📚 Critical Path Analysis for IB & OCR Maths: Key Concepts Explained | IB OCR 数学：关键路径分析 考点精讲
Critical path analysis (CPA) is a powerful technique in decision mathematics used to plan and schedule complex projects. It helps you identify which tasks are critical and cannot be delayed without affecting the overall project duration. This topic appears in both IB Mathematics (Applications and Interpretation HL/SL) and OCR A-Level Maths (Decision 1). Mastering CPA requires a clear understanding of activity networks, precedence tables, forward and backward passes, float calculations, and interpreting the critical path.
关键路径分析（CPA）是决策数学中用于规划和调度复杂项目的一种强大工具。它能帮助识别哪些任务至关重要、不能延迟，否则会影响整个项目的工期。该主题同时出现在 IB 数学（应用与解释 HL/SL）和 OCR A-Level 数学（决策 1）中。掌握 CPA 需要清晰理解活动网络、先后关系表、前向法和后向法、浮动时间计算以及解释关键路径。
1. What is Critical Path Analysis? | 什么是关键路径分析？
Critical path analysis is a project management method that models a sequence of activities, their durations, and their dependencies. The objective is to determine the shortest possible completion time for the entire project and to pinpoint the tasks that directly influence that timing.
关键路径分析是一种项目管理方法，对一系列活动、持续时间和依赖关系进行建模。其目标是确定整个项目的最短可能完成时间，并找出直接影响该时间的任务。
Every project can be broken down into discrete activities, each requiring time and resources. Some activities can be carried out concurrently, while others must follow a strict order. CPA uses network diagrams to represent these relationships, making it easier to visualise bottlenecks and schedule resources efficiently.
每个项目都可以分解为独立的活动，每一项都需要时间和资源。一些活动可以同时进行，而另一些则必须遵循严格的先后顺序。CPA 用网络图表示这些关系，便于直观发现瓶颈并进行高效的资源调度。
In IB and OCR exam questions, you may be asked to draw an activity-on-node network, perform forward and backward passes, calculate floats, identify the critical path, and construct a Gantt chart. A firm grasp of the underlying logic is essential for accuracy and speed.
在 IB 和 OCR 考试中，你可能需要绘制活动节点图、执行前向和后向计算、计算浮动时间、识别关键路径并构建甘特图。牢固掌握底层逻辑对于准确和快速解题至关重要。
2. Activity Networks and Precedence Tables | 活动网络与先后关系表
A precedence table is the starting point for any CPA problem. It lists each activity, its duration, and the activities that must be completed immediately before it can begin (immediate predecessors).
先后关系表是任何 CPA 问题的起点。它列出每个活动、其持续时间以及必须在开始前立即完成的活动（紧前活动）。
Dependencies are typically finish-to-start: a successor cannot start until its predecessor has finished. In an activity-on-node (AoN) representation, each activity is shown as a box (node) containing key information, and arrows indicate dependencies.
依赖关系通常是完成-开始型：后继活动必须等到前驱活动完成后才能开始。在活动节点图（AoN）表示法中，每个活动显示为一个包含关键信息的方框（节点），箭头指示依赖关系。
Sometimes a dummy activity (duration zero) is needed to clarify dependencies in activity-on-arrow diagrams, but for AoN networks required in IB and OCR, dummies are generally not used. However, careful linking of nodes is required to avoid illogical relationships.
有时在箭线图中需要使用虚活动（持续时间为零）来理清依赖关系，但在 IB 和 OCR 所要求的节点图中通常不使用虚活动。不过仍需要仔细连接节点，避免出现不合逻辑的关系。
Example precedence table: Activity A (5 days) has no predecessors; B (4 days) depends on A; C (3 days) depends on A; D (2 days) depends on B and C.
示例先后关系表：活动 A（5 天）无紧前活动；B（4 天）依赖于 A；C（3 天）依赖于 A；D（2 天）依赖于 B 和 C。
3. Drawing an Activity-on-Node Diagram | 绘制活动节点图
Start by drawing a single start node (often a hollow circle or node labelled ‘Start’) if there is more than one initial activity. However, many exam questions begin with a single activity with no predecessors, so you can place it immediately.
首先，如果有多个初始活动，可画一个单独的起始节点（通常是一个空心圆或标记为“开始”的节点）。不过，许多考题从没有紧前活动的单一活动开始，因此可以直接放置。
Each activity is represented by a rectangular node divided into sections. A standard layout includes: the activity label (top-left), duration (top-right), earliest start time EST (bottom-left) and earliest finish time EFT (bottom-right). Later, you will add latest start (LST) and latest finish (LFT) during the backward pass.
