Locating the Optimal Point | 定位最优点

📚 Locating the Optimal Point | 定位最优点

In Edexcel A-Level Decision Mathematics, linear programming is used to find the best possible outcome, such as maximum profit or minimum cost, when resources are limited. ‘Locating the optimal point’ means identifying the exact coordinates in the feasible region where the objective function takes its optimum value. This article explains both the objective line sliding method and the vertex testing method, with worked examples and exam tips.

在 Edexcel A-Level 决策数学中，线性规划用于在资源受限时寻找最佳结果，例如最大利润或最小成本。’定位最优点’是指在可行域中确定目标函数取得最优值的准确坐标。本文介绍目标线平移法和顶点检验法，并提供例题与考试技巧。

1. The Linear Programming Problem | 线性规划问题

A linear programming problem involves decision variables, an objective function, and a set of linear constraints. The objective function is the expression we aim to maximise or minimise, such as profit P = 3x + 2y. The constraints are linear inequalities that represent limited resources, and non-negativity restrictions x ≥ 0, y ≥ 0 are usually assumed.

线性规划问题包含决策变量、目标函数和一组线性约束条件。目标函数是我们要最大化或最小化的表达式，例如利润 P = 3x + 2y。约束条件表示有限资源的线性不等式，通常还假设非负限制 x ≥ 0, y ≥ 0。

In Edexcel questions, the variables often stand for numbers of products or amounts of materials, so the final answer must make practical sense. You may be asked to find the optimal point, state the optimal value, or adjust the objective function to test different scenarios.

在 Edexcel 考题中，变量通常代表产品数量或材料用量，因此最终答案必须符合实际意义。你可能需要求最优点、写出最优值，或调整目标函数来测试不同情景。

2. Feasible Region and Constraints | 可行域与约束条件

Each linear constraint defines a half-plane. For example, x + y ≤ 10 includes every point on one side of the line x + y = 10. The feasible region is the intersection of all these half-planes. It is the set of all points that satisfy every constraint simultaneously.

每个线性约束定义了一个半平面。例如，x + y ≤ 10 包含直线 x + y = 10 一侧的所有点。可行域是所有半平面的交集，即同时满足所有约束条件的点的集合。

When drawing the feasible region, shade out excluded areas or clearly label the allowed side. In a well-posed problem, the feasible region is a convex polygon. Its vertices, also called corner points, are essential because the optimal value is found at one of them for a linear objective function.

绘制可行域时，应涂掉不满足的区域或清楚标出允许的一侧。在良好的问题中，可行域是一个凸多边形。其顶点（也称角点）非常关键，因为线性目标函数的最优值总是出现在某个顶点处。

3. Objective Function and Objective Line | 目标函数与目标线

Suppose the objective is to maximise P = ax + by. For a fixed value c, the line ax + by = c is called an objective line or profit line. Every point on this line gives the same value P = c. Different values of c produce parallel objective lines with the same gradient.

假设目标是最大化 P = ax + by。对于某个固定值 c，直线 ax + by = c 称为目标线或利润线。该线上每个点都给出相同的 P = c 值。不同的 c 值产生平行且梯度相同的目标线。

Objective line: ax + by = c

目标线：ax + by = c

The gradient of the objective line is -a/b (provided b ≠ 0). This gradient controls the direction in which the line slides as the objective value increases or decreases.

目标线的梯度为 -a/b（假设 b ≠ 0）。该梯度决定了随着目标值增大或减小，直线平移的方向。

4. The Vertex Theorem | 顶点定理

If the feasible region is bounded, the fundamental theorem of linear programming states that the maximum and minimum values

Published by TutorHao | A-Level Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）