📚 KS3 Maths: Critical Path Analysis – Key Points Revision | KS3 数学：关键路径分析 考点精讲
Critical path analysis is a powerful tool for planning projects, helping you schedule tasks, identify bottlenecks, and work out the shortest possible completion time. This revision guide covers all the essential concepts you need to master for KS3 maths: from drawing activity networks to calculating earliest start times, latest start times, float, and finding the critical path. You'll learn how to break down a project, spot dependent activities, and ensure everything runs on time.
关键路径分析是一种强大的项目规划工具，能帮你安排任务、识别瓶颈，并计算出最短完成时间。这份复习指南涵盖了KS3数学需要掌握的所有核心概念：从绘制活动网络图，到计算最早开始时间、最晚开始时间、浮动时间，再到找出关键路径。你将学会如何分解项目、识别依赖活动，并确保一切按时完成。
1. What Is Critical Path Analysis? | 什么是关键路径分析？
Critical path analysis (CPA) is a method that uses a network diagram to model a project made up of individual tasks. It identifies the longest chain of dependent activities, called the critical path, which determines the minimum time needed to complete the entire project. Any delay on the critical path directly causes a delay to the whole project.
关键路径分析（CPA）是一种使用网络图对由独立任务组成的项目进行建模的方法。它找出最长的依赖活动链，称为关键路径，这条路径决定了完成整个项目所需的最短时间。关键路径上的任何延误都会直接导致整个项目延误。
CPA is widely used in construction, software development, and event planning. By calculating earliest and latest start times for each activity, you can see which tasks have spare time (float) and which are time-critical.
CPA被广泛用于建筑、软件开发和活动策划。通过计算每项活动的最早和最晚开始时间，你可以看出哪些任务有富余时间（浮动时间），哪些是时间紧迫的关键任务。
2. Activities and Durations | 活动与持续时间
An activity is a single task that takes time and uses resources. Every activity has a duration – the length of time required to complete it. Durations are usually given in hours, days, or weeks, and must be known before you can analyse the network. For example, 'Paint the wall' might have a duration of 4 hours.
活动是单个耗时且占用资源的任务。每项活动都有一个持续时间 – 完成它所需的时间长度。持续时间通常以小时、天或周为单位给出，必须在分析网络之前就知道。例如，“刷墙”可能持续4小时。
In a network diagram, activities are often drawn as nodes (rectangles or circles) containing the activity name and its duration. Arrows between nodes show the order in which activities must happen. Sometimes activities are written on the arcs themselves, but at KS3 we usually use the activity-on-node representation.
在网络图中，活动通常画成节点（矩形或圆形），内部写有活动名称和持续时间。节点之间的箭头表示活动必须发生的顺序。有时活动会写在弧线上，但在KS3阶段，我们通常使用节点表示法。
3. Precedence (Dependencies) | 前置关系（依赖）
Many activities cannot start until one or more previous activities have finished. These are called dependencies or precedence relationships. For instance, you cannot put on a roof until the walls are built. In a network, a directed arrow from Activity X to Activity Y means Y depends on X.
许多活动必须等一个或多个前置活动完成后才能开始。这被称为依赖或前置关系。例如，必须先砌好墙才能盖屋顶。在网络中，从活动X指向活动Y的有向箭头表示Y依赖X。
A clear way to describe dependencies is to list the immediate predecessors for each activity. This helps you build the network correctly without missing any logical links. Always check whether multiple activities feed into one task – that task cannot start until all its predecessors are complete.
清晰描述依赖关系的方法是列出每项活动的直接前置活动。这有助于正确构建网络，避免遗漏任何逻辑连接。务必检查是否有多个活动汇入一项任务——该任务必须等所有前置活动都完成后才能开始。
4. Drawing an Activity-on-Node Network | 绘制节点活动网络图
Start by placing the first activity (often labelled 'Start' or 'A') on the left. Then, following the precedence table, draw arrows to activities that can begin next. Each activity appears only once, and every dependency must be shown. Use a final node labelled 'Finish' if needed, to mark the project end.
首先把第一个活动（常标记为“开始”或“A”）放在左侧。然后，按照前置关系表，画出指向接下来可以开始的活动的箭头。每个活动只出现一次，每条依赖关系都必须表示出来。如果需要，可使用标记为“结束”的最终节点来标示项目的完成。
A typical KS3 question might give you a table like this:
| Activity | Duration (days) | Predecessors |
|---|---|---|
| A | 3 | – |
| B | 4 | A |
| C | 2 | A |
| D | 1 | B, C |
Draw node A first, then from A draw two arrows to nodes B and C. From B and C, both arrows point to D. The network is now complete. It is a good habit to check that every activity (except the start) has at least one incoming arrow.
首先画出节点A，然后从A画两个箭头分别指向节点B和C。从B和C各画一个箭头指向D。网络图就完成了。养成检查每项活动（除起始活动外）至少有一条入边的好习惯。
5. Calculating Earliest Start Times (EST) | 计算最早开始时间（EST）
The earliest start time (EST) is the soonest an activity can begin, assuming all previous activities start as early as possible. It is found by a forward pass: move from left to right across the network. The EST of the first activity is zero. For any other activity, the EST is the maximum of (EST of a predecessor + its duration) for all its predecessors.
