We study a feedforward supply network that involves assembly operations. We compute optimal stock levels which minimize inventory costs and maintain stockout probabilities below given desirable levels (service-level constraints). To that end, we develop large deviations approximations for inventory costs and service level constraints and formulate the stock level selection problem as a nonlinear programming problem which can be solved using standard techniques. This results in significant computational savings when compared to exhaustive search using simulation. Our distributional assumptions are general enough to include temporal dependencies in the demand and production processes. We leverage the solution of the inventory control problem in the design of supply contracts under explicit service-level constraints.