A-Level CCEA Maths: Linear Programming Key Points Revision | A-Level CCEA 数学：线性规划考点精讲

📚 A-Level CCEA Maths: Linear Programming Key Points Revision | A-Level CCEA 数学：线性规划考点精讲

Linear programming is a powerful optimisation tool within the CCEA A-Level Mathematics Decision module. It enables you to find the best possible outcome – such as maximum profit or minimum cost – in a situation modelled by linear relationships. This article distils the core concepts, graphical techniques, and exam strategies you need to master linear programming for CCEA.

线性规划是 CCEA A-Level 数学决策模块中一种强大的优化工具。它能让你在由线性关系建模的情境中，找到最佳可能结果——例如最大利润或最小成本。本文提炼了你为应对 CCEA 考试所需掌握的线性规划核心概念、图解技巧和应试策略。

1. Understanding Linear Programming | 认识线性规划

Linear programming deals with problems where we seek to maximise or minimise a linear objective function, subject to a set of linear inequalities called constraints. All decision variables are usually required to be non-negative. Typical applications include resource allocation, production planning, and diet problems.

线性规划处理的是在一组称为约束条件的线性不等式下，寻求最大化或最小化线性目标函数的问题。所有决策变量通常要求非负。典型的应用包括资源分配、生产计划以及饮食搭配问题。

The word ‘linear’ indicates that both the objective function and constraints involve only the first power of the decision variables – no products of variables, no powers, and no trigonometric functions. The ‘programming’ here refers to planning, not computer programming.

‘线性’一词意指目标函数和约束条件都只包含决策变量的一次方——没有变量之积、没有幂次、没有三角函数。这里的 ‘programming’ 指的是规划，而非计算机编程。

2. Formulating the Problem – Key Elements | 问题建模 – 关键要素

To set up a linear programming problem, you must identify three fundamental components: the decision variables, the objective function, and the constraints. A clear definition of variables at the start is critical – for example, ‘Let x be the number of chairs produced and y be the number of tables produced.’

要建立线性规划问题，你必须明确三个基本组成部分：决策变量、目标函数和约束条件。在开始时就清晰地定义变量至关重要——例如，’设 x 为生产的椅子数量，y 为生产的桌子数量’。

The objective function is the quantity you wish to optimise, written as a linear expression in terms of the decision variables. Constraints arise from limited resources, market demands, or contractual obligations, and are expressed as linear inequalities or equalities.

目标函数就是你希望优化的量，以决策变量的线性表达式写出。约束条件来源于有限的资源、市场需求或合同义务，并以线性不等式或等式表达。

Always include the non-negativity constraints (x ≥ 0, y ≥ 0) unless the context naturally allows negative values. For CCEA examinations, the problem statement will usually provide all the numerical data; your task is to translate the words into precise mathematical inequalities.

除非情境自然允许负值，否则务必包含非负约束 (x ≥ 0, y ≥ 0)。在 CCEA 考试中，题目通常会给出所有数值数据；你的任务就是将文字转化为精确的数学不等式。

3. Decision Variables, Objective Function & Constraints | 决策变量、目标函数与约束

Consider a furniture workshop that makes two types of units: standard (x) and deluxe (y). Each standard unit gives a profit of £40, each deluxe £60. The objective is to maximise total profit:

考虑一家家具工坊生产两种产品：标准型 (x) 和豪华型 (y)。每件标准型利润为 40 英镑，每件豪华型为 60 英镑。目标是最大化总利润：

Maximise Z = 40x + 60y

This is the objective function. Suppose assembling a standard unit takes 2 hours and a deluxe unit 3 hours, with 120 hours available per week. Finishing takes 1 hour for standard and 2 hours for deluxe, with 80 hours available. These give the constraints:

这就是目标函数。假设组装一件标准型需 2 小时，豪华型需 3 小时，每周可用 120 小时。精加工标准型需 1 小时，豪华型需 2 小时，可用 80 小时。由此得出约束：

2x + 3y ≤ 120
x + 2y ≤ 80
x ≥ 0, y ≥ 0

Every inequality is derived directly from the resource limitations. In CCEA papers, you might also encounter constraints such as ‘at least twice as many standards as deluxes’ which translates to x ≥ 2y, or a minimum production requirement.

