📚 Edexcel Mathematics: Linear Programming – Key Points Revision | Edexcel 数学：线性规划 考点精讲
Linear programming (LP) is a vital topic in Edexcel Decision Mathematics (D1). It equips students with the skill to model real-world constraints as linear inequalities, construct a feasible region, and locate the optimal solution – either by sliding an objective profit line or by evaluating corner points. This article recaps all essential concepts, methods, and exam techniques you need to master LP problems under the Edexcel specification.
线性规划（LP）是 Edexcel 决策数学（D1）的重要考点。它训练学生将现实中的约束条件建模为线性不等式，构建可行域，并通过移动目标等利润线或计算顶点坐标来找到最优解。本文将梳理所有核心概念、解题方法和应试技巧，帮助你彻底掌握 Edexcel 大纲下的线性规划问题。
1. Introduction to Linear Programming | 线性规划简介
Linear programming is a method to achieve the best outcome (such as maximum profit or minimum cost) in a mathematical model whose requirements are represented by linear relationships. In D1, all variables are non‑negative, constraints are linear inequalities, and the objective function is a linear expression to be maximised or minimised.
线性规划是一种在数学模型中寻求最佳结果（如最大利润或最低成本）的方法，其要求由线性关系表示。在 D1 中，所有变量非负，约束条件是线性不等式，目标函数是待最大化或最小化的线性表达式。
The standard form of an LP problem involves decision variables (often x, y), a set of linear constraints, and an objective function like P = 3x + 4y (maximise) or C = 5x + 2y (minimise). The solution must satisfy all constraints simultaneously; the set of points that do is called the feasible region.
线性规划问题的标准形式包含决策变量（通常为 x 和 y）、一组线性约束条件，以及一个目标函数，例如 P = 3x + 4y（最大化）或 C = 5x + 2y（最小化）。解必须同时满足所有约束条件，满足所有约束的点集称为可行域。
The goal is to find the point(s) within the feasible region that optimise the objective function. Because all relationships are linear, the optimal value will always occur at a corner (vertex) of the feasible region, provided the region is bounded and a unique optimum exists.
目标是在可行域内找到使目标函数最优的点。由于所有关系都是线性的，最优值总是出现在可行域的顶点处，前提是该区域有界且存在唯一最优解。
2. Formulating Constraints as Linear Inequalities | 将约束条件表示为线性不等式
Real-world scenarios are translated into inequalities. Common phrases: “at least” becomes ≥, “at most” becomes ≤, “no more than” → ≤, “a minimum of” → ≥. Always define your variables clearly; for example, let x be the number of type‑A widgets and y be the number of type‑B widgets. Then write constraints for resource limits, demand requirements, or time restrictions.
实际问题必须转化为不等式。常见表述：“至少”变为 ≥，“最多”变为 ≤，“不超过” → ≤，“至少需要” → ≥。要始终清晰地定义变量；例如，设 x 为产品 A 的数量，y 为产品 B 的数量。然后根据资源限制、需求量或时间限制写出约束条件。
Typical constraints include raw material usage: 2x + 3y ≤ 480 (material units). Labour hours: 5x + 2y ≤ 600. Market conditions: x ≥ 20, y ≤ 100. Non‑negativity: x ≥ 0, y ≥ 0. Remember to simplify inequalities where possible (e.g., dividing by a common positive factor), but never round decimal coefficients if they come from exact data.
典型约束条件包括原材料消耗：2x + 3y ≤ 480（材料单位）。人工工时：5x + 2y ≤ 600。市场条件：x ≥ 20，y ≤ 100。非负条件：x ≥ 0，y ≥ 0。记住尽可能化简不等式（例如同除以正公因数），但如果系数来自精确数据，绝不要对小数系数进行四舍五入。
When formulating, ensure each constraint is linear. If a condition involves “the number of A must be at least twice the number of B”, write x ≥ 2y. Do not confuse the direction of the inequality; test a simple point to check.
在列式时，要确保每个约束都是线性的。如果条件涉及“A 的数量必须至少是 B 数量的两倍”，应写为 x ≥ 2y。不要混淆不等号方向，可代入一个简单点进行检验。
3. Drawing the Feasible Region | 绘制可行域
Start by drawing a coordinate grid with suitable scales. Because x ≥ 0 and y ≥ 0, only the first quadrant is needed. For each inequality, first draw the boundary line as a solid line for ≤ or ≥ (inclusive) and a dashed line for < or > (strict). In D1, solid lines are typical because constraints usually allow equality.
