Gantt charts are widely used tools in project management to visualize schedules and task sequences. They help in planning, coordinating, and tracking specific tasks within a project timeline. Understanding the mathematical principles behind Gantt charts can improve their effective use and accuracy.

Task Duration and Scheduling

Each task in a Gantt chart is represented by a bar that spans from its start date to its end date. The length of this bar corresponds to the task's duration. Mathematically, if S is the start time and E is the end time, then the duration D is calculated as:

D = E - S

Task Sequencing and Dependencies

Tasks often depend on the completion of other tasks. To model this, dependencies are represented through constraints. If task B depends on task A, then the start time of task B must be after the end of task A:

SB ≥ EA

Scheduling Calculations

Scheduling involves calculating start and end times based on dependencies and durations. For example, if task A starts at time SA and has duration DA, then its end time is:

EA = SA + DA

Similarly, for task B dependent on task A:

SB = EA

These calculations ensure tasks are scheduled sequentially or in parallel, based on dependencies, optimizing project timelines.