IB Math: Critical Path Analysis Key Concepts | IB 数学：关键路径分析考点精讲

📚 IB Math: Critical Path Analysis Key Concepts | IB 数学：关键路径分析考点精讲

Critical Path Analysis (CPA) is a powerful project management tool used to schedule complex tasks and identify the minimum time needed to complete a project. Within the IB Mathematics curriculum, particularly in the Applications and Interpretation course, CPA appears as a practical application of graph theory. Understanding how to construct activity networks, compute earliest and latest start times, and determine the critical path not only helps you score well in exams but also equips you with a transferable skill for real-life planning.

关键路径分析（CPA）是一种强大的项目管理工具，用于安排复杂任务并确定完成项目所需的最短时间。在IB数学课程中，尤其是“应用与解释”方向，CPA作为图论的实际应用出现。理解如何构建活动网络、计算最早和最晚开始时间以及确定关键路径，不仅有助于在考试中取得高分，还能让你掌握一种可迁移到现实生活中的规划技能。

1. Project Activities and Precedence | 项目活动与前置关系

Before any network can be drawn, a project must be broken down into individual activities. Each activity has a duration and may depend on the completion of other activities. These dependencies are called precedence constraints. For example, you cannot paint a wall (activity B) until you have built it (activity A). Precedence is often recorded in a table showing each activity, its duration, and its immediate predecessors.

在绘制任何网络之前，必须将项目分解为单独的活动。每项活动都有一个持续时间，并且可能依赖于其他活动的完成。这些依赖关系称为前置约束。例如，在砌好墙（活动A）之前，你不能粉刷墙壁（活动B）。前置关系通常记录在表格中，显示每项活动、其持续时间以及紧前活动。

Activity: A specific task; Duration: time required (e.g., hours, days); Predecessor: an activity that must finish before this one can start.
活动：一项具体的任务；持续时间：所需时间（例如小时、天）；紧前活动：必须在此活动开始前完成的活动。

In IB exam problems, you are typically given a precedence table. Always check for activities that share the same predecessors or can run in parallel. This helps when later constructing the network and removing redundant dependencies.

在IB考试题目中，通常会给出一个前置关系表。务必检查是否有共有相同紧前活动或可以并行进行的活动。这有助于后续构建网络并去除冗余依赖。

2. Activity-on-Node Representation | 节点表示活动法

IB Mathematics predominantly uses the activity-on-node (AON) method. In this representation, each node (usually a rectangle) represents an activity. Arrows (directed edges) show the dependencies between activities. The network is directed and acyclic, meaning there can be no loops. A project has a single start activity and a single end activity; if not, dummy start or end nodes are added with zero duration.

IB数学主要使用节点表示活动法（AON）。在这种表示中，每个节点（通常是一个矩形）代表一项活动。箭头（有向边）表示活动之间的依赖关系。网络是有向且无环的，这意味着不能有循环。项目只有一个开始活动和一个结束活动；如果不是，就需要添加持续时间为零的虚拟开始或结束节点。

Node format: each node is divided into sections showing activity name, duration, earliest start time (EST), earliest finish time (EFT), latest start time (LST), and latest finish time (LFT). In IB, you often fill these during forward and backward passes.
节点格式：每个节点被分为几个部分，显示活动名称、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）。在IB中，你通常在进行正向和反向计算时填写这些值。

A typical node diagram looks like:

一个典型的节点图如下所示：

EST | Duration | EFT
—————————–
LST | Activity | LFT

Correctly drawing the AON network from a precedence table is the first step. Always place activities that have no predecessors after the start node, and ensure every arrow correctly reflects a prerequisite.

根据前置关系表正确绘制AON网络是第一步。总是将没有紧前活动的活动放在开始节点之后，并确保每条箭头正确反映前置条件。

3. Forward Pass: Earliest Start and Earliest Finish Times | 正向计算：最早开始和最早完成时间

The forward pass calculates the earliest time each activity can begin and end, assuming all previous activities start as soon as possible. You move from the start node (with EST = 0) to the finish node. For an activity, the EST is the maximum of the EFTs of all its immediate predecessors. Its EFT is then EST + duration.

