📚 Mastering Linear Programming for AQA A-Level Mathematics | AQA A-Level 数学线性规划考点精讲
Linear programming is a cornerstone of AQA A-Level Mathematics (Decision 1), enabling you to solve optimisation problems where profit, cost, or some other quantity must be maximised or minimised under a set of linear constraints. Mastering this topic means being able to translate a worded situation into inequalities, graph a feasible region, and apply graphical or algebraic methods to locate the optimal solution – often with integers involved. This revision guide walks you through every key exam point, from formulating variables to interpreting your final answer in context.
线性规划是 AQA A-Level 数学（决策1）的核心内容，用于在给定一组线性约束条件下最大化或最小化利润、成本等量。掌握该专题意味着能够将文字情境转化为不等式、绘制可行域图形，并运用几何或代数方法找到最优解——通常涉及整数。本复习指南带你逐一攻克所有关键考点，从定义变量到结合情境解读最终答案，助你自信应考。
1. What is Linear Programming? | 什么是线性规划？
Linear programming (LP) is a mathematical technique for finding the best possible result in a situation modelled by linear relationships. Every LP problem contains three essential ingredients: decision variables (the quantities you can control, typically x and y), an objective function (the expression to maximise or minimise, such as profit P = 5x + 8y), and a set of constraints written as linear inequalities. The constraints carve out a feasible region – tive function is linear, the optimum always lies on the boundary of this region, usually at a vertex.
线性规划是一种在由线性关系构建的模型中寻找最佳结果的数学技术。每个线性规划问题包含三个基本要素：决策变量（可以控制的量，通常用 x 和 y 表示）、目标函数（需要最大化或最小化的表达式，例如利润 P = 5x + 8y），以及写成线性不等式的一组约束条件。约束条件围出一个可行域——满足所有限制（包括非负性 x ≥ 0, y ≥ 0）的点的集合。由于目标函数是线性的，最优解总是出现在该区域的边界上，通常在某个顶点处。
2. Formulating the Problem: Variables and Constraints | 建模：变量与约束条件
The first exam skill is translating a word problem into precise mathematical statements. Begin by defining exactly what x and y represent, including units (e.g. number of tables and chairs produced per day). Express every limitation as an inequality. For example, if a workshop has 240 kg of metal and each table uses 12 kg while each chair uses 8 kg, the constraint is 12x + 8y ≤ 240. Labour hours, storage limits, and minimum order requirements are similarly encoded. Always include the non-negativity constraints x ≥ 0, y ≥ 0 unless the context forbids zero production.
第一项考试技能是将文字题转化为精确的数学表述。先清晰定义 x 和 y 所代表的含义及单位（例如每天生产的桌子和椅子数量）。将每项限制表示为不等式。例如，若车间有 240 kg 金属，每张桌子耗用 12 kg，每把椅子耗用 8 kg，则约束为 12x + 8y ≤ 240。劳动时长、存储容量、最低订单要求等也类似地编码。除非情境禁止零产量，否则务必包含非负约束 x ≥ 0, y ≥ 0。
A typical objective function might read: Maximise P = 100x + 150y. In the AQA exam, ensure every number has the correct units and that all constraints are linear – no x² terms, no products of variables, and no fractional powers.
典型的目标函数可能为：最大化 P = 100x + 150y。在 AQA 考试中，要确保每个数字都有正确的单位，而且所有约束都是线性的——不能出现 x² 项、变量的乘积或分数次幂。
| Resource / 资源 | Available / 可用量 | Per table (x) / 每张桌子 (x) | Per chair (y) / 每把椅子 (y) | Constraint / 约束 |
|---|---|---|---|---|
| Metal (kg) / 金属 (kg) | 240 | 12 | 8 | 12x + 8y ≤ 240 |
| Labour (hr) / 劳动 (小时) | 140 | 6 | 5 | 6x + 5y ≤ 140 |
| Storage / 存储 | 50 items | 1 | 1 | x + y ≤ 50 |
3. Drawing the Feasible Region: Graphical Inequalities | 绘制可行域：图形不等式
Once constraints are written, plot them on a coordinate grid. For each inequality of the form ax + by ≤ c, draw the boundary line ax + by = c d line if you ever encounter strict < or > (rare in D1). Decide which half-plane satisfies the inequality by testing a point (0,0) if the line does not pass through the origin. Shade the unwanted region – the side that does not satisfy the inequality. The clear, unshaded polygon that remains is the feasible region.
