Critical Path Analysis | 关键路径分析

📚 Critical Path Analysis | 关键路径分析

Critical path analysis is a network-based project management technique that helps identify the sequence of tasks which determine the minimum duration of a project. By examining precedence relationships and activity durations, you can find the critical path, calculate float times, and schedule resources efficiently. This guide covers all key concepts required for the AQA A-Level Mathematics specification, including activity networks, early and late event times, critical activities, total float, and Gantt charts.

关键路径分析是一种基于网络的项目管理技术，帮助识别决定项目最短工期的一系列任务。通过研究先后关系和活动时长，你能找到关键路径、计算浮动时间并高效调度资源。本指南覆盖了 AQA A-Level 数学规范中的所有核心概念，包括活动网络、最早和最晚事件时间、关键活动、总浮动以及甘特图。

1. Activity Networks and Precedence Tables | 活动网络与先后关系表

A project consists of a set of activities, each requiring a certain duration and depending on the completion of other activities. A precedence table lists each activity, its duration, and the activities that must be completed immediately before it. From this table, we construct an activity-on-node network, where nodes represent the start and end of activities and arcs represent the activities themselves.

一个项目由一系列活动组成，每一项活动需要一定的时间，并依赖于其他活动的完成。先后关系表列出了每项活动、所需时长以及紧前活动。从这个表可以构建节点表示活动的网络（活动节点法），其中节点代表事件的开始与结束，弧代表活动本身。

Nodes are usually numbered, and each activity is represented by a directed arc from its start node to its end node.
节点通常编号，每项活动由从开始节点指向结束节点的有向弧表示。
Dummy activities (duration 0, shown as a dashed arc) are used to maintain logical precedence when two activities share some but not all predecessors, avoiding multiple arcs between the same pair of nodes.
虚活动（时长为 0，用虚线弧表示）用于当两个活动共享部分（但不是全部）紧前活动时保持逻辑先后关系，避免同一对节点间有多条弧。

To draw an activity network correctly, ensure there is a single start node and a single end node. Each activity appears exactly once. Dummy activities are added to preserve precedence when needed.

要正确绘制活动网络，确保只有一个开始节点和一个结束节点。每个活动只出现一次。当需要维护先后关系时，可添加虚活动。

2. Early Event Times and Forward Pass | 最早事件时间与前推法

The early event time (EET) of a node is the earliest time at which all activities leading into that node can be completed. It is calculated by performing a forward pass from the start node, whose EET is set to 0. For each subsequent node, take the maximum of (preceding node’s EET + activity duration) over all incoming arcs.

节点的最早事件时间（EET）是所有到达该节点的活动能够完成的最早时间。它通过从开始节点进行前推计算，开始节点的 EET 设为 0。对于后续每个节点，取所有入弧的（前一个节点的 EET + 活动时长）中的最大值。

Formula: EET(node) = max{EET(predecessor) + duration(activity) for each incoming activity}

公式：EET(节点) = max{ EET(前驱节点) + 活动时长 | 每条入弧 }

The forward pass reveals the earliest possible project duration, which equals the EET at the end node.

前推法揭示了项目可能的最短工期，即结束节点的 EET。

3. Late Event Times and Backward Pass | 最晚事件时间与回溯法

The late event time (LET) of a node is the latest time at which an event must be completed to avoid delaying the entire project. Starting from the end node, whose LET equals its EET (the project duration), work backwards. For each node, take the minimum of (preceding node’s LET minus activity duration) over all outgoing arcs.

节点的最晚事件时间（LET）是为了不延误整个项目，该事件必须完成的最晚时间。从结束节点开始（其 LET 等于其 EET，即项目工期），向后计算。对于每个节点，在所有出弧中取（后继节点的 LET 减去活动时长）的最小值。

Formula: LET(node) = min{LET(successor) – duration(activity) for each outgoing activity}

公式：LET(节点) = min{ LET(后继节点) – 活动时长 | 每条出弧 }

The forward and backward passes together determine the float available for each activity.

