📚 A-Level Edexcel D2 Question Types Analysis | A-Level Edexcel 决策数学 2 题型解析
Decision Mathematics 2 (D2) is the more advanced optional unit in the Edexcel A-Level Further Mathematics specification. It builds directly on D1 and demands deeper algorithmic thinking, precise modelling, and a thorough understanding of optimality proofs. In this article, we systematically break down every major question type that appears in D2 examinations, with paired English and Chinese explanations to support bilingual learners. Each section identifies the core technique, common pitfalls, and the exact wording Edexcel examiners use to signal which method is required.
决策数学 2（D2）是 Edexcel A-Level 进阶数学中更具挑战性的选修单元，它直接建立在 D1 内容之上，要求学生具备更深入的算法思维、精确的建模能力以及对最优性证明的透彻理解。本文系统梳理了 D2 考试中出现的每一种核心题型，以中英双语对照讲解，帮助双语学习者掌握关键方法。每一节都会点明核心技巧、常见陷阱，以及 Edexcel 考官常用的问题措辞，让你瞬间识别该用哪种方法。
1. Critical Path Analysis with Resource Histograms | 含资源直方图的关键路径分析
D2 extends basic critical path analysis by introducing resource leveling and resource histograms. A typical question provides a precedence table with activity durations and the number of workers each activity requires, then asks you to construct a resource histogram for a given schedule, or to minimise the maximum number of workers needed while respecting a project’s overall duration. You must first draw the activity network, find earliest start times and latest finish times, and then produce a cascade (Gantt) chart. From the cascade chart, you count workers per unit time to build the histogram. Edexcel often phrases this as ‘Draw a resource histogram for the project on the grid provided, assuming each activity starts as early as possible.’ If the question later asks ‘By how much can the start of activity X be delayed without affecting the project duration?’, you revert to float calculations.
D2 在基本关键路径分析的基础上加入了资源平滑（resource leveling）和资源直方图。典型题目会给出包含活动工期及每项活动所需工人数的先后关系表，然后要求你画出某一调度方案下的资源直方图，或在项目总工期不变的前提下最小化所需工人的最大数量。你必须先绘制活动网络图，求出最早开始时间和最迟完成时间，再画出级联（甘特）图。从甘特图中逐一时间段统计工人数量即可构建直方图。Edexcel 常见的措辞是：“假设每一项活动都尽早开始，在给出的网格上画出该项目的资源直方图。” 如果题目接着问“在不影响项目工期的条件下，活动 X 的开始时间最多可推迟多少？”，你就要回到时差计算。
2. Dynamic Programming – Maximin and Minimax | 动态规划 – 极大极小与极小极大
Dynamic programming appears in D2 either as a ‘maximin’ problem (often in game-like investment scenarios) or as a ‘minimax’ problem (typical of allocation or routing decisions). You are expected to set up a table where rows represent stages and columns represent states. At each stage, you work backwards from a known terminal condition, applying the principle of optimality. For maximin, you choose the strategy that gives the largest of the minimum possible outcomes; for minimax, you minimise the maximum possible cost. Edexcel questions usually provide a network or a payoff matrix and ask ‘Use dynamic programming to determine the optimal strategy.’ You must show all working in tabular form, clearly labelling the action that achieves the optimal value at each state. If the question uses words like ‘worst-case scenario’ or ‘guarantee a minimum return,’ it signals a maximin approach. Conversely, ‘minimise the maximum possible delay’ indicates minimax.
