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On Elaboration for Paired Comparisons Matrix Conformity Estimation for the Analytic Hierarchy Process

Keywords:

N.P. Vasilyev, A.M. Spiridonov


The article is devoted to paired comparisons matrix (PCM) conformity estimation used in the Analytic Hierarchy Process (AHP). The authors have conducted a number of computational experiments targeted at getting random index (RI) values for different sizes of PCM (from 3 to 200). The received numeric results of the experiments are tabulated, and linear regression models for RI and PCM size dependency are also represented (for the PCM sizes more than 50). The paper consists of an introduction, four parts and a conclusion. Besides, it contains three pictures, four tables, and a list of nine bibliographic sources. The introduction defines the goal and tasks of the research – to elaborate the AHP’s PCM conformity evaluation by means of validly conducted computational experiments targeted to get RI values for different PCM sizes. The first part is dedicated to a survey of computational procedures making a basis for the AHP. The second part contains a number of formulated AHP evaluations problems. The main is that modern literature gives contradictory RI values, which makes the reliability of those values somewhat questionable. Moreover, the confidence level of those values is unknown. The existing values correspond to PCM sizes from 3 to 15 only. The third part describes computational experiments concerning RI results and substantiates the received RI values reliability. There has been developed special software for the PCM Monte Carlo simulation and for calculations. This program is free for use and anybody can repeat our experiments. The experiments results are presented in two tables. The first table corresponds to PCM simulation case by using a linear congruent pseudo-random number generator (PRNG), while the second table represents PCM simulation with PRNG earlier developed by one of the authors. That PRNG has better statistical properties than the linear congruent PRNG. Each table row has the following values: (1) PCM size, (2) a number of processed PCMs, (3) RI, (4) RI variance, (5) confidence level. Both the tables are filled up with numeric values for the PCM sizes from 3 to 53 in succession, and separately for 60, 70, 80, 90, 100, 120 and 200. The forth part is dedicated to experiment results analysis. It proves that the obtained estimations are somewhat stricter than those described in modern literature. Moreover, the use of PRNG with more statistical qualities results in more exact estimations. On the base of the tabulated values there have been developed two linear regression models for RI: one for linear congruent PRNG modeling variant and the other – for modeling with author’s PRNG. In conclusion, all conducted researches on RI estimation are summed up.
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