📚 Decision Mathematics 1 Question Types Analysis | 决策数学1 题型解析
Decision Mathematics 1 (D1) is a core applied module in A-level Further Mathematics that equips students with fundamental algorithms and modelling techniques used to solve real-world problems such as optimisation, routing, scheduling, and resource allocation. Mastering the typical question types and knowing exactly what the examiner expects in each scenario is essential for achieving top marks. This guide breaks down every major question type, highlights common pitfalls, and provides strategic revision advice.
决策数学1（D1）是A-level进阶数学中的核心应用模块，要求学生掌握基本算法与建模技术，以解决优化、路线规划、调度和资源分配等实际问题。熟练掌握典型题型，并准确理解考官的得分要求，是取得高分的关键。本指南逐一分解主要题型，指出常见易错点，并提供有效的复习策略。
1. Sorting Algorithms and Binary Search | 排序算法与二分搜索
Sorting questions typically present an unordered list and ask you to apply the bubble sort or quick sort algorithm for a given number of passes, or until the list is fully sorted. For bubble sort, you must show every comparison and swap clearly, while for quick sort, you need to indicate the pivot choice and sublists at each stage.
排序题通常会给出一个无序列表，要求你执行指定次数的冒泡排序或快速排序，或一直排序到完全有序。对于冒泡排序，必须清晰展示每一次比较和交换；对于快速排序，则需标明每次选择的枢轴及各阶段产生的子列表。
A complete bubble sort question might also ask for the number of comparisons and swaps. Remember that after the k-th pass, the k largest elements are in their final positions. For quick sort, always state the pivot and then list the resulting sublist after partitioning; repeat until all sublists are of length one.
完整的冒泡排序题可能还会要求计算比较次数与交换次数。记住，第k遍之后，前k个最大元素已处于最终位置。对于快速排序，必须声明枢轴，然后列出划分后的子列表；不断重复，直至所有子列表长度均为1。
Binary search is tested on an ordered list. You must show the middle item chosen at each step and how the search interval narrows, terminating when the target is found or the interval is empty. Always count the number of comparisons, as this is a common mark.
二分搜索在有序列表上进行考查。需要展示每一步所选中间项，以及搜索区间如何缩小，直至找到目标或区间为空时终止。务必统计比较次数，这通常是明确的得分点。
Mid = ⌊(Low + High) / 2⌋
2. Bin Packing Algorithms | 装箱算法
Bin packing problems involve packaging items of given sizes into bins of fixed capacity, and you are required to apply the first-fit, first-fit decreasing, or full-bin algorithm. Always list the items in the order given (or sorted order for FFD) and show the bins filling up step by step.
装箱问题要求将给定尺寸的物品装入容量固定的箱子中，通常需要应用首次适应、降序首次适应或满箱算法。务必按给定顺序（FFD则先排序）列出物品，并逐步展示箱子填充过程。
A typical exam question asks for the number of bins used by each method and sometimes the optimal solution. When using first-fit decreasing, sort items into descending order first. Do not forget to redraw bins clearly; crossed-out figures still count as part of the working.
典型的考题会要求给出每种方法使用的箱子数，有时还会询问最优解。使用降序首次适应时，首先要将物品按降序排列。不要忘记清楚地画出箱子；划掉的数字仍然作为解题过程的一部分。
Full-bin packing tries to form bins that are exactly full from off-cut items. This heuristic is often used to find a lower bound or improve a solution. You can sometimes combine items to achieve a better result than standard algorithms.
满箱算法尝试用边角料物品拼出恰好装满的箱子。这个启发式方法常用于寻找下界或改进解。有时你能通过组合物品获得比标准算法更好的结果。
3. Graphs and Networks Fundamentals | 图与网络基础
Questions on graph theory require precise language: vertices, edges, degree, path, cycle, tree, and isomorphism. You must be able to state definitions, determine whether a graph is Eulerian (all even degrees) or semi-Eulerian (exactly two odd degrees), and construct adjacency and distance matrices.
图论题目要求使用精确的术语：顶点、边、度、路径、圈、树以及同构图。需要能够陈述定义，判断一个图是欧拉图（所有度均为偶数）还是半欧拉图（恰好有两个奇数度），并能构建邻接矩阵和距离矩阵。
A common task is to list the degree sequence of a graph or determine the number of edges from the sum of degrees using the Handshaking Lemma: Sum of degrees = 2 × number of edges. Also be prepared to check for subgraphs and complements.
