Critical Path Analysis | 关键路径分析

📚 Critical Path Analysis | 关键路径分析

Critical Path Analysis (CPA) is a powerful project management tool that helps you schedule complex tasks efficiently. By constructing an activity network and performing forward and backward passes, you can identify which activities must not be delayed, and which ones can ‘float’ without affecting the overall project completion time. This topic appears regularly on OCR GCSE Mathematics papers, often carrying high marks in the Decision section. Mastering it not only boosts your exam performance but also develops logical thinking you will use in real‑life planning.

关键路径分析是一种强大的项目管理工具，可帮助你高效地安排复杂任务。通过构建活动网络并进行向前和向后计算，你能够确定哪些活动绝对不能延误，哪些活动可以“浮动”而不影响整个项目完成时间。该主题在 OCR GCSE 数学试卷中经常出现，通常在决策部分占有高分值。掌握它不仅能提高考试成绩，还能培养你在实际生活中用得上的逻辑思维能力。

1. Activity Networks and Precedence | 活动网络与前导关系

An activity network is a diagram showing the order in which tasks must be completed. Each activity is represented as an arrow, and each node (circle) represents an event – a point in time when one or more activities start or finish. The network must respect dependencies: if activity B cannot start until activity A has finished, A precedes B. Understanding these dependencies is the first step towards drawing a correct network.

活动网络是一种图示，显示任务必须完成的顺序。每项活动用箭头表示，每个节点（圆圈）代表一个事件——即一个或多个活动开始或结束的时间点。网络必须遵守依赖关系：如果活动 B 必须等活动 A 完成后才能开始，那么 A 前导于 B。理解这些依赖关系是绘制正确网络的第一步。

Dependencies are most clearly presented in a precedence table. This table lists each activity, its duration, and any immediate predecessors. For example, activity C may depend on both A and B being finished. The network diagram translates this information into a visual form, making it easy to apply algorithms to find the critical path.

依赖关系最清楚地呈现在前导表中。该表列出了每项活动、其持续时间以及任何直接前导活动。例如，活动 C 可能依赖于 A 和 B 的完成。网络图将这些信息转化为可视化形式，使应用算法寻找关键路径变得容易。

2. Reading and Constructing a Precedence Table | 阅读并构建前导表

A typical precedence table has three columns: Activity, Duration, and Immediately Preceding Activities. For instance:

一个典型的前导表有三列：活动、持续时间、紧前活动。例如：

Activity	Duration (days)	Preceding Activities
A	3	–
B	4	–
C	2	A
D	5	A, B
E	1	C, D

From the table, A and B can start immediately because they have no predecessors. C can start once A finishes. D requires both A and B to be completed. E depends on both C and D. Making a sketch of this table helps avoid mistakes when drawing the network.

从表中可以看出，A 和 B 没有前导活动，可以立即开始。C 可以在 A 完成后开始。D 需要 A 和 B 都完成。E 依赖于 C 和 D 两者。在绘制网络之前先草拟这个表格有助于避免错误。

3. Drawing Activity‑on‑Arc Networks | 绘制弧活动网络图

In OCR GCSE the convention used is activity‑on‑arc (AOA), where each activity is represented by a directed edge (arc), and nodes are numbered. The project start is a single node (usually labelled 1) and the finish is a final node. Arcs are labelled with the activity letter and its duration. For the table above, first place node 1, from which activities A and B emerge towards separate nodes. A leads to node 2, B leads to node 3. Activity C goes from node 2 (end of A) to a new node. Since D needs both A and B, a dummy activity is needed. We will discuss dummies next.

在 OCR GCSE 中，采用的惯例是弧活动网络图，其中每项活动用有向边（弧）表示，节点编号。项目开始是单个节点（通常标记为 1），结束是最后一个节点。弧线上标注活动字母及其持续时间。对于上表，首先放置节点 1，活动和 A 和 B 从该节点发出，分别指向不同节点。A 通向节点 2，B 通向节点 3。活动 C 从节点 2（A 的结束）指向一个新节点。由于 D 需要 A 和 B 两者，需要引入虚活动。我们接下来讨论虚活动。

Draw arcs as straight arrows. Label the duration near the arrow. Nodes are numbered logically from left to right. The final node must have all preceding activities converging into it. Double‑check that every dependency in the table is satisfied. A well‑drawn network is the foundation for the time calculations.

