📚 The Float of an Activity: Total, Free and Independent Float | 活动的浮动时间：总浮动、自由浮动与独立浮动
In critical path analysis, the float of an activity refers to the amount of time by which the activity can be delayed without affecting the overall project duration or the start of subsequent activities. Understanding float is crucial for effective project scheduling and resource allocation, because it tells a project manager exactly where there is spare time and where there is none.
在关键路径分析中，活动的浮动时间是指该活动可以延迟而不会影响整个项目工期或后续活动开始的时间量。理解浮动时间对于有效的项目调度和资源分配至关重要，因为它能告诉项目经理哪些地方有时间余量、哪些地方完全没有。
This article covers the whole topic from first principles: the terminology of earliest and latest times, the forward and backward passes, the three types of float used in Edexcel Decision Mathematics 1 (total, free and independent float), a fully worked network example, and the exam technique needed to score full marks.
本文将从头讲清整个主题：最早与最迟时间的术语体系、正向与反向推算、Edexcel 决策数学 1 中使用的三种浮动时间（总浮动、自由浮动、独立浮动）、一个完整的网络图例题，以及拿满分所需的考试技巧。
1. What Float Means | 浮动时间的含义
Float, also called slack, is the flexibility available in scheduling an activity. It is measured in the same units as the activity durations, normally days, hours or weeks. An activity with zero float is critical: any delay to it will delay the whole project. An activity with positive float can absorb delay, which allows a manager to shift resources, level workloads or absorb minor slippage without moving the completion date.
浮动时间（或称时差）是活动排程中可用的灵活性，其单位与活动持续时间相同，通常是天、小时或周。浮动时间为零的活动是关键活动：对它的任何延误都会导致整个项目延期。具有正浮动时间的活动可以吸收延误，使管理者能够在不改变完工日期的前提下转移资源、平衡工作量或消化轻微的进度滑落。
Three different measures of float are examined at A Level, and each answers a slightly different question.
A Level 阶段会考查三种不同的浮动时间，每一种回答的问题略有不同。
- Total float - how long an activity can be delayed without delaying the project, assuming all other activities are scheduled as early as possible.
- Free float - how long an activity can be delayed without delaying the earliest start of any immediately following activity.
- Independent float - how long an activity can be delayed when predecessors finish as late as possible and successors start as early as possible.
- 总浮动时间——假设其他所有活动都尽早安排时，该活动可以延迟而不拖延整个项目的最长时间。
- 自由浮动时间——该活动可以延迟而不影响任何紧后活动最早开始时间的最长时间。
- 独立浮动时间——当前置活动尽可能晚完成、后续活动尽可能早开始时，该活动可以延迟的时间。
If a calculation breaks this chain, there is an arithmetic error somewhere in the network.
这三种浮动时间始终满足不等式：独立浮动 ≤ 自由浮动 ≤ 总浮动。如果计算结果破坏了这一关系，说明网络中的某处存在计算错误。
2. Basic Terminology: Earliest and Latest Times | 基本术语：最早与最迟时间
To calculate float, we first determine four quantities for each activity: the earliest start time (EST), earliest finish time (EFT), latest start time (LST) and latest finish time (LFT). These are derived from forward and backward passes through the network diagram, and every float formula is built from them.
要计算浮动时间，我们首先要为每个活动确定四个量：最早开始时间（EST）、最早完成时间（EFT）、最迟开始时间（LST）和最迟完成时间（LFT）。它们通过对网络图进行正向推算和反向推算得出，而所有浮动时间公式都由这四个量构成。
The earliest start time of an activity is the earliest moment at which it can begin, determined by the longest path from the start of the project to that activity. The latest finish time is the latest moment at which the activity can finish without delaying the project, and it is found by tracing backwards from the project end.
活动的最早开始时间是该活动可以开始的最早时刻，由从项目起点到该活动的最长路径决定。最迟完成时间是该活动可以完成而不延误项目的最晚时刻，通过从项目终点向后追溯得出。
| Term | 术语 | Meaning | 含义 | Formula | 公式 |
| EST | Earliest moment the activity can start | 活动可以开始的最早时刻 | max of predecessors' EFT | 各紧前活动 EFT 的最大值 |
| EFT | Earliest moment it can finish | 活动可以完成的最早时刻 | EFT = EST + Duration |
| LFT | Latest it can finish without delaying the project | 不延误项目的最晚完成时刻 | min of successors' LST | 各紧后活动 LST 的最小值 |
| LST | Latest it can start without delaying the project | 不延误项目的最晚开始时刻 | LST = LFT − Duration |
On an activity-on-node diagram these four numbers are usually written in the four quadrants of the node box, with the activity letter and duration in the middle. Reading the box correctly is often the difference between a correct and an incorrect answer.