每个活动用一个分为若干部分的矩形节点表示。标准布局包括：活动标签（左上角）、持续时间（右上角）、最早开始时间 EST（左下角）和最早完成时间 EFT（右下角）。之后，在反向推算时再添加最晚开始时间 LST 和最晚完成时间 LFT。
Arrows connect activities according to the precedence table. The arrow goes from the predecessor to the successor. Ensure that no activity is left disconnected and that all dependencies are correctly represented. Multiple arrows can converge on a node.
根据先后关系表用箭头连接活动。箭头从前驱活动指向后继活动。确保没有活动未被连接，所有依赖关系均已正确表示。多条箭头可以汇聚于一个节点。
In the example: Start → A (5) ; A → B (4) and A → C (3) ; B → D (2) and C → D (2) ; then D → Finish.
示例中：开始 → A (5)；A → B (4) 且 A → C (3)；B → D (2) 且 C → D (2)；然后 D → 结束。
4. Forward Pass: Earliest Start and Earliest Finish Times | 前向法：最早开始与最早完成时间
The forward pass calculates the earliest possible time each activity can start and finish, assuming the project begins at time zero.
前向法计算每个活动可能的最早开始和最早完成时间，并假设项目从零时刻开始。
For the first activity, EST = 0. Then EFT = EST + duration. For any subsequent activity, EST = maximum EFT of all its immediate predecessors. If there is more than one predecessor, you must take the largest value to ensure all predecessors are complete.
对于第一个活动，EST = 0。然后 EFT = EST + 持续时间。对于任何后续活动，EST = 所有紧前活动 EFT 的最大值。如果有多个紧前活动，必须取最大值，以确保所有前驱活动均已完成。
Apply the rule systematically from left to right across the network. Record the values in the appropriate sections of each node.
从左到右在网络中系统应用此规则，将数值记录在每个节点的相应区域。
In our example: A: EST=0, EFT=5. B: EST=5, EFT=9. C: EST=5, EFT=8. D: predecessors B (9) and C (8), so EST = max(9,8)=9, EFT = 9+2=11. Project duration = 11 days.
在我们的例子中：A: EST=0, EFT=5。B: EST=5, EFT=9。C: EST=5, EFT=8。D: 前驱 B (9) 和 C (8)，因此 EST = max(9,8)=9，EFT = 9+2=11。项目工期 = 11 天。
5. Backward Pass: Latest Start and Latest Finish Times | 后向法：最晚开始与最晚完成时间
The backward pass determines the latest possible times activities can start and finish without delaying the project. You work backwards from the end node, setting the project’s minimum completion time as the LFT for all terminal activities.
后向法确定活动在不延误项目的前提下最晚可以开始和完成的时间。从结束节点开始逆向计算，将所有终止活动的 LFT 设为项目的最短完成时间。
For the last activity, LFT = project duration. Then LST = LFT – duration. For preceding activities, LFT = minimum LST of all immediate successors. This ensures that no successor is forced to start late.
对于最后一个活动，LFT = 项目工期。然后 LST = LFT – 持续时间。对于前驱活动，LFT = 所有紧后活动 LST 的最小值。这确保不会导致任何后继活动被迫推迟开始。
Continue moving right to left through the network, filling in LST and LFT for each node.
继续从右向左遍历网络，填写每个节点的 LST 和 LFT。
In our example: D: LFT=11, LST=9. B: successor D LST=9, so LFT=9, LST=5. C: LFT=9, LST=6. A: successors B (LST=5) and C (LST=6), so LFT = min(5,6)=5, LST=0.
在我们的例子中：D: LFT=11, LST=9。B: 后继 D LST=9，因此 LFT=9，LST=5。C: LFT=9，LST=6。A: 后继 B (LST=5) 和 C (LST=6)，因此 LFT = min(5,6)=5，LST=0。
6. Calculating Float: Total Float and Free Float | 计算浮动时间：总浮动与自由浮动
Total float is the amount of time an activity can be delayed without affecting the overall project duration. It is calculated as: Total Float = LFT – EFT or equivalently LST – EST. Both formulas always yield the same result.
