Open Journal Systems


Orrin Cooper, Guoqing Liu


When designing an ANP model it is important to acknowledge and properly address whether the elements in the model are dependent on or independent of each other. The decision maker must perform criteria cluster weighting comparisons individually for the criteria clusters in each column of the Supermatrix to correctly model when the criteria and alternatives are dependent on one another to accurately capture the dependence. Failing to recognize that the criteria in a criteria cluster in one column of the Supermatrix is not necessarily equal in weight to the criteria in that same criteria cluster but in another column can lead to misrepresented rankings in the final priorities. In the extreme case, it can remove all dependence from an ANP model. Two models are used to demonstrate this unintended effect on the final priorities, and also demonstrate a crucial contribution that this effect is independent of the tangibility of the criteria considered. In the third model, the solution is discussed and implemented. A proof is provided in the appendix. This criteria cluster weighting approach further extends the applicability of the ANP to additional decisions when a decision maker wishes to represent a fully-dependent ANP decision.


Dependence; intangible elements; criteria weights

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Arbel, A. (1983). A university budget problem: A priority-based approach. Socio-Economic Planning Sciences, 17(4), 181-189. Doi:

Belton, V., & Gear, T. (1983). On a short-coming of Saaty's method of analytic hierarchies. Omega, 11(3), 228-230. Doi: 10.1016/0305-0483(83)90047-6

Choo, E. U., Schoner, B., & Wedley, W. C. (1999). Interpretation of criteria weights in multicriteria decision making. Computers & Industrial Engineering, 37(3), 527-541. Doi:

Dyer, (1990). Remarks on the Analytic Hierarchy Process. Management Science, 36(3), 249-258. Doi:

Dyer, J. W. R. E. (1985). A critique of the analytic hierarchy process. Technical Report, 84/85-4-24, (Department of Management, The University of Texas at Austin).

Harker, P. T. (1987). Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical Modelling, 9(11), 837-848. Doi: 10.1016/0270-0255(87)90503-3

Harker, P. T., & Vargas, L. G. (1990). Reply to "Remarks on the Analytic Hierarchy Process" by Dyer. Management Science, 36(3), 269-273. Doi:

Hefnaway, A. E. & Mohammed, A. S. (2014). Review of different methods for deriving weights in the Analytic Hierarchy Process. International Journal of the Analytic Hierarchy Process, 6(1), 92-123. Doi:

Lipovetsky S., (2011) Priority eigenvectors in Analytic Hierarchy/Network Processes with outer dependence between alternatives and criteria, International Journal of the Analytic Hierarchy Process, 3(2), 172-179. Doi:

Lipovetsky, S. (2011). An interpretation of the AHP global priority as the eigenvector solution of an ANP supermatrix. International Journal of the Analytic Hierarchy Process, 3(1), 70-78. Doi:

Lipovetsky S., (2013). Supermatrix eigenproblem and interpretation of priority vectors in Analytic Network Process., International Journal of the Analytic Hierarchy Process, 5 (1), 105-113. Doi:

Lipovetsky, S. & Conklin W. (2015). AHP priorities and the Markov-Chapman-Kolmogorov steady-states probabilities. International Journal of the Analytic Hierarchy Process, 7(2), 349-363. Doi:

Saaty (1977a). A scaling method for priorities in hierarchical structures. Journal of mathematical psychology, 15(3), 234-281. Doi:

Saaty (1977b). The Sudan Transport Study. Interfaces, 8(1), 37-57. Doi:

Saaty (1986). Axiomatic foundation of the analytic hierarchy process. Management Science, 32(7), 841-855. Doi:

Saaty (1990). An exposition on the AHP in reply to the paper "Remarks on the Analytic Hierarchy Process". Management Science, 36(3), 259-268. Doi:

Saaty (1996). The Analytical Hierarchy Process (2nd ed. Vol. 1). Pittsburgh: RWS Publications. Doi:

Saaty (1996) Decision making with dependence and feedback: The Analytic Network Process. Pittsburgh: RWS Publications. Doi: 10.4018/978-1-59140-702-7.ch018

Saaty (2005). Theory and applications of the Analytic Network Process: Decision making with benefits, opportunities, costs, and risks. Pittsburgh, PA: RWS Publications. Doi: 10.1007/978-3-540-92828-7_4

Saaty (2008a). The analytic hierarchy and analytic network measurement processes: Applications to decisions under risk. European Journal of Pure and Applied Mathematics,1,122-196.

Saaty, (2008b). Relative measurement and its generalization in decision making – why pairwise comparisons are central in mathematics for the measurement of intangible factors: The Analytic Hierarchy/Network Process, RACSAM, 102, 251–318. Doi: 10.1007/BF03191825

Saaty (2011). Aligning the measurment of tangibles with intagibles and not the converse. International Journal of the Analytic Hierarchy Process, 3(1),79-87. Doi:

Saaty & Cillo, B. (2008). The Encyclicon: A Dictionary of complex decisions using the Analytical Network Process (Vol. 2). Pittsburgh, PA: RWS Publications. Doi: 10.1007/978-1-4614-7279-7_1

Saaty & Vargas, (2006). Decision making with the analytic network process: Economic, political, social and technological applications with benefits, opportunities, costs and risks (Vol. 95). Springer.

Saaty (2016). Five wasy to combine tangibles with intangibles. International Journal of the Analytic Hierarchy Process, 8(2), 372-381. Doi:

Schoner, B., Wedley, W. C., & Choo, E. U. (1993). A unified approach to AHP with linking pins. European Journal of Operational Research, 64(3), 384-392. . Doi: 10.1016/0377-2217(93)90128-A.

Wallenius, J., Dyer, Fishburn, P. C., Steuer, R. E., Zionts, S., & Deb, K. (2008). Multiple Criteria Decision Making, Multiattribute Utility Theory: Recent accomplishments and what lies ahead. Management Science, 54(7), 1336-1349. Doi: 10.1287/mnsc.1070.0838

Wedley, W. C. (2013). AHP/ANP before, present and beyond. Paper presented at the ISAHP 201, Kuala Lumpar, Malaysia.

Wedley, W. C., & Choo, E. U. (2001). A unit interpretation of multi-criteria ratios. Paper presented at the Proceedings of the Sixth International Symposium on the Analytic Hierarchy Process, Berne, Switzerland.

Wedley, W. C., & Choo, E. U. (2011). Multi‐Criteria ratios: What is the unit? Journal of Multi‐Criteria Decision Analysis, 18(3-4), 161-171. Doi: 10.1002/mcda.463

Zahedi, F. (1986). The analytic hierarchy process—a survey of the method and its applications. interfaces, 16(4), 96-108. Doi:

Zahir, S. (2007). A new approach to understanding and finding remedy for rank reversals in the additive Analytic Hierarchy Process. Paper presented at the Administrative Sciences Association of Canada (ASAC).