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Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices Kristóf Ábele-Nagy Eötvös Loránd University (ELTE); Corvinus University of Budapest (BCE) Sándor Bozóki Computer and Automation Research Institute, Hungarian Academy of Sciences (MTA SZTAKI); Corvinus University of Budapest (BCE)

15 December, 2010

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Outline Pairwise comparison matrix Incomplete pairwise comparison matrix Eigenvalue optimization Cyclic coordinates Newton’s method in one variable Newton’s method in higher dimensions

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Given n objects with weights w1, w2, w3, . . . , wn. The pairwise comparison matrix is defined as follows:

        1

w1 w2 w1 w3

. . .

w1 wn w2 w1

1

w2 w3

. . .

w2 wn w3 w1 w3 w2

1 . . .

w3 wn

. . . . . . . . . ... . . .

wn w1 wn w2 wn w3

. . . 1         ,

where

wij > 0, wij = 1 wji , wij = wikwkj.

for any i, j, k indices.

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In real decision situations, weights are unknown, but pairwise comparisons can be made:

A =         1 a12 a13 . . . a1n a21 1 a23 . . . a2n a31 a32 1 . . . a3n

. . . . . . . . . ... . . .

an1 an2 an3 . . . 1         ,

where

aij > 0, aij = 1 aji .

for i, j = 1, . . . , n. The aim is to determine the weight vector

w = (w1, w2, . . . , wn) ∈ Rn

+.

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In the Eigenvector Method (EM) the approximation wEM

• f w is defined by

AwEM = λmaxwEM,

where λmax denotes the maximal eigenvalue, also known as Perron eigenvalue, of A and wEM denotes the the right-hand side eigenvector of A corresponding to λmax. By Perron’s theorem, wEM is positive and unique up to a scalar multiplication. The most often used normalization is

n

• i=1

wEM

i

= 1.

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Saaty defined the inconsistency ratio as CR =

λmax−n n−1

RIn

,

where λmax is the Perron eigenvalue of the complete pairwise comparison matrix given by the decision maker, and RIn is defined as λmax−n

n−1 , where λmax is an average

value of the Perron eigenvalues of randomly generated

n × n pairwise comparison matrices.

It is well known that λmax ≥ n and equals to n if and only if the matrix is consistent, i.e., the transitivity property holds. It follows from the definition that CR is a positive linear transformation of λmax. According to Saaty, larger value of CR indicates higher level

• f inconsistency and the 10%-rule (CR ≤ 0.10) separates

acceptable matrices from unacceptable ones.

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Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements)

A =         1 a12 − . . . a1n 1/a12 1 a23 . . . − − 1/a23 1 . . . a3n

. . . . . . . . . ... . . .

1/a1n − 1/a3n . . . 1         .

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Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements)

A =         1 a12 x1 . . . a1n 1/a12 1 a23 . . . − 1/x1 1/a23 1 . . . a3n

. . . . . . . . . ... . . .

1/a1n − 1/a3n . . . 1         .

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Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements)

A =         1 a12 x1 . . . a1n 1/a12 1 a23 . . . xd 1/x1 1/a23 1 . . . a3n

. . . . . . . . . ... . . .

1/a1n 1/xd 1/a3n . . . 1         ,

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Incomplete pairwise comparison matrix ( = pairwise comparison matrix with missing elements)

A =         1 a12 x1 . . . a1n 1/a12 1 a23 . . . xd 1/x1 1/a23 1 . . . a3n

. . . . . . . . . ... . . .

1/a1n 1/xd 1/a3n . . . 1         ,

where x1, x2, . . . , xd ∈ R+.

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Based on the idea above, Shiraishi, Obata and Daigo considered the eigenvalue optimization problems as follows. In case of one missing element, denoted by x, the

λmax(A(x)) to be minimized: min

x>0 λmax(A(x)).

In case of more than one missing elements, arranged in vector x, the aim is to solve

min

x>0 λmax(A(x)).

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Graph representation of a pairwise comparison matrix Given A incomplete pairwise comparison matrix of size

n × n. Graph G = (V, E) is defined as follows: V = {1, 2, . . . , n} E = {e(i, j) | aij (and aji) are given and i = j}

Special case: all the comparisons are given, the corresponding graph is Kn.

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Graph representation of a pairwise comparison matrix

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Theorem (B., Fülöp, Rónyai, 2010): The optimal solution of the eigenvalue minimization problem

min

x>0 λmax(A(x)).

is unique if and only if the graph G corresponding to the incomplete pairwise comparison matrix is connected. If graph G corresponding to the incomplete pairwise comparison matrix is connected, then by using the exponential parametrization x1 = ey1, x2 = ey2, . . . xd = eyd, the eigenvalue minimization problem is transformed into a strictly convex optimization problem.

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Example

Q =    1 2 x 1/2 1 4 1/x 1/4 1    . λmax(Q(x)) and, by using the exponential scaling x = et, λmax(Q(et)) are plotted.