将弧线画成直线箭头。在箭头附近标注持续时间。节点按从左到右的顺序合理编号。最终节点必须是所有前导活动汇聚的地方。仔细检查表格中的每一项依赖关系是否都已满足。绘制良好的网络是时间计算的基础。

4. Dummy Activities and Logical Dependencies | 虚活动与逻辑依赖

A dummy activity is an arc with zero duration, drawn as a dashed arrow. Its purpose is purely logical – it represents a dependency without consuming time. Dummies are essential when two activities share some, but not all, predecessors. For example, activity D depends on A and B, while C depends only on A. If we simply drew D from the same node as C, we would incorrectly imply D needs only A. To fix this, we finish A at node 2, start C from node 2, and use a dummy from node 2 to node 3 (the end of B). Then D can start from node 3, correctly depending on both A and B via the dummy.

虚活动是持续时间为零的弧线，用虚线箭头表示。其目的纯粹是逻辑性的——它代表不消耗时间的依赖关系。当两个活动共享部分（而非全部）前导活动时，虚活动至关重要。例如，活动 D 依赖于 A 和 B，而 C 仅依赖于 A。如果简单地从与 C 相同的节点画出 D，我们将错误地暗示 D 仅需要 A。为了修正这个问题，让 A 结束于节点 2，C 从节点 2 开始，并使用从节点 2 到节点 3（B 的结束）的虚活动。然后 D 可以从节点 3 开始，通过虚活动正确依赖于 A 和 B 两者。

Remember: a dummy has duration 0 and does not affect the timing calculations. It only establishes the correct precedence. In OCR exams, you may be asked to insert dummies or identify why a dummy is needed. Always check that the network structure avoids wrong dependencies.

请记住：虚活动的持续时间为 0，不会影响时间计算。它仅用于建立正确的前导关系。在 OCR 考试中，你可能会被要求插入虚活动，或者解释为什么需要虚活动。请始终检查网络结构是否避免了错误的依赖关系。

5. Forward Pass – Calculating Earliest Event Times | 向前计算 – 计算最早事件时间

The forward pass finds the earliest time each event (node) can be reached. You start at node 1 with time 0. Then move through the network from left to right. For each activity going from node i to node j, the earliest time at node j is the maximum of (earliest time at i + duration). Where multiple arcs enter a node, take the largest value. We record Earliest Start Time (EST) inside a box next to each node, or write the value above the node. In OCR notation, it is often called early event time, denoted Eᵢ for node i.

向前计算得出每个事件（节点）能够到达的最早时间。从节点 1 开始，时间为 0。然后从左到右遍历网络。对于从节点 i 指向节点 j 的每项活动，节点 j 的最早时间等于（节点 i 的最早时间 + 持续时间）中的最大值。当有多条弧线进入一个节点时，取最大值。我们在每个节点旁的方框内记录最早开始时间，或在节点上方写下数值。在 OCR 的符号体系中，常称为最早事件时间，节点 i 记为 Eᵢ。

For example, if node 2 is reached from node 1 via activity A (duration 3), E₂ = 0 + 3 = 3. If node 4 is entered by activities C (from node 2, duration 2) and D (from node 3, duration 5), and E₂ = 3, E₃ = 4 (from B), then possible times: 3+2=5, 4+5=9. Therefore E₄ = max(5,9) = 9. The forward pass ends at the final node, giving the minimum project completion time.

例如，若从节点 1 通过活动 A（持续时间 3）到达节点 2，则 E₂ = 0 + 3 = 3。若节点 4 由活动 C（来自节点 2，持续时间 2）和 D（来自节点 3，持续时间 5）进入，且 E₂ = 3，E₃ = 4（来自 B），则可能的时间为：3+2=5，4+5=9。因此 E₄ = max(5,9) = 9。向前计算在最终节点结束，得出项目的最短完成时间。

6. Backward Pass – Calculating Latest Event Times | 向后计算 – 计算最晚事件时间

The backward pass determines the latest time each event can occur without delaying the whole project. Begin at the final node n, setting its latest finish time Lₙ equal to its earliest time Eₙ. Then work backwards from right to left. For an activity from i to j, the latest time at node i is the minimum of (Lⱼ – duration). Where multiple arcs leave a node, take the smallest value. Latest event times are often placed in a circle or semicircle below the node.

向后计算确定每个事件在不延误整个项目的前提下可以发生的最晚时间。从最终节点 n 开始，将其最晚完成时间 Lₙ 设定为等于最早时间 Eₙ。然后从右向左倒推。对于从 i 到 j 的一项活动，节点 i 的最晚时间等于（Lⱼ – 持续时间）中的最小值。当多条弧线离开一个节点时，取最小值。最晚事件时间通常放在节点下方的一个圆圈或半圆内。

Continuing the example, if final node is node 5 with E₅ = 12, then L₅ = 12. Suppose activity E goes from node 4 to 5 with duration 1. Then L₄ = 12 – 1 = 11. If another activity also leaves node 4, we compare. The backward pass reveals the flexibility in the schedule; any node where Eᵢ = Lᵢ is on the critical path.

