The need for flexible service provision has dramatically increased in recent years due to the increased penetration of variable energy resources, as has the need to ensure fair access to service provision from an increasingly diverse array of resources. In response to these needs, this study develops a new analytic optimization formulation for the clearing of day-ahead markets based on swing contracts for the combined flexible provision of energy and reserve services. This new optimization formulation is a mixed integer linear programming (MILP) problem that can be solved using standard MILP solution software. A numerical day-ahead market example is presented to illustrate the potential of this new optimization formulation for real-world implementation.