📚 Critical Path Analysis for A-Level WJEC Mathematics: Essential Revision | A-Level WJEC 数学：关键路径分析 考点精讲
Critical Path Analysis (CPA) is a powerful decision mathematics tool used to model and manage complex projects. By breaking a project into individual activities with known durations and dependencies, CPA allows you to determine the minimum overall completion time, identify which activities must not be delayed, and schedule resources efficiently. For WJEC A-Level Mathematics, mastering this topic means being able to construct activity networks, perform forward and backward passes, calculate floats, interpret Gantt charts, and apply your skills to scheduling problems. This revision guide covers every essential concept with clear explanations and exam-focused tips.
关键路径分析（CPA）是决策数学中用于建模和管理复杂项目的强大工具。通过将项目分解为具有已知持续时间和依赖关系的若干活动，CPA 能够确定最短的总完工时间、识别哪些活动不容延迟，并高效地安排资源。对 WJEC A-Level 数学而言，掌握这一主题意味着能够构建活动网络、执行正推和逆推计算、计算浮动时间、解释甘特图，并将这些技能应用于调度问题。本复习指南将覆盖所有核心概念，提供清晰的解释和紧扣考点的技巧。
1. Introduction to Critical Path Analysis | 关键路径分析简介
Critical Path Analysis helps project managers answer three fundamental questions: How long will the project take? Which tasks directly affect the completion date? How much flexibility exists in non-critical tasks? The technique models a project as a network of activities, where each activity has a duration and depends on other activities finishing before it can start. The sequence of activities that determines the overall project duration is called the critical path. Any delay on this path directly delays the entire project.
关键路径分析帮助项目经理回答三个基本问题：项目需要多长时间？哪些任务直接影响到完成日期？非关键任务有多少灵活性？该技术将项目建模为一个活动网络，其中每个活动有一个持续时间，并依赖于其他活动完成后才能开始。决定整体项目工期的活动序列称为关键路径。该路径上的任何延误都会直接导致整个项目延期。
In WJEC questions, you may be given a table of activities, their durations, and their immediate predecessors. From this information, you must construct an activity network and perform the necessary calculations. The standard notation used is Activity-on-Node (AON), where each node represents an activity and contains fields for recording earliest and latest start/finish times.
在 WJEC 的试题中，你可能会拿到一个包含活动、持续时间及其直接前导活动的表格。根据这些信息，你必须构建一个活动网络并执行必要的计算。标准记法采用节点表示法（AON），其中每个节点代表一个活动，并包含用于记录最早／最迟开始与完成时间的区域。
2. Activity-on-Node (AON) Representation | 节点表示法（AON）
In AON networks, each activity is represented by a rectangular node divided into sections. A standard WJEC node layout has spaces for the activity name (top left), duration (top right or below), and calculated times for earliest start, earliest finish, latest start, and latest finish. Arrows between nodes indicate precedence constraints — directing which activities must be completed before others can begin.
在 AON 网络中，每个活动由一个划分为若干部分的矩形节点表示。标准的 WJEC 节点布局包含活动名称（左上）、持续时间（右上或下方），以及计算得到的最早开始、最早完成、最迟开始和最迟完成时间格。节点之间的箭头表示前后顺序约束——指明哪些活动必须先完成，后面的活动才能开始。
Sometimes the node is shown as:
有时节点表示为：
| Activity letter | Duration |
| EST | LST |
| EFT | LFT |
where EST = earliest start time, LST = latest start time, EFT = earliest finish time, LFT = latest finish time. You will fill these in through the forward and backward passes.
其中 EST = 最早开始时间，LST = 最迟开始时间，EFT = 最早完成时间，LFT = 最迟完成时间。你将通过正推和逆推填入这些数值。
3. Drawing Precedence Networks | 绘制前导网络
To construct a precedence network from a table, first identify all activities and their dependencies. A common approach is to start with activities that have no predecessors (these link to the project start) and then add subsequent activities, ensuring arrows only point from predecessors to successors. You may need to introduce a dummy start node and a dummy end node to tie together multiple starting or ending activities, but in AON format this is rarely necessary because dependencies are shown by direct arrows. However, for clarity, many WJEC diagrams include labelled activity nodes connected in the logical order.
