📚 Critical Path Analysis for AS Mathematics | AS 数学：关键路径分析考点精讲
Critical Path Analysis (CPA) is a project management technique used to schedule a set of tasks and identify the minimum time needed to complete a project. In AS Mathematics, particularly within the Decision 1 module, you are expected to model projects using activity networks, perform forward and backward passes, calculate floats, and determine the critical path. This article covers all essential exam points with clear bilingual explanations.
关键路径分析是一种用于安排一系列任务并确定项目完成所需最短时间的项目管理技术。在AS数学（通常在决策数学D1模块）中，你需要使用活动网络对项目建模，执行向前和向后遍历，计算浮动时间，并确定关键路径。本文通过清晰的中英双语讲解，覆盖所有必要的考点。
1. What is Critical Path Analysis? | 什么是关键路径分析？
Critical Path Analysis breaks a project down into activities, each having a duration and dependencies. By modelling these as a network, we can find the earliest and latest times that each activity can start and finish without delaying the overall project. The ‘critical path’ is the longest sequence of dependent activities from start to finish – any delay on this path directly delays the project.
关键路径分析将项目分解为若干活动，每个活动都有持续时间与依赖关系。通过将这些活动建模为网络，我们可以找出每个活动在不延误整个项目情况下的最早和最晚开始与结束时间。“关键路径”是从开始到结束的、依赖关系最长的活动序列——这条路径上的任何延迟都会直接影响项目的总工期。
2. Activity Networks and Precedence Tables | 活动网络与前导表
An activity network is a diagram that shows the order in which tasks must be completed. In AS Decision Maths we mainly use Activity on Node (AoN) networks, where each node (box) represents an activity containing its name and duration. Arrows show the dependencies – an arrow from A to B means A must finish before B can start. The project data is usually given in a precedence table.
活动网络是一种显示任务完成顺序的图。在AS决策数学中，我们主要使用节点活动网络(AoN)，其中每个节点（方框）代表一个活动，包含活动名称和持续时间。箭线表示依赖关系——从A指向B的箭头意味A必须完成后B才能开始。项目数据通常以前导表的形式给出。
A simple precedence table might look like this:
一个简单的前导表可能如下所示：
| Activity | Duration | Predecessors |
|---|---|---|
| A | 3 | – |
| B | 4 | A |
| C | 2 | A |
| D | 5 | B, C |
From this table we can build the network. Notice that A has no predecessors, so it is at the start. B and C both depend on A. Activity D depends on both B and C, so D can only begin after both have finished.
根据此表我们可以构建网络。注意A没有前驱活动，所以位于开始处。B和C都依赖于A。活动D同时依赖于B和C，因此D只能在B和C都完成后开始。
3. Dummy Activities in Network Diagrams | 网络图中的虚活动
Sometimes a dependency cannot be drawn using only activity arrows without breaking the network’s logic. A dummy activity (often shown as a dashed arrow) is used to represent such logical constraints. It has zero duration and is not a real task, but it ensures correct ordering. In AoN networks, dummies may appear as arrows between nodes that clarify unique dependencies.
有时仅靠活动箭线会破坏网络逻辑，无法正确表示依赖关系。虚活动（通常用虚线箭头表示）就用于表示这种逻辑约束。它的持续时间为零，不是真实任务，但能确保正确的顺序。在AoN网络中，虚活动可以表现为节点之间阐明特定依赖的箭头。
For example, if B depends on A, C depends on A, and D depends on both B and C, but E depends only on B while F depends on B and C, you might need a dummy to avoid implying that F depends on C through a redundant arrow. The key rule: every activity must be uniquely identified, and dependencies must not create unnecessary logic chains.
例如，若B依赖于A，C依赖于A，D依赖于B和C，而E只依赖于B，F依赖于B和C，你可能需要虚活动以避免通过冗余箭头暗示F也依赖于C。关键规则：每个活动必须能被唯一识别，依赖关系不得产生不必要的逻辑链条。
4. Forward Pass: Earliest Start and Finish Times | 向前遍历：最早开始与完成时间
The forward pass calculates the earliest start time (EST) and earliest finish time (EFT) for each activity, working from the start to the end of the network. The EST of the first activity is 0. For any activity, its EST is the maximum EFT of all its immediate predecessors. Then EFT = EST + duration.
