A consulting firm has eight projects under contract. The firm has $48$ consultants available for assignment to these projects. Denote by $x_j$ the number of consultants assigned to project $j.$ The firm estimates the following profit functions $P_j(x_j)=a_j{x}_{j}b{j}^j[$$$]$ for each project, with estimated constant values $a_j$ and $b_j$ as shown below.

$\backslash$table$[[$Project$,a_j,b_j$