We are interested in minimizing a linear function over the set of weakly efficient solutions of a linear multiobjective problem. Here, we build a convex cone such that every vector belonging to this cone corresponds to one or more weakly efficient solutions. We show that each weakly efficient solution is associated with a vector in this cone. Moreover, we show that if the data vector of the linear objective function belongs to the building cone, then the original problem is equivalent to a linear programming one. Through the building cone, we present necessary and sufficient conditions for the existence of an optimal solution of the original problem. Further, we propose an algorithm to solve, if possible, the original problem or to check its infeasibility. © 2007 Taylor & Francis.