Pairwise comparison matrices and efficient weight vectors Sndor - - PowerPoint PPT Presentation

Pairwise comparison matrices and efficient weight vectors

Sándor BOZÓKI

Institute for Computer Science and Control Hungarian Academy of Sciences (MTA SZTAKI), Corvinus University of Budapest December 14, 2016

Efficiency – p. 1/32

Multi Criteria Decision Making Analytic Hierarchy Process Criterion tree Pairwise comparison matrix

Efficiency – p. 2/32

Criterion tree

Efficiency – p. 3/32

Pairwise comparison matrix

In practical problems

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

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

an1 an2 an3 . . . 1         ,

is given, where for any i, j = 1, . . . , n indices

aij > 0, aij =

1 aji.

The aim is to find the w = (w1, w2, . . . , wn)⊤ ∈ Rn

+ weight

vector.

Efficiency – p. 4/32

Weighting methods Eigenvector Method (Saaty): Aw = λmaxw. Least Squares Method (LSM): min

n

• i=1

n

• j=1
• aij − wi

wj 2

n

• i=1

wi = 1, wi > 0, i = 1, 2, . . . , n. Logarithmic Least Squares Method (LLSM): min

n

• i=1

n

• j=1
• log aij − log wi

wj 2

n

• i=1

wi = 1, wi > 0, i = 1, 2, . . . , n.

Efficiency – p. 5/32

       1 1 4 9 1 1 7 5 1/4 1/7 1 4 1/9 1/5 1/4 1        , wEM =        0.404518 0.436173 0.110295 0.049014        , w∗ =        0.436173 0.436173 0.110295 0.049014        wEM

i

wEM

j

• =

       1 0.9274 3.6676 8.2531 1.0783 1 3.9546 8.8989 0.2727 0.2529 1 2.2503 0.1212 0.1124 0.4444 1        w′

i

w′

j

• =

       1 1 3.9546 8.8989 1 1 3.9546 8.8989 0.2529 0.2529 1 2.2503 0.1124 0.1124 0.4444 1        .

Efficiency – p. 6/32

       1 1 4 9 1 1 7 5 1/4 1/7 1 4 1/9 1/5 1/4 1        , wEM =        0.404518 0.436173 0.110295 0.049014        , w∗ =        0.436173 0.436173 0.110295 0.049014        wEM

i

wEM

j

• =

       1 0.9274 3.6676 8.2531 1.0783 1 3.9546 8.8989 0.2727 0.2529 1 2.2503 0.1212 0.1124 0.4444 1        w′

i

w′

j

• =

       1 1 3.9546 8.8989 1 1 3.9546 8.8989 0.2529 0.2529 1 2.2503 0.1124 0.1124 0.4444 1        .

Efficiency – p. 7/32

       1 1 4 9 1 1 7 5 1/4 1/7 1 4 1/9 1/5 1/4 1        , wEM =        0.404518 0.436173 0.110295 0.049014        , w∗ =        0.436173 0.436173 0.110295 0.049014        wEM

i

wEM

j

• =

       1 0.9274 3.6676 8.2531 1.0783 1 3.9546 8.8989 0.2727 0.2529 1 2.2503 0.1212 0.1124 0.4444 1        w′

i

w′

j

• =

       1 1 3.9546 8.8989 1 1 3.9546 8.8989 0.2529 0.2529 1 2.2503 0.1124 0.1124 0.4444 1        .

Efficiency – p. 8/32

       1 1 4 9 1 1 7 5 1/4 1/7 1 4 1/9 1/5 1/4 1        , wEM =        0.404518 0.436173 0.110295 0.049014        , w∗ =        0.436173 0.436173 0.110295 0.049014        wEM

i

wEM

j

• =

       1 0.9274 3.6676 8.2531 1.0783 1 3.9546 8.8989 0.2727 0.2529 1 2.2503 0.1212 0.1124 0.4444 1        w′

i

w′

j

• =

       1 1 3.9546 8.8989 1 1 3.9546 8.8989 0.2529 0.2529 1 2.2503 0.1124 0.1124 0.4444 1        .

Efficiency – p. 9/32

The multi-objective optimization problem is as follows:

min

xi > 0 ∀i

• aij − xi

xj

• i=j

Efficiency – p. 10/32

Efficiency (Pareto optimality)

Let A = [aij]i,j=1,...,n be an n × n pairwise comparison matrix and w = (w1, w2, . . . , wn)⊤ be a positive weight vector. Definition: weight vector w is called efficient, if there exists no positive weight vector w′ = (w′

1, w′ 2, . . . , w′ n)⊤ such that

• aij − w′

i

w′

j

• ≤
• aij − wi

wj

• for all 1 ≤ i, j ≤ n,
• akℓ − w′

k

w′

ℓ

• <
• akℓ − wk

wℓ

• for some 1 ≤ k, ℓ ≤ n.

