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Incomplete Pairwise Comparison Matrices in Multi-Attribute Decision Making S. Bozki*, J. Flp*, L. Rnyai** * Research Group of Operations Research and Decision Systems, Laboratory on Engineering and Management Intelligence, Computer and

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Incomplete Pairwise Comparison Matrices in Multi-Attribute Decision Making

  • S. Bozóki*, J. Fülöp*, L. Rónyai**

*Research Group of Operations Research and Decision Systems, Laboratory

  • n Engineering and Management Intelligence, Computer and Automation

Research Institute, Hungarian Academy of Sciences, Budapest, Hungary

**Informatics Laboratory, Computer and Automation Research Institute,

Hungarian Academy of Sciences, Budapest, Hungary

**Institute of Mathematics, Budapest University of Technology and

  • Economics. Budapest, Hungary
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Outline

  • Pairwise

comparison matrix

  • Incomplete pairwise

comparison matrix

  • Eigenvalue

minimization

  • Incomplete Logarithmic Least Squares
  • Algorithms
  • Example
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Pairwise comparison matrix

It is one of the most often applied technique in Multi-Attribute Decision Making for weighting the criteria and evaluation of the alternatives. Pairwise comparison matrix is a basic concept of the Analytic Hierarchy Process (AHP) developed by Thomas Saaty in 1977.

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Pairwise comparison matrix

A (complete) pairwise comparison matrix A = [aij ]i,j=1,…,n is a matrix of size n×n with the properties as follows: aij > 0; aii = 1; aij = 1/aji for i,j=1,…,n, where aij is the numerical answer given by the decision maker for the question “How many times Criterion i is more important than Criterion j?” The weighting problem is to find the n-dimensional positive weight vector w = (w1 ,w2 ,…,wn )T such that the appropriate ratios of the components of w reflect,

  • r, at least, approximate all the aij values (i,j=1,…,n), given by

the decision maker.

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Incomplete pairwise comparison matrix

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Incomplete pairwise comparison matrix

  • Harker

(1987)

  • Kwiesielewicz

(1996)

  • Shiraishi, Obata, Daigo

(1997, 1998, 2002)

  • van Uden

(2002)

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Undirected graph representation

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Directed graph representation

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Eigenvalue minimization problem

min { λmax (A(x)) | x > 0 }

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Theorem 1: The optimal solution of the eigenvalue minimization problem is unique if and only if the graph G orresponding to the incomplete pairwise comparison matrix is connected.

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Theorem 2: The function λmax (A(x)) attains its minimum and the optimal solutions constitute an (s–1)-dimensional affine set, where s is the number of the connected components in the graph corresponding to the incomplete pairwise comparison matrix.

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Incomplete LLSM

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Theorem 3: The optimal solution of the incomplete LLSM problem is unique if and only if graph corresponding to the incomplete pairwise comparison matrix is connected. Moreover, the solution is explicitly written.

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An 8x8 incomplete example

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An 8x8 incomplete example

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An 8x8 incomplete example

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Conclusions

  • The connectedness of the graph is necessary

and sufficient for a unique solution of both eigenvalue minimization and logarithmic least squares problems for incomplete pairwise comparison matrices.

  • Algorithms are fast and simply implemented.
  • Applicable in practice.
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Main references 1

  • S. Bozóki, J. Fülöp, and L. Rónyai, “On optimal completions of incomplete

pairwise comparison matrices,” Working Paper 2009-1, Research Group of Operations Research and Decision Systems,Laboratory

  • n Engineering and

Management Intelligence,Computer and Automation Research Institute, Hungarian Academy of Sciences, 30 pages, 2009.

  • B. Aupetit, and C. Genest, “On some useful properties of the Perron

eigenvalue

  • f a positive reciprocal matrix in the context of the analytic

hierarchy process. ,” European Journal of Operational Research, vol. 70;

  • pp. 263–268, 1993.
  • G. Crawford G, and C. Williams, “A note on the analysis of subjective

judgment matrices,” Journal of Mathematical Psychology, vol. 29, pp. 387– 405, 1985.

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Main references 2

  • P.T. Harker, “Incomplete pairwise

comparisons in the analytic hierarchy process,” Mathematical Modelling, vol. 9, no. 11, pp. 837–848, 1987

  • J.F.C. Kingman, “A convexity property of positive matrices,”

The Quarterly Journal of Mathematics, vol. 12; pp. 283–284, 1961.

  • M. Kwiesielewicz, “The logarithmic least squares and the generalised

pseudoinverse in estimating ratios,” European Journal of Operational Research, vol. 93, pp. 611–619, 1996.

  • T.L. Saaty, The analytic hierarchy process. New York: McGraw-Hill,

1980.

  • S. Shiraishi, T. Obata, and M. Daigo, “Properties of a positive reciprocal

matrix and their application to AHP,” Journal of the Operations Research Society of Japan, vol. 41, no. 3, pp. 404–414, 1998.

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Thank you for attention.

bozoki@sztaki.hu http://www.sztaki.hu/~bozoki