📚 Linear Programming in A-Level Edexcel Mathematics | A-Level Edexcel 数学：线性规划 考点精讲
Linear Programming (LP) is a powerful mathematical technique used to find the best outcome in a model whose requirements are represented by linear relationships. In the Edexcel A-Level Mathematics specification, particularly within the Decision Mathematics 1 module, you are expected to formulate real‑world problems as linear programs, sketch feasible regions, and determine optimal solutions using graphical methods. This article walks you through every key concept, from decision variables to integer solutions, with clear explanations and exam‑style tips.
线性规划是一种强大的数学方法，用于在需求由线性关系表示的模型中寻找最优结果。在Edexcel A-Level数学考试中，特别是在决策数学1模块里，你需要将现实问题转化为线性规划模型，画出可行域，并使用图解法确定最优解。本文将从决策变量到整数解，逐一梳理每个关键概念，并提供清晰的解释和应试技巧。
1. Introduction to Linear Programming | 线性规划简介
Linear programming deals with optimising (maximising or minimising) a linear objective function subject to a set of linear constraints. The constraints typically represent limited resources such as time, money, raw materials, or production capacities. All variables must be non‑negative in standard LP problems. The term ‘linear’ means that both the objective function and all inequalities involve only first‑degree terms – no squares, products, or trigonometric functions.
线性规划处理的是在一组线性约束条件下优化（最大化或最小化）一个线性目标函数的问题。这些约束通常代表有限的资源，如时间、资金、原材料或生产能力。在标准的LP问题中，所有变量必须是非负的。“线性”意味着目标函数和所有不等式只包含一次项——没有平方项、乘积项或三角函数。
A standard LP problem can be expressed as: maximise (or minimise) P = ax + by, subject to constraints like cx + dy ≤ e, fx + gy ≥ h, and x, y ≥ 0. In Edexcel exams, you will primarily work with two decision variables so that the feasible region can be drawn on a two‑dimensional graph.
一个标准的线性规划问题可以表示为：最大化（或最小化）P = ax + by，满足约束条件如 cx + dy ≤ e、fx + gy ≥ h，以及 x, y ≥ 0。在Edexcel考试中，你主要处理两个决策变量，这样就可以在二维图上画出可行域。
2. Decision Variables and Constraints | 决策变量与约束
Decision variables represent the quantities we can control, such as the number of units of product A to produce (x) and product B to produce (y). They must be clearly defined at the start of any modelling question. For example: ‘Let x be the number of chairs made per day, and let y be the number of tables made per day.’ Always state the units and ensure they are non‑negative.
决策变量代表我们可以控制的量，例如要生产的产品A的数量（x）和产品B的数量（y）。在任何建模题目开始时，必须明确给它们下定义。例如：“设x为每天生产的椅子数量，y为每天生产的桌子数量。”务必说明单位，并确保它们是非负的。
Constraints are linear inequalities derived from the limitations in the problem. Each constraint should be written with the variables on the left‑hand side and the constant on the right. For instance, ‘each chair requires 2 hours of labour and each table requires 3 hours. The total labour available is 60 hours’ becomes 2x + 3y ≤ 60. Remember to include the non‑negativity constraints x ≥ 0, y ≥ 0 unless otherwise specified.
约束条件是从题目中的限制条件推导出的线性不等式。每个约束应写成变量在左边、常数在右边的形式。例如，“每把椅子需要2小时人工，每张桌子需要3小时人工，总可用工时为60小时”可写为2x + 3y ≤ 60。除非另有说明，务必包含非负约束 x ≥ 0, y ≥ 0。
3. Drawing the Feasible Region | 画出可行域
The feasible region is the set of all points (x, y) that satisfy every constraint simultaneously. To draw it, treat each inequality as an equation and plot the corresponding straight line. Use a solid line for ≤ or ≥ (inclusive) and a dashed line for < or > (exclusive, though rarely used in Edexcel D1). Then shade the unwanted region or clearly indicate the feasible side – the Edexcel convention is usually to shade out the region that is not required, leaving the feasible region unshaded.
可行域是同时满足所有约束条件的所有点(x, y)的集合。要画出可行域，将每个不等式看作等式，绘制出相应的直线。对于≤或≥（包含边界）使用实线，对于<或>（不包含边界，虽然在Edexcel D1中很少见）使用虚线。然后标出不可行区域或明确指出可行侧——Edexcel的惯例通常是涂掉不需要的区域，让可行域保持空白。
After drawing all constraint lines, the feasible region is usually a convex polygon (possibly unbounded). Label the lines with their equations and indicate the coordinates of corner points if they are easy to read. In exam questions, you may be asked to shade the region that satisfies a set of inequalities; always double‑check by testing a point such as (0,0) in each inequality.
