High-entropy alloys, with N elements and compositions {cν = 1,N} in competing crystal structures, have large design spaces for unique chemical and mechanical properties. Here, to enable computational design, we use a metaheuristic hybrid Cuckoo search (CS) to construct alloy configurational models on the fly that have targeted atomic site and pair probabilities on arbitrary crystal lattices, given by supercell random approximates (SCRAPs) with S sites. Our Hybrid CS permits efficient global solutions for large, discrete combinatorial optimization that scale linearly in a number of parallel processors, and linearly in sites S for SCRAPs. For example, a four-element, 128-site SCRAP is found in seconds—a more than 13,000-fold reduction over current strategies. Our method thus enables computational alloy design that is currently impractical. We qualify the models and showcase application to real alloys with targeted atomic short-range order. Being problem-agnostic, our Hybrid CS offers potential applications in diverse fields.