正向计算确定每项活动可以开始和结束的最早时间，假设所有前置活动都尽早开始。你从开始节点（EST = 0）移动到结束节点。对于一项活动，EST是其所有紧前活动的EFT中的最大值。然后，其EFT = EST + 持续时间。

If an activity has no predecessors, its EST is 0.
如果活动没有紧前活动，其EST为0。
For multiple predecessors, EST = max(EFT of predecessors).
对于多个紧前活动，EST = 紧前活动EFT的最大值。

Forward pass example: Activity A (duration 4) -> B (duration 5) depends on A; A’s EFT = 4, so B’s EST = 4, EFT = 9. If activity C (duration 3) also depends on A but starts after A, then C’s EST = 4, EFT = 7.

正向计算示例：活动A（持续时间4）-> B（持续时间5）依赖于A；A的EFT = 4，因此B的EST = 4，EFT = 9。如果活动C（持续时间3）也依赖于A但必须在A之后开始，那么C的EST = 4，EFT = 7。

Always compute the forward pass systematically, one node at a time, following the arrows. The EFT of the final activity gives the project’s minimum completion time.

始终按照箭头的方向、逐个节点系统地计算正向通行。最终活动的EFT给出项目的最短完成时间。

4. Backward Pass: Latest Start and Latest Finish Times | 反向计算：最晚开始和最晚完成时间

The backward pass determines the latest time each activity can start and finish without delaying the entire project. You begin at the final activity, setting its LFT equal to its EFT (the project’s minimum duration) and its LST = LFT – duration. Then move in reverse, from right to left. For an activity, the LFT is the minimum of the LSTs of all activities that directly follow it. The LST is then LFT – duration.

反向计算确定每项活动在不延误整个项目的前提下可以开始和结束的最晚时间。你从最后一项活动开始，将其LFT设置为等于其EFT（项目的最短持续时间），其LST = LFT – 持续时间。然后反向移动，从右向左。对于一项活动，LFT是其所有直接后继活动的LST中的最小值。然后LST = LFT – 持续时间。

For the end activity, LFT = EFT.
对于结束活动，LFT = EFT。
For multiple successors, LFT = min(LST of successors).
对于多个后继活动，LFT = 后继活动LST的最小值。

Backward pass example: Suppose final activity D has EFT=12, LFT=12, duration 2 -> LST=10. Preceding activity B (duration 5) leads only to D, so B’s LFT = D’s LST = 10; B’s LST = 10-5 = 5.

反向计算示例：假设最后活动D的EFT=12，LFT=12，持续时间2 -> LST=10。紧前活动B（持续时间5）只指向D，因此B的LFT = D的LST = 10；B的LST = 10-5 = 5。

The backward pass requires careful attention when activities have multiple successors. Always take the smallest LST among them to avoid delaying the project.

当活动有多个后继时，反向计算需要仔细注意。始终取它们中最小的LST，以避免延误项目。

5. 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 overall project duration. It is calculated as Total Float = LST – EST or equivalently LFT – EFT. Both formulas must give the same result if computed correctly. An activity with zero total float is critical; any delay in a critical activity directly delays the project finish.

总浮动时间是一项活动可以延迟而不影响整个项目工期的时间量。计算公式为：总浮动时间 = LST – EST，或等价地 LFT – EFT。如果计算正确，两个公式必须给出相同结果。总浮动时间为零的活动是关键活动；关键活动的任何延误都会直接导致项目完工延迟。

If EST=4, LST=4, total float = 0 → critical.
如果EST=4，LST=4，总浮动=0 → 关键。
If EST=4, LST=7, total float = 3 → activity can be delayed by up to 3 units.
如果EST=4，LST=7，总浮动=3 → 活动最多可延迟3个时间单位。

Total float is a measure of scheduling flexibility. IB questions often ask you to calculate this value and then use it to determine which activities are critical.

总浮动时间是衡量进度灵活性的指标。IB题目经常要求你计算该值，然后用它来确定哪些活动是关键活动。

6. The Critical Path and Its Significance | 关键路径及其意义

The critical path is the longest path through the network in terms of total duration. It determines the minimum project completion time. All activities on this path have zero total float. A project can have multiple critical paths if several paths share the same maximum length. Identifying the critical path allows managers to focus resources on those activities that cannot slip.

