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Binary variables $Y_i$ are introduced to model the activation status of each machine. They take the value 1 if machine $i$ is used and 0 otherwise.

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.

Binary variables $Y_i$ are introduced to model the activation status of each machine. They take the value 1 if machine $i$ is used and 0 otherwise.

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.

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.