📚 A-Level Edexcel Maths: Critical Path Analysis Revision | A-Level Edexcel 数学：关键路径分析 考点精讲
Critical Path Analysis (CPA) is a core topic in the Edexcel A-Level Decision Mathematics module. It provides a systematic method for planning, scheduling and managing complex projects made up of interdependent tasks. Mastering CPA allows you to determine the minimum project duration, identify the activities that cannot be delayed without affecting the overall timeline, and efficiently allocate limited resources efficiently.
关键路径分析 (CPA) 是爱德思 A-Level 决策数学模块中的核心课题。它为规划、调度和管理由相互依赖任务构成的复杂项目提供了一套系统方法。掌握关键路径分析，可以帮助你确定最短项目工期，识别那些一旦延迟就会影响整体进度的活动，并高效分配有限资源。
This revision guide covers the full Edexcel specification content: activity-on-node and activity-on-arc networks, precedence tables, forward and backward passes, float calculations, Gantt (cascade) charts, resource levelling and smoothing, and crashing. Every section pairs the English explanation with the Chinese equivalent so that you can revise the terminology and the technique at the same time.
本篇复习指南覆盖爱德思考纲的全部内容：活动节点图与活动弧图、前导表、前向遍历与后向遍历、浮动时间计算、甘特图（级联图）、资源均衡与资源平滑，以及赶工。每一节都将英文讲解与中文对照配对呈现，方便你同时复习术语与技术细节。
1. What is Critical Path Analysis? | 什么是关键路径分析？
Critical Path Analysis is a project management technique that models a project as a network of activities. Each activity has a duration and a set of precedence constraints - it cannot start until certain other activities are complete. The purpose of CPA is to find the longest path through the network, called the critical path, which determines the shortest possible completion time for the project.
关键路径分析是一种项目管理技术，它将项目建模为由活动构成的网络。每个活动都有持续时间，并带有一组前置约束——即必须等待某些其他活动完成后才能开始。CPA 的目的是找出网络中的最长路径，也就是关键路径，它决定了项目的最短可能完成时间。
Activities on the critical path have zero float (slack). This means any delay to a critical activity will directly delay the whole project by exactly the same amount. Activities that are not critical have spare time, so they can be moved within their float to smooth resource usage without extending the project.
关键路径上的活动浮动时间（松弛时间）为零。这意味着关键活动一旦延误，整个项目就会以相同的天数被推迟。非关键活动则拥有富余时间，因此可以在其浮动时间范围内移动，用来平滑资源使用，而不会延长项目总工期。
In the Edexcel Decision Mathematics specification you will be expected to construct networks from precedence tables, perform forward and backward passes, calculate earliest and latest times, identify float, draw Gantt charts (cascade diagrams), and carry out resource levelling and resource smoothing.
在爱德思决策数学考纲中，你需要能够根据前导表构建网络图、执行前向遍历和后向遍历、计算最早与最晚时间、识别浮动时间、绘制甘特图（级联图），并进行资源均衡与资源平滑。
2. Precedence Tables and Activity-on-Node Networks | 前导表与活动节点图
The starting point of almost every CPA question is a precedence table. It lists each activity, its duration, and the activities that must be finished before it can begin. In an Activity-on-Node (AoN) network, each activity is drawn as a node (usually a box or circle), and arrows show the direction of dependency.
几乎每一道关键路径分析题目的起点都是一张前导表。表中列出每项活动、它的持续时间，以及必须在它开始前完成的活动。在活动节点图 (AoN) 中，每项活动画成一个节点（通常是方框或圆圈），箭头表示依赖关系的方向。
Two special nodes are usually added: a start node with duration 0, connected to every activity with no predecessors, and an end node with duration 0, receiving arrows from every activity with no successors. This makes the forward and backward passes much easier to apply systematically.
通常还会加入两个特殊节点：一个工期为 0 的开始节点，连接到所有没有前置活动的活动；以及一个工期为 0 的结束节点，接收所有没有后续活动的活动指向它的箭头。这样可以让前向遍历和后向遍历更容易系统地执行。
For each node you should record the activity letter, the duration, the earliest start time (EST), the earliest finish time (EFT), the latest start time (LST) and the latest finish time (LFT). A clear layout prevents arithmetic errors later in the question.
