📚 GCSE WJEC Maths: Critical Path Analysis – Exam Focus | GCSE WJEC 数学：关键路径分析考点精讲
Critical Path Analysis (CPA) enables project managers to schedule activities logically, identify the minimum project completion time, and discover which tasks cannot be delayed without affecting the deadline. In the WJEC GCSE Mathematics specification, CPA questions appear regularly in the ‘Applications of Mathematics’ or ‘Decision Mathematics’ component. You will be expected to construct activity-on-node networks, perform forward and backward passes, calculate total float, and interpret the critical path and cascade (Gantt) charts.
关键路径分析（CPA）让项目管理者能够逻辑地安排各项活动，找出项目最短完成时间，并确定哪些任务不能延误、否则会影响最终期限。在 WJEC GCSE 数学考纲中，CPA 题目经常出现在“数学应用”或“决策数学”部分。你需要掌握构建节点式活动网络、进行前向与后向遍历、计算总浮动时间，并解释关键路径以及瀑布图（甘特图）。
1. What is Critical Path Analysis? | 什么是关键路径分析？
Critical Path Analysis is a mathematical modelling technique used to plan complex projects. It breaks a project down into smaller activities, determines their logical order, and calculates the earliest and latest time each activity can start. By doing this, the project’s critical path — the sequence of activities that directly dictates the overall duration — is identified.
关键路径分析是一种用于规划复杂项目的数学建模技术。它将一个项目分解成多个较小的活动，确定其逻辑顺序，并计算每项活动可以开始的最早与最晚时间。由此，便可识别项目的关键路径——即直接决定总工期的活动序列。
In WJEC exams, you will often be given a table of activities, their durations, and their immediate predecessors. Your task is to turn this information into a network diagram and then analyse it.
在 WJEC 考试中，你通常会拿到一张包含活动、持续时间和紧前活动的表格。你需要将这些信息转化为网络图并进行分析。
2. Activity-on-Node Networks: Nodes and Dependencies | 节点式活动网络：节点与依赖关系
WJEC uses the activity-on-node representation. Each activity is represented by a node (a rectangle or circle), and arrows show dependencies. The node typically contains the activity name or code, and sometimes its duration. An arrow from Activity P to Activity Q indicates that Q can only start after P is finished.
WJEC 采用节点式活动网络表示法。每项活动用一个节点（矩形或圆形）表示，箭头表示依赖关系。节点内通常标有活动名称或代号，有时也标持续时间。从活动 P 指向活动 Q 的箭头表示 Q 只能在 P 完成后才能开始。
A project always has a single start node and a single end node. If multiple activities have no predecessors, they all begin from the start. If multiple activities have no successors, they all feed into the end node. This prevents ‘dangling’ activities.
一个项目只有一个开始节点和一个结束节点。若多项活动没有紧前活动，它们都从开始节点出发。若多项活动没有后续活动，它们都汇聚到结束节点。这样可以避免出现“悬空”活动。
3. Constructing a Precedence Table | 构建前导表
A precedence table lists each activity, its duration, and the activities that must be completed immediately before it. This table is the starting point for drawing the network. Look carefully at the ‘predecessors’ column to understand the logic.
前导表列出了每项活动、其持续时间以及必须在它之前立即完成的活动。这个表格是绘制网络图的出发点。要仔细分析“紧前活动”一栏，以理解其中的逻辑。
For example, if Activity D requires both B and C to be finished, the predecessors column for D will show B and C. In the network, arrows must come from both B and C to D.
例如，如果活动 D 需要 B 和 C 都完成才能开始，那么 D 的紧前活动栏中会显示 B 和 C。在网络图中，必须有箭头分别从 B 和 C 指向 D。
4. Drawing the Activity-on-Node Network | 绘制节点式活动网络图
Begin by placing a ‘Start’ node on the left. Draw nodes for activities that have no predecessors, connecting them from Start. Work through the table systematically, adding nodes and arrows as you go. Always check that every dependency is represented. If an activity depends on two or more predecessors, all those preceding activities must converge with arrows into the new node.
首先在左侧放置一个“开始”节点。为没有紧前活动的活动绘制节点，并从 Start 节点用箭头连接它们。系统地根据表格逐步推进，依次添加节点和箭头。务必确认每一个依赖关系都已被表示出来。如果一项活动依赖于两个或更多紧前活动，那么所有这些前置活动都要用箭头汇聚到该新节点。
To avoid crossing arrows, sketch the diagram lightly in pencil, arranging nodes so that dependencies flow generally from left to right. The final layout should be neat and easy to read, as clarity is awarded marks in exams.
为了避免箭头交叉，可以先用铅笔轻轻画出草图，将节点排列得使依赖关系大体从左向右流动。最终的布局应当整洁、易读，因为在考试中清晰度是给分的依据之一。
5. Forward Pass: Calculating Earliest Start Times (EST) | 前向遍历：计算最早开始时间
The forward pass determines the earliest possible time each activity can begin, assuming all preceding activities start as early as possible. Set the EST of the Start node to 0. For any other node, its EST equals the maximum of (EST of predecessor + predecessor’s duration). This ensures the activity cannot start until all immediate predecessors are finished.
