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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.

An equality constraint is used because the customer order requires exactly 1000 units to be supplied. The problem specifies that the order must be met precisely, meaning neither underproduction (shortfall) nor overproduction (surplus) is permitted.

Conditions: The problem involves fulfilling a specific customer order.; The order quantity is fixed and must be met exactly.; Overproduction or underproduction is not allowed or desired.

An equality constraint is used because the customer order requires exactly 1000 units to be supplied. The problem specifies that the order must be met precisely, meaning neither underproduction (shortfall) nor overproduction (surplus) is permitted.

Conditions: The problem involves fulfilling a specific customer order.; The order quantity is fixed and must be met exactly.; Overproduction or underproduction is not allowed or desired.

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.

The total cost objective function is constructed by summing the variable production costs and the fixed setup costs. Variable costs are calculated by multiplying the unit variable cost of each machine by the number of units produced on that machine ($X_i$).

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.

The total cost objective function is constructed by summing the variable production costs and the fixed setup costs. Variable costs are calculated by multiplying the unit variable cost of each machine by the number of units produced on that machine ($X_i$).

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.

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.