IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 70 Vol. 3 Issue 1 2011 ISSN 1936-6744 AN INTERPRETATION OF THE AHP GLOBAL PRIORITY AS THE EIGENVECTOR SOLUTION OF AN ANP SUPERMATRIX Stan Lipovetsky stan.lipovetsky@gfk.com GfK Custom Research North America 8401 Golden Valley Rd., Minneapolis, MN 55427, USA ABSTRACT Not only the local priority vectors but also the global synthesized priority vector for the alternatives can be obtained from the eigenproblem solution for the ANP supermatrix. This global priority vector retains the main property of any AHP vector – to present the mean dominance of each of the items over the others with respect to the goal of the hierarchy. Keywords: AHP local and global vectors, eigenproblem of supermatrix, ANP supermatrix. “When making a decision of minor importance, I have always found it advantageous to consider all the pros and cons. In vital matters, however, such as the choice of a mate or a profession, the decision should come from the unconscious, from somewhere within ourselves. In the important decisions of personal life, we should be governed, I think, by the deep inner needs of our nature.” Sigmund Freud. 1. Introduction In the article, An Interpretation of the AHP Eigenvector Solution for the Lay Person, in the previous issue of this journal (Lipovetsky, 2010), we showed how the priority vector for an Analytic Hierarchy Process (AHP) pairwise comparison judgment matrix can be derived through an iterative process and that the vector obtained in this way is an eigenvector of the original judgment matrix. Similarly an eigenproblem solution can be obtained for an AHP problem, formulated in the supermatrix of the Analytic Network Process (ANP) rather than as a hierarchy; it gives the overall relative priorities of the alternatives with respect to the goal, and also the synthesized relative priorities of the alternatives with respect to all the other nodes in the model as well. 2. The Hierarchical Synthesis Process Synthesizing local priority vectors into global priority vectors for an AHP model can be easily performed by multiplying the matrix Z comprised of the local eigenvectors of the alternatives with respect to the criteria, times a column vector X of criteria priorities with respect to the goal. This is equivalent to performing additive hierarchical composition. Rob Typewritten Text http://dx.doi.org/10.13033/ijahp.v3i1.90 IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 71 Vol. 3 Issue 1 2011 ISSN 1936-6744 Consider the example shown in Table 1 of “Choosing the best house” (Saaty, 1996, pp. 26-31). This classical AHP example has also been considered in (Saaty & Kearns, 1985, ch.3; Saaty and Vargas, 1994, ch.1). This data was also used for testing some new techniques in (Lipovetsky, 1996, 2005; Lipovetsky & Tishler, 1999; Lipovetsky & Conklin, 2002). In this example three houses are evaluated with respect to 8 criteria. The top row in Table 1 contains the weights of the criteria obtained by pairwise comparing the criteria with respect to the goal. This row vector may be labeled 'X (where the prime denotes transposition) so that X is a column vector. The matrix Z is the 3 × 8 matrix containing the local priority vectors of the alternatives with respect to the criteria in Table 1. The last column contains the product of ZX, the global priority vector. As Z is 3 × 8 and the X is 8 × 1, ZX is a column vector of order 3 × 1. Table 1 Local and global priority vectors for “Choosing the best house” (Saaty, 1996) size yard trans- port facil- ities neigh- borhd condi- tion age finan- ce Criteria weights .1730 .0540 .1881 .0175 .0310 .0363 .1669 .3332 Global vector house A .7536 .6738 .2331 .7466 .7536 .2000 .3333 .0719 .3338 house B .1811 .1007 .0545 .0601 .0653 .4000 .3333 .6491 .3365 house C .0653 .2255 .7124 .1933 .1811 .4000 .3333 .2790 .3296 The ANP supermatrix for an AHP model that includes