A company manufacturing three different products $(P1,P2,P3)$ uses three assembly lines $,A3)$ for production. $P1$ is produced in $A1,P2$ is produced in $A2$ and $P3$ is produced in $A3.$ The company makes $$2$ profit out of one unit of $P1,$$$4$ profit out of one unit of $P2$ and $$6$ profit out of one unit of $P3.$ The setup costs for $A1,A2$ and $A3$ are $$300,$$$500$ and $$900$ respectively and paid only if the associated line is used. $3$ units of raw material are needed to manufacture one unit of $P1,5$ units of raw material are needed to manufacture one unit of $P2$ and $7$ units of raw material are needed to manufacture one unit of P$3.$ A total of $4200$ units of raw material are available. Define your decision variables and formulate an IP to maximize company's profits with following additional restrictions. $(20$ P$)$

If $A1$ is opened, then $A2$ must be opened.

Either A$1$ or A$2$ must be opened.

At least two assembly line must be opened.