IB OCR Mathematics: Linear Programming – Key Points | IB OCR 数学：线性规划考点精讲

📚 IB OCR Mathematics: Linear Programming – Key Points | IB OCR 数学：线性规划考点精讲

Linear programming is a powerful mathematical technique used to find the optimal outcome – maximum profit or minimum cost – under a set of constraints. For IB (AI and AA) and OCR A-level exams, you need to master graphing inequalities, identifying the feasible region, and solving objective functions. This guide breaks down every core concept you’ll face in the exam.

线性规划是一种强大的数学方法，用于在一组约束条件下找到最优结果——最大利润或最小成本。针对 IB（AI 与 AA）和 OCR A-level 考试，你需要掌握绘制不等式图形、确定可行域以及求解目标函数。本指南将逐一剖析考试中会遇到的每一个核心概念。

1. Introduction: What is Linear Programming? | 引言：什么是线性规划？

Linear programming (LP) is a method to optimise a linear objective function subject to linear equality and inequality constraints. It is widely applied in business, industry, and logistics. The key components are decision variables, constraints, the objective function, and the feasible region.

线性规划（LP）是一种在满足线性等式和不等式约束的条件下优化线性目标函数的方法。它广泛应用于商业、工业和物流领域。其关键组成部分包括决策变量、约束条件、目标函数和可行域。

In IB and OCR questions, you will typically formulate a problem from a worded scenario, then find the maximum or minimum value of an expression like P = 2x + 3y.

在 IB 和 OCR 的考题中，你通常需要根据文字描述的场景建立模型，然后求出类似 P = 2x + 3y 的表达式的最大值或最小值。

2. Decision Variables and Objective Function | 决策变量与目标函数

The decision variables are the unknowns we are solving for, e.g., x = number of product A, y = number of product B. The objective function is a linear expression like Profit = ax + by that we want to maximise or minimise. It must be written in terms of the decision variables.

决策变量是我们要求解的未知数，例如 x = 产品 A 的数量，y = 产品 B 的数量。目标函数是一个线性表达式，如 Profit = ax + by，我们要使它最大化或最小化。它必须用决策变量来表示。

In exam questions, always define your variables clearly: “Let x be the number of …” can earn method marks even before you start solving.

在考题中，务必清晰地定义变量：“设 x 为……的数量” 即使在你开始解题之前就能帮助你获得方法分。

Example: Maximise P = 5x + 7y, where x and y are units produced.
示例：最大化 P = 5x + 7y，其中 x 和 y 为生产的单位数。

3. Formulating Linear Constraints | 建立线性约束条件

Constraints are linear inequalities derived from the problem’s limits: resource availability, time, budget, minimum or maximum requirements. They always involve the decision variables. Common signs: ≤, ≥. Non-negativity constraints x ≥ 0, y ≥ 0 are almost always included.

约束条件是从问题中的限制条件得出的线性不等式：资源可用量、时间、预算、最低或最高要求。它们总是包含决策变量。常见的不等号有 ≤ 和 ≥。非负约束条件 x ≥ 0、y ≥ 0 几乎总是包含在内。

When constructing constraints, be careful: “at most”, “no more than” translate to ≤; “at least”, “must exceed” translate to ≥. Always check unit consistency.

在建立约束条件时要小心：“至多”、“不超过”用 ≤ 表示；“至少”、“必须超过”用 ≥ 表示。始终检查单位的一致性。

Phrase	Inequality	中文
Total time ≤ 40 hours	3x + 2y ≤ 40	总时间 ≤ 40 小时
At least twice as many A as B	x ≥ 2y	A 至少是 B 的两倍

4. Graphing the Constraints and Shading | 绘制约束条件与阴影标示

Plot each inequality on a coordinate plane. First draw the boundary line (solid for ≤ or ≥, dashed for < or >). Then shade the allowed region. The intersection of all shaded regions forms the feasible region. For clarity, many IGCSE/IB/OCR marking schemes expect only the feasible region to be clearly indicated.

在坐标平面上绘制每个不等式。首先画出边界线（对 ≤ 或 ≥ 用实线，对 < 或 > 用虚线）。然后标示出满足条件的区域。所有阴影区域的交集形成可行域。为清晰起见，许多 IGCSE/IB/OCR 的评分标准只要求明确标示出可行域。

Use the origin (0,0) to test the half-plane unless the line passes through the origin. Common mistake: shading the wrong side. Always test a point.