每一个不等式都直接源于资源限制。在 CCEA 试卷中，你还可能遇到诸如’标准型至少是豪华型的两倍’这样的约束，这转化为 x ≥ 2y，或者最低产量要求。

4. Graphical Method – Drawing Constraints | 图解法 – 绘制约束条件

With two decision variables, the feasible region can be represented on a Cartesian plane. Each linear inequality is drawn as a line, and the side satisfying the inequality is shaded. First, plot the line by turning the inequality into an equation. For 2x + 3y = 120, find the intercepts: when x = 0, y = 40; when y = 0, x = 60.

当有两个决策变量时，可行域可以表示在笛卡尔坐标平面上。每条线性不等式被画成一条直线，并给满足不等式的一侧涂色。首先，将不等式化为等式来绘制直线。对于 2x + 3y = 120，找出截距：当 x = 0 时，y = 40；当 y = 0 时，x = 60。

Use a ruler and sharp pencil. Decide the unwanted region by testing a point – often (0,0) if it is not on the line. If the test point satisfies the inequality, shade the opposite side; if not, shade the side containing the test point. Many CCEA mark schemes accept either shading the unwanted region or the feasible region, but consistent indication is essential.

使用直尺和尖细铅笔。通过测试一个点——通常是 (0,0)，如果不在直线上——来判断不需要的区域。若测试点满足不等式，涂另一侧；若不满足，涂包含测试点的一侧。许多 CCEA 评分方案既可涂出不可行区域也可涂出可行区域，但保持一致的指示至关重要。

Label each line with its equation. After drawing all constraints, the unshaded (or clearly marked) area where all constraints overlap is the feasible region. It is a convex polygon if the constraints are all linear.

给每条直线标上方程。画出所有约束后，所有约束条件重叠的未涂色（或明确标记的）区域就是可行域。若所有约束都是线性的，可行域将是一个凸多边形。

5. Identifying the Feasible Region | 确定可行域

The feasible region contains all possible combinations of the decision variables that satisfy every constraint simultaneously. Its boundaries are segments of the constraint lines. In a bounded problem, the region is a closed polygon; unbounded regions occur in some minimisation problems.

可行域包含了所有同时满足每一个约束条件的决策变量组合。其边界是各约束直线的一部分。在有界问题中，区域是一个封闭的多边形；无界区域出现在某些最小化问题里。

Always check that the region is correct by verifying a point inside it against all inequalities. For the example above, the point (20,10) gives 2(20)+3(10)=70 ≤ 120 and 20+20=40 ≤ 80, so it lies inside. Clearly mark the vertices of the feasible polygon, as they will be used to find the optimal solution.

务必通过用可行域内的一点验证所有不等式，来确认区域正确。在上述例子中，点 (20,10) 得到 2(20)+3(10)=70 ≤ 120 且 20+20=40 ≤ 80，因此它在区域内。清晰地标出可行多边形的顶点，因为它们将用于寻找最优解。

If you accidentally shade the wrong side, the entire solution may be invalid. A common CCEA exam technique is to lightly shade the unwanted regions and then outline the feasible region prominently.

如果你不小心涂错了侧边，整个解可能无效。一个常见的 CCEA 考试技巧是轻轻涂掉不可行区域，然后醒目地勾勒出可行域。

6. Vertex Method for Optimal Solution | 顶点法求最优解

The fundamental theorem of linear programming states that if an optimal solution exists, it occurs at a vertex (corner) of the feasible region. Therefore, to find the solution, evaluate the objective function at every vertex of the feasible polygon.

线性规划的基本定理指出，若存在最优解，它一定发生在可行域的一个顶点（角点）上。因此，要求解，需要计算目标函数在可行多边形每一个顶点处的值。

For the furniture example, the vertices are (0,0), (0,40), (60,0) and the intersection of the two constraint lines. Solve the simultaneous equations:

对于家具例子，顶点为 (0,0), (0,40), (60,0) 以及两条约束直线之交点。解联立方程：

2x + 3y = 120
x + 2y = 80

Multiply the second equation by 2: 2x + 4y = 160. Subtract the first: (2x+4y) – (2x+3y) = 160 – 120 → y = 40. Then x = 80 – 2(40) = 0. So the intersection is (0,40), which is already a vertex. This means the constraints are such that the deluxe line intercept coincides. Alternatively, a different set of numbers would give a distinct intersection vertex.

