Easy How-To: Calculate Late Start & Finish Times


Easy How-To: Calculate Late Start & Finish Times

The latest possible time an activity can begin without delaying the project’s completion is the Late Start (LS). Similarly, the Late Finish (LF) represents the latest possible time an activity can be completed without affecting the overall project duration. These values are determined through a backward pass analysis of the project schedule, starting from the project’s designated end date.

Understanding these schedule parameters offers significant advantages. They provide project managers with flexibility in resource allocation and task scheduling. By identifying the degree of leeway available within each activity, resources can be strategically deployed to critical tasks. Furthermore, knowledge of these calculated values aids in proactive risk management, facilitating the identification of potential delays and the implementation of mitigation strategies to keep the project on track.

Calculating these values requires knowledge of the project schedule, activity durations, and dependencies. The following sections will detail the process, providing a step-by-step guide for determining the Late Start and Late Finish times for project activities.

1. Backward Pass

The Backward Pass is the core process in determining the Late Start and Late Finish times within project schedule management. This analytical technique commences from the project’s designated completion date and proceeds backward through the network diagram, considering activity durations and dependencies. The Late Finish for the final activity is initially set equal to the project’s end date. Subsequently, the Late Start for this activity is calculated by subtracting the activity’s duration from its Late Finish. This process is iteratively repeated for all preceding activities.

The impact of the Backward Pass is that it reveals the allowable slippage for each activity without jeopardizing the overall project timeline. For example, if an activity has a duration of 5 days and its successor’s Late Start is day 15, the Late Finish for the preceding activity would be day 15, and its Late Start would be day 10. Without a thorough Backward Pass, the Late Start and Late Finish dates may be underestimated, leading to inefficient resource allocation. Conversely, an overestimated Late Start can lead to overconfidence and the potential for unforeseen delays.

In conclusion, the Backward Pass is not merely a calculation; it is the fundamental mechanism for establishing the Late Start and Late Finish times, crucial metrics for project scheduling and control. Recognizing its importance allows project managers to proactively manage resources, mitigate risks, and ultimately increase the likelihood of project success. Any inaccuracy in Backward Pass execution directly affects the validity of the Late Start and Late Finish, potentially leading to misguided project decisions.

2. Activity Duration

The duration of an activity is a foundational input in the calculation of the Late Start and Late Finish times. Accurate estimation of activity durations is, therefore, critical for generating a realistic and reliable project schedule. Underestimation or overestimation can significantly skew the results and lead to mismanagement of resources.

  • Impact on Backward Pass

    In the backward pass, the activity duration is directly subtracted from the Late Finish to determine the Late Start. A longer duration will result in an earlier Late Start, reflecting the need to commence the activity sooner. Conversely, a shorter duration allows for a later Late Start, providing greater flexibility in scheduling. For example, if an activity must finish by day 20 and has a duration of 5 days, its Late Start is day 15. An inaccurate duration of 3 days would incorrectly place the Late Start at day 17, potentially causing schedule conflicts.

  • Influence on Float/Slack

    Activity duration influences the total float or slack available for an activity. If an activity’s duration is shorter than initially estimated, the extra time translates to float. This buffer provides the project manager with leeway in scheduling and resource allocation. However, an inflated duration can mask the true amount of float, leading to a false sense of urgency and potentially misallocated resources. Activities on the critical path have minimal or no float; therefore, an inaccurate duration estimate on the critical path has a direct impact on the projects end date. A critical path activity requiring longer time, as against its initial duration, directly stretches the projects timeline.

  • Effect on Resource Allocation

    The estimated duration impacts resource allocation decisions. Activities with longer durations typically require a greater allocation of resources. Inaccurate duration estimates can lead to either under-resourcing, causing delays, or over-resourcing, leading to wasted resources and increased costs. For example, a software development task initially estimated to take 10 days might be allocated two developers. If the actual duration is only 5 days, the project manager could reallocate one developer to another task, optimizing resource utilization.

  • Interplay with Dependencies

    An activitys duration, in conjunction with its dependencies, determines the Late Start and Late Finish of its predecessors. If Activity B depends on the completion of Activity A, Activity As duration directly affects Activity Bs Late Start. For example, if Activity A has a duration of 7 days and Activity B has a Late Start of day 15, Activity As Late Finish must be day 15, and its Late Start would be day 8. Incorrectly estimating Activity As duration will propagate errors through the network diagram, affecting the Late Start and Late Finish calculations of subsequent activities.

