The John Equipment company that produces refrigerator engines needs to procure refrigerant gases from an external supplier. It is common that refrigerant gases are ordered and received in boxes of $20$kg and are costly to keep in inventory. The company estimates that the inventory carrying cost of each box is equivalent to $25\%$ of its cost per year.

Based on its target production, the company needs $2,600$ boxes of refrigerant gases every year. There are currently two suppliers that the company can procure its refrigerant gases from. Both suppliers sell at the same unit price of $$1,200/$box$,$ however they differ in their ordering and delivery policies:

Supplier A that is located in Australia is a big supplier and delivers the ordered amount on the same day, and it charges $$1,000$ for each order irrespective of the amount ordered.

Supplier B that is located in South Islan is a fairly smaller supplier and usually do not have refrigerant gases in stock for instant delivery. It can produce and deliver only $100$ boxes per week, but it does eventually deliver the entire ordered amount. This supplier charges $$1,500$ for each order irrespective of the amount ordered.

Assume $50$ weeks in a year.

a$)$ What are the optimal order quantity and total procurement cost if the company choose Supplier A$?[5$ marks$]$

b$)$ What are the optimal order quantity and total procurement cost if the company choose Supplier B$?[5$ marks$]$

c$)$ One of the board members of the company makes this argument that because both suppliers offer the same price, but Supplier A has lower ordering cost and it is also faster in delivering, it is the obvious choice. The company board finds this sensible$.$ Based on your calculations in parts $($a$)$ and $($b$),$ do you agree with this choice? If not, how would you justify your choice and convince the board that your choice is more sensible $($in simple words that the board members would understand$)?$