Ford and GM carry spare parts for their dealers at a third party warehouse in the Upper Peninsula $($UP$)$ of Michigan. Demand for Ford spare parts is $100$ units per month while demand for GM parts is $120$ per month. Each spare part costs $$100$ and both companies have a holding cost of $20$ percent. Currently, each company uses a separate truck to ship these parts. Each truck has a fixed cost of $$500.$ What is the optimal order size and frequency for Ford? For GM$?$ What is the annual ordering and holding cost for each company?

A $3$rd party logistics provider has offered to combine shipments for each of the two companies on a single truck. This will increase the cost of each truck to $$600.$ If the two companies agree to the joint shipment, what is the optimal order frequency and size? What is the annual ordering and holding cost for the two companies combined? Should Ford and GM accept the $3$rd party$$s proposal? How should they divide the fixed cost per truck among themselves?