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?

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.”

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.’”

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

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

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).

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

Analytic Hierarchy Process

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

For Size: ToyHonCitWeights Toy Hon Cit 1/31/ For Aesthetics: ToyHonCitWeights Toy Hon 1/ Cit 1/31/2 1.15

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 Hon Cit

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. )

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

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).

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):

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

Close?

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.)

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).

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