$5\mathrm{.}7$ Ravindran et al$.,1987)$ A company manufacturers three products A$,$ B and $C.$ Each unit of product A requires $1h$ of engineering service, $10h$ of direct labor, and $3lb$ of material. To produce one unit of product $B$ requires $2h$ of engineering, $4h$ of direct labor, and $2lb$ of material. Each unit of product $C$ requires $1h$ of engineering, $5h$ of direct labor, and $1lb$ of material. There are $100h$ of engineering, $700h$ of direct labor, and $400lb$ of materials available. The cost of production is a nonlinear function of the quantity produced as shown in Table $5\mathrm{.}16.$

Given the unit selling prices of products $A,B,$ and $C$ as $$12,9,$ and $7$ respectively, formulate a linear mixed integer program to determine the optimal production schedule that will maximize the total profit.

TABLE $5\mathrm{.}16$

Data for Exercise $5\mathrm{.}7$

$\backslash$table$[[$Product A$,$Product B$,$Product C$],[\backslash$table$[[$Production$],[($Units$)]],\backslash$table$[[$Unit$],[$Cost $($$$)$