10.2190/3272-5507-6531-8352 3272550765318352 3 Alternative Forms of Aggregation in the Analytic Hierarchy Process: Ordered Weighted Averaging Operators 49 68 20070614 20070614 20070614 20070614 3272550765318352.pdf http://baywood.metapress.com/link.asp?target=contribution&id=3272550765318352 1 P. N. Smith University of Queensland, Australia The analytic hierarchy process (AHP) methodology was introduced by Thomas Saaty and has had numerous applications in a wide range of contexts. In a common three-level hierarchy, AHP involves the aggregation of criterion importance weights (or priorities) and the performance scores (or priorities) of alternatives as the sum of the products of weights and scores for each alternative. The multiplicative AHP (MAHP) involves the multiplicative aggregation of performance scores raised to the power of the criterion weights and has been seen by many individuals (notably, Freerk Lootsma and John Barzilai) as an alternative more desirable structure. It is suggested that alternative forms of aggregation of performance scores and criterion weights might be more useful, in particular the ordered weighted averaging (OWA) operator introduced by Ronald Yager. The choice of weights in an OWA operator may be guided by a linguistic quantifier involving the importance weights associated with each criterion. The geometric ordered weighted averaging (GOWA) operator is also considered as a possibility for aggregation in the MAHP. An example is given relating to the location of a Green Bridge Link (for public transport and non-motorized modes of transport) in Brisbane, Queensland, Australia. The four approaches (alternatives) to the identified site of the Green Bridge (Dutton Park) are a Full Busway, Boggo Road, Cornwall Street, and Kent Street.

T. L. Saaty, Measuring the Fuzziness of Sets, Journal of Cybernetics, 4, pp. 57-68, 1974.

T. L. Saaty, A Scaling Method for Priorities in Hierarchical Structures, Journal Mathematical Psychology, 15, pp. 234-281, 1977.

T. L. Saaty, Exploring the Interface Between Hierarchies, Multiple Objectives and Fuzzy Sets, Fuzzy Sets, Fuzzy Sets and Systems, 1, pp. 57-68, 1978.

T. L. Saaty, The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation, McGraw-Hill, New York, 1980.

T. L. Saaty, How to Make a Decision: The Analytic Hierarchy Process, Interfaces, 24, pp. 19-43, 1994.

O. S. Vaidya and S. Kumar, Analytic Hierarchy Process: An Overview of Applications, European Journal of Operational Research, 169, pp. 1-29, 2006.

F. Zahedi, The Analytic Hierarchy Process—A Survey of the Method and its Applications, Interfaces, 16, pp. 96-108, 1986.

J. P. Shim, Bibliographical Research on the Analytic Hierarchy Process (AHP), Socio-Economic Planning Sciences, 23, pp. 161-167, 1989.

R. R. Yager, On Ordered Weighted Averaging Aggregation Operators in Multicriteria Decision Making, IEEE Transactions on Systems, Man and Cybernetics, 18, pp. 183-190, 1988.

P. N. Smith, Numeric Ordered Weighted Averaging Operators: Possibilities for Environmental Project Evaluation, Journal of Environmental Systems, 28, pp. 175-191, 2000-2001.

P. N. Smith, Linguistic Ordered Weighted Averaging Operators: Possibilities for Environmental Project Evaluation, Journal of Environmental Systems, 28, pp. 257-269, 2000-2001.

E. H. Forman and S. I. Gass, The Analytic Hierarchy Process—An Exposition, Operations Research, 49, pp. 469-486, 2001.

G. A. Miller, The Magical Number Seven, Plus or Minus Two: Some Limits to Our Capacity for Processing Information, The Psychological Review, 63, pp. 81-97, 1956.

T. L. Saaty and M. S. Ozdemir, Why the Magic Number Seven Plus or Minus Two, Mathematical and Computer Modelling, 38, pp. 233-244, 2003.

A. Basilevsky, Applied Matrix Algebra in the Statistical Sciences, North-Holland, New York, 1983.

F. A. Lootsma, Saaty's Priority Theory and the Nomination of a Senior Professor in Operations Research, European Journal of Operational Research, 4, pp. 380-388, 1980.

