📚 Resource Futures | 资源调度与资源直方图
In Edexcel A-Level Decision Mathematics (D1), the topic often labelled ‘Resource Futures’ refers to project scheduling with limited resources. You need to understand how activities, precedence constraints, critical paths and resource availability interact, and how to draw resource histograms and Gantt charts.
在 Edexcel A-Level 决策数学（D1）中，常标为 ‘资源未来’ 的主题实际上是指有限资源下的项目调度。你需要理解活动、先后关系、关键路径与资源可用性如何相互作用，并学会绘制资源直方图和甘特图。
1. What is Resource Scheduling? | 什么是资源调度？
Resource scheduling is the process of assigning limited resources (workers, machines, rooms) to project activities over time while respecting precedence constraints and minimising the overall project duration.
资源调度是指将有限资源（工人、机器、房间）在时间上分配给项目活动，同时遵守先后约束并尽量缩短项目总工期的过程。
A resource schedule shows exactly when each activity starts and finishes, and how many units of each resource are required in each time period.
资源调度表准确显示每项活动的开始和结束时间，以及每个时间段需要多少单位的每种资源。
In Edexcel D1, you will often be given a precedence table and a resource limit, then asked to produce a feasible schedule or a resource histogram.
在 Edexcel D1 中，通常会给出先后关系表和资源限制，然后要求你制定一个可行的调度方案或绘制资源直方图。
2. Activities, Precedence and Durations | 活动、先后关系与工期
An activity is a task that takes time and uses resources. A precedence table lists each activity, its duration and its immediate predecessors. For example, activity B cannot start until activity A is complete.
活动是一项需要时间并占用资源的任务。先后关系表列出了每项活动、其工期及其紧前活动。例如，活动 B 必须在活动 A 完成后才能开始。
The key notation includes:
关键记号包括：
- Activity label: A, B, C, … (活动编号)
- Duration: d(A) or a number of days (工期)
- Immediate predecessors: activities that must finish first (紧前活动)
Below is a typical precedence table for a small project:
下面是一个小项目的典型先后关系表：
| Activity | Duration (days) | Predecessors | Workers needed |
|---|---|---|---|
| A | 2 | None | 2 |
| B | 4 | A | 3 |
| C | 3 | B | 2 |
| D | 2 | B | 1 |
| E | 3 | A | 1 |
This table tells you, for example, that activity B lasts 4 days, needs 3 workers, and cannot begin until activity A has finished.
这张表告诉你，例如活动 B 持续 4 天，需要 3 名工人，并且在活动 A 完成后才能开始。
3. Critical Path Analysis Recap | 关键路径分析回顾
Before tackling resource scheduling, you must be able to construct an activity network and calculate earliest start times (EST), latest finish times (LFT) and total float.
在处理资源调度之前，你必须能够构建活动网络并计算最早开始时间（EST）、最晚完成时间（LFT）和总浮动时间。
The total float of an activity is given by the formula:
一项活动的总浮动时间由以下公式给出：
Total float = LFT − duration − EST
Activities with zero total float are critical. The longest path through the network is the critical path, and its length is the minimum project duration.
总浮动时间为零的活动是关键活动。网络中的最长路径是关键路径，其长度即为项目的最短工期。
In the table above, the critical path is A → B → C with total duration 2 + 4 + 3 = 9 days. Any delay to A, B or C will delay the whole project.
在上表中，关键路径是 A → B → C，总工期为 2 + 4 + 3 = 9 天。A、B 或 C 的任何延误都会推迟整个项目。
4. Resource Histograms | 资源直方图
A resource histogram is a bar chart showing the number of workers (or other resources) needed in each time period if every activity starts at its earliest start time.
资源直方图是一个条形图，显示如果每项活动都从其最早开始时间开始，每个时间段所需的工人（或其他资源）数量。
To draw a resource histogram: list each day or hour on the horizontal axis, count which activities are active in that period, and add their resource requirements.
