In project management, there are four main types of dependencies or relationships between tasks, often represented using the acronyms FS, SS, FF, and SF. Here's a brief explanation of each:

1. Finish-to-Start (FS): This is the most common type of dependency. It means that Task B cannot start until Task A finishes. Task A's completion triggers the start of Task B.

2. Start-to-Start (SS): In this relationship, Task B cannot start until Task A starts. Both tasks can occur simultaneously, but Task B cannot begin until Task A begins.

3. Finish-to-Finish (FF): With this dependency, Task B cannot finish until Task A finishes. The completion of Task A triggers the completion of Task B. This type of relationship is useful when two tasks need to be completed at the same time.

4. Start-to-Finish (SF): This is the least common type of dependency. It means that Task B cannot finish until Task A starts. Task A's initiation triggers the completion of Task B. This type of relationship is often used in situations where Task B needs to be completed within a specific time frame after Task A starts.

Understanding and properly managing these dependencies is crucial for creating realistic project schedules and ensuring that tasks are executed in the correct sequence to achieve project objectives efficiently.

Lead and lag are concepts used to adjust the timing of these relationships.

1. Lead: A lead allows the successor task to start before its predecessor task has finished. It means that there's an overlap between the two tasks. For example, if Task B has a lead of two days on Task A, Task B can start two days before Task A finishes.

2. Lag: A lag, on the other hand, inserts a delay between the finish of the predecessor task and the start of the successor task. It means that there's a waiting period between the two tasks. For instance, if Task B has a lag of three days on Task A, Task B can start three days after Task A finishes.

In project management, particularly in scheduling and dependency management, there are various types of relationships between tasks. "Hard logic," "soft logic," and "external dependency" are terms used to describe different types of dependencies between tasks:

1. Hard Logic: Hard logic, also known as "mandatory" or "internal" dependency, represents a relationship between tasks that is inflexible and strictly enforced. It means that the start or finish of one task is directly tied to the start or finish of another task. Hard logic dependencies are typically inherent to the nature of the work being performed and cannot be altered without changing the project scope or requirements.

2. Soft Logic: Soft logic, also referred to as "preferential" or "discretionary" dependency, represents a relationship between tasks that is more flexible and discretionary. Unlike hard logic dependencies, soft logic dependencies are not mandatory and are based on the preferences or best practices of the project team. They allow for some degree of flexibility in scheduling and sequencing tasks. Soft logic dependencies are often used when there are multiple valid ways to complete a project and when specific task sequences are not critical to project success.

3. External Dependency: An external dependency is a relationship between tasks where the completion of one task is dependent on factors outside the project's control. These factors can include deliverables from external vendors, regulatory approvals, or input from stakeholders external to the project team. External dependencies can introduce uncertainty and risk into a project schedule because the project team may have limited influence or control over the timing or outcome of these dependencies.

Understanding and managing these different types of dependencies is essential for creating realistic project schedules, identifying critical paths, and mitigating risks in project execution.

Analogous estimation for projects, also known as top-down estimation, involves using historical data from similar projects to estimate the duration, cost, or other parameters of a new project.

Parametric estimation is a project estimation technique that uses historical data and statistical relationships to predict project parameters such as duration, cost, or effort. Unlike analogous estimation, which relies on finding similar projects, parametric estimation uses mathematical models to estimate project parameters based on specific project characteristics or variables. 

Three-point estimation is a technique used in project management to estimate the time, cost, or other variables of a project task. It involves determining three types of estimates for each task: the optimistic estimate (O), the most likely estimate (M), and the pessimistic estimate (P). These estimates are then used to calculate a weighted average to arrive at a more accurate estimate.

PERT (Program Evaluation and Review Technique) is a statistical method used in project management to analyze and represent the tasks involved in completing a given project. PERT incorporates three-point estimation to calculate the expected duration or cost of a task. The formula for calculating the PERT estimate (E) is:

E= (O+4M+P​)/6

This formula places more weight on the most likely estimate, giving it four times the weight of the optimistic and pessimistic estimates.

The Triangular distribution is another statistical distribution used in project management for three-point estimation. It assumes that the values between the optimistic, most likely, and pessimistic estimates are equally distributed, forming a triangle when plotted on a graph. The formula for calculating the Triangular distribution estimate is simply the average of the three estimates:

E= (O+M+P​)/3

Both PERT and the Triangular distribution are used to provide more accurate estimates of project tasks compared to single-point estimates. PERT is preferred when there is significant uncertainty and variability in the estimates, as it provides a more weighted average, while the Triangular distribution is simpler and assumes a uniform distribution between the three points. The choice between them depends on the specific needs and characteristics of the project being estimated.

