Edexcel Maths: Critical Path Analysis Revision Notes | Edexcel 数学：关键路径分析考点精讲

📚 Edexcel Maths: Critical Path Analysis Revision Notes | Edexcel 数学：关键路径分析考点精讲

Critical Path Analysis (CPA) is a powerful project management tool within Edexcel Decision Mathematics. It helps you model complex projects, identify the minimum completion time, and pinpoint the activities that cannot be delayed without affecting the entire schedule. Mastering CPA means you can systematically determine earliest and latest event times, calculate total float, and extract the critical path – skills that are regularly examined and directly applicable to scheduling problems.

关键路径分析（CPA）是 Edexcel 决策数学中强大的项目管理工具。它能帮助你对复杂项目建模，确定最短完成时间，并找出那些一旦延迟就会影响整体进度的活动。掌握 CPA 意味着你可以系统地计算最早和最晚时间，算出总浮动时间，并提取关键路径——这些技能经常出现在考试中，也直接适用于现实中的调度问题。

1. Core Concepts of Critical Path Analysis | 关键路径分析的核心概念

A project consists of a set of activities that must be completed, with some activities depending on the completion of others. Each activity has a duration and cannot be interrupted once started. The aim is to represent the project as a network, find how long the project will take, and discover which activities are critical – meaning any delay in them will delay the whole project.

一个项目由一系列必须完成的活动组成，有些活动需要依赖其他活动完成后才能开始。每个活动都有一个持续时间，一旦开始就不能中断。目标是将项目表示为网络图，计算项目总耗时，并找出关键活动——这些活动一旦延迟，整个项目就会被推迟。

Activity: A task that takes time and resources. Represented by a node or an edge, depending on the convention.
活动： 一项需消耗时间和资源的工作。根据惯例，可以用节点或边来表示。
Precedence Table: A list of activities showing their durations and immediate predecessors.
优先关系表： 列出活动及其持续时间与直接前导活动。
Dummy activity: Used in activity-on-arc diagrams to preserve logic; not needed in Edexcel’s preferred activity-on-node method.
虚活动： 用在箭线图中以保持逻辑关系；在 Edexcel 常采用的节点表示法中不需要。

2. Activity Networks and Precedence | 活动网络与优先关系

Edexcel exams primarily use the activity-on-node (AON) representation, where each node represents an activity and arcs show dependencies. You will often be asked to draw the network from a precedence table. Always start from a single start node and finish at a single end node to avoid multiple sources or sinks. Check pairs of activities that share the same predecessors but are independent – you may need to introduce a dummy activity only if using activity-on-arc; in AON, simply draw parallel arrows.

Edexcel 考试主要采用节点表示法（AON），每个节点代表一个活动，箭头表示依赖关系。你常需要根据优先关系表画出网络图。务必从一个唯一的起始节点出发，结束于一个唯一的终止节点，避免出现多个源点或汇点。检查是否有活动共享相同的前导但又互相独立——只有在箭线图中才可能需要引入虚活动；在 AON 中，只需画出平行箭头即可。

The network must be a directed acyclic graph. Loops are not allowed because they would imply an activity must finish before itself, which is impossible. Once drawn, the network provides a visual overview of the project’s logical flow and is the foundation for time analysis.

网络必须是有向无环图。不允许出现回路，因为回路的含义是活动要在自身完成之前完成，这不可能。网络一旦画好，就能清晰展示项目逻辑流，并为时间分析奠定基础。

3. Forward Pass: Earliest Times | 前向扫描：最早时间

The forward pass calculates the earliest start time (EST) and earliest finish time (EFT) for each activity. Begin from the start node with EST = 0. For any activity, EST is the maximum of the EFTs of all its immediate predecessors. Then EFT = EST + duration. If an activity has no predecessors, its EST is 0. The project’s minimum completion time is the maximum EFT at the end node.

前向扫描计算每个活动的最早开始时间（EST）和最早完成时间（EFT）。从起始节点的 EST=0 开始。对任一活动，EST 等于它所有直接前导活动的 EFT 中的最大值。然后 EFT = EST + 持续时间。如果某活动没有前导，其 EST 就是 0。项目的最短完成时间就是终点节点的最大 EFT。

EST(activity) = max{ EFT(predecessors) }

EST(活动) = max{ 前导活动的 EFT }

EFT = EST + duration

EFT = EST + 持续时间

Use a systematic approach: process nodes in topological order, from start to finish. Recording both EST and EFT directly on the node is a common exam requirement, often by inserting numbers into a standard box layout.

