Linear Programming: Key Exam Points Explained | 线性规划：考点精讲

📚 Linear Programming: Key Exam Points Explained | 线性规划：考点精讲

Linear programming is a powerful mathematical technique used to find the best possible outcome, such as maximum profit or minimum cost, under a set of linear constraints. For IB and CIE mathematics students, mastering the graphical method and understanding how to formulate and interpret problems is essential for success in this topic. This article covers the key examination points, step-by-step solution strategies, and common pitfalls, supported by worked examples and exam-style advice.

线性规划是一种强大的数学技术，用于在一组线性约束下寻找最优结果，例如最大利润或最小成本。对于IB和CIE数学学生来说，掌握图解法并理解如何建模和解读问题是掌握这个主题的关键。本文涵盖了核心考点、逐步解题策略和常见误区，并配以例题和考试导向的建议。

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

Linear programming (LP) is a method for determining the optimal value (maximum or minimum) of a linear function, called the objective function, subject to a set of constraints expressed as linear inequalities. It is widely applied in business, economics, and engineering to allocate limited resources efficiently.

线性规划（LP）是一种确定线性函数（目标函数）在满足一系列线性不等式约束下的最优值（最大值或最小值）的方法。它广泛应用于商业、经济和工程领域，以高效分配有限资源。

The standard form of a linear programming problem involves decision variables, a linear objective function, and a system of linear constraints, often including non-negativity restrictions (e.g., x ≥ 0, y ≥ 0).

线性规划问题的标准形式包括决策变量、线性目标函数以及线性约束的系统，通常包含非负限制（例如 x ≥ 0，y ≥ 0）。

2. Formulating a Linear Programming Problem | 线性规划问题的建模

Formulation is the process of translating a real-world scenario into mathematical language. You must identify the decision variables (e.g., number of products A and B), write the objective function (e.g., profit = 5x + 8y), and list all constraints (e.g., material limits, labour hours) as linear inequalities.

建模是将现实场景转化为数学语言的过程。你必须识别决策变量（例如产品A和B的数量），写出目标函数（例如利润 = 5x + 8y），并将所有约束（例如原材料限制、工时）列为线性不等式。

In IB and CIE exams, questions often provide a description. Common steps include: define x and y clearly, derive inequalities from limiting factors, and state objective function. Always include non-negativity constraints unless otherwise stated.

在IB和CIE考试中，题目通常会提供一段描述。常见步骤包括：清晰地定义x和y，从限制条件推导不等式，并陈述目标函数。除非另有说明，始终包含非负约束。

Constraint description 约束描述	Inequality 不等式
At most 20 units of x	x ≤ 20
Total production ≤ 80	x + y ≤ 80
Labour hours: 2x + 3y ≤ 180	2x + 3y ≤ 180

3. Graphical Method: The Feasible Region | 图解法：可行域

The graphical method is used when there are two decision variables. Plot each constraint as a straight line on the coordinate plane, then shade the region that satisfies the inequality. The feasible region is the intersection of all these half-planes, typically a convex polygon.

当有两个决策变量时使用图解法。在坐标平面上将每个约束画成直线，然后给满足不等式的区域涂上阴影。可行域是所有半平面的交集，通常是一个凸多边形。

To graph an inequality like 3x + 2y ≤ 120, first draw the line 3x + 2y = 120. Choose a test point, e.g., (0,0); if it satisfies the inequality, shade the half-plane containing the origin. Label the feasible region clearly.

要画出 3x + 2y ≤ 120 这样的不等式，首先画出直线 3x + 2y = 120。选择一个测试点，例如 (0,0)；如果满足不等式，就涂上包含原点的半平面。清晰地标记可行域。

4. Objective Function and Optimal Solutions | 目标函数与最优解

The objective function, e.g., P = 4x + 5y, represents the quantity to be maximised or minimised. In the graphical approach, the optimum occurs at a vertex (corner point) of the feasible region, unless the objective function is parallel to a constraint line, leading to multiple optimal solutions.

目标函数，例如 P = 4x + 5y，代表需要最大化或最小化的量。在图解法中，最优值出现在可行域的顶点（角点），除非目标函数与某条约束线平行，从而导致多个最优解。

Slide a line of constant profit (isoprofit line) parallel to itself until it just touches the feasible region. The last point of contact gives the maximum; the first point of contact gives the minimum, if minimising.

将一条等利润线（等值线）平行移动，直到它与可行域刚好接触。最后一个接触点给出最大值；如果是求最小值，第一个接触点就是最优解。

5. Finding the Optimal Point: Corner Point Principle | 寻找最优解：顶点原理

The corner point principle states that for a linear programming problem with a bounded feasible region, the optimal value of the objective function will be attained at one of the vertices. You can evaluate the objective function at every corner point to find the optimal value.

