📚 IGCSE OCR Maths: Critical Path Analysis - Key Points Review | IGCSE OCR 数学：关键路径分析考点精讲
Critical Path Analysis (CPA) is a powerful project management tool that helps to schedule activities efficiently. In IGCSE OCR Mathematics, you are required to construct activity networks, carry out forward and backward passes to calculate earliest and latest event times, determine the float for each activity, and identify the critical path which governs the minimum completion time of a project.
关键路径分析 (CPA) 是一种强大的项目管理工具，用于高效地安排各项活动。在 IGCSE OCR 数学考试中，你需要构建活动网络，通过前向扫描和后向扫描计算最早和最晚事件时间，确定各项活动的浮动时间，并找出决定项目最短工期的关键路径。
1. What Is Critical Path Analysis? | 什么是关键路径分析？
Critical Path Analysis involves breaking a project down into individual activities, each with a given duration and a set of predecessor activities that must be completed before it can begin. The aim is to find the longest path through the network, which determines the shortest possible project duration.
关键路径分析将一个项目分解为若干独立活动，每一项活动都有给定的持续时间以及一些必须在它开始前完成的前置活动。目的是找出网络中最长的一条路径，它决定了项目的最短可能工期。
The critical path consists of activities that have zero float — any delay in these activities will directly delay the whole project. Understanding which activities are critical allows managers to allocate resources more effectively.
关键路径由总浮动时间为零的活动构成——这些活动一旦延误就会直接推迟整个项目。了解哪些活动是关键活动，可以帮助管理者更有效地分配资源。
2. Activity-on-Arc Networks: Nodes and Activities | 活动在弧网络：节点与活动
In IGCSE OCR Maths, we predominantly use the activity-on-arc (AOA) representation. Here, each activity is represented by a directed arc (an arrow), and nodes (circles) represent events — the start or finish of one or more activities. The duration of an activity is written along the arc.
在 IGCSE OCR 数学中，我们主要使用活动在弧 (AOA) 表示法。在这种方法里，每个活动用一条有向弧（箭头）表示，节点（圆圈）表示事件——即一项或多项活动的开始或结束。活动的持续时间写在弧上。
Each event node is numbered for identification. The network must have a single start node (source) and a single end node (sink). Arrows show the logical dependencies: an activity cannot start until all the arcs entering its start node have been completed.
每个事件节点都被编号以便识别。整个网络必须具有唯一的起始节点（源）和唯一的结束节点（汇）。箭头表示逻辑上的依赖关系：一项活动必须等到所有指向其起始节点的弧都完成后才能开始。
3. Precedence Tables and Dummy Activities | 先后关系表与虚活动
A precedence table lists each activity, its duration, and its immediate predecessors. Sometimes the dependencies are such that an additional dummy activity is required to preserve logical relationships without consuming any time. A dummy activity is drawn as a dashed arrow and has a duration of zero.
先后关系表列出了每一项活动、它的持续时间以及直接前驱活动。有时，依赖关系需要引入额外的虚活动才能在不消耗时间的前提下保持正确的逻辑关系。虚活动用虚线箭头绘制，持续时间为零。
For example, if activity D depends on both A and B, but C depends only on A, we need to ensure that the start of D waits for B as well. Dummy activities prevent a situation where an arc incorrectly ties two events that should not both precede a particular activity.
例如，如果活动 D 依赖于 A 和 B，而活动 C 只依赖于 A，那么我们需要确保 D 的开始也要等待 B 完成。虚活动可以防止一条弧错误地将两项本不该同时作为前置条件的事件连接到某个活动。
4. Drawing Activity Networks | 绘制活动网络
Follow these steps to draw an AOA network from a precedence table:
按照以下步骤从先后关系表绘制 AOA 网络：
List all activities and identify those with no predecessors — they can start from the initial node.
列出所有活动，找出没有前驱的那些——它们可以从起始节点直接出发。
Create event nodes for the start and end of each activity, ensuring that each activity has a unique pair of start and end nodes.
为每个活动的开始和结束创建事件节点，确保每项活动具有唯一的起始和结束节点对。
Where multiple activities share the same start–end pair, insert dummy activities to split the event or to enforce the correct dependencies.
若有多项活动共享相同的起始－结束节点对，须插入虚活动来拆分事件或强制执行正确的依赖关系。
Number all nodes carefully from left to right, and draw the ork as a neat directed graph.
