I am working on optimizing a manufacturing based dataset which consists of a huge number of controllable parameters. The goal is to attain the best run settings of these parameters.
I familiarized myself with several predictive algorithms while doing my research and if I say, use Random Forest to predict my dependent variable to understand how important each independent variable is, is there a way to extract the final equation/relationship the algorithm uses?
There is no general way to get an interpretable equation from a random forest, explaining how your covariates affect the dependent variable. For that you can use a different model more suitable, e.g., linear regression (perhaps with kernel functions), or a decision tree. Note that you can use one model for prediction, and one model for descriptive analysis - there's no inherent reason to stick with a single model.
use Random Forest to predict my dependent variable to understand how important each independent variable is
Understanding how important each dependent variable, does not necessarily mean you need the question in the title of your question, namely getting the actual relationship. Most random forest packages have a method quantifying how much each covariate affected the model over the train set.