📚 Scheduling Diagrams: Gantt Charts and Resource Histograms | 调度图：甘特图与资源直方图
Scheduling diagrams are visual tools that transform a project’s activity network into a time‑based plan. For Edexcel A‑Level Decision Mathematics, you must be able to construct both Gantt (cascade) charts and resource histograms, then use them to analyse float, identify the critical path and optimise resource usage. This article covers everything from basic construction to resource smoothing, equipping you to handle any scheduling question that appears on your exam.
调度图是把项目活动网络转化为基于时间的计划的视觉工具。对于 Edexcel A‑Level 决策数学，你必须既能画出甘特（瀑布）图，也能画出资源直方图，然后用它们分析时差、识别关键路径并优化资源使用。本文从基础绘制方法讲到资源平滑，帮助你应对试卷上可能出现的任何调度类题目。
1. Why We Need Scheduling Diagrams | 为什么我们需要调度图
An activity‑on‑node precedence network shows logical dependencies, but it does not tell you when activities can actually take place or how many workers are needed at a given time. Scheduling diagrams translate the network into a calendar, making it possible to allocate resources and monitor progress.
节点式前导网络图展示了活动间的逻辑依赖关系，但它不能告诉你在什么时间可以开展哪些活动，也无法显示某一时刻需要多少人力。调度图将网络转化为时间表，使资源分配和进度监控成为可能。
Two main types of diagram appear in Edexcel Decision Maths papers: the Gantt chart (sometimes called a cascade chart) and the resource histogram. Together they form the core of the scheduling topic.
Edexcel 决策数学试卷中出现的主要是两类图：甘特图（有时称瀑布图）和资源直方图。它们共同构成了调度专题的核心。
2. Constructing a Gantt Chart from a Precedence Table | 由前导关系表绘制甘特图
A Gantt chart shows each activity as a horizontal bar. The length of the bar represents the activity duration, and its position along the time axis reflects the earliest start time (EST) and earliest finish time (EFT) when all activities are scheduled at their earliest possible time. To build one, you first need a completed precedence table that includes EST and EFT for every activity.
甘特图把每项活动表示为一条水平横道，横道长度代表活动工期，它在时间轴上的位置反映活动按最早可能时间排程时的最早开始时间 (EST) 和最早结束时间 (EFT)。绘制之前，你需要有一张完整的包含每项活动 EST 和 EFT 的前导关系表。
Draw a horizontal time axis with a suitable scale. For each activity, draw a bar from its EST to its EFT. Label the bar with the activity letter or name. If an activity has total float, it is common to show the float as a dotted or shaded extension beyond the EFT up to the latest finish time (LFT), though in many Edexcel questions the simple bar from EST to EFT is sufficient.
画一条带合适刻度的水平时间轴。对每一项活动，从其 EST 到 EFT 画一根横道，并标上活动字母或名称。如果活动拥有总时差，通常会把时差部分用虚线或阴影从 EFT 延伸到最迟结束时间 (LFT)，不过很多 Edexcel 题目只要求画 EST 到 EFT 的实线横道。
3. Identifying Float and the Critical Path from a Gantt Chart | 在甘特图中识别时差与关键路径
Once the Gantt chart is drawn for the earliest‑start schedule, the critical activities are those whose bars have no float – meaning they cannot be delayed without affecting the project’s minimum completion time. The critical path is the chain of such activities passing from start to finish.
当甘特图按照最早开始方案绘制完成后，那些没有时差的活动棒条就是关键活动——也就是说，它们一旦延迟会影响整个项目的最短完成时间。关键路径就是从开始到结束、由这类活动串连而成的路径。
If you have also marked latest start times or float on the bars, any activity whose extended (dotted) region has zero length lies on the critical path. Gantt charts make it visually obvious which activities overlap in time and where delays can be absorbed.
如果你在横道上还标注了最迟开始时间或时差，那么任何虚线延伸段长度为零的活动都处在关键路径上。甘特图能直观地反映出哪些活动在时间上重叠，以及哪些地方可以吸收延误。
4. Reading the Project Duration and Overlaps | 读取项目工期与活动重叠
The far‑right end of the longest bar gives the project’s completion time, which should equal the EFT of the sink node in the precedence network. The chart also reveals how many activities run concurrently during any time window, a feature directly useful for resource planning.
