  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.
