Question $3.(25$ points$)$: Midville Manufacturing assembles heavy duty materials handling carts to meet demand of $500$ units in the first quarter of each year, $1200$ in the second, $1000$ in the third, and $300$ in the fourth. Elementary components, which consist of wheels, steering yokes, and carrying platforms, are first assembled separately. Then each steering yoke is equipped with $4$ wheels to form the front$-$end subassembly. Finally, front$-$end subassemblies are combined with a carrying platform and $8$ additional wheels at the rear to complete the cart. Using $\backslash ($ j$=1\backslash )$ for steering yokes, $\backslash ($ j$=2\backslash )$ for wheels, $\backslash ($ j$=3\backslash )$ for platforms, $\backslash ($ j$=4\backslash )$ for front$-$end assemblies, and $\backslash ($ j$=5\backslash )$ for finished carts, the following table shows the estimated value of each element $($in dollars$)$ and the factory hours required to assemble it$.$

Components, subassemblies, and finished carts produced in any quarter may be used or shipped in the same quarter or held over as inventory at $5\backslash \%$ per quarter interest on the held value. Midville seeks a plan that minimizes these holding costs while conforming to the factory production capacity, $1250$ hours per quarter. Midville has no starting inventory.

a$)$ Formulate an LP model to choose a production plan using the decision variables $\backslash ($ x$\_\{$j$,$ q$\}\backslash )$ to represent the number of units of element $\backslash ($ j $\backslash )$ produced in quarter $\backslash ($ q $\backslash )$ and $\backslash ($ h$\_\{$j$,$ q$\}\backslash )$ to represent the number of units of element $\backslash ($ j $\backslash )$ held in inventory at the end of quarter $\backslash ($ q $\backslash )$ for $\backslash ($ j$=1,\backslash$ldots$,5\backslash )$ and $\backslash ($ q$=1,\backslash$ldots$,4\backslash )$ for $\backslash ($ x $\backslash )$ and $\backslash ($ q$=0,\backslash$ldots$,4\backslash )$ for $\backslash ($ h $\backslash )($to more easily encode the starting inventory constraint$).$ Your model should include a system of main constraints for production capacity and an additional system for each element $\backslash ($ j $\backslash )$ to enforce material balance.

b$)$ Implement and solve the linear program from part a in JuMP. Display the values of all decision variables. $($The optimal value is $266,250).$