最长横道的最右端给出项目的总工期，这个数值应与前导网络中汇节点的 EFT 相等。图表还揭示了在任何时间窗口内同时运行的活动数量，这一特性对资源规划格外有用。
When multiple activities appear in a vertical slice of the chart, they demand resources at the same moment. By scanning the chart from left to right you can quickly find peaks in concurrency that may require extra workers.
当图表某一竖直切面上出现多条横道时，就意味着这些活动在同一时刻都会占用资源。从左到右扫描图表，你就能快速发现并发峰值，这些峰值可能需要额外增加人力。
5. Introducing the Resource Histogram | 资源直方图基础
A resource histogram is a bar chart where the horizontal axis is time (in the same units as the Gantt chart) and the vertical axis is the number of workers, machines or other resources required. It is constructed directly from the earliest‑start Gantt chart for a given resource type.
资源直方图是一种柱状图，横轴为时间（与甘特图单位一致），纵轴为所需的工人、机器或其他资源的数量。它是基于某一类型资源的甘特图（最早开始方案）直接绘制而成的。
To build the histogram, divide the project timeline into convenient time intervals (often days). For each interval, add up the resource demands of every activity that is active during that interval. Plot the total as a vertical bar for that period.
绘制直方图时，将项目时间轴分成合适的时间段（通常以天为单位）。对每一时段，统计该时段内所有正在进行中的活动的资源需求量之和，并在对应位置画出相应高度的柱条。
6. Step‑by‑Step Construction of a Resource Histogram | 资源直方图分步绘制法
Suppose a project involves four activities A, B, C and D. Their ESTs, durations and worker requirements are known from the schedule. Activity A runs from day 0 to 3 and needs 2 workers; B runs from day 0 to 2, needs 3 workers; C from day 2 to 6, needs 4 workers; D from day 3 to 7, needs 2 workers.
假设一个项目包含四项活动 A、B、C、D，它们的 EST、工期和所需工人数已通过排程知道。活动 A 从第 0 天运行到第 3 天，需要 2 名工人；B 从第 0 天运行到第 2 天，需要 3 名工人；C 从第 2 天运行到第 6 天，需要 4 名工人；D 从第 3 天运行到第 7 天，需要 2 名工人。
For day 0‑ ctive: (2 + 3) = 5 workers. Day 2, B is still active in the early part (if we assume whole‑ ting), but typical questions treat whole days; on day 2, A and B and C might all be active? plicity, exam questions usually let you list the daily number. Let’s assume activity B finishes at the end of day 1, so day 2 only A and C: 2+4=6 workers. Continue this way, drawing a bar for each day.
在 0–1 天，A 和 B 同时进行：2+3=5 名工人。第 2 天，若 B 在第 1 天结束时完工，那么第 2 天只有 A 和 C：2+4=6 名工人。以此类推，给每一天画一根柱状条。实际答题时务必遵循题目给定的计数规则。
Always read the question carefully: some papers use a “start‑of‑day” convention where an activity with duration d occupies d full days beginning at the start of its first day. Adopting a consistent approach is key.
务必仔细读题：有些试卷采用“日初”约定，即工期为 d 的活动从第一天开始时占用 d 个整天。保持方法一致至关重要。
7. Resource Smoothing and Levelling | 资源平滑与资源均衡
After drawing the earliest‑start resource histogram, you will often see sharp peaks and troughs, which are expensive and inefficient. Resource levelling (or smoothing) tries to obtain a more even profile without extending the project’s minimum completion time.
画完最早开始资源直方图后，你常会看到剧烈的高峰和低谷，这种波动既昂贵又低效。资源均衡（或称平滑）试图在不延长项目最短工期的前提下，得到一个更平缓的分布图。
Activities that possess total float can be shifted rightwards within their float window. By adjusting the start times of non‑critical activities, you can lower the peak resource demand. In an exam, you may be asked to draw a revised Gantt chart and its associated histogram that reduces the maximum number of workers.
