A cosine maximization method for the priority vector derivation in AHP

The derivation of a priority vector from a pair-wise comparison matrix (PCM) is an important issue in the Analytic Hierarchy Process (AHP). The existing methods for the priority vector derivation from PCM include eigenvector method (EV), weighted least squares method (WLS), additive normalization method (AN), logarithmic least squares method (LLS), etc. The derived priority vector should be as similar to each column vector of the PCM as possible if a pair-wise comparison matrix (PCM) is not perfectly consistent. Therefore, a cosine maximization method (CM) based on similarity measure is proposed, which maximizes the sum of the cosine of the angle between the priority vector and each column vector of a PCM. An optimization model for the CM is proposed to derive the reliable priority vector. Using three numerical examples, the CM is compared with the other prioritization methods based on two performance evaluation criteria: Euclidean distance and minimum violation. The results show that the CM is flexible and efficient.

[1]  A. G. Lockett,et al.  Judgemental modelling based on geometric least square , 1988 .

[2]  S. Greco,et al.  Multiple Criteria Hierarchy Process with ELECTRE and PROMETHEE , 2013 .

[3]  John A. Keane,et al.  A heuristic method to rectify intransitive judgments in pairwise comparison matrices , 2012, Eur. J. Oper. Res..

[4]  J. Barzilai.  Deriving weights from pairwise comparison matrices , 1997 .

[5]  Ying-Ming Wang,et al.  Priority estimation in the AHP through maximization of correlation coefficient , 2007 .

[6]  Gang Kou,et al.  An improved statistical approach for consistency test in AHP , 2013, Ann. Oper. Res..

[7]  Yi Peng,et al.  Ensemble of Software Defect Predictors: an AHP-Based Evaluation Method , 2011, Int. J. Inf. Technol. Decis. Mak..

[8]  T. Saaty.  Eigenvector and logarithmic least squares , 1990 .

[9]  T. Saaty,et al.  The Analytic Hierarchy Process , 1985 .

[10]  B. Golany,et al.  A multicriteria evaluation of methods for obtaining weights from ratio-scale matrices , 1993 .

[11]  E. Blankmeyer.  Approaches to consistency adjustment , 1987 .

[12]  William E. Stein,et al.  The harmonic consistency index for the analytic hierarchy process , 2007, Eur. J. Oper. Res..

[13]  Luis G. Vargas.  The consistency index in reciprocal matrices: Comparison of deterministic and statistical approaches , 2008, Eur. J. Oper. Res..

[14]  Noel Bryson,et al.  A Goal Programming Method for Generating Priority Vectors , 1995 .

[15]  Bojan Srdjevic,et al.  Combining different prioritization methods in the analytic hierarchy process synthesis , 2005, Comput. Oper. Res..

[16]  Gang Kou,et al.  A statistical approach to measure the consistency level of the pairwise comparison matrix , 2014, J. Oper. Res. Soc..

[17]  Stan Lipovetsky,et al.  Robust estimation of priorities in the AHP , 2002, Eur. J. Oper. Res..

[18]  R. Kalaba,et al.  A comparison of two methods for determining the weights of belonging to fuzzy sets , 1979 .

[19]  Gerard Salton,et al.  The SMART Retrieval System—Experiments in Automatic Document Processing , 1971 .

[20]  María Teresa Lamata,et al.  Consistency in the Analytic Hierarchy Process: a New Approach , 2006, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[21]  S French,et al.  Multicriteria Methodology for Decision Aiding , 1996 .

[22]  Margaret H. Dunham,et al.  Data Mining: Introductory and Advanced Topics , 2002 .

[23]  Gang Kou,et al.  A heuristic approach for deriving the priority vector in AHP , 2013 .

[24]  P. Yu,et al.  Eigenweight vectors and least-distance approximation for revealed preference in pairwise weight ratios , 1985 .

[25]  Kweku-Muata Osei-Bryson,et al.  Generating consensus priority point vectors: a logarithmic goal programming approach , 1999, Comput. Oper. Res..

[26]  Saul I. Gass,et al.  Singular value decomposition in AHP , 2004, Eur. J. Oper. Res..

[27]  José María Moreno-Jiménez,et al.  A Bayesian priorization procedure for AHP-group decision making , 2007, Eur. J. Oper. Res..

[28]  T. Saaty.  Fundamentals of Decision Making and Priority Theory With the Analytic Hierarchy Process , 2000 .

[29]  Ludmil Mikhailov,et al.  A fuzzy programming method for deriving priorities in the analytic hierarchy process , 2000, J. Oper. Res. Soc..

[30]  M. T. Lamata,et al.  A new measure of consistency for positive reciprocal matrices , 2003 .

[31]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[32]  M. Singh,et al.  Comparison analysis of methods for deriving priorities in the analytic hierarchy process , 1999, IEEE SMC'99 Conference Proceedings. 1999 IEEE International Conference on Systems, Man, and Cybernetics (Cat. No.99CH37028).

[33]  Sajjad Zahir,et al.  Geometry of decision making and the vector space formulation of the analytic hierarchy process , 1999, Eur. J. Oper. Res..

[34]  Luis G. Vargas,et al.  Comparison of eigenvalue, logarithmic least squares and least squares methods in estimating ratios , 1984 .

[35]  Hideo Tanaka,et al.  Interval priorities in AHP by interval regression analysis , 2004, Eur. J. Oper. Res..

[36]  Bojan Srdjevic,et al.  Bi-criteria evolution strategy in estimating weights from the AHP ratio-scale matrices , 2011, Appl. Math. Comput..

[37]  Measuring Salivary Cortisol.  Comparison of Two Methods for , 2002 .

[38]  Thomas L. Saaty,et al.  Marketing Applications of the Analytic Hierarchy Process , 1980 .

[39]  G. Crawford,et al.  A note on the analysis of subjective judgment matrices , 1985 .

[40]  Bruce L. Golden,et al.  Linear programming models for estimating weights in the analytic hierarchy process , 2005, Comput. Oper. Res..

[41]  Yi Peng,et al.  Evaluation of Classification Algorithms Using MCDM and Rank Correlation , 2012, Int. J. Inf. Technol. Decis. Mak..

[42]  G. Crawford.  The geometric mean procedure for estimating the scale of a judgement matrix , 1987 .

[43]  Ramakrishnan Ramanathan,et al.  Data envelopment analysis for weight derivation and aggregation in the analytic hierarchy process , 2006, Comput. Oper. Res..

[44]  José María Moreno-Jiménez,et al.  The geometric consistency index: Approximated thresholds , 2003, Eur. J. Oper. Res..

[45]  Yi Peng,et al.  FAMCDM: A fusion approach of MCDM methods to rank multiclass classification algorithms , 2011 .

[46]  G. Psacharopoulos,et al.  Linear Programming Models , 1987 .

[47]  Michael McGill,et al.  Introduction to Modern Information Retrieval , 1983 .

[48]  Yi Peng,et al.  Support A simple method to improve the consistency ratio of the pairwise comparison matrix in ANP , 2011 .

[49]  T. L. Saaty.  A Scaling Method for Priorities in Hierarchical Structures , 1977 .

[50]  S. Greco,et al.  Extreme ranking analysis in robust ordinal regression , 2012 .

[51]  Burton J. Bogitsh,et al.  A COMPARISON OF TWO METHODS , 1975 .