Efficiency – p. 11/32

An efficient weight vector cannot be improved such that every element of the pairwise comparison matrix is approximated at least as good, and at least one element is approximated strictly better.

Efficiency – p. 12/32

Test of efficiency

Given pairwise comparison matrix A and weight vector w,

• ur goal is check whether w is efficient.

Efficiency – p. 13/32

Let vi = log wi, 1 ≤ i ≤ n, and bij = log aij, 1 ≤ i, j ≤ n,

I =

• (i, j)
• aij < wi

wj

• J =
• (i, j)
• aij = wi

wj , i < j

• min
• (i,j)∈I

−sij yj − yi ≤ −bij

for all (i, j) ∈ I,

yi − yj + sij ≤ vi − vj

for all (i, j) ∈ I,

yi − yj = bij

for all (i, j) ∈ J,

sij ≥ 0

for all (i, j) ∈ I,

y1 = 0

Variables are yi, 1 ≤ i ≤ n and sij ≥ 0, (i, j) ∈ I.

Efficiency – p. 14/32

min

• (i,j)∈I

−sij yj − yi ≤ −bij

for all (i, j) ∈ I,

yi − yj + sij ≤ vi − vj

for all (i, j) ∈ I,

yi − yj = bij

for all (i, j) ∈ J,

sij ≥ 0

for all (i, j) ∈ I,

y1 = 0

Theorem (Bozóki, Fülöp, 2016): The optimum value of the linear program above is at most 0 and it is equal to 0 if and only if weight vector w is efficient. Denote the optimal solution to the LP above by

(y∗, s∗) ∈ Rn+|I|. If weight vector w is inefficient, then weight

vector exp(y∗) is efficient and dominates w internally.

Efficiency – p. 15/32

Pairwise Comparison Matrix Calculator

The efficiency of a weight vector can be tested at

pcmc.online

If the weight vector is found to be inefficient, then a dominating efficient weight vector is provided. PCMC deals with incomplete pairwise comparison matrices, too.

Efficiency – p. 16/32

Characterization of efficiency

Definition: Let A = [aij]i,j=1,...,n ∈ PCMn and

w = (w1, w2, . . . , wn)⊤ be a positive weight vector. Directed

graph (V, −

→ E )A,w is defined as follows: V = {1, 2, . . . , n} and − → E =

• arc(i → j)
• wi

wj ≥ aij, i = j

• .

Theorem (Blanquero, Carrizosa and Conde, 2006): Weight vector w is efficient if and only if (V, −

→ E )A,w is

strongly connected, that is, there exist directed paths from i to j and from j to i for all pairs of i = j nodes.

Efficiency – p. 17/32

A =        1 2 6 2 1/2 1 4 3 1/6 1/4 1 1/2 1/2 1/3 2 1        , wEM =        6.01438057 4.26049429 1 2.0712416        XEM =        1 1.41 6.01 2.90 0.71 1 4.26 2.06 0.1663 0.23 1 0.48 0.34 0.49 2.07 1        wEM =        6.01438057 4.26049429 1 2.0712416       

Efficiency – p. 18/32

Efficiency of the principal right eigenvector

Efficiency – p. 19/32

Special cases

Efficient principal right eigenvector: simple perturbed PCM double perturbed PCM Inefficient principal right eigenvector:

PCM with arbitrarily small inconsistency

Numerical examples

Efficiency – p. 20/32

       1 1 4 9 1 1 7 5 1/4 1/7 1 4 1/9 1/5 1/4 1        , wEM =        0.404518 0.436173 0.110295 0.049014        , w∗ =        0.436173 0.436173 0.110295 0.049014        wEM

i

wEM

j

• =

       1 0.9274 3.6676 8.2531 1.0783 1 3.9546 8.8989 0.2727 0.2529 1 2.2503 0.1212 0.1124 0.4444 1        w′

i

w′

j

• =

       1 1 3.9546 8.8989 1 1 3.9546 8.8989 0.2529 0.2529 1 2.2503 0.1124 0.1124 0.4444 1        .

Efficiency – p. 21/32

Fichtner’s metric

Theorem (Fichtner, 1984) Let d : PCMn × PCMn → R be as follows:

d(A, B) def =

• n
• i=1
• wEM(A)

i

− wEM(B)

i

2 + |λmax(A) − λmax(B)| 2(n − 1) + +χ(A, B) |λmax(A) + λmax(B) − 2n| 2(n − 1) ,

where

χ(A, B) =

• if A = B,

1

if A = B. Then, d is a metric in PCMn with the following properties:

Efficiency – p. 22/32

Fichtner’s metric

(a) for every A ∈ PCMn, XEM(A) is the optimal solution of the problem min{d(A, X)|X is consistent}; (b)

min{d(A, X)|X is consistent} = d(A, XEM(A)) = λmax(A)−n

n−1

.

Optimality with respect to a nice objective function does not exclude inefficiency. Note that Fichtner’s metric is not continuous, nor a monotonic increasing function of

• aij − xi

xj

• .

