This thesis describes the refinement of an Architecture Flow Optimization (AFO) tool for naval surface ship design, specifically focusing on the development of new network and matrix-based methods for AFO formulation and their application in Concept Development. The AFO tool analyzes and optimizes the flow of energy through the ship's Vital Components (VCs) interfacing with a Ship Synthesis and Product Model (SSM), ensuring that all physical and operational constraints are satisfied while minimizing system cost across multiple intact and damaged operational scenarios. The total ship system is described by physical and logical architectures in a network structure comprised of vital component nodes and arcs. These elements form the basis of a linear system of equations in matrix form, the manipulation of which relies heavily on linear algebra and matrix operations. The matrix system of equations is solved using linear programming with a significant improvement in computational efficiency. The solution supports the sizing of individual vital components and the refinement of system logical architecture. It also provides the basic AFO engine necessary to support future refinement of a dynamic architecture flow optimization (DAFO) with the computational speed necessary for rapid solution of dynamic mission scenarios insuring optimized and feasible warfighting reconfiguration, with and without damage. / Master of Science / This thesis describes the refinement of an Architecture Flow Optimization (AFO) tool for naval surface ship design, specifically focusing on the development of new network and matrix-based methods for AFO formulation and their application in naval ship Concept Development processes. The Architecture Flow Optimization tool analyzes and optimizes the flow of energy through the ship's Vital Components (VCs). The AFO tool completes this task by interfacing with a Ship Synthesis and Product Model (SSM), ensuring that all of the ship's physical and operational constraints are satisfied. This is done while minimizing the ship system cost across multiple intact and damaged operational scenarios. The total ship system is described by physical and logical architectures in a network structure comprised of vital components (nodes) and their connections (arcs). These elements form the basis of a linear system of equations in matrix form, the manipulation of which relies heavily on linear algebra and matrix operations. The matrix system of equations is solved using a linear programming algorithm with a significant improvement in computational speed. The solution provided from the optimization supports the sizing of individual vital components and the refinement of the ship system logical architecture. It also provides the basic AFO engine necessary to support future refinement of a dynamic architecture flow optimization (DAFO) with the computational speed necessary for rapid solution of dynamic mission scenarios insuring optimized and feasible warfighting reconfiguration, with and without damage.