采用系统方法：按拓扑顺序处理节点，从开始到结束。考试中常见的要求是将 EST 和 EFT 直接记录在节点上，通常填入标准方框布局。

4. Backward Pass: Latest Times | 后向扫描：最晚时间

The backward pass determines the latest finish time (LFT) and latest start time (LST) without extending the project duration. Start from the end node: set its LFT equal to the project completion time (or to the maximum EFT if there is a single end). For any activity, LFT is the minimum of the LSTs of all its immediate successors. Then LST = LFT – duration.

后向扫描确定在不延长总工期的前提下，每个活动的最晚完成时间（LFT）和最晚开始时间（LST）。从终点节点开始：设定其 LFT 等于项目完成时间（如果只有一个终点，则等于最大 EFT）。对任一活动，LFT 等于它所有直接后继活动的 LST 中的最小值。然后 LST = LFT – 持续时间。

LFT(activity) = min{ LST(successors) }

LFT(活动) = min{ 后继活动的 LST }

LST = LFT – duration

LST = LFT – 持续时间

Process nodes in reverse topological order, from finish to start. All ending activities that have no successors should have their LFT set to the project duration. If multiple end nodes exist, all must have LFT = project duration.

按逆拓扑顺序处理节点，从结束到开始。所有没有后继的结束活动，其 LFT 应设为项目持续时间。如果存在多个终点节点，必须全部设定 LFT = 项目持续时间。

5. Float: Total Float, Free Float, and Independent Float | 浮动时间：总浮动、自由浮动与独立浮动

Float (slack) measures how much an activity can be delayed without affecting the project schedule. The most important type for Edexcel is total float. Total float shows the maximum delay possible for an activity without delaying the overall project completion.

浮动时间（松弛时间）衡量一项活动能延迟多久而不影响项目进度。对 Edexcel 最重要的是总浮动时间。总浮动时间表示一项活动在不妨碍整个项目完成的前提下可以延迟的最大时间。

Total Float = LST – EST = LFT – EFT

总浮动 = LST – EST = LFT – EFT

Free float is the delay possible without affecting the earliest start of any successor. It is calculated as minimum EST of successors minus EFT of the current activity. Independent float is the delay possible if all predecessors finish as late as possible and all successors start as early as possible, but it is rarely examined in depth.

自由浮动是指在不影响任何后继活动最早开始的前提下可以延迟的时间。计算方法为后继活动的最小 EST 减去本活动的 EFT。独立浮动则假设所有前导都尽可能晚完成、所有后继都尽可能早开始，是一种理论上的延迟余量，但考试较少深入考查。

An activity with total float equal to zero is critical. Any delay in a critical activity will directly extend the project’s minimum completion time. The sequence of critical activities forms the critical path.

总浮动为零的活动就是关键活动。关键活动的任何延迟都会直接延长项目的最短完成时间。由关键活动组成的序列就是关键路径。

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

A critical path is a continuous chain of critical activities from the start node to the end node where the sum of durations equals the project duration. A project may have more than one critical path. All critical paths share the property that each activity on them has zero total float. To find the critical path, mark all activities with total float = 0, then trace a path from start to finish using only these activities.

关键路径是从起始节点到终止节点的一条连续关键活动链，其持续时间之和等于项目持续时间。一个项目可能有多条关键路径。每条关键路径上的全部活动总浮动均为零。要找出关键路径，先标记所有总浮动为 0 的活动，然后仅用这些活动构建一条从起始到结束的路径。

The critical path(s) must be stated clearly in the answer, for example: A – C – F – H. The exam often asks for the critical path and its length (the project duration), and subsequently for the effect of delays on the critical path.

答案中必须清楚写明关键路径，例如：A – C – F – H。考试常要求写出关键路径及其长度（项目持续时间），并进一步要求分析延迟对关键路径的影响。

7. Gantt Charts (Cascade Charts) and Scheduling | 甘特图（级联图）与调度

Edexcel requires you to construct a Gantt chart (also called a cascade chart) showing activities scheduled at their earliest start times. The chart is a horizontal bar chart where each activity is represented by a bar from its EST to EFT. The length of the bar equals the activity duration. Activities can be placed on separate rows; critical activities are often highlighted. The Gantt chart visualises the project timeline and resource usage when durations are fixed.

Edexcel 要求你构建甘特图（又称级联图），按照最早开始时间安排活动。甘特图是水平条形图，每个活动用一个从 EST 到 EFT 的长条表示。条的长度等于活动持续时间。活动可放在不同行；关键活动通常被突出显示。甘特图在工期固定时可视化项目时间线和资源使用情况。

When you draw a Gantt chart, ensure the time axis is scaled correctly and each bar is labelled with the activity letter or name. Floating activities (with total float > 0) can be shown with an extension to indicate the possible delay, often as a dotted or greyed-out bar after the solid earliest bar.