常见任务包括列出图的度序列，或利用握手引理由度总和推算边数：度总和 = 2 × 边数。还要能检验子图与补图。
∑ deg(v) = 2|E|
4. Minimum Spanning Trees | 最小生成树
To find a minimum spanning tree (MST), you can apply Prim’s algorithm (starting from any vertex, adding the cheapest connecting edge) or Kruskal’s algorithm (sorting edges by weight, adding without forming cycles). Both require clear steps and a statement of the order of edge selection.
寻找最小生成树（MST）可使用普里姆算法（从任一顶点出发，添加权重最小的连接边）或克鲁斯卡尔算法（将边按权重排序，不构成环则添加）。两种方法都需要清晰步骤并说明边的选择顺序。
In Prim’s algorithm on a matrix, you cross out columns or rows systematically. In Kruskal’s, you must redraw or list edges in ascending weight order, explicitly rejecting any edge that would create a cycle. A follow-up question might ask for the total weight or its use in a network.
在矩阵上使用普里姆算法时，需要系统地划去列或行。在克鲁斯卡尔算法中，必须按权重递增顺序重画或列出边，并明确拒绝任何会形成环的边。随后可能会要求给出总权重或其在实际网络中的意义。
5. Dijkstra’s Shortest Path Algorithm | Dijkstra最短路径算法
Dijkstra’s algorithm finds the shortest distance from a start node to all others. You need to draw boxes at each vertex showing the working values, final label, and order of permanent labelling. Box values typically consist of the working distance and the preceding vertex.
Dijkstra算法用于找出从起始节点到所有其他节点的最短距离。需要在每个顶点绘制方框，显示临时值、最终标号及永久标号的次序。方框内通常包含工作距离和前驱顶点。
After completing the algorithm, you must write down the shortest path by tracing backwards. A second part of the question often duplicates the network with a modified edge weight and asks for the effect on the shortest path without completely redoing the algorithm.
算法完成后，必须通过反向追踪写出最短路径。题目的后半部分常复制网络并修改某条边的权重，要求说明对最短路径的影响，而无需完全重做算法。
d(v) = min( d(u) + w(u,v) )
6. Route Inspection Problem | 路线检查问题
The route inspection (Chinese postman) problem requires a closed walk traversing every edge at least once. For a network with all even vertices, the total length is simply the sum of all edge weights. When odd vertices exist, you must pair them to minimise the repeated distance.
路线检查（中国邮递员）问题要求找到一条至少遍历每条边一次的闭合回路。对于所有顶点度均为偶数的网络，总路径长度就是所有边权重之和。当存在奇度顶点时，必须将它们配对以最小化重复距离。
To solve, list odd-degree vertices, find all possible pairings (often just one or two), calculate the shortest path lengths between each pair using Dijkstra or given distances, and choose the minimum total pairing length. Then add this repeated distance to the total weight.
解题步骤：列出奇度顶点，找出所有可能的配对（通常只有一或两种），利用Dijkstra算法或给定距离计算每对之间的最短路径长度，选择总配对长度最小的方案。然后将该重复距离加到总权重上。
7. Travelling Salesman Problem | 旅行商问题
The travelling salesman problem (TSP) seeks a minimum-weight Hamiltonian cycle. Exact solution for a small complete graph can be found by listing all distinct tours. However, questions usually test heuristic algorithms: nearest neighbour, and finding a lower bound by deleting a vertex and using MST + two edges.
旅行商问题（TSP）要求找到最小权重的哈密顿环。对于小型完全图，可通过列出所有不同的路径来精确求解。但考题通常考查启发式算法：最近邻法，以及通过删除一个顶点并利用MST加两条边来求下界。
For the nearest neighbour algorithm, start at the specified vertex, go to the nearest unvisited vertex, and repeat; finally return to start. You must write the tour and its total length. A better upper bound may be obtained by trying different start points or using an improvement heuristic.
对于最近邻算法，从指定顶点出发，前往最近的未访顶点，重复直至所有点被访问，最后返回起点。必须写出完整的路径及其总长度。尝试不同起点或使用改进启发式可获得更优的上界。
A typical lower bound is found by removing a vertex, finding an MST on the remaining vertices, then adding the two cheapest edges from the removed vertex. Check all vertices for the best (largest) lower bound.
下界的常见求法是：删除某个顶点，在剩余顶点上找到一棵MST，然后加上从被删顶点出发的两条最便宜边。对所有顶点尝试，取最佳（最大）下界。
8. Critical Path Analysis | 关键路径分析
Critical path analysis involves activity networks (activity on node or on arc). You must be able to construct a precedence table and a corresponding network, checking for uniqueness. Then perform forward and backward passes to find earliest and latest event times, floats, and the critical path.
