Why must capacity constraints be linked to binary variables in the fixed cost problem?

Conditions

• Machines have fixed setup costs.
• Production is bounded by machine capacity.
• Binary variables indicate machine activation.
• The model must prevent production on inactive machines.

Reasoning, step by step

1. Identify the simple capacity constraint $X_i \le C_i$.
2. Recognize that this constraint does not account for machine activation status.
3. Introduce the binary variable $Y_i$ for activation.
4. Modify the constraint to $X_i \le C_i Y_i$.
5. Verify that if $Y_i=0$, then $X_i \le 0$, forcing $X_i=0$.
6. Verify that if $Y_i=1$, then $X_i \le C_i$, restoring the capacity limit.

Example

For machine 1 with capacity 400, the linked constraint is $X_1 \le 400Y_1$. If $Y_1=0$ (machine off), $X_1 \le 0$, so no production occurs. If $Y_1=1$ (machine on), $X_1 \le 400$, allowing full capacity production.

Common misconceptions

• Thinking that simple capacity constraints are sufficient when fixed costs exist.
• Believing that the objective function alone prevents production on inactive machines.
• Confusing the capacity constant $C_i$ with the binary variable $Y_i$.

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