拥有总时差的活动可以在其时差范围内向右平移。通过调整非关键活动的开始时间，你能够压低资源需求峰值。考试中可能会要求你画出修改后的甘特图及对应的直方图，以降低最大工人数。
8. Scheduling with Limited Resources | 有限资源下的调度
When the available resources are capped, a feasible schedule must respect that constraint even if it means delaying some activities beyond their EST. Such problems appear in Edexcel papers, often asking you to complete a Gantt chart under a fixed worker limit, e.g. “maximum 4 workers available”.
当可用资源存在上限时，可行的排程必须遵守这个约束，即使这意味着将某些活动推迟到 EST 之后。Edexcel 试卷中会出现此类问题，常要求你在固定工人上限下完成甘特图，如“最多可用 4 名工人”。
The typical algorithm: list activities in an agreed priority order (often based on EST, LFT or float). At each decision point, assign the next eligible activity as soon as enough resources become available. Sketch the bars sequentially, respecting the worker cap.
典型算法是：按约定的优先级顺序列出活动（通常基于 EST、LFT 或时差）。在每一个决策点，只要资源足够，就安排下一个符合条件的活动尽早开始。依次画出横道，同时遵守工人人数上限。
This manual process mimics real‑world resource‑constrained scheduling and tests your ability to think algorithmically while keeping correct precedence relationships.
这种手工调度模拟了现实世界中资源受限的排程过程，既能考验你的算法思维，又要求你始终保持正确的前导关系。
9. Common Mistakes to Avoid | 常见错误避坑指南
- Missing float visualisation: Drawing a Gantt chart that only shows earliest times is correct, but if the question asks for latest times or float, you must extend the bars. Always label float clearly. / 缺少时差可视化：只画最早时间的甘特图本身没有错，但如果题目要求显示最迟时间或时差，就必须延伸横道，并清晰标注时差。
- Counting overlap incorrectly: When building a histogram, forgetting that an activity might be active for only a few hours on a particular day can result in an over‑estimate. Respect the exact start and finish times. / 重叠计数错误：绘制直方图时，如果忽略某活动在当天可能只进行数小时，就会导致高估。要尊重精确的起止时刻。
- Destroying the critical path: When shifting activities to smooth resources, never move a critical activity. Only non‑critical activities can be rescheduled. / 破坏关键路径：平移活动以平滑资源时，绝对不能移动关键活动。只有非关键活动可以重新排程。
- Ignoring precedence: Shifting a non‑critical activity must not cause its predecessors to be delayed or its successors to be forced to start early. Always check the network logic. / 忽视前导关系：平移非关键活动不能导致它的前驱活动延迟，也不能强迫它的后继活动提前开始。务必核对网络逻辑。
10. Edexcel Exam‑Style Practice Approach | Edexcel 题型训练方法
Typical Edexcel Decision Maths 1 questions present a precedence table and ask you to draw a Gantt chart for the earliest‑start schedule. Part (b) often asks you to draw a resource histogram. Part (c) might give a resource constraint and require a new Gantt chart. Part (d) may ask you to calculate the minimum project duration under the constraint.
典型的 Edexcel 决策数学 1 考题会给出一张前导关系表，要求你画出最早开始方案的甘特图。第 (b) 部分常要求画出资源直方图。第 (c) 部分可能给出资源限制并要求新的甘特图。第 (d) 部分可能要求计算该限制下的最短项目工期。
To master these multi‑step items, practise with small projects first (3‑5 activities) and then move to more complex ones with 7‑9 activities. Always keep your working tidy; use squared paper for Gantt charts in the exam. The cascade chart can be drawn directly beneath the resource histogram to help visualise the cause of peaks.
要想掌握这类多步骤题目，先练习 3–5 项活动的小项目，再过渡到 7–9 项活动的较复杂项目。答题务必保持卷面整洁；考试时用方格纸绘制甘特图。把瀑布图画在资源直方图的正下方，有助于看清峰值产生的原因。
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