$[25$ pts$.]$ Four projects are available for investment. The projects require the cash flows and yield the net present values $($NPV$)($in millions$)$ shown in the following table.

$\backslash$table$[[$Project id$.,\backslash$table$[[$Cash outflow at time $0],[($million Lira$)]],\backslash$table$[[$NPV$],[($million Lira$)]]],[1,8,12],[2,8,11],[3,6,8],[4,5,6]]$

If $20$ million Lira is available for investment at time $0,$ find the investment plan that maximizes NPV$.$ All investments are required to be $0$ or $1($fractional investment values are not permitted$).$

a$.$ Formulate the mathematical model. $($Write the decision variables, objective function and the constraints.$)[10$ pts$.]$

b$.$ Find the optimal solution by using Branch and Bound method $($Draw the branch and bound tree clearly, write also lower bounds$($LB$))($Left branches $x_i=0,$ right branches $x_i=1)$

pts$.].$