对每个节点，你都应记录：活动代号、持续时间、最早开始时间 (EST)、最早完成时间 (EFT)、最晚开始时间 (LST) 和最晚完成时间 (LFT)。清晰的排版可以避免后续计算中的算术错误。
3. Activity-on-Arc Networks and Dummy Activities | 活动弧图与虚活动
In an Activity-on-Arc (AoA) network, activities are represented by arcs (arrows) and events are represented by nodes (circles). The node label represents the moment at which all incoming activities finish and all outgoing activities can begin. Because events are numbered, a network must never contain two activities that share the same start event and the same end event.
在活动弧图 (AoA) 中，活动用弧（箭头）表示，事件用节点（圆圈）表示。节点编号代表某一时刻：所有流入的活动在此完成，所有流出的活动可以在此开始。由于事件需要编号，网络中绝不能存在两项共享相同起始事件和相同终止事件的活动。
A dummy activity is a dashed arrow with zero duration and zero resource requirement. It is used for two reasons: to maintain unique activity identification, and to express a logical dependency that the network would otherwise not capture.
虚活动是一条虚线箭头，持续时间为零，也不消耗资源。它有两个用途：一是保持活动标识的唯一性，二是表达一种网络原本无法体现的逻辑依赖关系。
Dummies carry no time, so they do not affect the project duration. However they do affect earliest and latest times because they transfer precedence information between events. Forgetting a dummy is one of the most common causes of lost marks in AoA questions.
虚活动不占用时间，因此不会影响项目总工期。但它们会在事件之间传递前置关系，从而影响最早时间和最晚时间。遗漏虚活动是活动弧图题目中最常见的失分原因之一。
4. The Forward Pass: Earliest Times | 前向遍历：最早时间
The forward pass calculates the earliest possible time at which each activity can start and finish, working from the start of the network towards the end. The earliest start time of any activity with no predecessors is 0.
前向遍历用于计算每项活动可能开始和完成的最早时间，遍历方向是从网络起点走向终点。任何没有前置活动的活动，其最早开始时间均为 0。
EST(activity) = max{ EFT(all immediate predecessors) }
EFT(activity) = EST(activity) + duration(activity)
Where an activity has more than one predecessor, you must take the maximum of their earliest finish times, because the activity cannot begin until all of them are complete. The project duration is the maximum EFT in the whole network, which is simply the EFT of the end node.
当一项活动有多个前置活动时，必须取它们最早完成时间的最大值，因为该活动必须等所有前置活动都完成后才能开始。项目总工期即整个网络中最大的 EFT，也就是结束节点的 EFT。
Always show your working on the diagram itself. Examiners award method marks for correctly labelled nodes even when a later arithmetic slip changes the final answer.
务必在图上直接写出计算过程。即使之后出现算术失误导致最终答案改变，考官仍会为标注正确的节点给出方法分。
5. The Backward Pass: Latest Times | 后向遍历：最晚时间
The backward pass calculates the latest times at which activities can start and finish without delaying the project. It works from the end of the network back towards the start. The latest finish time of the final activity is set equal to the project duration.
后向遍历用于计算在不延误项目的前提下，各项活动最晚可以开始和完成的时间。遍历方向是从网络终点回到起点。最后一项活动的最晚完成时间等于项目总工期。
LFT(activity) = min{ LST(all immediate successors) }
LST(activity) = LFT(activity) − duration(activity)
Where an activity has several successors, take the minimum of their latest start times. This ensures that no successor is forced to start later than it can afford to.
当一项活动有多个后续活动时，取其最晚开始时间的最小值。这样才能保证没有任何后续活动被迫晚于它所能承受的时间开始。
Compare the forward and backward results node by node. If EST equals LST for an activity, that activity lies on the critical path. If the backward pass produces a number smaller than the forward pass value at any node, you have made an arithmetic error and should recheck the network.
逐节点比较前向与后向结果。如果某活动的 EST 与 LST 相等，该活动就位于关键路径上。如果后向遍历在任何节点上得到比前向遍历更小的数值，说明出现了算术错误，应重新检查网络。
6. Float and the Critical Path | 浮动时间与关键路径
Float (also called slack) is the amount of time an activity can be delayed without delaying the project. Total float is the standard measure required by Edexcel and can be computed in two equivalent ways.
