📚 IB Math: Linear Programming Key Points | IB 数学：线性规划 考点精讲
Linear programming is a powerful optimization tool that sits at the heart of many IB Mathematics problems, especially in the Applications and Interpretation course, and occasionally in Analysis and Approaches. It enables you to model real-world constraints and find maximum or minimum values of a linear function over a polygonal feasible region. In the IB exam, you can expect questions that require formulating inequalities, drawing feasible regions, identifying objective functions, testing vertices, interpreting integer solutions, and even exploring the effects of changing a constraint. This article systematically walks you through every essential concept, complete with examples and exam tips, to ensure you master linear programming for your IB assessment.
线性规划是 IB 数学（尤其是应用与解释课程，分析与方法课程也会涉及）中一项强大的优化工具。它使你能够将现实约束建模并在线性可行域上寻找最大值或最小值。在 IB 考试中，你可以预期到需要列出不等式、绘制可行域、确定目标函数、检验顶点、解释整数解，甚至探索约束变化的影响的题目。本文系统地带你走过每一个核心概念，配以例题和备考提示，确保你为 IB 评估完全掌握线性规划。
1. What is Linear Programming? | 什么是线性规划？
Linear programming (LP) is a mathematical method for determining the best possible outcome (such as maximum profit or minimum cost) in a given model whose requirements are represented by linear relationships. The term “programming” here does not refer to computer code but to planning. An LP model consists of decision variables, a linear objective function to optimize, and a set of linear inequalities or equations known as constraints. In the IB syllabus, the focus is mainly on two-variable problems, which allow a graphical solution approach and a clear geometric interpretation of the feasible region.
线性规划（Linear Programming, LP）是一种数学方法，用于在一个由线性关系表示要求的给定模型中确定最佳结果（如最大利润或最小成本）。这里的“规划”一词不是指计算机编程，而是指计划。一个线性规划模型由决策变量、待优化的线性目标函数，以及一组称为约束条件的线性不等式或等式组成。在 IB 大纲中，主要关注两个变量的问题，这样可以采用图解法，并对可行域有清晰的几何解释。
2. Key Components: Decision Variables, Objective Function, Constraints | 关键组成部分：决策变量、目标函数、约束条件
In any linear programming problem, you start by identifying the decision variables, often denoted as x and y (or x₁ and x₂). These represent the quantities you can control, such as the number of units to produce. The objective function is a linear expression like P = 3x + 2y (to maximize) or C = 5x + 4y (to minimize). Constraints are linear inequalities that restrict the values of the variables. Typical constraints include resource limitations (e.g., 2x + y ≤ 100), non‑negativity (x ≥ 0, y ≥ 0), and market demands (e.g., x ≤ 50). All constraints together define the feasible region, which is the set of all points that satisfy every constraint simultaneously.
在任何线性规划问题中，你首先要识别决策变量，通常用 x 和 y（或 x₁ 和 x₂）表示。它们代表你可以控制的量，例如要生产的件数。目标函数是一个线性表达式，比如 P = 3x + 2y（求最大化）或 C = 5x + 4y（求最小化）。约束条件是限制变量值域的线性不等式。典型的约束包括资源限制（如 2x + y ≤ 100）、非负条件（x ≥ 0, y ≥ 0）以及市场需求（如 x ≤ 50）。所有约束一起定义了可行域，即同时满足每一个约束的所有点的集合。
3. Formulating a Linear Programming Problem | 构建线性规划问题
The first critical skill in IB exams is translating a word problem into a precise mathematical model. Read the problem carefully to define the variables explicitly, for example: “Let x be the number of units of Product A, and y the number of units of Product B.” e profit or minimise cost. Next, write each constraint as a linear inequality. Look for keywords like “at most,” “at least,” “a maximum of,” “not more than,” re all coefficients are constants and the relationships are linear. Always include the non‑negativity constraints x ≥ 0, y t explicitly allows negative values (which is extremely rare in standard problems). Double‑check that you have not omitted any conditions mentioned in the problem.