动态规划在 D2 中以“极大极小”（maximin）或“极小极大”（minimax）的形式出现，前者常见于博弈类投资问题，后者则常出现在分配或路径决策中。你需要建立一张表格，行表示阶段，列表示状态。在每个阶段，从已知的终点条件出发倒推，应用最优性原理。对于极大极小，你选择能使最小可能收益达到最大的策略；对于极小极大，则是使最大可能成本达到最小。Edexcel 的题目通常会给出一个网络或收益矩阵，要求“用动态规划确定最优策略”。你必须以表格形式展示所有推演过程，并清楚标明达到各状态最优值的动作。如果题目中出现“最坏情况”或“保证最低回报”等词，那就意味着极大极小；而“使最大可能延误最小化”则指向极小极大。
3. Game Theory – Graphical Solution for Mixed Strategies | 博弈论 – 混合策略的图解法
When a two-player zero-sum game has no saddle point, D2 questions often require finding optimal mixed strategies graphically. The standard set-up involves a payoff matrix for player A: if A has two strategies, you can plot expected payoffs against the probability p of choosing A1, drawing a line for each of B’s strategies. The lower envelope (for maximin) or upper envelope (for minimax) gives the optimal p. Edexcel questions may ask: ‘Use a graphical method to find the optimal strategy for player A and the value of the game.’ You must correctly draw and label axes (p from 0 to 1, payoff on vertical axis), plot the lines, and clearly mark the intersection that defines the maximin point. If the question instead provides a matrix and says ‘Formulate this as a linear programming problem for player A,’ then you must translate it into an LP with constraints before solving, which is a hybrid topic frequently examined.
当两人零和博弈没有鞍点时，D2 题目通常要求用图解法求混合策略最优解。标准设定是：给出参与人 A 的收益矩阵，如果 A 有两个策略，你可以以选择 A1 的概率 p 为横轴，画出对应 B 各个策略的期望收益直线。下包络线（极大极小情形）或上包络线（极小极大情形）的交点决定了最优的 p 值。Edexcel 题目会问：“用图解法求参与人 A 的最优策略及博弈值。” 你必须正确画出并标注坐标轴（p 从 0 到 1，纵轴为收益），绘出直线，并清楚标出决定极大极小点的交点。如果题目给出矩阵后说“将参与人 A 的问题表示成一个线性规划”，你就需要先把它转化为带约束的线性规划再求解，这是一种常考的混合题型。
4. Network Flows – Maximum Flow and Minimum Cut | 网络流 – 最大流与最小割
Maximum flow problems in D2 typically involve a directed network with capacities on arcs, a single source, and a single sink. The core task is to find the maximum flow using the labelling procedure (also called flow augmentation). You start with an initial feasible flow (often zero) and iteratively search for flow-augmenting paths, adjusting forward and backward flows while respecting residual capacities. Edexcel expects you to label each vertex with (predecessor, ± flow adjustment) and to clearly show each augmentation step. The final step is to identify a minimum cut and verify that the cut capacity equals the maximum flow, which proves optimality. Questions often say: ‘Use the labelling procedure to find the maximum flow from S to T. State the minimum cut and prove that your flow is maximal.’ Be meticulous with notation: distinguish between capacity, actual flow, and residual capacity.
D2 中的最大流问题通常涉及带容量约束的有向网络、一个源点和一个汇点。核心任务是利用标号法（也称流增广算法）找出最大流。你需要从一个初始可行流（通常为零流）出发，反复寻找增广路径，在遵守剩余容量的前提下调整正向流和逆向流。Edexcel 要求你在每个顶点标注 （前驱, ±流量调整量） 并清晰展示每一步增广过程。最后还要找出一条最小割，并验证割容量等于最大流，从而证明最优。题目常这样表述：“用标号法求从 S 到 T 的最大流量，给出最小割并证明你的流量是最大的。” 务必注意符号的精确性：区分容量、实际流量和剩余容量。
5. Hungarian Algorithm for Assignment Problems | 指派问题的匈牙利算法
The Hungarian algorithm finds the minimum cost assignment for an n × n cost matrix. D2 questions may present a table of costs and ask: ‘Use the Hungarian algorithm to find the minimum cost assignment. State the assignment and the total cost.’ The standard procedure requires row reduction, column reduction, covering zeros with the minimum number of lines, and augmenting the matrix if the number of lines is less than n. Edexcel expects you to show each matrix after reduction, the pattern of covering lines, and the steps where you subtract the smallest uncovered element and add it to double-covered entries. A common variant asks for a maximisation assignment: in that case you first subtract all entries from a large number (often the largest entry) to convert profit to relative cost. The phrase ‘maximum profit’ is your cue to convert.
