Edexcel决策数学D2真题详解：最短路径与网络流 | D2 Decision Math Solutions Guide

🧮 Edexcel 决策数学 D2 (Decision Mathematics 2) 全面解析

引言 / Introduction

决策数学 (Decision Mathematics) 是 Edexcel A-Level 数学中最独特的模块。D2 深入探索运筹学 (Operational Research) 的核心算法——从动态规划 (Dynamic Programming) 到网络流 (Network Flows)，这些知识不仅用于考试，在计算机科学、物流管理和经济学中也有广泛应用。

Decision Mathematics is one of the most distinctive modules in Edexcel A-Level Maths. D2 dives deep into the core algorithms of Operational Research — from Dynamic Programming to Network Flows. These concepts aren’t just for exams; they’re widely applied in computer science, logistics, and economics.

🎯 五大核心知识点 / Five Core Topics

1. 动态规划 / Dynamic Programming

动态规划是 D2 的开篇重点，通过最优性原则 (Principle of Optimality) 将复杂问题分解为一系列递推子问题。无论是最大/最小化问题还是分配问题，掌握状态转移方程 (recurrence relation) 是关键。

Dynamic Programming kicks off D2. Using the Principle of Optimality, complex problems are broken into recursive subproblems. Whether maximisation, minimisation, or allocation — mastering the recurrence relation is key.

2. 网络流与最大流最小割定理 / Network Flows & Max-Flow Min-Cut

这是 D2 中最具挑战性的主题之一。你需要掌握：① 标注法 (labelling procedure) 寻找增广路径 (augmenting path)；② 最大流最小割定理 (Max-Flow Min-Cut Theorem)——网络中最大流的值等于最小割的容量。理解反向边 (back edges) 在流调整中的作用至关重要。

One of D2’s most challenging topics. You must master: ① the labelling procedure to find augmenting paths; ② the Max-Flow Min-Cut Theorem — the value of the maximum flow equals the capacity of the minimum cut. Understanding back edges in flow adjustment is critical.

3. 运输问题 / Transportation Problems

运输问题是线性规划 (Linear Programming) 的特殊形式。先用西北角法 (North-West Corner Rule) 或最小成本法 (Least Cost Method) 求初始可行解，再用踏脚石法 (Stepping-Stone Method) 或改进分配法 (MODI) 优化至最优解。

Transportation problems are a special case of Linear Programming. Start with the North-West Corner Rule or Least Cost Method for an initial feasible solution, then optimize using the Stepping-Stone Method or MODI method.

4. 分配问题与匈牙利算法 / Assignment Problems & Hungarian Algorithm

分配问题可视为运输问题的特例（供给=需求=1）。匈牙利算法 (Hungarian Algorithm) 是标准解法：行归约 → 列归约 → 用最少线覆盖所有零 → 增广矩阵直至得到最优分配。这个算法在考试中出镜率极高。

Assignment problems are a special case of transportation (supply = demand = 1). The Hungarian Algorithm is the standard method: row reduction → column reduction → cover all zeros with minimum lines → augment until optimal. This appears frequently in exams.

5. 博弈论与零和博弈 / Game Theory & Zero-Sum Games

D2 引入博弈论的基础概念：支付矩阵 (payoff matrix)、纯策略 (pure strategy)、混合策略 (mixed strategy)、鞍点 (saddle point)。用图解法或线性规划法求解 2×n 或 m×2 博弈的最优混合策略。

D2 introduces foundational game theory: payoff matrices, pure strategies, mixed strategies, saddle points. Use graphical methods or linear programming to solve optimal mixed strategies for 2×n or m×2 games.

💡 学习建议 / Study Tips

• ✅ 算法必须”手熟”——每天练习一个 D2 算法 / Algorithms need muscle memory — practice one D2 algorithm daily
• ✅ 画出清晰的有向图/网络图再开始计算 / Draw clear digraphs/network diagrams before calculating
• ✅ 重点攻克动态规划的递归表和表格填充 / Focus on dynamic programming recursion tables and table-filling
• ✅ 用 past papers 计时模拟真实考试环境 / Use past papers with timed conditions to simulate the real exam
• ✅ 对照 Heineann Solutionbank 验证每一步 / Cross-check each step against the Heinemann Solutionbank

📚 更多 Edexcel 数学资源 / More Edexcel Maths Resources

我们的题库覆盖 Edexcel D1、D2、C1-C4、FP1-FP3、M1-M5、S1-S4 全部模块。无论你在备考 AS 还是 A2，海量真题 + 详细解析助你冲刺 A*。

Our bank covers all Edexcel modules: D1, D2, C1-C4, FP1-FP3, M1-M5, S1-S4. Whether you’re studying AS or A2, our extensive past papers and detailed solutions will help you aim for that A*.

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