In summary, the accuracy of activity duration estimates is paramount to the integrity of Late Start and Late Finish calculations. These calculations, in turn, are fundamental to effective project planning, resource management, and schedule control. Therefore, project managers must employ robust estimation techniques and regularly review and revise duration estimates throughout the project lifecycle to minimize discrepancies and maintain schedule accuracy.

3. Dependencies

Dependencies, the defined relationships between project activities, are a fundamental element in determining the Late Start and Late Finish times. These relationships dictate the order in which activities must be performed and directly influence the scheduling flexibility of individual tasks. Incorrectly defining or neglecting dependencies can lead to flawed Late Start and Late Finish calculations, ultimately jeopardizing project timelines.

  • Finish-to-Start (FS) Dependencies

    The most common type, Finish-to-Start, mandates that one activity must be completed before its successor can begin. In the context of Late Start and Late Finish, the Late Start of the successor activity is directly influenced by the Late Finish of its predecessor. If Activity A (predecessor) must finish before Activity B (successor) can start, the Late Finish of Activity A cannot exceed the Late Start of Activity B. For example, if Activity B’s Late Start is Day 15, then Activity A’s Late Finish must be Day 15 at the latest. An inaccurate dependency definition here would miscalculate available float and potentially delay Activity B.

  • Start-to-Start (SS) Dependencies

    Start-to-Start dependencies dictate that one activity cannot begin until another has started. This relationship impacts the Late Start times of both activities. The Late Start of the successor is constrained by the Late Start of its predecessor, accounting for any defined lag or lead time. If Activity C cannot start until Activity D has started, and there is a 2-day lag, Activity C’s Late Start must be at least 2 days after Activity D’s Late Start. Failing to account for this lag during the backward pass would result in an incorrect Late Start calculation for Activity C.

  • Finish-to-Finish (FF) Dependencies

    Finish-to-Finish dependencies require that one activity cannot be completed until another has been completed. This type of dependency impacts the Late Finish times. The Late Finish of the successor activity is linked to the Late Finish of its predecessor, potentially limiting the scheduling flexibility of the successor. If Activity E cannot finish until Activity F has finished, then the Late Finish of Activity E cannot precede the Late Finish of Activity F. Inaccurate assessment of this dependency type can lead to an overestimation of available float and schedule compression of Activity E.

  • Impact on the Critical Path

    Dependencies, especially those along the critical path, have the most significant impact on Late Start and Late Finish. Activities on the critical path have zero or minimal float, meaning any delay in a predecessor activity directly affects the Late Start and Late Finish of its successors, potentially delaying the entire project. Therefore, precise definition and management of dependencies on the critical path are crucial for maintaining the project schedule and accurately calculating Late Start and Late Finish times. Misidentification of critical path dependencies can cause cascading schedule issues.

In summary, a thorough understanding and accurate representation of dependencies are essential for determining the Late Start and Late Finish of project activities. The type of dependency, along with any associated lag or lead times, directly influences the backward pass calculations and ultimately affects the project schedule. Recognizing and managing these relationships is paramount for effective project planning and execution.

4. Total Float

Total Float, also known as slack, represents the amount of time an activity can be delayed without impacting the project’s overall completion date or violating a schedule constraint. It is a critical factor interwoven with the determination of Late Start and Late Finish, serving as an indicator of schedule flexibility and a buffer against potential delays.

  • Calculation and Interpretation

    Total Float is typically calculated as the difference between the Late Finish and Early Finish (or the Late Start and Early Start) of an activity. A positive Total Float indicates that the activity can be delayed without affecting the project end date. A Total Float of zero signifies that the activity is on the critical path, leaving no room for delay. A negative Total Float means the activity is already behind schedule, and corrective actions are needed. For example, if an activity has an Early Finish of day 10 and a Late Finish of day 15, its Total Float is 5 days. This activity can be delayed by up to 5 days without delaying the project. In project management, Total Float shows the degree of flexibility in scheduling each activity. Project managers rely on accurate Total Float calculations to properly allocate resources and decide which activities can be adjusted if unforeseen problems occur.