匈牙利算法用于求 n×n 成本矩阵的最小成本指派。D2 题目可能给出一个成本表格并问：“用匈牙利算法求最小成本指派。写出指派方案及总成本。” 标准步骤包括：行归约、列归约、用最少直线覆盖所有零元素，若直线数小于 n 则进行矩阵增广（将未被覆盖的最小元素减去并加到双重覆盖的位置）。Edexcel 希望你展示每个归约后的矩阵、覆盖线的图样以及增广步骤。一个常见的变式是求最大收益的指派：此时你应先用一个大数（通常是表中的最大数）减去所有元素，将利润转化为相对成本。看到“最大利润”这个词就应立即这么做。
6. Transportation Problem – Stepping-Stone Method and Optimality | 运输问题 – 踏脚石法与最优性检验
The transportation problem appears in D2 as an extension of the basic formulation introduced in D1. Questions may provide a cost matrix with supply and demand margins, and an initial solution found by the north-west corner method or the minimum-cost entry method. You must then test for optimality using the stepping-stone method, computing improvement indices for all unused routes. Edexcel often asks: ‘Find the improvement index of cell (X, Y) and explain its significance’ or ‘Perform one iteration of the stepping-stone method to obtain an improved solution.’ You need to show the closed path, the sign pattern (+ and −), the quantity transferred, and the updated tableau. After an iteration, recalculate all improvement indices to confirm whether the new solution is optimal. Remember: in a degenerate solution (fewer than m+n−1 occupied cells), you must introduce a zero-rated cell as a stepping stone.
运输问题在 D2 中是 D1 中基本模型的应用延伸。题目会给出带有供需量的成本矩阵以及用西北角法或最小成本法求出的初始解。接着你要用踏脚石法检验最优性，计算所有未使用路径的改进指数。Edexcel 常这样问：“求空格 (X, Y) 的改进指数并解释其意义”，或者“进行一步踏脚石法迭代，求出一个改进后的解”。你需要画出闭回路，标注正负号（+ 和 −），标出转移量，并写出更新后的运输表。迭代完成后，重新计算所有改进指数以确认新解是否达到最优。注意：若出现退化解（有数字的格子少于 m+n−1 个），你必须引入一个为零的中心空格作为踏脚石。
7. Linear Programming – Two-Phase Simplex and Duality | 线性规划 – 两阶段单纯形法与对偶
While D1 covers graphical LP and the basic simplex with slack variables, D2 introduces the two-phase simplex method with artificial variables, inequality constraints of mixed directions, and variables that are unrestricted in sign. A typical D2 problem might ask: ‘Use the two-phase simplex method to maximise P = 3x + 2y subject to x + y ≥ 4, x − y ≤ 2, x ≥ 0, y ≥ 0.’ You must construct the Phase I objective to minimise the sum of artificial variables, solve to reach a feasible basis, then revert to the original objective in Phase II. Edexcel expects clear reading of the simplex tableau, including correct pivot selection and row operations. Duality is also examined: you may be given a primal problem and asked to ‘Write down the dual LP’ and use the dual optimal tableau to find the primal solution without resolving.
