2020 Volume 76 Issue 3 Pages 223-235
This study provides an efficient method for solving two-dimensional FO model (Fujita and Ogawa, 1982). We first formulate the FO model as a quadratic programming problem (QP). Applying Benders’ decomposition, we then decompose the QP into a “master problem” and a “sub-problem.” The former obtains the firm’s location, while the latter obtains the household’s commute pattern (i.e. O-D demand for all possible pair of locations). Since the number of the unknown variables of the sub-problem thus should be the square order of the number of locations, it is extremely difficult to solve the sub-problem in a straightforward manner for the case with 10,000 (= 100 × 100) locations. In this article, we thus clarify that the sub-problem is equivalent to the Hitchcock’s transportation problem. This enables us to develop an efficient solution method, which reduces the number of the basic variables to the order of the number of the locations. Finally, some numerical examples exhibit that the proposed method can solve a large-scale FO model with more than 20,000 locations within a practical time.