C. Ragsdale, Spreadsheet Modelling and Decision Analysis (4th Edition), South-Western College Publishing, Cincinnati, 2003.

W. L. Winston and S. C. Albright, Practical Management Science: Spreadsheet Modelling and Applications (2nd Edition), Duxbury Press, Belmont, California, 2002.

G. Crawford and C. Williams, A Note on the Analysis of Subjective Judgment Matrices, Journal of Mathematical Psychology, 29, pp. 387-405, 1985.

J. Barzilai, W. D. Cook, and B. Golany, Consistent Weights for Judgements Matrices of the Relative Importance of Alternatives, Operations Research Letters, 6, pp. 131-134, 1987.

J. Barzilai, Deriving Weights from Pairwise Comparison Matrices, Journal of the Operational Research Society, 48, pp. 1226-1232, 1997.

N. V. Kumar and L. S. Ganesh, A Simulation-Based Evaluation of the Approximate and the Exact Eigenvector Methods Employed in AHP, European Journal of Operational Research, 95, pp. 656-662, 1996.

T. L. Saaty and G. Hu, Ranking by Eigenvector Versus Other Methods in the Analytic Hierarchy Process, Applied Mathematical Letters, 11, pp. 121-125, 1998.

T. L. Saaty, Decision-Making with the AHP: Why is the Principal Eigenvector Necessary, European Journal of Operational Research, 145, pp. 85-91, 2003.

T. L. Saaty and L. Vargas, Comparison of Eigenvalue, Logarithmic Least Squares and Least Squares Methods in Estimating Ratios, Mathematical Modelling, 5, pp. 309-324, 1984.

J. Fichtner, On Deriving Priority Vectors from Matrices of Pairwise Comparisons, Socio-Economic Planning Sciences, 20, pp. 341-345, 1986.

F. Zahedi, A Simulation Study of Estimation Methods in the Analytic Hierarchy Process, Socio-Economic Planning Sciences, 20, pp. 347-354, 1986.

J. Křovák, Ranking Alternatives—Comparison of Different Methods Based on Binary Comparison Matrices, European Journal of Operational Research, 32, pp. 86-95, 1987.

J. M. Hihn, Evaluation Techniques for Paired Ratio Comparison Matrices in a Hierarchical Decision Model, in W. Eichhorn (ed.), Measurement in Economics, Physica-Verlag, Heidelberg, pp. 269-288, 1988.

R. E. Jensen, An Alternative Scaling Method for Priorities in Hierarchical Structures, Journal of Mathematical Psychology, 28, pp 317-332, 1984.

S. I. Gass and T. Rapcsák, Singular Value Decomposition in AHP, European Journal of Operational Research, 154, pp. 573-584, 2004.

B. Srdjevic, Combining Different Prioritisation Methods in the Analytic Hierarchy Process Synthesis, Computers & Operations Research, 32, pp. 1897-1919, 2005.

J. I. Peláez and M. T. Lamata, A New Measure of Consistency for Positive Reciprocal Matrices, Computers & Mathematics with Applications, 46, pp. 1839-1845, 2003.

F. A. Lootsma and H. Schuijt, The Multiplicative AHP, SMART and ELECTRÉ in a Common Context, Journal of Multi-Criteria Decision Analysis, 6, pp. 185-196, 1997.

F. A. Lootsma, Fuzzy Logic for Planning and Decision Making, Kluwer, Dordrecht, 1997.

F. A. Lootsma, Multi-Criteria Decision Analysis via Ratio and Difference Judgement, Kluwer, Dordrecht, 1999.

O. I. Larichev, Comments on Barzilai and Lootsma, Journal of Multi-Criteria Decision Analysis, 6, p. 166, 1997.

P. Korhonen, Comments on Barzilai and Lootsma, Journal of Multi-Criteria Decision Analysis, 6, pp. 167-168, 1997.

L. G. Vargas, Comments on Barzilai and Lootsma: Why the Multiplicative AHP Is Invalid: A Practical Counterexample, Journal of Multi-Criteria Decision Analysis, 6, pp. 169-170, 1997.