前推和回溯共同决定了每个活动可用的浮动时间。

4. Total Float and Critical Activities | 总浮动与关键活动

Total float for an activity is the amount of time its start can be delayed without affecting the overall project duration. It is calculated as:

活动的总浮动是指其开始可以延迟而不影响整体项目工期的时间量。其计算公式为：

Total Float = LET(end node) – EET(start node) – Duration

总浮动 = 结束节点的 LET – 开始节点的 EET – 活动时长

An activity with total float equal to zero is critical; it lies on the critical path. Any delay in a critical activity will directly delay the project. The critical path is the longest path through the network in terms of total duration, and there may be more than one.

总浮动为零的活动是关键活动，位于关键路径上。关键活动的任何延迟都会直接推迟整个项目。关键路径是按总时长计算网络中最长的路径，可能不唯一。

5. Identifying the Critical Path | 识别关键路径

To identify the critical path, compute EETs and LETs for every node, then list activities with zero total float. Trace these activities from start to end. Alternatively, list all possible paths and compute their lengths; the one(s) with the greatest length is the critical path.

要识别关键路径，先计算每个节点的 EET 和 LET，然后列出总浮动为零的活动，从开始追踪到结束。也可以列出所有可能路径并计算长度，最长的那条就是关键路径。

Sometimes the critical path can be highlighted on the network diagram by a bold line or a different colour.

有时在网络上可以用粗线或不同颜色标出关键路径。

Example: If activities A(4), B(5), C(3), D(2) are connected, with A→B→C on one path and A→D on another, compute EETs and LETs to find that A, B, C are critical.

例如：若活动 A(4)、B(5)、C(3)、D(2) 相连，一条路径为 A→B→C，另一条为 A→D，计算 EET 和 LET 可发现 A、B、C 是关键活动。

6. Free Float and Interfering Float | 自由浮动与干扰浮动

Besides total float, an activity may have free float: the time by which its start can be delayed without affecting the early start of any successor activity. Free float = EET(successor) – EET(start) – Duration, assuming a single successor. Interfering float is the difference between total float and free float, representing the delay that does affect successors but not the overall project end.

除了总浮动，活动还有自由浮动：在不影响任何后续活动最早开始时间的前提下，开始可以延迟的时间。对于单一后继，自由浮动 = EET(后继) – EET(开始) – 时长。干扰浮动是总浮动与自由浮动之差，代表会影响后续活动但不推迟项目结束的延迟部分。

These float types are useful for resource levelling and understanding schedule flexibility.

这些浮动类型对于资源平衡和理解调度灵活性十分有用。

7. Gantt Charts and Resource Scheduling | 甘特图与资源调度

A Gantt chart is a horizontal bar chart that shows activities against time. Each activity is represented by a bar, positioned according to its early start and end times (or scheduled times). Critical activities are often highlighted, and floats can be shown by extending the bar to the latest start or by a secondary bar in a different shading.

甘特图是一种水平条形图，展示活动随时间的变化。每个活动用一条横条表示，位置根据最早开始和结束时间（或计划时间）放置。关键活动通常会被突出显示，浮动时间可通过将横条延伸到最晚开始时间或使用不同阴影的第二个条来表示。

Resource histograms can be derived from Gantt charts to show the number of workers or machines required each day, helping to smooth resource demand and avoid over-allocation.

资源直方图可以从甘特图中得出，显示每天所需的工人或机器数量，有助于平衡资源需求并避免过度分配。

8. Scheduling with Limited Resources | 有限资源下的调度

When resources are constrained, activities may need to be delayed beyond their early start times in order to level demand. This process, called resource levelling, uses float information to reschedule non-critical activities without extending the project duration. If the resource constraint is severe, the project duration may need to be extended.

当资源受限时，活动可能不得不推迟到其最早开始时间之后，以平衡需求。这个过程称为资源平衡，利用浮动信息重新安排非关键活动而不延长项目工期。如果资源限制非常严格，项目工期可能需要延长。

A common exam task is to draw a resource histogram from a given schedule and then adjust start times to reduce peaks while keeping the project on time.

