15-18, 2011, SORRENTO,

Transcription

1 The Eleventh International Symposium on the Analytic Hierarchy Process Dr. Dmitriy BORODIN Prof. Viktor GORELIK Bert Van Vreckem ir. Wim De Bruyn Production Information Systems Lab (University College Ghent), Belgium Dorodnicyn Computing Center of Russian Academy of Sciences, Russia June 15-18, 2011, SORRENTO, Naples, ITALY

2 This presentation is dedicated to our beloved colleague and friend Dr. Irina ZOLTOEVA ( ) who got very significant results in data approximation for the multicriteria decision making problems. In particular, she researched the AHP theory and developed a data approximation technique for decomposition of the main criteria (top of a hierarchy) into particular criterion (bottom of the hierarchy) by expressing them through the middle criterion (by using two types of coefficients values of which are obtained from a decision-maker) and approximating the inconsistent system of equations by a consistent one.

3 1. Introduction: Data Approximation 2. Pairwise comparison matrix in AHP: a reminder 3. Two PCM approximation techniques 4. Short overview of PCM consistency-driven approaches 5. Conclusions and Future work 6. References

4 Terms: Approximation, Correction, Fitting If for some reason a problem is inconsistent, we can try and force it to be consistent. What s the Motivation? A mathematical model is a rough reflection of a reality! Assumptions, human factor, etc. reduce the accuracy For example, if a set of linear equations doesn t have a solution, it can be approximated by a consistent set by minor correction of its elements

5 The following set of equations doesn t have a solution A , b The correction matrix and the solution vector x

6 1. Total Least Squares for parameter estimation S. Van Huffel (Belgium, 1987) and S. Van Huffel J. Vandewalle (1987, 1988,, 2011) 2. Optimal multiparameter approximation of inconsistent sets of linear equations I. Eremin, V. Mazurov, N. Astafyev (Russia, 1983), A. Vatolin (1984,, 1992) 3. Approximation of linear programming models V. Gorelik (Russia, 1998), V. Erokhin (2001), O. Muravyeva (2003), I. Zoltoeva (2004), R. Pechonkin (2006)

7 A decision maker provides as much information about a solution method (weights, bounds, etc.) as he/she could but without requiring consistency. Then inconsistency (a PCM is an excellent example of inconsistency) can be removed by correcting the decision-maker estimations in a minimum way (by an approximation criteria)

8 The AHP provides us with an effective method to present human evaluations numerically. The main role in this method belongs to a pairwise comparison matrix. It represents expert judgments for this or that object or event: Each element aij of the matrix shows the importance of object A i in comparison with A j, The judjments are given according to a special scale. Once the matrix is filled in, an importance of each criteria must be evaluated. The weight coefficients do exist if a ij a jk =a ik for all i,j,k (particularly, a ij =1/a ji )

9 for j 1 Comparisons are written down in the d table 1 (matrix) according to m i j Mat the chosen scale. Only the first half of the T if i j j i matrix should be filled-in m by a decision-maker. For each criteria i j Mat there T i j otherwise is a separate matrix. A w1 w1 w1. w1 w2 w nc1 w w w ( a ) w ij n n 1 w2 w n.... w w w. w1 w2 w n n n n C( dmat) for i 1d m C( 9T empc1) By the construction method this matrix is square, positive and reciprocal wi w j is a comparison between elements i and j

10 As it is proved, a positive reciprocal matrix is consistent if and only if λ max = n, where λ max is a maximum eigenvalue of the matrix and n is its dimension. Only in this case the eigenvector gives the precise values of the criteria weights. But usually the pairwise comparison matrix is inconsistent ie aija jk a, ik i j is true not for all i, j, k {1,2,..., r} And the values of criteria weights derived from such matrix can t be considered reliable.

11 We formulate the approximation problem as follows (where H is a correction matrix) 2 H min, so that (A H) is consistent, ie ( a h )( a h ) a, h where belongs to H. ij ij jk jk ik ik Two approximation methods were developed : h ij 1. Using the method of Lagrange multipliers. The following problem of correcting the elements of the pairwise comparison matrix is being investigated: r1 i1 2 h min, (a ij ij h ij)(a jk h jk) aik hik ji And the Lagrange multiplier method is applied as follows: r1 2 L L h ( a h a h h h h ) ij ijk ij jk jk ij ik ij jk h ij a h ijk jk ijk jk i1 ji i, j, k h k j 2 ( ) 0 ij

12 r1 1 ji 2 ij i ij ij j i h min, W (a h )W, i, j i, W 1. We express the PCM coefficients by their definition through the criteria weights and then formulate the correction problem 1 a a a a a r1, r x x2.. 0r( r1) / 2.. e 1.. x( r( r1) / 2) 1 Thus we have the correction problem: inconsistent linear system (represented as a matrix) with a spare structure

13 The problem is to find matrix H* so that system (A+H*)x=e is consistent and the following * condition is true The correction matrix H: 0 h h h H h h r1,r H min H p r ( r1) r ( r1) ( 1) r H,(AH)xe,xR r 23 2r r1,r p r(r1) 2 ( h, h,..., h, h,..., h,..., h ) R, And we have an unconstrained minimization problem: r(,x) p min. I. Zoltoeva (2006) proved that this problem could be solved by the TLN algorithm. If p=2 the problem is deduced to a set of algebraic linear equations (least squares method). If p equals infinity it s a linear programming problem,x

14 1) Simplified methods of constructing the matrix (V. Nogin, 2004) etalon method Problem: the information is not objective because all elements are compared only with one ( etalon ) and not with each other 2) Interval estimations (V. Podinovsky) Problem: this method provides reasonably good results but it demands to perform two estimations min and max (bounds), which gives an expert some additional work and reduces the accuracy of the final result by the confusion of two estimates of the same object 3) Approximation approach (E. Dopazo and González-Pachón, 2003).

15 1) PCM approximation is an interesting and promising approach 2) Further research and development needed -> next slide

16 1) Saaty s index-driven approximation A PCM is approximated to the necessary value of the CR 2) Numerical examples on real matrices 3) Sensitivity analysis 4) Approximation for group decision making and ANP 5) Development of the approximation software for efficient testing

17 Van Huffel S. Analysis of the total least squares problem and its use in parameter estimation // PhD thesis, Dept. of Electr. Eng., K.U.Leuven, Belgium, June Van Huffel S., Vandewalle J. Subset selection using the total least squares approach in collinearity problems with errors in the variables // Linear Algebra and its Applications, 1987, Vol. 88/89, pp Gorelik, V., Zoltoeva I. Matrix correction of inconsistent linear systems with the spare matrix structure // Journal for Modeling, decomposition and optimization of complex dynamic processes. Moscow, pp Saaty, T. 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. Rev. R. Acad. Cien. Serie A. Mat. VOL. 102 (2), 2008, pp Hansen P.C. Regularization Tools. A Matlab package for analysis and Solution of Discrete ill-posed problems. Department of Mathematical Modelling, Technical University of Denmark, June September 2001, 109p. Pechenkin R. The Total least Norm Algorithm Modified by Considering the Error on The Rigth-Hand Side of Overdetermined Linear Systems. Journal of Electrical Engineering, vol. 51, NO. 12/s, 2003, P. 6-9.

18 Thank you for your attention! Dmitriy BORODIN