前向遍历用来确定每项活动可能开始的最早时间，假设所有前置活动都尽早开始。将 Start 节点的 EST 设为 0。对于其他任何节点，其 EST 等于（前置节点的 EST + 该前置节点的持续时间）中的最大值。这确保了活动只有在所有直接紧前活动都完成后才能开始。
EST(node) = max{EST(predecessor) + duration(predecessor)} for all immediate predecessors
EST(节点) = 所有直接紧前活动的 max{EST(紧前活动) + 持续时长}
Record the calculated EST values on the network diagram, usually in the top-left corner of each node. The EST of the End node gives the minimum project duration.
将计算得到的 EST 值标注在网络图上，通常写在每个节点内部的左上角。End 节点的 EST 就是项目的最短完成时间。
6. Backward Pass: Calculating Latest Start Times (LST) | 后向遍历：计算最晚开始时间
The backward pass calculates the latest time each activity can start without delaying the entire project. Begin at the End node, setting its LST equal to its EST (the project duration). Work backwards through the network. For any node, its LST equals the minimum of (LST of successor − duration of the current activity).
后向遍历计算的是在不延误整个项目的前提下，每项活动可以开始的最晚时间。从 End 节点开始，将其 LST 设为与 EST 相等（即项目总工期）。然后逆向推进网络。对于任何节点，其 LST 等于所有直接后续节点的（后续节点 LST − 当前活动时长）中的最小值。
LST(node) = min{LST(successor) − duration(node)} for all immediate successors
LST(节点) = 所有直接后续活动的 min{LST(后续节点) − 本活动时长}
Write the LST value in the top-right corner of each node. If a node has no successors (the End node), its LST is simply the project duration.
将 LST 值写在每个节点内部的右上角。如果某节点没有后续活动（即 End 节点），其 LST 就等于项目总工期。
7. Total Float and Its Significance | 总浮动时间及其意义
Total float (or total slack) measures how much an activity can be delayed without affecting the overall project deadline. It is defined as the difference between the latest start time and the earliest start time: Float = LST − EST. An activity with zero float is critical; any delay in it will directly push back the project completion date.
总浮动时间（总时差）衡量一项活动可以延误多久而不影响项目最终期限。它定义为最晚开始时间与最早开始时间之差：浮动时间 = LST − EST。浮动时间为零的活动就是关键活动；它的任何延误都会直接导致项目完工日期推迟。
Activities with a positive float can be delayed by up to that amount without delaying the project. This information is essential for resource scheduling, as managers can shift non-critical tasks to level workloads.
具有正浮动时间的活动可以在不造成项目延误的前提下推迟相应的时间。这一信息对资源调度至关重要，因为管理者可以通过移动非关键任务来均衡工作负载。
8. Identifying the Critical Path | 识别关键路径
The critical path is the continuous chain of activities from Start to End where every activity has total float equal to zero. In other words, for all nodes on this path, LST equals EST. There may be more than one critical path if multiple routes share the same zero-float characteristic. The critical path determines the overall project duration.
关键路径是从 Start 到 End 的一条连续活动链，链上每项活动的总浮动时间均为零。换言之，这条路径上的所有节点都满足 LST = EST。如果多条路径都具备零浮动特征，就可能存在多条关键路径。关键路径决定了项目总工期。
In a WJEC exam, you might be asked to ‘state the critical path’ using activity letters, for example A → C → E → F. Always confirm by checking that each link is dependency-compatible and that no alternative zero-float route exists.
在 WJEC 考试中，可能会要求你“写出关键路径”，通常用活动字母表示，如 A → C → E → F。务必通过检查每一步依赖关系和确认没有其他零浮动路线来验证你的结果。
9. Interpreting the Critical Path and Project Duration | 解释关键路径与项目总工期
Once the analysis is complete, you need to interpret the results in context. State clearly the minimum time needed to finish the project, which is the EST (and LST) of the End node. Name the critical activities and explain that any delay in these will increase the project length.
分析完成后，你需要结合具体情景进行解释。要明确说出项目完成所需的最短时间，即 End 节点的 EST（也即 LST）。列出关键活动，并解释这些活动中的任何延误都会延长项目工期。
For example: ‘The project will take 23 days. The critical path is B → D → G → H. If activity D is delayed by 2 days, the entire project will be delayed by 2 days because D has zero float.’
例如：“项目需要 23 天完成。关键路径是 B → D → G → H。如果活动 D 延误 2 天，整个项目将延误 2 天，因为 D 的浮动时间为零。”
10. Gantt Charts (Cascade Charts) from Network Analysis | 由网络分析绘制甘特图（瀑布图）
A Gantt chart (referred to as a cascade chart by some WJEC materials) provides a visual timeline of activities. Each activity is represented by a horizontal bar placed according to its EST, and its length equals its duration. The float can be shown by an extended light bar or a dotted extension from the LST position.
