📚 Edexcel AS and A Level Further Mathematics Decision Mathematics 2 Common Pitfalls | Edexcel AS与A Level进阶数学决策数学2易错点总结
Decision Mathematics 2 (D2) extends algorithmic and modelling techniques from D1. While many concepts are logical, students repeatedly lose marks due to subtle mistakes in execution and interpretation. This article pinpoints the most common pitfalls in D2 and shows how to avoid them.
决策数学2（D2）在D1的基础上进一步拓展了算法与建模技巧。虽然许多概念逻辑清晰，但学生在计算与解释中常因细节错误而丢分。本文梳理D2中最常见的易错点，并说明如何避免。
1. Transportation Problems: Misinterpreting Degeneracy | 运输问题：退化情形的误判
A basic feasible solution in a transportation problem is degenerate when the number of occupied cells is less than (m + n – 1). Students often overlook degeneracy and proceed with the MODI method, which fails because not enough ui and vj values can be determined. This leads to incorrect shadow costs or endless loops. Always insert a zero epsilon allocation in an unoccupied but independent cell to resolve degeneracy before checking optimality.
当运输问题的初始基可行解占用的格子数少于 (m + n – 1) 时，解是退化的。学生常忽略退化情形，直接使用 MODI 法，导致无法求出全部 ui 和 vj，进而产生错误检验数或陷入死循环。在检查最优性之前，务必在一个未占用且独立的格子中插入零分配 ε，以消除退化。
2. MODI Method: Sign Errors in Improvement Indices | MODI法：检验数符号错误
After computing ui and vj, the improvement index for an unoccupied cell (i, j) is cij – (ui + vj). Many students mistakenly calculate (ui + vj) – cij, which gives the wrong sign. Remember: a negative index means cost can be reduced by entering that cell. If you reverse the sign, you might select a cell that actually increases cost or fail to spot an improving cell. Always double-check the formula and focus on minimisation context.
求出 ui 与 vj 后，未占用格子 (i, j) 的检验数（改进指数）应为 cij – (ui + vj)。很多学生误算为 (ui + vj) – cij，导致符号错误。记住：负的检验数表示通过进入该格可降低总费用。如果符号弄反，可能会选入一个实际上增加费用的格，或漏掉可改进的格。请务必核对公式，牢记目标是最小化。
3. Assignment Problems: Handling Unbalanced and Maximisation Cases | 指派问题：不平衡与最大化情形的处理
The Hungarian algorithm assumes a balanced minimisation problem. When the cost matrix is unbalanced, students forget to add dummy rows/columns with zero costs. For maximisation (e.g. profit), a common mistake is converting by subtracting all entries from a large number but then misapplying the algorithm by still treating zeros as minimum. Instead, convert maximisation to minimisation by subtracting each element from the maximum element in the matrix. Also, after adding dummy rows, ensure that the final assignment does not include dummy jobs/workers; real tasks must be assigned.
匈牙利算法默认处理平衡的最小化问题。当费用矩阵不平衡时，学生常忘记添加虚拟行或列并赋零成本。对于最大化问题（如利润），常见错误是用一个大数减去所有元素后仍将零当作最小值来操作。正确做法是用矩阵中的最大元素减去每个元素，转化为最小化问题。此外，添加虚拟行后，要确保最终指派不包含虚拟任务，所有实际任务必须被分配。
4. Dynamic Programming: Incorrect State Definition | 动态规划：状态定义错误
In dynamic programming, defining the state incorrectly is a critical error. The state must capture all information needed to make optimal decisions from that stage onward. For example, in a resource allocation problem, the state is the remaining amount of resource, not the stage index. Students often confuse stage and state, or omit variables such as the number of items considered. This leads to incorrect recurrence relations. Always check that your state is compact but sufficient for the next stage’s transition.
动态规划中，状态定义错误是致命失误。状态必须包含从该阶段起做出最优决策所需的全部信息。例如在资源分配问题中，状态是剩余资源量，而不是阶段序号。学生常混淆阶段与状态，或遗漏已考虑的物品数量等变量，从而建立错误的递推关系。请务必确保状态简洁但充分，能够反映向下一阶段的转移。
5. Network Flows: Missing Backward Arcs in Augmenting Paths | 网络流：增广链中遗漏反向弧
When finding a flow-augmenting path from source to sink in a network, any arc can be used in the forward direction if it has spare capacity, and in the backward direction if it already carries flow. A major pitfall is ignoring backward arcs, which can block finding the maximum flow. For example, after some initial flow, a reverse arc might be needed to redirect flow and increase overall throughput. Always label residual capacities on both forward and backward arcs and consider both when searching for an augmenting path via breadth-first search.
