Critical Path Analysis | 关键路径分析

📚 Critical Path Analysis | 关键路径分析

Critical Path Analysis (CPA) is an essential project management tool that enables you to determine the minimum time required to complete a project, identify the activities that directly influence the overall duration, and plan efficient use of resources. In the OCR A-Level Mathematics specification (Decision), CPA appears as a structured problem-solving topic that combines logical thinking with algorithmic processes. Mastery of this topic can secure a significant number of marks due to its routine, procedural nature.

关键路径分析是一种重要的项目管理工具，它能让你确定完成项目所需的最短时间，识别直接影响总工期的活动，并规划资源的高效利用。在 OCR A-Level 数学（决策模块）中，关键路径分析作为一个结构清晰的解题主题，将逻辑思维与算法过程结合在一起。掌握这一专题可以确保获得大量分数，因为它具有常规的操作流程。

1. Introduction to Critical Path Analysis | 关键路径分析简介

A project is broken down into a collection of activities, each with a given duration and a set of precedence requirements (which activities must be finished before it can start). The two main questions that CPA answers are: What is the earliest possible completion time of the whole project? And which activities cannot be delayed without pushing back this completion time?

项目会被分解为一系列活动，每项活动有给定的持续时间以及一组先后顺序要求（即哪些活动必须先完成，它才能开始）。关键路径分析主要回答两个问题：整个项目最早何时可以完成？哪些活动的延迟会使整个项目延期？

In decision mathematics, we model these projects using activity networks, perform a forward pass to calculate earliest times, a backward pass to determine latest times, and then compute floats to distinguish critical from non‑critical activities.

在决策数学中，我们使用活动网络对项目进行建模，通过前推法计算最早时间，利用后推法确定最迟时间，再计算浮动时间来区分关键活动与非关键活动。

2. Building Activity-on-Node Networks | 构建节点式活动网络

OCR Decision Mathematics primarily uses activity‑on‑node (AoN) networks. In this representation, each activity is drawn as a rectangle, often containing the activity label, and its duration is written beside it (commonly on the right). Arrows between rectangles indicate the dependency: an arrow from activity A to activity B means A must be finished before B can start.

OCR 决策数学主要采用节点式活动网络。在这种表示法中，每项活动画成一个矩形，通常包含活动标签，并在其旁边（一般是右侧）标明持续时间。矩形之间的箭头表示依赖关系：从活动 A 指向活动 B 的箭头意味着 A 必须完成后 B 才能开始。

Every network begins with a single start node and ends with a single end node. The start node has no incoming activities; the end node has no outgoing ones. When reading a precedence table, you connect each activity to those that immediately depend on it. For example, if the table shows: A (duration 5, no predecessors), B (duration 4, no predecessors), C (duration 2, predecessor A), D (duration 1, predecessors A and B), you would draw A and B emerging from the start, then draw an arrow from A to C, and arrows from both A and B into D. Finally, connect all final activities to the end node.

每个网络都从一个起始节点开始，以一个结束节点收尾。起始节点没有引入活动，结束节点没有引出活动。在阅读紧前关系表时，你要把每项活动与那些直接依赖于它的活动连接起来。例如，若关系表显示：A（持续时间5，无紧前），B（持续时间4，无紧前），C（持续时间2，紧前 A），D（持续时间1，紧前 A 和 B），你会画出从起始分别指向 A 和 B，然后从 A 指向 C，从 A 和 B 共同指向 D。最后将所有末端活动连至结束节点。

Precedence table → AoN network: start → A(5), B(4); A → C(2); A,B → D(1); C,D → end

3. Forward Pass: Earliest Start Times | 前推法：最早开始时间

The forward pass calculates the earliest time each activity can start – denoted EST (earliest start time) – and the earliest time the project can finish. The start node is assigned time 0. For any other activity j, its EST is the maximum of (EST of each immediate predecessor + duration of that predecessor). This ensures that all preceding activities have been completed.