Efficiency – p. 23/32

Simple perturbed PCM

Consider a consistent matrix:

A =         1 x1 x2 . . . xn−1

1 x1

1

x2 x1

. . .

xn−1 x1 1 x2 x1 x2

1 . . .

xn−1 x2

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

1 xn−1 x1 xn−1 x2 xn−1

. . . 1         ∈ PCMn,

then perturb a single element and its reciprocal. The perturbation is realized by a multiplication by δ > 0, δ = 1, while the reciprocal element is divided by δ.

Efficiency – p. 24/32

Simple perturbed PCM: wEM is efficient Aδ =         1 δx1 x2 . . . xn−1

1 δx1

1

x2 x1

. . .

xn−1 x1 1 x2 x1 x2

1 . . .

xn−1 x2

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

1 xn−1 x1 xn−1 x2 xn−1

. . . 1         ∈ PCMn.

Theorem (Ábele-Nagy, Bozóki, 2016): The principal right eigenvector of a simple perturbed pairwise comparison matrix is efficient. Proof is based on the explicit formulas of wEM.

Efficiency – p. 25/32

Double perturbed PCM (n ≥ 4)

             1 δx1 γx2 x3 . . . xn−1

1 δx1

1

x2 x1 x3 x1

. . .

xn−1 x1 1 γx2 x1 x2

1

x3 x2

. . .

xn−1 x2 1 x3 x1 x3 x2 x3

1 . . .

xn−1 x3

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

1 xn−1 x1 xn−1 x2 xn−1 x3 xn−1

. . . 1                           1 δx1 x2 x3 . . . xn−1

1 δx1

1

x2 x1 x3 x1

. . .

xn−1 x1 1 x2 x1 x2

1 γ x3

x2

. . .

xn−1 x2 1 x3 x1 x3 x2 γx3

1 . . .

xn−1 x3

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

1 xn−1 x1 xn−1 x2 xn−1 x3 xn−1

. . . 1             

Efficiency – p. 26/32

Double perturbed PCM: wEM is efficient

Theorem (Ábele-Nagy, Bozóki, Rebák, 2016): The principal right eigenvector of a double perturbed pairwise comparison matrix is efficient. Proof is based on the explicit formulas of wEM and the characterization of efficiency by a strongly connected digraph.

Efficiency – p. 27/32

A(p, q) =              1 p p p . . . p p 1/p 1 q 1 . . . 1 1/q 1/p 1/q 1 q . . . 1 1

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

1/p 1 1 1 . . . 1 q 1/p q 1 1 . . . 1/q 1              ,

• Proposition. (Bozóki, 2014):

Let q be positive and q = 1. Then wEM is internally inefficient, therefore inefficient. Furthermore, CR inconsistency can be arbitrarily small if q is close enough to

1.

Efficiency – p. 28/32

Weak efficiency

Definition: weight vector w is called weakly efficient, if there exists no positive weight vector w′ = (w′

1, w′ 2, . . . , w′ n)⊤

such that

• aij − w′

i

w′

j

• <
• aij − wi

wj

• for all 1 ≤ i = j ≤ n.

Theorem (Bozóki, Fülöp, 2016): The principal eigenvector of a pairwise comparison matrix is weakly efficient.

Efficiency – p. 29/32

Main references 1/2

Blanquero, R., Carrizosa, E., Conde, E. (2006): Inferring efficient weights from pairwise comparison matrices, Mathematical Methods of Operations Research 64(2):271–284 Conde, E., Pérez, M.d.l.P .R. (2010): A linear optimization problem to derive relative weights using an interval judgement matrix, European Journal of Operational Research 201(2):537–544 Fichtner, J. (1984): Some thoughts about the Mathematics

• f the Analytic Hierarchy Process, Report 8403, Universität

der Bundeswehr München, Fakultät für Informatik, Institut für Angewandte Systemforschung und Operations Research, Werner-Heisenberg-Weg 39, D-8014 Neubiberg, F .R.G.

Efficiency – p. 30/32

Main references 2/2

Ábele-Nagy, K., Bozóki, S. (2016): Efficiency analysis of simple perturbed pairwise comparison matrices, Fundamenta Informatica, 144(3-4):279–289 Ábele-Nagy, K., Bozóki, S., Rebák, Ö. (2016): Efficiency analysis of double perturbed pairwise comparison matrices, under review, arXiv:1602.07137 Bozóki, S. (2014): Inefficient weights from pairwise comparison matrices with arbitrarily small inconsistency, Optimization, 63(12):1893–1901. Bozóki, S., Fülöp, J. (2016): Efficient weight vectors from pairwise comparison matrices, under review, arXiv:1602.03311 Bozóki, S., Fülöp, J., Németh, Z., Prill, M. (2016): Pairwise Comparison Matrix Calculator. Available at pcmc.online

Efficiency – p. 31/32

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

Efficiency – p. 32/32