最早开始时间（EST）是指假设所有前置活动都尽早开始的情况下，一项活动能够开始的最早时间。它通过前向遍历求得：从左到右扫描网络。第一个活动的EST为零。对于任何其他活动，EST等于其所有前置活动的（EST + 该前置活动的持续时间）中的最大值。
In formula terms:
EST(current) = max { EST(predecessor) + duration(predecessor) }
公式表示为：
EST(当前活动) = max { EST(前置活动) + 持续时间(前置活动) }
depends on B and C: from B, 3 + 4 = 7; from C, 3 + 2 = 5. The maximum is 7, so EST(D) = 7.
以上例而言：A的EST = 0。B依赖于A，所以EST(B) = 0 + 3 = 3。C依赖于A，EST(C) = 0 + 3 = 3。D依赖于B和C：从B来，3 + 4 = 7；从C来，3 + 2 = 5。最大值为7，因此EST(D) = 7。
The earliest project completion time is the EST of a final 'Finish' node, or the EST of the last activity plus its duration. Here, D's duration is 1, so the project finishes at 7 + 1 = 8 days.
项目的最早完成时间是最终“结束”节点的EST，或者是最后一项活动的EST加上其持续时间。在这里，D的持续时间为1，因此项目在第8天完成。
6. Calculating Latest Start Times (LST) | 计算最晚开始时间（LST）
The latest start time (LST) is the latest an activity can begin without delaying the entire project, assuming all subsequent activities start as late as possible. It is found by a backward pass from right to left. Begin at the end of the network: the LST of the final activity (or finish node) equals the project's earliest completion time.
最晚开始时间（LST）是指在不延误整个项目的前提下，一项活动可以开始的最晚时间，假设所有后续活动都尽可能晚开始。它通过从右向左的后向遍历求得。从网络末端开始：最终活动（或结束节点）的LST等于项目的最早完成时间。
The LST of any other activity is the minimum of (LST of a successor − duration of the current activity) over all its immediate successors.
LST(current) = min { LST(successor) − duration(current) }
任何其他活动的LST等于其所有直接后继活动的（后继活动的LST − 当前活动的持续时间）中的最小值。
LST(当前活动) = min { LST(后继活动) − 持续时间(当前活动) }
ST(C) = 7 − 2 = 5. For A: two successors B (LST 3) and C (LST 5). LST(A) = min(3 − 3, 5 − 3) = min(0, 2) = 0.
对前例应用后向遍历：最后一项活动D。项目在第8天完成，所以LST(D) = 8 − 1 = 7。对于B：LST(D) = 7，因此LST(B) = 7 − 4 = 3。对于C：LST(C) = 7 − 2 = 5。对于A：有两个后继B（LST 3）和C（LST 5）。LST(A) = min(3 − 3, 5 − 3) = min(0, 2) = 0。
7. Float (Slack) | 浮动时间（时差）
Float (or slack) measures how long an activity can be delayed without affecting the overall project duration. It is calculated simply as:
Float = LST − EST
浮动时间（或称时差）衡量一项活动在不影响项目总工期的前提下可以延迟多长时间。它的计算公式很简单：
浮动时间 = LST − EST
If the float is zero, the activity is critical – re critical), while C has float = 5 − 3 = 2 days. C can be delayed by up to 2 days without affecting the finish date.
如果浮动时间为零，该活动就是关键活动——任何延误都会使整个项目延后。如果浮动时间为正，活动就有一定的富余时间。在例子中，A、B和D的浮动时间均为0（它们是关键活动），而C的浮动时间 = 5 − 3 = 2天。C最多可延迟2天而不会影响完成日期。
Float is extremely useful for resource management: you can move non-critical tasks to balance workloads without endangering the deadline.
浮动时间对资源管理非常有用：你可以调整非关键任务来平衡工作负荷，而不威胁截止日期。
8. Identifying the Critical Path | 识别关键路径
The critical path is the continuous chain of activities through the network that have zero float. It is always the longest path from start to finish in terms of total duration, and it dictates the minimum project completion time. There can be more than one critical path if multiple paths share the same maximum length.
关键路径是网络中由浮动时间为零的活动组成的连续链。就总时长而言，它始终是从开始到结束的最长路径，并决定了项目的最短完成时间。如果多条路径具有相同的最大长度，则可以有多条关键路径。
rk, A → B → D is the critical path, with total duration 3 + 4 + 1 = 8 days. The alternative path A → C → D lasts 3 + 2 + 1 = 6 days, so it is not critical.
要找出关键路径，需进行前向和后向遍历，然后突出显示每个EST = LST（因此浮动时间 = 0）的活动。在我们的网络中，A → B → D是关键路径，总时长为3 + 4 + 1 = 8天。另一条路径A → C → D持续3 + 2 + 1 = 6天，所以不是关键路径。
When presenting your solution, clearly state the critical path activities in order and the total project duration. Always double-check the network to confirm that no other sequence of activities has a longer summed duration.
