This paper studies six different AHP (Analytic Hierarchy Process) approaches and suggests that the features of the approaches are classified by application circumstances in order to contribute to the applicability and quality usage of the AHP. Our study investigates the hierarchical principles and characteristics of the AHP, and historical debates on the AHP evaluation in which the six approaches have been involved. One of six approaches is an ANP (Analytic Network Process) application that is directly connected to AHP usage. The application differences among the six approaches are validated with a plain example. Then, the four circumstances of AHP applications are classified by two dimensions: the first dimension is whether or not the importance (weights) of criteria is independent of restrictively setting alternatives, and the second dimension is whether or not preference (priorities) of alternatives is independent of adding alternative(s) to or removing alternative(s) from the considering set of alternatives. Then featuring way of weighting criteria is classified. We suggest the distinguishing manners and describe the implications of the AHP application. Finally, we discuss rank reversal and multiplicative AHP.