In this chapter, we introduce the optimization problem for transformation pathways in a residential building stock and develop corresponding mixed-integer linear programming (MILP) models. We begin by outlining the multi-layer structure of the problem, which consists of three interdependent levels: resource allocation at the building stock level, building efficiency pathways, and building operations. Each level has distinct decision-making time resolutions and is crucial for determining optimal transformation pathways. We then define parameters and sets used throughout our models before presenting a comprehensive MILP model that captures these elements. The objective is to evaluate trade-offs between economic factors, represented by net present value (NPV), and ecological impacts, captured by total global warming (TGW), thereby facilitating an integrated approach to optimizing residential energy systems.