关键路径是网络中总持续时间最长的路径。它决定了项目的最短完成时间。此路径上的所有活动总浮动时间为零。如果多条路径具有相同的最大长度，项目可以有多条关键路径。识别关键路径使管理者能够将资源集中在那些不能延误的活动上。

To find the critical path, list all activities with total float = 0. Trace from start to finish along these activities.
要找到关键路径，列出所有总浮动时间为零的活动。从开始到结束沿着这些活动追溯。

Example: In a simple project, if activities A – C – E – G have duration sum 20 and all have float 0, and any other path sums to less than 20, then A–C–E–G is the critical path and 20 is the minimum completion time.

示例：在一个简单项目中，如果活动A – C – E – G的持续时间总和为20且全部浮动为0，而任何其他路径总和小于20，则A–C–E–G为关键路径，20是最短完成时间。

7. Interpreting and Using Float Information | 解读和运用浮动时间信息

Float information is not just a number; it tells the project manager how much leeway exists for each sub-task. Total float is shared among activities on the same non-critical chain. IB questions may ask you to recompute the schedule if an activity is delayed by a certain amount, or to determine whether a given delay affects the critical path and the project finish.

浮动时间不仅仅是一个数字；它告诉项目经理每个子任务有多少余地。总浮动时间在同一个非关键链上的活动之间是共享的。IB题目可能会要求，如果某项活动延误一定时间，重新计算进度安排，或者判断给定的延误是否会影响关键路径和项目完成时间。

If an activity has total float of 4 and a delay of 3 occurs, the project finish remains unchanged. A delay of 5, however, changes the critical path and extends the project by (5-4)=1 unit. This reasoning is commonly examined.

如果一项活动总浮动为4，发生3个单位的延误，项目完成时间不变。然而，延误5个单位会改变关键路径，并使项目延长（5-4）=1个单位。这种推理经常被考查。

Additionally, IB may mention “free float” – the time an activity can be delayed without affecting the early start of its immediate successors. Although free float is less emphasized than total float, it is a useful concept for scheduling.

此外，IB可能会提到“自由浮动时间”——一项活动可以延误而不影响其紧后活动最早开始的时间。虽然自由浮动不如总浮动强调得多，但它是进度安排中的一个有用概念。

8. Dealing with Multiple Predecessors and Successors | 处理多个紧前和紧后活动

Complex networks often involve activities that must wait for several others to finish, or that feed into multiple subsequent tasks. When an activity has two or more predecessors, its EST is the maximum of their EFTs (as seen in forward pass). When an activity has multiple successors, its LFT is the minimum of their LSTs (backward pass).

复杂网络经常涉及必须等待多个其他活动完成的活动，或者输送到多个后续任务的活动。当一项活动有两个或更多紧前活动时，其EST是它们EFT的最大值（如正向计算所示）。当一项活动有多个后继时，其LFT是它们LST的最小值（反向计算）。

Scenario	Forward Pass Rule	Backward Pass Rule
Multiple predecessors	EST = max(all preceding EFTs)	N/A
Multiple successors	N/A	LFT = min(all succeeding LSTs)

These rules ensure the network logic holds: an activity cannot start until all prerequisites are met, and must finish in time to allow the most constrained successor to start on its latest start time.

这些规则确保网络逻辑成立：一项活动在所有前提条件满足之前无法开始，并且必须及时完成，以使最受约束的后继活动能够在其最晚开始时间开始。

9. Gantt Charts (Cascade Charts) | 甘特图（级联图）

IB often pairs critical path analysis with Gantt charts, also known as cascade charts. A Gantt chart visualises the schedule by showing each activity as a horizontal bar positioned according to its EST (for earliest start schedule) or LST (for latest start schedule). It gives an intuitive view of overlaps and resource requirements.

IB经常将关键路径分析与甘特图（也称为级联图）结合起来考查。甘特图通过将每项活动显示为水平条形图，根据其EST（最早开始进度）或LST（最晚开始进度）定位，直观地展示进度安排。它提供了关于重叠和资源需求的直观视图。

Bars are drawn with length proportional to activity duration. The start of the bar is at EST or LST.
条形图的长度与活动持续时间成正比。条形的起点在EST或LST处。

Given a network, you may be asked to sketch a Gantt chart for the earliest start times and identify times when certain resources (e.g., a worker) are needed. This connects CPA to resource levelling.

给定一个网络，你可能需要绘制最早开始时间的甘特图，并指出某些资源（例如工人）需要的时间段。这将CPA与资源平衡联系起来。

10. Resource Levelling and Scheduling | 资源平衡与调度

Critical path analysis identifies the time constraint, but real projects also have resource limits (e.g., number of workers available). Resource levelling involves shifting non-critical activities within their float to smooth the demand for resources over time. IB problems may present a resource histogram and ask how to adjust the schedule to avoid over-allocation.

