Bi-criteria lexicographical minimization problems with the makespan as the primary objective and the total machine assignment costs as the secondary objective have been recently introduced to the scheduling research, and polynomial time (r+1,1)-approximation algorithms have been suggested for their solution, where 1<r<2 is the performance ratio of an approximation algorithm for P||Cmax. We improve these results by presenting a polynomial time (1.5r−1,1)-approximation algorithm for the additive cost type. Then, we introduce a problem of minimizing the total machine assignment cost over the Δ-approximate solutions of the makespan minimization problem. We prove that this new problem is strongly NP-hard and pseudo-polynomially non-approximable in general. A polynomial time approximation algorithm with a guaranteed approximation ratio is presented for the additive cost type and bounded ratio between the maximal and minimal machine costs. An O(mn2k) time dynamic programming algorithm is also presented, where k is the fixed number of distinct job processing times.