A private equity firm is considering five competing projects in which to invest in the upcoming quarter. The firm needs to decide how to allocate its available capital based upon the combination of projects $($denoted as A to E$)$ selected to maximize returns $($based upon net present value $($NPV$)).$ Table $1$ below presents the capital requirements and the NPV for each project, along with the associated risk $($given as a percentage of the initial investment$).$ The company has $$43$ million in capital to allocate, with the goal of having an average associated risk of no more than $5\%.$ There are some additional constraints to be met: $($i$)$ if project $B$ is selected, then project $E$ is also selected; $($ii$)$ one of the two projects A and $C$ must be selected but not both; $($iii$)$ at least one of projects $A,B,$ and $D$ is selected.

Table $1$: Project Data

$\backslash$table$[[$Project$,$NPV $($M$$),$Risk $(\%),$Capital $($MS$)],[$A$,19,4,14],[$B$,22,5,10],[$C$,24,6,12],[$D$,27,7,15],[$E$,21,5,13]]$

The RISK constraint may be stated as:

$$

$x_B=x_E$

$4x_A-5x_B-6x_C+7x_D-5x_E=5$

$-x_A+x_C-2x_D=0$

Xa $+$XC $=1$