📚 Linear Programming for CCEA IGCSE Maths | CCEA IGCSE 数学线性规划考点精讲
Linear programming is a crucial topic in the CCEA IGCSE Mathematics syllabus. It equips you with the skills to solve real-world optimisation problems, such as maximising profit or minimising cost, under given constraints. This revision guide breaks down the essential concepts and techniques you need to master, including formulating inequalities, graphing feasible regions, and applying the objective function method.
线性规划是 CCEA IGCSE 数学大纲中的一个关键主题。它使你具备解决现实世界优化问题的技能，例如在给定约束条件下最大化利润或最小化成本。本复习指南分解了您需要掌握的基本概念和技巧，包括建立不等式、绘制可行域以及应用目标函数方法。
1. What is Linear Programming? | 什么是线性规划？
Linear programming is a mathematical method for determining the best possible outcome, such as maximum profit or minimum cost, in a situation described by linear relationships. The ‘programming’ refers to planning rather than computer code, and it is used extensively in business, industry and logistics.
线性规划是一种在由线性关系描述的情形中确定最佳可能结果（如最大利润或最低成本）的数学方法。’规划’指的是计划而非计算机编程，它广泛应用于商业、工业和物流领域。
In CCEA IGCSE exams, linear programming problems always involve two decision variables, usually named x and y. The problem will ask you to maximise or minimise a linear objective function while respecting a set of linear inequality constraints.
在 CCEA IGCSE 考试中，线性规划问题总是涉及两个决策变量，通常记为 x 和 y。题目会要求你在遵守一组线性不等式约束的同时，最大化或最小化一个线性目标函数。
2. Defining Decision Variables | 定义决策变量
Your first task is to read the problem carefully and define what x and y represent. For example, x might be the number of chairs produced, while y stands for the number of tables. Write this clearly as: ‘Let x be the number of chairs and y be the number of tables.’ This definition is essential for the following steps and is often credited in mark schemes.
你的首要任务是仔细阅读题目，并定义 x 和 y 代表什么。例如，x 可能代表生产的椅子数量，而 y 代表桌子数量。请清楚写明：’设 x 为椅子数量，y 为桌子数量。’ 这个定义对于后续步骤至关重要，并且评分方案中通常会给出分数。
Always choose variables that make sense in context and remember to state them at the very beginning of your solution. This sets a strong foundation for formulating constraints correctly.
始终选择在情境中有意义的变量，并记得在解答的开头就予以说明。这为正确建立约束条件奠定了坚实基础。
3. Formulating Constraints as Inequalities | 将约束表示为不等式
Constraints are the limitations provided by the problem, such as available materials, labour hours, or budget. Each constraint must be written as a linear inequality involving x and y. For instance, if a chair needs 2 hours of labour and a table needs 3 hours, with a maximum of 60 hours available, the constraint is: 2x + 3y ≤ 60.
约束是题目提供的限制条件，例如可用的材料、工时或预算。每个约束必须写成关于 x 和 y 的线性不等式。例如，如果一把椅子需要 2 小时工时，一张桌子需要 3 小时，且最多可用 60 小时，则约束为：2x + 3y ≤ 60。
Pay close attention to the wording: ‘at least’, ‘no more than’, ‘a minimum of’, and ‘cannot exceed’ all indicate inequality signs. Use ≤ or ≥ accordingly, and if the problem says ‘strictly less than’, use <, though this is rare in IGCSE. Make sure you do not confuse the direction of the inequality.
仔细留意措辞：’至少’、’不超过’、’最少’ 和 ‘不能超出’ 都指示了不等号。相应地使用 ≤ 或 ≥，如果题目说 ‘严格小于’，则用 <，尽管这在 IGCSE 中不常见。一定不要混淆不等号的方向。
4. The Non-negativity Constraints | 非负约束
In virtually every IGCSE linear programming problem, the variables represent real quantities that cannot be negative. Therefore, you must always include the constraints x ≥ 0 and y ≥ 0. Sometimes the problem may explicitly state these, but even if it does not, you should include them as part of your constraint list.
在几乎每个 IGCSE 线性规划问题中，变量代表的是不能为负的实际数量。因此，你必须始终包含约束 x ≥ 0 和 y ≥ 0。有时题目可能会明确说明，但即使没有，你也应将它们列入约束清单。
These non-negativity restrictions define the first quadrant of the coordinate plane and form two of the boundary lines. Forgetting them is a common error that can cost you marks in the feasibility and optimisation stages.
