📚 AQA Maths: Linear Programming Exam Tips | AQA 数学：线性规划 考点精讲
Linear programming (LP) is a cornerstone of AQA Decision Mathematics, where you learn to maximise or minimise a linear objective function under a set of linear constraints. Exam success demands precise formulation, accurate graphical work, and a methodical approach to vertex testing and integer solutions. This revision guide brings together the essential concepts, worked examples, and common pitfalls to help you score full marks.
线性规划 (Linear Programming) 是 AQA 决策数学的基石，你将在给定的线性约束条件下，学习如何最大化或最小化一个线性目标函数。在考试中取得高分，需要精准的建模能力、准确的图解技巧以及有条理地检验顶点和整数解。这份复习指南汇集了核心概念、典型例题和常见失分点，助你拿下满分。
1. What is Linear Programming? | 什么是线性规划？
Linear programming deals with problems where a quantity (profit, cost, time) needs to be optimised while respecting limitations expressed as linear inequalities. It appears widely in logistics, manufacturing, and resource allocation. In AQA exams, you will typically work with two variables so that a graphical solution is possible.
线性规划解决的是在遵守线性不等式所表示的约束条件下，对某个量（如利润、成本、时间）进行优化的问题。它在物流、制造和资源分配领域应用广泛。在 AQA 考试中，通常只涉及两个变量，便于用图解法求解。
2. Formulating the Problem | 建立问题模型
The first exam skill is translating a real-world scenario into mathematical inequalities. Start by defining your decision variables clearly, for example: let x be the number of chairs produced, and y be the number of tables. Then express the objective (maximise profit P = 20x + 30y) and each constraint (e.g. assembly time 2x + 5y ≤ 40).
第一项考试技巧是把现实情境转化为数学不等式。首先要清晰地定义决策变量，比如：设 x 为生产的椅子数量，y 为桌子数量。然后写出目标函数（例如最大化利润 P = 20x + 30y）和各个约束条件（如装配时间 2x + 5y ≤ 40）。
Key phrases that indicate constraint types:
表明约束类型的关键短语：
| Phrase | Inequality |
|---|---|
| at most, no more than | ≤ |
| at least, minimum, no less than | ≥ |
| exactly, must equal | = |
| non‑negative (cannot produce negative items) | x ≥ 0, y ≥ 0 |
中文关键词：最多/不超过 (≤)，至少/最少 (≥)，恰好/等于 (=)，非负约束 (x ≥ 0, y ≥ 0)。考试中漏掉非负约束是最常见的扣分点。
3. Graphing the Constraints | 绘制约束条件图形
Each linear inequality defines a half‑plane. To sketch it, draw the boundary line first: use a solid line for ≤ or ≥, and a dashed line for strict inequalities (< or >). A quick way to decide which side to shade is to test the origin (0,0) – if it satisfies the inequality, shade the opposite side so that the unshaded region represents the feasible set.
每一个线性不等式定义了一个半平面。画图时，先画出边界直线：≤ 或 ≥ 用实线，严格不等式 < 或 > 用虚线。判断阴影区域的一个快捷方法是把原点 (0,0) 代入检验——如果原点满足不等式，就将另一侧涂上阴影，这样留下的无阴影区域才表示可行域。
Example: For 2x + 3y ≤ 12, draw the line through (6,0) and (0,4). Test (0,0) gives 0 ≤ 12, so shade the side not containing (0,0). For exam accuracy, use a ruler and label axes clearly.
例如：对于 2x + 3y ≤ 12，先画出经过 (6,0) 和 (0,4) 的直线。检验原点 0 ≤ 12 成立，则应把不含原点的一侧涂上阴影。为确保考试作图准确，请务必使用直尺并清晰标注坐标轴。
4. The Feasible Region | 可行域
The feasible region is the unshaded polygon (often a quadrilateral or pentagon) that satisfies all constraints simultaneously. Sometimes the region is unbounded or even empty when constraints contradict each other. You must clearly label the region R on your graph or, if asked, leave it as the unshaded area.
可行域是同时满足所有约束条件的无阴影多边形（通常是四边形或五边形）。当约束条件互相矛盾时，可行域可能无界甚至为空。根据要求，你需要在图上标出区域 R，或保留为无阴影区域。
Always list the vertices of the feasible region exactly – you may need to solve simultaneous equations to find intersection points. Use brackets and show coordinates clearly.