将第二式乘以 2：2x + 4y = 160。减去第一式：(2x+4y) – (2x+3y) = 160 – 120 → y = 40。然后 x = 80 – 2(40) = 0。因此交点为 (0,40)，这已经是一个顶点。这意味着约束使得豪华型直线的截距恰好重合。若用另一组数字，则可得到一个独特的交点顶点。

Evaluate Z at each vertex. You can present this in a neat table:

在每个顶点处计算 Z。你可以用整齐的表格呈现：

Vertex (x, y)	Z = 40x + 60y
(0,0)	0
(60,0)	2400
(0,40)	2400

The maximum profit is £2400, achieved at two vertices and thus along the entire edge connecting them. This indicates multiple optimal solutions, a situation you should mention in CCEA answers when it appears.

最大利润为 2400 英镑，在两个顶点处达到，因此在连接它们的整条边上均可实现。这表明存在多个最优解，出现这种情况时你在 CCEA 答案中应当予以说明。

7. Integer Solutions and Real-World Context | 整数解与现实背景

In many practical situations, the decision variables must be integers – you cannot produce 2.7 chairs. If the optimal vertex has non-integer coordinates, you must apply integer programming reasoning. Simply rounding to the nearest integer may give an infeasible or non-optimal point.

在许多实际情境中，决策变量必须为整数——你不可能生产 2.7 把椅子。如果最优顶点具有非整数坐标，你必须应用整数规划的推理。简单地四舍五入可能会得到一个不可行或非最优的点。

For CCEA, you are usually asked to find the integer point that maximises or minimises the objective within the feasible region. First, plot the vertex and then test the integer lattice points nearby, staying inside the feasible region. Slide an objective function line slightly to find the best integer point.

在 CCEA 考试中，通常要求你找出在可行域内使目标函数最大化或最小化的整数点。首先，画出顶点，然后测试其附近的整数格点，同时保持在可行域内。可以轻微滑动目标函数直线来找到最佳整数点。

When the non-integer optimal is (8.2, 5.7), test points like (8,5), (8,6), (9,5), (9,6) but check all constraints. The integer optimum may not be an adjacent integer point, so a systematic approach or drawing objective function contours helps.

当非整数最优解为 (8.2, 5.7) 时，测试诸如 (8,5), (8,6), (9,5), (9,6) 等点，但要检查所有约束。整数最优解未必是邻近的整数点，因此系统的方法或画出目标函数等值线会有所帮助。

8. Slack and Surplus Variables | 松弛变量与剩余变量

Slack variables are added to a ‘≤’ constraint to convert it into an equation, representing unused resource. Surplus variables are subtracted from a ‘≥’ constraint to represent excess over a minimum requirement. For 2x + 3y ≤ 120, the slack variable s₁ is defined by:

松弛变量被加到 ‘≤’ 约束中以将其转化为等式，代表未使用的资源。剩余变量则从 ‘≥’ 约束中减去，表示超过最低要求的超出量。对于 2x + 3y ≤ 120，松弛变量 s₁ 定义为：

2x + 3y + s₁ = 120, s₁ ≥ 0

At the point (20,10), s₁ = 120 – 2(20) – 3(10) = 50, meaning 50 hours of assembly time are unused. These variables are not always required in graphical solutions, but understanding them aids sensitivity analysis and the simplex method should you progress further in decision mathematics.

在点 (20,10) 处，s₁ = 120 – 2(20) – 3(10) = 50，这意味着有 50 小时的组装时间未被使用。在图解法中，这些变量并非总是必需，但理解它们有助于灵敏度分析，以及未来进一步学习单纯形法时打下基础。

In the CCEA graphical context, you might be asked to calculate the value of a slack at the optimum. Simply substitute the optimum coordinates into the original inequality and find the remaining resource. This shows the extent to which a constraint is binding – a binding constraint has zero slack.