首先绘制具有合适刻度的坐标网格。由于 x ≥ 0 且 y ≥ 0，只需画出第一象限。对于每个不等式，先画出边界线：≤ 或 ≥ 用实线（包括边界），< 或 > 用虚线（严格不等）。在 D1 中，由于约束条件通常允许取等，实线为常见情况。
To draw the line, find its intercepts. For 3x + 2y = 36, set x=0 ⇒ y=18; set y=0 ⇒ x=12. Plot these points and extend the line. Label each line with its equation. Choose a test point (usually (0,0) unless it lies on the line) to determine which side of the line satisfies the inequality. Shade the unwanted region for each constraint – this is a standard Edexcel technique to keep the feasible region unshaded and clearly visible. The intersection of the unshaded half‑planes is the feasible region.
画线时，先求截距。对于 3x + 2y = 36，令 x=0 得 y=18；令 y=0 得 x=12。标出这些点并延长直线，用方程标记每条线。选择检验点（通常为 (0,0)，除非它恰好在线上）来确定满足不等式的半平面。根据 Edexcel 的惯用画法，对于每个约束，将不满足的一侧涂上阴影，这样可行域保持无阴影且清晰可见。所有未涂阴影的半平面交集即为可行域。
Sometimes an inequality like x ≥ 2 means you shade the left side of the vertical line x = 2. After shading all unwanted regions, the unshaded polygon (often bounded by axes and constraint lines) is the feasible region. Label its vertices clearly because they will be needed for the vertex method.
有时像 x ≥ 2 这样的不等式，需要将垂直线 x = 2 的左侧涂上阴影。在涂完所有不满足区域后，未涂阴影的多边形（通常由坐标轴和约束线围成）即为可行域。清晰地标出它的顶点，因为在顶点法中需要用到。
4. Locating the Feasible Region Correctly | 正确确定可行域位置
Accuracy is crucial. Use graph paper or draw axes neatly. Mark scales uniformly. For an inequality like y ≤ 2x, the boundary line passes through the origin with slope 2. Choose a test point not on the line, e.g., (0,10). Because 10 ≤ 2(0) is false, shade the region that does not contain (0,10). In this case, shade above the line.
绘图的准确性至关重要。使用坐标纸或整齐地画出坐标轴。刻度要均匀。对于 y ≤ 2x 这样的不等式，边界线经过原点且斜率为 2。选择不在直线上的检验点，例如 (0,10)。由于 10 ≤ 2(0) 不成立，因此将不包含 (0,10) 的区域涂上阴影。此时应在线条上方涂阴影。
When shading multiple inequalities, it is easy to make a mistake by shading the wrong side. Double‑check each constraint. A useful check: the feasible region must be a convex polygon. If it appears non‑convex or contains a “hole” where shading overlaps incorrectly, re‑examine your test points.
当对多个不等式进行阴影涂布时，很容易因涂错一侧而出错。要逐一检查每个约束条件。一个有用的检验方法：可行域必须是凸多边形。如果区域看起来是非凸的，或者由于阴影重叠错误而出现“空洞”，就需要重新检查检验点。
Edexcel examiners expect the feasible region to be clearly indicated, often by labelling it R. Do not over‑shade; use light pencil shading or diagonal lines that do not obscure the vertices. The unshaded region should stand out.
Edexcel 阅卷人要求清晰标出可行域，通常用 R 标注。不要过度涂阴影；使用轻铅笔涂影或斜线，避免遮盖顶点。未涂阴影的区域应当醒目。
5. Objective Function and Profit Lines | 目标函数与等利润线
The objective function, such as P = 5x + 3y, represents the quantity to maximise or minimise. To solve graphically, we can draw lines of constant profit, called iso‑profit lines (or iso‑cost lines for minimisation). Rewrite the objective equation for a chosen value of P, e.g., 5x + 3y = 30, and draw this line. All points on this line give the same profit 30.
