📚 Linear Programming for CIE A-Level Mathematics | 线性规划考点精讲
Linear programming is a powerful technique used to optimise a linear objective function subject to a set of linear constraints. In the CIE A-Level Mathematics syllabus, this topic sits within the Decision Mathematics or Pure Mathematics components and requires students to formulate problems, graph inequalities, identify feasible regions, and find optimal solutions. Mastery of this chapter not only secures marks in exams but also builds a foundation for real-world applications in business, logistics, and engineering.
线性规划是一种强大的数学方法，用于在一组线性约束条件下优化一个线性目标函数。在 CIE A-Level 数学考试中，这个主题属于决策数学或纯数部分，要求学生掌握建立数学模型、绘图表示不等式、确定可行域并求出最优解。扎实掌握本章内容不仅能确保考试得分，还能为未来在商业、物流和工程等领域的应用打下坚实基础。
1. Introduction to Linear Programming | 线性规划简介
Linear programming (LP) deals with maximising or minimising a linear function, such as profit or cost, while respecting limitations expressed as linear inequalities. The key components are decision variables, constraints, and an objective function. A typical LP problem might ask: “A factory produces two products; how many of each should be made to maximise profit, given machine hours and material limits?”
线性规划（LP）旨在最大化或最小化一个线性函数（例如利润或成本），同时满足由线性不等式表示的约束条件。其核心组成部分包括决策变量、约束条件和目标函数。一个典型的 LP 问题可能是：“一家工厂生产两种产品，在给定的机器工时和材料限制下，每种产品应生产多少才能最大化利润？”
- Decision variables represent the quantities we control, e.g., x = number of product A, y = number of product B.
- 决策变量代表我们控制的数量，如 x = 产品 A 的数量，y = 产品 B 的数量。
- Constraints are linear inequalities formed from resource limits or minimum requirements.
- 约束条件是由资源限制或最低要求形成的线性不等式。
2. Formulating the Linear Programming Problem | 构建线性规划问题
To construct an LP model, first identify the variables: usually x and y for two products or activities. Write each constraint as a linear inequality: for example, 2x + 3y ≤ 60 for machine hours. Don’t forget non-negative constraints: x ≥ 0, y ≥ 0. The objective is an expression like Z = 5x + 4y, to be maximised or minimised.
构建线性规划模型时，首先确定变量：通常用 x 和 y 表示两种产品或活动。将每个约束写成线性不等式：例如，2x + 3y ≤ 60 表示机器工时限制。不要忘记非负约束：x ≥ 0, y ≥ 0。目标函数是一个表达式，如 Z = 5x + 4y，需要最大化或最小化。
| Component | Example | 组件 | 示例 |
|---|---|---|---|
| Decision variables | x, y | 决策变量 | x, y |
| Constraints | 2x + 3y ≤ 60 | 约束条件 | 2x + 3y ≤ 60 |
| Objective | Maximise Z = 5x + 4y | 目标 | 最大化 Z = 5x + 4y |
3. Graphing the Constraints | 绘制约束条件图像
Each linear inequality is drawn as a straight line on a coordinate plane. Replace the inequality sign with an equals sign to plot the boundary line: e.g., 2x + 3y = 60. Use a solid line for ≤ or ≥, and a dashed line for < or > (though strict inequalities are rare in CIE LP). Shade the unwanted region for each inequality. The intersection of all unshaded areas gives the feasible region.
每一个线性不等式都在坐标平面上绘制成一条直线。将不等号替换为等号即可画出边界线，如 2x + 3y = 60。如果是不等式 ≤ 或 ≥，使用实线；如果是 < 或 >，使用虚线（尽管 CIE 线性规划中严格不等式很少见）。对每个不等式，将不希望包含的一侧涂上阴影。所有未涂阴影区域的交集即为可行域。
x + 2y ≤ 10, 3x + y ≤ 15, x ≥ 0, y ≥ 0
Always label your axes and boundary lines. In an exam, a well-drawn, clearly labelled graph can earn method marks even if the final answer is slightly off.
一定要给坐标轴和边界线作标注。考试中，即使最终答案稍有偏差，一张绘制清晰、标注明确的图像也能获得方法分。
4. Identifying the Feasible Region | 确定可行域
The feasible region is the set of all points (x, y) that satisfy every constraint simultaneously. It is typically a convex polygon bounded by the constraint lines and axes. In some problems, the region may be unbounded. CIE questions often require you to shade the interior of the feasible region or clearly label its vertices.
