📚 Critical Path Analysis for IB & CCEA Mathematics | IB & CCEA 数学：关键路径分析考点精讲
Critical Path Analysis (CPA), also called Critical Path Method (CPM), is a cornerstone of decision mathematics in both the IB Mathematics: Applications and Interpretation course and the CCEA Decision Maths module. It equips you with the skills to model real‑world projects, schedule activities efficiently, and guarantee on‑time completion. In this revision article we break down every key concept – from activity networks and float calculations to Gantt charts and resource levelling – and provide a fully worked exam‑style example to consolidate your understanding.
关键路径分析（CPA，又称关键路径法 CPM）是 IB 数学：应用与解释课程以及 CCEA 决策数学模块中决策数学的基石。它让你具备为现实项目建模、高效安排活动并保证按时完成的能力。在这篇复习文章中，我们将逐一拆解每个重要概念——从活动网络图和浮动时间计算到甘特图和资源均衡——并提供一个完整的考试题型示例来巩固你的理解。
1. Activity Networks and Precedence Tables | 活动网络图与先行关系表
An activity network is a directed graph that visualises the logical order of tasks. In the activity‑on‑arc (AOA) representation used in IB and CCEA exams, each directed arc stands for an activity, and each vertex (node) represents an event – the start or finish of one or more activities. The project is first described by a precedence table that lists every activity, its duration, and its immediate predecessors.
活动网络图是一个有向图，它将任务的逻辑顺序可视化。在 IB 和 CCEA 考试所使用的弧表示活动（AOA）表示法中，每条有向弧代表一个活动，每个顶点（节点）代表一个事件——一个或多个活动的开始或结束。项目首先由一个先行关系表来描述，该表列出了所有活动、其持续时间以及它的直接前驱活动。
The start node is conventionally numbered 1, and the finish node receives the largest number. Each arc is labelled with the activity letter and its duration, e.g. A(5). Arrows must strictly follow precedence; no activity can begin until all of its predecessors are complete.
按照惯例，起始节点编号为 1，结束节点使用最大的编号。每条弧上标注活动字母及其持续时间，例如 A(5)。箭头必须严格遵守优先关系；任何活动在所有前驱活动完成之前都不能开始。
2. Nodes, Arcs and Dummy Activities | 节点、弧与虚活动
Nodes are usually shown as circles enclosing their event number. A network is built by drawing an arc for each activity and connecting them according to dependencies. When two activities share the same start and end events, or when a dependency exists without a direct activity, we insert a dummy activity. A dummy is drawn as a dashed arrow and carries a duration of zero – it consumes no time or resources but ensures correct logical relationships.
节点通常表示为包含事件编号的圆。构建网络时需要为每个活动画一条弧并根据依赖关系将它们连接起来。当两个活动拥有相同的起始和结束事件，或者存在没有直接活动的依赖关系时，我们需要插入虚活动。虚活动用虚线箭头表示，持续时间为零——它不消耗时间或资源，但保证了正确的逻辑关系。
For example, if activity C depends on A alone but activity D depends on both A and B, a dummy can be used between A and the start of D to prevent C from being incorrectly linked to B.
例如，如果活动 C 仅依赖于 A，而活动 D 同时依赖于 A 和 B，则可以在 A 和 D 的起始事件之间使用虚活动，以防止 C 被错误地关联到 B。
3. Forward Pass – Earliest Start Times (EST) | 前推法 – 最早开始时间 (EST)
The forward pass calculates the earliest possible time each event can be reached. Set the EST of the start node to 0. For any other node j, the EST is the maximum over all paths leading into j of the sum (EST of its predecessor + activity duration).
前推法计算每个事件可以到达的最早可能时间。将起始节点的 EST 设为 0。对于任意其他节点 j，EST 等于所有到达 j 的路径上的（前驱节点的 EST + 活动持续时间）之和的最大值。
ESTⱼ = max{EST₁ + t₁ⱼ, EST₂ + t₂ⱼ, …}
Carry out the forward pass from left to right across the network, recording the EST in the top half of each node. The EST of the final node gives the minimum project duration.
在网络图中从左向右执行前推，将 EST 记录在每个节点的上半部分。最终节点的 EST 就是项目的最短持续时间。
4. Backward Pass – Latest Start Times (LST) | 后推法 – 最晚开始时间 (LST)
The backward pass determines the latest time each event can occur without delaying the entire project. Begin at the finish node, setting its LST equal to its EST (the project duration). For any node i, work backwards: LST(i) = min{ LST(j) – duration(i→j) } for all activities leaving i.
后推法确定每个事件在不延误整个项目的情况下可以发生的最晚时间。从结束节点开始，将其 LST 设为与其 EST 相等（即项目持续时间）。对于任意节点 i，逆向计算：LST(i) = min{ LST(j) – 持续时间(i→j) }，对所有从 i 出发的活动取最小值。
LST₁ = min{LSTⱼ – t₁ⱼ}
Write the LST in the bottom half of each node. A correctly calculated network will give LST = EST = 0 at the start node.
将 LST 写在每个节点的下半部分。正确计算的网络会在起始节点处得到 LST = EST = 0。
5. Total Float and Critical Activities | 总浮动时间与关键活动
Total float measures how much an activity can be delayed without affecting the overall project deadline. For an activity going from node i to node j, total float = LST(j) – EST(i) – duration. Activities with zero total float are called critical; any delay to a critical activity directly postpones the project finish.
总浮动时间衡量一个活动可以延迟多久而不影响整个项目的截止日期。对于从节点 i 到节点 j 的活动，总浮动时间 = LST(j) – EST(i) – 持续时间。总浮动时间为零的活动称为关键活动；对任何关键活动的延误都会直接推迟项目完成时间。
In exam problems you are often required to complete a table with earliest start time, latest start time and total float for all activities. Remember: EST of an activity is simply the EST of its start node, and LST of an activity is LST(start node).