一定要准确列出可行域的各个顶点——你可能需要通过解联立方程组来求交点。用括号表示坐标并清晰写出。
5. The Objective Function & Optimal Vertex | 目标函数与最优顶点
Once the feasible region is drawn, you can optimise the objective function. In a two‑variable linear programme, the optimal solution always occurs at a vertex (corner) of the feasible region, provided the region is bounded and the objective is linear. Use vertex testing: evaluate the objective at every vertex and pick the largest for maximisation or the smallest for minimisation.
一旦画出可行域，便可优化目标函数。在双变量线性规划中，只要可行域有界且目标函数是线性的，最优解必定出现在可行域的某个顶点（角点）上。采用顶点检验法：把每一个顶点的坐标代入目标函数计算，最大化问题取最大值，最小化问题取最小值。
Alternatively, use the sliding line method: draw an objective line (e.g. 20x + 30y = 600) and move it parallel until it leaves the feasible region. The last point it touches is the optimum. Both methods are accepted in AQA.
另一种方法是等值线滑移法：画一条目标函数平行的直线（如 20x + 30y = 600），然后平行移动，直至即将离开可行域，最后接触的点就是最优点。两种方法在 AQA 考试中均可接受。
6. Maximisation Worked Snippet | 最大化问题示例片段
Suppose we maximise P = 5x + 8y subject to x + y ≤ 7, 2x + y ≤ 10, x ≥ 0, y ≥ 0. The vertices are (0,0), (5,0), (3,4) and (0,7). Evaluating P gives 0, 25, 47, 56 respectively. So the maximum is 56 at (0,7). Always state the values of x and y that give the optimum.
假设最大化 P = 5x + 8y，约束为 x + y ≤ 7, 2x + y ≤ 10, x ≥ 0, y ≥ 0。顶点为 (0,0), (5,0), (3,4), (0,7)。P 值分别为 0, 25, 47, 56，因此最大值为 56，出现在 (0,7)。考试中务必写出最优解对应的 x 和 y 值。
7. Minimisation Problems | 最小化问题
When minimising, the objective line moves in the opposite direction. For example, minimise C = 3x + 2y with constraints that give a bounded region. Vertex testing remains reliable: simply pick the smallest objective value. Remember that for unbounded feasible regions, a minimum may still exist if the coefficients are positive; check the specification wording.
求解最小化问题时，等值线向相反方向移动。例如最小化 C = 3x + 2y，约束构成有界区域。顶点检验法依然可靠：只需找出最小的目标函数值。注意，若可行域无界，只要目标系数为正，最小值仍可能存在；务必审清题意。
8. Integer Solutions | 整数解
Many LP problems require x and y to be integers, e.g. number of cars. If the optimal vertex has fractional coordinates, you must search for the best integer point inside the feasible region. A common method is to draw a grid, list integer points near the fractional optimum, and test them. Do not just round the fractional vertex – that might be outside the region.
许多线性规划问题要求 x 和 y 为整数（如汽车数量）。如果最优顶点的坐标是分数，你必须在可行域内寻找最佳整数点。常用方法是画网格，列出靠近分数最优解的整数点并逐一检验。切勿直接对分数顶点四舍五入，因为四舍五入后的点可能落在可行域外。
Tip: After finding the optimal integer point, always verify it satisfies all constraints and gives the best objective compared with neighbouring integer points.
提示：找到最优整数点后，务必验证它满足所有约束，并且与相邻整数点相比确实给出了最优值。
9. Special Cases: Unboundedness and Infeasibility | 特殊情况：无界与无解
If the feasible region is unbounded and you are maximising with positive coefficients, the objective can increase without limit – the problem has no finite solution. In such a case, you should state “unbounded solution”. Conversely, if constraints conflict (e.g. x ≥ 5 and x ≤ 3), there is no feasible region at all; answer that the problem is infeasible.
若可行域无界，且你要最大化一个系数为正的目标函数，则目标值可以无限增大——问题没有有限解。此时应注明“解无界”。相反，如果约束条件相互矛盾（例如 x ≥ 5 和 x ≤ 3），可行域根本不存在，应回答该问题不可行。
AQA questions sometimes include a ‘no feasible region’ scenario; always sketch the lines accurately to spot conflicts.