inner-dependent criteria has been considered by Saaty (1994, pp. 245-6; 1996, pp. 132-133; 2010, p. 190) and is shown below (with a first row of zeros for the goal): ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = IZ YXW 0 0 000 , (1) Our house model can be arranged into a supermatrix W with components X, Y and Z in the form shown in equation (1) from (Saaty, 1996, p. 97). The supermatrix W of our house hierarchy consists of a goal with n criteria and m alternatives. The top row of zeros contains the priorities of the goal node with respect to all the other nodes in the model. The goal is a source node that connects only to other nodes and not from them, so it has only zero priorities with respect to the other nodes. The X component is an n × 1 column vector of weights derived by pairwise comparing the criteria with respect to the goal, Y is an n × n order matrix of priority vectors derived for inner-dependent criteria, Z is an m × n matrix of priority vectors of the alternatives with respect to the criteria, and for a hierarchy, it is necessary to include an identity matrix I of order m × m in the right hand bottom corner of W , thus the matrix W is of the order m + n +1. This formulation leads to a supermatrix W of the 12th order, shown in Table 2. If there were subcriteria, the supermatrix would need to be expanded to accommodate them and all the rows would no longer be stochastic. Any column can be made stochastic by normalizing it. The vector X is the 8 × 1 vector of criteria priorities with respect to the Goal (blue); Y is an 8 × 8 matrix of priority vectors, one for each of the criteria in the case IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 72 Vol. 3 Issue 1 2011 ISSN 1936-6744 where the criteria are inner dependent (shown in yellow), but Y contains only zeros as for a hierarchy where the criteria are not inner dependent; Z is the 3 × 8 matrix of priority vectors of alternatives with respect to the criteria (in green); and I is the 3 × 3 identity matrix (pink). Following Saaty’s approach of raising this matrix to powers until it reaches a stable solution (at the second power in this example), we obtain the global priority vector, with respect to the goal, in the first column in Table 3. It is the same as the results in Table 1 obtained using hierarchic composition. The results for hierarchic composition coincide with the vector product ZX. IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 73 Vol. 3 Issue 1 2011 ISSN 1936-6744 Table 2 The supermatrix W containing the local priority vectors for the house hierarchy GOAL CRITERIA ALTERNATIVES Goal Size Yard Transprt Facilities Nghbrhd Condition Age Finance House A House B House C GOAL Goal 0 0 0 0 0 0 0 0 0 0 0 0 Size 0.173 0 0 0 0 0 0 0 0 0 0 0 Yard 0.054 0 0 0 0 0 0 0 0 0 0 0 Transprt 0.1881 0 0 0 0 0 0 0 0 0 0 0 CRITERIA Facilities 0.0175 0 0 0 0 0 0 0 0 0 0 0 Nghbrhd 0.031 0 0 0 0 0 0 0 0 0 0 0 Condition 0.0363 0 0 0 0 0 0 0 0 0 0 0 Age 0.1669 0 0 0 0 0 0 0 0 0 0 0 Finance 0.3332 0 0 0 0 0 0 0 0 0 0 0 House A 0 0.7536 0.6738 0.2331 0.7466 0.7536 0.2000 0.3333 0.0719 1 0 1 ALTERNATIVES House B 0 0.1811 0.1007 0.0545 0.0601 0.0653 0.4000 0.3333 0.6491 0 1 0 House C 0 0.0653 0.2255 0.7124 0.1933 0.1811 0.4000 0.3333 0.2790 0 0 0 IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 74 Vol. 3 Issue 1 2011 ISSN 1936-6744 Table 3 The limit supermatrix W for the house example; the global priority vector for the alternatives is shown in blue GOAL CRITERIA ALTERNATIVES Goal Size Yard Transprt Facilities Nghbrhd Condition Age Finance House A House B House C GOAL Goal 0 0 0 0 0 0 0 0 0 0 0 0 Size 0 0 0 0 0 0 0 0 0 0 0 0 Yard 0 0 0 0 0 0 0 0 0 0 0 0 Transprt 0 0 0 0 0 0 0 0 0 0 0 0 CRITERIA Facilities 0 0 0 0 0 0 0 0 0 0 0 0 Nghbrhd 0 0 0 0 0 0 0 0 0 0 0 0 Condition 0 0 0 0 0 0 0 0 0 0 0 0 Age 0 0 0 0 0 0 0 0 0 0 0 0 Finance 0 0 0 0 0 0 0 0 0 0 0 0 House A 0.3339 0.7536 