Correct answer: A$.$ Critical Path Method $($CPM$)$

The Critical Path Method $($CPM$)$ is a project scheduling technique that prioritizes tasks based on their importance to the project's completion. It entails determining the longest sequence of jobs in a project, called the "critical path," which determines the project's lowest potential time. Tasks on this path have no slack time, thus any delay in these tasks will cause the entire project to be delayed.

Due dates and slack times are critical components in CPM$.$ Slack time, also known as float, is the amount of time that a task can be delayed without affecting future tasks or the project's completion deadline. Tasks with the least slack time are prioritized since their timely completion is critical for keeping the project on track. CPM$,$ by focusing on the critical path, assists project managers in determining which tasks demand immediate attention and which can wait if required.

Explanation:

The method employs a mathematical approach to determine the earliest and latest potential start and finish times for each activity, allowing for more effective resource allocation and prioritization. It is especially effective in large projects with various activities and dependencies, where there is a high risk of delays and overruns.

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Other Options Are Incorrect:

B$.$ Gantt Chart: A Gantt chart is a visual project management tool that illustrates project tasks on time. While it is great for keeping track of project timelines and progress, it does not automatically prioritize tasks based on due dates and slack times. Gantt charts illustrate when a task is scheduled to start and finish, but they do not provide an in$-$depth study of task dependencies or critical routes. They are more of a tool for tracking and presenting information than they are for deciding task prioritizing.

C$.$ PERT Diagram: The Program Evaluation and Review Technique $($PERT$)$ is a method for analyzing and representing the tasks involved in project completion. To compute the entire project length, PERT estimates the shortest, longest, and most likely durations for activities. While it considers task durations and dependencies in the same way as CPM does, it is more focused on time estimation and less on task prioritizing based on due dates and slack times. PERT focuses on scenario analysis and probability rather than direct job prioritizing.

D$.$ First$-$Come, First$-$Served $($FCFS$)$: First$-$Come, First$-$Served is a basic scheduling system in which activities are completed in the order in which they arrive or are requested. This technique does not take into account task due dates or slack hours. It is a simplistic, non$-$prioritized strategy that is usually utilized in basic queuing systems and is not appropriate for complicated project management when jobs have varying degrees of urgency and impact on the entire project timeline. FCFS lacks the analytical depth required to properly manage a project, particularly when it comes to prioritizing tasks based on criticality or deadlines.

Explanation:

Gantt Charts, on the other hand, while valuable for managing project schedules, do not automatically prioritize jobs based on slack times. They are more of a tool for visual depiction. The Program Evaluation and Review Technique $($PERT$)$ is more concerned with time estimation and probability analysis than with direct task prioritizing. Finally, First$-$Come, First$-$Served $($FCFS$)$ is a simple, non$-$prioritized scheduling system that does not take job urgencies or deadlines into account, making it inappropriate for sophisticated project management.

Final solution

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Correct answer: A$.$ Critical Path Method $($CPM$)$

The correct scheduling technique that prioritizes tasks based on their due dates and slack times is the Critical Path Method $($CPM$).$