How does the linking constraint $X_i \le C_i Y_i$ enforce the on/off production rule?

Conditions

• $X_i$ is the production quantity on machine $i$.
• $Y_i$ is the binary activation indicator for machine $i$.
• $C_i$ is the maximum capacity of machine $i$.
• $X_i \ge 0$ is enforced.

Reasoning, step by step

1. Start with the constraint $X_i \le C_i Y_i$.
2. Case 1: Assume $Y_i = 1$ (machine active). Substitute to get $X_i \le C_i \cdot 1$, so $X_i \le C_i$.
3. Case 2: Assume $Y_i = 0$ (machine inactive). Substitute to get $X_i \le C_i \cdot 0$, so $X_i \le 0$.
4. Apply the non-negativity constraint $X_i \ge 0$.
5. Conclude that in Case 2, $X_i$ must be 0.
6. Verify that the constraint correctly models the logical dependency.

Example

For machine 1 with capacity 400, the constraint is $X_1 \le 400Y_1$. If $Y_1=1$, $X_1 \le 400$. If $Y_1=0$, $X_1 \le 0$, and since $X_1 \ge 0$, $X_1=0$.

Common misconceptions

• Thinking that $Y_i$ only records usage without affecting the production bound.
• Believing that $X_i$ can be negative when $Y_i=0$.
• Assuming the constraint works without the non-negativity condition on $X_i$.

Watch the explanation

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