What is the optimal production plan and machine usage for the three-machine example?

Conditions

• The model includes three machines with specific costs and capacities.
• The demand is exactly 1000 units.
• The objective is to minimize total cost.
• The solution is derived from solving the integer linear program.

Reasoning, step by step

1. Solve the integer linear programming model.
2. Identify the values of the continuous variables $X_i$.
3. Identify the values of the binary variables $Y_i$.
4. Verify that the total production equals the demand.
5. Verify that production on each machine does not exceed its capacity.
6. Interpret the binary values to determine which machines are active.

Example

The displayed solution is $X_1=0, X_2=550, X_3=450$ and $Y_1=0, Y_2=1, Y_3=1$. This means machine 1 is off, machine 2 produces its full capacity of 550, and machine 3 produces 450 out of its 600 capacity.

Common misconceptions

• Assuming all machines are used in the optimal solution.
• Thinking that machine 1 is used because it has the lowest variable cost (it is not, due to fixed costs and capacity trade-offs).
• Confusing the production quantity $X_i$ with the activation status $Y_i$.

Watch the explanation

Explore next

• How does the linking constraint $X_i \le C_i Y_i$ enforce the on/off production rule?
• Why must capacity constraints be linked to binary variables in the fixed cost problem?
• What other kinds of constraints can be modeled with binary variables?