Total locational surplus for facility users distributed continuously along a network

This paper proposes an appropriate index for evaluating the degree to which facilities are desirable to their users under realistic assumptions. We derive an analytical expression for the index and discuss its calculation. Throughout this paper, we make the following three assumptions: (1) the distance from a user to a facility is measured by the shortest path, (2) facility users are continuously distributed along a network, and (3) the choice behavior of the facility users follows the Huff model. Under the third assumption, the index, called the ‘total locational surplus,’ enables us to conduct evaluations based on the concept of consumer surplus in microeconomics. We argue that this index is considered the most reliable among possible indices. First, we express the total locational surplus under our three assumptions as a definite integral. We then express a part of the integrand in the definite integral as an infinite series of exponential functions, which allows us to derive the infinite series expression of the definite integral. The infinite series expression enables us to calculate an approximate value for the total locational surplus using the first several terms of the series. We describe the computational procedure for calculating this approximation and evaluate its time complexity. We also utilize the procedure to evaluate the spatial configuration of some shopping centers in the northern part of Japan.