📚 Mastering Decision Mathematics 1: Core Concepts and Problem-Solving Techniques | 决策数学1核心知识点精讲与解题方法
Decision Mathematics 1 (D1) forms a vital part of many further mathematics qualifications, introducing students to algorithms, networks, and optimisation techniques that underpin modern computing and logistics. This article dissects the key topics commonly assessed in D1 exam papers, combining concise explanations with practical problem-solving approaches to help you master each concept efficiently.
决策数学1（D1）是许多进阶数学课程的重要组成部分，向学生介绍支撑现代计算与物流的算法、网络和优化技术。本文剖析了D1考卷中常见的关键知识点，结合简明解释和实用的解题方法，帮助你高效掌握每一个概念。
1. Algorithms and Their Efficiency | 算法及其效率
An algorithm is a finite sequence of step-by-step instructions designed to solve a specific problem. In D1, you need to understand how to describe algorithms using flowcharts or written steps. The efficiency of an algorithm is often measured by the number of operations required as the input size n grows. Common orders of complexity include constant time O(1), linear O(n), and quadratic O(n²).
算法是一组有限步骤的指令序列，用于解决特定问题。在D1中，你需要会用流程图或文字步骤描述算法。算法的效率通常通过输入规模n增大时所需的操作次数来衡量。常见的复杂度包括常数阶O(1)、线性阶O(n)和平方阶O(n²)。
For example, a linear search algorithm checks each item in a list one by one, taking at most n comparisons. Its worst-case efficiency is O(n). In contrast, the bubble sort algorithm repeatedly compares adjacent items and swaps them if they are in the wrong order; it requires roughly n²/2 comparisons in the worst case, giving O(n²). Understanding these differences helps you justify which algorithm is more suitable for a given data set.
例如，线性搜索算法逐个检查列表中的元素，最多需要n次比较。其最坏情况效率为O(n)。而冒泡排序算法反复比较相邻元素并在顺序错误时交换它们，最坏情况下约需n²/2次比较，即O(n²)。理解这些差异有助于你判断哪种算法更适合给定的数据集。
2. Sorting Algorithms: Bubble, Shuttle, and Quick Sort | 排序算法：冒泡、穿梭和快速排序
Sorting algorithms rearrange a list into ascending or descending order. Bubble sort makes multiple passes through the list, comparing adjacent pairs and swapping if necessary. After the first pass, the largest element ‘bubbles’ to the end. The process repeats for the remaining unsorted portion until no swaps occur.
排序算法将列表重新排列为升序或降序。冒泡排序对列表进行多轮扫描，比较相邻元素并在需要时交换。第一轮后最大元素会“冒泡”到末尾。对剩余未排序部分重复直到没有交换发生。
The shuttle sort, also known as insertion sort, builds a sorted sublist by taking each element and inserting it into the correct position within the already sorted part. Quick sort selects a pivot element and partitions the list into two sublists: items less than the pivot and items greater than the pivot. It then recursively sorts each sublist. On exam papers you may be asked to show the list after each pass or partitioning step, so careful step-by-step working is essential.
穿梭排序又称插入排序，通过将每个元素插入到已排序部分的正确位置来构建有序子列表。快速排序选择一个主元（pivot），将列表分割为两个子列表：小于主元的元素和大于主元的元素。然后递归地对子列表排序。考卷上可能要求你展示每一轮或每次分割后的列表，因此仔细写出步骤至关重要。
3. Binary Search and Search Algorithms | 二分查找与搜索算法
Binary search is an efficient algorithm for finding an item in a sorted list. It works by repeatedly dividing the search interval in half. You compare the target value with the middle element; if not equal, you discard the half that cannot contain the target and continue on the remaining half. The maximum number of comparisons is log₂n, making it O(log n).
二分查找是在有序列表中查找元素的高效算法。它不断将搜索区间对半分。你将目标值与中间元素比较；若不相等，则丢弃不可能包含目标的一半，继续在另一半中搜索。最大比较次数为log₂n，因此复杂度为O(log n)。
In D1 questions, you often need to apply binary search to a given list and record the pivot values tested at each stage. You might also be asked to trace a binary search tree and determine how many comparisons are required to locate (or fail to locate) a specific item. It is crucial to clearly show the lower and upper bounds at each step.