目标函数，例如 P = 5x + 3y，表示要最大化或最小化的量。通过图解法求解时，我们可以画出等利润线（或最小化问题中的等成本线）。选择一个 P 值，改写目标方程为 5x + 3y = 30，并画出该直线。该线上所有点都给出相同的利润 30。
By drawing several parallel lines for increasing values of P (or shifting a ruler parallel to the gradient of the objective function), we can determine the last point of contact with the feasible region before leaving it. For maximisation, slide the line in the direction of increasing profit; the final vertex touched gives the optimum. For minimisation, slide in the opposite direction. This approach is often called the ruler method or objective line method.
通过画出若干条不同 P 值的平行线（或用直尺沿目标函数的梯度方向平移），我们可以确定直线在离开可行域前与区域的最后接触点。对于最大化问题，沿利润增加的方向平移直线；最后接触的顶点即为最优点。对于最小化问题，则向相反方向平移。这种方法通常被称为直尺法或目标直线法。
The gradient of the objective function is constant. For P = 5x + 3y, the gradient is −5/3. All iso‑profit lines share this gradient. Even if you do not draw them, you must know how to calculate the exact coordinate of the optimal point by solving simultaneous equations for the intersecting constraint lines.
目标函数的梯度是恒定的。对于 P = 5x + 3y，梯度为 −5/3。所有等利润线都具有相同的梯度。即便你没有画出它们，也必须知道如何通过联立相交约束线的方程来精确计算出最优点的坐标。
6. Corner Point Theorem (Vertex Method) | 顶点法（顶点定理）
The foundation of graphical LP is the Corner Point Theorem: if a linear programming problem has an optimal solution, it will occur at a corner (vertex) of the feasible region. Therefore, we can simply list all vertices of the feasible region, calculate the value of the objective function at each, and pick the best one. This is the vertex method.
图解线性规划的基础是顶点定理：如果线性规划问题存在最优解，那么该解一定出现在可行域的某个角点（顶点）上。因此，我们可以直接列出可行域的所有顶点，分别计算目标函数的值，并选出最佳值。这就是顶点法。
To find vertices, solve pairs of boundary line equations simultaneously. For example, the intersection of 2x + y = 20 and x + 3y = 30 gives a vertex. Use algebraic methods (substitution or elimination) to find exact coordinates; fractions are acceptable. Then evaluate P = 5x + 3y at this point and at all other vertices. The largest value gives the maximum; the smallest gives the minimum (if applicable).
为求出顶点，需要成对地联立边界线方程。例如，联立 2x + y = 20 和 x + 3y = 30 可解出一个顶点。使用代数方法（代入法或消元法）求出精确坐标；允许有分数。然后在该点及所有其他顶点处计算 P = 5x + 3y 的值。最大值对应最大利润；最小值对应最小成本（如果适用）。
Even when using the vertex method, it is good practice to confirm that the optimal vertex indeed satisfies all constraints (especially non‑negativity) and that the point lies within the feasible region, not just on its boundary lines. The vertex method is robust, but it must be backed by accurate coordinates.
即使使用顶点法，也应按惯例确认最优顶点确实满足所有约束条件（特别是非负条件），且该点位于可行域内部，而不仅仅是边界线上。顶点法非常稳健，但必须以准确的坐标为基础。
7. Solving LP Problems Graphically | 图解法求解线性规划问题
A step‑by‑step strategy works best: (1) Define variables. (2) Formulate constraints and objective. (3) Draw axes and boundary lines. (4) Shade unwanted regions to reveal the feasible region R. (5) Identify the coordinates of all vertices of R. (6) Use either ruler method or vertex method to find the optimum. (7) Write a concluding statement answering the original question in context, including the value of the objective and the corresponding variable values.
最佳策略是逐步进行：(1) 定义变量。(2) 列出约束条件和目标函数。(3) 画出坐标轴和边界线。(4) 涂上不满足区域的阴影，显示出可行域 R。(5) 确定 R 的所有顶点坐标。(6) 使用直尺法或顶点法找到最优解。(7) 写出结论，结合题意回答问题，包括目标函数值和相应的变量取值。
If the question asks for integer solutions (e.g., number of items must be whole numbers), careful attention is needed. The optimal vertex may have non‑integer coordinates; in that case we must investigate integer points near that vertex, testing each against constraints and objective value. This is common in D1 exams.