IB 考试中第一个关键技能是将文字题转化为精确的数学模型。仔细阅读题目，明确地定义变量，例如：“令 x 为产品 A 的件数，y 为产品 B 的件数。”然后提取目标：最大化利润或最小化成本。接着，将每个约束写成线性不等式。留意“至多”、“至少”、“最多”、“不超过”等关键词，它们指示 ≤ 或 ≥。确保所有系数是常量且关系是线性的。除非上下文明确允许负值（这在标准问题中极罕见），始终要包括非负约束 x ≥ 0, y ≥ 0。再次检查你是否遗漏了题目中提到的任何条件。
4. Graphing the Feasible Region | 绘制可行域
For two‑ ion is a polygon (possibly unbounded) on the Cartesian plane. To graph it, treat each inequality as an equality to draw its boundary line. Choose a suitable scale so that all intercepts and intersections are clearly visible. Use a dashed line for strict inequalities ( < or > ) and a solid line for non‑ that satisfies each inequality, typically by testing the origin (0,0) if it does not lie on the line. The overlapping shaded area, which satisfies all constraints, is the feasible region. Label the axes, the boundary lines with their equations, and the vertices with their coordinates. In IB exams, neat and accurate graphing is essential, often worth several marks.
对于两个变量的问题，可行域是直角坐标平面上的一个多边形（可能无界）。要绘制它，将每个不等式当作等式处理来画出边界线。选择合适的比例尺，使所有截距和交点都清晰可见。严格不等式（ < 或 > ）使用虚线，非严格不等式（ ≤ 或 ≥ ）使用实线。然后给满足每个不等式的区域涂上阴影，通常通过检验原点 (0,0) 是否在线外来判定。所有约束条件均满足的叠合阴影区域就是可行域。给坐标轴、带方程的边界线以及带坐标的顶点加上标签。在 IB 考试中，清晰准确的作图至关重要，通常值好几分。
5. Finding the Optimal Solution – Corner Point Method | 求最优解——顶点法
According to the fundamental theorem of linear programming, if a maximum or minimum value of the objective function exists, it occurs at one of the vertices (corner points) of the feasible region, or along an entire edge of the region. The corner point method involves evaluating the objective function at each vertex. First, find the coordinates of all vertices by solving the simultaneous equations of intersecting boundary lines. Then substitute each vertex into the objective function. The vertex that gives the highest value (for a maximization problem) or the lowest value (for minimization) is the optimal solution. If two adjacent vertices give the same optimal value, every point on the edge connecting them is also optimal, which is a case of multiple optimal solutions.
根据线性规划的基本定理，如果目标函数的最大值或最小值存在，它必然出现在可行域的顶点（角点）处，或沿着区域的一整条边出现。顶点法就是计算目标函数在每个顶点处的值。首先，通过解边界线交点对应的联立方程求出所有顶点的坐标。然后将每个顶点代入目标函数。给出最大值（对于最大化问题）或最小值（对于最小化问题）的顶点就是最优解。如果两个相邻顶点给出相同的最优值，那么连接这两点的边上的每个点都是最优解，这就是多重最优解的情况。
6. Working with the Objective Function Line | 使用目标函数线
An alternative, geometric approach is to draw an iso‑profit or iso‑cost line. For a maximization problem, set the objective function equal to some constant k, e.g., 3x + 2y = 60, and draw this line. Then slide this line parallel to itself in the direction of increasing (or decreasing) k while keeping it within the feasible region. The last point the line touches before leaving the feasible region is the optimal vertex. This method offers a visual check and is helpful when you need to justify why a vertex is optimal without computing all vertex values. In IB exams, you may be asked to sketch such a line, especially when partial credit is available for reasoning.
