Many fundamental problems in scientific computing have more than one solution method. It is not uncommon for alternative solution methods to represent different tradeoffs between solution cost and reliability. Furthermore, the performance of a solution method often depends on the numerical properties of the problem instance and thus can vary dramatically across application domains. In such situations, it is natural to consider the construction of a multi-method composite solver to potentially improve both the average performance and reliability. In this paper, we provide a combinatorial framework for developing such composite solvers. We provide analytical results for obtaining an optimal composite from a set of methods with normalized measures of performance and reliability. Our empirical results demonstrate the effectiveness of such optimal composites for solving large, sparse linear systems of equations.