Monthly demand for a certain part is $10,000$ units. The part is produced in

batches and its manufacturing costs are estimated to be $8\mathrm{.}00.$ Holding costs are

$20\%$ of piece costs. Currently the production equipment used to produce the part

is also used to produce $19$ other parts with similar usage and cost data $($assume

the data to be identical for the purposes of this problem$).$ Changeover time

between batches of different parts is now $4.$ Ohours and cost of downtime on the

equipment is $\frac{250}{?}$ hour. A proposal has been submitted to fabricate a fast$-$acting

slide mechanism that would permit the changeover to be completed in just $10$

minutes. The cost to fabricate and install the mechanism is $125,000.$

$(1)$ Demonstrate whether or not the cost is justified by savings in total annual

inventory cost that would be achieved by reducing the economic batch

quantity from its current value based on a $4-$hour set$-$up to the new value

based on a $10-$minute set$-$up

$(7$ marks$)$

$(2)$ How many months of savings are required to payoff the $$125,000$

investment?

$(3$ marks$)$

Please use the EOQ model in Question $2$ above