在D1题目中，你通常需要对给定列表应用二分查找，并记录每个阶段测试的中间值。你还可能被要求追踪二分搜索树，并确定定位（或未能定位）某个特定项所需的比较次数。关键是要清楚地显示每一步的下界和上界。
4. Graphs and Networks: Terminology and Representation | 图与网络：术语和表示
A graph consists of vertices (nodes) connected by edges (arcs). In D1, you encounter simple graphs, directed graphs (digraphs), weighted graphs, and trees. Important terms include degree (number of edges meeting at a vertex), path, cycle, and connectedness. You must be able to represent a graph using an adjacency matrix or an adjacency list.
图由顶点（节点）和边（弧）组成。在D1中，你会遇到简单图、有向图、加权图和树。重要术语包括度（一个顶点所连接的边数）、路径、回路和连通性。你必须能够用邻接矩阵或邻接表来表示图。
An adjacency matrix is a square matrix where the entry in row i and column j indicates the number of edges between vertex i and vertex j (or the weight if weighted). For an undirected graph, the matrix is symmetric. A tree is a connected graph with no cycles; a spanning tree of a graph includes all vertices and is a tree. These fundamentals lay the groundwork for network algorithms such as shortest path and route inspection.
邻接矩阵是一个方阵，其中第i行第j列的项表示顶点i和顶点j之间的边数（或带权重的边的权重）。对于无向图，矩阵是对称的。树是没有回路的连通图；图的生成树包含所有顶点且是一棵树。这些基础知识为最短路径和路径检查等网络算法奠定基础。
5. Shortest Path: Dijkstra’s Algorithm | 最短路径：Dijkstra算法
Dijkstra’s algorithm finds the shortest path from a start node to every other node in a weighted graph where all weights are non-negative. It uses temporary labels that are updated as the algorithm explores. You begin at the start node with a permanent label of 0. At each step, you select the unvisited node with the smallest temporary label, make its label permanent, and update the temporary labels of its neighbours by adding the edge weight.
Dijkstra算法用于在所有权重非负的加权图中找出从起始节点到其他各节点的最短路径。它使用临时标签，在探索过程中更新。你从起始节点开始，赋予永久标签0。每一步选择具有最小临时标签的未访问节点，将其标签永久化，并通过加上边权来更新其邻居的临时标签。
In exam problems, you are often asked to fill in a table showing the order of permanent labelling, the working values at each vertex, and the final shortest distances. You may also need to trace back the shortest path route. Common errors include forgetting to update temporary labels when a shorter path is found, so always check all neighbouring vertices carefully.
在考试问题中，常常让你填写表格，展示永久标签的顺序、每个顶点的计算值以及最终最短距离。你还可能需要回溯出最短路径的路线。常见错误包括当发现更短路径时忘记更新临时标签，因此务必仔细检查所有相邻顶点的状态。
6. Critical Path Analysis: Activity Networks and Float | 关键路径分析：活动网络与浮动时间
Critical path analysis (CPA) is used to plan and schedule projects. Activities are represented on an activity-on-arc network, where arcs represent activities and vertices represent events (milestones). Each activity has a duration, and the network respects precedence constraints. The earliest start time (EST) and latest finish time (LFT) are calculated for each event using forward and backward passes.
关键路径分析（CPA）用于规划项目进度。活动在活动-弧线网络中表示，弧代表活动，顶点代表事件（里程碑）。每个活动有时间长度，网络遵守先后关系约束。通过前向扫描和后向扫描计算每个事件的最早开始时间（EST）和最晚结束时间（LFT）。
Total float for an activity is the amount of time it can be delayed without delaying the whole project: LFT at end node – EST at start node – duration. Activities with zero total float lie on the critical path. The critical path is the longest path through the network and determines the minimum project completion time. D1 questions often require you to draw the network, complete a precedence table, and determine the critical activities.