如果题目要求整数解（例如，物品数量必须为整数），就需要特别留意。最优顶点的坐标可能不是整数；此时必须考察该顶点附近的整数点，逐一检验是否满足约束并计算目标值。这在 D1 考试中十分常见。
8. Integer Solutions and Special Cases | 整数解与特殊情况
When the decision variables must be integers, the optimum may not lie exactly on a vertex. The feasible region becomes a set of discrete points. After finding the continuous optimum, list integer coordinate pairs in the vicinity (e.g., using a grid on the graph) and test them. Check that each integer pair satisfies every constraint. The one yielding the best objective value is the integer optimal solution.
当决策变量必须为整数时，最优解可能并不恰好位于顶点上。可行域变成一组离散的点。找到连续最优解后，列出其附近的整数坐标对（例如，借助图形上的网格），并逐个检验。确保每个整数对都满足所有约束条件。给出最佳目标值的点就是整数最优解。
Special cases can appear: infeasible problems occur when no point satisfies all constraints – the feasible region is empty. This arises from contradictory constraints. Unbounded feasible regions may lead to infinite profit if the objective can increase without limit; in practice, realistic constraints bound the solution. Another case: multiple optimal solutions happen when an iso‑profit line coincides with a boundary of the feasible region; then every point on that boundary segment is optimal, and the problem has infinitely many solutions. You should be able to recognise and explain these scenarios.
可能出现特殊情况：当没有任何点能同时满足所有约束时，问题无可行解——可行域为空，这通常源于相互矛盾的约束条件。当可行域无界且目标值可以无限增大时，可能出现无界解，但实际问题中约束条件总会使解有界。另一种情况是多重最优解：当等利润线与可行域的一条边界重合时，该边界线段上的每一个点都是最优的，问题有无穷多解。你应能识别并解释这些情况。
9. Interpreting the Solution in Context | 结合实际解释解
Getting the numerical answer is not the end. You must interpret it. For a maximisation problem, state: “The maximum profit is £P when producing x units of A and y units of B.” If some resource is left unused, you can calculate the slack by substituting the optimum into resource constraints and finding the difference between the limit and the amount used.
得到数值结果并非终点，还需对其进行解释。对于最大化问题，应说明：“当生产 A 产品 x 件、B 产品 y 件时，可获得最大利润 £P。”如果存在未使用的资源，可以将最优解代入资源约束，计算出剩余量（即限制量与使用量之差），这便是松弛量。
Slack is valuable information: if a constraint is not tight (i.e., the optimal point does not lie on that boundary line), it means the corresponding resource is not fully utilised. You might be asked to suggest how to use the spare capacity or to determine shadow prices, although shadow prices are explored more fully in further D1 topics such as simplex or sensitivity analysis. For graphical LP, identifying slack is sufficient.
松弛量提供了有价值的信息：如果某个约束并非紧约束（即最优点不在该边界线上），就意味着相应的资源未被充分利用。题目可能会建议如何利用剩余产能，或让你确定影子价格，不过影子价格在 D1 的后续内容（如单纯形法或灵敏度分析）中会更深入地探讨。在图解线性规划中，能够识别松弛量即可。
10. Exam Tips and Common Pitfalls | 考试技巧与常见错误
Top exam tips: (1) Always define variables precisely at the start – “Let x be the number of …” (2) Draw large, clear graphs; use at least one-third of a page for the grid. (3) Shade the unwanted regions as per Edexcel convention unless instructed otherwise. (4) Label all lines and vertices. (5) Show all calculations for vertex coordinates even if you read them from the graph. (6) If using the ruler method, draw at least one iso‑profit line and indicate the direction of increase with an arrow. (7) Write a final answer as a full sentence.
考试高分技巧：(1) 始终在开始时精确定义变量——“设 x 为……的数量”。(2) 绘制大而清晰的图形；为坐标格至少留出三分之一页面。(3) 除非另有说明，按照 Edexcel 惯例涂上不满足区域的阴影。(4) 标注所有直线和顶点。(5) 即使能从图中读出顶点坐标，也须展示所有计算过程。(6) 若使用直尺法，至少画出一条等利润线，并用箭头标明利润增加的方向。(7) 用完整句子写出最终答案。
Common pitfalls: confusing ≤ with < when drawing solid/dashed lines; shading the feasible region instead of the unwanted region (leading to confusion in complex diagrams); arithmetic errors when solving simultaneous equations; forgetting non‑negativity constraints; and, particularly, neglecting integer requirements when they are explicitly stated. Review every constraint after formulation to ensure the inequality sign matches the context.