总浮动是指一项活动可以延迟而不会影响整个项目工期的时间量。计算公式为：总浮动 = LFT – EFT，或等效地 LST – EST。两种公式结果始终相同。
Free float is the amount of time an activity can be delayed without affecting the earliest start of any subsequent activity. It is defined as: Free Float = EST(successor) – EFT(activity). For multiple successors, take the smallest EST.
自由浮动是指一项活动可以延迟而不会影响任何后续活动最早开始的时间量。其定义为：自由浮动 = 后继活动的 EST – 本活动的 EFT。若有多个后继，则取最小的 EST。
Activities on the critical path have zero total float. A common exam mistake is to confuse total and free float or to forget that free float cannot exceed total float for the same activity.
关键路径上的活动总浮动为零。考试中常见错误是混淆总浮动与自由浮动，或忘记同一活动的自由浮动不能超过总浮动。
In our example: A: total float = 0; B: total float = 0; C: total float = LFT(9)-EFT(8)=1; free float for C = EST(D) – 8 = 9-8=1; D: total float = 0.
在我们的例子中：A: 总浮动 = 0；B: 总浮动 = 0；C: 总浮动 = LFT(9)-EFT(8)=1；C 的自由浮动 = EST(D) – 8 = 9-8=1；D: 总浮动 = 0。
7. Identifying the Critical Path | 识别关键路径
The critical path consists of all activities that have zero total float. These activities determine the project’s minimum duration. Any delay in a critical activity directly extends the project completion time.
关键路径由所有总浮动为零的活动构成。这些活动决定了项目的最短工期。任何关键活动的延迟都会直接延长项目完成时间。
To identify the critical path, trace a continuous chain from start to finish through activities with total float = 0. In our example, the critical path is A – B – D, with a duration of 5+4+2 = 11 days. There may be more than one critical path; all must be monitored.
要识别关键路径，可以从开始到结束追踪一条由总浮动为零的活动组成的连续链条。在我们的例子中，关键路径为 A – B – D，工期为 5+4+2 = 11 天。可能存在多条关键路径，全部都需要监控。
It is essential to write the critical path clearly on your diagram or answer paper, often listing the activity letters in order. Examiners award marks for correct identification.
必须在图或答卷上清楚地写出关键路径，通常按顺序列出活动字母。考官会对正确识别给分。
8. Gantt Charts (Cascade Charts) | 甘特图（级联图）
A Gantt chart is a horizontal bar chart showing activities against time. For CPA, you can use the earliest start times to draw a basic Gantt chart, and then add floats as broken lines to show flexibility.
甘特图是一种水平条形图，显示活动随时间的变化。对于 CPA，可以使用最早开始时间绘制基本甘特图，然后用虚线添加浮动时间以显示灵活性。
To construct the chart, draw a time axis. For each activity, draw a solid bar from its EST to EFT. Then, for non-critical activities, extend a dotted line from the EFT to the LFT to represent total float. Free float can be shown as the portion of the dotted line before the first successor’s EST.
要构建甘特图，先画一个时间轴。对于每个活动，从 EST 到 EFT 画一个实心条。然后，对于非关键活动，从 EFT 到 LFT 画一条虚线表示总浮动。自由浮动可以显示为第一个后继活动 EST 之前的虚线部分。
In exams, you may be asked to draw a Gantt chart given the ESTs and floats, or to read one and infer activity durations and dependencies. Practice translating between the network and the Gantt chart to avoid confusion.
在考试中，可能要求根据提供的 EST 和浮动时间绘制甘特图，或阅读甘特图推断活动持续时间和依赖关系。练习网络图与甘特图的相互转换，避免混淆。
9. Resource Levelling and Scheduling | 资源均衡与调度
Resource levelling aims to minimise fluctuations in resource usage over the project’s duration by shifting non-critical activities within their float. This does not change the project end date.
资源均衡旨在通过在浮动时间内移动非关键活动，将项目周期内资源使用的波动降至最低。这不会改变项目的结束日期。
You may be given a resource histogram showing the number of workers required if all activities start at their earliest times. The task is to reduce peaks by delaying some activities, using their total float. This often involves moving activities to later times or splitting them (if allowed).
你可能会得到一个资源直方图，显示所有活动都按最早开始时间开始所需的工人数。任务是利用总浮动延迟某些活动，从而降低峰值。这通常涉及将活动向后移动或拆分（如果允许）。
A typical exam question will provide a precedence table and a target resource limit, asking you to devise a schedule that does not exceed that limit while keeping the same critical path duration.