另一种几何方法是绘制等利润线或等成本线。对于最大化问题，将目标函数设为一个常数 k，如 3x + 2y = 60，并画出这条直线。然后，平行移动这条线，使其随着 k 的增加（或减小）方向移动，同时保持在可行域内。该直线在离开可行域前最后接触的点即为最优顶点。这种方法提供了直观的验证，当你需要证明某顶点为何最优而不必计算所有顶点的函数值时很有帮助。在 IB 考试中，你可能需要绘出这样一条线，尤其是在推理部分可获得部分分数时。
7. Special Cases: Unbounded, Infeasible, and Redundant Constraints | 特殊情况：无界、不可行与冗余约束
Not all linear programming problems have a unique finite solution. If the feasible region is unbounded in the direction of optimization, the objective function can increase or decrease without limit, and the problem has no bounded optimal solution. In contrast, if the constraint set is contradictory (e.g., x ≥ 10 and x ≤ 5), the feasible region is empty, making the problem infeasible. Sometimes a constraint does not affect the feasible region at all; it is called a redundant constraint because it is automatically satisfied by all points that meet the other constraints. IB exam questions may ask you to identify which constraint is redundant or to explain why no feasible solution exists.
并非所有的线性规划问题都有唯一的有限解。如果可行域在优化方向上无界，目标函数可以无限增大或减小，那么该问题就没有有界最优解。相反，如果约束集合相互矛盾（例如 x ≥ 10 而 x ≤ 5），可行域为空，问题即为不可行。有时某个约束完全不影响可行域；它被称为冗余约束，因为所有满足其他约束的点都自动满足它。IB 考题可能会让你指出哪条约束是冗余的，或解释为什么不存在可行解。
8. Integer Solutions in Context | 实际问题中的整数解
Many real‑world linear programming problems require decision variables to be integers—you cannot produce 2.7 cars or hire 3.5 workers. The IB syllabus typically handles this by first finding the optimal continuous solution and then testing nearby integer points within the feasible region. You must ensure the integer points still satisfy all constraints. The optimal integer solution is not always obtained by simply rounding the continuous coordinates; sometimes you need to check a small grid of integer points around the optimum vertex. A table of candidate integer points with their objective values is an excellent way to present your working. Always interpret your integer solution in the context of the problem, for example: “Produce 2 units of Product A and 5 units of Product B for a maximum profit of $440.”
很多现实中的线性规划问题要求决策变量为整数——你不可能生产 2.7 辆汽车或雇用 3.5 个工人。IB 大纲通常的处理方式是先求出连续最优解，然后在可行域内检验其附近的整数点。你必须确保这些整数点仍然满足所有约束。最优整数解并不总是通过简单地将连续坐标四舍五入得到；有时你需要检查最优顶点周围的一个小的整数点网格。用一张列出候选整数点及其目标函数值的表格是展现解题过程的好方法。始终在题目的背景下解释你的整数解，例如：“生产 2 件产品 A 和 5 件产品 B，可获得最大利润 440 美元。”
9. Sensitivity Analysis: Shadow Price and Ranges | 敏感性分析：影子价格与范围
Sensitivity analysis examines how changes in the constraints or objective coefficients affect the optimal solution. The shadow price (or dual value) of a resource is the improvement in the objective function per extra unit of that resource, within a certain allowable range. In a two‑variable problem solved graphically, you can find the shadow price by increasing the right‑hand side of a binding constraint by 1 unit and recalculating the optimal value of the objective function. The allowable range for a constraint is the interval in which the shadow price remains valid. Similarly, for an objective coefficient, you can find the range of optimality—the limits within which the current optimal vertex remains optimal. IB questions may ask you to interpret what a shadow price means in context or to analyse the effect of a small change.