浮动时间（也称松弛时间）是指一项活动在不延误项目的前提下可以推迟的时间量。总浮动时间是爱德思要求的标准指标，有两种等价的计算方式。
Total float = LFT − EFT = LST − EST
The critical path is the sequence of activities with zero total float, running from the start node to the end node. It is the longest path through the network, and its length equals the minimum project duration. A network may contain more than one critical path if two routes share the same total length.
关键路径就是总浮动时间为零的活动序列，从开始节点一直连到结束节点。它是网络中耗时最长的路径，其长度等于项目的最短工期。如果两条路径总长度相同，网络中可能存在不止一条关键路径。
in{ EST of successors } − EFT of the activity. Free float is always less than or equal to total float. Interfering float is the difference between total float and free float.
自由浮动时间是指在不影响任何后续活动最早开始时间的前提下可推迟的时间量：自由浮动 = 后续活动 EST 的最小值 − 本活动的 EFT。自由浮动总是小于或等于总浮动。干扰浮动则等于总浮动减去自由浮动。
7. Gantt Charts and Cascade Diagrams | 甘特图与级联图
A Gantt chart (often called a cascade diagram in Edexcel Decision Maths) shows activities as horizontal bars plotted against a time axis. Critical activities are normally drawn as solid bars, while non-critical activities are drawn as hollow or dashed bars extending to the limit of their total float.
甘特图（在爱德思决策数学中常称为级联图）把每项活动画成沿时间轴的横向条形。关键活动通常画成实心条，非关键活动则画成空心条或虚线延伸条，一直延伸到其总浮动时间的上限。
To draw a correct Gantt chart: first complete the forward and backward passes, then plot each activity starting at its earliest start time. Shade the bar up to its earliest finish time, then indicate the float with a lighter extension up to its latest finish time.
绘制正确甘特图的步骤是：先完成前向与后向遍历，然后从每项活动的最早开始时间起画出条形。将条形涂色至最早完成时间，再用较浅的延伸段表示浮动时间，直到最晚完成时间。
Gantt charts are extremely useful for reading off the number of activities running on any given day, which feeds directly into resource histogram questions. They also make it easy to check whether a proposed re-scheduling is feasible.
甘特图非常适合用来读出某一天正在进行多少项活动，这直接服务于资源柱状图的题目。它们也能让你轻松检验某个重新排期的方案是否可行。
8. Resource Histograms, Levelling and Smoothing | 资源柱状图、资源均衡与资源平滑
A resource histogram shows the total resource requirement on each day of the project, obtained by adding together the resource needs of every activity in progress. Steep peaks in the histogram indicate inefficient scheduling, since resources must be hired or released rapidly.
资源柱状图显示项目每一天的总资源需求量，方法是把当天正在进行的所有活动的资源需求相加。柱状图中陡峭的峰值意味着调度效率低下，因为资源必须被快速雇用或释放。
Resource levelling means delaying non-critical activities within their float so that the peak resource demand is reduced, even if this prolongs the project. Resource smoothing means re-scheduling non-critical activities within their float to make the histogram as even as possible while keeping the project duration unchanged.
资源均衡是指在浮动时间范围内推迟非关键活动，以降低资源的峰值需求，即使这会延长项目工期。资源平滑则是在浮动时间范围内重新安排非关键活动，使柱状图尽可能平稳，同时保持项目总工期不变。
In practice, resource smoothing never changes the critical path or the project duration. Resource levelling may extend the duration, because the available resource is genuinely insufficient to complete the work within the original timescale.
在实际操作中，资源平滑绝不会改变关键路径或项目工期。资源均衡则可能延长工期，因为可用资源确实不足以在原定时间内完成全部工作。
9. Crashing and the Time-Cost Trade-off | 赶工与时间-成本权衡
Crashing means reducing the duration of an activity by allocating extra resources, at an additional cost. The cost slope of an activity measures how much it costs to shorten that activity by one unit of time.
赶工是指通过投入额外资源来缩短某项活动的持续时间，并因此产生额外成本。活动的成本斜率衡量的是将该活动缩短一个时间单位所需的费用。
Cost slope = (Crash cost − Normal cost) ÷ (Normal time − Crash time)
To shorten a project at minimum cost, always crash an activity on the critical path with the lowest cost slope. After each crash you must recalculate the network, because the critical path may change. Only crashing critical activities can reduce the overall project duration.
