IGCSE WJEC Mathematics: Critical Path Analysis - Essential Revision | IGCSE WJEC 数学：关键路径分析考点精讲

📚 IGCSE WJEC Mathematics: Critical Path Analysis – Essential Revision | IGCSE WJEC 数学：关键路径分析考点精讲

Critical Path Analysis (CPA) is a powerful Decision Mathematics topic within the IGCSE WJEC Mathematics A specification. It gives you the tools to plan complex projects, identify the tasks that directly influence the overall duration, and schedule resources efficiently. This exam-focused revision guide breaks down every concept you need to master, from constructing activity-on-arc networks and performing forward/backward passes to calculating floats and drawing Gantt charts. Understanding CPA thoroughly will not only boost your exam confidence but also equip you with a practical skill widely used in engineering, construction, and business management.

关键路径分析（CPA）是 IGCSE WJEC 数学 A 课程决策数学部分的重要主题。它帮助你规划复杂项目、识别直接影响总工期的任务，并高效调度资源。本考点精讲逐一剖析你需要掌握的每个概念，从构建箭线图网络、进行正向与反向计算，到求算浮动时间和绘制甘特图。透彻掌握 CPA 不仅能增强考试信心，还能让你掌握一项在工程、建筑和商业管理中广泛应用的实际技能。

1. Understanding Critical Path Analysis | 理解关键路径分析

Critical Path Analysis is a logical method for determining the minimum completion time of a project. It models a project as a network of activities, where some activities can be carried out simultaneously while others must wait for predecessors to finish. The core aim is to pinpoint the critical path – the sequence of dependent tasks that dictates the project’s overall duration.

关键路径分析是一种确定项目最短完成时间的逻辑方法。它将项目建模为由活动组成的网络，其中部分活动可同时进行，而另一些则必须等待前导活动完成。其核心目标是找出关键路径——即决定项目总工期的一系列依赖任务。

CPA was developed in the 1950s for industrial and defence planning, but its principles are now universal. In the WJEC IGCSE exam, you will typically be given a precedence table listing activities, their durations, and immediate predecessors. Your job is to transform this data into a fully analysed network showing earliest and latest event times, floats, and the critical path.

CPA 于 20 世纪 50 年代为工业和国防规划而开发，但其原理如今已普及。在 WJEC IGCSE 考试中，通常会给出一个前导表，列出活动、工期和前导活动。你的任务是将这些数据转化为一个完整分析过的网络，显示最早和最晚事件时间、浮动时间以及关键路径。

Mastering CPA requires practice in drawing networks correctly, because a single mistake in logic can lead to incorrect times. Always check that the network respects all dependencies and uses dummies only when necessary.

掌握 CPA 需要练习正确绘制网络，因为逻辑上的一个小错误就可能导致时间计算错误。务必检查网络是否尊重所有依赖关系，并仅在必要时使用虚活动。

2. Activity-on-Arc (AoA) Networks | 箭线图 (AoA) 网络

In an AoA network, each activity is represented by an arrow (arc) connecting two numbered circles called events or nodes. The length of the arrow does not represent duration; instead, the duration is written above or below the arrow. Events mark the start or completion of activities.

在箭线图中，每项活动用连接两个编号圆圈的箭头（弧）表示，圆圈称为事件或节点。箭头的长度并不代表工期；工期标注在箭头上方或下方。事件表示活动的开始或完成。

Every AoA network must have a single start node (source) and a single end node (sink). Activities that have no predecessors emerge from the source, and activities that are not required by any subsequent activity converge at the sink. This ensures the network models the entire project as one connected system.

每个箭线图网络必须有一个起始节点（源）和一个终止节点（汇）。无前导的活动从源点出发，不被任何后续活动需要的活动汇聚于汇点。这确保了网络将整个项目建模为一个连通的系统。

Dummy activities are a distinctive feature of AoA networks. They are drawn as dashed arrows with a duration of zero. A dummy does not consume time or resources; it is used purely to maintain correct logical dependencies or to prevent two activities from starting and ending at the same pair of nodes, which would cause ambiguity in identification.

虚活动是箭线图的一个显著特征。它们用虚线箭头绘制，工期为零。虚活动不消耗时间或资源，仅用于维护正确的逻辑依赖关系，或防止两项活动起始和结束于同一对节点而造成标识不清。

3. Rules for Drawing AoA Networks | 绘制箭线图的规则

Before you draw, list all activities with their immediate predecessors and durations. Identify activities with no predecessors – they will start at the source node. Then add each subsequent activity, ensuring that it is connected after all its predecessors.

在绘制之前，列出所有活动及其前导活动和工期。找出无前导的活动——它们将从源节点出发。然后依次添加后续活动，确保每项活动连接在其所有前导活动之后。

A common mistake is to forget that an activity may depend on several other activities. In such cases, you may need to merge paths using dummies. For example, if activity C depends on A and B, but A and B do not necessarily end at the same event, you must introduce a dummy to make both A and B finish before C starts.

一个常见错误是忘记某项活动可能依赖于多个其他活动。此时，你可能需要用虚活动合并路径。例如，若活动 C 依赖于 A 和 B，但 A 和 B 不一定在同一个事件结束，你就必须引入虚活动，使 A 和 B 都在 C 开始前完成。

Number events sequentially from left to right. The event number at the tail of an arrow must be lower than the number at the head. Event numbers do not need to be consecutive, but they must increase along every path. This numbering convention makes it easier to refer to activities by their start and end events.