  • Relationship to Late Start and Late Finish

    The Late Start and Late Finish are fundamentally derived from the concept of Total Float. The Early Start and Early Finish are determined through a forward pass analysis. The backward pass, which calculates Late Start and Late Finish, uses the project completion date and activity durations, while factoring in the Total Float to determine the latest possible times for each activity. The Late Start and Late Finish represent the boundaries within which an activity can occur without jeopardizing the project timeline. A larger Total Float provides more flexibility in setting the Late Start and Late Finish. An activity with zero Total Float has its Late Start equal to its Early Start and its Late Finish equal to its Early Finish.

  • Resource Allocation and Prioritization

    Total Float guides resource allocation and task prioritization. Activities with minimal or no Total Float (those on the critical path) demand higher priority and often require more resources to ensure timely completion. Activities with significant Total Float offer greater flexibility and can be scheduled around resource constraints or competing priorities. If a software development project, there are activities in the critical path and other activities that can be delayed by some time without putting the project timeline in risk. For instance, activities like designing the database structure may be in the critical path, and cannot be delayed, while generating project documentation can have some amount of Total Float.

  • Impact of Schedule Changes

    Any change to the project schedule, such as modifications to activity durations or dependencies, will affect Total Float and, consequently, the Late Start and Late Finish times. If an activity’s duration increases, it can reduce the Total Float of subsequent activities and potentially shift activities onto the critical path. Conversely, if an activity’s duration is shortened, it can increase the Total Float of subsequent activities, providing greater scheduling flexibility. Therefore, project managers must continuously monitor and recalculate Total Float and Late Start/Late Finish in response to any schedule changes to maintain accurate project control.

In summary, Total Float is inextricably linked to the determination of Late Start and Late Finish. It provides a measure of scheduling flexibility, informs resource allocation decisions, and helps project managers proactively manage risks. Accurate calculation and interpretation of Total Float are essential for maintaining project control and ensuring timely project completion. The insights that derived from the above is used by project managers to decide how best to deal with activities while keeping projects on track.

5. Project Deadline

The Project Deadline serves as the anchor point for backward pass calculations, directly impacting the determination of Late Start and Late Finish times. This fixed date initiates the process of establishing the latest permissible completion times for each project activity. Without a clearly defined Project Deadline, the backward pass cannot be effectively executed, rendering Late Start and Late Finish values meaningless. For instance, in a construction project with a contractual completion date, the Late Finish of the final activity is initially set to this deadline. This constraint then cascades backward through the network diagram, influencing the allowable completion times for all preceding activities. A shifting or undefined Project Deadline introduces uncertainty into the scheduling process, making accurate Late Start and Late Finish determination impossible.

The implications of a mismanaged Project Deadline extend beyond mere schedule calculations. Consider a software development project where the release date is tied to a marketing campaign. If the actual development time exceeds the initial estimates, potentially pushing the Project Deadline, the marketing campaign must be adjusted accordingly, incurring additional costs and potentially diminishing its impact. Similarly, in a manufacturing setting, a missed Project Deadline for delivering a product to a customer can result in financial penalties, damage to reputation, and loss of future business. Therefore, the Project Deadline serves as both a constraint and a driver, influencing the allocation of resources and the management of risks to ensure timely project completion.

In summary, the Project Deadline is the cornerstone upon which Late Start and Late Finish calculations are based. Its accuracy and stability are essential for effective project scheduling and control. While challenges may arise from unforeseen circumstances, a proactive approach to managing the Project Deadline, including contingency planning and continuous monitoring of progress, is crucial for mitigating risks and ensuring that the project stays on track. Failure to do so can lead to inaccurate Late Start and Late Finish times, ultimately undermining the project’s chances of success.

6. Critical Path

The critical path identifies the sequence of activities that determine the minimum possible duration of a project. Activities on the critical path possess zero or near-zero float. This means any delay in these activities directly extends the project’s completion date. Consequently, the accurate determination of Late Start and Late Finish times for critical path activities is of paramount importance.

The calculation of Late Start and Late Finish for critical path activities differs subtly from non-critical activities due to the float constraint. On the critical path, the Early Start is equal to the Late Start, and the Early Finish is equal to the Late Finish. Therefore, the focus shifts to rigorous monitoring and control to prevent deviations from the planned schedule. For example, in a construction project, if pouring the foundation is a critical path activity and is delayed by one day, the Late Start and Late Finish of all subsequent critical path activities are shifted by one day, impacting the overall project deadline. Similarly, in software development, a delay in coding a critical module directly impacts the integration and testing phases, affecting the project’s release date.