Bottom-up estimation is a technique used in project management to estimate the duration, cost, or resource requirements of individual project components. Instead of relying on high-level estimates for the entire project, bottom-up estimation breaks down the project into smaller, more manageable components or tasks and estimates each one individually. These individual estimates are then aggregated to provide a comprehensive estimate for the entire project.

The process of bottom-up estimation typically involves the following steps:

1. Identifying Work Packages: Break down the project scope into smaller, more manageable work packages or tasks. These tasks should be specific, measurable, and clearly defined.

2. Estimating Work Package Details: For each work package, gather detailed information about the resources required, the duration or effort needed to complete the task, and any other relevant factors.

3. Estimating Resources and Duration: Estimate the resources (such as labor, materials, equipment) required for each work package and the duration or effort needed to complete it. This estimation can be based on historical data, expert judgment, or other estimation techniques.

4. Aggregating Estimates: Once estimates are determined for all work packages, aggregate them to calculate the total duration, cost, or resource requirements for the entire project. This aggregation can be done by summing up the estimates for individual work packages.

5. Reviewing and Adjusting: Review the aggregated estimates to ensure they are realistic and comprehensive. Adjustments may be made based on feedback from subject matter experts, changes in project scope, or other factors.

Bottom-up estimation offers several advantages:

• Accuracy: By estimating each task individually, bottom-up estimation tends to produce more accurate overall estimates compared to top-down approaches.

• Detail-Oriented: It provides a detailed breakdown of project requirements, which can help in planning and resource allocation.

• Granular Control: It allows project managers to identify critical tasks and allocate resources more effectively.

However, bottom-up estimation can be time-consuming and resource-intensive, especially for large and complex projects. It also requires a thorough understanding of project requirements and dependencies to ensure accurate estimates for each task. Despite these challenges, bottom-up estimation is often preferred for its accuracy and detail-oriented approach, particularly in projects where precision is crucial.

1. Contingency Reserves: Contingency reserves are funds, time, or resources allocated for known risks that have been identified and analyzed during the project planning phase. These are specifically set aside to address risks that are anticipated but uncertain in terms of their impact or occurrence. Contingency reserves are there to cover the cost or schedule impacts of identified risks. They act as a buffer to absorb the effects of known risks without requiring a change in the project baseline. Contingency reserves are typically managed by the project manager and are incorporated into the project's overall budget and schedule. The amount allocated to contingency reserves is based on risk analysis and is often determined using techniques such as quantitative risk analysis or expert judgment.

2. Management Reserves: Management reserves are funds, time, or resources set aside for unforeseen events or circumstances that are not identified during project planning. Unlike contingency reserves, management reserves are not specifically tied to known risks but are rather general buffers for any unexpected issues that may arise. Management reserves provide flexibility to the project manager and stakeholders to address unforeseen risks or changes in project scope, requirements, or conditions. They are intended to cover unknown unknowns—risks that were not identified during the planning phase. Management reserves are typically controlled by senior management or the project sponsor rather than the project manager. Access to management reserves usually requires formal approval and may involve a change request process.

In summary, contingency reserves are for identified risks, while management reserves are for unidentified risks or other unforeseen events. Both are essential for effective risk management and project control, providing a level of protection against uncertainties and helping to ensure that projects can adapt and succeed despite challenges.

Critical path is the longest path in a project schedule, determining the project's duration. 

Float or slack(Total Float or Total Slack) refers to the amount of time a task can be delayed without affecting the project's schedule. 

Tasks on the critical path cannot be delayed without delaying the project's completion date. Any delay in a task on the critical path will directly impact the project's overall timeline.

Critical Path Analysis (CPA) is a project management technique used to identify the longest sequence of dependent tasks and activities required to complete a project. It helps project managers understand which tasks are critical and cannot be delayed without delaying the entire project.

Here's how it typically works:

1. Task Identification: List all the tasks required to complete the project, along with their durations and dependencies.

2. Dependency Mapping: Identify the dependencies between tasks. Some tasks can only start once others are completed, while some can happen concurrently.

3. Estimation of Durations: Estimate the time required to complete each task. This could be based on historical data, expert judgment, or other estimation techniques.

4. Network Diagram: Construct a network diagram showing the sequence of tasks and their dependencies. This helps visualize the flow of work.

5. Critical Path Determination: Calculate the earliest start and finish times for each task, as well as the latest start and finish times. The critical path consists of the tasks with zero slack or float, meaning any delay in these tasks will delay the entire project.

