects with many activities.
5. Progress in terms of physical percentage is judged subjectively.
6. No time–cost trade-off analysis.

These limitations led to milestone charts and later network techniques (CPM, PERT).

Linked bar chart

A bar chart with arrows linking dependent activities — shows logic partially; modern software displays Gantt charts linked to network logic.

Milestone chart

A milestone is a significant event (point in time, zero duration) — e.g. "foundation completed", "roof slab cast".

• A milestone chart is an improvement over a bar chart in which milestones are marked on bars, dividing long activities into sub-activities with key events.
• Helps monitoring long activities and showing some sequence within an activity.
• Still does not show interdependencies between milestones of different activities — which led to network diagrams (milestones connected by arrows became the forerunner of PERT networks).

Monitoring progress on bar charts

• Planned bar vs actual bar drawn beneath or shaded.
• A status (time-now) line drawn at the reporting date; activities behind the line are behind schedule, ahead are ahead of schedule.
• Percentage completion marked on bars.

Line of balance (LOB)

A technique for repetitive projects — multi-storey buildings (floor by floor), housing units, pipelines, highways.

• Plots cumulative units completed (vertical axis) against time (horizontal axis) for each activity/crew — each crew's progress is a line whose slope = production rate.
• Parallel lines indicate balanced crews moving continuously from unit to unit; converging lines indicate conflicts (crews catching up).
• Helps maintain continuity of work and identify bottlenecks.

Critical Path Method (CPM)

CPM was developed in the late 1950s (by DuPont with Remington Rand) for planning and scheduling projects with deterministic (known) activity durations. It is activity-oriented and widely used in construction, where durations can be estimated from experience; it also forms the basis of time–cost trade-off (crashing).

Network terminology

Types of network diagrams

Activity on arrow (AOA)

• Activities on arrows, events as nodes (circles); activity identified by $(i,j)$ — tail and head events.
• Needs dummies for correct logic.

Activity on node (AON) / precedence diagram

• Activities in nodes (boxes); arrows show dependencies only.
• No dummies needed; easier to draw and modify; used by most software.
• Allows relationships FS, SS, FF, SF with lags/leads.

Rules for drawing AOA networks

1. Each activity is represented by one arrow only.
2. No two activities may have the same tail and head events — use a dummy.
3. There should be one initial and one final event (in a basic network).
4. Arrows flow from left to right; no looping (circular logic).
5. No dangling — every activity except the last must have a successor, and every event except the first must have a predecessor.
6. Avoid redundant dependencies.
7. The length of the arrow has no significance (not to scale).

Fulkerson's rule for numbering events

1. Number the initial event 1.
2. Delete all arrows emerging from numbered events; this creates new initial events — number them 2, 3, …
3. Repeat until the final event is numbered.

Result: for every activity $(i,j)$, $i<j$ (head number greater than tail number).

Common errors

• Looping (cycling) — endless loop of activities.
• Dangling — an activity disconnected from the end event.
• Redundancy — unnecessary dummy or link where dependency is already implied.

Time computations

Floats

• Relationship: $IF\le FF\le TF$.
• Critical activities have zero total float (when the project's scheduled completion equals its earliest completion). Negative float indicates the schedule cannot meet a target date.
• The critical path joins critical activities from start to end; there may be more than one critical path.

Worked example 1 — AOA network

Worked example 2 — using a dummy

Precedence diagram relationships (AON)

AON node boxes typically show ES, EF, LS, LF, duration and total float.

Advantages of CPM

• Shows interdependencies; identifies critical activities needing close control.
• Computes floats — flexibility for resource allocation.
• Basis for crashing, resource levelling, cost control and updating.
• Effect of delays on the project can be analysed.

Frequently tested points

• CPM: deterministic durations, activity-oriented, used for time–cost trade-off; PERT: probabilistic, event-oriented.
• Dummy: zero time, zero resources, dashed arrow — logic or unique identification.
• Fulkerson's rule numbers events so that $i<j$ for every activity.
• Errors: looping, dangling, redundancy.
• $E_j=\max (E_i+t)$; $L_i=\min (L_j-t)$.
• TF = L_j − E_i − t; FF = E_j − E_i − t; IF = E_j − L_i − t; interfering float = TF − FF = head slack.
• IF ≤ FF ≤ TF; critical activities TF = 0; critical path = longest path; may be several.
• AON needs no dummies; FS, SS, FF, SF relationships with lags.