Generalized p-median problem for the optimal distribution of facilities

Abstract

The generalized p-median problem is related with the task to find an optimal distribution of p facilities for given heterogeneous population distribution. In the real world, the meaning of what is optimal strongly depends on the type and the purpose of the facility: the optimal distribution of fire stations may not be the same as the ones for hospitals, coffee shops, and schools. We start from the variational approach in Gastner and Newman (Phys Rev E 74:016117, 2006) and generalize the objective function in such a way that the accessibility of a facility is an algebraic function of distance with an exponent $\beta$. Applying the second-order variational approach, we find that the extremum of the objective function corresponds to either a minimum or a maximum, depending on the value of the exponent $\beta$. We find that the optimal distribution of public (private) facilities minimizes (maximizes) the objective function. We also perform simulations for the real population distribution of South Korea to validate our results from the variational approach.