We consider a high volume assemble-to-order system with mulitple products and components. Our objective is to maximize infinite horizon expected discounted profit. We show that optimal product prices and component production capacity result in utilization near 100%, so the system is in heavy traffic. We further show that heavy traffic remains the optimal operating regime when customer orders must be assembled within a maximum delay, and component production can be expedited at some additional cost. In heavy traffic, the system exhibits a reduction in problem dimensionaltiy. The limiting diffusion approximation has dimension equal to the number of components (rather than the number of components plus the number of products). We use this insight to propose discrete review policies for sequencing product orders for assembly in both of the aforementioned models. When delay constraints are present, we additionally provide a policy for expediting components at discrete review time points. We show our discrete review policies are asymptotically optimal in heavy traffic.