某项活动的总浮动时间是指它可被推迟而不影响整个项目的时间量：终节点的LFT – 起节点的EST – 持续时间。总浮动时间为零的活动位于关键路径上。关键路径是网络中耗时最长的路径，决定项目的最短完成时间。D1题目通常要求你绘制网络图，完成先后关系表，并确定关键活动。
7. Linear Programming: Graphical Method | 线性规划：图解法
Linear programming (LP) involves maximising or minimising a linear objective function subject to a set of linear inequalities (constraints). In two variables, the feasible region is a convex polygon. The optimal solution occurs at a vertex of this region. You graph the constraints, identify the feasible region, and then test the objective function at each vertex, or use an objective line to find the optimal point.
线性规划（LP）涉及在一组线性不等式（约束条件）下最大化或最小化线性目标函数。在两个变量情况下，可行域是一个凸多边形。最优解出现在该区域的一个顶点处。你画出约束条件，确定可行域，然后在每个顶点处检验目标函数值，或使用目标直线来找到最优解点。
You also need to be able to deal with integer solutions when the problem requires discrete quantities. If the optimal vertex has non-integer coordinates, you must test integer points within the feasible region near that vertex. LP questions often combine contextual phrasing: ‘at least twice as many of x as y’ translates to x ≥ 2y. Careful labelling of the graph and clear working are essential for full marks.
当问题要求离散量时，你还需要处理整数解。如果最优顶点坐标不是整数，你必须检验该顶点附近可行域内的整数点。线性规划题常结合现实情境表述：“x的数量至少为y的两倍”可转化为x ≥ 2y。清晰地标注图形和写出步骤对获得满分至关重要。
8. Transportation Problems: Finding an Initial Solution | 运输问题：求初始解
The transportation problem aims to minimise the cost of distributing goods from several sources (supply) to several destinations (demand). The problem is represented by a cost matrix. You first use the North-West Corner rule to find an initial feasible solution, then improve it using the stepping-stone method or the method of multipliers (MODI).
运输问题旨在最小化将货物从多个来源（供应）运往多个目的地（需求）的总成本。问题用成本矩阵表示。你首先使用西北角法求出一个初始可行解，然后通过踩石子法或乘数法（MODI）进行优化。
In D1, emphasis is often on obtaining an initial solution and demonstrating improvement. For the North-West Corner rule, you start at the top-left cell and allocate as much as possible up to the supply or demand limit, then move right or down accordingly. You must show your allocation table clearly. Other methods such as the least-cost method may also be used to get a better starting solution.
在D1中，重点通常是得到初始解并展示优化过程。使用西北角法时，你从左上角单元格开始，尽可能多地分配运量，直到达到供应或需求上限，然后相应地向右或向下移动。你必须清楚地展示分配表。其他方法如最小成本法也可用于获得更好的初始解。
9. Matchings in Bipartite Graphs | 二分图中的匹配
In a bipartite graph, the vertex set is partitioned into two sets, and edges only connect vertices from different sets. A matching is a set of edges where no two share a vertex. A maximum matching contains as many edges as possible. The Hungarian algorithm or the alternating path algorithm can be used to find a maximum matching by starting with an initial matching and then applying augmenting paths.
在二分图中，顶点集被划分为两个集合，边只连接不同集合的顶点。匹配是一组边，其中没有两条边共享顶点。最大匹配包含尽可能多的边。可以使用匈牙利算法或交替路径算法，通过初始匹配并应用增广路径来求最大匹配。
An alternating path alternates between edges in the matching and edges not in the matching. An augmenting path is an alternating path that starts and ends at unmatched vertices; by changing the status of edges along this path, the size of the matching increases by one. D1 problems often ask you to find a complete matching (if one exists) or to state why no further improvement is possible.