要根据表格构建前导网络，首先识别所有活动及其依赖关系。常见的方法是先画出没有前导的活动（它们连接到项目开始），然后添加后续活动，确保箭头只从前导活动指向后继活动。你可能需要引入虚拟开始节点和虚拟结束节点来连接多个起始或结束活动，但在 AON 格式中这很少必要，因为依赖关系已通过箭头直接显示。不过为清晰起见，许多 WJEC 图表仍包含按逻辑顺序连接的活动节点。
Make sure your diagram respects every precedence rule exactly. Double-check that no activity starts before all its predecessors are complete. In the exam, neatness counts — use a pencil and ruler, and label nodes clearly.
确保你的图表完全遵循每一条前后顺序规则。再次检查是否所有活动都等前导活动完成后才开始。在考试中，卷面整洁很重要——使用铅笔和尺子，并清晰地标记节点。
4. Forward Pass: Earliest Start Times | 正推法：最早开始时间
The forward pass moves from left to right across the network to calculate the earliest possible start and finish times for each activity. For an activity with no predecessors, set EST = 0. Its earliest finish time is EFT = EST + duration. For subsequent activities, EST is the maximum of the EFTs of all its immediate predecessors. Formally:
正推法从左到右遍历网络，计算每个活动可能的最早开始和最早完成时间。对于没有前导的活动，设 EST = 0。其最早完成时间为 EFT = EST + 持续时间。对于后续活动，EST 取所有直接前导活动 EFT 的最大值。公式为：
ESTᵢ = max { EFTₐ : a is a predecessor of i }
EFTᵢ = ESTᵢ + durationᵢ
Continue through the network until you reach the final node(s). The overall project duration is the maximum EFT among all final activities. Record these values in the appropriate cells on your diagram.
继续推进网络直至最终节点。项目总工期是所有最终活动中最大的 EFT。在图中相应格子里记录这些数值。
Example: Activity A (duration 5) has no predecessors, so EST=0, EFT=5. Activity B depends on A; its EST = 5, duration 3 → EFT=8. If activity C also depends on A, its EST = 5, duration 4 → EFT=9. If activity D depends on both B and C, its EST = max(8,9) = 9, and you continue.
示例：活动 A（历时 5）无前导，因此 EST=0，EFT=5。活动 B 依赖于 A；其 EST=5，历时 3 → EFT=8。如果活动 C 也依赖于 A，其 EST=5，历时 4 → EFT=9。若活动 D 同时依赖于 B 和 C，其 EST = max(8,9) = 9，依此类推。
5. Backward Pass: Latest Start Times | 逆推法：最迟开始时间
The backward pass starts from the end of the network and works right to left to find the latest possible times activities can start and finish without delaying the project. First, set the LFT of all final activities equal to the project duration (the maximum EFT from the forward pass). Then, for any activity, the latest start time is LST = LFT − duration. The LFT for earlier activities is the minimum of the LSTs of all its immediate successors:
逆推法从网络末端开始，从右向左计算，以找出活动在不延误项目的前提下最迟可以开始和完成的时间。首先，将所有最终活动的 LFT 设为项目工期（正推法得到的最大 EFT）。然后，对于任一活动，其最迟开始时间为 LST = LFT − 持续时间。前导活动的 LFT 取其所有直接后继活动 LST 的最小值：
LFTᵢ = min { LSTⱼ : j is a successor of i }
LSTᵢ = LFTᵢ − durationᵢ
Continue until you reach the starting activities. At the start, the EST and LST for the initial activities should both be zero; any discrepancy indicates an arithmetic error.
继续回溯，直至起始活动。在起点处，初始活动的 EST 和 LST 应均为零；任何不一致都表明有计算错误。
Example: If project duration is 15, then a final activity E (duration 3) has LFT=15, LST=12. If activity D precedes E, its LFT = LST(E)=12. If duration of D is 4, its LST=8, and so on.
示例：若项目工期为 15，则最终活动 E（历时 3）的 LFT=15，LST=12。若活动 D 是 E 的前导，其 LFT = LST(E)=12。若 D 的历时为 4，其 LST=8，依此类推。
6. Identifying the Critical Path | 确定关键路径
An activity is critical if its total float is zero — meaning any delay in its start will delay the entire project. Critical activities form one or more continuous paths from start to finish. To identify them, compare EST and LST (or EFT and LFT) for each activity. If EST = LST (and hence EFT = LFT), the activity is critical. Highlight these on your network. The critical path is the longest path through the project in terms of total duration; multiple critical paths can exist if there are ties.
