Heuristics to Improve Efficiency of Solution Algorithm of Path Flow Estimator

Abstract

The path flow estimator (PFE) proposed by Bell and Shield in 1997 is an important demand estimator in ground transportation. The PFE is formulated as a nonlinear optimization model and solved by the iterative balancing solution algorithm given by Bell in 1995. This paper presents several theoretical results about the solution algorithm. In particular, the solution algorithm is shown to maximize the dual problem of the PFE sequentially along the directions defined by unit vectors. A closed-form relationship was identified between the increase in the objective of the dual problem and the optimal step length along each direction. In addition, three heuristics were developed to improve the computational efficiency of the solution algorithm. The first heuristic reduced the computational effects that could not yield significant increases to the objective value of the dual. The second reduced the probability of yielding marginal increases to the objective value of the dual. The third sped up convergence by partially avoiding zigzagging (i.e., approaching the optimal solution in small steps along perpendicular directions). The heuristics could be applied simultaneously with each other or with the methods developed to determine better initial values. The three heuristics were applied to a large-sized network, and the computational results were generally consistent with the theoretical results.