AHP-Group Decision Making: A Bayesian Approach Based on Mixtures for Group Pattern Identification

This paper proposes a Bayesian estimation procedure to determine the priorities of the Analytic Hierarchy Process (AHP) in group decision making when there are a large number of actors and a prior consensus among them is not required. Using a hierarchical Bayesian approach based on mixtures to describe the prior distribution of the priorities in the multiplicative model traditionally used in the stochastic AHP, this methodology allows us to identify homogeneous groups of actors with different patterns of behaviour for the rankings of priorities. The proposed procedure consists of a two-step estimation algorithm: the first step carries out a global exploration of the model space by using birth and death processes, the second concerns a local exploration by means of Gibbs sampling. The methodology has been illustrated by the analysis of a case study adapted from a real experiment on e-democracy developed for the City Council of Zaragoza (Spain).

[1]  Manuel Salvador,et al.  Group preference structures in AHP–group decision making , 2005 .

[2]  T. Saaty,et al.  Procedures for Synthesizing Ratio Judgements , 1983 .

[3]  José María Moreno-Jiménez,et al.  A linkage between the Analytic Hierarchy Process and the Compromise Programming Models , 2002 .

[4]  E. Forman,et al.  Aggregating individual judgments and priorities with the analytic hierarchy process , 1998, Eur. J. Oper. Res..

[5]  John Geweke,et al.  Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments , 1991 .

[6]  José María Moreno-Jiménez,et al.  Multicriteria procedural rationality on SISDEMA , 1999, Eur. J. Oper. Res..

[7]  José María Moreno-Jiménez,et al.  Reciprocal distributions in the analytic hierarchy process , 2000, Eur. J. Oper. Res..

[8]  Raimo P. Hämäläinen,et al.  Analyzing AHP-matrices by regression , 2003, Eur. J. Oper. Res..

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

[10]  Alfonso Mateos,et al.  Monte Carlo Simulation Techniques in a Decision Support System for Group Decision Making , 2005 .

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

[12]  M. Stephens.  Bayesian analysis of mixture models with an unknown number of components- an alternative to reversible jump methods , 2000 .

[13]  José María,et al.  A Spreadsheet Module for Consistent Consensus Building in AHP-Group Decision Making , 2005 .

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

[15]  P. Green,et al.  On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion) , 1997 .

[16]  Sylvia Richardson,et al.  Markov Chain Monte Carlo in Practice , 1997 .

[17]  R. Ramanathan,et al.  Group preference aggregation methods employed in AHP: An evaluation and an intrinsic process for deriving members' weightages , 1994 .

[18]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

[19]  Thomas L. Saaty,et al.  Group Decision Making and the AHP , 1989 .

[20]  José María Moreno-Jiménez,et al.  Local stability intervals in the analytic hierarchy process , 2000, Eur. J. Oper. Res..

[21]  T. Saaty.  The Analytic Network Process , 2001 .

[22]  Patrick T. Harker,et al.  The Analytic hierarchy process : applications and studies , 1989 .

[23]  G. Casella,et al.  Rao-Blackwellisation of sampling schemes , 1996 .

[24]  Simon French,et al.  Web-enabled strategic GDSS, e-democracy and Arrow's theorem: A Bayesian perspective , 2007, Decis. Support Syst..

[25]  Luis G. Vargas,et al.  A probabilistic study of preference structures in the analytic hierarchy process with interval judgments , 1993 .

[26]  José María Moreno Jiménez,et al.  METODOLOGÍA CIENTÍFICA EN VALORACIÓN Y SELECCIÓN AMBIENTAL , 2001 .

[27]  Zhi-Ping Fan,et al.  A compromise weight for multi-criteria group decision making with individual preference , 2000, J. Oper. Res. Soc..

[28]  A. Brix.  Bayesian Data Analysis, 2nd edn , 2005 .

[29]  Indrani Basak,et al.  The categorical data analysis approach for ratio model of pairwise comparisons , 2001, Eur. J. Oper. Res..

[30]  John Aitchison,et al.  The Statistical Analysis of Compositional Data , 1986 .

[31]  Heikki Mannila,et al.  Principles of Data Mining , 2001, Undergraduate Topics in Computer Science.

[32]  Luis G. Vargas.  Reciprocal matrices with random coefficients , 1982 .

[33]  José María Moreno-Jiménez,et al.  E-democracy and knowledge: A multicriteria framework for the new democratic era , 2003 .

[34]  J. Fichtner.  On deriving priority vectors from matrices of pairwise comparisons , 1986 .

[35]  P. Green,et al.  Corrigendum: On Bayesian analysis of mixtures with an unknown number of components , 1997 .

[36]  Indrani Basak,et al.  Probabilistic judgments specified partially in the analytic hierarchy process , 1998, Eur. J. Oper. Res..

[37]  Jyrki Kangas,et al.  Analysing uncertainties of interval judgment data in multiple-criteria evaluation of forest plans , 1998 .

[38]  Jyrki Kangas,et al.  Uncertainty in Expert Predictions of the Ecological Consequences of Forest Plans , 1996 .

[39]  D. Rubin,et al.  Inference from Iterative Simulation Using Multiple Sequences , 1992 .

[40]  C. Genest,et al.  A statistical look at Saaty's method of estimating pairwise preferences expressed on a ratio scale , 1994 .

[41]  M. Urmeneta,et al.  El pesar en el proceso analítico jerárquico 1 , 2000 .

[42]  J. González-Pachón,et al.  Distance-based consensus methods: a goal programming approach , 1999 .

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

[44]  Bodo Glaser.  Fundamentals of Decision Making , 2002 .

[45]  Jyrki Kangas,et al.  Analyzing uncertainties in experts' opinions of forest plan performance , 1997 .