📚 Linear Programming: Key Concepts for IB AQA Mathematics | 线性规划考点精讲
Linear programming is a powerful mathematical technique used to find the best possible outcome in a model where requirements are expressed as linear equations or inequalities. In your IB AQA Mathematics course, you will learn to formulate real-world problems, graph constraints, identify the feasible region, and determine the optimal solution using graphical methods. This guide covers all the essential concepts you need to master for exam success.
线性规划是一种强大的数学技术，用于在模型要求表示为线性方程或不等式的情况下找到最佳可能结果。在 IB AQA 数学课程中，您将学习构建实际问题模型、绘制约束条件、识别可行域，并通过图解法确定最优解。本指南涵盖了您为考试成功需要掌握的所有核心概念。
1. What is Linear Programming? | 什么是线性规划？
Linear programming (LP) is a method of optimisation used to achieve the maximum profit or minimum cost given limited resources. The word “linear” means that both the objective function and all constraints are linear functions of the decision variables (e.g., x and y). In two-variable problems, LP is solved by drawing graphs and finding the best point within a shaded region.
线性规划是一种优化方法，用于在资源有限的情况下实现利润最大化或成本最小化。’线性’一词意味着目标函数和所有约束条件都是决策变量（如 x 和 y）的线性函数。在双变量问题中，线性规划通过画图并在涂色区域内寻找最佳点来求解。
LP has widespread applications in logistics, finance, agriculture and manufacturing. The IB AQA syllabus focuses on formulating problems, graphing, interpreting solutions, and dealing with special cases like integer variables.
线性规划广泛应用于物流、金融、农业和制造业。IB AQA 教学大纲侧重于问题建模、绘图、解的解释以及处理整数变量等特殊情况。
2. Formulating the Problem | 问题建模
The first step is to translate a verbal description into mathematical language. You must define your decision variables clearly, state the objective function (what you want to maximise or minimise), and list all constraints. Standard notation uses x and y for the variables, and non-negativity constraints (x ≥ 0, y ≥ 0) should always be included unless otherwise stated.
第一步是将文字描述转化为数学语言。您必须清楚地定义决策变量，陈述目标函数（您希望最大化或最小化的量），并列出所有约束条件。标准符号使用 x 和 y 作为变量，除非另有说明，否则应始终包含非负约束（x ≥ 0, y ≥ 0）。
Consider a typical problem: A toy manufacturer produces two types of dolls, Star (x) and Moon (y). Each Star doll requires 2 hours of assembly and 1 hour of painting, while each Moon doll needs 1 hour of assembly and 3 hours of painting. The factory has 100 assembly hours and 90 painting hours available. Profit per Star doll is £40 and per Moon doll is £30. Find the production mix that maximises profit.
考虑一个典型问题：一家玩具制造商生产两种玩偶，星星（x）和月亮（y）。每个星星玩偶需要 2 小时组装和 1 小时喷漆，而每个月亮玩偶需要 1 小时组装和 3 小时喷漆。工厂有 100 个组装工时和 90 个喷漆工时可用。星星玩偶的利润为 40 英镑，月亮玩偶的利润为 30 英镑。求出使利润最大化的生产组合。
The formulation becomes:
建模如下：
Maximise Z = 40x + 30y
Subject to: 2x + y ≤ 100 (assembly)
x + 3y ≤ 90 (painting)
x ≥ 0, y ≥ 0
3. Graphing Constraints | 图示约束条件
Each linear inequality divides the coordinate plane into two half-planes. To graph a constraint like 2x + y ≤ 100, first draw the boundary line 2x + y = 100. Use a solid line if the inequality includes equality (≤ or ≥) and a dashed line for strict inequalities (< or >). Then choose a test point, usually (0,0), to determine which side to shade. If (0,0) satisfies the inequality, shade the side containing the origin.
