📚 Linear Programming: Key Points for IB CCEA Mathematics | IB CCEA 数学：线性规划考点精讲
Linear programming is a powerful decision-making tool that appears frequently in IB and CCEA mathematics courses. It allows us to optimise a given objective – such as profit or cost – under a set of linear constraints. This article distils the essential concepts, techniques, and exam pitfalls so that you can approach any linear programming question with confidence.
线性规划是一种强大的决策工具，频繁出现在IB和CCEA数学课程中。它让我们能够在一组线性约束条件下优化给定目标（如利润或成本）。本文提炼了基本概念、解题技巧和考试中常见的误区，让你自信应对任何线性规划题目。
1. What Is Linear Programming? | 什么是线性规划？
Linear programming (LP) is a method for determining the best possible outcome – such as maximum profit or minimum cost – in a mathematical model whose requirements are described by linear equations and inequalities.
线性规划（LP）是一种在数学模型（其需求由线性方程和不等式描述）中确定可能最佳结果（如最大利润或最小成本）的方法。
In exam contexts, a typical LP problem involves decision variables, constraints that limit these variables, and an objective function that we want to optimise. All relationships must be linear; this means no x², xy, or 1/x terms.
在考试中，典型的线性规划问题包括决策变量、限制这些变量的约束条件，以及我们要优化的目标函数。所有关系式都必须是线性的；这意味着不能出现x²、xy或1/x项。
IB and CCEA syllabuses expect you to convert a written scenario into algebraic inequalities, graph the feasible region, and find the optimal solution using the corner-point method. Understanding the real-world meaning of your answer is just as important as the calculation.
IB和CCEA的教学大纲要求你将文字情境转化为代数不等式，画出可行域，并使用顶点法求出最优解。理解答案的现实意义与计算过程同样重要。
2. Formulating a Linear Programming Problem | 建立线性规划模型
The first step is identifying the decision variables. For example, let x be the number of chairs produced and y be the number of tables produced. Always define your variables clearly, stating their units and any non-negativity constraints such as x ≥ 0, y ≥ 0.
第一步是确定决策变量。例如，设x为生产的椅子数量，y为生产的桌子数量。务必清晰地定义变量，说明其单位以及非负约束，如x ≥ 0、y ≥ 0。
Next, translate each restriction into a linear inequality. A resource constraint like “a chair needs 2 hours of labour and a table needs 3 hours; total labour must not exceed 120 hours” becomes 2x + 3y ≤ 120. Pay attention to direction: ‘at most’ means ≤, ‘at least’ means ≥.
接下来，将每条限制转化为线性不等式。像“一把椅子需要2小时工时，一张桌子需要3小时；总工时不得超过120小时”这样的资源约束转化为2x + 3y ≤ 120。注意不等号方向：“至多”用≤，“至少”用≥。
The objective function is what you aim to maximise or minimise. If each chair yields a profit of £40 and each table £30, the function is P = 40x + 30y (maximise). Always state clearly whether you are maximising or minimising.
目标函数是你想要最大化或最小化的量。如果每把椅子利润为40英镑，每张桌子利润为30英镑，则函数为P = 40x + 30y（最大化）。务必明确表述是最大化还是最小化。
3. Graphing Linear Inequalities | 绘制线性不等式
To graph an inequality such as 3x + 2y ≤ 12, first draw the boundary line 3x + 2y = 12. Use a solid line for ≤ or ≥, and a dashed line for strict < or >. In most LP problems, solid lines are standard because equalities are included.
绘制3x + 2y ≤ 12这类不等式时，首先画出边界线3x + 2y = 12。对于≤或≥使用实线，对于严格的不等号<或>使用虚线。在大多数线性规划问题中，因为包含等号，实线是常规做法。
Decide which side of the line to shade. Choose a test point not on the line – the origin (0,0) is easiest if it does not lie on the line. Substitute into the inequality; if the inequality holds, shade the side containing the test point. In the context of multiple constraints, shade the unwanted region or the feasible region as directed by your exam board; CCEA often prefers shading the unwanted region, while IB may ask for clear labelling of the feasible set. Always follow the mark scheme instructions.