考试中常见的任务是，根据给定调度绘制资源直方图，然后调整开始时间以降低峰值，同时保持项目按时完成。

9. Dummy Activities in Detail | 虚活动详解

Dummy activities have zero duration and are used for logical purposes only. They are essential when two activities share some but not all predecessors, or when an activity depends on another indirectly. Without dummies, the network may imply false dependencies or have parallel arcs that are not allowed.

虚活动时长为零，仅用于逻辑目的。当两个活动共享部分（而非全部）紧前活动，或者一个活动间接依赖于另一个活动时，虚活动至关重要。没有虚活动，网络可能会暗示错误的依赖关系，或出现不允许的平行弧。

For example, if activity C depends on A, and D depends on A and B, then a dummy from the end of A to the start of D (via a node) ensures D waits for A without making C dependent on B.

例如，若活动 C 依赖于 A，D 依赖于 A 和 B，那么从 A 的末端到 D 的始端（通过一个节点）添加虚活动，确保 D 等待 A，同时不会让 C 依赖于 B。

10. Converting between Node and Arc Representations | 节点表示与弧表示的转换

Although the AQA specification primarily uses activity-on-node (where nodes are events and arcs are activities), some students encounter activity-on-arrow diagrams where arrows represent activities and nodes are events. In activity-on-arrow, dummy activities appear as dashed arrows with zero duration, used to maintain proper logic.

虽然 AQA 规范主要使用活动节点法（节点是事件，弧是活动），一些学生也会遇到活动弧法，其中箭头代表活动，节点是事件。在活动弧法中，虚活动显示为零时长的虚线箭头，用于维护正确的逻辑。

You should be able to interpret both representations and convert a precedence table into either network type. The underlying principles of forward and backward passes remain the same.

你应该能够理解两种表示法，并能将先后关系表转换为任何一种网络类型。前推和回溯的基本原理保持一致。

11. Exam Strategy: Critical Path Analysis | 考试策略：关键路径分析

In AQA A-Level exams, typical questions ask you to draw a network from a precedence table (including dummies), perform a forward and backward pass to complete a diagram showing EETs and LETs, identify the critical path and minimum project duration, and calculate total float for given activities. Later parts may require a Gantt chart, a resource histogram, or scheduling decisions.

在 AQA A-Level 考试中，典型题目要求你根据先后关系表绘制网络（包括虚活动），完成前推和回溯以标出 EET 和 LET，识别关键路径和项目最短工期，并计算给定活动的总浮动时间。后续可能要求绘制甘特图、资源直方图或做出调度决策。

Always check your network for a single start and end node.
始终检查网络是否有单一的开始和结束节点。
Use systematic forward and backward passes, recording EETs in a box at each node and LETs in a box underneath.
进行系统性的前推和回溯，将 EET 记在每个节点的方框中，LET 记在下方的方框中。
Show total float calculations clearly: LET(end) – EET(start) – Duration.
清楚写出总浮动计算：LET(结束) – EET(开始) – 时长。
On a Gantt chart, use early start times by default; floats can be indicated by dashed extensions.
在甘特图中默认使用最早开始时间；浮动可用虚线延伸来表示。

12. Common Pitfalls and How to Avoid Them | 常见错误与避免方法

Mistakes often arise from incorrectly adding dummy activities, mixing up forward and backward pass rules, forgetting to use the maximum on forward pass or minimum on backward pass, and miscalculating total float by using the wrong node times. In resource levelling, students may shift activities without considering all successors’ constraints, thus creating new dependencies accidentally.

常见错误包括：错误添加虚活动、混淆前推和回溯规则、忘记前推用最大值而回溯用最小值，以及用错节点时间计算总浮动。在资源平衡中，学生可能移动活动而未考虑所有后继约束，意外产生新的依赖关系。

To avoid these, practice drawing networks for various precedence tables, and double-check each dummy’s logic. When calculating floats, always label EET and LET boxes clearly, and verify that critical activities form a continuous chain from start to end.

为避免这些错误，多练习为不同先后关系表绘制网络，并反复检查每个虚活动的逻辑。计算浮动时，始终清晰标出 EET 和 LET 框，并验证关键活动是否形成从开始到结束的连续链。

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Critical Path Analysis | 关键路径分析