画出所有约束线后，可行域通常是一个凸多边形（可能是无界的）。给每条线标上方程，如果角点坐标容易读取，也标出来。在考试题目中，可能要求你着色表示满足一组不等式的区域；务必通过检验一个点（如(0,0)）来复核每个不等式。
4. The Objective Function | 目标函数
The objective function is a linear expression that we aim to maximise or minimise, usually in terms of profit, cost, or time. It is written as P = ax + by (or C = ax + by for cost). In the graphical method, this function is represented by a family of parallel lines, each corresponding to a different value of P. The gradient of these lines is determined by the coefficients a and b.
目标函数是我们希望最大化或最小化的线性表达式，通常以利润、成本或时间为单位。它写作P = ax + by（如果是成本则写作C = ax + by）。在图解法中，这个函数由一族平行线表示，每条线对应不同的P值。这些线的斜率由系数a和b确定。
To find the optimal point, we need to locate the vertex of the feasible region where the objective function attains its maximum or minimum value. If you are maximising, you want to push the line as far as possible in the direction of increasing P; if minimising, you push it in the opposite direction.
要找到最优点，我们需要确定可行域中使目标函数达到最大值或最小值的顶点。如果是最大化，就要让直线朝着P增大的方向尽可能远地平移；如果是最小化，则朝相反方向。
5. The Method of Sliding Lines | 等值线法
The sliding line method involves drawing one line of the objective function, often passing through the origin if convenient, and then sliding a ruler parallel to it across the feasible region. The last vertex touched before leaving the region gives the optimal solution. For maximisation, slide in the direction that increases P; for minimisation, slide in the direction that decreases P.
等值线法是先画出目标函数的一条线（如果方便，通常过原点），然后将一把直尺平行于该线在可行域上滑动。在离开可行域前最后接触的顶点即为最优解。对于最大化问题，沿着使P增大的方向滑动；对于最小化问题，则沿着使P减小的方向滑动。
To determine the correct direction, evaluate the gradient. For example, if P = 3x + 2y, the line has slope −3/2. Increasing P means shifting the line upwards and to the right. Alternatively, you can test two corners of the feasible region to see which gives the larger P – this is essentially the vertex testing method.
要确定正确的方向，先求出斜率。例如，如果P = 3x + 2y，该线的斜率为−3/2。增大P意味着将直线向右上方平移。或者，也可以测试可行域的两个角点，看哪个给出更大的P——这实际上就是顶点检验法。
6. Vertex Testing Method | 顶点检验法
Since the objective function is linear and the feasible region is convex, the optimal solution (if it exists) will always occur at a corner (vertex) of the feasible region. Therefore, an alternative to sliding lines is simply to list all vertices of the feasible region, calculate the objective function value at each one, and pick the best. This is often quicker when the feasible region has only a few vertices, and it avoids graphical inaccuracies.
由于目标函数是线性的，且可行域是凸的，最优解（如果存在）总是出现在可行域的某个角点（顶点）。因此，替代等值线滑动法的一个办法是，直接列出可行域的所有顶点，计算每个顶点处的目标函数值，然后选出最优解。当可行域只有少数几个顶点时，这种方法通常更快，而且避免了作图误差。
To find the coordinates of a vertex, solve the simultaneous equations of the two lines that intersect at that point. Be careful with vertices that lie on the axes – those are easier to identify. Once you have all vertices, substitute into P (or C) and compare. You must show your working clearly in exams to gain method marks.
要求出某个顶点的坐标，解出在该点相交的两条直线的方程组。对于在坐标轴上的顶点要留心——这些更容易识别。获得所有顶点后，代入P（或C）进行比较。在考试中，你必须清晰地展示计算过程，才能拿到方法分。
7. Unique and Multiple Optimal Solutions | 唯一最优解与多重最优解
Most LP problems have a single optimal vertex. However, if the objective function is parallel to one of the binding constraints, there may be infinitely many optimal solutions along that edge. In this case, any point on the line segment between two optimal vertices yields the same optimal value. When answering an exam question, you may be asked to give all optimal solutions or simply the coordinates of two vertices that define the segment.
大多数线性规划问题只有一个最优点。但如果目标函数与某个起作用的约束平行，则沿着那条边可能存在无限多个最优解。在这种情况下，两个最优顶点之间的线段上的任意点都会产生相同的最优值。在回答考题时，你可能需要给出所有最优解，或者只是给出构成该线段的两个顶点的坐标。
When multiple optimal solutions exist, the sliding line method will show the objective line coinciding with a boundary of the feasible region. The problem might then ask you to state a range of integer solutions as well. Always check the wording: ‘Find the optimal solution’ may require a single point, whereas ‘State the optimal solutions’ could require the general form.
当存在多重最优解时，等值线法会显示目标直线与可行域的某条边界重合。题目随后可能还会要求你给出一系列整数解。始终要仔细审题：“求最优解”可能需要给出一个点，而“写出最优解”则可能要求给出一般形式。
8. Integer Solutions and Integer Programming | 整数解与整数规划
In many real‑world problems, decision variables must be whole numbers – you cannot produce 2.7 tables. When the optimal vertex does not have integer coordinates, you need to find the best integer solution within the feasible region. The Edexcel syllabus expects you to search for integer points near the optimal vertex using a systematic approach.