每个线性不等式将坐标平面分成两个半平面。要绘制像 2x + y ≤ 100 这样的约束，首先画出边界线 2x + y = 100。如果不等式包含等号（≤ 或 ≥），使用实线；对于严格不等式（< 或 >），使用虚线。然后选择一个测试点，通常是 (0,0)，以确定涂色哪一侧。如果 (0,0) 满足不等式，则涂色包含原点的这一侧。
In our toy example, we plot 2x + y = 100 (x-intercept 50, y-intercept 100) and x + 3y = 90 (x-intercept 90, y-intercept 30), both as solid lines because of the ≤ signs. The non-negativity constraints restrict us to the first quadrant. The feasible region is where all shaded areas overlap.
在我们的玩具例子中，我们画出 2x + y = 100（x 截距 50，y 截距 100）和 x + 3y = 90（x 截距 90，y 截距 30），因为带有 ≤ 符号，所以都用实线。非负约束将我们限制在第一象限。可行域是所有涂色区域重叠的部分。
4. Identifying the Feasible Region | 确定可行域
The feasible region is the set of all points (x,y) that satisfy every constraint simultaneously. It is usually a convex polygon in the first quadrant. If any constraint makes the region empty, the problem has no feasible solution. The optimal solution will be found on the boundary of this region, not in its interior, because the objective function is linear.
可行域是所有同时满足每个约束条件的点 (x,y) 的集合。它通常是第一象限中的一个凸多边形。如果任何约束导致区域为空，则问题没有可行解。最优解将位于此区域的边界上，而非内部，因为目标函数是线性的。
For our problem, the feasible region is a quadrilateral with vertices at (0,0), (0,30), (45,0) and the intersection point of the two lines. To find the exact intersection, solve 2x + y = 100 and x + 3y = 90 simultaneously. Solving gives x = 42, y = 16. So the vertices are (0,0), (0,30), (42,16) and (50,0)? Check: (50,0) is on 2x+y=100, but does it satisfy painting? 50 + 0 = 50 ≤ 90, yes. So vertices are (0,0), (0,30), (42,16), (50,0). Actually (45,0) from x+3y=90, but (45,0) fails assembly: 2*45+0=90≤100, okay. Need to carefully determine all boundary intersections. Let us correct: line 1: 2x+y=100, x-int 50, y-int 100. line 2: x+3y=90, x-int 90, y-int 30. The axes intersections: (0,100) not in feasible due to line 2; (90,0) not due to line 1? 2*90+0=180>100, so excluded. So vertices are (0,0) where both non-negative; (0,30) where line2 hits y-axis; (50,0) where line1 hits x-axis; and the intersection of the two lines (42,16). All satisfy constraints. Feasible region is convex quadrilateral with these four vertices. The (0,0), (0,30), (42,16), (50,0).
在我们的问题中，可行域是一个四边形，顶点为 (0,0)、(0,30)、(50,0) 和两条线的交点。为找到确切交点，联立求解 2x + y = 100 和 x + 3y = 90。解得 x = 42, y = 16。因此顶点为 (0,0)、(0,30)、(42,16) 和 (50,0)。
5. Objective Function and Level Lines | 目标函数和等值线
The objective function Z = ax + by associates a value with every point in the feasible region. A level line (or isoprofit line) is a line along which Z takes a constant value, e.g., Z = 40x + 30y. By rearranging, we get y = -(a/b)x + (Z/b), which has a fixed slope. As Z increases, the line shifts parallelly. The direction of increase can be found by calculating the gradient of the objective function.
目标函数 Z = ax + by 为可行域中的每个点关联一个值。等值线（或等利润线）是 Z 取常数值的直线，例如 Z = 40x + 30y。通过整理，我们得到 y = -(a/b)x + (Z/b)，它具有固定的斜率。随着 Z 的增加，直线平行移动。可以通过计算目标函数的斜率找到增加方向。
For our example, Z = 40x + 30y has slope -4/3. We can draw several level lines, e.g., for Z = 0, 1200, 1800, and observe which corner is the last one touched as the line moves up and to the right while still intersecting the feasible region. The best profit will occur at a vertex.