绘制资源直方图的方法：横轴列出每一天或每一小时，统计该时段正在进行哪些活动，并将它们的资源需求相加。
Using the example table with earliest start scheduling, days 1-2 have activity A only (2 workers). Days 3-6 have activity B (3 workers) and days 3-5 also have activity E (1 worker), giving 4 workers on days 3-5. Days 7-8 have C and D, giving 2 + 1 = 3 workers; day 9 has C only (2 workers).
以上表为例，按最早开始时间调度：第 1-2 天只有活动 A（2 名工人）。第 3-6 天有活动 B（3 名工人），且第 3-5 天还有活动 E（1 名工人），因此第 3-5 天需要 4 名工人。第 7-8 天有 C 和 D，需要 2 + 1 = 3 名工人；第 9 天只有 C（2 名工人）。
This histogram peaks at 4 workers. If only 3 workers are available, the schedule is not feasible without adjustment.
该直方图的峰值为 4 名工人。如果只有 3 名工人可用，这个调度方案在没有调整的情况下是不可行的。
5. Scheduling with Limited Resources | 有限资源下的调度
If the resource histogram exceeds the available number of workers, some activities must be delayed. You must decide which activities to delay while keeping the project as short as possible.
如果资源直方图超出可用的工人数量，就必须推迟某些活动。你需要决定推迟哪些活动，同时尽可能保持项目工期最短。
A simple rule is to prioritise critical activities because delaying them extends the overall project. Non-critical activities can use their float to be moved without delaying the project.
一个简单的规则是优先安排关键活动，因为推迟它们会延长总工期。非关键活动可以利用其浮动时间移动而不延误项目。
In the example, activity E is non-critical. Its total float is 9 − 3 − 2 = 4 days. Therefore E can be delayed by up to 4 days without affecting the 9-day project completion.
在该例中，活动 E 是非关键活动。其总浮动时间为 9 − 3 − 2 = 4 天。因此 E 最多可以推迟 4 天而不影响 9 天的项目完工。
If we delay E to start on day 7, then days 7-9 have C (2 workers) plus E (1 worker) = 3 workers. The resource peak is now 3 workers, so the schedule is feasible with only 3 workers.
如果我们将 E 推迟到第 7 天开始，那么第 7-9 天有 C（2 名工人）加上 E（1 名工人）= 3 名工人。资源峰值现在为 3 名工人，因此只有 3 名工人时该调度方案是可行的。
6. Gantt Charts | 甘特图
A Gantt chart displays activities as horizontal bars on a time axis. It can be drawn using earliest start times or after resource-levelling has been applied.
甘特图用水平条形在时间轴上显示活动。它可以使用最早开始时间绘制，也可以在资源均衡后绘制。
Floating activities are often shown with a solid bar for the core activity and a dashed extension for the float.
有浮动的活动通常用实心条形表示核心活动，用虚线延伸表示浮动时间。
For the adjusted schedule, the Gantt chart would show A on days 1-2, B on days 3-6, C on days 7-9, D on days 7-8, and E on days 7-9. The critical activities A, B, C are all solid bars because no float is used.
对于调整后的调度方案，甘特图将显示 A 在第 1-2 天，B 在第 3-6 天，C 在第 7-9 天，D 在第 7-8 天，E 在第 7-9 天。关键活动 A、B、C 都用实心条形表示，因为没有使用浮动时间。
7. Resource Leveling and Smoothing | 资源均衡与平滑
Resource levelling aims to remove resource overloads by moving non-critical activities within their float. Smoothing aims to make resource usage as even as possible without extending the critical path.
资源均衡旨在通过在其浮动范围内移动非关键活动来消除资源超出。平滑旨在使资源使用尽可能均匀，而不延长关键路径。
When smoothing, you may move an activity to a later start time to fill a valley in the histogram, as long as the critical path is not affected.
进行平滑时，你可以将活动移至较晚的开始时间，以填补直方图中的低谷，只要关键路径不受影响。
In our example, delaying E from day 3 to day 7 is a levelling move because it removes the overload on days 3-5 and keeps the project at 9 days.