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Algorithms for solving the eigenvalue minimization problem

min

x>0 λmax(A(x)).

cyclic coordinates with Matlab’s function fminbnd cyclic coordinates univariate Newton’s method multivariate Newton’s method

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Method of cyclic coordinates

M(x) =                1 5 3 7 6 6 1/3 1/4 1/5 1 x1 5 x2 3 x3 1/7 1/3 1/x1 1 x4 3 x5 6 x6 1/7 1/5 1/x4 1 x7 1/4 x8 1/8 1/6 1/x2 1/3 1/x7 1 x9 1/5 x10 1/6 1/3 1/x5 4 1/x9 1 x11 1/6 3 1/x3 1/6 1/x8 5 1/x11 1 x12 4 7 1/x6 8 1/x10 6 1/x12 1               

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Method of cyclic coordinates Let x(k)

i

denote the value of xi in the k-th step of the iteration, which has d (in the example, d = 12) substeps for each k. For k = 0 : Let the initial points be equal to 1 for every variable:

x(0)

i

:= 1 (i = 1, 2, . . . , d).

while

max

i=1,2,...,dxk i − xk−1 i

> T x(k)

i

:= arg min

xi λmax(M(x(k) 1 , . . . , x(k) i−1, xi, x(k−1) i+1

, . . . , x(k−1)

d

)), i = 1, 2, . . . , d

next k end while

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Method of cyclic coordinates Focus on min

xi λmax(M(x(k) 1 , . . . , x(k) i−1, xi, x(k−1) i+1

, . . . , x(k−1)

d

))

Matlab’s function fminbnd solves is directly and fast. Univariate Newton’s method can be also applied.

Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 19/39

min

x>0 λmax(A(x))

Let x = et and L(t) = λmax(et).

tn+1 = tn − L′(tn) L′′(tn) = tn −

∂λmax(x) ∂x ∂2λmax(x) (∂x)2

· etn + ∂λmax(x)

∂x

.

By Harker, formal derivatives ∂λmax(x)

∂x

and ∂2λmax(x)

(∂x)2

are known.

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Harker’s formula for the first derivative ∂λmax(x)

∂x

∂λmax(A) ∂aij

• i > j
• =
• [y(A)ix(A)j] − [y(A)jx(A)i]

[aij]2

• where vectors x(A), y(A) are the right-hand side and

left-hand side eigenvectors of A, respectively. Normalization y(A)Tx(A) = 1 is applied.

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Harker’s formula for the second derivative

∂2λmax(A) ∂aij∂akl = (x(A)y(A)T)liQ+

jk + (x(A)y(A)T)jkQ+ li

− (x(A)y(A)T)kiQ+

jl + (x(A)y(A)T)jlQ+ ki

[akl]2 − (x(A)y(A)T)ljQ+

ik + (x(A)y(A)T)ikQ+ lj

[aij]2 + (x(A)y(A)T)klQ+

il + (x(A)y(A)T)ilQ+ kj

[aij]2[akl]2

if i = k or j = l,

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Harker’s formula for the second derivative

∂2λmax(A) ∂aij∂akl = 2(x(A)y(A)T)ij [aij]3 + 2(x(A)y(A)T)jiQ+

ii

−2 (x(A)y(A)T)iiQ+

jj + (x(A)y(A)T)jjQ+ ii

[aij]2 +2 (x(A)y(A)T)ijQ+

ij

[aij]4

if i = k and j = l, where Q = λmax(A)I − A and Q+ denotes the Moore-Penrose pseudoinverse of Q.

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Multivariate Newton’s method

tn+1 = tn − [HL(tn)]−1∇L(tn).

Newton’s method in eigenvalue optimization for incomplete pairwise comparison matrices – p. 24/39

Computational results: All methods mentioned above are fast enough for typical matrices in multi-attribute decision making; in the talk’s example (8 × 8 matrix, 12 variables, 20 cycles with fminbnd: 0.3 seconds, 1-variable Newton: 5.3 seconds); as a test, in a 150 × 150 matrix, ∼ 10000 variables, 1 cycle takes 1.5 hours with fminbnd.

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Applications of the results: A generalization of the Eigenvector Method for the incomplete case

CR-inconsistency can be computed during the filling in

process, as soon as a connected graph is given User may get an automatic warning in case of misprints, detected as a high jump in CR-inconsistency.

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Questions: How many comparisons are needed, if less than

n(n − 1)/2?

Thresholds for warning the user? Other inconsistency indices, presented by Attila Poesz

• n Monday (session 2A).

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References 1/2 Ábele-Nagy, K. [2010]: Incomplete pairwise comparison matrices in multi-attribute decision making, Master’s Thesis, Eötvös Loránd University. Bozóki, S., Fülöp, J., Rónyai, L. [2010]: On optimal completions of incomplete pairwise comparison matrices, Mathematical and Computer Modelling, 52, pp. 318–333. Harker, P .T. [1987]: Incomplete pairwise comparisons in the analytic hierarchy process. Mathematical Modelling, 9(11),

• pp. 837–848.

Harker, P .T. [1987]: Derivatives of the Perron root of a positive reciprocal matrix: with application to the Analytic Hierarchy Process. Applied Mathematics and Computation, 22, pp. 217–232.

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References 2/2 Saaty, T.L. [1977]: A scaling method for priorities in hierarchical structures, Journal of Mathematical Psychology, 15, pp. 234–281. Saaty, T.L. [1980]: The analytic hierarchy process, McGraw-Hill, New York. Shiraishi, S., Obata, T., Daigo, M. [1998]: Properties of a positive reciprocal matrix and their application to AHP . Journal of the Operations Research Society of Japan, 41(3), pp. 404–414. Shiraishi, S., Obata, T. [2002]: On a maximization problem arising from a positive reciprocal matrix in AHP . Bulletin of Informatics and Cybernetics, 34(2), pp. 91–96.

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Thank you for attention. bozoki@sztaki.hu http://www.sztaki.hu/∼bozoki

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