写好约束条件后，在坐标网格上绘制它们。对于形如 ax + by ≤ c 的每个不等式，画出边界线 ax + by = c。≤ 或 ≥ 用实线，遇到严格不等号 < 或 >（D1 中罕见）时用虚线。通过代入一个测试点（若直线不经过原点，常选 (0,0)）来判断哪个半平面满足不等式。涂抹不满足要求的区域——即涂抹错误的一侧。最终留下的清晰、无涂抹的多边形就是可行域。
Always label your axes, mark the scale clearly, and write the coordinates of key points where lines intersect. In AQA questions, accuracy is vital; use a ruler and sharp pencil, and allow enough space for the region to be read easily.
务必标注坐标轴，清晰标出刻度，并写出直线交点的坐标。在 AQA 考题中，准确性至关重要；使用直尺和削尖的铅笔作图，并留出足够空间以使区域清晰可读。
- Draw each line as an accurate straight line using two well-spaced points./ 用两个间距合适的点精确画出每条直线。
- Test shading with an easy point to avoid shading the feasible region by accident./ 用一个简单的点测试涂抹方向，避免无意中将可行域涂抹掉。
4. The Objective Function and Objective Line Method | 目标函数与目标线法
0x + 150y. Choose a convenient multiple of the coefficients, such as 600, to draw the line 100x + 150y = 600. This is the objective line. Now imagine sliding this line parallel to itself in the direction that improves P – for maximisation, move in the direction of increasing P; for minimisation, move to decrease P. The last point of the feasible region that the line touches before leaving it entirely is the optimum.
要定位最优解，为目标函数选取一个试验值，例如 P = 100x + 150y。选择一个便于计算的系数公倍数，比如 600，画出直线 100x + 150y = 600。这就是目标线。然后想象将该直线平行滑动，朝着改进 P 的方向移动——最大化问题朝 P 增大方向移动，最小化问题朝 P 减小方向移动。在直线即将完全脱离可行域之前，最后接触的那个点即为最优解。
The direction of improvement can be checked by evaluating P at a test point relative to the objective line, or by noting that coefficients are positive, so moving north-east increases P.
可以通过相对于目标线测试某个点处的 P 值来检验改进方向，或者注意到系数均为正，因此向东北方向移动会使 P 增加。
Example: Maximise P = 3x + 2y, choose line 3x + 2y = 12 and slide north-east / 示例：最大化 P = 3x + 2y，选择直线 3x + 2y = 12 并向东北方向移动
5. Vertex Testing Method: Corner Point Principle | 顶点检验法：角点原理
Because the objective function is linear, the optimal value (if it exists and the feasible region is bounded) will occur at a vertex of the feasible region. List every vertex by solving pairs of intersecting boundary lines. Compute the value of the objective function at each vertex. For maximisation, choose the vertex with the highest P; for minimisation, the lowest. This method is particularly robust when the feasible region has many edges, and it avoids the need for precise sliding.
由于目标函数是线性的，最优值（如果存在且可行域有界）将出现在可行域的某个顶点上。通过解出每对相交边界直线来列出所有顶点。计算每个顶点处的目标函数值。对于最大化问题，选择使 P 最大的顶点；对于最小化问题，选择使 P 最小的顶点。当可行域有很多条边时，这种方法尤其可靠，且无需精确滑动即可求解。
- carefully: e.g. intersection of 12x + 8y = 240 and 6x + 5y = 140 gives (x, y) = (10, 8)./ 仔细解联立方程：例如 12x + 8y = 240 和 6x + 5y = 140 的交点为 (x, y) = (10, 8)。
- Reject any vertex that does not satisfy all inequalities, including non-negativity./ 舍去任何不满足所有不等式（含非负约束）的顶点。
50y to get 0, 2000, 2200, 2625 → optimum at (0,17.5) but check integer restrictions later.
若顶点为 (0,0), (20,0), (10,8), (0,17.5)，计算 P = 100x + 150y 得到 0, 2000, 2200, 2625 → 最优解在 (0,17.5)，但后面需检查整数限制。
6. Finding Optimal Integer Solutions | 寻找最优整数解
Many AQA linear programming questions involve discrete products, so x and y must be whole numbers. If the optimal vertex does not have integer coordinates, look at integer points on or just inside the boundary near that vertex. Create a small search area: draw the objective line through the fractional optimum and examine all integer points that lie within the feasible region and are close to that line. Evaluate P at each integer point and pick the best.