以我们的例子为例，Z = 40x + 30y 的斜率为 -4/3。我们可以画几条等值线，例如 Z = 0、1200、1800，并观察当线向上向右移动且仍与可行域相交时，最后触及的角点是哪一个。最佳利润将出现在一个顶点处。
6. Finding the Optimal Solution | 寻找最优解
The maximum value of the objective function over a bounded feasible region occurs at one of the vertices. Therefore, the most efficient method is to list all vertices, calculate the Z value at each, and select the largest (or smallest) as required. This is called the corner point method and is the standard approach in IB exams.
在有界可行域上，目标函数的最大值出现在其中一个顶点处。因此，最有效的方法是列出所有顶点，计算每个顶点处的 Z 值，并根据需要选择最大（或最小）值。这称为角点法，是 IB 考试中的标准方法。
Using our vertices:
利用我们的顶点：
| Vertex (x,y) | Z = 40x + 30y |
|---|---|
| (0,0) | 0 |
| (0,30) | 900 |
| (42,16) | 40×42 + 30×16 = 1680 + 480 = 2160 |
| (50,0) | 2000 |
The maximum Z is 2160 at (42,16). Therefore, the manufacturer should produce 42 Star dolls and 16 Moon dolls to achieve a profit of £2160.
最大 Z 值为 2160，位于 (42,16)。因此，制造商应生产 42 个星星玩偶和 16 个月亮玩偶，以获得 2160 英镑的利润。
7. Optimal Solution at a Vertex | 最优解在顶点
The corner point theorem is a fundamental principle in linear programming. Because the feasible region is convex and the objective function is linear, the optimum always occurs at an extreme point (vertex). If the feasible region is unbounded, a maximum may not exist, but if it does, it will still be at a vertex or along an edge. This theorem justifies the vertex-testing method.
角点定理是线性规划的一个基本原理。由于可行域是凸的且目标函数是线性的，最优解总是出现在极点（顶点）处。如果可行域无界，最大值可能不存在，但如果存在，它仍将位于顶点或某条边上。该定理证明了顶点测试法的合理性。
There are some cases where multiple vertices yield the same optimal value, leading to infinitely many solutions along an edge. We will explore this later. But the vertex method remains a reliable computational tool.
在某些情况下，多个顶点产生相同的最优值，从而导致沿某条边存在无穷多个解。我们稍后将探讨这一点。但顶点法仍然是一个可靠的计算工具。
8. Redundant Constraints | 冗余约束
Not all constraints affect the feasible region. A redundant constraint is one whose boundary line lies entirely outside the feasible region and does not touch it. Removing it would not change the shape or size of the feasible region. For instance, if we had an extra condition x + 2y ≤ 200 in our toy problem, it would be redundant because the existing constraints already restrict the region to a much smaller area.
并非所有约束都会影响可行域。冗余约束是指其边界线完全位于可行域之外且不与可行域接触的约束。移除它不会改变可行域的形状或大小。例如，如果在我们的玩具问题中加入一个额外条件 x + 2y ≤ 200，它将是冗余的，因为现有约束已经将区域限制在一个小得多的范围内。
While redundant constraints do not affect the final solution, they should still be listed during formulation. On a graph, they are often obvious because their shaded region covers the entire feasible region. In exams, you might be asked to identify redundant constraints.
虽然冗余约束不影响最终解，但在建模过程中仍应列出。在图上，它们通常很明显，因为它们的涂色区域覆盖了整个可行域。考试中，您可能会被要求识别冗余约束。
9. Multiple Optimal Solutions | 多重最优解
When the level line of the objective function is parallel to one of the binding constraints, the optimal value is achieved by every point along that edge of the feasible region. This situation is known as multiple or alternative optimal solutions. To detect this, check if the slope of the objective function matches the slope of any constraint boundary.
当目标函数的等值线与某个起作用约束平行时，沿可行域那条边的每个点都能达到最优值。这种情况称为多重或替代最优解。要检测这一点，检查目标函数的斜率是否与任何约束边界的斜率匹配。
For example, if the objective were Z = 20x + 10y in a problem with constraint 2x + y = 50 (slope -2), the objective slope -2 matches, and the entire edge from one vertex to another could be optimal. In this case, you would state that all points on the segment between those vertices give the same optimum.
例如，如果在一个含有约束 2x + y = 50（斜
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