在网络中寻找从源点到汇点的增广链时，任何弧在正向有剩余容量时可使用，在反向若已承载流量也可使用。常见的重大遗漏是忽视反向弧，这会阻碍找到最大流。例如，在初始流分配后，可能需要使用反向弧重新路由流量以提高总通过量。务必在正向和反向弧上标注剩余容量，并用广度优先搜索同时考虑两类弧。
6. Simplex Method: Choosing the Wrong Pivot Column | 单纯形法：选择错误的入基变量
In maximising linear programming with the simplex method, the entering variable is the one with the most negative coefficient in the objective row (for maximisation). Students occasionally pick a positive coefficient because they remember ‘largest’ from some other variant, or they fail to recognise that in minimisation the rule is opposite. In two-phase or Big M, mixing rules for artificial variables is another source of error. Always verify whether you are maximising or minimising, and follow the correct sign rule.
在单纯形法最大化问题中，入基变量是目标行中系数最负的变量。学生有时因为记成“最大”而选择正系数，或未意识到最小化问题的规则正好相反。在两阶段法或大M法中，混淆人工变量的处理规则也是常见错误。务必先确认目标是最大化还是最小化，并遵循正确的符号规则。
7. Two-Stage Simplex: Forgetting to Remove Artificial Variables | 两阶段单纯形法：忘记移除人工变量
After completing Phase I of two-stage simplex, all artificial variables must be dropped from the tableau, and the original objective function reinstated. A frequent mistake is leaving an artificial variable in the basis when its value is zero, or forgetting to eliminate its column before starting Phase II. If an artificial remains in the basis at zero level, you must pivot to replace it with a legitimate variable; otherwise the Phase II solution may be invalid. Always check that the basis contains only original slack/surplus and real variables before Phase II.
完成两阶段单纯形法的第一阶段后，必须将所有人工变量从表中移除，并恢复原始目标函数。常见错误是当某个人工变量取值为零时仍保留在基中，或在第二阶段开始前忘记删除其列。如果人工变量以零水平留在基中，必须通过旋转用合法变量将其替换，否则第二阶段的解可能无效。务必在第二阶段前检查基变量仅包含原始的松弛/剩余变量和实际决策变量。
8. Game Theory: Miscalculating Expected Payoffs in Mixed Strategies | 博弈论：混合策略期望收益计算错误
When computing the expected payoff for a mixed strategy game (with no saddle point), students often misassign probabilities. For a 2×2 zero-sum game, let player A use strategies A1, A2 with probabilities p and 1–p. The expected payoffs for B’s pure strategies are calculated, but errors arise from mixing up ‘A’s payoff’ and ‘B’s payoff’, or forgetting that payoffs are given from A’s perspective. Use the payoff matrix consistently, and set the expected payoff equations equal if B is indifferent. Then solve for p and verify the value. The graphical method also suffers from plotting the wrong lines when the payoff matrix is not directed.
在计算混合策略期望收益时（无鞍点），学生经常混淆概率分配。对2×2零和博弈，设玩家A以概率 p 和 1–p 使用策略 A1、A2。需要针对B的纯策略计算期望收益，但常会搞错“A的收益”与“B的收益”，或忘记支付矩阵是以A为视角给出的。要一致地使用支付矩阵，当B无差异时令期望收益方程相等，解出 p 并验证博弈值。在使用图解法时，若支付矩阵方向弄错，也容易画出错误直线。
9. Critical Path Scheduling: Resource Histograms and Gantt Charts Confusion | 关键路径排程：资源直方图与甘特图的混淆
In resource levelling and scheduling, students often misinterpret Gantt charts (which show activity timing and float) with resource histograms (which show resource usage over time). A common pitfall is using the wrong scale or mixing up the activity’s total float with its independent float when shifting activities to reduce peaks. When adjusting start times, ensure precedence constraints are not violated. Also, when reading from a cascade chart, remember that activities cannot be split unless stated.
在资源均衡与排程中，学生常混淆甘特图（显示活动时间与浮动）与资源直方图（显示随时间变化的资源使用量）。常见错误是使用错误的比例尺，或在为削峰而移动活动时，将总浮动与自由浮动混用。调整开始时间时，必须确保不违反前置关系。同时，从阶梯图读取时，除非题目说明，否则活动不可拆分。
10. Queuing Theory: Applying Formulae Without Checking Steady-State Conditions | 排队论：未检查稳态条件就应用公式
The standard M/M/1 queue formulae for Lq, Wq, etc. are only valid when the traffic intensity ρ = λ/μ < 1. Students often plunge into calculation without checking this condition. If ρ ≥ 1, the queue is unstable and expected queue length becomes infinite—standard formulae fail. Moreover, when using Little's Law L = λW, ensure that λ and W are consistent in units (per hour, minutes), and that effective arrival rate is used if some customers are blocked. Check assumptions: single server, exponential inter-arrival and service times, infinite population.
标准 M/M/1 排队公式（如 Lq、Wq 等）仅在服务强度 ρ = λ/μ < 1 时成立。学生往往未检查此条件就直接计算。当 ρ ≥ 1 时，队列不稳定，期望队长趋于无穷，标准公式失效。此外，使用利特尔法则 L = λW 时，应确保 λ 与 W 单位一致（每小时、每分钟），并考虑是否需使用有效到达率（如有顾客被拒绝）。还需核实假设：单服务台、到达间隔和服务时间服从指数分布、无限总体等。
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