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Capacity constraints must be linked to binary variables to ensure logical consistency between machine activation and production. Simple capacity limits ($X_i \le C_i$) allow production even if the machine is not activated ($Y_i=0$), which would incur variable costs without paying the fixed setup cost.

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.

Capacity constraints must be linked to binary variables to ensure logical consistency between machine activation and production. Simple capacity limits ($X_i \le C_i$) allow production even if the machine is not activated ($Y_i=0$), which would incur variable costs without paying the fixed setup cost.

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.

The optimal solution uses machines 2 and 3, while machine 1 remains unused. The production quantities are $X_1=0$, $X_2=550$, and $X_3=450$.

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.

The optimal solution uses machines 2 and 3, while machine 1 remains unused. The production quantities are $X_1=0$, $X_2=550$, and $X_3=450$.

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.

The linking constraint $X_i \le C_i Y_i$ enforces the on/off production rule by scaling the capacity limit $C_i$ with the binary activation variable $Y_i$. When the machine is active ($Y_i=1$), the constraint becomes $X_i \le C_i$, allowing production up to the full capacity.

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.

The linking constraint $X_i \le C_i Y_i$ enforces the on/off production rule by scaling the capacity limit $C_i$ with the binary activation variable $Y_i$. When the machine is active ($Y_i=1$), the constraint becomes $X_i \le C_i$, allowing production up to the full capacity.

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.