A-Level WJEC Mathematics: Linear Programming Key Concepts | A-Level WJEC 数学：线性规划 考点精讲

📚 A-Level WJEC Mathematics: Linear Programming Key Concepts | A-Level WJEC 数学：线性规划 考点精讲

Linear programming is a powerful optimisation tool in WJEC A-Level Mathematics, enabling you to maximise or minimise a linear function subject to linear inequality constraints. This topic typically appears in the decision mathematics or applied modules, where two-variable problems are solved graphically. Mastering the formulation, graphing, and vertex testing is essential for high marks.

线性规划是 WJEC A-Level 数学中强大的优化工具，使你能在线性不等式约束下最大化或最小化一个线性函数。该主题通常出现在决策数学或应用模块中，其中涉及两变量问题的图形解法。掌握建模、画图及顶点检验是取得高分的关键。

1. Introduction to Linear Programming | 线性规划简介

Linear programming (LP) deals with the problem of allocating limited resources to achieve the best outcome. In A-Level contexts, it usually involves two decision variables, such as the number of products A and B to produce, with constraints on materials, labour, or time. The objective is to maximise profit or minimise cost, expressed as a linear equation.

线性规划 (LP) 处理如何分配有限资源以获得最佳结果的问题。在 A-Level 课程中，它通常包含两个决策变量，例如生产产品 A 和 B 的数量，受原材料、劳动力或时间限制。目标通常用线性方程表示，旨在最大化利润或最小化成本。

The standard LP problem consists of an objective function Z = ax + by (to maximise or minimise) and a set of linear inequalities such as x ≥ 0, y ≥ 0 and constraints like 2x + y ≤ 10. The solution is found within the feasible region defined by these inequalities.

标准的线性规划问题由目标函数 Z = ax + by（求最大值或最小值）和一组线性不等式组成，如 x ≥ 0, y ≥ 0 以及 2x + y ≤ 10 等约束。解存在于这些不等式所定义的可行区域中。

2. Formulating Constraints | 构建约束条件

The first step is to translate a word problem into mathematical inequalities. Identify the decision variables (usually x and y), then write each limitation as a linear inequality. For example, if a factory has 40 hours of machine time and product A requires 5 hours while product B requires 2 hours, the constraint is 5x + 2y ≤ 40. Always include non-negativity constraints x ≥ 0, y ≥ 0, as negative production makes no sense.

第一步是将文字问题转化为数学不等式。确定决策变量（通常为 x 和 y），然后将每种限制写成线性不等式。例如，如果某工厂有 40 小时的机器时间，产品 A 需要 5 小时，产品 B 需要 2 小时，则约束条件为 5x + 2y ≤ 40。务必包含非负约束 x ≥ 0, y ≥ 0，因为负产量没有意义。

Pay close attention to keywords: ‘at most’ means ≤, ‘at least’ means ≥, ‘exactly’ means =. Some problems may also involve integer requirements, but in pure LP formulation, variables are usually considered continuous unless specified.

务必注意关键词：’不超过’ 对应 ≤，’至少’ 对应 ≥，’恰好’ 对应 =。有些问题也可能需要整数约束，但在纯线性规划建模中，除非特别说明，变量通常被视为连续取值。

3. Graphing Inequalities and Feasible Region | 绘制不等式及可行域

To graph a linear inequality, first draw the boundary line by treating it as an equation (e.g., 2x + 3y = 6). Use a solid line for ≤ or ≥, and a dashed line for strict < or > (though strict ones rarely appear in WJEC LP). Next, choose a test point not on the line, often (0,0), and substitute into the inequality. If the test point satisfies it, shade the half-plane containing that point; otherwise shade the other side.

绘制线性不等式时，首先将其视为等式画出边界线（如 2x + 3y = 6）。对于 ≥ 或 ≤ 使用实线，对于严格的 < 或 > 使用虚线（尽管 WJEC 线性规划中很少出现严格不等式）。然后选择不在直线上的测试点，常用 (0,0)，代入不等式。若测试点满足不等式，则着色包含该点的半平面；否则着色另一侧。

The feasible region is the intersection of all shaded half-planes and must satisfy every constraint simultaneously. In WJEC exam papers, you are often given a partially drawn graph and asked to complete the shading or to label the feasible region, so practise reading and drawing these diagrams accurately.

可行区域是所有着色半平面的交集，必须同时满足每个约束条件。在 WJEC 试卷中，通常给出部分绘制的图形，要求完成着色或标出可行区域，因此要练习准确读懂并绘制此类图形。

4. Identifying the Objective Function | 确定目标函数

The objective function expresses the quantity to be optimised, usually profit or cost. It is written as Z = ax + by, where a and b are constants derived from the problem. For a maximisation problem, you aim to find the largest possible Z within the feasible region; for minimisation, the smallest Z.

目标函数表达需要优化的量，通常是利润或成本，写作 Z = ax + by，其中 a、b 是由问题导出的常数。最大化问题要在可行域内找到最大的 Z；最小化问题则找到最小的 Z。

When you are asked to interpret the objective line, note that lines of the form ax + by = k are parallel for different k values. Moving the line parallel to itself in the direction that increases (or decreases) Z will help identify the optimal point. This ‘ruler method’ is sometimes used in conjunction with vertex testing.

当需要解释目标函数直线时，应注意不同 k 值的直线 ax + by = k 是互相平行的。将该直线沿垂直于自身的某一方向平移，使 Z 增大（或减小），有助于找到最优点。这种 ‘直尺法’ 有时与顶点法结合使用。

5. Finding Optimal Solutions: Vertex Method | 寻找最优解：顶点法

The fundamental theorem of linear programming states that if an optimal solution exists, it will occur at a vertex (corner) of the feasible region. Therefore, you can find the optimum by evaluating the objective function at every vertex. The vertex giving the maximum (or minimum) Z is the optimal solution.

线性规划的基本定理指出，若存在最优解，它一定出现在可行区域的某个顶点（角点）处。因此，可以通过计算每个顶点处的目标函数值来

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