甘特图（某些 WJEC 教材中称为瀑布图）提供了活动的可视化时间线。每个活动用一个水平长条表示，其位置根据 EST 确定，长度等于活动的时间长。浮动时间可以用从 LST 处延伸的浅色长条或虚线延长部分来表示。
To draw a cascade chart, first list activities in order of their EST. Draw a time scale on the horizontal axis. For each activity, draw a solid bar starting at EST. Then, if the activity has float, extend the bar with a dashed or empty box to the LST + duration point. Critical activities have no such extension.
要绘制瀑布图，首先按 EST 顺序列出活动。在横轴上画出时间刻度。对每一项活动，从 EST 处画一个实心长条。然后，如果该活动有浮动时间，就用虚线或空白矩形条将其延伸至 LST + 持续时间处。关键活动没有这种延伸部分。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One frequent mistake is forgetting to merge multiple predecessors when drawing the network. If an activity depends on, say, B and C, you must draw arrows from both B and C into it. Simply linking them in series will alter the logic and lose marks.
一个常见错误是在绘制网络图时忘记合并多个紧前活动。如果一项活动依赖于 B 和 C，就必须把 B 和 C 的箭头都汇聚到该项活动。仅仅将它们串联起来会改变逻辑并导致失分。
Another pitfall is miscalculating LST during the backward pass. Always use the minimum of (successor LST − current duration) and check that you are subtracting the correct duration. Do not confuse EST and LST when calculating float; Float = LST − EST, never the reverse.
另一个易错点是在后向遍历中算错 LST。始终使用（后续活动 LST − 当前活动时长）的最小值，并确认减去了正确的时长。计算浮动时间时不要混淆 EST 和 LST；浮动时间 = LST − EST，绝不能反过来。
When drawing cascade charts, label the axes and clearly distinguish between solid and dotted parts. Numerous marks are allocated to interpretation, so practise writing clear answers that explain the impact of a delay on a critical activity versus a non-critical one.
绘制瀑布图时，应标注坐标轴，并清晰区分实线和虚线部分。很多分数都分配给解释能力，因此要练习写出清晰的回答，说明延误对关键活动与非关键活动分别有何影响。
12. Worked Example: Putting It All Together | 典型例题解析：综合运用
Consider the following project: Activity A takes 3 days, no predecessor. B takes 5 days, after A. C takes 2 days, after A. D takes 6 days, after B and C. E takes 4 days, after C. F takes 3 days, after D and E.
考虑以下项目：活动 A 需要 3 天，无紧前活动；B 需要 5 天，须在 A 之后；C 需要 2 天，须在 A 之后；D 需要 6 天，须在 B 和 C 之后；E 需要 4 天，须在 C 之后；F 需要 3 天，须在 D 和 E 之后。
Forward pass: EST(A)=0. EST(B)=0+3=3. EST(C)=0+3=3. EST(D)=max(3+5, 3+2)=max(8,5)=8. EST(E)=3+2=5. EST(F)=max(8+6,5+4)=max(14,9)=14. Project duration = 14+3=17 days.
前向遍历：EST(A)=0。EST(B)=0+3=3。EST(C)=0+3=3。EST(D)=max(3+5, 3+2)=max(8,5)=8。EST(E)=3+2=5。EST(F)=max(8+6,5+4)=max(14,9)=14。项目总工期 = 14+3=17 天。
Backward pass: LST(F)=17, so LST(F)=17−3=14. LST(D)=14−6=8. LST(E)=14−4=10. LST(B)=8−5=3. LST(C) must be min(8−2=6 from D, 10−2=8 from E)=6. LST(A)=min(3−3=0 from B, 6−3=3 from C)=0.
后向遍历：LST(F)=17，故 LST(F) 本身 = 14。LST(D)=14−6=8。LST(E)=14−4=10。LST(B)=8−5=3。LST(C) 必须取 min(8−2=6 来自 D, 10−2=8 来自 E)=6。LST(A)=min(3−3=0 来自 B, 6−3=3 来自 C)=0。
Float values: A: 0−0=0; B: 3−3=0; C: 6−3=3; D: 8−8=0; E: 10−5=5; F: 14−14=0. Critical path: A → B → D → F. Activities C and E have float and are not critical.
浮动时间值：A:0−0=0；B:3−3=0；C:6−3=3；D:8−8=0；E:10−5=5；F:14−14=0。关键路径：A → B → D → F。活动 C 和 E 有浮动时间，不是关键活动。
| Activity | Duration | EST | LST | Total Float |
|---|---|---|---|---|
| A | 3 | 0 | 0 | 0 |
| B | 5 | 3 | 3 | 0 |
| C | 2 | 3 | 6 | 3 |
| D | 6 | 8 | 8 | 0 |
| E | 4 | 5 | 10 | 5 |
| F | 3 | 14 | 14 | 0 |
This step-by-step approach mirrors what is expected in a WJEC examination. Practise reproducing similar tables and clearly stating the critical path and project duration.
这种循序渐进的方法正是 WJEC 考试中所期望的。多练习填写类似表格，并清晰地给出关键路径和项目总工期。
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