📚 Critical Path Analysis Core Concepts for IB & CIE Mathematics | 关键路径分析：IB与CIE数学核心考点精讲
Critical Path Analysis (CPA) is a project management technique that helps you identify the longest sequence of dependent tasks, ensuring a project is completed in the shortest possible time. For IB and CIE Mathematics students, mastering CPA means understanding activity networks, earliest and latest start times, float, and how to draw Gantt charts. This guide breaks down every concept you need to ace your exam questions.
关键路径分析（CPA）是一种项目管理技术，用于识别依赖任务的最长序列，确保项目在最短时间内完成。对于 IB 和 CIE 数学学生来说，掌握 CPA 意味着理解活动网络、最早和最晚开始时间、时差以及甘特图的绘制。本文将分解你攻克考试题目所需的每一个概念。
1. What is Critical Path Analysis? | 什么是关键路径分析？
Critical Path Analysis models a project as a set of activities with given durations and dependencies. The critical path is the chain of activities that determines the overall project duration. Any delay on a critical activity directly delays the whole project.
关键路径分析将项目建模为一组有给定时长和依赖关系的活动。关键路径是决定整个项目工期的活动链。任何关键活动的延迟都会直接导致整个项目延期。
2. Activity Networks: AOA vs AON | 活动网络：箭线图与节点图
Two types of network diagrams appear in IB and CIE syllabi: Activity on Arrow (AOA) and Activity on Node (AON). In AOA, activities are represented by arrows, and nodes mark the start or finish of activities (events). In AON, each activity is a node, and arrows show precedence relationships.
IB 和 CIE 教学大纲中出现两种网络图：箭线图（AOA）和节点图（AON）。在 AOA 中，活动用箭线表示，节点标记活动的开始或结束（事件）。在 AON 中，每个活动是一个节点，箭线表示先后关系。
- Activity on Arrow (AOA): arrows = activities, nodes = events (e.g., start event 1, end event 2).
- 箭线图（AOA）：箭线 = 活动，节点 = 事件（如起始事件1，结束事件2）。
- Activity on Node (AON): nodes = activities, arrows = dependencies. This is more common in modern syllabi.
- 节点图（AON）：节点 = 活动，箭线 = 依赖关系。在现代考纲中更常见。
3. Precedence Tables and Dependencies | 前导关系表与依赖关系
Exam questions often give a precedence table listing each activity, its duration, and its immediate predecessors. For example, Activity B may depend on Activity A being finished; Activity C may depend on A and B. You must translate this table into a network diagram correctly.
考题通常会给出一个前导关系表，列出每项活动、持续时间及其紧前活动。例如，活动 B 可能依赖于活动 A 的完成；活动 C 可能依赖于 A 和 B。你必须正确地将此表转化为网络图。
Example: Activity A (dur. 3), B (dur. 4, depends on A), C (dur. 2, depends on A, B)
示例：活动 A (工期3), B (工期4, 依赖A), C (工期2, 依赖A、B)
4. Dummy Activities in AOA Networks | 箭线图中的虚工作
Dummy activities have zero duration and are used in AOA networks to maintain logical dependencies without altering the timeline. They are drawn as dashed arrows. Dummies prevent two activities from sharing the same start and end event numbers.
虚工作持续时间为零，在 AOA 网络图中用于保持逻辑依赖关系而不改变时间线。它们用虚线箭头表示。虚工作可避免两项活动共用相同的起始和结束事件编号。
For instance, if activity D depends only on B, but not on C, and both B and C follow A, a dummy is needed to separate the dependencies.
例如，如果活动 D 仅依赖于 B 而不依赖于 C，且 B 和 C 都跟在 A 之后，则需要虚工作来区分依赖关系。
5. Drawing the Network Diagram | 绘制网络图
Start from the source node (start event) and add activities in order of their dependencies. Ensure every activity has a unique identifier, and arcs do not cross unnecessarily. For AON, use labelled circles; for AOA, number the nodes sequentially.
从源节点（开始事件）开始，按照依赖顺序添加活动。确保每项活动有唯一标识符，且箭线不要不必要地交叉。对于 AON，使用带标签的圆圈；对于 AOA，按顺序为节点编号。
A simple AON for the earlier example: node A (3) → node B (4) → node C (2). This is sequential, so the network is a straight chain.
前例的简单 AON：节点 A (3) → 节点 B (4) → 节点 C (2)。这是顺序的，因此网络是一条直线链。
6. Forward Pass – Earliest Times | 正向推进——最早时间
The forward pass calculates the earliest start time (EST) for each activity. At the start node, EST = 0. For subsequent activities, EST = max{EST of all immediate predecessors + duration of that predecessor}.