前推法计算每项活动的最早开始时间（记为 EST）以及项目的最早完成时间。起始节点分配时间为 0。对于任何其他活动 j，其 EST 等于每个直接前驱活动的 EST 加上该前驱活动的持续时间后取最大值。这保证了所有前置活动都已完成。

EST(j) = max { EST(i) + duration(i) } for all i immediately preceding j

Working from left to right across the network, you fill in the EST for every node. The earliest project completion time is the EST of the end node (which equals the maximum of EST + duration among all terminal activities).

在网络中从左到右推进，为每个节点填上 EST。项目的最早完成时间就是结束节点的 EST（等于所有末端活动 EST + 持续时间中的最大值）。

Consider the previous example with EST(start)=0. For A: EST=0. For B: EST=0. For C: EST = max{EST(A)+5}=5. For D: EST = max{EST(A)+5, EST(B)+4}=max{5,4}=5. End node: EST = max{EST(C)+2, EST(D)+1}=max{5+2,5+1}=7. Hence the project cannot finish earlier than 7 days.

以前面的例子为例，EST(start)=0。A: EST=0，B: EST=0。C: EST = max{EST(A)+5}=5。D: EST = max{EST(A)+5, EST(B)+4}=max{5,4}=5。结束节点: EST = max{EST(C)+2, EST(D)+1}=max{7,6}=7。因此项目最早需要 7 天完成。

4. Backward Pass: Latest Start Times | 后推法：最迟开始时间

The backward pass finds the latest time each activity can start without delaying the overall project – denoted LST (latest start time). You begin at the end node, setting its LST equal to the project’s earliest finish time. Then, moving right to left, the LST of an activity i is the minimum of (LST of each immediate successor minus duration of i).

后推法用于寻找在不延误整个项目的前提下每项活动的最迟开始时间（记为 LST）。从结束节点开始，将其 LST 设为项目的最早完成时间。然后从右向左推进，活动 i 的 LST 等于每个直接后继活动的 LST 减去活动 i 的持续时间后取最小值。

LST(i) = min { LST(j) – duration(i) } for all j immediately following i

Applying the backward pass to the same network: End LST = 7. For C: LST = LST(end) – 2 = 5. For D: LST = LST(end) – 1 = 6. For A: LST = min{ LST(C) – 5, LST(D) – 5 } = min{5 – 5, 6 – 5} = min{0,1} = 0. For B: LST = min{ LST(D) – 4 } = 6 – 4 = 2. Notice that B has some flexibility while A is tightly constrained.

对该网络进行后推法：结束节点 LST = 7。C: LST = 7 – 2 = 5。D: LST = 7 – 1 = 6。A: LST = min{ LST(C) – 5, LST(D) – 5 } = min{5-5, 6-5} = min{0,1} = 0。B: LST = min{ LST(D) – 4 } = 6 – 4 = 2。注意 B 具有一定的灵活性，而 A 则十分紧迫。

5. Total Float and Critical Activities | 总浮动与关键活动

The total float of an activity is the amount of time it can be delayed without affecting the project completion time. It is calculated as the difference between its latest start time and its earliest start time (or equivalently, latest finish minus earliest finish).

一项活动的总浮动是指在不影响项目完成时间的前提下，它可以被延迟的时间量。总浮动等于最迟开始时间减去最早开始时间（或等价地，最迟完成时间减去最早完成时间）。

Total Float(i) = LST(i) – EST(i)

Activities with a total float of zero are called critical activities. Any delay in a critical activity will directly cause an equal delay to the project’s overall finish. In our example, total floats: A: 0 – 0 = 0 (critical), B: 2 – 0 = 2, C: 5 – 5 = 0 (critical), D: 6 – 5 = 1. Hence A and C are critical activities, while B and D have spare time.

总浮动为零的活动被称为关键活动。对关键活动的任何延迟都会直接导致项目整体完成时间的同等延迟。在我们的例子中，总浮动分别为：A: 0 – 0 = 0（关键），B: 2 – 0 = 2，C: 5 – 5 = 0（关键），D: 6 – 5 = 1。因此 A 和 C 是关键活动，而 B 和 D 有缓冲时间。

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

The critical path is a continuous chain of critical activities running from the start node to the end node. It represents the sequence of activities that dictates the overall project duration. There can be more than one critical path if several parallel chains each have zero total float throughout.

