While making a surprising observation linking Neyman (1934) sample allocation in probability sampling and the current method used to allocate the seats in the U.S. House of Representatives called equal proportions, Wright (2012) provides an exact optimal allocation [n₁, ..., n_h, ..., n_H] of the fixed overall sample size n among H strata under stratified random sampling that minimizes the sampling variance Var(T̂_Y) of an estimator of a total T̂_Y subject to the constraint n = Σ^H_h=1   n_h. The exact optimal allocation avoids the need to round to integer values, as is the case with Neyman allocation. Neyman allocation with rounded integers does not always lead to the optimal allocation. In this paper, we demonstrate a very easy extension and generalization of the result in Wright (2012) to the problem of finding an exact optimal allocation [n₁, ..., n_h, ..., n_H] to minimize the sampling variance subject to n = Σ^H_h=1   n_h and additional mixed constraint patterns 0 < a_h ≤ n_h ≤ b_h ≤ N_h, where n, N_h, a_h, and b_h are fixed integers and N_h is the size of the h^th stratum. Avoiding the costly tendency to round up to ensure minimum sampling variance, the exact optimal allocation is especially useful in applications where H is very large and there are minimum and maximum size constraints on the allocated sample sizes n_h, as is the case with the Census Bureau’s Service Annual Survey which has H = 391 sampling strata. The presented methods are what some might call “greedy” algorithms, and the methods solve a non-linear optimization problem with linear constraints over a space of integer values. While it’s common that greedy algorithms are easy to compute, they are not guaranteed to find the global optimum. Remarkably, the presented simple algorithms always find the global optimum.