A Weibit-Based Combined Distribution and Assignment Model: Equivalent Formulation and a Partial Linearization Solution Algorithm

The combined distribution and assignment (CDA) problem is considered in which both the destination choice and the route choice follow the multinomial weibit model rather than the conventional multinomial logit model. Existing CDA models typically adopt a hierarchical logit structure with independently and identically distributed Gumbel errors, whose fixed perception variance is unrealistic when trips of very different lengths are handled simultaneously. To circumvent this drawback, a hierarchical weibit choice structure is developed in which the perceived travel costs follow the Weibull distribution, making the perception variance route-specific and increasing with the trip length, and a weibit composite cost plays the role of the logsum in the logit-based CDA model. An equivalent mathematical programming formulation is constructed by combining a multiplicative Beckmann transformation with route-level and destination-level entropy terms, an exogenous destination utility term, and a path-size term, and its equivalence and uniqueness are proved. An equivalent variational inequality formulation accommodates further model extensions. A path-based partial linearization algorithm is developed whose search direction is obtained in closed form by the hierarchical weibit loading and whose step size follows Barzilai–Borwein strategies, including an adaptive scheme. Numerical experiments on the Winnipeg network demonstrate the computational performance of the algorithm and the behavioral differences from the logit-based model. A validation against the observed Winnipeg demand further shows that, once both models are calibrated to the observed mean trip length, the weibit-based model reproduces the observed demand pattern substantially more closely than its logit counterpart.