0.6738 0.2331 0.7466 0.7536 0.2000 0.3333 0.0719 1 0 0 ALTERNATIVES House B 0.3365 0.1811 0.1007 0.0545 0.0601 0.0653 0.4000 0.3333 0.6491 0 1 0 House C 0.3296 0.0653 0.2255 0.7124 0.1933 0.1811 0.4000 0.3333 0.2790 0 0 1 IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 75 Vol. 3 Issue 1 2011 ISSN 1936-6744 We shall now show a second method of obtaining the synthesized global priority vector. Instead of raising the supermatrix W of equation (1) to powers, we solve the transpose of this matrix in a straightforward manner for its left eigenvectors. The left eigenvectors of (2) coincide with the right eigenvectors of W but are obtained from the transposed matrix: λγγ =′W (2) For a hierarchy of only three levels, such as the house model, the columns of W are stochastic, that is, they sum to one. The left eigenvectors are needed for the transposed matrix because they correspond correctly to the calculations by its blocks for the needed product ZX that gives the synthesized global vector. The left eigenvectors and their eigenvalues of W ′ , the transpose of the matrix in Table 2, are given in Table 4. Three of the columns have eigenvalues equal to 1, and these columns contain the synthesized global (row) vector for the alternative houses in the Goal node row: House C = 0.3296, House B = 0.3365, and House A = 0.3339. In this solution the order of the columns is the reverse of the order of the rows. The synthesized result of the alternatives with respect to each criterion can also be read from the rows of Table 4. It is useful to note the following to clarify the problem. As is well known, Perron proved that every positive matrix has a largest real eigenvalue that is greater than the absolute value of any of the others (including the complex ones). Frobenius carried the work on to matrices with zeros in them, like our supermatrix in Table 2. He proved there could be more than one largest eigenvalue. And the supermatrix W ′ is a transposed stochastic one whose eigenvalues are all one. So when we transpose it in (2) it becomes a stochastic matrix with the rows equaling one, as in our example which has multiple eigenvalues of 1. Table 4 All twelve eigenvectors (columns) and associated eigenvalues (bottom row) for the supermatrix with the synthesized global priorities for the houses in row 1. House C House B House A Fin- anc e Ag- e Con- ditio n Neig- hbo- ur Fac- Ilitie s Tran- sport Ya- rd Si- ze Go- al Goal 0.3296 0.3365 0.3339 -1 -1 -1 -1 -1 -1 -1 -1 1 Size 0.0653 0.1811 0.7536 0 0 0 0 0 0 0 0 0 Yard 0.2255 0.1007 0.6738 0 0 0 0 0 0 0 0 0 Transpor t 0.7124 0.0545 0.2331 0 0 0 0 0 0 0 0 0 Facilities 0.1933 0.0601 0.7466 0 0 0 0 0 0 0 0 0 Neighbo ur 0.1811 0.0653 0.7536 0 0 0 0 0 0 0 0 0 Conditio n 0.4 0.4 0.2 0 0 0 0 0 0 0 0 0 Age 0.3333 0.3333 0.3333 0 0 0 0 0 0 0 0 0 Finance 0.279 0.6491 0.0719 0 0 0 0 0 0 0 0 0 HouseA 0 0 1 0 0 0 0 0 0 0 0 0 IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 76 Vol. 3 Issue 1 2011 ISSN 1936-6744 HouseB 0 1 0 0 0 0 0 0 0 0 0 0 HouseC 1 0 0 0 0 0 0 0 0 0 0 0 Eigen- Values 1 1 1 0 0 0 0 0 0 0 0 0 Inverting the arrangement of the three main eigenvectors to put the houses back into the proper order with which we started, and extracting the synthesis information we are interested in, we obtain Table 5. It shows the global synthesized priorities with respect to the Goal in the first row along with the global synthesized priorities of all the other nodes (in rows). The global priority vectors are the same as the original local priority vectors for a simple three level hierarchy. However, for a hierarchy of more than 3 levels, the synthesis is more interesting as the global priorities are not usually the same as the local priorities for the other nodes in the model. Table 5 Local and global priorities obtained from the main eigenvectors from the supermatrix for the “Choosing the best house” example 3. Conclusion