A bicriteria model for locating a semi-desirable facility in the plane

We consider the problem of locating a facility in the plane. Customers want the facility to be close to obtain cheaply the service offered by it. But customers also want the facility to be far away to avoid the pollution from it. We model the situation using two criteria: One is the well-known minisum criterion; in the other we want to minimize the weighted sum of Euclidean distances raised to a negative power. The second criterion is analyzed in some detail, and we state some properties of this part of the model. In the bicriteria model we minimize the weighted sum of the two criteria, with weights adding to one. We outline a method for determining the efficient frontier of facility locations, and provide an illustrative numerical example.