从左到右仔细为所有节点编号，并画出整洁的有向图。
5. Forward Pass: Earliest Event Times | 前向扫描：最早事件时间
The forward pass calculates the earliest event time (EET) for each node. Starting with the source node, set its EET to 0. For any subsequent node, the EET is the maximum of (EET of predecessor node + duration of connecting activity). If multiple arcs enter a node, take the largest value.
前向扫描用于计算每个节点的最早事件时间 (EET)。从源节点开始，将其 EET 设为 0。对于后续任一节点，其 EET 等于（前驱节点的 EET + 连接活动的持续时间）的最大值。如果有多条弧进入该节点，则取最大值。
Work from left to right through the network. The EET of the sink node gives the minimum project completion time, often called the project duration. Remember that dummy activities have zero duration but still contribute to the calculation.
在网络中从左到右逐步计算。汇节点的 EET 就是项目的最短完成时间，常称为项目工期。请记住，虚活动的持续时间为零，但仍然参与计算。
6. Backward Pass: Latest Event Times | 后向扫描：最晚事件时间
The backward pass determines the latest event time (LET) for each node without delaying the entire project. Start at the sink node and set its LET equal to the project duration (i.e., its EET). Move right to left: for any node, the LET is the minimum of (LET of successor node − duration of connecting activity).
后向扫描用于确定每个节点的最晚事件时间 (LET)，而不推迟整个项目。从汇节点开始，将其 LET 设为项目工期（即它的 EET）。从右向左移动：对于任一节点，LET 等于（后继节点的 LET − 连接活动的持续时间）的最小值。
If a node has several outgoing arcs, take the smallest result. This ensures that even the most restrictive path can finish on time. The difference between LET and EET at a node is called the node slack.
如果某节点有多条发出弧，则取最小的结果。这样可以确保即使是最严格的路径也能按时完成。一个节点的 LET 与 EET 之差称为节点松弛时间。
7. Float and Its Calculation | 总浮动时间及其计算
Float (total float) is the amount of time an activity can be delayed without affecting the overall project duration. For an activity going from node i to node j with duration d, the total float is given by:
浮动时间（总浮动时间）是指一项活动可以延误而不影响整个项目完工的时间量。对于从节点 i 到节点 j 且持续时间为 d 的活动，总浮动时间公式为：
Total Float = LET(j) − EET(i) − d
Activities on the critical path have a total float of zero. A positive float indicates spare time; a negative float (not expected at IGCSE level) signals that the project plan is already behind schedule.
关键路径上的活动总浮动时间为零。正浮动时间表示有富余时间；负浮动时间（IGCSE 阶段不要求）则表示项目进度已经滞后了。
8. Critical Path Identification | 关键路径的确定
The critical path is the continuous chain of activities (including any necessary dummy activities) from the start to the end node for which the float is zero. It is the longest path through the network in terms of duration and dictates the minimum possible project length.
关键路径是从起始节点到结束节点的连续活动链（包括任何必要的虚活动），且这些活动的总浮动时间为零。在持续时间方面，它是网络中最长的路径，并决定了可能的最短项目工期。
There can be more than one critical path, especially in complex networks. If so, all activities on any of these paths must be carefully monitored, because a delay in any critical activity will extend the project through that route.
关键路径可以不止一条，尤其是在复杂的网络中。如果存在多条关键路径，那么这些路径上的所有活动都必须仔细监控，因为任何一项关键活动的延误都会通过那条路径延长整个项目。
9. Worked Example Step-by-Step | 例题逐步讲解
Consider a project with the following activities:
考虑一个具有以下活动的项目：
| Activity | Duration | Predecessors | 中文 |
|---|---|---|---|
| A | 4 | — | 无 |
| B | 3 | — | 无 |
| C | 2 | A | A |
| D | 5 | A, B | A 和 B |
To construct the AOA network, start with node 1. Both A and B have no predecessors, so they leave node 1. A goes to node 2, B goes to node 3. Activity D depends on both A and B, so we connect node 2 to node 3 with a dummy activity (dashed arrow, duration 0). D then leaves node 3 and goes to node 4. Activity C depends only on A, so it leaves node 2 and also goes to node 4. Node 4 becomes the sink. Number the nodes: 1, 2, 3, 4.