In conclusion, the critical path significantly influences Late Start and Late Finish calculations by highlighting activities with minimal scheduling flexibility. The accurate identification of the critical path and meticulous monitoring of its activities are essential for maintaining project timelines. Effective project management demands a thorough understanding of the interplay between the critical path and the determination of Late Start and Late Finish times.

7. Network Diagram

The network diagram serves as a visual representation of the project schedule, delineating activities and their dependencies. This graphical depiction forms the basis for calculating Late Start and Late Finish times, providing a framework for understanding activity sequencing and critical path analysis.

  • Activity Sequencing and Dependencies

    The network diagram clearly illustrates the sequence in which activities must be performed, highlighting dependencies between them. This visual representation allows project managers to readily identify predecessor and successor relationships, which are essential for determining the Late Start and Late Finish times. For instance, if Activity B depends on the completion of Activity A, the diagram visually reinforces that Activity B’s Late Start is constrained by Activity A’s Late Finish. Without this clear visualization, accurately assessing these dependencies and their impact on Late Start and Late Finish calculations becomes significantly more challenging. In a software project, one module needs to be coded and tested before any integration and deployment can happen. The network diagram allows the project manager to figure out which ones need to happen and which ones can be delayed.

  • Critical Path Identification

    The network diagram facilitates the identification of the critical path, the sequence of activities that directly impacts the project’s completion date. Activities on the critical path have zero or near-zero float, making their Late Start and Late Finish times particularly sensitive. By visually representing the network, project managers can readily pinpoint these critical activities and prioritize their management. Delays in any critical path activity directly affect the project timeline; hence, their Late Start and Late Finish times require meticulous monitoring and control. For example, in a construction project, if pouring the foundation and erecting the structure are critical activities, their timely completion is paramount. The network diagram makes this dependency clear, informing resource allocation and risk mitigation strategies.

  • Float Visualization

    The network diagram provides a visual representation of the float or slack associated with each activity. This visual cue aids in understanding the degree of flexibility available in scheduling individual tasks. Activities with significant float can be delayed without affecting the project’s overall completion date. This information is crucial for optimizing resource allocation and managing potential delays. By visualizing float, project managers can make informed decisions about where to allocate resources and which activities can be adjusted in response to unforeseen circumstances. When the activities are mapped out in a project and you see the activities, as a project manager, you can see which activities need to have more people assigned to it so that the project doesnt get delayed and stays on course.

  • Backward Pass Support

    The network diagram supports the backward pass analysis, the process of determining Late Start and Late Finish times. By visually representing the activity sequence and dependencies, the diagram facilitates the systematic calculation of Late Start and Late Finish values, starting from the project’s completion date and working backward through the network. This visual aid ensures that all dependencies are properly considered and that the Late Start and Late Finish times are accurately calculated for each activity. The diagram provides a clear roadmap for the backward pass, minimizing the risk of errors and omissions. For instance, each task is written on the diagram and shows a date on it, this visual display allows the project manager to determine how far back each task needs to be finished and how far back the next task needs to be finished to make sure that all the tasks get completed so that the deadline does not get missed.

In summary, the network diagram is an indispensable tool for calculating Late Start and Late Finish times. Its visual representation of activity sequences, dependencies, critical paths, and float facilitates accurate schedule analysis and effective project control. Employing a well-constructed network diagram ensures that Late Start and Late Finish values are reliably determined, enabling project managers to proactively manage resources, mitigate risks, and ensure timely project completion.

8. Resource Constraints

Resource constraints, encompassing limitations on personnel, equipment, and funding, significantly influence the calculation of Late Start and Late Finish times. The availability of resources directly affects the duration of activities. If resources are limited, activity durations may extend, thereby shifting the Late Start and Late Finish times. For example, if a construction project has a limited number of skilled laborers, tasks requiring these laborers may take longer than initially planned, pushing back the Late Start and Late Finish of subsequent activities. Similarly, in software development, a shortage of experienced programmers can extend the coding phase, impacting the schedule of testing and deployment. Accurate assessment of resource availability is, therefore, crucial for generating realistic and reliable Late Start and Late Finish times.

