📚 A-Level Maths: Linear Programming Revision Guide | A-Level 数学：线性规划 考点精讲
Linear Programming is a powerful graphical technique used in Decision Mathematics to determine the optimal use of limited resources. Whether you are looking to maximise profit or minimise cost, this topic asks you to translate real-world constraints into algebraic inequalities, sketch the feasible region, and then apply a rigorous method to find the vertex that optimizes the objective function. This guide breaks down every critical step, from plotting constraints to handling unbounded regions and integer solutions.
线性规划是运筹学中一个强大的图形化技术，用于确定有限资源的最优分配。无论是寻求利润最大化还是成本最小化，这一考点都要求你将现实世界的限制条件转化为代数不等式，绘制出可行域，然后运用严谨的方法找到使目标函数最优的顶点。本文将详细拆解从绘制不等式到处理无界区域及整数解的每一个关键步骤。
1. Understanding the Core Concept | 理解核心概念
At its heart, Linear Programming involves optimizing (maximizing or minimizing) a linear objective function. This is subject to a set of linear constraints, typically represented as inequalities involving two decision variables (usually x and y). The solution is found visually by analyzing the feasible region where all constraints overlap.
归根结底，线性规划涉及对线性目标函数进行最优化（最大化或最小化）。这需要在一组线性约束条件下进行，这些约束条件通常被表示为涉及两个决策变量（通常是 x 和 y）的不等式。通过分析所有约束条件重叠的可行域，可以从图形上找到问题的解。
2. Key Terminology Breakdown | 关键术语分解
Before diving into the method, it is crucial to master the specific vocabulary. You must be able to identify the decision variables, sketch the constraints, and label the objective function accurately in your exam script.
在深入探讨方法之前，掌握特定术语至关重要。你必须在试卷上准确识别决策变量、绘制约束条件并标记目标函数。
| English Term | 中文术语 | Explanation |
|---|---|---|
| Objective Function | 目标函数 | The expression to be maximised (e.g., Profit) or minimised (e.g., Cost), usually in the form P = ax + by. |
| Constraints | 约束条件 | Linear inequalities that restrict the values of the decision variables (e.g., x + 2y ≤ 10). |
| Feasible Region (R) | 可行域 | The shaded or unshaded area where all constraints are satisfied simultaneously. |
| Decision Variables | 决策变量 | The unknowns (usually x, y) that represent quantities we can control. |
| Vertex/Vertices | 顶点 | The corner points of the feasible region where the optimal solution usually lies. |
3. Graphing Constraints & the Feasible Region | 绘制约束条件与可行域
To establish the feasible region, plot each linear inequality as if it were an equation. Use a solid line for ≤ or ≥, and a dashed line for strict < or > (though strict inequalities are rare in A-Level LP). Determine the unwanted region by testing a point (usually (0,0)). Shade out the side you do NOT want, leaving the feasible region (R) clear. Typically, x ≥ 0 and y ≥ 0 restrict the region to the first quadrant.
要确定可行域，将每个线性不等式当作方程来绘制直线。对于 ≤ 或 ≥ 使用实线，对于严格的 < 或 > 使用虚线（尽管在 A-Level 线性规划中严格不等式很少见）。通过测试一个点（通常是(0,0)）来确定不满足不等式的一侧。将不满足要求的区域涂上阴影，留下清晰的可行域 (R)。通常，x ≥ 0 和 y ≥ 0 将区域限制在第一象限。
4. The Objective Function and Profit Lines | 目标函数与等值线
Once the feasible region R is drawn, we introduce the objective function P = ax + by. To find the maximum value of P, we draw a ‘profit line’ (also called an isoprofit or isocost line) for a specific value of P, usually ax + by = 0. Parallel lines have the same gradient. To optimise, slide a ruler parallel to this line across the feasible region.
一旦画出了可行域 R，我们就引入目标函数 P = ax + by。为了找到 P 的最大值，我们为特定的 P 值绘制一条“利润线”（或称为等利润线/等成本线），通常是 ax + by = 0。 平行线具有相同的梯度。为了求最优解，用直尺平行于这条线在可行域上滑动。
For Maximisation: Slide the ruler as far from the origin as possible while still touching R.
最大化问题：将直尺向远离原点的方向移动，直到它刚好要离开 R。
For Minimisation: Slide the ruler as close to the origin as possible while still touching R.
最小化问题：将直尺向尽可能靠近原点的方向移动，同时保持与 R 接触。
5. Method 1: The Vertex Method | 方法一：顶点法
The Fundamental Theorem of Linear Programming states that the optimal solution must lie at a vertex of the feasible region, provided the region is bounded. List all the vertices of R. Calculate their exact coordinates by solving the simultaneous equations of the lines that intersect at those points. Plug each vertex into the objective function P to find the maximum or minimum value.
线性规划基本定理指出，只要可行域是有界的，最优解必须位于可行域的顶点处。列出 R 的所有顶点。通过求解在这些点相交的直线的联立方程，来计算各顶点的精确坐标。将每个顶点代入目标函数 P，以找出最大值或最小值。
6. Method 2: Sliding the Objective Line | 方法二：平移目标函数线
This visual method is often required to justify your answer in exams, especially when using graph paper. Draw a clear reference line: ax + by = 0. Keep your ruler perfectly parallel to this line. Move it to the highest extreme of R (maximising) or the lowest extreme (minimising). Identify the vertex where the ruler last touches R. This visual check verifies your algebraic results.