在呈现解答时，请清楚地按顺序列出关键路径活动以及总项目工期。务必再次检查网络，确认没有其他活动序列的总和持续时间更长。
9. Importance of the Critical Path | 关键路径的重要性
Knowing the critical path allows project managers to focus on the tasks that cannot slip. Any delay on a critical activity directly increases the project's overall completion time. Resources, such as extra workers or equipment, can be shifted from non-critical tasks to critical ones if they risk falling behind.
了解关键路径能让项目经理重点关注那些不能延误的任务。关键活动的任何延误都会直接增加项目整体完成时间。如果关键任务有落后的风险，可以将额外工人或设备等资源从非关键任务调拨过来。
Critical path analysis also helps with 'what-if' planning. By seeing how changes in certain durations affect the finish date, you can make better decisions about overtime, outsourcing, or resequencing work.
关键路径分析还有助于进行“假设”规划。通过观察某些持续时间的变化如何影响完成日期，你可以就是否加班、外包或重新排序工作做出更好的决策。
10. Common Mistakes and Tips | 常见错误与提示
One frequent mistake is forgetting to take the maximum when merging multiple predecessors for EST. Always compare all incoming paths and choose the largest value. Similarly, for LST during the backward pass, take the minimum of the options presented by different successors.
一个常见错误是在合并多个前置活动的EST时忘记取最大值。一定要比较所有入边路径并选择最大的值。同样，在后向遍历计算LST时，要从不同后继活动给出的选项中取最小值。
Another pitfall is drawing the network incorrectly. Double-check your arrows: each dependency in the precedence table must correspond to exactly one directed edge. Never add extra links that aren't stated. It can help to sketch a rough diagram first, then redraw neatly for the formal solution.
另一个陷阱是绘图错误。仔细检查箭头：前置关系表中的每条依赖关系必须恰好对应于一条有向边。绝不要添加未指明的额外连接。先画一个草图，然后再整洁地重绘作为正式解答，这会很有帮助。
Students sometimes misread the question and use the wrong unit of time, or forget to write the final project duration clearly. Always label nodes with both EST and LST, and show the float for each activity. Neatness and clarity earn marks in exams.
学生有时会误读题目而用错时间单位，或忘记清楚地写出最终项目工期。务必在节点上标出EST和LST，并显示每项活动的浮动时间。整洁清晰的书写在考试中能赢得分数。
11. Worked Example | 实例演练
Let's work through a full example to consolidate the steps. The project has four activities with the following data:
让我们通过一个完整示例来巩固步骤。该项目有四项活动，数据如下：
| Activity | Duration (days) | Predecessors |
|---|---|---|
| A | 3 | – |
| B | 4 | A |
| C | 2 | A |
| D | 1 | B, C |
4) and A (3) → C (2). Both B and C point to D (1). We will compute EST and LST in a table.
绘制网络：A (3) → B (4) 且 A (3) → C (2)。B和C都指向D (1)。我们将在表格中计算EST和LST。
B) = 0+3=3; EST(C) = 0+3=3; EST(D) = max(3+4, 3+2) = 7. Project completion = 7+1 = 8 days.
前向遍历：EST(A) = 0；EST(B) = 0+3=3；EST(C) = 0+3=3；EST(D) = max(3+4, 3+2) = 7。项目完成时间 = 7+1 = 8天。
T(D) = 8−1 = 7; LST(B) = 7−4 = 3; LST(C) = 7−2 = 5; LST(A) = min(3−3, 5−3) = min(0,2) = 0.
后向遍历：LST(D) = 8−1 = 7；LST(B) = 7−4 = 3；LST(C) = 7−2 = 5；LST(A) = min(3−3, 5−3) = min(0,2) = 0。
| Activity | Duration | EST | LST | Float |
|---|---|---|---|---|
| A | 3 | 0 | 0 | 0 |
| B | 4 | 3 | 3 | 0 |
| C | 2 | 3 | 5 | 2 |
| D | 1 | 7 | 7 | 0 |
project duration is 8 days. Notice that activity C has 2 days of float and can be delayed without affecting the finish date.
活动A、B和D的浮动时间为零，因此关键路径是A → B → D。项目总工期为8天。注意活动C有2天浮动时间，可以在不影响完成日期的前提下延迟。
This systematic approach will give full marks on most KS3 exam questions. Always present your working clearly: draw the network, tabulate EST and LST, calculate float, and state the critical path.
这种系统化方法能让你在大多数KS3考试题目中获得满分。务必清晰地展示解题过程：画出网络图，用表格列出EST和LST，计算浮动时间，并指出关键路径。
12. Summary and Exam Tips | 总结与考试技巧
To master critical path analysis, remember the four golden steps: (1) build the network using the precedence table; (2) perform a forward pass to find ESTs; (3) perform a backward pass to find LSTs; (4) compute float and identify the critical path. Practice with different network shapes – some will have multiple starting activities or several merging points.
要掌握关键路径分析，请记住四个黄金步骤：（1）使用前置关系表建立网络；（2）执行前向遍历以求得EST；（3）执行后向遍历以求得LST；（4）计算浮动时间并确定关键路径。多练习不同形状的网络——有些网络可能有多个
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