用原点 (0,0) 检验半平面，除非直线经过原点。常见错误：将阴影标在错误的一侧。务必用一个点进行检验。

Graph line, shade above for y ≥ mx + c, below for y ≤ mx + c

绘制直线，对于 y ≥ mx + c 向上方阴影，对于 y ≤ mx + c 向下方阴影

5. The Feasible Region and Its Vertices | 可行域及其顶点

The feasible region is the set of all points that satisfy every constraint simultaneously. It is usually a convex polygon. The optimal solution (if it exists and is unique) will always lie at one of the vertices (corner points) of this polygon. This is the corner-point principle.

可行域是同时满足所有约束条件的所有点的集合。它通常是一个凸多边形。最优解（如果存在且唯一）总是位于该多边形的某个顶点（角点）上。这就是顶点法则。

You must find the coordinates of all vertices. Solve pairs of boundary line equations simultaneously. Be systematic; list all intersections and then eliminate those that violate any constraint.

你必须求出所有顶点的坐标。联立一对边界线方程来求解。要有条理；列出所有交点，然后排除那些违反任一约束的点。

Vertex A: (0,0) – zero production
Vertex B: where 3x + 2y = 40 meets x = 0 → (0, 20)
顶点 A: (0,0) – 零生产
顶点 B: 直线 3x + 2y = 40 与 x = 0 的交点 → (0, 20)

6. Solving Graphically: The Objective Line Method | 图解法求解：目标直线法

Draw a line representing the objective function using a convenient constant, e.g., P = 5x + 7y = 35. Then slide this parallel line in the direction of increase (for maximisation) or decrease (for minimisation) until it touches the last point of the feasible region. That point gives the optimal solution.

画一条表示目标函数的直线，使用一个方便的常数值，例如 P = 5x + 7y = 35。然后平行移动这条直线，朝着增大的方向（最大化问题）或减小的方向（最小化问题）移动，直到它与可行域的最后一个点接触。那个点就是最优解。

Alternatively, and more reliably for exact values, evaluate the objective function at every vertex. Both methods are accepted in exams; the vertex method is particularly recommended when precise coordinates are obtained algebraically.

另一种方法，并且在需要精确值时更可靠的是，计算每个顶点上的目标函数值。两种方法在考试中都接受；当通过代数方法获得精确坐标时，尤其推荐使用顶点法。

P = 5x + 7y at A(0,0): 0; B(0,20): 140; C(8,8): 96; D(10,0): 50 → Maximum is 140 at B

P = 5x + 7y 在 A(0,0): 0; B(0,20): 140; C(8,8): 96; D(10,0): 50 → 在 B 点取得最大值 140

7. Special Cases: Unbounded, Infeasible, and Multiple Solutions | 特殊情况：无界、无解与多重解

If the feasible region is unbounded in the direction of optimisation, there may be no maximum (or no minimum). If no point satisfies all constraints, the problem is infeasible. If the objective line is parallel to a boundary of the feasible region, there could be infinitely many optimal solutions along that edge.

如果可行域在优化方向上无界，则可能没有最大值（或没有最小值）。如果没有任何点能满足所有约束，则该问题无解。如果目标直线与可行域的一条边界平行，则沿着那条边可能会有无穷多个最优解。

IB and OCR sometimes ask you to explain why a solution is not possible or why many solutions exist. You must be able to identify these cases from the graph or algebra.

IB 和 OCR 有时会要求你解释为什么某解不可行或为什么存在多个解。你必须能够从图中或通过代数识别这些情况。

8. Integer Programming and Discrete Constraints | 整数规划与离散约束

In many realistic problems, decision variables must be whole numbers – you cannot produce 3.7 tables. This introduces integer programming. If the continuous optimum has non-integer coordinates, test integer points near the optimum within the feasible region. The integer optimum may not be the nearest whole number point.

在许多实际问题中，决策变量必须是整数——你不能生产 3.7 张桌子。这就引入了整数规划。如果连续最优解具有非整数坐标，则在可行域内测试最优解附近的整数点。整数最优解可能并非最近的那个整数点。

Typical exam task: “Find the maximum number of cakes and pies if they must be in whole numbers.” Plot, list feasible integer points near the optimum, and test each in the objective function.

典型的考试任务是：“如果蛋糕和馅饼的数量必须是整数，求最大数量。” 画出图形，列出最优解附近可行的整数点，并将每个点代入目标函数进行检验。

9. Sensitivity and Shadow Price (Extension) | 敏感度与影子价格（拓展）

For higher-tier IB and some OCR modules, you may discuss the shadow price – the rate of improvement in the objective function if a constraint is relaxed by one unit. This is the value of an extra resource. Graphically, it relates to the gradient of the constraint line at the binding vertex.

