Given a number of objectives, is there any quantitative method through which I can determine what objectives are more crucial than others
You may recognize some objective as "more important" and some as "less important" but it doesn't mean that there's any objective method to determine this importance. Instead, it may depend on a real-world application standing behind the optimization problem. You can assign weights to particular objectives or solve the problem in many stages.
Or, if you mean by "crucial" large influence of a particular objective on the "overall performance", you may consider normalizing objectives to [0,1] range of output values and maximizing the sum of them subject to the constraints of the problem. The larger is the value of a particular objective for the optimal solution, the more "crucial" this objective can be considered.
Just a quick idea, not based on any literature, somehow inspired by Nicolas Dupin's answer. :)
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