虽然 D1 涵盖了图解法线性规划和带松弛变量的单纯形法，D2 进一步引入带人工变量的两阶段单纯形法、混合方向的不等式约束以及符号无限制的变量。典型的 D2 问题可能是：“用两阶段单纯形法最大化 P = 3x + 2y，约束为 x + y ≥ 4, x − y ≤ 2, x ≥ 0, y ≥ 0”。你需要构造第一阶段目标函数（使人工变量之和最小化），解出一个可行基，然后在第二阶段恢复原目标。Edexcel 要求你清晰解读单纯形表，包括正确选择主元以及行变换。对偶理论也是考点之一：题目可能给出一个原始问题，要求你“写出对偶线性规划”，并利用对偶最优表直接得出原始解，无需再从头求解。
8. Allocating Resources – Knapsack and Binary Decision Variables | 资源分配 – 背包问题与二进制决策变量
D2 exam papers occasionally feature a knapsack-style problem where items have a weight (or cost) and a value, and you must select a subset to maximise total value under a weight constraint. This is solved either by dynamic programming or, for small instances, by forming an integer linear programming model. Edexcel favours the modelling route: ‘Formulate this as a linear programming problem. Define your decision variables clearly.’ You typically use binary variables xᵢ where xᵢ = 1 if item i is selected and 0 otherwise. The constraint is Σ(weightᵢ × xᵢ) ≤ capacity. Additional constraints can model mutual exclusivity (‘exactly one of project A and B must be chosen’) or dependency (‘if project C is chosen, project D must also be chosen’). You are not usually required to solve the resulting integer LP by branch-and-bound, but you should be able to set up the model precisely.
D2 试卷中偶尔会出现背包类问题：物品具有重量（或成本）和价值，要求从中选出若干物品，在总重量约束下最大化总价值。这类问题可用动态规划求解，但在小规模情形下，更常见的做法是建立整数线性规划模型。Edexcel 偏爱建模路线：“将此问题表示为一个线性规划问题，并明确定义决策变量。” 通常使用二进制变量 xᵢ，xᵢ = 1 表示选中物品 i，否则为 0。约束为 Σ(重量ᵢ × xᵢ) ≤ 容量。还可以加入互斥约束（“项目 A 和 B 中恰好选一个”）或依赖约束（“若选项目 C，则必须选项目 D”）。一般不需要用分支定界法去求解，但必须能够准确写出模型。
9. Minimum Connector and Shortest Path Extensions | 最小生成树与最短路径的进阶变体
Although these algorithms are D1 staples, D2 remixes them with resource constraints or alternative objectives. For example, you may be asked to find the shortest path from A to B on a network where some arcs can be traversed only once (route inspection crossover) or where a certain node must be included. Edexcel might phrase it as: ‘Find the minimum route that starts at A, passes through X, and ends at B.’ You then break the problem into two shortest-path subproblems: A to X and X to B, using Dijkstra’s algorithm twice. For minimum spanning trees, a sophisticated D2 question can ask you to apply Prim’s or Kruskal’s algorithm on a distance matrix rather than a drawn network, and then interpret the resulting tree in a real-world context like cabling or pipeline layout. Always watch out for directed arcs: Prim’s works only on undirected networks.
尽管这些算法是 D1 的核心内容，D2 往往会在其中加入资源约束或改变优化目标。例如，要求找出从 A 到 B 的最短路径，但某些弧只能经过一次，或者必须经过某一指定节点。Edexcel 可能这样提问：“求出一条从 A 出发，经过 X 并到达 B 的最短路程。” 此时你可以把问题拆成两个最短路径子问题：A 到 X 以及 X 到 B，分别使用 Dijkstra 算法。对于最小生成树，高层次的 D2 题目可能要求你在距离矩阵（而非图形）上应用普里姆或克鲁斯卡尔算法，并解释由此得到的树在布线或管道铺设等现实场景中的意义。要特别注意有向弧的问题：普里姆算法仅适用于无向网络。
10. Resource Histogram and Scheduling with Gantt Charts | 甘特图调度与资源直方图综合分析
A full D2 scheduling problem typically combines precedence constraints with a fixed number of workers. You are asked to schedule the activities so that the project is completed in a given time with the minimum number of workers, or to construct a Gantt chart when activities are scheduled by some heuristic (e.g., latest start time prioritisation). Edexcel questions often provide a partially completed Gantt chart and ask you to fill in the remaining activities, respecting precedence and resource limits. The phrase ‘By considering the activities on the critical path, explain why the project cannot be completed with only two workers’ requires you to demonstrate that at some point, three activities that need a worker each must happen simultaneously. You should be able to translate between a resource histogram and the Gantt chart: a peak in the histogram indicates a time period where many activities overlap.
