We approach the problem of trait introgression as an optimization challenge with clearly defined objectives in 3 different scenarios that largely capture the introgression problem in a diploid outcrossing selfing-tolerant species: 1) the introgression of a single allele into a recipient background via backcrossing, 2) the introgression of multiple alleles from one line into a recipient line, and 3) the introgression of several alleles from multiple lines into the background of a recipient line. For each of the 3 cases, we present a mathematical formulation, based on optimization principles from Operations Research (OR), defining the objectives to be optimized, decision variables and constraints of the introgression problem. We then use simulation, with genome size and reproductive biology based on maize, to estimate the probability of achieving a set of breeding goals. Algorithms from OR and combinatorics are used to optimize selections. Finally, Pareto response surfaces are presented for each of the 3 cases to concisely show the tradeoffs between objectives. With this systematic approach of defining quantifiable objectives, translating the objectives into a mathematical model, then building simulation models that allow for analysis of the tradeoffs between objectives, we show a way forward where plant breeding has a deeper engagement with applied mathematics.