Critical Path Analysis | 关键路径分析

📚 Critical Path Analysis | 关键路径分析

Critical path analysis (CPA) is a fundamental decision mathematics technique used in Edexcel A Level. It helps project managers model complex tasks, identify the minimum completion time, and decide which activities can be delayed safely.

关键路径分析（CPA）是 Edexcel A Level 决策数学中的基础技术。它帮助项目管理者对复杂任务进行建模，确定最短完成时间，并判断哪些活动可以安全延期。

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1. What is Critical Path Analysis? | 什么是关键路径分析？

Critical path analysis is a network-based method for planning and scheduling projects. A project is broken down into individual activities, each with a duration and a set of precedence constraints. The main goal is to find the sequence of activities that determines the overall project duration.

关键路径分析是一种基于网络的规划和排程方法。一个项目被分解为若干独立活动，每项活动都有持续时间和一组先后关系约束。主要目标是找出决定项目总工期的那条活动序列。

In Edexcel Decision Mathematics, you will be expected to construct an activity network from a precedence table, carry out forward and backward passes, calculate floats, identify the critical path, and produce scheduling diagrams such as Gantt charts.

在 Edexcel 决策数学中，你需要根据先后关系表构建活动网络，进行正向和反向遍历，计算时差，确定关键路径，并绘制甘特图等排程图。

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2. Activity Networks: Precedence Tables and Dummies | 活动网络：先后关系表与虚活动

A project is modelled by an activity-on-arc network. In this model, nodes represent events, which are the start or finish of one or more activities, while arcs represent the activities themselves. Each arc is labelled with the activity name and its duration.

项目使用弧表示活动的网络来建模。在这个模型中，节点代表事件，即一项或多项活动的开始或结束，而弧代表活动本身。每条弧都标有活动名称及其持续时间。

A precedence table lists each activity, its duration, and any other activities that must be completed before it can begin. For example, if activity C depends on activity A, then C can only start after A has finished. This dependency is shown directly by the network.

先后关系表列出每项活动、持续时间以及必须在它开始前完成的其他活动。例如，如果活动 C 依赖于活动 A，那么 C 只能在 A 完成后才能开始。这种依赖关系直接体现在网络中。

Dummy activities are drawn as dashed arcs with zero duration. They are used to preserve logical dependencies when two activities share some, but not all, predecessors. Dummies do not consume time or resources.

虚活动用虚线弧表示，持续时间为零。当两项活动共享部分而非全部前导活动时，虚活动用于保持逻辑依赖关系。虚活动不消耗时间或资源。

• Use one source node and one sink node. | 使用一个源节点和一个汇节点。
• Do not cross arcs unnecessarily. | 尽量避免弧线交叉。
• Use dummies only when needed to show dependency. | 仅在需要表示依赖关系时使用虚活动。

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3. Early Event Times: Forward Pass | 最早事件时间：正向遍历

EET = 0. Work through the network from left to right, following the direction of the arcs.

正向遍历计算每个节点的最早事件时间 EET。源节点的 EET 始终为 0。沿着弧的方向，从左到右遍历网络。

For each activity joining node i to node j with duration d(i, j), the earliest event time at node j is given by:

对于连接节点 i 和节点 j、持续时间为 d(i, j) 的每项活动，节点 j 的最早事件时间为：

EET(j) = max { EET(i) + d(i, j) }

When several activities enter the same node, choose the largest value. This is because the event cannot occur until every preceding activity has been completed. The EET at the sink node gives the minimum project duration.

当多项活动进入同一节点时，选择最大值。这是因为直到所有前导活动都完成，该事件才能发生。汇节点的 EET 给出项目的最短工期。

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4. Late Event Times: Backward Pass | 最晚事件时间：反向遍历

The backward pass calculates the latest event time, LET, for each node. Start at the sink node, where the LET is usually set equal to the project duration found from the forward pass. Work from right to left through the network.

反向遍历计算每个节点的最晚事件时间 LET。从汇节点开始，汇节点的 LET 通常设为正向遍历得到的项目总工期。从右到左遍历网络。

For each activity joining node i to node j, the latest event time at node i is given by:

对于连接节点 i 和节点 j 的每项活动，节点 i 的最晚事件时间为：

LET(i) = min { LET(j) - d(i, j) }

When several activities leave a node, choose the smallest value. This ensures that every successor activity can finish without delaying the entire project. If the sink node LET is not specified, it should be set equal to the sink node EET.

当多项活动离开同一节点时，选择最小值。这样可以确保每项后续活动都能在不延误整个项目的情况下完成。如果未指定汇节点的 LET，应将其设为汇节点的 EET。

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5. Identifying the Critical Path | 确定关键路径

An activity is critical if it has zero total float. The total float of an activity from node i to node j is calculated as LET(j) - EET(i) - duration(i, j). If this value is 0, the activity is critical and cannot be delayed without delaying the whole project.

如果一项活动的总时差为零，它就是关键活动。从节点 i 到节点 j 的活动总时差计算公式为 LET(j) - EET(i

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