📚 Further Maths Decision Maths 1: Exam Question Analysis | 进阶数学 决策数学1 题型解析
Decision Mathematics 1 (D1) is a core module in many A-level Further Mathematics specifications, focusing on algorithms, graph theory, linear programming, and critical path analysis. This article breaks down the most common exam-style question types, offering step-by-step strategies and key insights to help you tackle them with confidence.
决策数学1（D1）是许多A-level进阶数学大纲中的核心模块，专注于算法、图论、线性规划和关键路径分析。本文拆解最常见的考试题型，提供分步策略和关键见解，帮助你从容应对。
1. Sorting Algorithms | 排序算法
Sorting algorithms are frequently tested: you may be asked to perform one pass of bubble sort, complete a shuttle sort, or demonstrate quick sort on a list. Each algorithm has a specific procedure and efficiency. Bubble sort compares adjacent pairs; shuttle sort (also known as insertion sort) inserts each element into the sorted section; quick sort selects a pivot and partitions.
排序算法经常被考查：可能要求执行一次冒泡排序、完成穿梭排序或演示快速排序。每种算法都有特定步骤和效率。冒泡排序比较相邻对；穿梭排序（即插入排序）将每个元素插入已排序部分；快速排序选取枢轴并分区。
Exam tip: Keep a clear record of the number of comparisons and swaps, as marks are often allocated for these counts. For bubble sort, after the first pass the largest element ‘bubbles’ to the end. For quick sort, ensure you correctly list sublists after partitioning and apply the algorithm recursively.
考试提示：清晰记录比较和交换次数，这些计数往往有分值。冒泡排序中，第一趟后最大元素“冒泡”到最后。快速排序要确保在分区后正确列出子列表，并递归应用算法。
Example: Given the list [6, 3, 8, 5, 2], one pass of bubble sort: compare 6 and 3 → swap (3,6,8,5,2); compare 6 and 8 → no swap; compare 8 and 5 → swap (3,6,5,8,2); compare 8 and 2 → swap (3,6,5,2,8). End of pass 1. Comparisons: 4, swaps: 3.
示例：给定列表 [6, 3, 8, 5, 2]，一次冒泡排序的趟：比较6和3→交换 (3,6,8,5,2)；比较6和8→不交换；比较8和5→交换 (3,6,5,8,2)；比较8和2→交换 (3,6,5,2,8)。第1趟结束。比较次数：4，交换次数：3。
2. Bin Packing Algorithms | 装箱算法
Bin packing problems appear regularly. You are given items of various sizes and bin capacity, and must apply heuristics: first-fit, first-fit decreasing, and sometimes full-bin packing. First-fit places each item into the first available bin that has enough space. First-fit decreasing sorts items in descending order first, which usually improves the solution.
装箱问题经常出现。给定不同大小的物品和箱子容量，需应用启发式算法：首次适应、降序首次适应，有时还有满箱装箱。首次适应将每个物品放入第一个有足够空间的箱子。降序首次适应先将物品降序排序，通常会改进结果。
Full-bin combinations identify groups of items that exactly fill a bin, reducing waste. In exam questions, you must display the bins’ contents clearly, often in a table. The number of bins used is compared against the theoretical minimum (total size ÷ bin capacity).
满箱组合是找出恰好填满一个箱子的物品组，减少浪费。考试中需要清楚地展示各箱子的内容，通常以表格呈现。所用箱子数量常与理论最小值（总大小÷箱容量）比较。
Common pitfalls: Failing to sort correctly for first-fit decreasing, or missing a full-bin opportunity. When drawing bins, list items in the order they are placed.
常见陷阱：降序首次适应中排序错误，或错过满箱组合。绘制箱子时，按放置顺序列出物品。
3. Graph Theory Terminology | 图论术语
Decision 1 expects you to know precise definitions: graph, vertex, edge, degree, path, cycle, tree, spanning tree, simple graph, directed graph, and network. Questions may ask you to identify whether a graph is simple, or to find a minimum connector. A tree is a connected graph with no cycles; a spanning tree connects all vertices using some edges.
决策1要求掌握精确的定义：图、顶点、边、度、路径、环、树、生成树、简单图、有向图和网络。问题可能要求判断一个图是否为简单图，或找出最小连接器。树是无环的连通图；生成树使用部分边连接所有顶点。
You might be given a list of vertices and edges and asked to draw the graph, or to complete a table of orders (degrees). Remember the handshaking lemma: sum of degrees = 2 × number of edges. In a weighted graph, the sum of edge weights matters for optimisation problems.
可能给定顶点和边列表，要求画出图或完成度数表。记住握手引理：度数之和 = 2 × 边数。在加权图中，边权总和对优化问题很重要。
4. Minimum Spanning Trees | 最小生成树
Two classical algorithms are Kruskal’s and Prim’s. Kruskal’s algorithm selects edges in increasing weight order, avoiding cycles, until n−1 edges are chosen. Prim’s algorithm starts from any vertex and grows a tree by adding the cheapest edge connecting the tree to a new vertex.
