Title: Extended p-median problems for balancing service efficiency and equality

Abstract: This article deals with the location problem for balancing the service efficiency and equality. In public service systems, some people may feel envy in case that they need longer travel distance to access services than others. The strength of the envy can be measured by comparing one's travel distance to service facility with a threshold distance. Using the total envy function, four extended p-median problems are proposed for trade-off between service efficiency and equality. Five analytical properties of the new problems are mathematically proven. The new problems were tested on three sets of well-designed instances. The experimentation shows that the equality measures, such as the standard deviation, the mean absolute deviation, and the Gini coefficient between travel distances, can be substantially improved by minimizing the travel cost and the spatial envy. The experimentation also shows that, when the service supply is given in terms of the number of facilities, the service equality can be considerably improved by slightly increasing the travel distance. When the service supply is increased in terms of the number of facilities, both the service efficiency and spatial equality can be significantly improved.