常见错误：画线时混淆 ≤ 和 < 的实/虚线；把阴影涂在了可行域而不是不满足区域（导致复杂图形中产生混淆）；联立解方程时出现计算错误；忘记非负约束条件；特别需要注意的是，当题目明确要求整数解时却忽略了这一要求。列式完成后要逐条核对每个约束，确保不等号方向与题意相符。
| Concept | Key Point | 概念 | 关键点 |
| Decision Variables | Usually x, y; must be defined | 决策变量 | 通常为 x、y；必须定义 |
| Constraints | Linear inequalities; solid/dashed lines | 约束条件 | 线性不等式；实/虚线 |
| Feasible Region | Unshaded convex polygon | 可行域 | 未涂阴影的凸多边形 |
| Objective | P = ax + by; iso‑profit gradient | 目标函数 | P = ax + by；等利润线梯度 |
| Optimum | At a vertex; test all vertices | 最优解 | 位于顶点；检验所有顶点 |
Checklist for Edexcel D1 linear programming
Edexcel D1 线性规划核对清单
11. Worked Example: Maximising Profit | 典型例题：利润最大化
Problem: A factory produces two products, X and Y. Each unit of X requires 2 hours on machine A and 1 hour on machine B; each unit of Y requires 1 hour on machine A and 3 hours on machine B. Machine A is available for 20 hours, machine B for 30 hours. The profit is £50 per unit of X and £40 per unit of Y. Find the numbers to maximise profit.
问题：某工厂生产两种产品 X 和 Y。每件 X 需在机器 A 上加工 2 小时、机器 B 上 1 小时；每件 Y 需在机器 A 上加工 1 小时、机器 B 上 3 小时。机器 A 可用 20 小时，机器 B 可用 30 小时。每件 X 利润为 50 英镑，每件 Y 利润为 40 英镑。求能使利润最大化的产品数量。
Let x = number of X, y = number of Y. Constraints: (1) 2x + y ≤ 20 (machine A), (2) x + 3y ≤ 30 (machine B), (3) x ≥ 0, y ≥ 0. Objective: Maximise P = 50x + 40y. The feasible region has vertices at (0,0), (0,10), (6,8) – from solving 2x+y=20 and x+3y=30 – and (10,0). Evaluating P:
设 x = X 的数量，y = Y 的数量。约束条件：(1) 2x + y ≤ 20（机器 A），(2) x + 3y ≤ 30（机器 B），(3) x ≥ 0，y ≥ 0。目标：最大化 P = 50x + 40y。可行域的顶点为 (0,0)、(0,10)、(6,8)——由 2x+y=20 和 x+3y=30 联立解得——以及 (10,0)。计算 P 值：
P(0,0) = 0; P(0,10) = 400; P(6,8) = 620; P(10,0) = 500.
Maximum profit is £620 when x = 6 and y = 8. The solution is integer, so no further adjustment is needed. Both machines are fully utilised (the point lies on both constraint lines). This illustrates a typical D1 question.
最大利润为 620 英镑，此时 x = 6，y = 8。解为整数，所以无需进一步调整。两台机器均被充分利用（该点位于两条约束线上）。这展示了一道典型的 D1 题型。
12. Final Summary and Revision Focus | 最后总结与复习重点
Linear programming in Edexcel D1 is highly structured. Master the translation from a word problem to inequalities, accurate graph plotting, shading conventions, and the vertex or ruler method. Ensure you can handle integer points when required, and always present your final answer with clarity. Practice plenty of past‑paper questions to become fluent in identifying the correct vertices and interpreting the outcomes.
Edexcel D1 的线性规划章节结构清晰。务必掌握将文字题转化为不等式、准确绘图、规范的阴影标注，以及顶点法或直尺法。要能在需要时处理整数点，并始终清晰地给出最终答案。多练习历年真题，熟练识别正确的顶点并解释结果，才能在考试中游刃有余。
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