如果一项活动的总浮动时间为零——意味着其开始时间的任何延迟都会拖延整个项目——那么该活动就是关键活动。关键活动构成从起点到终点的一条或多条连续路径。要识别它们，请比较每个活动的 EST 与 LST（或 EFT 与 LFT）。若 EST = LST（从而 EFT = LFT），则该活动为关键活动。在网络中醒目标出这些活动。关键路径在总工期上是项目中最长的一条路径；若存在并列情况，可能会有多条关键路径。
In WJEC questions you will often be asked to state the critical path(s) explicitly, e.g., A – C – F – H, and to give the project duration. Always check your arithmetic: the sum of durations along the critical path must equal the overall project time.
在 WJEC 试题中，你经常会被要求明确写出关键路径，例如 A – C – F – H，并给出项目工期。请务必检查计算：沿关键路径的持续时间之和必须等于总项目时间。
7. Calculating Total Float | 计算总浮动时间
Total float (total slack) measures the maximum amount of time an activity can be delayed without affecting the overall project completion date. It is calculated as:
总浮动时间（总时差）衡量在不影响项目整体完成日期的前提下，一项活动可以延迟的最大时间量。计算公式为：
Total Float = LST − EST or LFT − EFT
The two expressions are equivalent because LFT − EFT = (LST + duration) − (EST + duration) = LST − EST. Thus, total float is the same whether you use start or finish times.
这两个表达式是等价的，因为 LFT − EFT = (LST + 持续时间) − (EST + 持续时间) = LST − EST。因此，无论使用开始时间还是完成时间，总浮动时间都相同。
Activities with positive total float can be delayed by up to that amount without delaying the project. Zero total float identifies critical activities. In WJEC problems, you may also be asked to compute independent float or interfering float, so be aware of these terms, but total float is the primary focus.
具有正总浮动时间的活动可以在不延误项目的情况下推迟最多等于浮动时间的时间。零总浮动时间标识关键活动。在 WJEC 问题中，你可能会被要求计算独立浮动时间或干涉浮动时间，因此要熟悉这些术语，但总浮动时间是主要重点。
8. Interpreting Float and Critical Activities | 理解浮动时间与关键活动
Float information is crucial for project management. Delaying a critical activity by even one day will push back the project finish. Non-critical activities have a buffer, but using up buffer on one activity may reduce the float available to subsequent activities. This is why critical path analysis helps managers decide where to allocate extra resources — if you need to shorten the project duration, you must reduce the duration of one or more critical activities (project crashing).
浮动时间信息对项目管理至关重要。哪怕将关键活动延迟一天，都会推迟项目完成。非关键活动有缓冲时间，但在一个活动上用尽缓冲可能会减少后续活动可用的浮动时间。这就是关键路径分析能够帮助管理者决定额外资源分配方向的原因——若需缩短项目工期，你必须减少一项或多项关键活动的持续时间（项目快进）。
When interpreting networks, always relate float to real-world implications. For example, if an activity has total float of 2 days, its start can be delayed by up to 2 days without affecting the project deadline, provided no other delays occur. However, if a preceding activity uses some float, the available float for later activities may shrink.
在解读网络图时，始终要将浮动时间与实际意义联系起来。例如，如果一项活动的总浮动时间为 2 天，则其开始时间可最多推迟 2 天而不影响项目截止时间，前提是没有其他延迟发生。然而，若前导活动使用了部分浮动时间，后续活动的可用浮动时间可能会变少。
9. Gantt Charts (Cascade Charts) | 甘特图（瀑布图）
WJEC often asks you to draw a Gantt chart (also called a cascade chart) based on the results of your network analysis. A Gantt chart represents each activity as a horizontal bar on a time scale, positioned at its earliest start time and lasting for its duration. Critical activities are usually drawn in one colour or shading, with non-critical activities shown in another. Floating time can be indicated as a dotted extension or a separate bar at the end.
WJEC 经常要求你根据网络分析结果绘制甘特图（也称瀑布图）。甘特图以水平条形图的样式将每项活动绘制在时间轴上，条形图的起点为最早开始时间，长度对应其持续时间。关键活动通常用一种颜色或阴影绘制，非关键活动则用另一种。浮动时间可用虚线延伸段或条形图末尾的另一段来表示。
To draw a Gantt chart, first list activities and their ESTs and durations. Draw a horizontal time axis; then, for each activity, draw a bar from EST to EST+duration. Label each bar clearly. You may be asked to illustrate a particular resource schedule or demonstrate when an activity could be shifted within its float.
