The State of West Virginia legislature is considering passing a bill that would provide Opioid Crisis Centers (OCCs) to the state's residents. The goal is to have the offices located so the residents can travel to them in a minimal amount of time. It is agreed that the residents should be able to visit the OCC either within the county that they live in or in the county adjacent to their county of residence, their neighboring county. Provided is a link with an adjacency matrix, which provides information on adjacent counties. A 1 in the matrix shows that two counties are adjacent (or touching one another). Notice that all the diagonals have a 1 in them; this indicates that a county is adjacent to itself. For example, Barbour County is adjacent to itself and six other counties (Upshur, Randolph, Tucker, Preston, Taylor, Harrison).

WV Counties Name County Assignment.xls

WV Counties Uncolored.jpg

4a. Formulate and solve a binary integer programming problem to determine the minimum number of counties required to meet the constraints above. List the name of the counties that will satisfy the solution and specify how many OCCs are required.

Once you have determined the minimum number of counties, create a measure of redundancy by adding up all counties adjacent to counties with offices. For example, the theoretical minimum value is 55; this would occur if each county were only adjacent to one county with an office, basically, no county is adjacent to another county. The actual minimum redundancy is higher than this theoretical minimum. However, this gives you a reference to see if your redundancy calculation seems reasonable. Remember, this is always an important part of the model-building process. Do your answers seem reasonable?

4b. Formulate a new optimization model that will maximize the measure of redundancy, what is that value, and in what counties should the offices be located to maximize this value while maintaining the required minimum number of offices.

4c. Formulate a new optimization model that will minimize the measure of redundancy, what is that value, and in what counties should the offices be located to minimize this value while maintaining the required minimum number of offices.

4d. Create a map highlighting the counties with the offices in them with blue and the counties adjacent with green for the arrangement with the highest redundancy.

4e. Create a map highlighting the counties with the offices in them with blue and the counties adjacent with green for the arrangement with the lowest redundancy.