The Rugger Corporation is a Seattle$-$based R&D company that recently developed a new type of fiber substrate that is waterproof and resists dirt. Several carpet manufacturers in northeast Georgia want to use Rugger as their sole supplier for this new fiber. The locations of the carpet manufacturers are summarized in the following table:

Rugger expects to make $130,75,90,$ and $80$ deliveries to the carpet producers in Dalton, Rome, Canton, and Kennesaw, respectively. The company wants to build its new plant in the location that would minimize the annual shipping miles. However, Rugger also wants to be within $50$ miles of each of the new customers so that it will be easy to provide on$-$site technical support for any production problems that may occur.

a$.$Formulate an NLP model for this problem.

b$.$Implement your model in a spreadsheet and solve it$.$

c$.$What is the optimal location for the new plant? How many annual shipping miles are associated with this solution?