对于 IB 高等级和某些 OCR 模块，你可能需要讨论影子价格——即约束条件放宽一个单位时目标函数值改善的速率。这是额外资源的价值。在图形上，这与在起作用的顶点处的约束线斜率有关。

Although not always required, understanding that the objective function changes linearly near the optimum helps in scenario analysis. The binding constraints are those that pass through the optimal vertex.

虽然并非总是要求，但理解目标函数在最优解附近呈线性变化有助于情景分析。起作用约束是指那些通过最优顶点的约束。

10. Exam Strategy and Common Pitfalls | 应试策略与常见陷阱

Always start by defining variables, listing given data, and writing constraints in a neat format. Draw axes with a scale large enough to show all intercepts. Label lines and the feasible region. Show clearly how you obtained vertex coordinates. In the answer, state the optimal values of the variables and the optimal objective value in the context of the question.

答题时始终从定义变量、列出已知数据并把约束条件整齐书写开始。以足够大的比例绘制坐标轴，以显示所有截距。标注直线和可行域。清晰地展示你是如何获得顶点坐标的。在答案中，根据题意陈述变量的最优值以及最优目标函数值。

Common mistakes: forgetting non-negativity constraints, shading incorrectly, misreading “at least half”, and not checking integer solutions when required. Also, always refer back to the original units (e.g., pounds, hours).

常见错误：忘记非负约束、错误标示阴影、误读“至少一半”以及在不要求整数解时未进行检查。此外，始终要回归原单位（例如，英镑、小时）。

11. Worked Example: A Simple Production Model | 实例讲解：一个简单的生产模型

A factory makes chairs (x) and tables (y). Each chair needs 2 hours on machine A and 1 hour on machine B. Each table needs 1 hour on A and 3 hours on B. Machine A is available for 10 hours, machine B for 12 hours. Profit is £3 per chair and £4 per table. Maximise profit.

一家工厂生产椅子 (x) 和桌子 (y)。每把椅子需要在机器 A 上加工 2 小时、机器 B 上加工 1 小时。每张桌子需要在机器 A 上加工 1 小时、机器 B 上加工 3 小时。机器 A 可用 10 小时，机器 B 可用 12 小时。利润为每把椅子 £3、每张桌子 £4。最大化利润。

Constraints: 2x + y ≤ 10, x + 3y ≤ 12, x ≥ 0, y ≥ 0. Vertices: (0,0), (0,4), (5,0), and intersection of 2x+y=10 and x+3y=12 → (3.6,2.8). Profit P = 3x+4y gives 0, 16, 15, and 3(3.6)+4(2.8)=10.8+11.2=22. So maximum continuous profit £22 at (3.6,2.8). If integer required, test (3,2)=17, (4,2)=20, (3,3)=21, (4,1)=16. Integer max is £21 at (3,3).

约束条件：2x + y ≤ 10, x + 3y ≤ 12, x ≥ 0, y ≥ 0。顶点为：(0,0)、(0,4)、(5,0)，以及 2x+y=10 与 x+3y=12 的交点 (3.6,2.8)。利润 P = 3x+4y 得出 0, 16, 15, 及 3(3.6)+4(2.8)=10.8+11.2=22。所以连续最大利润在 (3.6,2.8) 处为 £22。若要求整数解，检验 (3,2)=17, (4,2)=20, (3,3)=21, (4,1)=16。整数最大利润在 (3,3) 处为 £21。

12. Summary and Quick Reference | 总结与快速参考

Linear programming is a systematic method: define variables, write constraints and objective, graph feasible region, find vertices, test optimality. The optimal solution lies at a vertex. Pay attention to integer requirements and special cases. Practise with real exam questions to master the graphical accuracy and interpretation.

线性规划是一种系统性的方法：定义变量，写出约束条件和目标函数，绘制可行域，求出顶点，检验最优性。最优解位于顶点处。要注意整数要求和特殊情况。通过练习真实考题来掌握图形的准确性和解读能力。

Key Concept	Must Remember	核心概念	必须记住
Feasible region	Intersection of all constraints, convex	可行域	所有约束的交集，凸集
Objective function	Maximise or minimise, evaluate at vertices	目标函数	最大化或最小化，在顶点处求值
Integer solutions	Test nearby integer points within feasible region	整数解	在可行域内测试附近的整数点

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IB OCR Mathematics: Linear Programming – Key Points | IB OCR 数学：线性规划 考点精讲