$[35$ 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$[[\backslash$table$[[$Project$],[$id$.]],\backslash$table$[[$Cash outflow at time$],[$O$($million Lira$)]],\backslash$table$[[$NPV$],[($million Lira$)]]],[1,4,40],[2,7,42],[3,5,25],[4,3,12]]$

If $10$ 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.$)[5$ pts$.]$

b$.$ Find the optimal solution by using Branch and Bound method $($Draw the branch and bound tree clearly. $($Left branches $x_1=0,$ right branches $x_1=1)[20$ pts$.].$