顶点原理指出，对于一个有界可行域的线性规划问题，目标函数的最优值会在某个顶点取得。你可以在每一个角点计算目标函数的值，从而找到最优值。

To find corner points, solve the simultaneous equations of the intersecting lines. For example, intersection of x + y = 50 and 2x + y = 80 gives x = 30, y = 20. Then evaluate P = 3x + 4y at each vertex; the largest (or smallest) value is the optimum.

为了找到角点，解相交直线的方程组。例如，x + y = 50 和 2x + y = 80 的交点为 x = 30，y = 20。然后在每个顶点计算 P = 3x + 4y 的值；最大（或最小）值即为最优。

6. Maximization and Minimization Problems | 最大化与最小化问题

Maximisation problems seek the highest possible value of the objective function, such as profit or output. The feasible region’s boundaries are determined by constraints like resources, and the optimal solution is the vertex farthest along the direction of increasing objective function.

最大化问题寻找目标函数的最大可能值，如利润或产量。可行域的边界由资源等约束决定，最优解是沿着目标函数增大方向最远的顶点。

Minimisation problems, like cost minimisation, often have a minimum at the vertex closest to the origin or along a boundary when sliding the objective function line inward. Always check the direction of optimisation: for P = ax + by, maximise means moving the line ax + by = k in the direction of increasing k.

最小化问题，如成本最小化，通常在向内移动目标函数线时，最优解出现在最靠近原点的顶点或边界上。始终检查优化方向：对于 P = ax + by，最大化意味着沿着 k 增大的方向移动直线 ax + by = k。

7. Special Cases: No Solution, Unbounded, Multiple Optima | 特殊情况：无解、无界、多重最优解

A problem has no feasible solution if the constraints are contradictory, e.g., x ≥ 10 and x ≤ 5. The feasible region is empty. In exams, recognise this from an impossible intersection.

如果约束条件相互矛盾，例如 x ≥ 10 且 x ≤ 5，则问题没有可行解。可行域为空。在考试中，要从不可能的交集来识别这种情况。

An unbounded feasible region occurs when constraints do not enclose a finite area, and the objective function can increase or decrease indefinitely. Maximising in an unbounded region may not be possible; exam questions usually ensure the optimum is finite.

当约束条件没有围成有限区域，并且目标函数可以无限增大或减小时，就会出现无界可行域。在无界区域内最大化可能无法求得有限解；考试题目通常确保最优解是有限的。

Multiple optimal solutions arise when the objective function line is parallel to one of the binding constraint lines. In such cases, every point on that edge of the feasible region is optimal. State the range of solutions or two extreme optimal vertices.

当目标函数线与其中一条有效约束线平行时，会出现多个最优解。在这种情况下，可行域该边上的每个点都是最优的。要陈述解的范围或两个极值最优顶点。

8. Integer Linear Programming | 整数线性规划

When decision variables must be whole numbers (e.g., number of cars, people), the problem becomes an integer programming problem. Graphically, the optimal integer solution may not simply be the rounded continuous optimum; you must test integer points near the optimal vertex.

当决策变量必须是整数时（例如汽车、人数），问题就变成了整数规划问题。在图解法中，最优整数解可能并非直接取连续最优解的四舍五入值；你必须测试最优顶点附近的整数点。

IB and CIE may ask for the optimal integer coordinates within the feasible region. List all integer lattice points near the boundary and evaluate the objective function at each to find the best integer solution. Be careful: rounding each coordinate down or up may leave the feasible region.

IB和CIE可能要求找出可行域内的最优整数坐标。列出边界附近的所有整数格点，分别计算目标函数的值，以找到最佳整数解。注意：对每个坐标向下或向上取整可能会超出可行域。

9. Common Pitfalls and Exam Tips | 常见错误与应考提示

Many students lose marks by incorrectly shading the feasible region. Always test with (0,0) and shade the side that satisfies the inequality. Label lines with equations and indicate the feasible region clearly, often marked as ‘R’.

许多学生因错误地涂阴影表示可行域而丢分。始终用 (0,0) 检验，并涂上满足不等式的半平面。用方程标记直线，并清楚地标明可行域，通常标记为“R”。

Forgetting non-negativity constraints (x ≥ 0, y ≥ 0) is a frequent error. Unless stated otherwise, variables represent quantities that cannot be negative. Include them to ensure a realistic feasible region.

忘记非负约束（x ≥ 0, y ≥ 0）是常见错误。除非另有说明，变量代表的数量不能为负。要包含它们以确保现实的可行域。

When using the corner point method, list all vertices and calculate objective values correctly. Double-check coordinates by solving simultaneous equations carefully, and present the final solution in the context of the problem, with correct units.

