Reciprocal transitive matrices over abelian linearly ordered groups: Characterizations and application to multi-criteria decision problems

doi:10.1016/j.fss.2014.07.005

Fuzzy Sets and Systems

Volume 266, 1 May 2015, Pages 33-46

https://doi.org/10.1016/j.fss.2014.07.005 Get rights and content

Abstract

We consider reciprocal matrices over an abelian linearly ordered group; in this way we provide a general framework including multiplicative, additive and fuzzy matrices. In a multi-criteria decision making context, a pairwise comparison matrix $A = (a_{i j})$ is a reciprocal matrix that represents a useful tool for determining a weighting vector w for a set X of decision elements; but, when A is inconsistent, the weighting vector, usually proposed in literature, may provide a ranking on X that does not agree with the expressed preference intensities $a_{i j}$ , thus, it is unreliable. We analyze a condition of transitivity for a reciprocal matrix $A = (a_{i j})$ over an abelian linearly ordered group, that, whenever A is a pairwise comparison matrix, allows us to state a qualitative dominance ranking on X and obtain ordinal evaluation vectors; in this way, we get a first tool for checking the reliability of a weighting vector. We also provide tools to check the transitivity.

Introduction

The quantitative pairwise comparisons are a useful tool for estimating the relative weights on a set $X = {x_{1}, x_{2}, . . ., x_{n}}$ of decision elements such as criteria or alternatives. Pairwise comparisons can be modelled by a quantitative preference relation on X: $A : (x_{i}, x_{j}) \in X \times X \to a_{i j} = A (x_{i}, x_{j}) \in R$ where $a_{i j}$ quantifies the preference intensity of $x_{i}$ over $x_{j}$ . When the cardinality of X is small, $A$ can be represented by the Pairwise Comparison Matrix (PCM) $A = (\begin{matrix} a_{11} & a_{12} & . . . & a_{1 n} \\ a_{21} & a_{22} & . . . & a_{2 n} \\ . . . & . . . & . . . & . . . \\ a_{n 1} & a_{n 2} & . . . & a_{n n} \end{matrix}) .$

In literature, several kinds of PCMs are considered: if $a_{i j} \in] 0, + \infty [$ represents a preference ratio, then $A$ is a multiplicative preference relation and A a multiplicative PCM; if $a_{i j} \in R =] - \infty, + \infty [$ represents a preference difference, then $A$ is an additive preference relation and A an additive PCM; if $a_{i j} \in [0, 1]$ reflects a preference degree, then $A$ is called fuzzy preference relation [11], [14] and, as a consequence, A can be called fuzzy PCM.

A condition of reciprocity is assumed for PCMs, for which the information provided by $a_{i j}$ , can be exactly deduced from $a_{j i}$ .

If the expert is fully coherent when expresses his preferences, then the PCM satisfies the consistency condition; several kinds of consistency are proposed in literature: multiplicative consistency [19], additive consistency [1], fuzzy additive consistency (called additive transitivity in [12], [21]), and fuzzy multiplicative consistency (called multiplicative transitivity in [12], [20]).

Many authors have studied the problem of the inconsistency of a PCM, for instance: Saaty [18] proposes a consistency index defined in terms of the principal eigenvalue; Barzilai [1] proposes the relative error; Peláez and Lamata [16] propose a measure of consistency based on the determinant of the PCM; Bortot and Marques Pereira [4] propose a measure of dominance inconsistency in the framework of Choquet integration; Brunelli and Fedrizzi [5] present five axioms aimed at characterizing inconsistency indices.

In order to unify different approaches to the PCMs and remove some drawbacks, in [6] the authors introduce PCMs over an abelian linearly ordered group (alo-group) $G = (G, ⊙, \leq)$ . In this general context, conditions of ⊙-reciprocity and ⊙-consistency are proposed. In [6], if $G$ is divisible, then a mean operator $m_{⊙}$ is associated with ⊙, and a ⊙-consistency index (see [9] for its properties) is defined.

The preference relation $A$ induces a qualitative preference structure on X [8]: $x_{i} ≻ x_{j} \Leftrightarrow a_{i j} > e, x_{i} \sim x_{j} \Leftrightarrow a_{i j} = e,$ where $x_{i} ≻ x_{j}$ means “ $x_{i}$ preferred to $x_{j}$ ” and $x_{i} \sim x_{j}$ means “ $x_{i}$ indifferent to $x_{j}$ ”, and e is the identity of the group operation ⊙.

By relations ≻ and ∼, we obtain the relation ≿ defined as follows: $x_{i} ≿ x_{j} \Leftrightarrow (x_{i} ≻ x_{j} o r x_{i} \sim x_{j}) \Leftrightarrow a_{i j} \geq e .$ If $x_{i} ≿ x_{j}$ , then we say “ $x_{i}$ is weakly preferred to $x_{j}$ ”.

The transitivity of the relations in (2) and (3) is the minimal logical requirement and a fundamental principle that preference relations should satisfy; the transitivity is in fact acyclic about the alternatives or criteria ranking. Unfortunately, if the PCM is not ⊙-consistent, then it may happen that these relations are not transitive; thus, the need to consider PCMs that ensure the transitivity of relations in (2) and (3).

In this paper, we focus on ⊙-reciprocal matrices over a divisible alo-group $G = (G, ⊙, \leq)$ and, in this general context, we introduce the notion of ⊙-transitivity that is equivalent to weak transitivity introduced in [21] for fuzzy preference relations, and generalizes the transitivity given in [2] and [3] for multiplicative matrices, under the assumption of no indifference (i.e. $a_{i j} \neq 1$ for $i \neq j$ ). The ⊙-transitivity ensures transitivity of the relations in (2) and (3); thus, it allows us to determine the actual dominance ranking $x_{i_{1}} ≿ x_{i_{2}} ≿ . . . ≿ x_{i_{n}}$ on X and ordinal evaluation vectors for this ranking

For conditions ensuring transitive collective decisions, see [15].