要绘制甘特图，首先列出活动及其 EST 与持续时间。画出水平时间轴；然后为每个活动从 EST 到 EST+持续时间画一个条形。清晰地标记每个条形。你可能需要说明特定的资源调度，或演示某项活动如何在其浮动范围内进行移动。
Gantt charts are particularly useful for visualizing parallel activities and for resource levelling exercises. They give an immediate picture of project progress and slack.
甘特图对于并行活动的可视化以及资源平衡练习特别有用。它们能够直观地展示项目进展和时差。
10. Resource Levelling and Scheduling | 资源平衡与调度
Projects often have limited resources, such as workers or machines, and cannot have too many activities running simultaneously. Resource levelling aims to minimise the peak resource requirement by shifting non-critical activities within their floats without extending the project duration. You start with an initial resource histogram showing resource usage over time when all activities start at their EST. Then you identify periods where demand exceeds supply and delay selected non-critical activities (up to their total float) to smooth the resource profile.
项目通常资源有限，如工人或机器，不能同时开展过多活动。资源平衡的目标是通过在其浮动范围内移动非关键活动，在不超过项目工期的前提下，将资源需求峰值降到最低。你首先创建一个初始资源直方图，显示当所有活动均从其 EST 开始时的资源使用情况随时间的变化。然后识别需求超过供给的时段，并将选定的非关键活动推迟（最多推迟其总浮动时间）以平滑资源曲线。
WJEC questions may ask you to draw a resource histogram or to suggest a revised schedule that resolves an over-allocation. Always check that after levelling, no activities start before their predecessors and no critical activities are delayed.
WJEC 试题可能会要求你绘制资源直方图，或提出解决超分配的修正计划。务必检查在平衡后，没有任何活动早于其前导活动开始，且没有关键活动被推迟。
11. Exam Tips for WJEC | WJEC 考试技巧
– Always read the precedence table carefully and highlight ‘must follow’ relationships. Misreading predecessors is a common mistake that affects the entire network.
– 始终仔细阅读前导关系表，并标出“必须紧随”的关系。误读前导活动是影响整个网络的常见错误。
– Summarise your critical path and project duration with a clear statement. Exam questions often award marks just for stating these correctly.
– 用清晰的语句总结关键路径和项目工期。考试中仅仅正确陈述这些内容通常就能得分。
– Keep your network neat. Use a pencil so you can correct errors. Illegible diagrams may lose marks.
– 保持网络图整洁。使用铅笔以便修正错误。潦草的图可能会失分。
– Show all working in the node cells. Even if your final numbers are wrong, method marks are available for correct forward/backward pass logic.
– 在节点格中展示所有运算过程。即使最终数字有误，正确的前向/后向算法逻辑仍可获得方法分。
– For Gantt charts, ensure bars align exactly with the time scale and activities are identifiable at a glance.
– 对于甘特图，确保条形与时间轴精确对齐，活动一目了然。
– When levelling resources, redraw the histogram or clearly state the new start times and the reduced peak demand.
– 平衡资源时，重新绘制直方图，或明确说明新的开始时间和降低后的峰值需求。
– Double-check that the duration along the critical path sums to the project completion time; this acts as an arithmetic check.
– 再次检查关键路径上的工期之和是否等于项目完成时间；这可以作为算术检验。
12. Summary | 总结
Critical Path Analysis transforms complex scheduling into a systematic, solvable model. By mastering AON networks, forward and backward passes, float calculations, and Gantt chart representation, you equip yourself to handle any WJEC decision mathematics problem in this area. Remember: the critical path is the sequence of zero-float activities that dictates project length. Use the forward pass to find earliest times, the backward pass for latest times, subtract to obtain floats, and always verify your work. With practice, these steps become second nature, helping you secure top marks and gain a skill that is genuinely useful beyond the exam room.
关键路径分析将复杂的调度转换为一个系统且可求解的模型。通过掌握 AON 网络、正推和逆推计算、浮动时间计算以及甘特图表示法，你将有能力应对任何 WJEC 决策数学中这一领域的问题。记住：关键路径是决定项目长度的零浮动活动序列。用正推法求最早时间，逆推法求最迟时间，相减得到浮动时间，并始终验证工作。通过练习，这些步骤将变得得心应手，助你取得高分，并掌握一项在考场之外真正有用的技能。
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