正向推进计算每项活动的最早开始时间 (EST)。在起始节点，EST = 0。对于后续活动，EST = max{所有紧前活动的 EST + 该紧前活动的持续时间}。
EST(activity) = max { EST(predecessor) + tpredecessor }
EST(活动) = max { EST(前导活动) + t前导活动 }
Using the example: EST(A)=0, EST(B)=0+3=3, EST(C)=3+4=7. The project’s earliest finish time = 7+2=9.
使用示例：EST(A)=0, EST(B)=0+3=3, EST(C)=3+4=7。项目的最早完成时间 = 7+2=9。
7. Backward Pass – Latest Times | 反向推进——最晚时间
The backward pass determines the latest start time (LST) for each activity, starting from the end node with LST equal to the earliest finish time. For preceding activities, LST = min{LST of all immediate successors – duration of the activity itself}.
反向推进确定每项活动的最晚开始时间 (LST)，从结束节点开始，其 LST 等于最早完成时间。对于前驱活动，LST = min{所有紧后活动的 LST – 该活动本身的持续时间}。
LST(activity) = min { LST(successor) – tactivity }
LST(活动) = min { LST(后继活动) – t活动 }
For the chain example: LST(C)=9-2=7, LST(B)=7-4=3, LST(A)=3-3=0. All zero float – the whole chain is critical.
对于链式示例：LST(C)=9-2=7, LST(B)=7-4=3, LST(A)=3-3=0。所有活动时差为零，整条链都是关键路径。
8. Total Float and Critical Activities | 总时差与关键活动
Total float is the amount of time an activity can be delayed without delaying the whole project. It is calculated as Total Float = LST – EST (or LFT – EFT). Activities with zero total float are critical; they must start and finish exactly on time.
总时差是一项活动可以延迟而不影响整个项目完工的时间量。计算公式为 总时差 = LST – EST（或 LFT – EFT）。总时差为零的活动是关键活动；它们必须准时开始和完成。
You may also encounter free float, but the total float is the main measure for determining the critical path.
你也可能会遇到自由时差，但总时差是确定关键路径的主要度量。
9. Finding the Critical Path | 确定关键路径
The critical path is the sequence of activities with zero total float. In large networks, you identify it by tracing back from the end node, following the activities where EST = LST for each node (AOA) or for each activity (AON).
关键路径是总时差为零的活动序列。在大型网络中，你通过从结束节点回溯，追踪每个节点（AOA）或每项活动（AON）满足 EST = LST 的路径来确定它。
In a more complex network with multiple parallel paths, the critical path is the longest path from start to finish, and its length gives the minimum project duration.
在具有多个并行路径的复杂网络中，关键路径是从起点到终点的最长路径，其长度即项目的最短工期。
10. Gantt (Cascade) Charts | 甘特图（瀑布图）
A Gantt chart is a horizontal bar chart that represents each activity’s start, duration, and float. Activities are listed vertically, and time runs horizontally. Critical activities are often shaded or marked differently, and float is shown as a lighter extension past the earliest finish.
甘特图是一种水平条形图，表示每项活动的开始、持续时间和时差。活动垂直排列，时间水平延伸。关键活动通常用阴影或不同标记，时差显示为超过最早完成时间的较浅延长线。
In many IB/CIE questions, you are asked to draw a cascade chart using the earliest start times, clearly indicating the critical path and floats.
在许多 IB/CIE 考题中，要求你使用最早开始时间绘制瀑布图，并清晰标示关键路径和时差。
11. Resource Smoothing Basics | 资源平滑基础
Resource smoothing attempts to minimise fluctuations in resource usage over the project timeline without extending the overall duration. You use float to shift non-critical activities so that resource demand stays as even as possible.
资源平滑试图在不延长总工期的前提下，最小化项目时间线上资源使用的波动。你利用时差来移动非关键活动，使资源需求尽可能保持平稳。
This is a common extension in higher-band exam questions; you must interpret a resource histogram and adjust activity start times accordingly.
这是高分考题中常见的拓展；你必须会解读资源直方图并相应调整活动的开始时间。
12. Exam Tips & Common Pitfalls | 考试技巧与常见错误
Always draw the network from the precedence table carefully, checking for missed dummy activities in AOA. When computing forward/backward passes, double-check that you take the maximum for EST and the minimum for LST. Remember that float cannot be negative; if it appears negative, you’ve made a calculation error.
务必根据前导关系表仔细绘制网络图，检查箭线图中是否有遗漏的虚工作。在计算正推/反推时，仔细核对 EST 取最大值、LST 取最小值。记住时差不能为负；如果出现负值，说明计算有误。
For Gantt charts, use a ruler‑like precision: mark the time axis evenly, label critical activities clearly, and show float as a distinct shading. Practise with past papers to get comfortable with the layout required by your exam board.
对于甘特图，要做到如尺子般精确：均匀标记时间轴，清晰标明关键活动，并用独特的阴影表示时差。通过往年真题练习，熟悉考试局要求的排版格式。
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