非负限制定义了坐标平面的第一象限，并构成两条边界线。忘记这些限制是一个常见错误，可能会使你在可行性和优化阶段失分。
5. Graphing Linear Inequalities | 绘制线性不等式
To graph an inequality such as 2x + y ≤ 40, first draw the line 2x + y = 40 as if it were an equation. Find two points (e.g., the intercepts) and draw a straight line using a ruler. Since the inequality includes equality (≤), the line should be solid. If the inequality were strict (<), you would draw a dashed line.
要绘制诸如 2x + y ≤ 40 的不等式，首先将 2x + y = 40 当作方程画出直线。找到两个点（例如截距），并用直尺画一条直线。由于该不等式包含等号 (≤)，直线应为实线。如果不等式是严格的 (<)，则应画虚线。
Now decide which side of the line represents the inequality. Pick a test point not on the line, usually (0,0) if the line does not pass through the origin. Substitute into the inequality: if it satisfies, shade the opposite side; if not, shade the side containing the point. CCEA often asks you to shade the region that is NOT required, leaving the feasible region clear. Always follow the specific instruction in the question.
现在确定直线的哪一侧代表该不等式。选取一个不在直线上的测试点，通常如果直线不经过原点就选 (0,0)。代入不等式：如果成立，涂掉相反一侧；如果不成立，则涂掉包含该点的一侧。CCEA 通常要求你涂掉不被需要的区域，从而留下清晰的可行域。始终遵循题目中具体的操作指令。
6. Identifying the Feasible Region | 确定可行域
The feasible region is the area that satisfies all the constraints simultaneously. After graphing each inequality and shading the excluded regions, the unshaded area (often labelled R) is your feasible region. It may be a convex polygon with straight edges, bounded or unbounded. In IGCSE problems, it is usually a closed quadrilateral or triangle.
可行域是同时满足所有约束条件的区域。在绘制出每个不等式并涂掉被排除的区域后，未被涂色的区域（通常标记为 R）就是你的可行域。它可能是一个具有直边的凸多边形，有界或无界。在 IGCSE 问题中，它通常是一个封闭的四边形或三角形。
Label the feasible region clearly with the letter R, as instructed. Double-check that every vertex lies within the axes (x ≥ 0, y ≥ 0) and that you have not accidentally shaded the wrong side of any line. Testing a point inside the intended region in all original inequalities is a quick way to verify correctness.
按照指示，用字母 R 清楚地标注可行域。仔细检查每个顶点是否都在坐标轴内 (x ≥ 0, y ≥ 0)，并且你没有意外涂错任何一条线的一侧。在目标区域内选取一点代入所有原始不等式进行检验，是验证正确性的快捷方法。
7. Finding the Corner Points (Vertices) | 寻找角点（顶点）
The optimal solution to a linear programming problem always occurs at a vertex (corner point) of the feasible region, provided the region is bounded and the objective function is linear. To find the vertices, solve the equations of the boundary lines in pairs. For example, the intersection of 2x + y = 40 and x + 3y = 60 can be found by simultaneous equations.
线性规划问题的最优解总是出现在可行域的顶点（角点），前提是该区域有界且目标函数是线性。要找到顶点，需要成对地解边界线方程。例如，2x + y = 40 和 x + 3y = 60 的交点可通过解联立方程组求得。
List all the vertices, which include the intersections of constraint lines and the points where lines meet the axes. If the feasible region is unbounded, you must check whether an optimum exists, but exam questions are typically bounded. Accurate coordinates are vital; any mistake here will lead to an incorrect optimum value.
列出所有顶点，包括约束线之间的交点以及线与坐标轴的交点。如果可行域无界，你必须检查最优解是否存在，但考题通常是有界的。精确的坐标至关重要；此处任何错误都将导致最优值出错。
8. The Objective Function | 目标函数
The objective function is the expression you want to maximise or minimise, such as profit P = 5x + 3y or cost C = 10x + 6y. It is given in the problem statement. The variables are the same x and y defined earlier. The objective function is always linear, meaning no squares or products of variables.
目标函数是你希望最大化或最小化的表达式，例如利润 P = 5x + 3y 或成本 C = 10x + 6y。它在题目陈述中给出。变量与之前定义的 x 和 y 相同。目标函数始终是线性的，因此不包含变量的平方或乘积。
In the CCEA IGCSE, you will not be asked to derive the objective function from a description; it is always provided. Your job is to evaluate it at the correct vertices to find the optimum. Sometimes you might see different letters like R for revenue or C for cost – just treat the letter as a label for the value to be optimised.
