The Core of Consistency in AHP-Group Decision Making

Abstract

This paper presents a new tool, the Consistency Consensus Matrix, designed to encourage the search for consensus in group decision making when using the Analytic Hierarchy Process (AHP). The procedure exploits one of the characteristics of AHP: the possibility of measuring consistency in judgement elicitation. Using two other tools, Preference Structures and Stability Intervals, we derive the Consistency Consensus Matrix that corresponds to the actor’s core of consistency. The performance analysis of the preference structure obtained from this matrix provides us with valuable information in search for knowledge. The new tool is illustrated by means of a case study adapted from a real-life experiment in e-democracy developed for the City Council of Zaragoza (Spain).

Appendix

1.1 Proof of Theorem 1

Notation

Let A = (a _ij ), i, j =  1,..., n, be a pairwise comparison matrix, ω =  (ω _i ) be its associated priority vector, e _ij the error obtained when estimating the judgement a _ij from the priority vector ω (e _ij = a _ij ω _j /ω _i ), A′ = (a _ij ′) the matrix obtained when modifying the original judgements, and t _ij the relative variations in the judgements (t _ij  = a _ij ′/a _ij ) and ρ _ij = t ^1/n _ij .

Lemma 1

Given A = (a _ij ), i, j = 1,..., n, and its variation A′ = (a _ij ′), the new values of the judgement errors when applying the row geometric mean method are:

$$ e'_{ij}=e_{ij}\rho_{ij}^n\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}} $$

(8)

Proof.

Immediate, on applying the definition of the priorities obtained with the RGMM.

Lemma 2

Given A = (a _ij ), i, j = 1,..., n, and its variation A′ = (a′ _ij ), the new value of the GCI is

$$ GCI'=\frac{1}{(n-1)(n-2)}\left[ \sum_{i,j=1}^n\log^2 e_{ij}t_{ij}- \frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n \log t_{ij}\right)^2\right] $$

(9)

Proof.

Applying the expression of the GCI and expression (8) we have:

$$GCI'=\frac{1}{(n-1)(n-2)}\sum_{i,j=1}^n\log^2 e'_{ij}=\frac{1}{(n-1)(n-2)}\sum_{i,j=1}^n\log^2 (e_{ij}\rho_{ij}^n\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}) $$

where $\rho_{ij}=\left(\frac{a'_{ij}}{a_{ij}}\right)^{1/n}.$ Developing the expression of the square of the logarithm:

$$\eqalign{ \nonumber GCI'=\frac{1}{(n-1)(n-2)}\sum_{i,j=1}^n\left(\log e_{ij}+\log\rho_{ij}^n\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}\right)^2\cr = \frac{1}{(n-1)(n-2)}\cr \quad\times\left( \sum_{i,j=1}^n\log^2e_{ij}+{\underbrace{\sum_{i,j=1}^n\log^2\rho_{ij}^n\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}}_{=A}} \quad+\quad\underbrace{2\sum_{i,j=1}^n\log e_{ij}\log\rho_{ij}^n\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}}_{=B} \right) }<!endaligned> $$

(10)

Let us now operate with the term A

$$ \eqalign{ A=\sum_{i,j=1}^n\log^2\rho_{ij}^n+ \underbrace{\sum_{i,j=1}^n\log^2\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}}_{=A_1}+ \underbrace{2\sum_{i,j=1}^n \log\rho_{ij}^n\log\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}}_{=A_2} }<!endaligned> $$

For A ₁ we have, after simple computations:

$$ \eqalign{ A_1=\sum_{i,j=1}^n\left(\log\prod_{k=1}^n\rho_{jk}-\log\prod_{k=1}^n\rho_{ik}\right)^2\cr =2n\sum_{i=1}^n \left(\sum_{k=1}^n\log\rho_{ik}\right)^2 -2\sum_{i=1}^n \left[\left(\log \prod_{k=1}^n\rho_{ik}\right)\left(\log \prod_{j < k=1}^n\rho_{jk}\rho_{kj}\right)\right] }<!endaligned> $$

As ρ _jk and ρ _kj are reciprocal, the last product equals 1, the logarithm is null and, therefore, all the second term of A ₁ equals 0. With all these, and changing the index k by j, we have:

$$ A_1=2n\sum_{i=1}^n \left(\sum_{j=1}^n\log\rho_{ij}\right)^2 $$

Let us now consider the value of A ₂

$$ \eqalign{ A_2=2n\left[\sum_{i,j=1}^n \left(\log\rho_{ij}\log\prod_{k=1}^n\rho_{jk}- \log\rho_{ij}\log\prod_{k=1}^n\rho_{ik}\right)\right]\cr =2n\left[\sum_{i,j=1}^n \log\rho_{ij}\log\prod_{k=1}^n\rho_{jk}+ \sum_{i,j=1}^n\log\rho_{ji}\log\prod_{k=1}^n\rho_{ik}\right]= 4n\sum_{i,j=1}^n \left(\log\rho_{ij}\log\prod_{k=1}^n\rho_{jk}\right)\cr =4n\sum_{j=1}^n \left(\log\prod_{k=1}^n\rho_{jk}\sum_{i=1}^n\log\rho_{ij}\right)= 4n\sum_{j=1}^n \left(\sum_{k=1}^n\log\rho_{jk}\sum_{i=1}^n\log\rho_{ij}\right)\cr =-4n\sum_{j=1}^n \left(\sum_{k=1}^n\log\rho_{jk}\sum_{i=1}^n\log\rho_{ji}\right)= -4n\sum_{j=1}^n \left(\sum_{i=1}^n\log\rho_{ji}\right)^2 }<!endaligned> $$

Swapping the indexes i and j:

$$ A_2=-4n\sum_{i=1}^n \left(\sum_{j=1}^n\log\rho_{ij}\right)^2 $$

With the values of A ₁ and A ₂ we have:

$$ A=\sum_{i,j=1}^n\log^2\rho_{ij}^n-2n\sum_{i=1}^n \left(\sum_{j=1}^n\log\rho_{ij}\right)^2 $$

Now, the term B

$$ \eqalign{ B=2\left(\sum_{i,j=1}^n\log e_{ij}\log\rho_{ij}^n+\sum_{i,j=1}^n\log e_{ij}\log\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}\right)\cr =2n\sum_{i,j=1}^n\log e_{ij}\log\rho_{ij}+ \underbrace{2\sum_{i,j=1}^n\log e_{ij}\log\prod_{k=1}^n\frac{\rho_{jk}}{\rho_{ik}}}_{B_1} }<!endaligned> $$

Finally, developing B ₁:

$$ \openup-2pt\eqalign{ B_1=2\sum_{i,j=1}^n\log e_{ij}\log\prod_{k=1}^n \rho_{jk}- 2\sum_{i,j=1}^n\log e_{ij}\log\prod_{k=1}^n\rho_{ik}\cr =2\sum_{i,j=1}^n\log e_{ij}\log\prod_{k=1}^n \rho_{jk}+ 2\sum_{i,j=1}^n\log e_{ji}\log\prod_{k=1}^n\rho_{ik} }<!endaligned> $$

Therefore:

$$ \eqalign{ B_1=4\sum_{i,j=1}^n\left[\log e_{ij}\log\prod_{k=1}^n \rho_{jk}\right]= 4\sum_{j=1}^n\left[\log\prod_{k=1}^n \rho_{jk}\left(\sum_{i=1}^n\log e_{ij}\right)\right] }<!endaligned> $$

As ${\prod_{j}e_{rj}=1}$ we have ∑ ⁿ _i=1 log e _ij  = 0 and then B ₁ = 0. The value of B will be:

$$B=2n\sum_{i,j=1}^n\log e_{ij}\log\rho_{ij}$$

Finally, returning to the value of GCI′ in expression (10), after simple computations:

$$\eqalign{(n-1)(n-2)GCI' =\sum_{i,j=1}^n\log^2e_{ij}+A+B\cr =\sum_{i,j=1}^n\left(\log e_{ij}+n\log\rho_{ij}\right)^2 -2n\sum_{i=1}^n \left(\sum_{j=1}^n\log\rho_{ij}\right)^2 }<!endaligned> $$

If we now consider ${t_{ij}=\frac{a'_{ij}}{a_{ij}}}$, log t _ij  = nlogρ _ij and then:

$$ GCI'=\frac{1}{(n-1)(n-2)}\left[ \sum_{i,j=1}^n\left(\log e_{ij}+\log t_{ij}\right)^2-\frac{2}{n} \sum_{i=1}^n \left(\sum_{j=1}^n\log t_{ij}\right)^2\right]\qquad \# $$

Proof.

If we make the variable change x _ij  = log t _ij , the condition imposed to demand that the increase in inconsistency would be less than Δ is:

$$ \frac{1}{(n-1)(n-2)}\left[ \sum_{i,j=1}^n\left(\log e_{ij}+x_{ij}\right)^2- \frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n x_{ij}\right)^2\right]\le GCI+\Delta $$

(11)

On determining the maximum value of δ so that if t _ij ∈ [1/δ, δ], the inconsistency does not increase its value above δ, it is equivalent to determining the maximum value of x* so that if x ∈ [−x*, x*] the inequality above is verified. In other words, we must determine the maximum ball (L_∞ norm) centred in the origin which verifies the inequality (11). The radius of this ball is determined by the closest point in the frontier to the origin in L_∞. In order to obtain the radius we must solve the optimisation problem (5). ${\qquad}$ #