📚 Linear Programming: Key Points for IB & WJEC Mathematics | 线性规划：IB & WJEC 数学考点精讲
Linear programming is a powerful optimisation technique used to find the best outcome—maximum profit or minimum cost—under a set of linear constraints. In IB and WJEC Mathematics, this topic bridges algebra, coordinate geometry and real-world decision‑making. Mastering it requires not only mechanical graph‑plotting skills but also a deep understanding of feasible regions, objective functions and special cases such as integer solutions or unbounded problems.
线性规划是一种强大的最优化方法，用于在一组线性约束条件下寻找最佳结果——如最大利润或最小成本。在 IB 和 WJEC 数学中，这一主题连接了代数、坐标几何与现实决策。要掌握它，不仅需要熟练的作图技巧，还需要深入理解可行域、目标函数以及特殊情况（如整数解或无界问题）。
1. What is Linear Programming? | 什么是线性规划？
Linear programming (LP) deals with maximising or minimising a linear objective function subject to linear inequalities (constraints). The variables are usually non‑negative, and the goal is to find the optimal solution that satisfies all conditions simultaneously.
线性规划涉及在多个线性不等式（约束条件）的限制下，最大化或最小化一个线性目标函数。变量通常非负，目标是找到同时满足所有条件的最优解。
2. Formulating the Problem | 建立数学模型
Start by defining the decision variables, typically x and y. Express the objective function (e.g. P = 3x + 5y) and translate each restriction into a linear inequality such as 2x + y ≤ 20 or x ≥ 0, y ≥ 0. Check that all units and quantities are consistent.
首先定义决策变量，通常为 x 和 y。写出目标函数（例如 P = 3x + 5y），并将每个限制条件转化为线性不等式，如 2x + y ≤ 20 或 x ≥ 0、y ≥ 0。确保所有单位和数量是一致的。
3. Graphing Linear Inequalities | 绘制线性不等式
Draw the boundary line (solid for ≤ or ≥, dashed for < or >) by finding intercepts or rearranging to y = mx + c. Test a point not on the line—usually (0,0)—to decide which side to shade. The intersection of all shaded regions forms the feasible region.
画出边界直线（≤ 或 ≥ 用实线，< 或 > 用虚线），可通过截距或改写为 y = mx + c 来绘制。选取直线外的一点（通常为 (0,0)）进行检验，确定该将哪一侧涂色。所有着色区域的交集就是可行域。
4. Identifying the Feasible Region | 确定可行域
The feasible region is the set of all points that satisfy every constraint simultaneously. It is often a convex polygon. In an unbounded problem, the region extends infinitely in some direction; extra care is then needed for optimisation.
可行域是所有同时满足全部约束条件的点构成的集合，通常是一个凸多边形。在无界问题中，区域会向某个方向无限延伸，此时需要特别注意如何求最优解。
5. The Corner‑Point Principle | 顶点原理
For a linear objective function over a closed bounded feasible region, the optimal value always occurs at a vertex (corner point) of the region. Therefore, list all vertices—solving simultaneous equations for their coordinates—and evaluate the objective function at each.
在有界闭可行域上的线性目标函数，其最优值总是在可行域的顶点（角点）处取得。因此，列出所有顶点坐标（通过解联立方程求出），并分别计算每个顶点的目标函数值。
6. Finding the Optimal Solution | 寻找最优解
Compute the objective value at every vertex. For maximisation, pick the largest value; for minimisation, the smallest. If two adjacent vertices give the same optimal value, all points on the edge between them are also optimal (multiple optimal solutions).
计算每个顶点的目标函数值。对于最大化问题，取最大值；对于最小化问题，取最小值。如果相邻两个顶点得到相同的最优值，那么这两点连线上的所有点都是最优解（多重最优解）。
7. Using Sliding Line Method | 使用滑动直线法
Draw a line representing the objective function (e.g. 3x + 5y = 0) and slide it parallel to itself. The last point of contact with the feasible region in the direction of improvement gives the optimum. This visual method is especially helpful in exam sketches.
画一条代表目标函数的直线（如 3x + 5y = 0），并将其平行移动。与可行域在改善方向上最后接触的点即给出最优解。这种直观方法在考试作图时特别有用。
8. Interpreting Shadow Values (Dual Variables) | 解释影子价格（对偶变量）
In resource‑based problems, the shadow value indicates how much the objective function would improve if one more unit of a scarce resource were available. It is found by noting the change in optimum when a constraint’s right‑hand side is increased by 1, within the allowable range.
在资源分配问题中，影子价格表示增加一单位稀缺资源时目标函数能改进多少。可在一个约束条件的右端项增加 1（在允许范围内），观察最优值的变化来求得。
9. Integer Programming Essentials | 整数规划要点
When variables must be whole numbers (e.g. numbers of cars or people), the continuous optimum may not be integer. Find the nearby integer points inside the feasible region and test them to get the best integer solution—this is often done by inspection or enumeration.
当变量必须为整数时（如汽车数或人数），连续最优解可能不是整数。应在可行域内找出附近的整点并逐一检验，以得到最佳整数解——通常通过观察或枚举来完成。
10. Dealing with Unbounded or Inconsistent Regions | 处理无界区域或无解情况
An unbounded feasible region may not have a maximum (or minimum) if the objective function can increase (or decrease) indefinitely. If constraints contradict each other, the feasible region is empty and the problem has no solution. Always check the direction of optimisation.
如果目标函数可以无限增大（或减小），无界可行域可能没有最大值（或最小值）。若约束条件互相矛盾，可行域为空，则问题无解。务必检查优化方向。
11. Sensitivity Analysis and Range of Optimality | 敏感性分析与最优区间
After finding an optimum, consider how changes to coefficients in the objective function or to the right‑hand side constants affect the solution. The range of optimality specifies the interval within which the current optimal vertex remains optimal, while shadow prices remain valid within the allowable increase/decrease.
找到最优解后，考虑目标函数系数或右端常数的变化如何影响解。最优区间给出了当前最优顶点保持最优的系数变化范围，而影子价格在允许的增减范围内有效。
12. Common Exam Pitfalls | 考试常见误区
Mistakes often arise from shading the wrong side of a line, failing to include non‑negativity constraints, misreading the optimisation direction, or rounding integer solutions incorrectly. Always label axes, shading and vertices clearly, and double‑check calculations by substitution.
常见错误包括：在直线错误的一侧涂色、遗漏非负约束、搞混最大化或最小化方向、或整数解舍入不当。务必清晰标注坐标轴、阴影和顶点，并通过代入原方程来验算。
Published by TutorHao | Linear Programming Revision Series | aleveler.com
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