$26\mathrm{.}4($A$)$ Monthly demand rate for a certain part is $5,000$ units. The part is produced in batches

and its manufacturing costs are $$8\mathrm{.}00.$ Holding cost is $20\%$ of piece cost. Currently the

production equipment used to produce this part is also used to produce $14$ other parts with

similar usage and cost data $($assume the data to be identical for purposes of this problem$).$

Changeover time between batches of the different parts is now $4\mathrm{.}0hr,$ and cost of downtime

on the equipment is $$\frac{200}{h}r.$ A proposal has been submitted to fabricate a fast$-$acting slide

mechanism that would permit the changeovers to be completed in just $10$min. Cost to fab$-$

ricate and install the slide mechanism is $$125,000.($a$)$ Is this cost justified by the savings in

total annual inventory cost that would be achieved by reducing the economic batch quantity

from its current value based on a $4-$hr setup to the new value based on a $10-$min setup?

$($b$)$ How many months of savings are required to pay off the $$125,000$ investment