使用顶点法时，列出所有顶点并正确计算目标函数值。通过仔细解方程组复核坐标，并在问题背景下给出最终解，附带正确的单位。

10. Worked Example | 例题解析

A factory produces two types of lamps: standard (x) and deluxe (y). Profit per unit is £40 for standard and £60 for deluxe. Production constraints: x ≤ 200, y ≤ 150, and 2x + 3y ≤ 480. Non-negativity: x ≥ 0, y ≥ 0. Find the maximum profit.

某工厂生产两种灯具：标准型 (x) 和豪华型 (y)。每台标准型利润为40英镑，豪华型为60英镑。生产约束：x ≤ 200，y ≤ 150，以及 2x + 3y ≤ 480。非负：x ≥ 0，y ≥ 0。求最大利润。

Step 1: Plot lines x=200 (vertical), y=150 (horizontal), and 2x+3y=480. Intersections: (0,0), (0,150), (200,0), (200, 80/3 ≈ 26.67) from 2(200)+3y=480 → y=26.67, but must check intersection with y=150: 2x+450=480 → x=15, so (15,150). Feasible region vertices: (0,0), (0,150), (15,150), (200,26.67), (200,0). Since x and y must be integers (lamps), we test integer points: (15,150), (16,149), (200,26) etc. Profit function P = 40x + 60y.

步骤1：画出直线 x=200（垂直线）、y=150（水平线）和 2x+3y=480。交点：(0,0)、(0,150)、(200,0)、(200, 80/3≈26.67) 来自 2(200)+3y=480 → y=26.67，但需检查与 y=150 的交点：2x+450=480 → x=15，因此 (15,150)。可行域顶点：(0,0)、(0,150)、(15,150)、(200,26.67)、(200,0)。由于 x 和 y 必须是整数（灯具），我们测试整数点：(15,150)、(16,149)、(200,26) 等。利润函数 P = 40x + 60y。

Step 2: Evaluate P at integer vertices: P(0,150)=9000; P(15,150)=40(15)+60(150)=600+9000=9600; P(200,26)=40(200)+60(26)=8000+1560=9560; P(200,0)=8000; P(0,0)=0. Check nearby integer points: (16,150) would give x=16>15 but y=150, does it satisfy 2x+3y ≤ 480? 2(16)+450=482 > 480, not feasible. Thus maximum is at (15,150) with profit £9600.

步骤2：计算整数顶点处的 P 值：P(0,150)=9000；P(15,150)=40×15+60×150=600+9000=9600；P(200,26)=40×200+60×26=8000+1560=9560；P(200,0)=8000；P(0,0)=0。检查附近的整数点：(16,150) 其中 x=16>15 但 y=150，是否满足 2x+3y ≤ 480？2(16)+450=482 > 480，不可行。因此最大利润在 (15,150)，利润为 9600 英镑。

11. Sensitivity Analysis and the Objective Line | 敏感性分析与目标函数线

In some advanced settings, students might be asked to determine the range of profit coefficients for which the current optimal solution remains optimal. This is known as sensitivity analysis. Graphically, it involves the slope of the objective function line relative to the active constraint lines.

在某些高级情境中，学生可能会被要求确定当前最优解保持最优的利润系数范围。这被称为敏感性分析。从图形上看，这涉及目标函数线的斜率相对于有效约束线的斜率。

For P = ax + by, the slope is -a/b. The optimal vertex remains optimal as long as this slope lies between the slopes of the two binding constraint lines at that vertex. For example, if the binding constraints have slopes -1 and -2/3, then the objective slope must be between these values, which restricts the ratio a/b.

对于 P = ax + by，斜率为 -a/b。只要该斜率位于该顶点处两条有效约束线的斜率之间，该最优顶点就保持最优。例如，如果有效约束的斜率为 -1 和 -2/3，则目标函数斜率必须介于这些值之间，从而限制 a/b 的比值。

This concept is examined in some CIE further pure topics or IB HL applications. Practice using the slope condition to set up inequalities and find allowable ranges for coefficients.

这个概念在部分CIE纯数进阶或IB HL应用中有考查。练习通过斜率条件建立不等式，求出系数的允许范围。

12. Summary and Final Advice | 总结与最后建议

Linear programming problems in IB and CIE exams reward careful graphical work, accurate solving of simultaneous equations, and clear presentation. Focus on shading the correct region, identifying all corner points, and interpreting the solution in context. For integer solutions, always verify feasibility after rounding. With systematic practice, this topic can become one of the most secure marks on your paper.

IB和CIE考试中的线性规划题目对细致的绘图、准确解方程组和清晰的表述给予奖励。重点在于正确涂出可行域、找出所有角点，并根据背景解读解。对于整数解，在取整后始终要验证可行性。通过系统练习，本专题可以成为试卷上最稳妥的得分点之一。

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Linear Programming: Key Exam Points Explained | 线性规划：考点精讲