许多 AQA 线性规划问题涉及离散产品，因此 x 和 y 必须为整数。如果最优顶点坐标不是整数，就查看该顶点附近位于边界上或刚好在其内的整数点。创建一个小的搜索区域：画出经过分数最优点的目标线，然后检查所有位于可行域内且靠近该线的整数点。计算每个整数点处的 P 值，并选择最佳者。
- If optimum is (2.4, 5.8), test points (2,5), (2,6), (3,5), (3,6) etc., checking feasibility./ 若最优解为 (2.4, 5.8)，则测试点 (2,5), (2,6), (3,5), (3,6) 等，并检验可行性。
- Never simply round; always verify the new point still satisfies all constraints./ 切勿简单四舍五入；务必验证新点仍满足所有约束。
7. Interpreting the Solution in Context | 在上下文中解读解
State which quantities to produce (12 tables and 15 chairs), what the maximum profit is (£100 × 12 + £150 × 15 = £ nfirm that all constraints are satisfied. If some resource is not fully used, identify the slack labour constraint was 6x + 5y ≤ 140 and with (12,15) gives 6(12)+5(15)=147 which violates – then the point is invalid. Always check.
如果没有最后的结论句，x = 12, y = 15 这样的答案就毫无意义。要说明应生产多少（12 张桌子和 15 把椅子），最大利润是多少（£100 × 12 + £150 × 15 = £3450），并确认所有约束均已满足。如果某些资源未用完，指出其松弛量。例如，劳动约束为 6x + 5y ≤ 140，代入 (12,15) 得 6(12)+5(15)=147，超出限制，则该点是无效的。务必检查。
In the AQA mark scheme, marks are often awarded for the interpretation line, so never skip it.
在 AQA 评分方案中，常常为解读陈述分配分数，因此千万不要遗漏。
8. Special Cases: Multiple Optimal Solutions and Unbounded Regions | 特殊情况：多重最优解与无界区域
When the objective line is parallel to a binding constraint, every point along that edge of the feasible region is optimal, yielding infinitely many solutions. Show that P has the same value at two adjacent vertices along that edge; any weighted average also works. If a maximisation problem has an unbounded feasible region (missing a necessary constraint), it may be impossible to reach a maximum because you can keep increasing P without bound. In exams, you simply state that the maximum is unbounded or does not exist. Minimisation on an unbounded region can still have a finite answer if the region is open in the correct direction.
当目标线与某个硬约束平行时，可行域的整条边界上的每一个点都是最优的，产生无限多解。证明沿该边界两个相邻顶点处的 P 值相等；任何加权平均也有效。若最大化问题的可行域无界（缺少必要的约束），则可能无法达到最大值，因为可以无限增大 P。考试中，只需说明最大值无界或不存在。如果可行域在正确方向上开放，无界区域上的最小化问题仍可能有有限解。
Example: Maximise x + y subject to x + 2y ≤ 10, x ≥ 0, y ≥ 0 and no upper bound on x – region is bounded in y but not in x. Maximum is unbounded as x → ∞ / 示例：最大化 x + y，约束为 x + 2y ≤ 10, x ≥ 0, y ≥ 0 且 x 无上限——区域在 y 上有界但在 x 上无界，最大值无界，因为 x → ∞。
9. Common Mistakes in Linear Programming | 线性规划常见错误
Many marks are lost on avoidable errors.
许多分数都因可避免的错误而丢失。
- Forgetting to state x ≥ 0, y ≥ 0. / 忘记声明 x ≥ 0, y ≥ 0。
- Not labelling axes or indicating a scale, making the graph ambiguous. / 未标注坐标轴或未标明刻度，导致图形模糊。
- Shading the wrong region or shading both sides. / 涂抹错误区域，或两侧都涂抹。
- Using a rounded vertex instead of exact fractions, which can mislead integer searches. / 使用近似顶点值而非精确分数，可能误导整数搜索。
- Misinterpreting 'at least' or 'at most' phrases, e.g. 'at least twice as many chairs as tables' becomes y ≥ 2x, not y ≤ 2x. / 误解“至少”或“至多”等表述，例如“椅子数量至少是桌子的两倍”应写为 y ≥ 2x 而非 y ≤ 2x。
- Attempting integer optimisation without explicitly testing all candidate points. / 未明确测试所有候选点就进行整数优化。
the logical sequence: variables → constraints → graph → optimal solution → interpretation.
确保解题步骤遵循逻辑顺序：变量 → 约束 → 图形 → 最优解 → 解读。
10. AQA Exam Strategy and Tips | AQA 考试策略与技巧
Linear programming usually appears as a structured, multi-part question in AQA Decision 1. Show clear algebraic working for constraint simplification and vertex calculation. On the graph, use a ruler for all lines, mark the feasible region clearly, and label vertices. If the question
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