Let x be the binary decision variable which takes the value $1$ if you assign Machine

$1$ to Department $1,$ and $0$ otherwise. Similarly, we define y as the binary decision variable which takes the value $1$ if Machine $2$ is assigned to Department $2,$ and $0$ otherwise. You want to decide the values of your decisions x and y depending on the flow between these machines:

$$ If you assign Machine $1$ to Department $1$ and Machine $2$ to Department $2,$ then the flow between these machines will be a$.$

$$ You either assign Machine $1$ to Department $1$ or Machine $2$ to Department $2$ but not both, then the total flow will be b$.$

$$ If you do not assign Machine $1$ to Department $1$ and Machine $2$ to Department $2,$ then the flow between these machines will be $0.$

Formulate a linear objective function and linear constraints which model these conditions.

consider the following cases:

x y Proxy

$11$ a

$10$ b

$01$ b

$000$

You should also introduce a single auxiliary variable.

$($the problem should be taken into account by considering the linearized quadratic assignment problem $($with its objective function and constraints$)$ by introducing a auxiliary variable$)$