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Engineering design often requires that one consider multiple criteria; pursuing just one objective can be insufficient. The process of optimizing a collection of objective functions is called multi-objective optimization, and it necessitates the use special optimization methods. Potential applications for even the most fundamental and common methods span a variety fields. Consequently, the significance of any contribution to this process is far reaching, with implications in terms of computational problem solving and in terms of conceptual decision-making. However, the study of fundamental methods has been deficient. Critical information concerning the significance of method-parameters, the effect of formulation variations, and guidelines for effective use has been unrevealed, and drawbacks of different methods have not been explored. This work responds to these deficiencies on three levels: 1) qualitative assessment, 2) analysis with consequent enhancements and guidelines, and 3) application to significant problems. New in-depth analysis reveals useful insight into how the methods work, fosters new methods, and provides method augmentation that improves performance.