两个经典算法是Kruskal和Prim。Kruskal算法按权重递增顺序选边，避开环，直至选出n−1条边。Prim算法从任意顶点出发，每次添加连接树与新顶点的最便宜边来扩展树。
In an exam, you might be asked to apply both algorithms on the same network to demonstrate they give the same minimum total weight. For Kruskal, list edges in order and delete a selected edge only if it does not create a cycle. For Prim, use a tabular format or a clear stepwise list of added vertices and edges.
考试中可能要求在同一网络上应用两种算法，说明它们得出相同的最小总权重。对Kruskal，按顺序列出边，仅当选出的边不产生环时才添加。对Prim，使用表格或清晰的逐步添加的顶点与边列表。
Key exam technique: Show all rejected edges for Kruskal, and for Prim indicate the order of vertex inclusion. State the total weight clearly.
关键考试技术：对Kruskal要展示所有被拒绝的边，对Prim要标明顶点加入的顺序。清晰写出总权重。
5. Dijkstra’s Shortest Path Algorithm | Dijkstra最短路径算法
Dijkstra’s algorithm finds the shortest path from a start node to all other nodes in a weighted graph with non-negative weights. It uses labels (temporary and permanent) updating the working values as better routes are found. The path is reconstructed by backtracking.
Dijkstra算法在边权非负的加权图中找出从起点到所有其他节点的最短路径。它使用标号（临时和永久），一旦发现更优路径就更新工作值。通过回溯重建路径。
In answer scripts, you must produce a complete labelling table or on-diagram boxes showing the order of finalisation and distance from start. Exam questions frequently then ask for the route itself, e.g., S → A → C → F → T. Always write the final path clearly, and sometimes you must state its length.
在答卷中，必须绘制完整的标号表或图上标签框，显示永久化顺序和距起点的距离。考试题常会接着要求写出路径本身，如 S → A → C → F → T。务必清晰写出最终路径，有时还须说明其长度。
Watch out for networks where multiple shortest paths exist; the algorithm can find any one. Also, ensure you do not forget to backtrack: follow ‘previous vertex’ pointers from destination back to start.
注意存在多条最短路径的网络；算法可以找出任意一条。同时切勿忘记回溯：从目的地沿“前一顶点”指针回到起点。
6. Critical Path Analysis (CPA) | 关键路径分析
Activity-on-node (AON) diagrams are standard. You will be given a table of activities, durations, and precedences. Build the network, then perform forward and backward passes to find earliest start times (EST) and latest finish times (LFT). Total float = LFT – duration – EST. Critical activities have zero total float.
节点活动图（AON）是标准形式。会给出包含活动、历时和紧前关系的表格。构建网络，然后执行前向和后向推算，求出最早开始时间（EST）和最晚完成时间（LFT）。总浮动 = LFT – 历时 – EST。关键活动总浮动为零。
Exam questions typically ask for: the minimum project completion time; the critical path(s); and the effect of delaying a specific activity. Be systematic: use a two-pass process, label each node clearly with EST and LFT. A neat diagram prevents errors.
考试题目通常要求：最小项目完成时间；关键路径；以及延迟某特定活动的影响。要有条理：使用两次推算过程，在每个节点上清晰标出EST和LFT。整洁的图可避免出错。
When an activity has incoming dummies or overlaps, draw carefully according to precedence constraints. Remember that dummy activities have zero duration and are used to maintain logic.
当活动有引入虚活动或重叠时，要根据紧前约束仔细绘制。记住虚活动历时为零，用于维护逻辑。
7. Linear Programming – Formulation and Modelling | 线性规划——建模
Linear programming (LP) problems in D1 often require you to define decision variables, write the objective function, and construct constraints from a word problem. Variables usually represent quantities of products or resources. Use clear notation such as x₁, x₂, with units stated.
决策1中的线性规划（LP）问题经常要求定义决策变量，写出目标函数，并根据文字题构建约束。变量通常代表产品数量或资源量。使用清晰记号如 x₁、x₂，并注明单位。
Constraints are linear inequalities, e.g. 2x₁ + 3x₂ ≤ 48. Always include non-negativity constraints (x₁, x₂ ≥ 0). The objective might be to maximise profit or minimise cost. Ensure that you convert all information faithfully; a small misinterpretation can lead to an entirely different feasible region.
约束为线性不等式，如 2x₁ + 3x₂ ≤ 48。务必包括非负约束（x₁, x₂ ≥ 0）。目标可能是最大化利润或最小化成本。确保忠实地转换所有信息；小的误解可能导致完全不同的可行域。
Tip: Check for ‘at least’, ‘at most’, and ratios. If a problem states ‘at least twice as many A as B’, translate as x_A ≥ 2x_B.
提示：检查“至少”、“至多”和比例。如果问题说“A的数量至少是B的两倍”，翻译为 x_A ≥ 2x_B。
8. Linear Programming – Graphical Solution | 线性规划——图解法
After formulating the LP, you draw the feasible region on a graph. Plot each constraint as a straight line, shade the unwanted region or outline the feasible polygon. Identify the coordinates of all vertices.