在 CCEA 的图解情境中，你可能会被要求计算最优解处的松弛量。只需将最优坐标代入原始不等式，求出剩余资源即可。这显示了某个约束是否为紧约束——紧约束的松弛量为零。

9. Sensitivity Analysis – Assessing Changes | 灵敏度分析 – 评估变化

Sensitivity analysis examines how the optimal solution changes if a parameter in the objective function or a resource availability is varied. While CCEA does not require deep simplex-based shadow pricing, graphical reasoning can answer simple ‘what-if’ questions.

灵敏度分析考察的是如果目标函数中的参数或资源可用量发生变化，最优解会如何改变。虽然 CCEA 不要求深入的基于单纯形的影子价格分析，但图解推理可以回答简单的’如果……会怎样’的问题。

If the coefficient of x in the objective function changes, the slope of the objective function line changes. You can test the range of this coefficient for which the current vertex remains optimal by finding the slopes of the binding constraints at that vertex. If the objective slope lies between the slopes of those binding lines, the optimum stays the same.

如果目标函数中 x 的系数发生变化，目标函数直线的斜率就会改变。你可以通过找出当前顶点处紧约束直线的斜率，来测试该系数保持当前顶点最优的取值范围。若目标直线的斜率落在那些紧约束直线斜率之间，最优点维持不变。

For resource changes, e.g. an increase in the right-hand side of 2x + 3y ≤ 120 to 125, the constraint line shifts outward. This may enlarge the feasible region and potentially move the optimum. Graphically, you can re-draw the new line, find the new intersection vertex, and recalculate Z. These explorations appear in CCEA questions that ask you to investigate the effect of a change.

对于资源变化，例如将 2x + 3y ≤ 120 的右侧增至 125，约束直线会向外移动。这可能扩大可行域，并可能改变最优解。在图形上，你可以重新画出新直线，找到新的交点顶点，并重新计算 Z。这种探究会出现在 CCEA 要求你研究变化影响的试题中。

10. Common Pitfalls and Examination Advice | 常见陷阱与考试建议

Many CCEA students lose marks by defining variables vaguely, such as ‘x = chairs’ instead of ‘x = number of chairs’. Always specify ‘Let x be the number of…’ with units. Another frequent error is forgetting non-negativity constraints in the formulation list – these must be explicitly stated.

许多 CCEA 学生因为定义变量模糊而丢分，比如写成 ‘x = chairs’ 而不是 ‘x = number of chairs’。务必明确 ‘设 x 为……的数量’ 并带单位。另一个常见错误是在列出的模型中遗漏非负约束——这些必须明确写出。

Graphically, inaccurate line drawing leads to wrong vertices. Use a sharp pencil, plot intercepts precisely, and always verify at least one point inside the feasible region. When shading, be consistent; if you shade feasible, mark it boldly; if you shade unwanted, leave the feasible region clear. Label the feasible region ‘R’.

图形方面，绘制直线不精确会导致顶点错误。使用尖细铅笔，准确标出截距点，并始终验证可行域内的至少一个点。涂色时要保持一致；如果涂可行区域就醒目地标出，如果涂不可行区域则让可行域保持清晰。将可行域标注为 ‘R’。

In integer linear programming, never round a fractional optimum directly without checking feasibility and optimality – the rounded point might violate a constraint. Show all tested integer points and the corresponding objective values. Finally, answer the question in context: if asked for a production plan, state ‘Produce 8 standard and 6 deluxe units’ rather than just giving (8,6).

在整数线性规划中，切勿在不检查可行性和最优性的情况下直接对小数最优解取整——舍入后的点可能违反约束。要展示所有测试过的整数点及其对应的目标值。最后，在上下文中回答问题：如果问的是生产计划，要说明 ‘生产 8 件标准型和 6 件豪华型’，而非仅仅给出 (8,6)。

Always reread the problem statement to assign the correct objective – maximising profit or minimising cost? A mistake in the objective sign or sense changes everything. Also, check if the question asks you to find the maximum value only, or both the value and the coordinates. Explicit final statements impress CCEA examiners.

始终重新阅读题设以确定正确的目标——是最大化利润还是最小化成本？目标正负号或优化方向上的错误会改变一切。同时，检查题目是只要求找最大值，还是既要求值又要求坐标。清晰明确的最终陈述会给 CCEA 考官留下好印象。

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A-Level CCEA Maths: Linear Programming Key Points Revision | A-Level CCEA 数学：线性规划 考点精讲