Integer linear programs (ILPs) and mixed integer programs (MIPs) often have multiple distinct optimal solutions, yet the widely used Gurobi optimization solver returns certain solutions at disproportionately high frequencies. This behavior is disadvantageous, as, in fields such as biomedicine, the identification and analysis of distinct optima yields valuable domain-specific insights that inform future research directions. In the present work, we introduce MORSE (Multiple Optima via Random Sampling and careful choice of the parameter Epsilon), a randomized, parallelizable algorithm to efficiently generate multiple optima for ILPs. MORSE maps multiplicative perturbations to the coefficients in an instance's objective function, generating a modified instance that retains an optimum of the original problem. We formalize and prove the above claim in some practical conditions. Furthermore, we prove that for 0/1 selection problems, MORSE finds each distinct optimum with equal probability. We evaluate MORSE using two measures; the number of distinct optima found in independent runs, and the diversity of the list (with repetitions) of solutions by average pairwise Hamming distance and Shannon entropy. Using these metrics, we provide empirical results demonstrating that MORSE outperforms the Gurobi method and unweighted variations of the MORSE method on a set of 20 Mixed Integer Programming Library (MIPLIB) instances and on a combinatorial optimization problem in cancer genomics.