确定直线的哪一侧需要涂色。选择一个不在直线上的检验点——如果原点(0,0)不在线上，用它就最简单。代入不等式；如果不等式成立，则将检验点所在的一侧涂色。对于多个约束条件，根据考局要求涂掉不可行区域或直接标出可行区域；CCEA通常倾向于涂掉不可行区域，而IB可能要求清晰标示可行域。务必遵循评分方案要求。
Example boundary line: 3x + 2y = 12 ⇒ Intercepts (4,0) and (0,6)
示例边界线：3x + 2y = 12 ⇒ 截距点 (4,0) 和 (0,6)
4. The Feasible Region | 可行域
The feasible region is the set of all points that satisfy every constraint simultaneously. On a graph, it is the unshaded (or shaded) polygonal area where all inequalities overlap.
可行域是同时满足所有约束条件的点的集合。在图上，它是所有不等式的重叠部分，呈现为未被涂色（或被涂色）的多边形区域。
If the problem is well-posed, the feasible region will be a convex polygon, often bounded. In many maximisation problems, the feasible region is a set of vertices connected by edges. You must be able to identify the coordinates of all vertices, either by reading from the graph or by solving simultaneous equations.
如果问题设置合理，可行域将是一个凸多边形，通常是有界的。在许多最大化问题中，可行域是一组由边相连的顶点。你必须能够通过从图上读取或解联立方程来确定所有顶点的坐标。
Never assume the axes intercepts are the only vertices. Intersections between the constraint lines often form the most critical corners, so accurate solving of simultaneous equations is essential.
绝不要假定坐标轴截距是仅有的顶点。约束线之间的交点常构成最关键的角点，因此精确地解联立方程至关重要。
5. Objective Function and Level Lines | 目标函数与等值线
The objective function can be represented by a family of parallel lines called level lines or iso-profit lines. For a function like P = 40x + 30y, set P equal to a constant to draw one line, e.g. 40x + 30y = 1200. All parallel lines have the same gradient, here -4/3.
目标函数可以用一组平行的等值线（或等利润线）表示。对于像P = 40x + 30y这样的函数，可令P等于某个常数来画出一条线，例如40x + 30y = 1200。所有平行线的梯度相同，本例中为 -4/3。
To find the optimal solution visually, slide a ruler parallel to these level lines across the feasible region in the direction of improvement. For maximisation, the optimum occurs at the last vertex the ruler touches before leaving the region; for minimisation, it is the first vertex.
为从图形上找出最优解，可将直尺与这些等值线平行，朝改进方向在可行域上滑动。对于最大化问题，最优点是直尺离开该区域前接触的最后一个顶点；对于最小化问题，则是第一个顶点。
Gradient of level lines = – (coefficient of x) / (coefficient of y) = –40/30 = –4/3
等值线的梯度 = – (x的系数)/(y的系数) = –40/30 = –4/3
6. The Corner-Point Method for Optimal Solution | 顶点法求最优解
Because the feasible region is a convex polygon, the optimal solution will always occur at a corner point (vertex) when the objective function is linear. This is the fundamental theorem of linear programming. Therefore, you can simply evaluate the objective function at each vertex and select the best value.
由于可行域是凸多边形，当目标函数为线性时，最优解总出现在角点（顶点）处。这是线性规划的基本定理。因此，你可以简单地评估每个顶点处的目标函数值并选取最优值。
List all vertices in a table. For each vertex, compute the objective value. Compare and identify the maximum or minimum. This method is robust and preferred in exams because it shows clear working and reduces the risk of graphical misreading.
将所有顶点列在表格里。对每个顶点计算目标值。比较并确定最大值或最小值。这个方法稳健可靠，在考试中较受欢迎，因为它展示了清晰过程并减少了图形误读的风险。
| Corner Point (x, y) | Value of P = 40x + 30y |
|---|---|
| (0, 0) | 0 |
| (0, 40) | 1200 |
| (30, 20) | 1800 |
| (40, 0) | 1600 |
The optimal solution is (30, 20) giving a maximum profit of 1800. Always state the solution in the context of the original problem, e.g. “produce 30 chairs and 20 tables for a profit of £1800”.
最优解为(30, 20)，最大利润为1800。始终要在原问题情境中陈述答案，例如“生产30把椅子和20张桌子可获得利润1800英镑”。
7. Special Cases: Unbounded, Infeasible, and Multiple Optima | 特殊情况：无界、不可行、多最优解
Unbounded feasible region: If the region is not fully enclosed and extends infinitely in the direction of improvement, no finite maximum exists. The problem is then said to be unbounded. In an exam, you can state that “the objective can be increased without limit”.
