Many engineering design problems are associated with computationally expensive and time-consuming simulations for design evaluation. In such problems, each candidate design should be selected carefully, even though it means extra algorithmic complexity. This study develops the Proximity-based Surrogate-Assisted Evolutionary Algorithm (PSA-EA) that aims at handling both single-objective and multi-objective computationally expensive problems. It controls the trade-off between exploration and exploitation by defining proximity and trust regions around high-fidelity solutions. The proximity measure aims to maximize the diversity of information about specific regions of the search space and to improve the goodness of the surrogate for future cycles simultaneously. The method employs an ensemble of metamodels and a parallel infill criterion. PSA-EA is evaluated and compared to a recently developed surrogate-assisted evolutionary algorithm on ten test problems. Thereafter, a case study involving a multi-objective design optimization of the cylinder head water jacket of a vehicle engine is presented and discussed. Online supplemental data for this article can be accessed at https://doi.org/10.1080/0305215X.2020.1808972.