Abstract An extension of the CSP optimization framework tailored to identify fair solutions to instances involving multiple optimization functions is studied. Two settings are considered, based on the maximization of the minimum value over all the given functions (M AX -M IN approach) and on its lexicographical refinement where, over all solutions maximizing the minimum value, those maximizing the second minimum value are preferred, and so on, until all functions are considered (LEX M AX -M IN approach). For both settings, the complexity of computing an optimal solution is analyzed and the tractability frontier is charted for acyclic instances, w.r.t. the number and the domains of the functions to be optimized. Larger islands of tractability are then identified via a novel structural approach, based on a notion of guard that is designed to deal with the interactions among constraint scopes and optimization functions.