AQA 考题有时会包含“无可行域”的情形；务必精确画线以识别矛盾。
10. Common Exam Mistakes | 常见失分点
- Forgetting non‑negativity constraints x ≥ 0, y ≥ 0.
- Using dashed lines for ≤/≥ or solid lines for strict inequalities.
- Shading the wrong side – always test a point, not just guess.
- Misreading the optimisation direction: maximise vs minimise.
- Rounding fractional integer solutions without checking the feasible region.
- Not explicitly writing the coordinates of the optimal point.
- Poor graph scaling leading to inaccurate intersection reading.
- 忘记写非负约束 x ≥ 0, y ≥ 0。
- ≤ 或 ≥ 用了虚线，或者严格不等式用了实线。
- 阴影区域选错——一定要代入检验点，不能凭感觉。
- 看错优化方向：混淆最大化与最小化。
- 对分数解简单四舍五入，而没有检验可行域。
- 没有明确写出最优点的坐标。
- 图形比例不当，导致交点读数不准。
11. Full Worked Example | 完整例题精讲
A factory produces two products A and B. Product A takes 2 hours on machine 1 and 3 hours on machine 2; product B takes 4 hours on machine 1 and 1 hour on machine 2. Machine 1 is available for 16 hours, machine 2 for 12 hours. Profit is £40 per A and £50 per B. Let x = number of A, y = number of B.
某工厂生产两种产品 A 和 B。产品 A 在机器 1 上耗时 2 小时，机器 2 上耗时 3 小时；产品 B 在机器 1 上耗时 4 小时，机器 2 上耗时 1 小时。机器 1 可用 16 小时，机器 2 可用 12 小时。利润分别为 A 每个 40 英镑，B 每个 50 英镑。设 x = A 的数量，y = B 的数量。
Constraints: 2x + 4y ≤ 16, 3x + y ≤ 12, x ≥ 0, y ≥ 0. Objective: Maximise P = 40x + 50y.
约束条件：2x + 4y ≤ 16, 3x + y ≤ 12, x ≥ 0, y ≥ 0。目标：最大化 P = 40x + 50y。
Graph: Line 2x+4y=16 passes through (8,0) and (0,4); line 3x+y=12 through (4,0) and (0,12). Feasible region vertices: (0,0), (4,0), (3,3) [intersection of 2x+4y=16 and 3x+y=12], (0,4).
图形：直线 2x+4y=16 过 (8,0) 和 (0,4)；直线 3x+y=12 过 (4,0) 和 (0,12)。可行域顶点：(0,0), (4,0), (3,3) [由 2x+4y=16 与 3x+y=12 联立解得], (0,4)。
| Vertex (x, y) | P = 40x + 50y |
|---|---|
| (0,0) | 0 |
| (4,0) | 160 |
| (3,3) | 270 |
| (0,4) | 200 |
Maximum profit is £270 when x = 3 units of A and y = 3 units of B. All constraints are met and the solution is integer without further search.
最大利润为 270 英镑，此时生产 3 个 A 和 3 个 B。所有约束满足，且解本身是整数，无需额外搜索。
12. Exam Readiness Checklist | 备考速查清单
- Define variables with units.
- List all constraints, including non‑negativity.
- Draw accurate graphs with a ruler; use shading convention correctly.
- Locate and label the feasible region.
- Find all relevant vertices by solving equations simultaneously.
- Evaluate the objective at each vertex or use the sliding line.
- State the optimal values clearly: ‘Maximum profit is £… when x = … , y = …’.
- For integer problems, test integer neighbours and confirm they stay inside R.
- Check for special cases and write a concluding sentence.
- 定义变量并标明单位。
- 列出所有约束，包含非负条件。
- 用直尺精确绘图；正确使用阴影约定。
- 找出并标出可行域。
- 通过联立方程求所有相关顶点。
- 在各顶点计算目标函数值，或使用等值线滑移法。
- 清晰表述最优结果：“最大利润为 £…，当 x = …，y = … 时”。
- 整数问题要检验相邻整数点，确保它们在可行域内。
- 注意特殊情况，并撰写结论句。
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