向前遍历从网络起点向终点方向计算每个活动的最早开始时间(EST)和最早完成时间(EFT)。第一个活动的EST为0。对于任何活动，其EST等于所有紧前活动EFT的最大值。然后EFT = EST + 持续时间。
Using the table above: A has EST=0, EFT=0+3=3. B depends on A, so EST(B)=3, EFT(B)=3+4=7. C depends on A, giving EST(C)=3, EFT(C)=3+2=5. D depends on B and C, so EST(D)=max(7,5)=7, EFT(D)=7+5=12. The project duration is the maximum EFT at the end: 12.
以上表为例：A的EST=0, EFT=0+3=3。B依赖于A, 所以EST(B)=3, EFT(B)=3+4=7。C依赖于A, 得EST(C)=3, EFT(C)=3+2=5。D依赖于B和C, 因此EST(D)=max(7,5)=7, EFT(D)=7+5=12。项目工期为终点活动EFT的最大值：12。
5. Backward Pass: Latest Start and Finish Times | 向后遍历：最晚开始与完成时间
The backward pass begins at the last activity of the project. The latest finish time (LFT) of the final activity is set equal to its EFT (or the project’s minimum completion time). Then we work backwards: for an activity, its LFT is the minimum of the latest start times (LST) of all activities that immediately follow it. The LST is then LFT – duration.
向后遍历从项目的最后一个活动开始。最终活动的最晚完成时间(LFT)设为其EFT（或项目最短工期）。然后我们反向推导：一个活动的LFT等于其所有紧后活动最晚开始时间(LST)的最小值。LST = LFT – 持续时间。
In the example: D has LFT=12, LST=12-5=7. For B: its only successor is D, so LFT(B)=LST(D)=7, LST(B)=7-4=3. For C: LFT(C)=LST(D)=7, LST(C)=7-2=5. For A: successors B and C, so LFT(A)=min(LST(B), LST(C))=min(3,5)=3, LST(A)=3-3=0. These values match the forward pass, confirming consistency.
示例中：D的LFT=12, LST=12-5=7。对于B：其唯一后继是D，因此LFT(B)=LST(D)=7, LST(B)=7-4=3。对于C：LFT(C)=LST(D)=7, LST(C)=7-2=5。对于A：后继为B和C，因此LFT(A)=min(LST(B), LST(C))=min(3,5)=3, LST(A)=3-3=0。这些值与前向遍历一致，验证了正确性。
6. Total Float and Free Float | 总浮动时间与自由浮动时间
Total float is the amount of time an activity can be delayed without delaying the whole project. It is calculated as: Total Float = LST – EST (or equivalently LFT – EFT). A large total float means the activity has scheduling flexibility.
总浮动时间是指一个活动可以延迟而不影响整个项目完成的时间量。计算公式为：总浮动 = LST – EST（等同于 LFT – EFT）。总浮动时间较大，意味着该活动在调度上有灵活性。
Free float is the delay possible without affecting the earliest start of any succeeding activity. Free float = EST(successor) – EFT – any lag, but for standard networks often free float = min(EST of all successors) – EFT. In many exam questions, only total float is required; free float is an extension.
自由浮动时间是指在不影响任何后继活动最早开始时间的情况下，该活动可以延迟的时间。自由浮动 = 后继活动的EST – EFT。在许多考试题中，只需计算总浮动时间，自由浮动是拓展内容。
For activity A: total float = 0-0=0; B: total float = 3-3=0; C: total float = 5-3=2; D: total float = 7-7=0. Activities with zero total float lie on the critical path.
活动A：总浮动 = 0-0=0；B：总浮动 = 3-3=0；C：总浮动 = 5-3=2；D：总浮动 = 7-7=0。总浮动为零的活动位于关键路径上。
7. Identifying the Critical Path | 识别关键路径
The critical path consists of all activities where total float = 0. These activities directly determine the project duration, so any delay to a critical activity postpones the entire project. In our example, the critical path is A → B → D, as these have zero floats. C can be delayed up to 2 time units without affecting the project end date.
