Deriving weights from a pairwise comparison matrix over an alo-group

Abstract

In this paper, at first, we provide some results on the group of vectors with components in a divisible Abelian linearly ordered group, the related subgroup of $\odot$-normal vectors, the relation of $\odot$-proportionality and the corresponding quotient group. Then, we apply the achieved results to the groups of reciprocal and consistent matrices over divisible Abelian linearly ordered groups; this allows us to deal with the problem of deriving a weighting ranking for the alternatives from a pairwise comparison matrix. The proposed weighting vector has several advantages; it satisfies, for instance, the independence of scale-inversion condition.

Appendix

In this section, we provide examples for deriving a weighting vector from a PCM (Examples 1, 2, 3) and for showing how our approach is able to deal with the independence of the scale-inversion condition (Example 4).

Example 1

Let $\{x_{1},x_{2},x_{3},x_{4}\}$ be a set of alternatives and

$$ A =\left(\begin{array}{llllllll} 1 & & 2 & & 4 & & 5 & \\ \frac{1}{2} & & 1 & & 2 & & \frac{5}{2} & \\ \frac{1}{4} & & \frac{1}{2} & & 1 & & \frac{5}{4} & \\ \frac{1}{5} & & \frac{2}{5} & & \frac{4}{5} & & 1 & \end{array}\right) $$

a multiplicative PCM encoding the preference intensities of the alternatives. $A$ is a consistent PCM because the columns are proportional with respect to the multiplication $\cdot$ (see Proposition 30): $\underline {a}^{k}=a_{1k}\cdot\underline {a}^{1},$ for $k = 2,3, 4.$ By Proposition 36, each column is an ordinal evaluation vector; thus the actual ranking is $x_{1} \succ x_{2} \succ x_{3} \succ x_{4}.$ The vector

$$ \begin{aligned} \underline {w}_{m_{\cdot}}(A)&=(m_{\cdot}( \underline {a}_{1}),m_{\cdot}( \underline {a}_{2}), m_{\cdot}( \underline {a}_{3}), m_{\cdot}( \underline {a}_{4}))\\ &=\left(\root{4}\of{40}, \root{4}\of{\frac{5}{2}}, \root{4}\of{\frac{5}{32}},\root{4}\of{\frac{8}{125}}\right) \end{aligned} $$

is the weighting vector; it is a $\odot$-normal vector, in fact $\prod_{i=1}^{4}m_{\cdot}( \underline {a}_{i})=1.$

We stress that $\underline {w}_{m_{\cdot}}(A)$ can be obtained by applying the $\odot$-normalization function $N$ to an arbitrary column, for instance by choosing $\underline {a}^{1},$ we have:

$$ \begin{aligned} N(\underline {a}^{1})&= \frac{\underline {a}^{1}}{m_{\cdot}( \underline {a}^{1})}\\ &=\underline {a}^{1} \cdot \root{4}\of{40}=(\root{4}\of{40}, \frac{1}{2}\root{4}\of{40},\frac{1}{4}\root{4}\of{40},\frac{1}{5}\root{4}\of{40})\\ &= \underline {w}_{m_{\cdot}}(A). \end{aligned} $$

Example 2

Let $\{x_{1},x_{2},x_{3},x_{4}\}$ be a set of alternatives and

$$ B= \left( \begin{array}{llll} 0 & -3 & -4 & -4 \\ 3 & 0 & -1 & -1 \\ 4 & 1 & 0 & 0 \\ 4 & 1 & 0 & 0 \\ \end{array} \right). $$

an additive PCM encoding the preference intensities of the alternatives. $B$ is a consistent PCM because the columns are proportional with respect to the addition $+$ (see Proposition 30): $\underline {b}^{k}=b_{1k}+\underline {b}^{1},$ for $k = 2,3, 4.$ By Proposition 36, each column is an ordinal evaluation vector; thus the actual ranking is $x_{3} \sim x_{4} \succ x_{2} \succ x_{1}.$ The vector

$$ \begin{aligned} \underline {w}_{m_{+}}(B)&=(m_{+}( \underline {b}_{1}),m_{+}( \underline {b}_{2}), m_{+}( \underline {b}_{3}), m_{+}( \underline {b}_{4}))\\ =&\left(-\frac{11}{4}, \frac{1}{4}, \frac{5}{4}, \frac{5}{4} \right) \end{aligned} $$

is the weighting vector; it is a $\odot$-normal vector, in fact $\sum_{i=1}^{4}m_{+}( \underline {b}_{i})=0.$

We stress that $\underline {w}_{m_{+}}(B)$ can be obtained by applying the $\odot$-normalization function $N$ to an arbitrary column, for instance by choosing $\underline {b}^{1},$ we have:

$$ \begin{aligned} N(\underline {b}^{1})&= \underline {b}^{1}- m_{+}(\underline {b}^{1})\\ &=\underline {b}^{1} - \frac{11}{4}=\left(-\frac{11}{4}, 3-\frac{11}{4},4-\frac{11}{4},4-\frac{11}{4}\right)\\ &=\underline {w}_{m_{+}}(B). \end{aligned} $$

