📚 The Float of an Activity | 活动的浮动时间
In critical path analysis, the float of an activity measures how much the activity can be delayed without affecting the project completion time or the start of subsequent activities. Understanding float is essential for identifying the critical path and for efficient resource scheduling.
在关键路径分析中，活动的浮动时间衡量该活动可以延迟多久而不影响项目完成时间或后续活动的开始。理解浮动时间对于识别关键路径和高效安排资源至关重要。
1. What is Float? | 什么是浮动时间？
Float, sometimes called slack, is the amount of time by which an activity can be delayed beyond its earliest start time without causing a delay to the project or to dependent activities.
浮动时间（有时称为时差）是指活动可以比其最早开始时间延迟多久，而不会导致项目或依赖活动延迟。
There are three types of float: total float, free float and independent float. Each gives a different level of flexibility.
浮动时间有三种类型：总时差、自由时差和独立时差。每种类型给出不同层次的灵活性。
An activity with zero total float lies on the critical path.
总时差为零的活动位于关键路径上。
2. Key Times in Critical Path Analysis | 关键路径分析中的关键时间
To calculate float, you must first find the earliest start time (EST), earliest finish time (EFT), latest start time (LST) and latest finish time (LFT) for each activity.
要计算浮动时间，必须首先求出每项活动的最早开始时间（EST）、最早完成时间（EFT）、最晚开始时间（LST）和最晚完成时间（LFT）。
EST is the soonest an activity can begin once all predecessors are complete. EFT = EST + duration.
EST 是所有前置活动完成后该活动可以开始的最早时间。EFT = EST + 持续时间。
LFT is the latest an activity can finish without delaying the project. LST = LFT − duration.
LFT 是在不延迟项目的前提下该活动可以完成的最晚时间。LST = LFT − 持续时间。
These four times are obtained from a forward pass and a backward pass through the activity network.
这四个时间通过对活动网络进行正推和逆推得到。
3. Total Float: Definition and Formula | 总时差：定义与公式
Total float is the maximum amount of time an activity can be delayed without delaying the whole project.
总时差是活动可以延迟的最大时间量，而不会延迟整个项目。
It is calculated using: Total Float = LFT − EFT, or equivalently Total Float = LST − EST.
计算公式为：总时差 = LFT − EFT，或等价地总时差 = LST − EST。
Both expressions always give the same value because EFT = EST + duration and LFT = LST + duration.
两种表达式总是给出相同的结果，因为 EFT = EST + 持续时间，LFT = LST + 持续时间。
If an activity has total float of 5 days, it can be delayed by up to 5 days without affecting the project finish date.
如果一项活动的总时差为5天，那么它可以延迟最多5天而不影响项目完成日期。
4. Interpreting Total Float | 解释总时差
Total float is shared with other activities on the same non-critical chain. Using all the float on one activity may reduce the float available for later activities.
总时差在同一非关键链上的其他活动之间是共享的。在一项活动上用掉全部时差，可能会减少后续活动可用的时差。
A positive total float means the activity is not on the critical path. A zero total float means the activity is critical.
总时差为正意味着该活动不在关键路径上。总时差为零意味着该活动是关键活动。
In some exam questions, total float can be negative if a deadline is earlier than the current earliest finish; however, for standard project networks it is usually zero or positive.
在某些考题中，如果截止日期早于当前最早完成时间，总时差可能为负；但在标准项目网络中，它通常为零或正数。
5. Free Float: Definition and Formula | 自由时差：定义与公式
Free float is the amount of time an activity can be delayed without delaying the earliest start of any immediately following activity.
自由时差是指活动可以延迟的时间量，而不延迟任何紧后活动的最早开始时间。
Free float is calculated as: Free Float = EST(next activity) − EFT(activity).
自由时差的计算公式为：自由时差 = 紧后活动的 EST − 本活动的 EFT。
Unlike total float, free float belongs to the activity itself and is not shared with other activities.
与总时差不同，自由时差属于活动本身，不与其他活动共享。
Free float can never exceed total float; it is often zero even when total float is positive.
自由时差永远不超过总时差；即使总时差为正，自由时差也常常为零。
6. Independent Float: Definition and Formula | 独立时差：定义与公式
Independent float is the amount of time an activity can be delayed regardless of the timing of preceding and following activities.
独立时差是指无论前置活动和后续活动的时间安排如何，活动都可以延迟的时间量。
It is calculated using: Independent Float = EST(next activity) − LFT(previous activity) − duration.
计算公式为：独立时差 = 紧后活动的 EST − 紧前活动的 LFT − 持续时间。
If this expression is negative, the independent float is taken as zero, because negative float has no practical meaning for a single activity.
