$($Locating DCs for Toy Stores$)$ A toy store chain operates $100$ retail stores throughout the United States. The company currently ships all products from a central distribution center $($DC$)$ to the stores, but it is considering closing the central DC and instead operating multiple regional DCs that serve the retail stores. It will use the uncapacitated facility location problem to determine where to locate DCs$.$ Planners at the company have identified $24$ potential cities in which regional DCs may be located. The file toy$-$sores.xlsx lists the longitude and latitude for all of the locations $($stores and DCs$),$ as well as the annual demand $($measured in pallets$)$ at each store and the fixed annual location cost at each potential DC location. Assume that transportation from DCs to stores costs $$1$ per mile, as measured by the great circle distance between the two locations.

a$.$ Formulate the associated facility location model with Abstract Formulation with the index format. $($Define the sets, parameters, decision variables, objective function, and constraints explicitly$).$ Provide clear explanations about the model.

b$.$ Solve the problem with Excel$-$Solver or AMPL optimization software, report the optimal cities to locate DCs in and the optimal total annual cost.