||Despite many advances in both optimization methods and computational fluid dynamics, the timely automatic selection and refinement, via physics-based and empirical methods, of "optimal" configurations of compression systems remains challenging. This is due, in part, to the large number of design parameters (with associated high computational cost) operating over wide ranges that can be non-smooth, if not discontinuous (to which many optimization algorithms, developed for smooth problems, are ill-suited). It is further complicated by the phasic nature of turbomachinery design and the associated need to balance the amount of time and computational resource devoted to selecting the most promising configurations with that expended in their refinement. This paper compares a number of combinations of a multi-fidelity approach for configuration selection with a high-fidelity method for design refinement. The system is tested on the aerodynamic design of a complete two-spool core compression system for a generic high bypass ratio turbofan. The resulting designs are obliged to meet familiar constraints for overall design point pressure rise and surge margin together with a number of mechanical constraints including maximum shaft speeds. Through the configuration phase, the number of stages and the duty split between the spools are permitted to change. It is shown that the performance of the design refinement phase is only a weak function of the preceding configuration phase provided that the latter is well into diminishing returns with respect to approaching a converged solution. It is hence shown possible to obtain equally good designs in around half the computational run-time by exploiting this weak dependence by effectively decoupling the configuration and refinement phases and starting the latter before the former has apparently finished. It is also shown that if either configuration or refinement is allowed to dominate the design process, inferior designs result. The best designs are associated with between half and three-quarters of the design effort being devoted to configuration selection.