画甘特图时，要确保时间轴刻度正确，每个条都标有活动字母或名称。有浮动的活动（总浮动 > 0）可用延伸段表示可能的延迟，通常是在实线最早条之后用虚线或灰色条表示。

8. Resource Histograms and Resource Levelling | 资源直方图与资源平衡

A resource histogram shows the number of workers (or amount of a resource) required on each day when all activities start as early as possible. The histogram is constructed by summing the resource requirements of all activities active on each day, then plotting the totals against time. The exam may ask you to interpret or draw such a histogram from a given Gantt chart with resource allocations.

资源直方图展示当所有活动都尽早开始时，每天所需的工人数量（或资源数量）。构建方法是对每天所有正在进行的活动的资源需求进行求和，然后将总需求按时间绘制成柱状图。考试可能要求根据已给的有资源分配的甘特图解读或绘制这样的直方图。

Resource levelling is the process of delaying non-critical activities (within their float) to smooth out resource demand and avoid peaks above a given limit. The ideal resource profile should be as even as possible. In exams, you may be asked to schedule activities subject to a maximum number of workers per day, using total float to shift activities so that the resource limit is not exceeded.

资源平衡是指在浮动时间允许范围内推迟非关键活动，使资源需求变得平滑，避免需求峰值超过给定限制。理想的资源曲线应尽可能平坦。考试中可能要求你在每天最多工人数量的约束下调度活动，利用总浮动时间平移活动，使不超过资源上限。

Resource profile: sum of workers on day d = Σ{ workers for activities i with ESTᵢ ≤ d < EFTᵢ }

资源曲线：第 d 天工人总数 = Σ{ 满足 ESTᵢ ≤ d < EFTᵢ 的活动 i 所需工人数 }

9. Algorithmic Summary for Full CPA Analysis | 关键路径分析完整步骤概览

The standard sequence of steps expected in an Edexcel CPA question is as follows: 1) Draw the activity network from the precedence table. 2) Carry out a forward pass to obtain EST and EFT. 3) Carry out a backward pass to obtain LFT and LST. 4) Calculate the total float for each activity. 5) Identify all critical activities (float = 0) and state the critical path(s). 6) Draw a Gantt chart (cascade chart) at earliest start times. If required, 7) produce a resource histogram for the earliest start schedule, and 8) perform resource levelling to meet a constraint.

Edexcel CPA 考题中要求的标准步骤顺序如下：1）根据优先表绘制活动网络。2）进行前向扫描，得到 EST 和 EFT。3）进行后向扫描，得到 LFT 和 LST。4）计算每个活动的总浮动时间。5）识别所有关键活动（浮动时间=0）并写出关键路径。6）按最早开始时间绘制甘特图（级联图）。如有需要，7）生成最早开始调度下的资源直方图，8）进行资源平衡以满足约束。

Memorising this workflow reduces the risk of missing marks. Always label nodes clearly with the activity name, duration, EST, EFT, LST, and LFT in the format given by the exam board. A typical node layout places EST and LST on the left, EFT and LFT on the right, and the duration in the centre.

记住这个工作流程可以减少丢分风险。务必按考试局给出的格式清晰标注节点：活动名称、持续时间、EST、EFT、LST、LFT。典型的节点布局把 EST 和 LST 放在左侧，EFT 和 LFT 放在右侧，持续时间放在中间。

10. Common Mistakes and How to Avoid Them | 常见错误及如何避免

One frequent mistake is misreading the precedence table, leading to an incorrect network. Always double-check the immediate predecessors column; do not add unnecessary dependencies. Another typical error is using the wrong formula for total float – students sometimes subtract EST from EFT, which only gives duration, not float. Use LST – EST or LFT – EFT consistently.

一个常见错误是误读优先表，导致网络图绘制错误。务必仔细核对直接前导列；不要添加不必要的依赖关系。另一个典型错误是使用错误的总浮动计算公式——常有同学用 EFT 减去 EST，得到的是持续时间而非浮动时间。应始终使用 LST – EST 或 LFT – EFT。

During the backward pass, forgetting to set the LFT of all end activities to the maximum EFT (project duration) is a serious mistake. Also, students sometimes wrongly assume the last activity in the list is the end node; check the network structure to identify all activities that have no successors.