从左到右按顺序为事件编号。箭头尾部的事件编号必须小于头部的事件编号。事件编号无需连续，但必须沿每条路径递增。这种编号惯例便于通过起止事件来指代活动。

4. Forward Pass: Early Event Times | 正向计算：最早事件时间

The forward pass calculates the earliest time each event can be reached, denoted as EET(j) for event j. Begin by setting EET(source) = 0. Then, for every event j, consider all incoming activities (i, j) and compute EET(i) + duration(i, j). The maximum of these values is EET(j).

正向计算求出每个事件可以到达的最早时间，记为 EET(j)。首先设定 EET(源点) = 0。然后，对每个事件 j，考虑所有进入活动 (i, j)，计算 EET(i) + 工期(i, j)。这些值的最大值即为 EET(j)。

EET(j) = max { EET(i) + d(i, j) } for all i preceding j

Work through the network from left to right, layer by layer. The forward pass reveals the earliest start time for each activity: the earliest start of activity (i, j) is simply EET(i). The project’s overall minimum duration is the early event time of the sink node, EET(sink).

从左到右逐层计算。正向计算揭示了每项活动的最早开始时间：活动 (i, j) 的最早开始时间就是 EET(i)。项目的最短总工期就是终止节点的最早事件时间 EET(sink)。

If an event has multiple incoming paths, you must take the maximum because the event cannot happen until all preceding activities are complete. This maximization step is crucial for determining realistic completion times.

如果某个事件有多条进入路径，必须取最大值，因为该事件必须等到所有前导活动完成后才能发生。这一最大化步骤对确定实际完成时间至关重要。

5. Backward Pass: Late Event Times | 反向计算：最晚事件时间

After establishing the project duration, the backward pass finds the latest time each event can occur without delaying the project. Denote the late event time of node i as LET(i). Set LET(sink) equal to the project deadline – if no deadline is given, use LET(sink) = EET(sink).

确定了项目工期后，反向计算求算在不延误项目的前提下，每个事件可以发生的最晚时间。用 LET(i) 表示节点 i 的最晚事件时间。设 LET(汇点) 等于项目截止时间——若未给定期限，则使用 LET(sink) = EET(sink)。

LET(i) = min { LET(j) − d(i, j) } for all j succeeding i

Work backwards from the sink. For each event i, examine all activities (i, j) leaving i, subtract the activity duration from LET(j), and take the minimum. The logic is: to finish all downstream activities by their latest allowed times, event i must occur no later than this minimum value.

从汇点开始反向进行。对每个事件 i，检查所有离开 i 的活动 (i, j)，用 LET(j) 减去活动工期，取最小值。其逻辑是：要使所有下游活动在其最晚允许时间前完成，事件 i 必须不晚于这个最小值发生。

Always verify that LET(source) = 0. If you obtain a positive value, a delay before the start is allowed; a result of zero confirms consistency. Backward pass values are essential for computing activity floats.

务必验证 LET(源点) = 0。若得到正值，说明允许在开始前有一定的延迟；结果为 0 则确认一致性。反向计算值是计算活动浮动时间的基础。

6. Total Float and its Calculation | 总浮动时间及其计算

Total float is the amount of time an activity can be delayed without affecting the project’s overall completion time. For an activity (i, j), total float is given by:

总浮动时间是指一项活动可以延迟而不影响项目总完成时间的时间量。对于活动 (i, j)，总浮动时间由下式给出：

Total Float = LET(j) − EET(i) − duration(i, j)

This formula directly uses the node times you have already computed. Alternatively, total float can be expressed as Latest Start Time (LST) minus Earliest Start Time (EST). LST of activity (i, j) = LET(j) − duration(i, j), and EST = EET(i).

该公式直接使用你已经计算出的节点时间。此外，总浮动时间也可表示为最晚开始时间 (LST) 减去最早开始时间 (EST)。活动 (i, j) 的 LST = LET(j) − 工期(i, j)，EST = EET(i)。

If total float equals zero, the activity is critical – any delay in it will extend the project. Positive float indicates the activity has some scheduling flexibility. Total float cannot be negative; if it appears negative in your calculation, check your backward pass values, because the project deadline may be tighter than the earliest possible finish.

如果总浮动时间为零，该活动就是关键活动——其任何延误都将延长项目。正值浮动表示活动具有一定的调度弹性。总浮动时间不能为负；若计算中出现负值，请检查反向计算值，因为可能项目截止时间比最早可能完成时间更紧张。

7. Identifying the Critical Path | 识别关键路径

The critical path is the continuous chain of critical activities (total float = 0) from the source to the sink. It is the longest path through the network in terms of total duration, but paradoxically it determines the shortest time in which the project can be completed.

关键路径是从源点到汇点、由总浮动时间为零的关键活动组成的连续链条。从总工期角度看，它是网络中最长的路径，但矛盾的是，它决定了项目能够完工的最短时间。

To highlight the critical path on your diagram, trace from the source, choosing only activities with zero total float. There may be more than one critical path; all will have exactly the same length. You should mark the critical path clearly, for example with double lines or by colouring the arrows.

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IGCSE WJEC Mathematics: Critical Path Analysis – Essential Revision | IGCSE WJEC 数学：关键路径分析 考点精讲