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Eigenvectors and Decision Making Analytic Hierarchy Process (AHP) Positive Symmetrically Reciprocal Matrices (PSRMs) and Transitive PSRMs Estimation methods -- How close? Close to what? Close in what sense?

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Analytic Hierarchy Process “ In the 1970’s the Egyptian government asked Tom Saaty, a pioneering mathematician with a fistful of awards, to help clarify the Middle East conflict. The Egyptians needed a coherent, analytical way to assess the pros and cons of their less than cozy strategic relationship with the Soviet Union. Saaty, a Wharton professor with a background in arms-control research, tackled the question with game theory.”

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Analytic Hierarchy Process “ The Egyptian government was pleased with his work (and eventually did ask the Russians to leave), but Saaty himself wasn’t satisfied with the process. He felt his conclusion was incomplete – that important but intangible information was left out of the final equation because game theory was too rigid. ‘I couldn’t use it to solve a real-life problem.’”

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Analytic Hierarchy Process Xerox(Software: BoeingExpert IBMChoice) Northrop Grumman US Steel Corning Governments of U.S., South Africa, Canada, and Indonesia

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Analytic Hierarchy Process Decision: Choose between a Toyota, Honda, or Citation (Chevrolet)

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Analytic Hierarchy Process One set of criteria: cost, dependability, size, and aesthetics First we want to assign numbers to these four criteria that represent their relative importance w 1, w 2, w 3, w 4. Cost Depen Size Aesth Cost1233 Dependability1/21 33 Size1/31/311/2 Aesthetics1/31/321 This matrix has maximum eigenvalue approximately 4.08 and eigenvector is (.44,.31,.10,.15).

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Analytic Hierarchy Process A n x n matrix x n x 1 λ scaler Ax = λx (λI-A)x = 0 det(λI-A) = 0

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Analytic Hierarchy Process

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Generate reciprocal matrices for each criterion, comparing the three types of cars pairwise on each criterion as shown below. For Dependability: ToyHonCitWeights Toy 1 1 3.43 Hon 1 1 3.43 Cit 1/31/3 1.14 For Cost: ToyHonCitWeights Toy 1 1 2.40 Hon 1 1 1.35 Cit 1/2 1 1.25

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For Size: ToyHonCitWeights Toy 1 3 1.43 Hon 1 1 3.14 Cit 1/31/3 1.43 For Aesthetics: ToyHonCitWeights Toy 1 4 3.63 Hon 1/4 1 2.22 Cit 1/31/2 1.15

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The weight for each criterion is then multiplied by the weight for each car within that criterion, and added across criteria as shown below: Cost Depen Size AesthComposite Priorities (.44) (.31) (.10) (.15) Toy.40.43.43.63.45 Hon.35.43.14.33.33 Cit.25.14.43.15.22

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Positive Symmetrically Reciprocal Matrices n x n matrix A a ij > 0 a ji = 1/a ij Transitive or consistent: a ik = a ij a jk w i /w j = w i /w k x w k /w j (The entries a ij represent the ratio w i /w j. )

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Cost Depen Size Aesth Costw 1 /w 1 w 1 /w 2 w 1 /w 3 w 1 /w 4 Dependabilityw 2 /w 1, w 2 /w 2 w 2 /w 3 w 2 /w 4 Sizew 3 /w 1 w 3 /w 2 w 3 /w 3 w 3 /w 4 Aestheticsw 4 /w 1 w 4 /w 2 w 4 /w 3 w 4 /w 4 w 1 /w 1 w 1 /w 2 w 1 /w 3 w 1 /w 4 w 1 w 1 w 2 /w 1, w 2 /w 2 w 2 /w 3 w 2 /w 4 w 2 = 4 w 2 w 3 /w 1 w 3 /w 2 w 3 /w 3 w 3 /w 4 w 3 w 3 w 4 /w 1 w 4 /w 2 w 4 /w 3 w 4 /w 4 w 4 w 4

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Positive Symmetrically Reciprocal Matrices Theorem (Perron-Frobenius): (Let A be non-negative and irreducible.) (1) A has a real positive simple (not multiple) eigenvalue λ max (Perron root) which is not exceeded in modulus by any other eigenvalue of A. (2) The eigenvector of A corresponding to the eigenvalue λ max has positive components and is essentially unique (to within multiplication by a constant). Theorem: A positive, reciprocal matrix is consistent if and only if λ max = n. (Saaty. J of Mathematical Psychology, 1977).

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Positive Symmetrically Reciprocal Matrices Theorem: Let R be an n x n positive reciprocal matrix with entries 1/S 1 and let λ max denote the largest eigenvalue of R in modulus, which is known to be real and positive. Then n < λ max < 1 + ½(n-1)(S + 1/S), the lower and upper bound being reached if and only if R is supertransitive or maximally intransitive, respectively. (Aupetit and Genest – European Journal of Operational Research, 1993):

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Positive Symmetrically Reciprocal Matrices Theorem:A transitive matrix is SR and has rank one. (Farkas, Rozsa, Stubnya, Linear Algebra Appl., 1999.)

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Close?

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Saaty: Limit of normalized row sums of powers of the matrix. μ =(λ max -n)/(n-1) If we consider inconsistencies in the powers of A, generated by cycles of length 1, 2, 3, … then the right eigenvector represents the dominance of one alternative over all others through these cycles. (Mathematics Magazine, 1987.)

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Close? Logarithmic Least Square Method Least Squares Another kind of eigenvector in another metric, the “max eigenvector” minimizes relative error (Elsner & vanden Driesscle, Linear Algebra and Its Applications, 2004).

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Finally Why do I care? What am I doing with these matrices and eigenvectors?

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