The impact of resource constraints extends beyond individual activity durations. These limitations often necessitate resource leveling, a process of adjusting activity start and finish times to optimize resource utilization. Resource leveling can alter the critical path, leading to changes in the Late Start and Late Finish of activities. For instance, if two activities require the same specialized equipment, and only one piece is available, one activity must be delayed. This delay affects its Late Start and Late Finish, as well as those of its successor activities. Understanding resource constraints is vital for project managers to make informed decisions about activity scheduling and resource allocation. Consider a project that needs a special crane to deliver big metal beams that will be used in a building construction. If the project only have access to a crane for one day out of the week, the project manager has to schedule this deliver on the day of the week where the crane is available, otherwise, there would be a great delay in the project.

In summary, resource constraints are a critical factor in the determination of Late Start and Late Finish times. Accurately accounting for resource limitations is essential for generating a realistic project schedule and effectively managing project timelines. Failure to address resource constraints can lead to inaccurate Late Start and Late Finish calculations, potentially jeopardizing project success. Understanding all this makes sure that project managers do not make promises that they cannot keep, or make plans that they cannot make reality.

9. Successors

In project scheduling, successor activities are those that cannot begin until their predecessor activities are completed. These sequential relationships are critical in determining the Late Start and Late Finish times. The Late Start of a successor activity is directly dependent on the Late Finish of its predecessor. Accurate identification and management of these relationships are essential for effective project planning and control.

  • Impact on Backward Pass

    The backward pass, used to calculate Late Start and Late Finish times, relies heavily on the defined successor relationships. The Late Finish of a successor activity is used to determine the Late Start of its predecessor. For example, if Activity B cannot start until Activity A is finished, the Late Start of Activity B constrains the Late Finish of Activity A. If Activity B has a Late Start of Day 10, then Activity A must finish by Day 10 at the latest. An incorrect definition of successor relationships would lead to inaccurate Late Start and Late Finish calculations, potentially jeopardizing the project timeline. Consider a bridge construction project where pouring the concrete foundation (Activity A) is a predecessor to erecting the steel structure (Activity B). If Activity B cannot start until Activity A is complete, the Late Start of erecting the steel structure constrains the Late Finish of pouring the concrete foundation.

  • Critical Path Implications

    Successor relationships on the critical path have the most significant impact on project completion. Activities on the critical path have zero or near-zero float, meaning any delay in a predecessor activity directly affects the Late Start and Late Finish of its successors, and, consequently, the overall project deadline. Accurate identification of successor relationships on the critical path is crucial for proactive risk management and schedule control. A delay in designing a critical software module, for instance, will directly impact the Late Start and Late Finish of the coding and testing phases, potentially delaying the project’s release. For another instance, a predecessor on the critical path might be designing a blue print before the construction can actually start. A delay on the blue print delay every other activity afterwards.

  • Float Calculation

    The Late Start and Late Finish times, in conjunction with the Early Start and Early Finish times, determine the total float or slack available for an activity. The presence and duration of successor activities influence the total float calculation. If an activity has numerous successor activities, its Late Finish time will be constrained by the earliest Late Start of its successors, resulting in less float. Conversely, an activity with few or no successors may have more float. Understanding these relationships is essential for optimizing resource allocation and managing potential delays. This is especially useful for project managers who needs to decide which activities can be delayed and which activities cannot be delayed, the ones on the critical path. For instance, consider an activity involving writing project documentation. If this activity has minimal successor activities, it may possess significant float, allowing for greater scheduling flexibility.

In conclusion, the concept of successor activities is intricately linked to the determination of Late Start and Late Finish times. Accurate identification and management of these relationships are essential for effective project scheduling, resource allocation, and risk management. The backward pass analysis relies heavily on the defined successor relationships to determine the Late Start and Late Finish times, highlighting the importance of a thorough understanding of activity dependencies.

Frequently Asked Questions

This section addresses common inquiries related to the determination of Late Start and Late Finish times in project schedule management. The goal is to clarify concepts and provide concise answers to frequently raised questions.

Question 1: What is the primary purpose of calculating Late Start and Late Finish times?

The primary purpose is to identify the latest possible dates an activity can begin and end without delaying the overall project completion date. This information assists in resource allocation, schedule risk assessment, and flexibility in task management.

