When the processing times of jobs are controllable, selected processing times affect both the manufacturing cost and the scheduling performance. A well known example for such a case that this paper specifically deals with is the turning operation on a CNC machine. Manufacturing cost of a turning operation is a nonlinear convex function of its processing time. In this paper, we deal with making optimal machine-job assignments and processing time decisions so as to minimize total manufacturing cost while the makespan being upper bounded by a known value, denoted as -constraint approach for a bicriteria problem. We then give optimality properties for the resulting single criterion problem. We provide alternative methods to compute cost lower bounds for partial schedules, which are used in developing an exact (branch and bound) algorithm. For the cases where the exact algorithm is not efficient in terms of computation time, we present a recovering beam search algorithm equipped with an improvement search procedure. In order to find improving search directions, the improvement search algorithm uses the proposed cost bounding properties. Computational results show that our lower bounding methods in branch and bound algorithm achieve a significant reduction in the search tree size that we need to traverse. Also, our recovering beam search and improvement search heuristics achieve solutions within 1% of the optimum on the average while they spent much less computational effort than the exact algorithm.