Example 3

Let $\{x_{1},x_{2},x_{3},x_{4}\}$ be a set of alternatives and

$$ C =\left( \begin{array}{llllllll} 0.5 & 0.6 & 0.7 & 0.8 \\ 0.4 & 0.5 & 0.609 & 0.727 \\ 0.3 & 0.391 & 0.5 & 0.632 \\ 0.2 & 0.273 & 0.368 & 0.5 \end{array} \right) $$

a fuzzy PCM encoding the preference intensities of the alternatives. $C$ is a consistent PCM because the columns are $\otimes$-proportional (see Proposition 30): $\underline {c}^{k}=c_{1k}\otimes \underline {c}^{1},$ for $k = 2,3, 4.$ By Proposition 36, each column is an ordinal evaluation vector; thus the actual ranking is $x_{1} \succ x_{2} \succ x_{3} \succ x_{4}.$ Moreover, by applying (8), we obtain the weighting vector:

$$ \begin{aligned} \underline {w}_{m_{\otimes}}(C)&=(m_{\otimes}( \underline {c}_{1}),m_{\otimes}( \underline {c}_{2}), m_{\otimes}( \underline {c}_{3}), m_{\otimes}( \underline {c}_{4}))\\ =&\left(0.65921, 0.56323, 0.45329, 0.32593 \right); \end{aligned} $$

$\underline {w}_{m_{\otimes}}(C)$ is a $\otimes$-normal vector, in fact, by applying (7), $\bigotimes_{i=1}^{4}m_{\otimes}( \underline {c}_{i})=0.5.$

We stress that $\underline {w}_{m_{\otimes}}(C)$ can be obtained by applying the $\odot$-normalization function $N$ to an arbitrary column, for instance by choosing $\underline {c}^{1},$ we have:

$$ \begin{aligned} N(\underline {c}^{1})&= \underline {c}^{1}\otimes (1-m_{\otimes}(\underline {c}^{1}))=\underline {c}^{1} \otimes 0.65921 \\ &=(0.5 \otimes0.65921, \;0.4 \otimes 0.65921,\\ & \quad\; 0.3 \otimes 0.65921,\; 0.2 \otimes 0.65921)\\ &=\underline {w}_{m_{\otimes}}(C). \end{aligned} $$

Example 4

Let $\{x_{1},x_{2},x_{3},x_{4}\}$ be a set of alternatives and

$$ D =\left(\begin{array}{llllllll} 1 & \frac{1}{4} & \frac{1}{2} & \frac{1}{2} \\ 4 & 1 & 2 & 1 \\ 2 & \frac{1}{2} & 1 & \frac{1}{4} \\ 2 & 1 & 4 & 1 \end{array}\right) $$

a multiplicative PCM encoding the preference intensities of the alternatives. As $d_{14}\neq d_{13}d_{34}, D$ is an inconsistent PCM and, by $\odot$-reciprocity, $D^{(-1)}=\frac{1}{D}$ is inconsistent too. The principal eigenvalue of $D$ and $\frac{1}{D}=D^{T}$ is $\lambda_{\rm max}=4.18552$ and the corresponding eigenvectors are

$$ \underline {w}_{\lambda_{\rm max}}(D)=(0.20177, 0.64411, 0.28131, 0.68211), $$

$$ \underline {w}_{\lambda_{\rm max}}\left(\frac{1}{D}\right)= (0.76963, 0.22028, 0.55202, 0.23327). $$

The eigenvectors $\underline {w}_{\lambda_{\rm max}}(D)$ and $\underline {w}_{\lambda_{\rm max}}(\frac{1}{D})$ provide different rankings, that are:

$$ x_{4}\succ x_{2}\succ x_{3} \succ x_{1}, $$

$$ x_{2}\succ x_{4}\succ x_{3} \succ x_{1}, $$

respectively. Differently, the geometric mean vectors associated to $D$ and $\frac{1}{D},$

$$ \underline {w}_{m_{.}} (D)=\left(\frac{1}{16}, 8, \frac{1}{4},8\right), \quad \underline {w}_{m_{.}} \left(\frac{1}{D}\right)=\left(16, \frac{1}{8}, 4,\frac{1}{8}\right) $$

provide the same ranking:

$$ x_{4}\sim x_{2}\succ x_{3} \succ x_{1}. $$

(43)

We stress that, the preference ratios, expressed by means of $D$ (or equivalently $\frac{1}{D}$), agree with ranking (43); indeed, $x_{4}\sim x_{2}$ according to $d_{24}=d_{42}=1, x_{2} \succ x_{3}$ according to $d_{23}>1, x_{3} \succ x_{1}$ according to $d_{31}>1$ and $x_{4} \succ x_{1}$ according to $d_{41}>1.$