在我们的例子中，将 E 从第 3 天推迟到第 7 天是一种均衡操作，因为它消除了第 3-5 天的资源超出，并使项目仍保持 9 天完成。
8. Float and Resource Impact | 浮动时间与资源影响
Total float is the maximum time an activity can be delayed without delaying the project. Independent float and interfering float are sometimes considered, but Edexcel D1 mainly uses total float.
总浮动时间是一项活动可延迟而不延误项目的最大时间。独立浮动和干扰浮动有时会考虑，但 Edexcel D1 主要使用总浮动时间。
When you delay an activity for resource reasons, its successor may lose float, so the critical path can change.
当你因资源原因推迟一项活动时，其后续活动可能会失去浮动时间，因此关键路径可能改变。
You should check after any delay that no new critical path has been created. In the example, delaying E does not create a new critical path because E uses all of its float and finishes at day 9, which is the project end.
在任何延迟之后，你应检查是否产生了新的关键路径。在该例中，推迟 E 不会产生新的关键路径，因为 E 恰好用完了其浮动时间并在第 9 天完工，正好是项目结束时间。
9. Worked Example: Building a Wall | 例题：砌墙项目
This worked example brings together the key ideas from scheduling, float and resource histograms.
这道例题将调度、浮动时间和资源直方图的关键概念整合在一起。
Project activities:
项目活动：
- A: dig foundation, 2 days, 2 workers, no predecessors (挖地基)
- B: lay bricks, 4 days, 3 workers, predecessor A (砌砖)
- C: build roof, 3 days, 2 workers, predecessor B (盖屋顶)
- D: paint, 2 days, 1 worker, predecessor B (油漆)
- E: landscaping, 3 days, 1 worker, predecessor A (景观美化)
Step 1: Find the critical path. The path A-B-C has length 2+4+3=9 days. The path A-B-D has length 2+4+2=8 days. The path A-E has length 2+3=5 days. So the critical path is A-B-C with 9 days.
步骤 1：找出关键路径。路径 A-B-C 的长度为 2+4+3=9 天。路径 A-B-D 的长度为 2+4+2=8 天。路径 A-E 的长度为 2+3=5 天。因此关键路径是 A-B-C，工期 9 天。
Step 2: Draw the resource histogram using earliest start times. We obtain 2 workers on days 1-2, 3 workers on day 6 only, 4 workers on days 3-5, 3 workers on days 7-8, and 2 workers on day 9.
步骤 2：使用最早开始时间绘制资源直方图。我们得到第 1-2 天 2 名工人，仅第 6 天 3 名工人，第 3-5 天 4 名工人，第 7-8 天 3 名工人，第 9 天 2 名工人。
Step 3: If only 3 workers are available, delay activity E by 4 days so it starts on day 7. The new resource peak is 3 workers.
步骤 3：如果只有 3 名工人可用，将活动 E 推迟 4 天，使其从第 7 天开始。新的资源峰值为 3 名工人。
Step 4: Draw the Gantt chart for the feasible schedule. Critical activities A, B, C remain fixed, D stays at days 7-8, and E moves to days 7-9.
步骤 4：绘制可行调度方案的甘特图。关键活动 A、B、C 保持固定，D 仍在第 7-8 天，E 移动到第 7-9 天。
10. Exam Tips for Edexcel D1 | Edexcel D1 考试技巧
In the exam, read the precedence table carefully and draw the activity network first. Label each node with EST and LFT. Check for dummies if two activities share the same start and end nodes.
考试时，仔细阅读先后关系表并先绘制活动网络。在每个节点标出 EST 和 LFT。如果两项活动有相同的开始和结束节点，检查是否需要虚活动。
When drawing a resource histogram, show a clear scale and label the axes. State any assumption about which activities are delayed.
绘制资源直方图时，明确标出刻度并给坐标轴加标签。说明你推迟了哪些活动以及假设。
Always use the exact time units from the question. If durations are in hours, use hours; if days
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