Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups

Characterizations of Consistent PairwiseComparison Matrices over Abelian LinearlyOrdered GroupsBice Cavallo,∗ Livia D’Apuzzo†

Department of Constructions and Mathematical Methods in ArchitectureUniversity of Naples, Federico II, Italy

We consider the framework of pairwise comparison matrices over abelian linearly ordered groups.We introduce the notion of �-proportionality that allows us to provide new characterizations ofthe consistency, efÞcient algorithms for checking the consistency and for building a consistentmatrix. Moreover, we provide a new consistency index. C© 2010 Wiley Periodicals, Inc.

1. INTRODUCTION

Let X = {x1, x2, . . . , xn} be a set of alternatives or criteria. An useful tool todetermine a weighted ranking on X is a pairwise comparison matrix (PCM)

A =

⎛⎜⎝

a11 a12 . . . a1na21 a22 . . . a2n. . . . . . . . . . . .

an1 an2 . . . ann

⎞⎟⎠ , (1)

where entry aij expresses how much the alternative xi is preferred to alternative xj .A condition of reciprocity is assumed for the matrixA = (aij ) in such a way that thepreference of xi over xj expressed by aij can be exactly read by using the elementaji .

Under a suitable condition of consistency for A = (aij ), X is totally orderedand the values aij can be expressed by means of the components wi and wj of asuitable vector w, that is called consistent vector for the matrix A = (aij ); then w

provides the weights for the elements of X.The shape of the reciprocity and consistency conditions depends on the different

meaning given to the number aij , as the following well-known cases show.

∗Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 25, 1035�1059 (2010)C© 2010 Wiley Periodicals, Inc.View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.20438

1036 CAVALLO AND D�APUZZO

Multiplicative case: aij ∈]0, +∞[ is a preference ratio and the conditions ofmultiplicative reciprocity and consistency are given respectively by

aji = 1

aij

∀ i, j = 1, . . . , n,

aik = aij ajk ∀ i, j, k = 1, . . . , n.

Aconsistent vector is a positive vectorw = (w1, w2, . . . , wn) verifying the conditionwi

wj= aij and so perfectly representing the preferences over X.

Additive case: aij ∈]− ∞, +∞[ is a preference difference and the conditionsof additive reciprocity and consistency are expressed as follows:

aji = −aij ∀ i, j = 1, . . . , n,

aik = aij + ajk ∀ i, j, k = 1, . . . , n.

A consistent vector is a vector w = (w1, w2, . . . , wn) verifying the condition wi −wj = aij .

Fuzzy case: aij ∈ [0, 1] measures the distance from the indifference that isexpressed by 0.5; the conditions of fuzzy reciprocity and consistency are

aji = 1− aij ∀ i, j = 1, . . . , n,

aik = aij + ajk − 0.5 ∀ i, j, k = 1, . . . , n.

A consistent vector is a vector w = (w1, w2, . . . , wn) verifying the condition wi −wj = aij − 0.5.

The multiplicative PCMs play a basic role in the analytic hierarchy process,a procedure developed by T.L. Saaty at the end of the 1970s.12−14 In Refs. 2�5, and 9, properties of multiplicative PCMs are analyzed in order to determine aqualitative ranking on the set of the alternatives and Þnd vectors representing thisranking. Additive and fuzzy matrices are investigated for instance by Barzilai1 andHerrera-Viedma et al.11

In the case of a multiplicative PCM, Saaty suggests that the comparisons ex-pressed in verbal terms have to be translated into preference ratios aij taking value in

S∗ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19}. Let us stress that the assumption

of the Saaty scale restricts the decision maker�s possibility to be consistent: indeed,if the decision maker (DM) expresses the following preference ratios aij = 5 andajk = 3, then he will not be consistent because aij ajk = 15 > 9. Similarly, for thefuzzy case, the assumption that aij ∈ [0, 1], restricts the possibility to realize thefuzzy consistency: indeed, if the DM claims aij = 0.9 and ajk = 0.8, then he willnot be consistent because aij + ajk − 0.5 = 1.7− 0.5 > 1.

International Journal of Intelligent Systems DOI 10.1002/int

CONSISTENT PAIRWISE COMPARISON MATRICES 1037

In order to unify the several approaches to PCMs and remove the above draw-backs, in Ref. 6 we introduce PCMs whose entries belong to an abelian linearlyordered group (alo-group) G = (G, �, ≤). In this way, the reciprocity and consis-tency conditions are expressed in terms of the group operation� and the drawbacksrelated to the consistency condition are removed; in fact the consistency conditionis expressed by aik = aij � ajk , where aij � ajk is an element ofG, for each choiceof aij , ajk ∈ G. As a nontrivial alo-group G = (G, �, ≤) has neither the greatestelement nor the least element (see Ref. 6), the Saaty set S∗ and the interval [0, 1],embodiedwith the usual order≤ onR, cannot be structured as alo-groups.Moreover:

� the assumption of divisibility for G allows us to introduce the notion of mean m�(a1, . . . ,an) of n elements and associate a mean vector wm� to a PCM;

� the introduction of a notion of distance dG , linked to the operation � in a divisiblealo-group G, allows us to provide a measure of consistency for a PCM over G: indeedthe consistency index IG(A) is deÞned as mean of the distances dG(aik, aij � ajk), withi < j < k.

In Ref. 7, we analyze some properties of a consistent PCM and provide al-gorithms, to check whether or not a matrix is consistent and to build a consistentmatrix by means of n − 1 comparisons. Moreover, we provide a new consistencyindex linked to the index IG(A) introduced in Ref. 6 but easier to compute.

In this paper, we extend the previous results and introduce the abelian groupGn = (Gn, �×) and the notion of �-proportionality, that allows us to:

� provide new characterizations of a consistent PCM;� introduce an equivalence relation onGn and a bijection between the quotient setGn/ ∼�and the set of the consistent PCMs;

� provide more efÞcient algorithms to check the consistency;� provide more efÞcient algorithms to build a consistent PCM.

The paper is organized as follows: Section 2 provides notations, deÞnitionsand some results useful in the next sections; starting from an alo-group G = (G, �,

≤), Section 3 introduces the abelian group Gn = (Gn, �×) and the notion of �-proportional vectors; Section 4 provides characterizations of the consistency interms of �-proportionality of rows or columns and algorithms to build consistentPCMs; furthermore, this section links the consistent PCMs to the related consistentvectors and provides the way to build consistent PCMs starting from a vector;Section 5 provides further characterizations that allows us to build consistent PCMsand to check the consistency in a more efÞcient way; Section 6 gives the consistencyindex for a PCM; Section 7 provides concluding remarks and directions for futurework.

2. PRELIMINARIES

Let G be a nonempty set provided with a total weak order ≤ and a binaryoperation � : G × G → G. Then G = (G, �, ≤) is called alo-group, if and only if



(G, �) is an abelian group and the the following implication holds:

a ≤ b ⇒ a � c ≤ b � c.

The above implication is equivalent to

a < b ⇒ a � c < b � c,

where < is the strict simple order associated to ≤.If G = (G, �, ≤) is an alo-group, then G is naturally equipped with the ordertopology induced by ≤ and the abelian group G × G is equipped with the relatedproduct topology. G is called a continuous alo-group if and only if the function� iscontinuous.Let G = (G, �, ≤) be an alo-group, then we assume that e denotes the identity ofG, x(−1) the symmetric of x ∈ G with respect to �, ