In this study, a new SUE model using the Weibull random error terms is proposed as an alternative to overcome the drawbacks of the multinomial logit (MNL) SUE model. A path-size weibit (PSW) model is developed to relax both independently and identically distributed assumptions, while retaining an analytical closed-form solution. Specifically, this route choice model handles route overlapping through the path-size factor and captures the route-specific perception variance through the Weibull distributed random error terms. Both constrained entropy-type and unconstrained equivalent MP formulations for the PSW-SUE are provided. In addition, model extensions to consider the demand elasticity and combined travel choice of the PSW-SUE model are also provided. Unlike the logit-based model, these model extensions incorporate the logarithmic expected perceived travel cost as the network level of service to determine the demand elasticity and travel choice. Qualitative properties of these minimization programs are given to establish equivalency and uniqueness conditions. Both path-based and link-based algorithms are developed for solving the proposed MP formulations. Numerical examples show that the proposed models can produce a compatible traffic flow pattern compared to the multinomial probit (MNP) SUE model, and these models can be implemented in a real-world transportation network.