继续上例，假设最终节点是节点 5，E₅ = 12，则 L₅ = 12。假设活动 E 从节点 4 到节点 5，持续时间为 1。那么 L₄ = 12 – 1 = 11。如果有另一活动也从节点 4 离开，需要比较。向后计算揭示了安排中的灵活性；任何满足 Eᵢ = Lᵢ 的节点都位于关键路径上。

7. Total Float and Activity Slack | 总浮动时间与活动松弛

Total float (TF) is the amount of time an activity can be delayed without affecting the project finish date. For an activity from node i to j with duration d, total float = Lⱼ – Eᵢ – d. For a node itself, the ‘node slack’ is Lᵢ – Eᵢ, but exam questions usually ask for the total float of activities. Activities with zero total float are critical; they cannot be delayed at all. Positive float means the activity has some spare time.

总浮动时间是指活动在不影响项目完成日期的前提下可以延迟的时间量。对于从节点 i 到 j、持续时间为 d 的活动，总浮动时间 = Lⱼ – Eᵢ – d。对于节点本身，“节点松弛”为 Lᵢ – Eᵢ，但考试题目通常要求计算活动的总浮动时间。总浮动时间为零的活动是关键活动，完全不能延误。正浮动时意味着活动有一些空闲时间。

Consider activity C with Eᵢ = 3, Lⱼ = 11, d = 2. Its total float = 11 – 3 – 2 = 6. So C could be delayed up to 6 days without affecting the overall project. This information is vital for resource allocation. When you are asked to explain the effect of a delay, you compare the delay with the total float. If delay > total float, the project completion time will increase by the excess.

考虑活动 C，Eᵢ = 3，Lⱼ = 11，d = 2。其总浮动时间 = 11 – 3 – 2 = 6。因此 C 可延迟最多 6 天而不影响整个项目。这一信息对于资源分配至关重要。当被要求解释某项延误的影响时，需要将延误时间与总浮动时间进行比较。若延误时间 > 总浮动时间，项目完成时间将增加超过的数量。

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

The critical path is the longest path through the network from start to finish, where all activities have zero total float. It determines the minimum project duration. To find it, after completing forward and backward passes, list all activities with total float = 0. Join them into one or more continuous routes from node 1 to the final node. Usually there is a single critical path, but sometimes there can be multiple critical paths.

关键路径是网络图中从起点到终点的最长路径，其中所有活动的总浮动时间均为零。它决定了项目的最短工期。要找到关键路径，在完成向前和向后计算后，列出所有总浮动时间为零的活动。将它们连接成一条或数条从节点 1 到最终节点的连续路线。通常只有一条关键路径，有时可能存在多条关键路径。

In the example, after calculations, activities B, D, and E may have zero float. So the critical path is B – D – E, and the project duration equals the sum of their durations: 4 + 5 + 1 = 10 days (if the final node time matches). Always state the critical path clearly in the exam, e.g. ‘The critical path is B–D–E with a project length of 10 days.’

在该示例中，经过计算，活动 B、D 和 E 的浮动时间可能为零。因此关键路径是 B – D – E，项目工期等于它们的持续时间之和：4 + 5 + 1 = 10 天（如果最终节点时间匹配）。在考试中要清晰地陈述关键路径，例如“关键路径为 B–D–E，项目长度为 10 天。”

9. Using Gantt Charts to Visualise the Schedule | 使用甘特图可视化调度

A Gantt chart (cascade chart) is often used alongside CPA to display activities over time. It shows each activity as a horizontal bar, positioned according to its earliest start time and duration. Critical activities are usually drawn in a different colour or shading. The chart also indicates the total float of non‑critical activities. In OCR exams, you may be asked to draw a simple Gantt chart for a small set of activities, given the earliest and latest start times. This demonstrates your ability to interpret the schedule visually.

甘特图（级联图）常与关键路径分析一起使用，以展示活动随时间分布的情况。它将每项活动显示为水平条，根据其最早开始时间和持续时间定位。关键活动通常以不同颜色和阴影绘制。甘特图还能显示非关键活动的总浮动时间。在 OCR 考试中，你可能会被要求为一小组活动绘制简单的甘特图，给出最早和最晚开始时间。这展示了你以可视化方式诠释计划的能力。

To draw a Gantt chart, create a horizontal time axis. For each activity, draw a bar from its earliest start time to earliest finish time. Extend a dashed or thinner bar from earliest finish to latest finish to represent the float. This helps project managers see where resources can be shifted.