无界可行域：如果区域未完全封闭并朝改进方向无限延伸，则不存在有限最大值。此类问题称为无界。在考试中，可以表述为“目标可以无限增加”。
Infeasible region: When constraints contradict each other, there is no point that satisfies all of them. The feasible region is empty. You must conclude “no solution” or “infeasible”. Check your algebra carefully – a common mistake is copying a sign incorrectly.
不可行区域：当约束条件互相矛盾时，没有点能同时满足所有约束。可行域为空。你必须得出“无解”或“不可行”的结论。仔细检查你的代数——常见的错误是抄错不等号。
Multiple optimal solutions: If the objective function has the same gradient as one of the boundary lines, every point on that boundary segment within the feasible region gives the same optimal value. Look for parallel lines: e.g. if P = 2x + 4y and a constraint is x + 2y ≤ 10, the boundary x + 2y = 10 is parallel to the level lines, so multiple optima occur. State that any point on that line segment between the two vertices is optimal.
多最优解：如果目标函数的梯度与某条边界线的梯度相同，那么可行域内该边界线段上的每一点都给出相同的最优值。判断平行关系：例如若P = 2x + 4y，且约束条件为x + 2y ≤ 10，则边界x + 2y = 10与等值线平行，因此出现多最优解。这时应说明该线段上两顶点之间的任何一点都是最优解。
8. Introduction to Integer Programming | 整数线性规划简介
When decision variables must be whole numbers – e.g. number of cars or people – you need integer solutions. The standard corner-point method might give fractional coordinates. In an IB or CCEA problem, you may be asked to find the best integer solution near the continuous optimum.
当决策变量必须为整数时（例如汽车或人的数量），就需要整数解。标准顶点法可能给出分数坐标。在IB或CCEA题目中，可能会要求你找出连续最优解附近的最佳整数解。
Technique: First solve the continuous LP problem. Then test integer points around the optimum vertex, ensuring they remain inside the feasible region. List the integer points and evaluate the objective function. The one with the best value is the integer optimum. Never simply round the vertex coordinates; a rounded point could lie outside the feasible region.
方法：先解出连续型线性规划问题。然后检验最优顶点附近的整数点，确保它们仍在可行域内。列出这些整数点并评估目标函数。拥有最优值的那个点就是整数最优解。绝不能只是将顶点坐标四舍五入；舍入后的点可能落在可行域之外。
In some CCEA decision mathematics papers, this is a separate style of question that requires listing all possibilities. Always show your testing steps and justify the final choice.
在某些CCEA决策数学试卷中，这是一类独立题型，需要列出所有可能情况。务必展示你的检验步骤并说明最终选择的理由。
9. Exam Tips and Common Pitfalls | 考试技巧与常见误区
Always write a clear definition of your variables at the start – marks are often allocated for this. Use correct notation such as “let x = … and y = …”. Never forget the non-negativity constraints x ≥ 0, y ≥ 0 unless the context explicitly excludes them.
始终在开头清晰定义变量——这通常有采分点。使用正确的符号，如“let x = … and y = …”。除非题意明确排除，否则千万别漏掉非负约束x ≥ 0, y ≥ 0。
Check the objective function direction carefully. A maximisation problem that you treat as minimisation will cost many marks. Highlight the words ‘maximise’ or ‘minimise’ in the question.
仔细确认目标函数的优化方向。把最大化问题当作最小化来处理会丢掉很多分数。在题目中将“最大”或“最小”圈出来。
When drawing constraint lines, use a ruler and label each line with its equation. Plot points carefully – accurate graphs are vital. If you miscalculate an intercept, your feasible region and subsequent table of vertices will be wrong. Double-check by substituting into two different equations.
绘制约束线时务必使用直尺，并将每条线标注其方程。准确描点——精确的图形至关重要。如果算错截距，你的可行域和后续顶点表都将错误。可以通过代入两个不同方程来双重检查交点。
Show the corner-point evaluation in a table. This not only earns method marks but also makes it easy to spot the optimum. Finally, write a concluding sentence that directly answers the question in context, e.g. “The maximum profit is £1800 when 30 chairs and 20 tables are produced.”
用表格展示顶点评估过程。这不仅能获得方法分，还便于找出最优值。最后，写一句总结性语句，直接回答题目情境中的问题，例如：“当生产30把椅子和20张桌子时，最大利润为1800英镑。”
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