关键路径是一条从起始节点到结束节点、由关键活动组成的连续链。它代表了决定项目总持续时间的那条活动序列。如果有多条并行链路上各项活动的总浮动都为零，则可能存在多条关键路径。

From the earlier calculations, the critical path consists of activities A → C. We can verify: start → A(5) → C(2) → end, with total float zero on both A and C. The non‑critical activities B and D show positive floats, meaning they can be rescheduled within their slack time. A common exam requirement is to list the critical path using activity letters and to state its overall length.

由前面的计算可知，关键路径由活动 A → C 组成。我们可以验证：起始 → A(5) → C(2) → 结束，A 和 C 的总浮动均为零。非关键活动 B 和 D 表现出正浮动，意味着它们可以在各自的松驰时间内重新安排。考试中常见的要求是用活动字母列出关键路径，并说明其总长度。

If a revised network yields multiple critical paths, the project is inherently more risky, because a delay on any of those paths will extend the project. Examiners often ask about the implication of multiple critical paths on project sensitivity.

如果修改后的网络产生多条关键路径，那么该项目的风险就更大，因为任何一条关键路径发生延迟都会延长项目。考官常常会问多条关键路径对项目敏感度的含义。

7. Project Scheduling with Gantt Charts | 用甘特图进行项目调度

Once earliest and latest start times are known, a Gantt chart (often called a cascade chart in OCR resources) can be drawn to show when each activity will take place. Activities are represented as horizontal bars spanning their duration. Critical activities are usually placed first at their EST (which equals their LST), while non‑critical activities can be shifted within their float to create a workable schedule.

一旦知道最早和最迟开始时间，就可以画出甘特图（在 OCR 资料中也常称作级联图）来显示每项活动将在何时进行。活动用横跨其持续时间的水平横条表示。关键活动通常先放在其 EST（等于 LST）的位置，而非关键活动可以在其浮动范围内移动，以做出可行的时间安排。

In a Gantt chart, the horizontal axis represents time (often in days). Activities are listed on the vertical axis. For each activity, you draw a bar from its scheduled start time to its scheduled finish time. Critical activities should be scheduled at their earliest to avoid unnecessary risk. Non‑critical activities can be positioned anywhere between their EST and LST, but choosing a later start may make the schedule more efficient in terms of resource usage.

在甘特图中，横轴表示时间（通常以天为单位），纵轴列出各项活动。为每项活动画一条横条，从其计划开始时间到计划结束时间。关键活动应安排在其最早时间以降低不必要的风险。非关键活动可以安排在 EST 和 LST 之间的任意位置，但选择较晚的开始时间可能使资源使用更高效。

8. Resource Histograms and Resource Smoothing | 资源直方图与资源平滑

Projects often have limited resources (workers, machines). A resource histogram displays the number of resources required on each day if all activities start at their earliest times. Where a peak demand exceeds the available level, we may need to delay non‑critical activities – this is called resource smoothing (or resource levelling).

项目通常具有有限的资源（工人、机器）。资源直方图显示了如果所有活动都在最早时间开始时，每一天所需的资源数量。当峰值需求超过可用水平时，我们可能需要延迟非关键活动，这称为资源平滑（或资源平衡）。

To construct a resource histogram, you first schedule all activities at EST. For each time unit, count the total resources needed (e.g. number of workers). Draw a bar chart with time on the horizontal axis and resource units on the vertical axis. If at any point the required resources exceed a given limit, you can shift non‑critical activities (by using some of their float) to reduce the peak. The goal is to stay within the resource constraint while keeping the project duration unchanged, if possible.

要构建资源直方图，首先将所有活动放在 EST 上排程。对每个时间单位，统计所需资源总量（例如工人数量）。画出以时间为横轴、资源数目为纵轴的条形图。如果某一点所需资源超出给定限制，你可以利用非关键活动的部分浮动进行平移，从而降低峰值。目标是在资源约束下尽可能保持项目工期不变。

After smoothing, the histogram often becomes flatter, but total duration remains the same as before. If it is impossible to keep the original duration, the project may need to be extended, which is determined by the smallest float sacrificed. OCR examination questions may ask you to re‑draw the Gantt chart and histogram after smoothing.

