Why are binary variables introduced in the fixed cost production problem?

Conditions

• The problem involves machines with fixed setup costs.
• Production quantities are continuous variables.
• The goal is to minimize total cost including both variable and fixed components.

Reasoning, step by step

1. Identify that fixed costs are incurred only if a machine is used.
2. Define a binary variable $Y_i$ for each machine.
3. Set $Y_i = 1$ if machine $i$ is active, and $Y_i = 0$ if inactive.
4. Multiply the fixed cost of each machine by its corresponding binary variable in the objective function.
5. Link the binary variable to the production quantity to enforce logical consistency.

Example

For three machines with fixed costs 300, 250, and 400, the binary variables $Y_1, Y_2, Y_3$ are defined. The fixed cost part of the objective function becomes $300Y_1 + 250Y_2 + 400Y_3$. If $Y_1=0$, the cost 300 is not added to the total.

Common misconceptions

• Thinking binary variables are only for counting discrete items.
• Assuming fixed costs are always paid regardless of machine usage.
• Confusing the binary activation variable $Y_i$ with the continuous production variable $X_i$.

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Explore next

• How do you formulate the objective function for the fixed cost production problem?
• What is the role of the linking constraint $X_i \le C_i Y_i$?
• Why can't we just use continuous variables for machine usage?