The small boatyard White Kayaks built $156$ epoxy kayaks last year. Each of these requires $12$ kg of epoxy, purchased in $6-$kg barrels at $$170$ each, with a freight cost of $$18$ per shipment plus $$1/$kg$.$ The annual carrying cost is estimated to $20\%$ of the value.

a$)$If the purchase policy is to buy $20$ barrels each time, how much time can be spent on administrative work if the order quantity is considered optimal? The boat$-$builder charges $$25/$hour$.$

b$)$If time spent is $30$ minutes, what would be the optimal order quantity?