一个完整的 D2 调度问题通常将先后关系与固定工人数结合起来。你需安排各项活动，使得项目在给定时间内完成，且所需工人数最少，或者根据某种启发式规则（如最迟开始时间优先）画出甘特图。Edexcel 常提供一个部分完成的甘特图，让你在遵守先后关系和资源限制的条件下填入剩余活动。“通过考虑关键路径上的活动，解释为何仅有 2 名工人时该项目无法完成” 这类问题，需要你证明在某些时刻，三个各需一名工人的活动必须同时进行。你需要能够在资源直方图和甘特图之间互相转换：直方图的峰值表示多个活动重叠的时间段。
11. Linear Programming Formulation from Worded Problems | 由文字问题建立线性规划模型
A significant portion of D2 marks comes from correctly translating a business or logistical scenario into an LP formulation. This goes beyond the D1 style by including integer or binary variables, multi-period constraints, and blending restrictions. For instance, ‘A factory produces two types of alloy, Alpha and Beta, using three metals. Each alloy must contain at least a certain percentage of each metal. There are limited daily availabilities of metals.’ You will define variables for the amounts of metals used in each alloy, and constraints for percentage composition (e.g., (amount of metal 1 in Alpha) ≥ 0.3 × (total weight of Alpha)). Edexcel often asks you to ‘Formulate, but do not solve, the linear programming problem.’ This tests precise algebraic translation. Be careful: non-linear terms such as products or ratios must be avoided; if they arise naturally, you might need to linearise through substitution or scaling.
D2 考试中有相当一部分分数来自将商业或物流场景准确转化为线性规划模型。这比 D1 的要求更高，常涉及整数或二进制变量、多阶段约束以及混合比例限制。例如：“一家工厂使用三种金属生产 Alpha 和 Beta 两种合金。每种合金中每种金属的含量必须达到最低百分比。每日可用金属量有限。” 你需要为每种合金中使用的每种金属量定义变量，并为成分百分比写出约束（如 Alpha 中金属 1 的量 ≥ 0.3 × Alpha 总重量）。Edexcel 经常要求“建立线性规划问题的模型，但不求解”。这恰恰考查精准的代数翻译能力。注意：必须避免出现乘积或比值等非线性项；如果自然出现，可能需要通过代换或缩放将其线性化。
12. Interpreting Simplex Tableaux and Post-Optimality | 单纯形表的解读与后最优性分析
D2 examinations frequently present a final simplex tableau and ask interpretive questions: ‘State the optimal solution and the value of the objective function.’ ‘How much would the coefficient of x₂ need to increase before it becomes worthwhile to produce x₂?’ ‘Which constraints are binding?’ This requires a deep understanding of reduced costs and shadow prices. In a maximisation problem, a non-basic variable with reduced cost −3 means the coefficient would need to increase by 3 to make it worth entering the basis. The shadow price of a resource, found in the slack or artificial variable column of the objective row, indicates the increase in the objective per unit increase in resource availability. Edexcel also tests your ability to detect alternative optimal solutions when a non-basic variable has a reduced cost of zero in the optimal tableau. The notation in a tableau follows strict D2 conventions: the objective row shows values with signs reversed, so interpret accordingly.
D2 考试常常给出最终单纯形表并要求解答如下问题：“给出最优解及目标函数值”“要使生产 x₂ 变得有利可图，其系数需要增加多少？”“哪些约束是紧的？” 这要求你透彻理解简约成本（reduced cost）和影子价格（shadow price）。在最大化问题中，某个非基变量的简约成本为 −3 意味着该变量在目标函数中的系数需要增加 3 才值得进入基。某一资源的影子价格体现在目标行中对应松弛或人工变量列的数字，它表示每增加一单位资源可用量目标函数的增长量。Edexcel 还会考察你识别多重最优解的能力：当最优表中某非基变量的简约成本为零时，即存在替代最优解。单纯形表中的符号遵循 D2 的严格约定：目标行数值显示的是相反符号的值，解读时务必小心。
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