交替路径在匹配中的边和不在匹配中的边之间交替。增广路径是一条从非匹配顶点开始并结束于非匹配顶点的交替路径；通过改变该路径上边的状态，匹配的大小增加1。D1题目常要求你找出完全匹配（如果存在），或说明为何无法进一步改进。
10. The Route Inspection Problem and Eulerian Graphs | 路径检查问题与欧拉图
The Route Inspection problem (also called the Chinese Postman problem) seeks the shortest route that traverses every edge in a network at least once, starting and ending at the same vertex. If the graph is Eulerian (all vertices have even degree), an Eulerian circuit exists and that is the optimal route. If there are odd-degree vertices, you must pair them up with additional routes of minimum total weight.
路径检查问题（又称中国邮递员问题）寻求一条遍历网络中每条边至少一次并回到起点的最短路线。如果图是欧拉图（所有顶点度数均为偶数），则存在欧拉回路，那便是最优路线。如果有奇数度顶点，则必须以最小总权重将它们配对增加额外路线。
To solve, you identify all odd vertices, find all possible pairings of these vertices, and for each pairing calculate the shortest distances between paired vertices (using Dijkstra). You then select the pairing that gives the minimum total extra distance. The length of the optimal route is the sum of all edge weights plus the minimum extra distance. This topic is highly algorithmic and frequently examined.
解题时，先找出所有奇数度顶点，列出这些顶点的所有可能配对，并对每对组合计算最短距离（用Dijkstra算法）。然后选择使额外总距离最小的配对。最优路线长度等于所有边权之和加上最小额外距离。这一主题算法性强，经常出现在考卷中。
11. Bin Packing Algorithms: First Fit and First Fit Decreasing
Bin packing problems involve assigning items of given sizes into bins of fixed capacity, with the aim of using as few bins as possible. The First Fit algorithm takes items in the order given and places each item into the first bin that has enough remaining capacity. First Fit Decreasing first sorts items into descending order, then applies First Fit.
装箱问题涉及将给定大小的物品装入固定容量的箱子中，目标是使用尽可能少的箱子。首次适应算法（First Fit）按照给定的物品顺序，将每个物品放入第一个有足够剩余容量的箱子。降序首次适应算法（First Fit Decreasing）先将物品按大小降序排列，再应用首次适应。
While neither algorithm guarantees an optimal solution, First Fit Decreasing often performs better. In D1, you may be asked to execute both algorithms on a data set and compare the number of bins used. You must show the bins and their contents at each stage. Understanding these heuristics helps with practical resource allocation problems.
尽管两种算法都不能保证最优解，但降序首次适应通常表现更好。在D1中，你可能需要对一个数据集运行两种算法并比较使用的箱子数量。你必须展示每个阶段各个箱子及其内容。理解这些启发式方法有助于解决实际资源分配问题。
12. Exam Techniques and Common Pitfalls | 考试技巧与常见误区
Decision Mathematics 1 exam questions are highly structured and reward clear, logical working. Always show all steps, label diagrams meticulously, and write down the values obtained at each iteration. In algorithms such as bubble sort or Dijkstra, partial marks are awarded for correct intermediate results even if the final answer is wrong.
决策数学1的考题结构严密，清晰的逻辑推导过程至关重要。务必展示所有步骤，细致标注图表，并写下每次迭代得到的数值。在冒泡排序或Dijkstra等算法中，即使最终答案错误，正确的中期结果也能得到部分分数。
Common pitfalls include misreading constraints in linear programming (e.g. confusing ‘≤’ with ‘≥’), forgetting to check for alternative optimal vertices, and not adjusting the temporary labels fully in Dijkstra when a shorter path is found. Practice interpreting written descriptions into mathematical formulations, and review past papers to become familiar with the exam style.
常见误区包括误读线性规划中的约束条件（如混淆≤和≥），忘记检查是否有其他最优顶点，以及在Dijkstra算法中发现更短路径时未能完全更新临时标签。要练习将文字描述转化为数学表达式，并复习往年试卷以熟悉考试风格。
Staying calm and methodical is key: read the whole question before starting, annotate the given network or data, and double-check your arithmetic. With consistent practice, the structured nature of D1 makes it a subject where methodical students can achieve high marks.
保持冷静和条理是关键：开始前通读全题，在提供的网络图或数据上做注释，并反复检查算术。通过持续练习，D1的结构化特性使其成为有条理的学生能够取得高分的科目。
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