$2.[25$ 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.

If $8$ million Lira is available for investment at time $0,$ find the investment plan that maximizes NPV$.$

a$.$ Formulate the mathematical model. $($Write the decision variales, obective function and the constraints.$)[5$ pts$.]$

b$.$ Find the optimal solution by using Branch and Bound method $($Draw the branch and bound tree clearly. $($Left branches $\backslash ($ x$\_\{$i$\}=0\backslash ),$ right branches $\backslash ($ x$\_\{$i$\}=1\backslash ))20$ pts$.\backslash (\backslash$left$.\backslash$right$]\backslash ).$