A mixed traffic assignment problem and its upper bound of efficiency loss

A variational inequality (vi) formulation of a mixed traffic assignment problem associated with user equilibrium (ue) player and altruistic players (ue-ap) is developed. the users governed by the ue player use the paths with minimum actual travel cost,while the users governed by altruistic players use the paths with minimum perceived travel cost,which is a linear combination of the actual and marginal travel costs. a nonlinear programming method is adopted to derive the efficiency loss of the ue-ap mixed equilibrium with polynomial cost function. finally,numerical examples are given to verify our analytical results. © 2014 xiao-jun yu and jiancheng long.