Abstract
A problem of a hierarchy structure optimization is considered. Hierarchical structures are widely used in the Analytic Hierarchy Process, conjoint analysis, and various other methods of multiple criteria decision making. The problem consists in finding a structure that needs a minimum number of pair comparisons for a given total number of the alternatives. For an optimal hierarchy, the minimum efforts are needed for eliciting data and synthesizing the local preferences across the hierarchy to get the global priorities or utilities. Special estimation techniques are developed and numerical simulations performed. Analytical and numerical results suggest optimal ways of priority evaluations for practical managerial decisions in a complex environment.
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References
F. S. Hillier and G. J. Lieberman, Introduction to Operations Research, Holden-Day, San Francisco, 1974.
M. Zeleny, Multiple Criteria Decision Making, McGraw Hill, New York, 1982.
R. E. Steuer, Multiple Criteria Optimization Theory, Computation, and Application, Wiley, New York, 1986.
A. S. Bryk and S. W. Raudenbush, Hierarchical Linear Models, Sage, London, 1992.
G. H. Tzeng, H. F. Wang, U. P. Wen, and P. L. Yu, Multiple Criteria Decision Making, Springer-Verlag, Amsterdam, 1994.
Y. Bar-Yam, Dynamics of Complex Systems, Addison-Wesley, Reading, MA, 1997.
T. L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980.
T. L. Saaty, Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process, RWS Publications, Pittsburgh, PA, 1994.
T. L. Saaty, Decision Making with Dependence and Feedback: the Analytic Network Process, RWS Publications, Pittsburgh, PA, 1996.
B. L. Golden, E. A. Wasil, and P. T. Harker, The Analytic Hierarchy Process: Applications and Studies, Springer, Berlin-Heidelberg, 1989.
R. F. Dyer and E. H. Forman, An Analytic Approach to Marketing Decisions, Prentice-Hall, Englewood Cliffs, NJ, 1991.
S. Lipovetsky, The synthetic hierarchy method: An optimizing approach to obtaining priorities in the AHP, European Journal of Operational Research, 1996, 93: 550–564.
S. Lipovetsky and A. Tishler, Interval estimation of priorities in the AHP, European Journal of Operational Research, 1999, 114: 153–164.
S. Lipovetsky and M. Conklin, Robust estimation of priorities in the AHP, European Journal of Operational Research, 2002, 137: 110–122.
S. Lipovetsky, Analytic hierarchy processing in Chapman-Kolmogorov equations, International Journal of Operations and Quantitative Management, 2005, 11: 219–228.
S. Lipovetsky, Global priority estimation in multiperson decision making, Journal of Optimization Theory and Applications, 2009, 140: 77–91.
R. Keeney and H. Raiffa, Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Wiley, New York, 1976.
F. A. Lootsma, Multi-Criteria Decision Analysis via Ratio and Difference Judgement, Kluwer Academic Publishers, Dordrecht, 1999.
V. Srinivasan, A conjunctive-compensatory approach to the self-explication of multiattributed preferences, Decision Sciences, 1988, 19: 295–305.
J. W. Moy, K. F. Lam, and E. U. Choo, Deriving the partial values in MCDM by goal programming, Annals of Operations Research, 1997, 74: 277–288.
S. B. Maurer, The king chicken theorems, Mathematics Magazine, 1980, 53: 67–80.
L. R. Foulds and F. Y. Partovi, Integrating the analytic hierarchy process and graph theory to model facilities layout, Annals of Operations Research, 1998, 82: 435–451.
D. Stauffer and J. S. Sa Martins, Asymmetry in hierarchy model of Bonabeau et al., Advances in Complex Systems, 2003, 6: 559–564.
K. Hausken and R. Cressman, Formalization of multi-level games, International Game Theory Review, 2004, 6: 195–221.
L. Hurvicz and S. Reiter, Designing Economics Mechanisms, Cambridge University Press, Cambridge, UK, 2006.
T. W. Leigh, D. B. Mackay, and J. O. Summers, Reliability and validity of conjoint analysis and self-explicated weights: A comparison, Journal of Marketing Research, 1984, 21: 456–462.
D. C. Miller, Handbook of Research Design and Social Measurement, SAGE Publication, Newbury-Park, CA, London, New Delhi, 1991.
M. Wedel and W. A. Kamakura, Market Segmentation: Concepts and Methodological Foundations, Kluwer Academic Publishers, Boston, 1998.
R. L. Andrews, A. Ansari, and I. Currim, Hierarchical Bayes versus finite mixture conjoint analysis models: A comparison of fit, prediction and partworth recovery, Journal of Marketing Research, 2002, 39: 87–98.
J. J. Louviere, D. A. Hensher, and J. D. Swait, Stated Choice Methods: Analysis and Applications, Cambridge University Press, Cambridge, UK, 2000.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, DC, 1970.
S. Lipovetsky and F. A. Lootsma, Generalized golden sections, repeated bisections and aesthetic pleasure, European Journal of Operational Research, 2000, 121: 213–216.
T. L. Saaty, Optimization in Integers and Related Extremal Problems, McGraw-Hill, New York, 1970.
H. S. Wilf, Some examples of combinatorial averaging, The American Mathematical Monthly, 1985, 92: 250–261.
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Lipovetsky, S. Optimal hierarchy structures for multi-attribute-criteria decisions. J Syst Sci Complex 22, 228–242 (2009). https://doi.org/10.1007/s11424-009-9159-5
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DOI: https://doi.org/10.1007/s11424-009-9159-5