敏感性分析探讨约束条件或目标系数的变化如何影响最优解。某项资源的影子价格（或对偶值）是指在某一允许范围内每增加一单位该资源所带来的目标函数的增量。在图形法求解的两变量问题中，可以通过将一个绑定约束的右端项增加 1 单位并重新计算目标函数的最优值来求出影子价格。某一约束的允许范围是影子价格保持有效的区间。类似地，对于目标函数的系数，你可以求出最优性范围——即当前最优顶点保持最优性的系数变化极限。IB 题目可能要求你解释影子价格在题意中的含义，或分析一个小变动产生的影响。
10. Using Technology: GDC and Software in IB | 技术应用：IB 中的图形计算器与软件
The IB encourages the use of a graphic display calculator (GDC) or approved software to solve linear programming problems, especially in papers where technology is allowed. You can graph inequalities, find intersection points, and even use built‑in LP solvers (available on some calculators) to obtain optimal solutions directly. However, you must still be able to show your working: state the decision variables, write down the full mathematical model, and interpret the results. When using technology, carefully transfer the graphs to your answer sheet if required, or provide screenshots and clear explanations. Familiarize yourself with your GDC’s inequality graphing mode and the “intersect” function to find vertices efficiently.
IB 鼓励使用图形计算器（GDC）或经批准的软件来求解线性规划问题，尤其是在允许使用技术的考卷中。你可以绘制不等式图形、求交点，甚至使用内置的线性规划求解器（部分计算器有此功能）直接获得最优解。但是，你仍需展示解题过程：说明决策变量，写出完整的数学模型，并解释结果。使用技术时，如果需要，应仔细地将图形转绘到答题纸上，或提供屏幕截图和清晰的解释。熟悉你的 GDC 的不等式绘图模式以及“求交”功能，以便高效地找到顶点。
11. Common IB Exam Pitfalls | IB 考试常见误区
Students often lose marks by forgetting to state non‑negativity constraints or misinterpreting “at least” and “at most.” Another frequent mistake is drawing the wrong boundary line – always check the intercepts and the direction of the inequality shading. Some candidates round integer solutions incorrectly and then fail to verify that the rounded point still lies in the feasible region. Additionally, when a question involves multiple optimal solutions, failing to mention that all points on the edge are optimal can cost valuable marks. Finally, ensure your final answer is given in the context of the problem, with proper units, and that you clearly state both the values of the variables and the optimal objective value.
学生常因忘记写出非负约束，或误解“至少”与“至多”的意思而丢分。另一个常见错误是画错边界线——务必检查截距和不等式涂色的方向。有些考生对整数解进行错误的舍入，然后没有验证舍入点是否仍在可行域内。此外，当题目涉及多重最优解时，若未提及边上的所有点都是最优解，会损失宝贵的分数。最后，确保你的最终答案是在问题背景下给出的，带有适当的单位，并清楚地同时说明变量的取值和最优目标值。
12. Summary and Exam Strategy | 总结与应考策略
To ace linear programming in the IB exam, follow a structured approach: define variables, build the model, graph accurately, determine vertices, evaluate the objective function, check for integer requirements, and interpret the results. Practise both with and without technology, because Paper 1 often requires graphical analysis by hand, while Paper 2 may involve GDC use. Regularly review the key vocabulary (feasible region, objective function, constraints, shadow price) and be ready to explain your reasoning in clear, logical sentences. Remember that while the algebra is straightforward, the conceptual depth—understanding what the solution means in context—is what distinguishes top‑level answers.
要在 IB 考试中攻克线性规划，请遵循结构化的步骤：定义变量、建立模型、精确作图、确定顶点、计算目标函数、检查整数要求、解释结果。既要借助技术练习，也要练习手工作图，因为试卷一常要求手工图形分析，而试卷二可能会用到 GDC。定期复习关键术语（可行域、目标函数、约束条件、影子价格），并准备好用清晰、有逻辑的语句来解释你的推理。请记住，虽然代数运算本身并不复杂，但概念深度——理解解在题意中的意义——才是区分高水平答案的关键。
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