For production issues, value chain is always a good start.
I would break down issue in the typical blocks of value chain, and investigate one by one where are the bottlenecks
1. Supply : Do we have enought raw material stock / or can we order enough raw materials from our supplier to increase our production ?
2. Production : How is the process operated and are we operating at full capacity at every step of the process ? Are there opportunity to increase capacity without material investments (increase work duration, teams, number of shift, production planning ) ? If we decide to invest in equipment what is the step of the process on the critical path to prioritize
3. Transport : if we are able to produce more, are we able to deliver more ?
Hope this helps