Abstract
The analytic hierarchy process (AHP) proposed by T.L.Saaty is a method for decision making that takes into consideration uncertain situations or multiple evaluation criteria. In AHP, a decision maker gives values of paired comparisons between evaluation criteria and substitutive propositions. Saaty proposed the eigen vector method (EV method) to calculate evaluation values called weights of the substitutive propositions from a given pair-wise comparison matrix. On the other hand, logarithmic least square error method (LLS method) has been proposed. It is a technique that calculates weights of the substitutive propositions based on statistical theory. In the LLS, a linear logarithm model assumes an identical variance for noises and the logarithmic least square error method is applied to calculate the weights. This supposition seems to be useful in many practical cases. However, a pair comparative value depends on decision makers. In other words, there may exist the case such that "the pair comparative value between substitutive propositions A and B is more reliable than that between B and C" arise in practice. That is, when a paired comparison matrix is made, the reliability of each comparative pair may be different in the matrix. In this case, it is more suitable that we assume the noises with different variances in the log-linear model. In the conventional LLS, even if the reliability of each comparative pair is different in the matrix it cannot be used for the analysis. In this paper, we show a new method that takes into consideration the reliability of decision makers for each comparative pair value. We propose the weight estimation method of AHP based on a weighted least squares error method where weights represent the conviction of decision maker. If a decision maker can give a precise comparative pair value, then the proposed method is theoretically sound. Furthermore, from a simulation experiment, we examined the properties of the proposed method. We show that a better estimator than LLS can be acquired even if weights that are different from the right values calculated from true variances of noises.
