Suppose that there are three green technologies you consider investing in and you want to select which green technologies to invest in$.$ If you choose a technology to invest, there is a fixed payment you need to make. Each technology has a profit and environmental benefit. The table below gives the information about these projects.

Technology $1$ Technology $2$ Technology $3$

Profit $105030$

Environment Benefit $701020$

Fixed Payment $304045$

While selecting which technologies to invest, you want to maximize the total profit and maximize the total environmental benefit; but you have a limited budget of $100,$ so you cannot pay more than $100$ in total fixed payments.

a$)$ Mathematically formulate a multi$-$objective binary programming model for the above technology selection problem. To do so$,$ clearly define the decision variables $($for binary variables, define what $0$ means and what $1$ means$),$ and then formulate the objective functions and the constraints using the decision variables.