An oil refinery needs $816$ barrels of a particular chemical annually. The chemical can be produced by the plant at a rate of $2000$ barrels per year. It is estimated that the production set$-$up cost is $$40,$ each barrel costs $$10$ to produce, and the holding cost is based on $40\%$ annual interest rate.

$($i$)$ Determine the optimal size of a production run for this chemical.

$($ii$)$ Determine the average annual cost of holding and set$-$up$.$

$($iii$)$ If this particular chemical can be purchased from an external supplier with the following prices shown in Table $4,$ determine the optimal order quantity and total cost. Suppose that the ordering cost is $$12.$

$($iv$)$ Based on the cost information about this particular chemical, the oil refinery has to make a choice between manufacturing it in$-$house and purchasing it from an external supplier. Which is the better choice $($determine the choice by the total cost calculations$)?$