在考试中，经常要求你填写一个包含所有活动的最早开始时间、最晚开始时间和总浮动时间的表格。请记住：活动的 EST 就是其起始节点的 EST，活动的 LST 就是起始节点的 LST。
6. Identifying the Critical Path | 确定关键路径
The critical path is the longest path through the network from start to finish, consisting entirely of critical activities (float = 0). To identify it, trace a route that follows activities where LST – EST – duration = 0. The sum of durations along this path equals the project’s minimum completion time.
关键路径是从开始到结束穿过网络的最长路径，完全由关键活动（浮动时间 = 0）组成。要确定它，沿着那些满足 LST – EST – 持续时间 = 0 的活动追踪路线。沿着这条路径的持续时间之和等于项目的最短完成时间。
Always present your critical path as a sequence of activity letters, e.g. A – C – F – H, and then state the total project duration. In IB and CCEA scripts, missing a critical activity or misidentifying a near‑critical path is a frequent loss of marks.
在答题时，总要给出关键路径的活动字母序列，例如 A – C – F – H，然后说明项目总工期。在 IB 和 CCEA 答卷中，遗漏关键活动或误判一条接近关键的路径是常见的失分点。
7. Gantt Charts (Cascade Charts) | 甘特图（级联图）
A Gantt chart is a horizontal bar chart that displays activities against time. Each activity is drawn from its earliest start time, with the bar length equal to its duration. Critical activities are often shaded differently. The chart visually reveals float: non‑critical bars have slack beyond their fixed portion, usually drawn as a dotted extension or a gap.
甘特图是一个水平条形图，按时间显示各个活动。每个活动从其最早开始时间开始绘制，条形的长度等于其持续时间。关键活动通常以不同的阴影表示。该图直观地展现了浮动时间：非关键条形在其固定部分之外有空闲，通常用虚线延长或留白来表示。
When constructing a cascade chart, first list activities in order of EST, draw bars, and then add float segments. This tool also helps with resource smoothing in later problems.
在绘制级联图时，首先按 EST 顺序列出活动，画出条形，然后添加浮动时段。该工具还有助于后续问题中的资源平滑处理。
8. Resource Levelling | 资源均衡调度
Projects often have limited resources (e.g. workers, machinery). Resource levelling shifts non‑critical activities within their float to reduce peak resource usage while keeping the project duration unchanged. You will be given a resource histogram and must re‑schedule activities so that the maximum daily resource demand is minimised.
项目往往受到有限资源（如工人、机器）的约束。资源均衡调度在保持项目工期不变的前提下，将非关键活动在其浮动时间内移动，以降低资源使用峰值。你会被给出一张资源直方图，并需要重新安排活动使每日资源需求的最大值降至最低。
The method: start with the earliest start schedule, plot the resource profile, then sequentially delay activities with the largest float and high resource need. Exam questions may ask for the final levelled schedule or the new resource peak.
方法如下：从最早开始时间计划开始，绘制资源分布图，然后依次推迟浮动时间最大且资源需求高的活动。考试问题可能会要求给出最终的均衡调度或新的资源峰值。
9. Step‑by‑Step Time Analysis Summary | 逐步时间分析总结
Below is a checklist for a complete time analysis – essential for any CPA exam question:
以下是全部分时间分析的检查清单，对于任何 CPA 考试题都是必不可少的：
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Draw the activity network (AOA) from the precedence table, inserting dummies as needed.
根据先行关系表绘制活动网络图（AOA），必要时插入虚活动。
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Perform a forward pass to compute EST for all events. Record values in the top half of nodes.
执行前推计算所有事件的 EST。将数值记录在节点的上半部。
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Perform a backward pass from the end node to find LST for all events. Mark in the bottom half.
从结束节点开始执行后推，计算所有事件的 LST。标记在节点的下半部。
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For each activity, calculate total float = LST(end) – EST(start) – duration. Identify critical activities (float 0).
对每个活动计算总浮动时间 = LST(结束节点) – EST(起始节点) – 持续时间。确定关键活动（浮动时间为 0）。
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Trace the critical path(s) and state project duration. Draw a Gantt chart or resource histogram as required.
追踪关键路径并说明项目工期。根据要求绘制甘特图或资源直方图。
10. Exam‑Style Worked Example | 考试题型精讲
The following worked example models a small project. Use it to check your understanding of every step.
下面的例题对一个小型项目进行建模。用它来检验你对每一步的理解。
Precedence table:
先行关系表：
| Activity 活动 |
Duration 持续时间 |
Predecessors 前驱 |
|---|---|---|
| A | 3 | – |
| B | 4 | A |
| C | 2 | A |
| D | 5 | B, C |
| E | 3 | C |
| F | 2 | D, E |
Network construction: Start node 1. Arc A goes to node 2. From node 2, arcs B (to node 3) and C (to node 4) are drawn. Because D requires both B and C, we use a dummy from 3 to 4 (zero duration). Then D goes from 4 to 5. E goes from 4 to 5 as well, but since we need separate arcs for activities, we can introduce another node. However, a simpler way: let C go to node 3, B go to node 3 also. Wait – typical AOA: A → 2, then B from 2 to 3, C from 2 to 4. D needs B and C, so we need a common event: dummy from 3 to 4 (or from 4 to 3) to merge. After merging at 4, D goes to 5. E depends only on C, so E from 4 to 5. Then F from 5 to 6. This inserts one dummy. Let’s formalise: nodes: 1 (start), 2 (end of A),
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