The paper is organized as follows: in Section 2, we introduce the notation and definitions used in the paper; in Section 3, we provide the notion of ⊙-transitivity; in Section 4, we assume that $A = (a_{i j})$ is a PCM associated with a set X of decision elements, and show that the ⊙-transitivity allows us to state the actual ranking on X; in Section 5, we analyze the notion of ⊙-transitivity and provide characterizations, that allow us to easily check the ⊙-transitivity, to state the actual ranking on X and to obtain ordinal evaluation vectors; finally, in Section 6 we provide concluding remarks and directions for future work.

Section snippets

Preliminaries

From now on, G will denote an open interval of the real line $R$ , ≤ the total order on G inherited from the usual order on $R$ , $G = (G, ⊙, \leq)$ a divisible alo-group, e the identity of $G$ , $x^{(- 1)}$ the symmetric of $x \in G$ with respect to ⊙, ÷ the inverse operation of ⊙ defined by $a \div b = a ⊙ b^{(- 1)}$ , $x^{(n)}$ , for a non-negative integer n, the (n)-power of $x \in G$ and $a^{(1 / n)}$ , that is the unique solution of the equation $x^{(n)} = a$ , the (n)-root of a (see [6]).

By definition of alo-group, the following implication holds: $a \leq b \Rightarrow a ⊙ c \leq b ⊙ c,$

⊙-transitive RMs

From now on, let us assume $A = (a_{i j})$ be an RM over $G$ .

Definition 3.1

$A = (a_{i j})$ is ⊙-transitive if and only if verifies the condition $a_{i j} \geq e, a_{j k} \geq e \Rightarrow a_{i k} \geq e .$

If $A = (a_{i j})$ is a fuzzy RM, then (16) corresponds to weak transitivity introduced in [21], that is: $a_{i j} \geq 0.5, a_{j k} \geq 0.5 \Rightarrow a_{i k} \geq 0.5,$ whereas if $A = (a_{i j})$ is a multiplicative RM, (16) generalizes the transitivity given in [2] and [3] for a multiplicative matrix, under the assumption $a_{i j} \neq 1$ for $i \neq j$ , that is: $a_{i j} > 1, a_{j k} > 1 \Rightarrow a_{i k} > 1 .$

Theorem 3.1

The following assertions are equivalent:

1.
$A = (a_{i j})$ is ⊙

⊙-transitive PCMs and actual ranking on X

Let $A = (a_{i j})$ be the ⊙-reciprocal PCM associated with the set $X = {x_{1}, x_{2}, . . ., x_{n}}$ of decision elements, and ≻, ∼ the binary relations on X described by (2). By the meaning given to $x_{i} \sim x_{j}$ in Section 1, we say that (10) is a condition of no indifference between distinct objects of X.

Proposition 4.1

The relation ≻ is asymmetric, the relation ∼ is reflexive and symmetric and $x_{i} ≻ x_{j} or x_{i} \sim x_{j} or x_{j} ≻ x_{i} \forall i, j \in {1, 2, \dots, n} .$ The relation ≿ is strongly complete, that is: $x_{i} ≿ x_{j} or x_{j} ≿ x_{i} \forall i, j \in {1, 2, \dots, n} .$

Proof

By (2) and (12), ≻ is asymmetric and ∼

Characterizations of the ⊙-transitivity

In this section, we show that, to check the ⊙-transitivity condition, it is enough to count, for each row of an RM, the number of components greater than e, or equal to e, and then verify whether a suitable relation among them is satisfied.

Let us denote with $| Y |$ the cardinality of a finite set Y, then we introduce the following integer numbers:

•
$n_{I} (a_{i}) = | {j \in {1, . ., n} : a_{i j} = e} | = | I (a_{i}) |$ ;
•
$n (a_{i}) = | {j \in {1, . ., n} : a_{i j} > e} |$ ;
•
$n_{0} (a_{i}) = | {j \in {1, . ., n} : a_{i j} \geq e} |$ .

a_{i} \in I (a_{i})

, then

n_{I} (a_{i}) \geq 1

and

n_{0} (a_{i}) = n (a_{i}) + n_{I} (a_{i}) > n (a_{i}) .

Conclusion and future work

We analyze a condition of ⊙-transitivity for a ⊙-reciprocal matrix $A = (a_{i j})$ over an alo-group, that allows us to state the actual ranking on a set of alternatives X, whenever $A = (a_{i j})$ is a PCM for X. We provide tools to check the ⊙-transitivity, and find the actual ranking and ordinal evaluation vectors.

Finally, we stress that although both matrix in Example 4.1 and matrix in Example 5.5 are ⊙-transitive and no ⊙-consistent, the ⊙-mean vector $w_{m_{⊙}} (A)$ is an ordinal evaluation vector only in Example

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Reciprocal transitive matrices over abelian linearly ordered groups: Characterizations and application to multi-criteria decision problems

Abstract

Introduction

Section snippets

Preliminaries

⊙-transitive RMs

⊙-transitive PCMs and actual ranking on X

Characterizations of the ⊙-transitivity

Conclusion and future work

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Eur. J. Oper. Res.

Fuzzy Sets Syst.

Comput. Math. Appl.

J. Math. Psychol.

Fuzzy Sets Syst.

Consistency measures for pairwise comparison matrices

J. Multi-Criteria Decis. Anal.

Ranking and weak consistency in the A.H.P. context

Decis. Econ. Finance

Weak consistency and quasi-linear means imply the actual ranking

Int. J. Uncertain. Fuzziness Knowl.-Based Syst.