关键路径由所有总浮动为0的活动组成。这些活动直接决定项目工期，因此关键活动的任何延迟都会导致整个项目延期。在例子中，关键路径为 A → B → D，它们浮动为零。C可以最多延迟2个时间单位而不影响项目结束日期。
It is important to check that the critical path forms a continuous chain from start to end. In some networks there can be multiple critical paths; all must have the same total duration and zero float throughout.
务必检查关键路径是否形成从起点到终点的连续链条。某些网络可能存在多条关键路径，所有路径需具有相同的总工期，且途中活动浮动均为零。
8. Cascade Charts (Gantt Charts) | 级联图（甘特图）
A cascade chart (or Gantt chart) is a horizontal bar chart that visualises the schedule of activities against time. Each activity is drawn as a bar, starting at its EST and ending at its EFT. The chart can also show floats and dependencies, helping project managers see which activities can be shifted without delaying the project.
级联图（即甘特图）是一种水平条形图，用于按时间展示活动的日程安排。每个活动绘制为一个条形，从它的EST开始，到EFT结束。该图还可以显示浮动时间和依赖关系，帮助项目经理看出哪些活动可以调整而不延误项目。
When drawing a Gantt chart from your CPA results, you should first plot activities on the critical path, then slot in non-critical activities at their earliest times. Often exam questions require you to schedule several workers (resource levelling) by moving non-critical tasks within their floats to avoid overlap.
根据关键路径分析结果绘制甘特图时，应首先标出关键路径上的活动，然后将非关键活动在其最早时间插入。考试题常常要求你在浮动范围内移动非关键任务以均衡资源（例如工人数量），避免重叠。
9. Interpreting Network Outcomes and Scheduling | 解读网络结果与调度
Once the network is fully analysed, you can answer practical questions: the minimum project completion time, which activities cannot be delayed, how many workers are needed at certain times, and how to replan if an activity takes longer than expected. The total float values guide rescheduling to resolve resource conflicts.
一旦网络分析完成，你可以回答实际问题：最短项目完成时间、哪些活动不可延迟、某些时段需要多少工人、若某活动耗时超出预期如何重新规划。总浮动值可用于指导重新调度以解决资源冲突。
For resource levelling, you often consider a histogram showing the number of workers required each day if all activities start at their EST. Then you shift non-critical activities within their total floats to smooth the demand below a given limit. This is a common exam task.
对于资源均衡，你通常需要考察每天所需工人数量的直方图（假设所有活动都在最早时间开始）。然后你在总浮动范围内平移非关键活动，使需求平滑到给定限制以下。这是常见的考题类型。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
Always double-check your precedence table when drawing the network. Missing a dependency or introducing a false one will propagate errors through the entire analysis. Draw the network neatly, label nodes clearly with activity names and durations, and use dummies only when necessary to maintain logical order.
绘制网络时务必再次核对前导表。遗漏一个依赖或引入错误依赖会使整个分析出错。网络要绘制整洁，节点上清晰标注活动名称与持续时间，只在必要时使用虚活动以保持逻辑顺序。
When calculating floats, remember that total float = LST – EST. Do not confuse with free float. For the backward pass, start from the end node with LFT equal to project duration. Make sure you take the minimum LST of successors when setting an activity’s LFT, not the maximum.
计算浮动时，记住总浮动 = LST – EST，不要与自由浮动混淆。向后遍历时，从结束节点开始，将LFT设为项目工期，且设置一个活动的LFT时应取其后继活动LST的最小值，而非最大值。
In scheduling diagrams, clearly indicate the critical activities. If you are asked to produce a worker schedule, show the final timeline with float adjustments. Practice past-paper questions of both AoN and AoA types if your specification requires it, and always label your final answer to the question asked.
在调度图中，要清晰标示关键活动。如果要求你制定工人日程，应展示经过浮动调整后的最终时间线。若考纲要求，请同时练习节点活动网络和箭线活动网络的历年试题，并始终按照题目要求标记最终答案。
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