F. A. Lootsma and J. Barzilai, Response to the Comments by Larichev, Korhonen and Vargas on Power Relations and Group Aggregation in the Multiplicative AHP and SMART, Journal of Multi-Criteria Decision Analysis, 6, pp. 171-174, 1997.

D. L. Olson, G. Fliedner, and K. Currie, Comparison of the REMBRANDT System with Analytic Hierarchy Process, European Journal of Operational Research, 82, pp. 522-539, 1995.

D. L. Olson, Decision AIDS for Selection Problems (Springer Series in Operations Research), Springer-Verlag, New York, 1996.

A. Stam and A. P. Duarte Silva, On Multiplicative Priority Rating Methods for the AHP, European Journal of Operational Research, 145, pp. 92-108, 2003.

J. J. Buckley, Fuzzy Hierarchical Analysis, Fuzzy Sets and Systems, 17, pp. 233-244, 1985.

J. J. Buckley, T. Feuring, and Y. Hayashi, Fuzzy Hierarchical Analysis Revisited, European Journal of Operational Research, 129, pp. 48-64, 2001.

H. Deng, Multicriteria Analysis with Fuzzy Pairwise Comparisons, International Journal of Approximate Reasoning, 21, pp. 215-231, 1999.

T.-S. Hsu, Public Transport System Project Evaluation Using the Analytic Hierarchy Process: A Fuzzy Delphi Approach, Transportation Planning and Technology, 22, pp. 229-246, 1999.

M. T. Lamata, Ranking of Alternatives with Ordered Weighted Averaging Operators, International Journal of Intelligent Systems, 19, pp. 473-482, 2004.

P. J. M. van Laarhoven and W. Pedrycz, A Fuzzy Extension of Saaty's Priority Theory, Fuzzy Sets and Systems, 11, pp. 229-241, 1983.

J. G. de Graan, Extensions to the Multiple Criteria Analysis of T. L. Saaty, Report National Institute of Water Supply, The Netherlands, 1980.

C. G. E. Boender, J. G. de Graan, and F. A. Lootsma, Multi-Criteria Decision Analysis with Fuzzy Pairwise Comparisons, Fuzzy Sets and Systems, 29, pp. 133-143, 1989.

D.-Y. Chang, Applications of the Extent Analysis Method on Fuzzy AHP, European Journal of Operational Research, 95, pp. 649-655, 1996.

M. Enea and T. Piazza, Project Selection by Constrained Fuzzy AHP, Fuzzy Optimization and Decision Making, 3, pp. 39-62, 2004.

Brisbane City Council (BCC), Green Bridge Link: Impact Assessment Study, September 2003.

R. R. Yager, OWA Aggregation over a Continuous Interval Argument with Application to Decision Making, IEEE Transactions on Systems, Man, and Cybernetics. Part B: Cybernetics, 34, pp. 1952-1963, 2004.

V. Torra, Weighted OWA Operators for Synthesis of Information, Proceedings of the Fifth International Conference on Fuzzy Systems, FUZZ-IEEE '96, New Orleans, Louisiana, pp. 966-971, 1996.

V. Torra, The Weighted OWA Operator, International Journal of Intelligent Systems, 12, pp. 153-166, 1997.

Z. S. Xu and Q. L. Da, An Overview of Operators for Aggregating Information, International Journal of Intelligent Systems, 18, pp. 953-969, 2003.

F. Chiclana, F. Herrera, and E. Herrera-Viedma, The Ordered Weighted Geometric Operator: Properties and Application in MCDM Problems, in Technologies for Constructing Intelligent Systems 2: Tools, B. B. Bouchon-Meunier, J. Gutiárrez-Rios, L. Mapdalena, and R. R. Yager (eds.), Series: Studies in Fuzziness and Soft Computing, Physica-Verlag, Heidelberg, pp. 173-184, 2002.

F. Herrera, E. Herrera-Viedma, and F. Chiclana, A Study of the Origin and Uses of the Ordered Weighted Geometric Operator, International Journal of Intelligent Systems, 18, pp. 689-707, 2003.