A general unconstrained optimization formulation for the combined distribution and assignment problem

• We provide a general unconstrained optimization formulation for the CDA problem.
• We prove the equivalence and uniqueness of the solution.
• We formulate two logit CDA models with closed-form solution.
• We develop a self-regulated averaging (SRA) descent algorithm.
• We demonstrate the proposed formulations and algorithm with a real network.

This paper proposes an alternate formulation for the combined distribution and assignment (CDA) problem, which seeks to determine consistent level-of-service and flow values of the trip distribution and traffic assignment steps. The CDA problem is modeled as a hierarchical travel choice problem based on random utility theory, which forms the basis for constructing as a general unconstrained optimization formulation. It has the flexibility to handle general probabilistic distributions (not just the Gumbel distribution) in a hierarchical travel choice structure. Qualitative properties of the general unconstrained CDA formulation are rigorously proved to ensure the equivalence and uniqueness of the solution. Particularly, the model is analyzed for two logit CDA models where the choice probability can be expressed in a closed form. The first logit CDA model with the independently and identically distributed (IID) Gumbel distribution is shown to be equivalent to several well-known existing CDA models. The second logit CDA model handles the independence assumption by accounting for the unobserved similarities among destinations in the destination choice level using a spatially correlated logit model and the route overlapping in the route choice level using the path size logit model. A descent direction algorithm with the self-regulated averaging (SRA) scheme is also developed for solving the unconstrained optimization formulation of two logit CDA models. Numerical experiments are conducted to demonstrate the features of the proposed general unconstrained CDA formulations and the computational performance of the descent SRA algorithm. The results reveal that route overlapping, destination similarity, congestion, and distribution errors can have a significant influence on the network equilibrium flow allocations.