为了构建 AOA 网络，从节点 1 开始。A 和 B 都没有前驱，因此它们从节点 1 出发。A 到达节点 2，B 到达节点 3。活动 D 同时依赖于 A 和 B，所以我们用一条虚活动（虚线箭头，持续时间为 0）连接节点 2 到节点 3。然后 D 从节点 3 出发到达节点 4。活动 C 只依赖于 A，因此它从节点 2 出发也到达节点 4。节点 4 成为汇节点。节点编号为：1, 2, 3, 4。
Now perform the forward pass:
现在执行前向扫描：
EET₁ = 0
EET₂ = EET₁ + 4 = 4
EET₃ = max( EET₁ + 3 = 3 , EET₂ + 0 = 4 ) = 4
EET₄ = max( EET₂ + 2 = 6 , EET₃ + 5 = 9 ) = 9
Project duration = 9. Next, the backward pass from node 4 with LET₄ = 9:
项目工期 = 9。接着从节点 4 进行后向扫描，LET₄ = 9：
LET₃ = LET₄ − 5 = 4
LET₂ = min( LET₄ − 2 = 7 , LET³ − 0 = 4 ) = 4
LET₁ = min( LET₂ − 4 = 0 , LET₃ − 3 = 1 ) = 0
Now calculate float for each activity using the formula TF = LET(j) − EET(i) − d:
现在用公式 TF = LET(j) − EET(i) − d 计算各项活动的浮动时间：
A (1→2): 4 − 0 − 4 = 0 → critical
A (1→2)：4 − 0 − 4 = 0 → 关键
B (1→3): 4 − 0 − 3 = 1
B (1→3)：4 − 0 − 3 = 1
Dummy (2→3): 4 − 4 − 0 = 0 → critical
虚活动 (2→3)：4 − 4 − 0 = 0 → 关键
C (2→4): 9 − 4 − 2 = 3
C (2→4)：9 − 4 − 2 = 3
D (3→4): 9 − 4 − 5 = 0 → critical
D (3→4)：9 − 4 − 5 = 0 → 关键
The critical path is therefore A → Dummy → D, or directly A– o 1 unit of time without delaying the overall project. Activity C has plenty of free time.
因此，关键路径为 A → 虚活动 → D，或直接表示为 A–D（持续时间 4 + 5 = 9）。活动 B 的浮动时间为 1，意味着它可以最多延误 1 个单位时间而不会推迟整个项目。活动 C 则有充裕的自由时间。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always draw a dummy activity when two or more activities share the same start and end nodes, or when a dependency would otherwise be incorrectly represented.
当两项或多项活动共享相同的起始和结束节点，或者依赖关系可能被错误表示时，务必画出虚活动。
Use a pencil and ruler — neatness counts. Nodes should be clearly numbered and arrows should not cross unnecessarily.
使用铅笔和直尺——整洁度很重要。节点应当清晰地编号，箭头应尽量避免不必要的交叉。
Check that every node (except the start and end) has at least one entering and one leaving arc.
检查每个节点（起始节点和结束节点除外）至少有一条进入弧和一条离开弧。
When doing the forward or backward pass, show your working on the network near each node to earn method marks.
在进行前向或后向扫描时，在每个节点附近展示计算过程，以便获得方法分。
Don't forget that dummy activities have zero duration and are included in the critical path calculations just like real activities.
不要忘记虚活动的持续时间为零，并且像真实活动一样参与关键路径计算。
Be careful with the float formula — it must be LET of the end node minus EET of the start node minus the activity duration. Mixing up EET and LET is a common error.
小心浮动时间公式——必须是结束节点的 LET 减去起始节点的 EET 再减去活动持续时间。混淆 EET 和 LET 是常见错误。
11. Reviewing Key Terminologies | 关键术语回顾
Recapping the essential vocabulary helps secure marks: earliest event time (EET), latest event time (LET), total float, critical path, dummy activity, and precedence. Being able to define these in clear mathematical language is often required in exam questions.
回顾基本词汇有助于稳拿分数：最早事件时间 (EET)、最晚事件时间 (LET)、总浮动时间、关键路径、虚活动和先后关系。能够用清晰的数学语言定义这些术语常常是试题的要求。
12. Final Practice Advice | 最后练习建议
Work through past paper questions systematically: draw the network, label EET and LET at each node, compute floats, and highlight the critical path. Time yourself to build speed and accuracy. Remember, the critical path may involve a dummy, so never ignore dashed arrows.
系统地练透过往真题：绘制网络，在各节点标出 EET 和 LET，计算浮动时间并标出关键路径。计时练习以提高速度和准确性。记住，关键路径可能包含虚活动，因此绝不能忽视虚线箭头。
Always verify that the project duration equals the sum of durations along the critical path. This simple check can catch arithmetic errors before you finish the paper.
务必检查项目工期是否等于关键路径上的持续时间总和。这一简单的核对能在你答完试卷前揪出计算错误。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图