关键路径分析涉及活动网络（节点型或箭线型）。需要能绘制前导关系表和相应的网络图，并检查唯一性。然后通过正向和反向递推求出最早和最迟事件时间、浮动时间以及关键路径。
Forward pass: for each activity, earliest start = max of earliest finish times of predecessors. Backward pass: latest finish = min of latest start times of successors. Total float = latest finish – duration – earliest start. Activities with zero total float are critical.
正向递推：每个活动的最早开始时间等于所有前驱活动最早结束时间的最大值。反向递推：最迟结束时间等于所有后继活动最迟开始时间的最小值。总浮动时间 = 最迟结束 – 持续时间 – 最早开始。总浮动为零的活动位于关键路径上。
Gantt (cascade) charts are often required, showing each activity scheduled at its earliest start time and the slack they have. A problem might ask for the minimum number of workers needed, which requires resource levelling.
通常要求绘制甘特（阶梯）图，按最早开始时间安排每个活动并显示其松弛时间。题目还可能要求找出最少所需工人数，这就需要资源调配。
9. Linear Programming | 线性规划
Linear programming questions present constraints as inequalities in terms of x and y. You must formulate the problem (if not given as inequalities already), draw the feasible region accurately, and test the objective function at all vertices, or use an objective line to find the optimal integer solution if required.
线性规划问题以关于x和y的不等式约束给出。如果未直接以不等式形式给出，需要自己建立模型。然后精确画出可行域，在所有顶点处检验目标函数，或根据需要使用目标函数直线寻找最优整数解。
When shading the feasible region, be consistent with the convention (usually shade out, or shade in – follow the exam board’s expectation). Label each line clearly and write its equation. If the variables must be integers, test integer coordinates around the vertex that optimises the fractional solution.
给可行域画阴影时，要遵循一致的习惯（通常画“不可行”区域或“可行”区域——按考试局要求执行）。清楚标出每条直线并写出方程。若变量要求取整数，需在使小数解最优的顶点附近检验整数坐标点。
Maximise P = 3x + 5y subject to 2x + y ≤ 20, x + 3y ≤ 30, x ≥ 0, y ≥ 0
10. Matchings and the Hungarian Algorithm | 匹配与匈牙利算法
Matching problems ask for a maximal or maximum matching in a bipartite graph. You may start with an initial matching and then use the alternating path algorithm to improve it until no further augmenting paths exist.
匹配问题要求找出二分图中的一个极大匹配或最大匹配。通常从一个初始匹配开始，然后利用交错路算法进行改进，直到不存在可增广路径为止。
The Hungarian algorithm (for optimising an allocation problem on a cost matrix) follows the steps: reduce rows, reduce columns, cover zeros with a minimum number of lines, and adjust the matrix by subtracting the smallest uncovered element. Repeat until an optimal assignment is achieved.
匈牙利算法（用于在代价矩阵上优化指派问题）步骤为：行归约、列归约、用最少条线覆盖所有零元素，并通过减去未被覆盖的最小元素来调整矩阵。重复直至得到最优指派。
For the alternating path algorithm on a bipartite graph, you must clearly mark matched edges and alternating vertices, and then exchange matched and unmatched edges along the augmenting path to increase the matching size by one.
在二分图上使用交错路算法时，必须清楚标记匹配边和交替顶点，然后沿可增广路径交换匹配边与非匹配边，使匹配基数增加1。
11. Flows in Networks | 网络流
Network flow problems involve pushing flow from a source to a sink. You need to understand capacities, flow, saturated edges, back capacity (the ability to send flow back along an edge), and the concept of a minimum cut.
网络流问题涉及从源点到汇点推送流量。需要理解容量、流量、饱和边、反向容量（沿边送回流量的能力），以及最小割的概念。
The maximum flow – minimum cut theorem states that the value of the maximum flow equals the capacity of the minimum cut. To find the maximum flow, you usually start with zero flow, use flow-augmenting paths, and update the residual network. Label each edge with flow/capacity.
最大流-最小割定理指出，最大流的值等于最小割的容量。要找出最大流，通常从零流开始，利用增流路径，并更新残差网络。每条边标注为 流/容量。
When constructing a cut after finding the maximum flow, list the edges crossing from the source side to the sink side, and verify that the total capacity equals the flow value. Be careful with edges that go backward across the cut – they are not included in the cut capacity under the standard definition but matter for flow conservation.
在找到最大流后构建割时，列出从源点侧到汇点侧交叉的边，并验证总容量等于流值。注意反向穿越割的边——按标准定义它们不计入割容量，但对流守恒有影响。
Flow value = capacity of minimal cut = sum of forward edges in cut
Published by TutorHao | Decision Mathematics 1 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导