在 CCEA IGCSE 考试中，不会要求你从描述中推导目标函数；它总是直接给出。你的工作是在正确的顶点处对其求值以找到最优解。有时你可能会看到不同的字母，如表示收入的 R 或表示成本的 C——只需将该字母视为待优化值的标签即可。
9. Optimisation: Maximising or Minimising | 优化：求最大值或最小值
To find the maximum or minimum value of the objective function, create a table of the vertices and their corresponding objective function values. Substitute the x and y coordinates into the objective expression. The largest value gives the maximum; the smallest gives the minimum.
要找出目标函数的最大值或最小值，可建立一个顶点表及其对应的目标函数值。将 x 和 y 坐标代入目标表达式。最大值即最大，最小值即最小。
| Vertex (x, y) | P = 5x + 3y |
|---|---|
| (0, 0) | 0 |
| (20, 0) | 100 |
| (10, 15) | 95 |
| (0, 20) | 60 |
From the table, the maximum P is 100 at (20, 0). Always state your final answer clearly in the context of the problem: “The maximum profit is £100, achieved by making 20 chairs and 0 tables.” CCEA examiners expect this conclusion.
从上表可知，最大 P 值为 100，出现在 (20, 0) 处。始终在问题背景下清楚写出最终答案：”最大利润为 £100，通过生产 20 把椅子和 0 张桌子实现。” CCEA 考官期望看到这样的结论。
10. Integer Solutions in Practical Problems | 实际问题中的整数解
Sometimes the optimal vertex gives non-integer coordinates, but the context demands whole numbers (e.g., you cannot produce 2.5 cars). In such cases, you must search for the optimal integer solution near that vertex. Use the method of testing integer points within the feasible region, often on the boundary lines close to the optimal non-integer vertex.
有时最优顶点的坐标不是整数，但现实情境要求整数（例如，你无法生产 2.5 辆车）。在这种情况下，你必须寻找该顶点附近的最优整数解。方法是测试可行域内靠近最优非整数顶点的边界线上的整数点。
For example, if the optimum is (15.3, 8.6), test (15, 8), (15, 9), (16, 8), and (16, 9) provided they satisfy all constraints. Calculate the objective function for each and select the best one. This process is specifically examined in CCEA and you must show all candidate points checked.
例如，如果最优解是 (15.3, 8.6)，则测试 (15, 8), (15, 9), (16, 8) 和 (16, 9) 等点，只要它们满足所有约束。计算每个点的目标函数并选择最佳点。CCEA 专门考查这一过程，你必须展示所有检查过的候选点。
11. Common Mistakes to Avoid | 应避免的常见错误
One of the most frequent errors is shading the wrong side of a line. Always use a test point and double-check. Another is omitting the x ≥ 0 and y ≥ 0 constraints, which can drastically change the feasible region and lose easy marks. Also, be careful when solving simultaneous equations – an arithmetic slip in the vertices can ruin the entire optimisation.
最常见的错误之一是涂错直线的一侧。始终使用测试点并仔细检查。另一个是遗漏 x ≥ 0 和 y ≥ 0 的约束，这会极大地改变可行域并丢失容易得到的分数。此外，在解联立方程时要小心——顶点处的计算失误可能毁掉整个优化过程。
Students sometimes evaluate the objective function at the wrong set of points, forgetting to consider all vertices of the feasible region. Remember that the maximum or minimum might occur at an intersection with an axis, not only at the intersections of two slanting lines. Finally, failing to interpret the result in context or not providing units can result in the final answer mark being withheld.
学生有时会在错误的点集上计算目标函数，忘记考虑可行域的所有顶点。请记住，最大值或最小值可能出现在与坐标轴的交点处，而不仅仅是两条斜线的交点。最后，未能结合具体情境解释结果或不提供单位可能导致最终答案得分被扣。
12. Exam Strategy and Tips | 应试策略与技巧
In the exam, start by reading the whole question and noting the specific instructions for shading. Use graph paper if provided, and always use a ruler for straight lines. Make a neat, large graph so that your feasible region and vertices are clear. Show all algebraic working for intersections, even if you can read them from the graph.
在考试中，先通读全题，并留意关于涂色的具体指示。如果提供坐标纸，请使用坐标纸，并始终用直尺画直线。绘制整洁、较大的图形，以便清晰展示可行域和顶点。即使你可以从图中读出交点，也要写出求交点的代数运算过程。
Label the axes clearly, mark the feasible region with an R, and list all vertices in a table. If an integer solution is needed, demonstrate the search systematically. Manage your time well – linear programming questions can be lengthy, but the method is very structured. With practice, you can secure high marks consistently.
清晰地标注坐标轴，用 R 标出可行域，并将所有顶点列表呈现。如果需要整数解，系统地展示搜索过程。合理安排时间——线性规划问题可能很长，但方法非常结构化。通过练习，你能持续获得高分。
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