可行域是同时满足所有约束条件的点 (x, y) 的集合。它通常是一个由约束线和坐标轴围成的凸多边形。在某些问题中，可行域可能无界。CIE 试题通常要求考生对可行域内部涂色或清晰标注其顶点。
- If a point lies on a boundary line, it still satisfies the corresponding inequality (for ≤, ≥).
- 如果一个点位于边界线上，它仍然满足相应的不等式（对 ≤, ≥ 而言）。
- Test a point (0,0) to determine which side of the line to shade, provided it does not lie on the line.
- 可用 (0,0) 测试来确定涂阴影的一侧，前提是该点不在边界线上。
5. The Objective Function | 目标函数
The objective function Z = ax + by represents the quantity to be maximised or minimised. Graphically, different values of Z correspond to a family of parallel lines called iso-profit or iso-cost lines. As Z varies, these lines slide across the feasible region.
目标函数 Z = ax + by 表示需要最大化或最小化的量。在图形上，不同的 Z 值对应一族平行线，称为等利润线或等成本线。随着 Z 值变化，这些直线在可行域上平移。
To find the optimal point, one can push the objective line parallelly until it is just about to leave the feasible region. The last point(s) it touches gives the optimal solution.
为了找到最优点，可以平行移动目标函数直线，直到它即将离开可行域。它所触及的最后一个点（或多个点）即为最优解。
6. Finding Optimal Solutions: Corner Point Method | 求最优解：顶点法
The fundamental theorem of linear programming states that if a linear programming problem has an optimal solution, it occurs at a vertex (corner point) of the feasible region. Therefore, a reliable method is to list all vertices, evaluate the objective function at each, and select the best value.
线性规划的基本定理指出，如果线性规划问题存在最优解，那么该解必定出现在可行域的顶点（角点）上。因此，一种可靠的方法是列出所有顶点，计算每个顶点处的目标函数值，并从中选出最优值。
- Vertices are found by solving pairs of boundary equations simultaneously.
- 顶点可通过联立求解边界线方程组得到。
- In an exam, you must show the coordinates of each vertex clearly.
- 考试中，必须清晰给出每个顶点的坐标。
Z = 5x + 4y
For example, if vertices are (0,0), (0,10), (4,6), (7,0), calculate Z at each: (0,0): Z=0; (0,10): Z=40; (4,6): Z=5(4)+4(6)=44; (7,0): Z=35. Maximum Z is 44 at (4,6).
举例来说，若顶点为 (0,0), (0,10), (4,6), (7,0)，分别计算 Z: (0,0): Z=0; (0,10): Z=40; (4,6): Z=5(4)+4(6)=44; (7,0): Z=35。最大 Z 为 44，位于 (4,6)。
7. Testing Vertices and Optimal Value | 测试顶点与最优值
Once vertices are known, substitute them into the objective function. The highest Z gives the maximum; the lowest Z gives the minimum. CIE mark schemes often award marks for correct evaluation and final statement. Remember to state the optimal value in context, e.g., “The maximum profit is £44, achieved by producing 4 units of A and 6 units of B.”
知道顶点坐标后，将其代入目标函数。Z 值最大即为最优最大值，最小即为最优最小值。CIE 评分标准通常对正确代入和结论给予分数。记得结合上下文表述最优值，例如：“最大利润为 44 英镑，通过生产 4 台 A 和 6 台 B 实现。”
If two vertices yield the same optimal Z, every point on the line segment joining them is also optimal, leading to multiple optimal solutions.
如果两个顶点得到相同的最优 Z 值，那么连接这两点的线段上的所有点都是最优解，这就出现了多解情况。
8. Special Cases: Unbounded, Infeasible, Multiple Solutions | 特殊情况：无界、无解、多解
An unbounded feasible region may not have a maximum if the objective can increase indefinitely. However, a minimum may still exist. Conversely, an infeasible region occurs when constraints contradict, leaving no overlapping area. CIE problems usually design feasible bounded regions, but you should recognise these exceptions.
无界可行域可能不存在最大值，因为目标函数可以无限增大，但仍可能存在最小值。相反，当约束条件相互矛盾、没有重叠区域时，便会出现无解可行域。CIE 试题通常设计有界的可行域，但你仍需了解这些特殊情况。
Multiple optimal solutions happen when the objective line is parallel to one of the constraint boundaries. In this case, all points along that edge are optimal. The answer must specify the range or general solution.
