Pattaya Specialty Foods can produce its famous meat pies at a rate of $1,650$ cases of $48$ mango pies each per day. The firm distributes the mango pies to regional stores and restaurants at a steady rate of $250$ cases per day. The cost of setup, cleanup, idle time in transition from other products to pies, etc., is $$320.$ Annual holding costs are $$11\mathrm{.}50$ per case. Assume $250$ days per year.

$($a$)$ Determine the optimum production run $($batch size$)$

$($b$)$ Determine the number of production runs per year.

$($c$)$ Determine maximum inventory.

$($d$)$ Determine total inventory$-$related $($setup and carrying$)$ costs per year $($rounded to the nearest dollar$).$