How do you construct the total cost objective function for the three-machine production problem?

Conditions

• There are multiple machines with distinct variable and fixed costs.
• $X_i$ represents the continuous production quantity on machine $i$.
• $Y_i$ represents the binary activation status of machine $i$.
• The goal is cost minimization.

Reasoning, step by step

1. Identify the variable cost per unit for each machine.
2. Identify the fixed cost for each machine.
3. Multiply each machine's variable cost by its production variable $X_i$.
4. Multiply each machine's fixed cost by its binary variable $Y_i$.
5. Sum all variable cost terms and all fixed cost terms.
6. Formulate the minimization objective: Min $\sum (v_i X_i + f_i Y_i)$.

Example

For machines with variable costs 2.39, 1.99, 2.99 and fixed costs 300, 250, 400, the objective function is: Min $2.39X_1 + 1.99X_2 + 2.99X_3 + 300Y_1 + 250Y_2 + 400Y_3$.

Common misconceptions

• Adding fixed costs directly without multiplying by $Y_i$.
• Multiplying variable costs by $Y_i$ instead of $X_i$.
• Maximizing the cost function instead of minimizing it.

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