📚 Workers’ Control: Linear Programming for Labour Planning | 工人控制：劳动力规划的线性规划方法
In Edexcel A-Level Mathematics, linear programming (LP) is introduced as a method for optimising a linear objective function subject to linear inequality constraints. The idea of workers' control can be translated into a mathematical question: how should a manager allocate a fixed number of workers, or worker-hours, between competing tasks so that output is maximised or cost is minimised? By modelling labour availability, skill requirements and production targets as inequalities, we can use graphs and algebraic tests to find the best feasible plan.
在 Edexcel A-Level 数学中，线性规划是一种在若干线性不等式约束下优化线性目标函数的方法。工人控制这一概念可以转化为一个数学问题：管理者应如何在相互竞争的任务之间分配固定数量的工人或工时，才能使产量最大化或成本最小化？通过将劳动力数量、技能要求和生产目标建模为不等式，我们可以使用图形和代数检验来找到最佳可行方案。
1. What Is Workers' Control in a Mathematical Model? | 数学模型中的工人控制指什么
In a business or factory problem, workers are a limited resource. Mathematically, workers' control means choosing the values of decision variables, usually numbers of workers assigned to different jobs, subject to constraints such as total hours, minimum staffing levels and budget. The phrase does not imply political control; it is simply a label for workforce allocation.
在商业或工厂问题中，工人是一种有限资源。从数学上看，工人控制意味着在总工时、最低人员配置和预算等约束下，选择决策变量的值，通常是分配到不同岗位的工人数量。这一术语并不表示政治控制；它只是劳动力分配的一个标签。
2. Defining Decision Variables | 定义决策变量
Start by defining clear variables. For example, let x be the number of workers assigned to Task A, and let y be the number of workers assigned to Task B. Both x and y must be non-negative integers in real workforce problems, but for graphical LP we often first treat them as continuous variables and then check integer points.
首先定义清晰的变量。例如，设 x 为分配到任务 A 的工人数，y 为分配到任务 B 的工人数。在实际劳动力问题中，x 和 y 必须是非负整数，但在图形线性规划中，我们通常先将其视为连续变量，然后再检查整数点。
3. From Verbal Constraints to Inequalities | 从文字约束到不等式
Each sentence in the problem can become an inequality. For instance, 'The total number of workers cannot exceed 60' gives x + y ≤ 60. 'Task A requires at least 10 workers' gives x ≥ 10. 'Each Task B worker needs 3 hours of supervision and only 120 hours are available' gives 2x + 3y ≤ 120 if Task A needs 2 hours.
题目中的每句话都可以变成一个不等式。例如，工人总数不能超过 60 人给出 x + y ≤ 60。任务 A 至少需要 10 名工人给出 x ≥ 10。如果任务 A 每名工人需要 2 小时监管，任务 B 需要 3 小时，而可用监管时间为 120 小时，则给出 2x + 3y ≤ 120。
- x + y ≤ 60 (total workers cannot exceed 60 / 总工人数不超过 60)
- x ≥ 10 (at least 10 workers on Task A / 任务 A 至少 10 名工人)
- y ≤ 2x (at most twice as many on B as on A / 任务 B 人数最多为任务 A 的两倍)
4. Labour Constraints and Resource Limits | 劳动力约束与资源限制
s 5x + 4y ≥ 200. This is a minimum constraint, so the feasible region lies above the line.
常见的约束是总工时。如果任务 A 的工人每小时贡献 5 个单位，任务 B 的工人每小时贡献 4 个单位，且必须至少生产 200 个单位，则生产约束为 5x + 4y ≥ 200。这是一个最低约束，因此可行区域位于该直线上方。
5. Non-Negativity and Realistic Domains | 非负约束与实际定义域
el exam questions, these are implied but should still be stated. If the variables represent numbers of workers, we also know x and y must be whole numbers, so we will later look at integer solutions.
工人数量不能为负，因此我们总是包含 x ≥ 0 和 y ≥ 0。在很多 Edexcel 考试题中，这些约束是隐含的，但仍应写出。如果变量表示工人数，我们还知道 x 和 y 必须是整数，因此稍后我们将讨论整数解。
6. Drawing the Feasible Region | 绘制可行域
e the unwanted region or the wanted region consistently; Edexcel often asks you to label the feasible region R.
将每个不等式绘制为一条直线。严格不等式 (< 或 >) 使用虚线，弱不等式 (≤ 或 ≥) 使用实线。始终一致地涂掉不需要的区域或标记所需区域；Edexcel 经常要求你标出可行区域 R。
7. The Objective Function: Maximising Output or Minimising Cost | 目标函数：最大化产出或最小化成本
To find the optimal point graphically, draw the objective line for a chosen value of P or C, and slide it parallel until it is just leaving the feasible region.
目标是一个线性表达式，例如 P = 40x + 60y（最大化利润）或 C = 80x + 100y（最小化成本）。为了通过图形找到最优点，画出选定 P 或 C 值的目标函数直线，并平行滑动，直到它刚好离开可行区域。
Maximise P = 40x + 60y subject to x + y ≤ 60, x ≥ 10, 2x + 3y ≤ 120
8. Vertex Testing and the Optimal Solution | 顶点检验与最优解
For a linear objective over a bounded feasible region, the optimum occurs at a vertex (corner). Find the coordinates of all vertices by solving pairs of boundary equations, then substitute into the objective function. The largest value gives the maximum; the smallest gives the minimum.
对于有界可行域上的线性目标函数，最优值出现在某个顶点（角点）。通过联立边界方程求出所有顶点的坐标，然后代入目标函数。最大值对应最大目标值；最小值对应最小目标值。
| Vertex | x | y | P = 40x + 60y |
|---|---|---|---|
| A | 10 | 10 | 1000 |
| B | 10 | 40/3 | 1200 |
| C | 30 | 10 | 180
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