A soft drink company has placed a vending machine that dispenses cans in the basement of a building. The machine holds c cans when full. The machine may be restocked at the beginning of any month at a cost of $r that covers labor and transportation. Demand for cans is quite predictable and has been forecast to be di with i = 1 corresponding to January, i = 2 corresponding to February, and so on. The company has a policy that requires a machine to be fully restocked at the beginning of January and August to support the start of the academic semesters. Every loss is estimated to cost the company $s.
Formulate a shortest path model whose solution would provide the company with the least cost-plan for stocking machine during the January to August time frame. Be sure to explain the meaning of all nodes, arcs, and arc lengths as well as any new notation you choose to introduce.