建立LP后，在图上绘制可行域。将每个约束画作直线，屏蔽不可行区域或勾勒出可行多边形。识别所有顶点的坐标。
The optimal solution occurs at a vertex of the feasible region. Use a ruler to draw the objective function line (e.g. profit line) and slide it parallel until it is about to leave the region, touching the last vertex. The coordinates of that vertex give the optimal solution. Substitute back to find the optimal value.
最优解出现在可行域的顶点上。用直尺绘制目标函数线（如利润线），平行移动直至即将离开区域，触及最后一个顶点。该顶点的坐标给出最优解。代回求出最优值。
Exam questions may require you to test integer solutions if variables must be integers. This involves checking points near the optimum vertex while staying inside the feasible region. Show all working clearly.
考试可能要求测试整数解（若变量必须为整数）。这需要检查最优顶点附近的点且满足可行域。清晰展示所有步骤。
9. Transportation Problems – Finding Initial Solutions | 运输问题——求初始解
A balanced transportation problem has supply equal to total demand. You are given a cost matrix. Two common initial solution methods: the North-West Corner rule and the Least Cost method. In an exam, you may need to apply both and compare costs.
平衡运输问题中总供给等于总需求。给定成本矩阵。两种常用初始解方法：西北角法则和最低成本法。考试中可能需要应用两者并比较成本。
North-West Corner starts at the top-left cell, allocating as much as possible without exceeding row supply or column demand, then moves right or down. Though mechanically simple, it ignores costs, so solutions may be far from optimal. The Least Cost method selects the cell with the smallest unit cost each time, which often gives a better starting solution.
西北角从左上角单元格开始分配，尽可能填满而不超出行供给或列需求，然后向右或向下移动。虽然机械简单，但忽略了成本，因此解可能离最优较远。最低成本法每次选择单位成本最小的单元格，通常给出更好的初始解。
Always check that the number of allocated cells equals m + n − 1 (non-degenerate). If not, add a zero allocation in a suitable independent cell to proceed with optimality testing.
始终检查已分配单元格数是否等于 m + n − 1（非退化）。若不是，在合适的独立单元格中添加零分配以进行最优性检验。
| Method | Pros | Cons |
|---|---|---|
| North-West Corner | Quick, systematic | Ignores cost |
| Least Cost | Cost sensitive | May need tie-breaking |
方法比较（见上表）
10. Transportation Problems – Optimality Testing | 运输问题——最优性检验
Stepping-stone or MODI (the latter is common in Decision 1) are used to check optimality. The MODI method calculates dual variables uᵢ and vⱼ for rows and columns using the allocated cells, then evaluates shadow costs for unallocated cells. If any improvement index (vᵢⱼ − costᵢⱼ) is positive (for minimisation), the solution can be improved.
跳石法或MODI法（后者在决策1中常见）用于检验最优性。MODI法通过已分配单元格计算行和列的对偶变量 uᵢ 和 vⱼ，然后评估未分配单元格的影子成本。若任何改进指数（vᵢⱼ − costᵢⱼ）为正（对最小化问题），则可改进。
To improve: form a loop from the entering cell through allocated cells, adding and subtracting θ alternately. The maximum θ is the smallest shipment in the subtracting cells. Adjust allocations and repeat until all improvement indices are ≤ 0.
改进方法：从进入单元格形成回路，通过已分配单元格，交替加减 θ。最大 θ 为减少单元格中的最小运量。调整分配并重复，直到所有改进指数 ≤ 0。
Exam accuracy depends on setting up initial tables neatly and performing arithmetic without error. Present the loop clearly, and state the new transportation cost after each iteration.
考试准确性取决于整洁设置初始表格、无错误运算。清晰展示回路，并在每次迭代后声明新的运输成本。
11. Matchings in Bipartite Graphs | 二分图匹配
Matching questions involve bipartite graphs where vertices are split into two sets, often representing tasks and workers or students and projects. The initial matching is built from a given bipartite graph, and then you find a maximal matching using an alternating path algorithm.
匹配问题涉及二分图，顶点分为两个集合，常表示任务与工人或学生与项目。根据给定的二分图构造初始匹配，然后使用交替路径算法找出最大匹配。
Starting from an unmatched vertex in the first set, attempt to construct an alternating path that ends at an unmatched vertex in the second set. If an augmenting path is found, flip the matching status along the path to increase the number of matched edges by one. Repeat until no augmenting path exists.
从第一个集合中未匹配的顶点出发，尝试构建一条交替路径，终止于第二个集合中未匹配的顶点。若找到增广路径，就沿路径翻转匹配状态，使匹配边数增加一。重复直至没有增广路径。
In exams, you must show the initial matching as a set of edges, then list the alternating paths tried, and finally state the maximal matching and the number of matched vertices. Be systematic: label vertices and draw clear diagrams.
考试中，必须将初始匹配表示为边集，然后列出尝试的交替路径，最后声明最大匹配和已匹配的顶点数。要条理清晰：标注顶点并画出清晰图表。
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