Early and Late Event Times | 最早与最晚事件时间

📚 Early and Late Event Times | 最早与最晚事件时间

In Edexcel A Level Decision Mathematics, early and late event times are central to critical path analysis. They allow you to determine how soon each stage of a project can begin and how late it can be left without delaying the whole project.

在 Edexcel A Level 决策数学中，最早与最晚事件时间是关键路径分析的核心。它们帮助你确定项目每个阶段最早可以何时开始，以及最晚可以推迟到何时而不延误整个项目。

1. Activity Networks: Events and Activities | 活动网络：事件与活动

In an activity-on-arc network, every node represents an event: a point in time when one or more activities finish and the next ones may begin. Arcs represent activities with fixed durations. The event at the tail of an arc must occur before that activity can start; the event at the head occurs only when all incoming activities are complete.

在“活动在弧上”的网络中，每个节点代表一个事件：即一项或多项活动完成、后续活动可以开始的时间点。弧表示具有固定工期的活动。弧尾的事件必须先发生，该活动才能开始；弧头的事件只有在所有指向该节点的活动都完成后才会发生。

2. Early vs Late Event Times | 最早与最晚事件时间

The earliest event time (EET) is the soonest possible time at which an event can be reached, given the durations and dependencies. The latest event time (LET) is the latest time an event can occur without delaying the overall project. Neither is a duration; both are clock times measured from the project start.

最早事件时间（EET）是考虑工期和依赖关系后，事件能够到达的最早时间。最晚事件时间（LET）是不延误整个项目的前提下，事件可以发生的最晚时间。二者都不是持续时长，而是从项目开始起计的时钟时间。

3. Earliest Event Time: Forward Pass | 最早事件时间：正向推进

The forward pass starts at the source node with EET = 0. For any other node j, consider every activity (i, j) entering j. Add its duration d(i,j) to EET(i). The earliest time for j is the maximum of these sums, because j cannot occur until all incoming activities have finished.

正向计算从源节点开始，令 EET = 0。对于任何其他节点 j，考虑所有进入 j 的活动 (i, j)，将 EET(i) 加上该活动工期 d(i,j)。节点 j 的最早时间是这些和的最大值，因为只有所有进入活动都完成后，j 才能发生。

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

4. Latest Event Time: Backward Pass | 最晚事件时间：逆向推进

The backward pass starts at the sink node. Set its LET equal to the project duration, normally the EET of the sink. For any node i, consider every activity (i, j) leaving i. Subtract its duration from LET(j). The latest time for i is the minimum of these differences, because if i occurs any later, at least one following activity will overrun.

逆向计算从汇节点开始。将汇节点的 LET 设为项目总工期，通常等于该节点的 EET。对于任何节点 i，考虑所有离开 i 的活动 (i, j)，用 LET(j) 减去活动工期。节点 i 的最晚时间是这些差值的最小值，因为如果 i 再晚发生，至少一个后续活动会超期。

LET(i) = min { LET(j) − d(i,j) : (i,j) leaves i }

If a question imposes a later target completion time, set LET(sink) equal to that target time instead of the earliest project duration. This changes the floats but not the forward pass values.

如果题目规定了更晚的目标完成时间，则将 LET(汇节点) 设为该目标时间，而不是最早项目工期。这会改变浮动时间，但不会改变正向计算值。

5. Notation, Source, Sink and Dummy Activities | 符号、源节点、汇节点与虚活动

Edexcel questions often number events 1, 2, 3, … with the source as the lowest number and the sink as the highest. A dummy activity has zero duration and is shown as a dashed arc; it represents a logical dependency only. In forward and backward passes, dummies are treated like any other activity with d = 0.

Edexcel 题目通常将事件编号为 1、2、3……，源节点编号最小，汇节点编号最大。虚活动的工期为 0，用虚线弧表示；它只表示逻辑依赖关系。在正向和逆向计算中，虚活动与 d = 0 的普通活动一样处理。

6. Worked Example: Activity Data | 示例：活动数据

The table below gives a small project. We will build the network, compute EETs, LETs, and identify the critical path. Activities are labelled A to F, with durations and immediate predecessors.

下表给出一个小型项目。我们将构建网络，计算 EET 和 LET，并确定关键路径。活动用 A 到 F 标注，并给出工期和紧前活动。

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