The eigenvectors in Tables 4 and 5 give the same results in the first row for the global synthesis vectors as that obtained using the regular AHP synthesis shown in Table 1. Thus, the global AHP priority can be found by solving the supermatrix’ eigenproblem for its left eigenvectors, which in a way similar to the local eigenvectors represents the mean dominance of each item over the others. REFERENCES Lipovetsky, S. (1996). The synthetic hierarchy method: an optimizing approach to obtaining priorities in the AHP. European Journal of Operational Research, 93, 550-564. Eigenvectors of the Supermatrix Names of nodes in hierarchy House A House B House C Goal .3338 .3365 .3296 Size .7536 .1811 .0653 Yard .6738 .1007 .2255 Transport .2331 .0545 .7124 Facilities .7466 .0601 .1933 Neighbor .7536 .0653 .1811 Condition .2000 .4000 .4000 Age .3333 .3333 .3333 Finance .0719 .6491 .2790 House A 1 0 0 House B 0 1 0 House C 0 0 1 eigenvalues λ 1 1 1 IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 77 Vol. 3 Issue 1 2011 ISSN 1936-6744 Lipovetsky, S., & Tishler, A. (1999). Interval estimation of priorities in the AHP. European Journal of Operational Research, 114, 153-164. Lipovetsky, S., & Conklin, M. (2002). Robust estimation of priorities in the AHP. European Journal of Operational Research, 137, 110-122. Lipovetsky, S. (2005). Analytic hierarchy processing in Chapman-Kolmogorov equations. International J. of Operations and Quantitative Management, 11, 219-228. Lipovetsky, S. (2010). An interpretation of the AHP eigenvector solution for the lay person. IJAHP, 2/2, 159-162. Saaty, T.L. (1994). Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process. Pittsburgh, PA: RWS Publications. Saaty, T.L. (1996). Decision Making with Dependence and Feedback: The Analytic Network Process. Pittsburgh, PA: RWS Publications. Saaty, T.L. (2010). Principia Mathematica Decernendi: Mathematical Principles of Decision Making: Generalization of the Analytic Network Process to Neural Firing and Synthesis. Pittsburgh, PA: RWS Publications. Saaty, T.L., & Kearns, K.P. (1985). Analytical Planning. New York: Pergamon Press. Saaty, T.L., & Vargas, L.G. (1994). Decision Making in Economic, Political, Social and Technological Environment with the Analytic Hierarchy Process. Pittsburgh, PA: RWS Publications. Appendix I. Additional bibliography Bernasconi M., Choirat C. and Seri R. (2010). The analytic hierarchy process and the theory of measurement. Management Science, 56, 699-711. Lipovetsky, S. (2008). Comparison among different patterns of priority vectors estimation methods. International J. of Mathematical Education in Science and Technology, 39, 301-311. Lipovetsky, S. (2009a). Global priority estimation in multiperson decision making. Journal of Optimization Theory and Applications, 140, 77-91. Lipovetsky, S. (2009b). Optimal hierarchy structures for multi-attribute-criteria decisions. Journal of Systems Science and Complexity, 22, 228-242. Saaty, T.L. (1977). A scaling method for priorities in hierarchical structures. Mathematical Psychology, 15, 234-281. Saaty, T.L. (1980). The Analytic Hierarchy Process. New York: McGraw-Hill. IJAHP ARTICLE: Lipovetsky/An Interpretation of the AHP Global Priority as the Eigenvector Solution of an ANP Supermatrix International Journal of the Analytic Hierarchy Process 78 Vol. 3 Issue 1 2011 ISSN 1936-6744 Saaty, T.L. (2000). Decision Making for Leaders. Pittsburgh, PA: RWS Publications. Saaty, T.L. (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, T.L. (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 Rev. R. Acad. Cien. Ser. A. Mat., 102, 251–318. Vaidya O.S. and Kumar S. (2006). Analytic hierarchy process: An overview of applications. European Journal of Operational Research, 169, 1-29. Wasil, E.A., & Golden, B.L. (2003). Celebrating 25 years of AHP-based decision making. Computers and Operations Research, 30, 1419-1420.