典型的考题会提供一个先后关系表和一个目标资源限制，要求你制定一个不超过该限制且保持相同关键路径工期的调度方案。
Always check that the new start times are within the available float and that no dependencies are violated. Present the final schedule using a Gantt chart or a time-phased activity list.
始终检查新的开始时间是否在可用浮动范围内，并且没有违反依赖关系。用甘特图或按时段分列的活动列表展示最终调度方案。
10. Exam Techniques and Common Pitfalls | 考试技巧与常见错误
Read the precedence table carefully. Look for activities that have multiple dependencies or are themselves prerequisites for several others. Misreading one dependency can ruin the entire network.
仔细阅读先后关系表。注意具有多个依赖项或自身是多个其他活动先决条件的活动。误读一个依赖关系可能毁掉整个网络。
Double-check forward and backward pass calculations. A small arithmetic error in one node cascades through the rest. Always verify that EST + duration = EFT, and LST + duration = LFT.
反复检查前向和后向计算。一个节点的小算术错误会级联影响其余部分。始终验证 EST + 持续时间 = EFT，以及 LST + 持续时间 = LFT。
Do not forget to include the project duration at the end of the forward pass, and use it as the LFT for the final activity in the backward pass.
不要忘记在前向计算结束时记录项目工期，并在后向计算中将其用作最终活动的 LFT。
When calculating free float, remember to take the minimum EST of all immediate successors. If there is a gap, free float is that gap; otherwise, free float is zero. Free float cannot be negative.
计算自由浮动时，记得取所有紧后活动 EST 的最小值。如果存在间隙，自由浮动即为该间隙；否则自由浮动为零。自由浮动不能为负。
Present your answer neatly. Clearly label EST, EFT, LST, LFT inside nodes. Write the critical path explicitly. For Gantt charts, use a ruler (or clear digital lines) and annotate floats.
整洁地呈现答案。在节点内清晰标注 EST、EFT、LST、LFT。明确写出关键路径。绘制甘特图时，使用直尺（或清晰的数字线条）并标注浮动时间。
11. Worked Example | 例题精讲
Consider a small project with the following precedence table:
考虑一个具有以下先后关系表的小型项目：
| Activity | Duration (days) | Predecessors |
| A | 5 | None |
| B | 4 | A |
| C | 3 | A |
| D | 2 | B, C |
Forward pass: A (0,5); B (5,9); C (5,8); D receives from B and C, so EST = max(9,8)=9; EFT = 11. Project duration = 11.
前向计算：A (0,5)；B (5,9)；C (5,8)；D 从 B 和 C 接收，因此 EST = max(9,8)=9；EFT = 11。项目工期 = 11。
Backward pass: D LFT=11, LST=9; B LFT=9, LST=5; C LFT=9, LST=6; A LFT = min(5,6)=5, LST=0.
后向计算：D LFT=11, LST=9；B LFT=9, LST=5；C LFT=9, LST=6；A LFT = min(5,6)=5, LST=0。
Floats: A: total=0; B: total=0; C: total=1, free= EST(D)-EFT(C)=9-8=1; D: total=0. Critical path: A–B–D.
浮动时间：A: 总浮动=0；B: 总浮动=0；C: 总浮动=1，自由浮动=1；D: 总浮动=0。关键路径：A–B–D。
A Gantt chart would show bars for A (0–5), B (5–9), C (5–8) with a dotted extension from 8 to 9 representing its total float, and D (9–11). The critical activities have no dotted extension.
甘特图将显示 A (0–5)、B (5–9)、C (5–8) 的条形，其中 C 有一条从 8 到 9 的虚线代表其总浮动，D (9–11)。关键活动没有虚线延伸。
12. Summary | 总结
Critical path analysis is a systematic method for planning projects, emphasising the identification of tasks that cannot be delayed. Master the forward and backward pass algorithms, float formulas, and the ability to translate between networks, tables, and Gantt charts.
关键路径分析是一种系统化的项目规划方法，重点在于识别不能延迟的任务。掌握前向和后向推算法、浮动时间公式，以及在网络图、表格和甘特图之间转换的能力。
Practice with a variety of precedence tables, including those with multiple starts or complex dependencies. In exams, stay calm, work methodically, and always verify your calculations. With these skills, CPA becomes a reliable source of marks in IB and OCR Mathematics.
多练习各种先后关系表，包括具有多个起点或复杂依赖关系的情况。考试中保持冷静，有条理地解题，并始终验证计算。掌握这些技能，CPA 将成为你在 IB 和 OCR 数学中稳定的得分点。
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