要以最低成本缩短项目，应始终选择关键路径上成本斜率最小的活动进行赶工。每次赶工后都必须重新计算网络，因为关键路径可能发生改变。只有对关键活动赶工，才能缩短项目总工期。
There is usually a limit to how far an activity can be crashed, known as the crash time. Crashing beyond that point is physically impossible, so the cost slope applies only within the normal-to-crash range.
某项活动通常存在可压缩的极限，称为极限时间（赶工时间）。超过该点进一步压缩是不可能的，因此成本斜率只在正常时间到赶工时间的区间内适用。
10. Worked Example: Full Network Analysis | 典型例题：完整网络分析
Consider a project with the following activities and precedence relationships. Activity A (duration 3) has no predecessors. B (4) follows A. C (2) follows A. D (5) follows B. E (3) follows B and C. F (2) follows D and E.
考虑一个具有下列活动及前置关系的项目。活动 A（工期 3）没有前置活动。B（4）在 A 之后。C（2）在 A 之后。D（5）在 B 之后。E（3）在 B 和 C 之后。F（2）在 D 和 E 之后。
| Activity | Duration | EST | EFT | LST | LFT | Total float |
| A | 3 | 0 | 3 | 0 | 3 | 0 |
| B | 4 | 3 | 7 | 3 | 7 | 0 |
| C | 2 | 3 | 5 | 7 | 9 | 4 |
| D | 5 | 7 | 12 | 7 | 12 | 0 |
| E | 3 | 7 | 10 | 9 | 12 | 2 |
| F | 2 | 12 | 14 | 12 | 14 | 0 |
es has zero total float. Activity C has the largest float, 4 days, so it is the first candidate for resource smoothing.
项目总工期为 14 天，由 F 的 EFT 给出。关键路径是 A → B → D → F，因为这些活动的总浮动均为零。活动 C 的浮动最大，为 4 天，因此它是资源平滑的首选对象。
4 − 2 = 2 days. Activity E has free float of 12 − 10 = 2 days and zero interfering float.
注意 C 在第 5 天完成，但 E 必须等到第 7 天（B 完成时）才能开始。因此 C 的自由浮动为 7 − 5 = 2 天，干扰浮动为 4 − 2 = 2 天。活动 E 的自由浮动为 12 − 10 = 2 天，干扰浮动为零。
11. Common Mistakes and Exam Technique | 常见错误与应试技巧
The most frequent errors in Edexcel CPA questions are: forgetting to take the maximum in the forward pass, forgetting to take the minimum in the backward pass, mislabelling the end node, and omitting a dummy activity in an AoA network.
爱德思 CPA 题目中最常见的错误包括：前向遍历时忘记取最大值、后向遍历时忘记取最小值、结束节点标注错误，以及在活动弧图中遗漏虚活动。
Other common slips include confusing total float with free float, assuming there is always only one critical path, and crashing a non-critical activity in a time-cost question. Always re-read the question to check whether it asks for levelling or smoothing, and whether the project duration must be preserved.
其他常见失误还包括：混淆总浮动与自由浮动、误以为关键路径永远只有一条，以及在时间-成本题中对非关键活动进行赶工。务必重读题目，确认它要求的是资源均衡还是资源平滑，以及项目工期是否必须保持不变。
Good exam technique: draw the network large and neat, label every node with all six values, state the critical path as a full sequence of letters, and always give units (days, hours, £) with numerical answers. Show the formula you are using before substituting numbers.
良好的应试技巧是：把网络图画得大而整洁，为每个节点标注全部六个数值，把关键路径写成完整的字母序列，并在数值答案后注明单位（天、小时、英镑）。在代入数字之前，先写出所用的公式。
12. Quick Reference Summary | 核心公式速查总结
Use this summary table as a final check before your exam. Every formula below appears regularly in Edexcel Decision Mathematics papers and should be memorised.
把这张汇总表作为考前最后的检查清单。下列每一个公式都经常出现在爱德思决策数学试卷中，应当熟记。
| Quantity | Formula |
| Earliest finish time | EFT
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