6. Slack Calculation: Calculate the slack or float for non-critical tasks. Slack is the amount of time a task can be delayed without delaying the project.

7. Monitoring and Controlling: Continuously monitor the progress of tasks along the critical path. If any delays occur, efforts should be focused on these critical tasks to ensure the project stays on track.

CPA helps project managers identify where to allocate resources and attention to ensure timely completion of the project. By focusing on critical tasks, they can reduce the risk of project delays and overruns. Additionally, it allows for better scheduling and resource management, optimizing project timelines and costs.

The Critical path calculation comprises three major activities

• forward pass;
• backward pass;
• float calculations.

The forward pass calculates the earliest times when activities can occur and the backward pass calculates the latest times. The difference between the two indicates the degree of flexibility there is in the performance of the activity and this is quantified as float.

Using a computer, the model as represented by the network diagram, can be adjusted and these calculations repeated time and time again. This allows different situations to be tested in what is often called 'what-if' analysis.

Each activity in the network diagram has the following elements:

The duration is the time required to complete the activity. The upper dates will indicate the earliest time the activity could be performed and the lower dates will indicate the latest time the activity could be performed.

The forward pass:

The forward pass then starts by placing 0 as the earliest start of activity A. Since its duration is 2 days its earliest finish is day 2. If A is finished by day 2 then that is the earliest start of B and D, and so the calculation continues.  The next point to note is activity F. This activity has two predecessors C and D. C's earliest finish is 8 but D's is 12. Therefore the earliest time that F can start is day 12.Following the calculation to its conclusion the earliest finish for G (and therefore the project) is 21 days.

The backward pass:

The earliest finish for the project is assumed to also be its latest finish. The process is then repeated, but in reverse.

If G's latest finish is day 21, and it takes 3 days, its latest start is therefore, day 18. This means that F and E must be finished by day 18 at the latest.

E's latest start is 14 and therefore D, perhaps, has a latest finish of 14. But D also has to be complete in time for F to start. The latest start of F is 12 and hence D has a latest finish of 12.

The same principle applies to A. Even though B doesn't have to start until 6, A must be finished by 2 because that is the latest time D can start.

Float  in a Critical Path Analysis (CPA), the concept of "float" or "slack" refers to the amount of time by which a non-critical task can be delayed without causing a delay in the overall project completion time.

In the critical path, where tasks have zero float, any delay in those tasks directly affects the project's completion date. However, tasks that are not on the critical path may have some flexibility in their start or finish times without affecting the overall project timeline. This flexibility is represented by the float.

Float is calculated by finding the difference between the earliest and latest start or finish times for a task. If this difference is zero, it means the task is on the critical path and has no float. If the difference is greater than zero, it indicates the amount of time the task can be delayed without affecting the project's completion date.

Understanding float is crucial for project managers because it allows them to prioritize tasks and allocate resources effectively. Tasks with more float may have less urgency, while those with less float or zero float require closer monitoring and management to ensure they stay on schedule and do not cause delays to the overall project timeline calculations:

Free float, also known as "free slack," refers to the amount of time that a specific task can be delayed without impacting the start of another dependent task. In other words, it's the amount of time a task can be delayed without affecting subsequent tasks on the same critical path. In the above diagram Activity C can be delayed by up to 4 days without having any knock on effect because F cannot start until day 12 anyway.

Calculating free float involves considering the dependencies between tasks. If a task has free float, it means there is some flexibility in its start or finish time without causing delays to tasks that come after it.

To calculate free float for a particular task:

1. Identify the task in question.
2. Determine all its immediate successor tasks (tasks that depend on its completion).
3. Find the earliest start time of the immediate successor tasks.
4. Subtract the earliest start time of the immediate successor tasks from the earliest finish time of the task in question.

The result is the free float for that task.

Free float is particularly useful for identifying tasks that can be delayed without impacting the overall project schedule. It helps project managers in scheduling and resource allocation decisions, allowing them to optimize project timelines effectively.

Example 1:  Find the critical path and the float of activity F and D in the project network diagram below.

No of paths and their length in the above project network diagram.

ABDH =14 days, ABEH = 11 days,  ACFGH = 13 days

The longest path is ABDH and the length is 14 days. Hence ABDH is critical path with the length 14 days.

Float of activity  :  D is 0 as this activity is on the critical path 

Float of activity  :  F is = 14 - 13 = 1 day

Example 2 :  Activity 1 has a duration of 20 days, Activity 2 of 10 days, Activity 3 of 5 days and Activity 4 of 6 days. What is the minimum total duration between the Milestones A and B?

The total duration between Milestones A and B = 20 + 10-2+3+6=37 days