Question 2: How does activity duration impact Late Start and Late Finish calculations?

Activity duration is a fundamental input. In the backward pass, the activity’s duration is subtracted from its Late Finish to determine the Late Start. Inaccurate duration estimates directly skew the results.

Question 3: How do dependencies influence Late Start and Late Finish?

Dependencies dictate the order in which activities must be performed. The Late Start of a successor activity is constrained by the Late Finish of its predecessor, necessitating accurate dependency definition.

Question 4: What is the significance of Total Float in relation to Late Start and Late Finish?

Total Float represents the time an activity can be delayed without impacting the project end date. The Late Start and Late Finish reflect the boundaries within which an activity can occur, considering its Total Float.

Question 5: How does the Project Deadline affect the Late Start and Late Finish calculations?

The Project Deadline serves as the anchor point for the backward pass analysis. It is the starting point from which Late Finish times are derived, directly impacting all subsequent calculations.

Question 6: What role does the critical path play in determining Late Start and Late Finish?

Activities on the critical path have zero or near-zero float. The Early Start equals the Late Start, and the Early Finish equals the Late Finish. Therefore, maintaining schedule adherence for critical path activities is paramount.

Understanding these core principles enables more accurate scheduling and better project control.

The next section will delve into practical applications of Late Start and Late Finish within various project management scenarios.

Calculating Late Start and Late Finish

The following tips offer practical guidance for ensuring accurate and effective determination of Late Start and Late Finish times. Implementation of these strategies enhances project schedule reliability and control.

Tip 1: Precisely Define Activity Durations Accurately estimate activity durations using historical data, expert judgment, and realistic assessments of resource availability. Consistent underestimation or overestimation introduces systematic errors into Late Start and Late Finish calculations.

Tip 2: Meticulously Document Activity Dependencies Clearly define and document all dependencies between activities, specifying the type of relationship (Finish-to-Start, Start-to-Start, Finish-to-Finish) and any associated lead or lag times. Neglecting dependencies leads to inaccurate Late Start and Late Finish times.

Tip 3: Thoroughly Validate the Project Network Diagram Ensure the project network diagram accurately reflects the sequence of activities and their interdependencies. A well-constructed diagram serves as the foundation for accurate Late Start and Late Finish calculations.

Tip 4: Regularly Review and Update the Schedule Project schedules are dynamic. Consistently review and update the schedule to reflect changes in activity durations, dependencies, resource availability, and project scope. Failure to update the schedule invalidates previously calculated Late Start and Late Finish times.

Tip 5: Prioritize Critical Path Activities Activities on the critical path demand heightened attention. Their Late Start and Late Finish times have minimal float, making them highly sensitive to delays. Implement proactive risk management strategies to prevent disruptions to critical path activities.

Tip 6: Implement Resource Leveling Techniques Optimize resource allocation to mitigate the impact of resource constraints on Late Start and Late Finish times. Resource leveling may necessitate adjustments to activity start and finish dates, affecting the overall schedule.

Tip 7: Employ Project Management Software Utilize project management software to automate Late Start and Late Finish calculations and facilitate schedule analysis. These tools provide visual representations of the schedule, enabling easier identification of critical path activities and potential scheduling conflicts.

These tips are aimed at improving project schedule accuracy and effectiveness. Employing these strategies can lead to better planning, risk mitigation, and resource allocation. Effective Late Start and Late Finish calculations result in more predictable project outcomes.

The subsequent section summarizes the key concepts discussed and offers a final perspective on this crucial aspect of project management.

Conclusion

This exploration of how to calculate late start and late finish underscores the critical role these metrics play in effective project management. Accurate determination of these values, through meticulous analysis of activity durations, dependencies, and resource constraints, provides project managers with the insights necessary for proactive schedule management and risk mitigation. Proper application of the backward pass, coupled with a thorough understanding of the critical path and available float, facilitates informed decision-making regarding resource allocation and task prioritization.

Mastering the techniques for calculating late start and late finish is therefore essential for any project management professional seeking to ensure project success. Continued refinement of these skills, through practical application and ongoing learning, will contribute to improved project outcomes and a greater likelihood of achieving desired objectives. The ability to precisely determine these schedule parameters enables proactive management, reducing the potential for costly delays and ultimately increasing the likelihood of project completion within defined constraints.