平滑后资源直方图通常会变得更加平坦，但总工期保持不变。如果无法保持原工期，项目可能需要延长，这取决于牺牲的最小浮动值。OCR 考试可能会要求你重新画出平滑后的甘特图和直方图。

9. Dummy Activities in Activity-on-Arc Networks | 活动在弧网络中的虚活动

Although OCR primarily uses activity‑on‑node networks, it is useful to be aware of dummy activities when they appear in context. In activity‑on‑arc (AoA) representation, activities are drawn as arcs (arrows) and nodes represent events (points in time). When two activities share the same start and end events but are logically distinct, a dummy activity (shown as a dashed arrow of zero duration) is used to preserve the correct dependencies without introducing extra work.

尽管 OCR 主要使用节点式网络，但在某些情景中了解虚活动仍然有用。在活动在弧的表示法中，活动画成弧线（箭头），节点代表事件（时间点）。当两项活动共享相同的开始和结束事件但在逻辑上不同时，需要使用虚活动（零持续时间的虚线箭头）来维持正确的依赖关系，而不引入额外的工作。

For instance, if activity D depends on A and B, while activity E depends only on B, an AoA network needs a dummy from the completion of B to the start of E to uniquely define the condition. In AoN, this problem does not arise because nodes explicitly separate activities. If an OCR question provides an AoA network, you must be able to interpret dummy arrows and spot logical errors such as redundant dummies or incorrect dependencies.

例如，若活动 D 依赖于 A 和 B，而活动 E 仅依赖于 B，那么在 AoA 网络中就需要从 B 的完成节点引出一条虚箭头到 E 的起始节点，以明确该条件。在 AoN 中不会出现这个问题，因为节点本身就分隔了各项活动。如果 OCR 试题给出了一个 AoA 网络，你必须能够解读虚箭头，并发现逻辑错误，如冗余虚活动或不正确的依赖关系。

10. Common Errors and Exam Tips | 常见错误与考试技巧

One common mistake in the forward and backward passes is forgetting to use the maximum or minimum functions correctly. When multiple routes converge on a node, EST takes the maximum of all incoming values; during the backward pass, LST takes the minimum of outgoing constraints. Always double-check each calculation, as a single error will propagate.

前推后推法中一个常见错误是不能正确使用最大值或最小值函数。当多条路径汇聚至一个节点时，EST 取所有引入值的最大值；后推法计算 LST 时，取各引出约束的最小值。务必仔细检查每一步计算，因为一个错误就会传导开来。

In resource smoothing, many students shift activities beyond their total float, inadvertently lengthening the project. Ensure that the new start time remains ≤ LST. Also, reconstruct the histogram from the smoothed schedule to confirm that the resource limit is satisfied throughout.

在资源平滑中，许多学生会把活动的开始时间移出其总浮动范围，无意中延长了项目。要保证新开始时间 ≤ LST。并且根据平滑后的排程重建直方图，以确认整个过程中资源限制得到满足。

When drawing networks, always include a clear start and end node. Label each node with its activity letter and duration. If you are given a precedence table, check for ‘no predecessors’ activities – they must start directly from the start node. Also, watch out for activities that appear as predecessors but are not themselves listed in the table; these are errors in the question or missing activities.

绘制网络时，务必包含明确的起始和结束节点，并在每个节点上标注活动字母和持续时间。如果给出紧前关系表，请检查“无紧前”的活动——它们必须直接从起始节点引出。同时要注意，某些活动作为紧前出现但自己不在活动列表中，这些是题目中的错误或遗漏的活动。

Finally, interpret the critical path in context. Questions may ask about the effect of a delay in a specific activity – if it is critical, the project completion will be delayed by exactly the same amount; if it has float, a delay less than its total float has no effect on the finish date. Being able to explain these implications clearly is a mark‑winning skill.

最后，要结合情境解释关键路径。题目可能会问某一特定活动的延迟会带来什么影响——如果是关键活动，项目完成日期会被同等延迟；如果该活动有浮动时间，那么小于其总浮动的延迟对完工日期没有影响。能清晰地解释这些含义是一项赚分技能。

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Critical Path Analysis | 关键路径分析