在许多实际问题中，决策变量必须是整数——你不能生产2.7张桌子。当最优顶点不是整数坐标时，你需要在可行域内寻找最佳整数解。Edexcel考纲要求你采用系统的方法，在最优顶点附近寻找整数点。
One method is to draw a grid over the feasible region and list all integer points in the vicinity of the non‑integer optimal vertex. Then test each candidate in the objective function and ensure that they still satisfy all constraints. Always state clearly why a point is the best integer solution, and show comparison of objective values.
一种方法是在可行域上画出网格，并列出非整数最优顶点附近的所有整数点。然后测试每个候选点是否满足所有约束条件，并比较目标函数值。务必清楚地说明为什么某个点是最佳整数解，并展示目标值的比较过程。
In some problems, a condition ‘x and y are integers’ is explicitly given; in others, the context implies it. Even if the LP gives an integer‑optimal vertex, you should confirm that it is indeed the best integer point – but there will be no need to search further.
在某些问题中，明确给出了“x和y为整数”的条件；而在其他问题中，语境暗示了这一点。即使线性规划本身给出了一个整数最优顶点，你仍应确认它确实是最佳整数点——但通常无需进一步搜索。
9. Modelling Real‑world Problems | 应用问题建模
The most challenging part of linear programming in Edexcel D1 is often translating a word problem into mathematical form. You must identify: (1) the decision variables with clear definitions, (2) the objective function (profit, cost, etc.), (3) all constraints from the given resources, and (4) any implicit non‑negativity or integer requirements.
Edexcel D1中线性规划最具挑战性的部分，往往是将文字题转化为数学形式。你必须找出：(1) 决策变量并给出清晰定义，(2) 目标函数（利润、成本等），(3) 根据给定资源得出的所有约束条件，以及 (4) 任何隐含的非负或整数要求。
A typical problem may involve blending ingredients, mixing chemicals, scheduling workers, cutting stock (like paper rolls), or transportation. The key is to translate phrases like ‘at most’, ‘at least’, ‘no more than’, ‘a minimum of’ into the correct inequality signs (≤ or ≥). Always re‑read the problem to ensure you haven’t missed a hidden constraint, such as a total production target or a ratio requirement.
典型问题可能涉及混合配料、化学物混合、工人排班、切割原材料（如纸卷）或运输。关键在于将“至多”、“至少”、“不超过”、“最少”等词语转化为正确的不等号（≤ 或 ≥）。务必重读题目以确保没有遗漏隐藏的约束，例如总产量目标或配比要求。
Once the LP is formulated, it can be solved graphically or by vertex testing. In an exam, you are often given a partially filled table of constraints – you need to interpret the text to complete the inequalities and then proceed to the graphical solution.
一旦建立了线性规划模型，就可以用图解法或顶点检验法求解。在考试中，通常会给出一个部分填好的约束表格——你需要根据文本解释来完成不等式，然后继续进行图解求解。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Many students lose marks by misinterpreting inequality signs. Remember: ‘at least 10’ means ≥ 10, while ‘a maximum of 10’ means ≤ 10. Also, when scaling axes, choose scales that make the feasible region clear; use graph paper if provided, but even on a blank grid, keep proportions accurate.
许多学生因误解不等号而失分。记住：“至少10个”意味着≥ 10，而“最多10个”意味着≤ 10。此外，在标记坐标轴时，要选择合适的比例，使可行域清晰可见；如果有坐标纸就用，但即使是空白网格也要保持比例准确。
Another common error is forgetting to state the non‑negativity constraints x ≥ 0, y ≥ 0. They may seem obvious, but the examiner expects to see them listed. When finding integer solutions, don’t just assume the nearest whole‑number point is optimal – you must check all integer points around the optimal vertex.
另一个常见错误是忘记列出非负约束 x ≥ 0, y ≥ 0。它们看似理所当然，但考官希望看到你写出来。在求整数解时，不要想当然认为最接近的整数点就是最优的——你必须检查最优顶点附近的所有整数点。
Finally, always present your answer in the context of the problem: ‘The maximum profit is £123, achieved by making 5 chairs and 8 tables.’ If there are multiple optimal solutions, state them clearly and mention that any point on the line segment between them also yields the same optimal value.
最后，始终要在题目背景下呈现答案：“最大利润为123英镑，是通过生产5把椅子和8张桌子实现的。”如果存在多个最优解，要明确说出来，并指出它们之间线段上的任何点都能得到相同的最优值。
Practice drawing inequalities quickly and accurately, and become familiar with the format of Edexcel D1 questions – they often combine linear programming with critical path analysis or graph theory in a single paper, so speed is essential.
练习快速而准确地画出不等式，并熟悉Edexcel D1的题型——考试中，线性规划常常和关键路径分析或图论组合在同一张试卷中，因此速度至关重要。
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