如果该表达式为负，独立时差取为零，因为负时差对单个活动没有实际意义。
Independent float is the most conservative measure and is rarely used in basic scheduling decisions, but exam boards may ask for it.
独立时差是最保守的度量，在基本调度决策中很少使用，但考试局可能会要求计算它。
7. Calculating Float from a Precedence Table | 从优先关系表计算时差
A precedence table lists activities, their durations and immediate predecessors. Use it to build the activity network, then carry out forward and backward passes.
优先关系表列出活动、持续时间及紧前活动。用它构建活动网络，然后进行正推和逆推。
Record EST, EFT, LST, LFT in a table. Then apply the float formulas to each activity.
在表格中记录 EST、EFT、LST、LFT。然后对每项活动应用时差公式。
For example, if activity B has duration 4, EST 6, EFT 10, LST 9, LFT 13, then total float = 13 − 10 = 3 days.
例如，若活动 B 持续时间为4，EST 为6，EFT 为10，LST 为9，LFT 为13，则总时差 = 13 − 10 = 3 天。
Always show your workings clearly, as method marks are awarded in Edexcel Decision Mathematics.
务必清晰展示计算步骤，因为 Edexcel 决策数学中会给予方法分。
8. Float and the Critical Path | 时差与关键路径
The critical path is the longest path through the network. All activities on this path have zero total float.
关键路径是穿过网络的最长路径。该路径上的所有活动总时差为零。
Therefore, total float is a quick way to confirm the critical activities: if total float = 0, the activity is critical.
因此，总时差是确认关键活动的快捷方法：如果总时差为零，该活动就是关键活动。
A critical path can be identified by listing all activities with zero total float and checking that they form a continuous chain from start to finish.
关键路径可以通过列出所有总时差为零的活动并检查它们是否形成从开始到结束的连续链来识别。
9. Using Cascade (Gantt) Charts to Show Float | 使用级联图（甘特图）表示时差
A cascade chart, or Gantt chart, displays each activity as a block starting at its earliest start time, with its total float shown as a dotted extension.
级联图（或甘特图）将每项活动显示为从最早开始时间开始的方块，并用虚线延伸表示总时差。
The solid block represents the activity duration, while the dotted line shows the maximum delay possible before the project is affected.
实心方块表示活动持续时间，虚线表示在项目受到影响之前可能的最大延迟。
Exam questions often ask you to draw a cascade chart and shade or label the float clearly.
考题经常要求绘制级联图，并清楚地标记或涂色表示时差。
10. Common Misconceptions and Exam Tips | 常见误区与考试技巧
Misconception 1: total float and free float are the same. They are not; total float may delay the project only when the whole chain is considered, while free float delays only the next activity.
误区1：总时差和自由时差是相同的。其实不同；总时差只有在考虑整条链时才会延迟项目，而自由时差只影响下一个活动。
Misconception 2: an activity with zero free float must be critical. This is false; zero free float can happen on non-critical activities if the next activity starts immediately after it.
误区2：自由时差为零的活动一定是关键活动。这是错误的；如果下一个活动紧随其后开始，非关键活动也可能自由时差为零。
Exam tip: always calculate all four times first, then apply the float formulas systematically to avoid arithmetic errors.
考试技巧：始终先计算四个时间，然后系统地应用时差公式，以避免计算错误。
11. Worked Example | 例题解析
Activity network: A(3) → B(4) and A(3) → C(2), with B and C both preceding D(5). The project start is at time 0.
活动网络：A(3) → B(4) 且 A(3) → C(2)，B 和 C 都是 D(5) 的紧前活动。项目从时间0开始。
Forward pass: EST(A)=0, EFT(A)=3. EST(B)=3, EFT(B)=7. EST(C)=3, EFT(C)=5. EST(D)=max(7,5)=7, EFT(D)=12.
正推：EST(A)=0，EFT(A)=3。EST(B)=3，EFT(B)=7。EST(C)=3，EFT(C)=5。EST(D)=max(7,5)=7，EFT(D)=12。
Backward pass: LFT(D)=12, LST(D)=7. LFT(B)=7, LST(B)=3. LFT(C)=7, LST(C)=5. LFT(A)=min(LST(B), LST(C)) = min(3,5)=3, LST(A)=0.
逆推：LFT(D)=12，LST(D)=7。LFT(B)=7，LST(B)=3。LFT(C)=7，LST(C)=5。LFT(A)=min(LST(B), LST(C)) = min(3,5)=3，LST(A)=0。
Total floats: A: 0; B:
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