在后向扫描中，忘记将所有结束活动的 LFT 设为最大 EFT（项目持续时间）是一个严重错误。此外，有同学习惯性地认为列表中的最后一个活动就是终点节点；应检查网络结构以找出所有没有后继的活动。

In scheduling with resource constraints, avoid moving an activity beyond its total float – this would delay the whole project. Always record the resource requirement per day and count overlapping activities carefully. When levelling, prioritise delaying activities with the largest float first, and try to keep the schedule as close to the earliest start as possible.

在资源约束下调度时，避免将活动移动超过其总浮动时间——这会导致整个项目延迟。务必记录每天的资源需求，仔细计数重叠的活动。进行平衡时，优先推迟浮动时间最大的活动，并尽量使调度接近最早开始时间。

11. Shortcuts for Efficiency in Exam Settings | 考场中的高效技巧

When drawing the network, use a clear layout, placing nodes in roughly left-to-right order following the flow of time. Label durations inside nodes as you read the table. For the forward pass, start from the source and work left to right; for the backward pass, work right to left. Perform the pass on all nodes even if some values seem obvious – this systematic approach eliminates arithmetic mistakes.

画网络图时，布局要清晰，大致按时间流向从左到右放置节点。读表时直接在节点内标注持续时间。前向扫描从源点开始从左到右进行；后向扫描从右到左进行。即使某些值看似明显，也要对所有节点执行扫描——这种系统化方法可消除计算错误。

If the question only asks for the critical path and project duration, you can skip redundant calculations: the longest path (by duration) from start to finish is the critical path. However, the full time analysis is often required. To save time, calculate float only for activities that appear non-critical at a glance – but always verify.

如果题目只要求关键路径和项目持续时间，你可以跳过冗余计算：从起点到终点持续时间之和最长的路径就是关键路径。然而，通常要求完整的时间分析。为节省时间，可只对那些一眼看去非关键的活动计算浮动时间——但一定要验证。

When resource levelling, draw a timeline with each day as a column, and fill in the workers required by each activity on the days it runs. This visual table helps avoid counting errors and makes it easy to shift activities by erasing and rewriting numbers in allowed intervals.

进行资源平衡时，画一个时间表，以每一天为一列，填入当天各活动所需的工人数。这种可视化表格有助于避免计数错误，并且方便在允许区间内通过擦写数字来平移活动。

12. Worked Mini-Example | 小型示例演练

**Precedence Table | 优先表**
Activity \| 活动	Duration (days) \| 工期	Predecessors \| 前导活动
A	4	–
B	5	–
C	3	A
D	6	A
E	2	B, C

Network construction: A and B start from the source. C and D depend only on A, so both follow A. E depends on B and C, so E receives arrows from B and C. Note that B and C are independent, so no dummy is needed in AON. The end node is the sink after D and E.

网络构建：A 和 B 从源点出发。C 和 D 仅依赖于 A，所以都在 A 之后。E 依赖于 B 和 C，所以 E 接收来自 B 和 C 的箭头。注意 B 和 C 是独立的，在 AON 法中无需虚活动。终点节点在 D 和 E 之后汇合。

Forward pass: A (EST 0, EFT 4); B (EST 0, EFT 5); C (EST 4, EFT 7); D (EST 4, EFT 10); E (EST max(5,7)=7, EFT 9). Project duration = max(10,9) = 10 days.

前向扫描：A (EST 0, EFT 4)；B (EST 0, EFT 5)；C (EST 4, EFT 7)；D (EST 4, EFT 10)；E (EST max(5,7)=7, EFT 9)。项目工期 = max(10,9) = 10 天。

Backward pass: Project LFT = 10. D (LFT 10, LST 4); E (LFT 10, LST 8); C (LFT min(LST of E)=8, LST 5); B (LFT min(LST of E)=8, LST 3); A (LFT min(LST of C=5, LST of D=4)=4, LST 0).

后向扫描：项目 LFT = 10。D (LFT 10, LST 4)；E (LFT 10, LST 8)；C (LFT min(E 的 LST)=8, LST 5)；B (LFT min(E 的 LST)=8, LST 3)；A (LFT min(C 的 LST=5, D 的 LST=4)=4, LST 0)。

Total float: A (0), B (3), C (1), D (0), E (1). Critical path: A – D (float 0 on both). Project duration 10 days. A delay in A or D would delay the project; a delay of up to 3 days in B would not affect overall completion.

总浮动：A (0), B (3), C (1), D (0), E (1)。关键路径：A – D（两者浮动均为 0）。项目工期 10 天。A 或 D 的延迟会推迟整个项目；B 甚至可延迟至多 3 天而不影响总完工时间。

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Edexcel Maths: Critical Path Analysis Revision Notes | Edexcel 数学：关键路径分析 考点精讲