In this paper we address two optimization problems arising in the design of genomic assays based on universal tag arrays. First, we address the universal array tag set design problem. For this problem, we extend previous formulations to incorporate antitag-to-antitag hybridization constraints in addition to constraints on antitag-to-tag hybridization specificity, establish a constructive upper bound on the maximum number of tags satisfying the extended constraints, and propose a simple greedy tag selection algorithm. Second, we give methods for improving the multiplexing rate in large-scale genomic assays by combining primer selection with tag assignment. Experimental results on simulated data show that this integrated optimization leads to reductions of up to 50% in the number of required arrays.