当目标函数直线与某约束边界平行时，便会出现多重最优解。此时，该边界上的所有点都是最优解。答案必须说明这一范围或一般解。
| Special Case | Interpretation | 特殊情况 | 解释 |
|---|---|---|---|
| Unbounded | No finite maximum, or min exists | 无界 | 无有限最大值，或存在最小值 |
| Infeasible | No solution satisfies all constraints | 无解 | 无任何解满足所有约束 |
| Multiple optima | Edge parallel to objective line | 多解 | 边界与目标线平行 |
9. Integer Programming Requirements | 整数规划要求
In many exam problems, the decision variables represent counts of items, so solutions must be integers. If the optimal vertex has non-integer coordinates, you need to test integer points near that vertex within the feasible region to find the best integer solution. Simply rounding the coordinates may not yield the optimal integer answer.
在许多考试问题中，决策变量表示物品的件数，因此解必须是整数。如果最优顶点的坐标为非整数，你需要在可行域内该顶点附近的整数点中进行测试，以找到最佳的整数解。仅仅将坐标四舍五入往往得不到最优整数答案。
Optimal vertex: (3.8, 2.4) → test (3,2), (3,3), (4,2), (4,3)
Always check all combinations that lie inside the feasible region and choose the point that gives the best objective value. The CIE mark scheme expects explicit testing of candidate integer points.
务必检查位于可行域内的所有组合，选择使目标函数最优的点。CIE 评分标准要求对候选整数点进行明确测试。
10. Sensitivity Analysis (Basic) | 灵敏度分析基础
While full sensitivity analysis is beyond the scope of most CIE syllabuses, you may encounter simple questions about changing a coefficient in the objective function or a right-hand side constant. The key is to understand how the slope of the objective line affects optimality. If the objective slope lies between the slopes of two binding constraints, the current optimal vertex remains optimal, though the value changes.
虽然全面的灵敏度分析超出了大多数 CIE 考纲范围，但你可能遇到简单的变化问题，例如改变目标函数中的系数或右侧常数。关键在于理解目标函数直线的斜率如何影响最优解。如果目标斜率位于两个起作用约束的斜率之间，则当前最优顶点保持最优，只是最优值会改变。
For instance, if Z = ax + by and the binding constraints have slopes m₁ and m₂, the optimal vertex stays the same as long as -a/b lies between m₁ and m₂. Such reasoning can be tested with “find the range of values for a coefficient so that the optimal solution remains unchanged”.
例如，若 Z = ax + by，起作用的约束斜率为 m₁ 和 m₂，则只要 -a/b 介于 m₁ 和 m₂ 之间，最优顶点就保持不变。这种推理可能以“求系数的取值范围，使得最优解不变”的形式进行考查。
11. Common Exam Pitfalls | 常见考试陷阱
Many students lose marks by misinterpreting inequality directions. Always read the wording carefully: “at most” means ≤, “at least” means ≥, “exceeds” means >, etc. Another pitfall is forgetting non-negativity constraints: x, y ≥ 0 must be included unless the problem says otherwise.
许多学生因误解不等式的方向而失分。务必仔细阅读文字：“最多”用 ≤，“至少”用 ≥，“超过”用 >，等等。另一个陷阱是忘记非负约束：除非题目另有说明，必须包含 x, y ≥ 0。
- Shading the wrong side of a line can make the entire graph useless. Double-check by testing a point like (0,0).
- 给直线错误的一侧涂阴影会导致整张图无效。请用 (0,0) 等测试点再次确认。
- Careless coordinate calculations when solving simultaneous equations lead to wrong vertices. Show working to gain method marks.
- 联立方程求解时的粗心计算会得到错误顶点。展示解题步骤以获取方法分。
Also, when integer solutions are required, failing to test neighbouring integer points is a direct loss of accuracy marks. Always answer in the context of the problem, including units and a final statement.
此外，当需要整数解时，没有测试相邻整数点会直接失去准确性分数。一定要结合题意回答，包括单位和最终结论句。
12. Summary and Tips | 总结与技巧
Linear programming in CIE A-Level Maths is highly structured: formulate, graph, shade, find vertices, evaluate, and select the optimum. Practice drawing accurate, labelled graphs quickly. Use the corner point method as it is systematic and minimises errors. For integer problems, list candidate points explicitly. Revise past papers focusing on the specific phrasing of constraints. With careful method and clear presentation, this topic becomes a reliable source of marks.
CIE A-Level 数学中的线性规划题型高度结构化：建立模型、绘图、涂影、求顶点、计算、选择最优值。练习快速绘制精准、清晰的带标注图线。使用顶点法，因为它系统且能减少错误。对于整数问题，明确列出候选点。复习历年真题，专注于约束条件的具体表述。通过严谨的方法和清晰的呈现，这个话题将成为可靠的得分点。
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