考试中常常要求使用这种可视化方法来证明你的答案，尤其是在使用坐标纸时。画一条清晰的参考线：ax + by = 0。保持直尺与此线完全平行。将其移动到 R 的最高点（最大化）或最低点（最小化）。识别直尺最后接触 R 的顶点。这种视觉检查可以验证你的代数结果。
7. Finding Intersection Points (Simultaneous Equations) | 寻找交点（解联立方程）
To secure method marks, you must show the algebraic working out used to find the vertices. If two lines intersect, you must solve them. For example, if constraints are x + y = 5 and 2x + 3y = 12, you can solve by elimination: multiply the first equation by 2, subtract from the second, or use substitution. Accurate coordinates are vital as an incorrect vertex leads to an incorrect optimum value.
为了确保拿到方法分，你必须展示寻找顶点所用的代数计算过程。如果两条直线相交，必须求解它们。例如，如果约束条件是 x + y = 5 和 2x + 3y = 12，你可以通过消元法来求解：将第一个方程乘以 2，再与第二个方程相减，或使用代入法。精确的坐标至关重要，因为错误的顶点会导致错误的最优值。
8. Special Case: Multiple Optimal Solutions | 特殊情况：多重最优解
This occurs when the objective function line is parallel to one of the constraint boundaries that form the edge of R. When sliding the profit line, it overlaps a whole segment of the feasible region rather than just a single vertex. In this case, any point on that line segment between the two vertices gives the same optimal value for P.
当目标函数线与构成 R 边界的某条约束线平行时，就会出现这种情况。在平移利润线时，它会与可行域的一条整边重叠，而不仅仅是一个顶点。在这种情况下，该线段上位于两个顶点之间的任何点都会为 P 提供相同的最优值。
9. Special Case: Unbounded Feasible Regions | 特殊情况：无界可行域
Sometimes the constraints do not fully enclose the region, leaving it open in one direction. If you are maximising an objective function and the region is unbounded, you might find that the maximum is infinite, which is usually a sign of a mistake in defining the constraints or an unfeasible real-world plan. For minimisation in an unbounded region, you can usually still find a finite optimal vertex.
有时约束条件没有完全封闭区域，导致区域在某个方向是敞开的。如果你在最大化一个目标函数且区域是无界的，你可能会发现最大值是无穷大的，这通常表明约束条件定义有误，或是现实计划不可行。对于无界区域的最小化问题，通常仍然可以找到一个有限的最优顶点。
10. Integer Solutions and Discrete Programming | 整数解与离散规划
If the context requires integer solutions (e.g., you cannot produce half a car or 0.7 of a person), you cannot just round the optimal vertex. You must look at the grid points near the vertex while ensuring they still lie strictly inside the feasible region R. Test the integer coordinates around the optimal vertex to find the maximum integer value of P without breaking any constraints.
如果实际背景要求整数解（例如，你不能生产半辆汽车），不能只是简单地对最优顶点进行四舍五入。你必须检查顶点附近的网格点，同时确保它们仍然严格位于可行域 R 内。测试最优顶点附近的整数坐标，以找到不违反任何约束条件下的 P 的最大整数值。
11. Real-World Applications in A-Level Exams | A-Level 考试中的实际应用
Exam questions typically frame the problem as a resource allocation scenario. You might be asked to maximise profit given limited machine hours, raw materials, or ingredients. You must carefully define your variables (e.g., “Let x be the number of Type-A cakes; let y be the number of Type-B cakes”). Be systematic: list the non-negativity constraints (x ≥ 0, y ≥ 0), formulate the resource constraints, define the objective, plot, and solve.
考题通常将问题设定为资源分配的场景。你可能会被要求计算在有限的机器工时、原材料或配料的情况下最大化利润。你必须仔细定义变量（例如，“设 x 为 A 型蛋糕的数量；设 y 为 B 型蛋糕的数量”）。需要有条不紊：列出非负性约束条件（x ≥ 0，y ≥ 0），制定资源约束条件，定义目标，绘图，然后求解。
12. Summary and Exam Mastery Tips | 总结与高分技巧
Linear Programming is one of the most structured and highly-scoring topics in the A-Level Maths applied paper if approached methodically. Always draw a clear, large diagram. Use a ruler for all lines. Label R clearly and annotate your vertices. Remember: no diagram is complete without clearly indicating the feasible region. Practice the sliding ruler technique physically, as it prevents parallax errors and confirms the correct vertex instantly.
线性规划是 A-Level 数学应用题卷中结构最严谨、最容易拿满分的考点之一，只要按部就班解题即可。一定要画出清晰、足够大的示意图。所有直线都要用直尺画。清晰地标出 R 并注释你的顶点。记住：没有清晰标明可行域 (R) 的图解是不完整的。多练习实际移动直尺的技巧，这可以防止视差错误，并快速确认正确的顶点。
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