关键路径分析确定了时间约束，但实际项目也有资源限制（例如可用工人数量）。资源平衡涉及在浮动时间内移动非关键活动，以平滑随时间变化的资源需求。IB题目可能给出资源直方图，并询问如何调整进度以避免分配过度。

Example: If only 3 workers are available and the Gantt chart based on ESTs requires 5 workers on day 4, you can delay low-float activities with high labour demands to later start times as long as total float is not exceeded. The critical path must remain untouched.

示例：如果只有3名工人可用，而基于EST的甘特图在第4天需要5名工人，你可以将浮动时间较长且需要大量劳动力的活动推迟到更晚的开始时间，只要不超过总浮动时间。关键路径必须保持不变。

This topic is a higher-order skill often tested in longer IA-style questions but also appears in Paper 1 or 2 for analysis.

这个主题是一项较高层次的技能，经常在较长的IA风格题目中测试，但也出现在试卷1或2中进行分析。

11. Dummy Activities and Network Adjustments | 虚拟活动与网络调整

While IB primarily uses activity-on-node networks, occasionally you encounter activity-on-arc diagrams (where activities are represented by arrows and nodes represent events). In these diagrams, dummy activities – shown as dashed arrows with zero duration – are used to maintain correct precedence logic without adding real work. A dummy may be required when two activities share some but not all predecessors.

尽管IB主要使用节点表示活动网络，偶尔也会遇到弧表示活动图（其中活动由箭头表示，节点表示事件）。在这些图中，虚拟活动——显示为虚线箭头且持续时间为零——用于在不增加实际工作的情况下维护正确的前置逻辑。当两项活动共享部分而非全部紧前活动时，可能需要虚拟活动。

If activity C depends on activity A only, and activity D depends on both A and B, but you need to show D’s dependency on B without implying C depends on B. A dummy from B to the start of D can solve this.
如果活动C只依赖于活动A，而活动D同时依赖于A和B，但你需要表示D对B的依赖而不暗示C也依赖B。从B到D起点的虚拟活动可以解决这个问题。

In AON, dummy activities are less common because every node is an activity; you simply draw arrows from all relevant predecessors. Nevertheless, spotting unnecessary dependencies is an exam skill: avoid drawing arrows directly between nodes that are already linked via another path.

在AON中，虚拟活动较少见，因为每个节点都是一项活动；你只需从所有相关的紧前活动引出箭头。然而，识别不必要的依赖关系是一项考试技能：避免在已经通过另一条路径连接起来的节点之间直接绘制箭头。

12. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often lose marks due to simple arithmetic errors in the forward/backward passes. Always double-check that you have selected the correct max or min values. Another common pitfall is misreading the precedence table, which leads to a fundamentally wrong network. Take time to verify that every dependency is represented by an arrow and no extra arrows are added.

学生常因正向/反向计算中的简单算术错误而失分。务必反复检查你是否选择了正确的最大值或最小值。另一个常见陷阱是误读前置关系表，这会导致网络从根本上错误。花时间验证每个依赖关系都由箭头表示，并且没有添加多余的箭头。

Tip 1: After drawing the network, count the number of arrows; it should match the number of predecessor relationships.
技巧1：绘制网络后，计算箭头数量；它应与前置关系的数量相匹配。
Tip 2: For forward pass, write EST and EFT lightly in pencil first; re-check max calculations.
技巧2：正向计算时，先用铅笔轻写EST和EFT；重新检查最大值计算。
Tip 3: The critical path must form a continuous chain from start to finish. If disconnected, there’s an error.
技巧3：关键路径必须形成从开始到结束的连续链条。如果断开，就有错误。

When asked to determine the effect of a delay, compare the delay amount to the total float of that activity. If the delay ≤ total float, no effect on project duration. Otherwise, the excess delay extends the project. Write your reasoning clearly to gain method marks.

当被要求确定延误的影响时，将延误量与该项活动的总浮动时间进行比较。如果延误 ≤ 总浮动时间，对项目工期无影响。否则，超出部分会延长项目。清晰写出你的推理以获得方法分。

Published by TutorHao | IB Math Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

IB Math: Critical Path Analysis Key Concepts | IB 数学：关键路径分析 考点精讲