在单代号网络图（活动在节点上）中，这四个数字通常写在节点框的四个象限内，活动代号与持续时间写在中央。能否正确读出节点框中的信息，往往决定了答案的对错。
3. The Forward Pass: Earliest Times | 正向推算：最早时间
The forward pass moves from left to right through the network, starting with the first activity at time zero. For every activity, the earliest finish time is the earliest start time plus the duration.
正向推算从网络左端开始、向右推进，第一个活动从时刻 0 开始。对每个活动而言，最早完成时间等于最早开始时间加上持续时间。
EFT = EST + Duration
When two or more activities merge into the same successor, that successor cannot start until all of its predecessors have finished, so we take the maximum of the predecessors' earliest finish times.
当两个或更多活动汇入同一个紧后活动时，该紧后活动必须等所有紧前活动都完成后才能开始，因此取各紧前活动最早完成时间的最大值。
EST(successor) = max [ EFT(predecessors) ]
The largest EFT reached at the end node of the network is the minimum project duration. It is not a sum of all activity durations, which is a very common misconception: activities running in parallel overlap, so only the longest path counts.
网络终点节点处得到的最大 EFT 就是项目的最短工期。它并不是所有活动持续时间之和，这是一个非常常见的误解：并行进行的活动在时间上重叠，因此只有最长的路径起决定作用。
4. The Backward Pass: Latest Times | 反向推算：最迟时间
The backward pass works from the end of the network back to the start. The latest finish time of the final activity equals the project duration found in the forward pass. For every activity, the latest start time is the latest finish time minus the duration.
反向推算从网络终点回到起点。最后活动的最迟完成时间等于正向推算得到的项目工期。对每个活动而言，最迟开始时间等于最迟完成时间减去持续时间。
LST = LFT − Duration
When a single activity feeds into two or more successors, it must finish in time for the most demanding of them, so we take the minimum of the successors' latest start times.
当一个活动有多个紧后活动时，它必须赶在最“苛刻”的紧后活动之前完成，因此取各紧后活动最迟开始时间的最小值。
LFT(activity) = min [ LST(successors) ]
In an activity-on-node network, dummy activities have zero duration; they are used purely to show logical dependency without consuming time. They still take part in both passes and can carry float like any other activity.
在单代号网络中，虚活动（dummy activity）的持续时间为零，它们仅用于表示逻辑依赖关系而不占用时间。它们同样参与正向与反向推算，并且和其他活动一样可能带有浮动时间。
5. Total Float | 总浮动时间
Total float is the amount of time an activity can be delayed without delaying the overall project completion date. It is the most widely used of the three measures and the one most often quoted in reports.
总浮动时间是指活动可以延迟而不影响整个项目完工日期的时间量。它是三种浮动时间中使用最广、在报告中引用最多的一种。
Total Float = LST − EST = LFT − EFT
Because the two expressions are algebraically identical, you can use whichever is quicker with the numbers you already have. If an activity has total float of three days and is delayed by two days, the project finish is unaffected, but the remaining float falls to one day.
由于这两个表达式在代数上完全等价，你可以用手上已有的数据选择计算更快的那一个。如果一个活动的总浮动为 3 天而它被延误了 2 天，项目完工日期不会受到影响，但剩余浮动降为 1 天。
Total float does not belong to one activity alone: it is shared along the chain of activities that make up the corresponding path. If an early activity consumes the float, later activities on the same path lose it. This is why experienced schedulers treat float as a shared resource rather than a personal allowance for one task.
总浮动并不独属于某一个活动：它由构成该路径的整条活动链共享。如果链上靠前的活动消耗了浮动，后面同链活动就失去了这部分浮动。这正是有经验的排程者把浮动视为共享资源、而非某项任务私有额度的原因。
6. Free Float | 自由浮动时间
Free float measures how long an activity can be delayed without delaying the earliest start of any immediately following activity. It is the "safe" part of the total float, because consuming it does not disturb any other activity at all.
自由浮动时间衡量的是活动可以延迟而不影响任何紧后活动最早开始时间的时间长度。它是总浮动中“安全”的部分，因为消耗它完全不会干扰其他任何活动。
Free Float = min [ EST(successors) ] − EFT(activity)
last activity in the network, the terminating time of the project replaces the successors' finishes at time 13 and the final activity has EFT 13, its free float is 13 − 13 = 0 days.
对于网络中的最后一个活动，用项目的结束时刻代替紧后活动的 EST。若项目在时刻 13 结束，而最后活动的最早完成时间为 13，则其自由浮动为 13 − 13 = 0 天。
The difference between total float and free float is called interfering float, because using it interferes with the start time of following activities.
总浮动与自由浮动之差称为干扰浮动（interfering float），因为动用它会影响后续活动的开始时间。
Interfering Float = Total Float − Free Float
7. Independent Float | 独立浮动时间
Independent float is the most conservative of the three measures. It answers the question: if every preceding activity
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