要绘制甘特图，创建一条水平时间轴。对每项活动，从最早开始时间到最早完成时间画一个条形。从最早完成时间到最晚完成时间用虚线或更细的条形表示浮动时间。这有助于项目经理看到资源可以调配的地方。

10. Interpreting Changes and Delays | 解析变化与延迟

Exam questions often describe a scenario where the duration of an activity changes, or an activity is delayed. You need to determine whether the project completion time is affected. To do this, check if the activity is on the critical path. If it is critical, any increase in its duration will increase the overall project time by the same amount (unless there is another path that becomes critical with the same duration). If it is non‑critical, compare the delay with the total float. Only the portion of the delay exceeding the total float increases the project length.

考试题目常常描述活动持续时间发生变化或某项活动出现延误的场景。你需要判断项目完成时间是否受影响。做法是检查该活动是否位于关键路径上。如果它是关键活动，其持续时间的任何增加都将导致整体项目时间增加相同的量（除非有另一条路径因相同持续时间也成为关键路径）。如果它是非关键活动，将延误时间与总浮动时间进行比较。只有超过总浮动时间的那部分延误才会增加项目长度。

For example, if activity C (total float 6 days) is delayed by 8 days, the project will be delayed by 8 – 6 = 2 days. You must show the reasoning clearly. Additionally, some questions ask you to explain the effect of overlapping activities or how to crash the project. Crash actions reduce durations at extra cost, but usually the focus in GCSE is on understanding the impact of delays and priority.

例如，如果活动 C（总浮动时间 6 天）延误了 8 天，项目将延迟 8 – 6 = 2 天。必须清晰地展示推理过程。此外，有些题目要求解释活动重叠的影响，或如何压缩项目工期。压缩措施是以额外成本缩短持续时间，但在 GCSE 中，重点通常是理解延误与优先级的影响。

11. Common Mistakes and How to Avoid Them | 常见错误及其避免方法

One frequent mistake is forgetting to include dummies where needed, leading to incorrect dependencies. Always compare the precedence table with your network to ensure no extra dependencies are created or missing. Another error is misapplying the forward/backward pass rules: using minimum instead of maximum in forward pass, or forgetting to take the smallest value in backward pass. Devise a checklist: ‘Forward pass – take MAX; backward pass – take MIN at nodes.’

一个常见的错误是忘记在需要的地方加入虚活动，导致依赖关系错误。务必对照前导表检查网络，确保没有制造多余的依赖关系，也没有遗漏。另一个错误是弄错向前/向后计算的规则：在向前计算时用了最小值而不是最大值，或者在向后计算时忘记取最小值。可以设计一个检查清单“向前计算取最大；向后计算在节点处取最小。”

Students also confuse total float with free float. At GCSE OCR level, only total float is required. Free float is the delay allowed without affecting any subsequent activity, but you do not need to calculate it. Stick to the formula TF = Lⱼ – Eᵢ – d. Also, when labelling nodes, use the correct early event times inside squares and late times inside circles if that is the convention in the exam. Check the question instructions.

学生还可能会混淆总浮动时间和自由浮动时间。在 OCR GCSE 级别，只需掌握总浮动时间。自由浮动时间是指不影响任何后续活动的情况下允许的延迟，但无需计算。坚持使用公式 TF = Lⱼ – Eᵢ – d。此外，在标注节点时，如果考试中采用某种惯例，最早事件时间放入方框，最晚时间放入圆圈，请遵循题目说明。

12. Exam Strategy and Practice Tips | 考试策略与练习建议

CPA questions are typically multi‑part, starting with drawing a network, adding dummies, performing passes, calculating floats, identifying the critical path, and then interpreting a delay. Allocate time according to marks. Use a ruler for the network; a messy diagram can lead to errors in reading durations. If you are provided with a precedence table, use it as your single source of truth.

关键路径分析题目通常包含多个部分：先绘制网络、加入虚活动、进行正反向计算、计算浮动时间、识别关键路径，然后解释一次延误。根据分值分配好时间。绘制网络时使用尺子；杂乱无章的图会导致读取持续时间时出错。如果给出了前导表，就把它作为唯一的准确信息来源。

Practice with past OCR papers to become familiar with the common formats. For example, a 7‑mark question might be split as: (a) draw network (3 marks), (b) give early and late event times (2 marks), (c) state critical path and length (2 marks). Another common variant is a cascade (Gantt) chart question where you interpret the chart to find float. Always double‑check your arithmetic; one small miscalculation can cascade through the entire answer.

多练习 OCR 历年真题，熟悉常见的题型格式。例如，一道 7 分的题目可能划分为 (a) 绘制网络（3 分）、(b) 给出最早和最晚事件时间（2 分）、(c) 陈述关键路径及长度（2 分）。另一种常见变体是级联（甘特）图题目，你需要解读图表找出浮动时间。始终复核你的计算；一个小小的计算错误可能会使整个答案连锁崩溃。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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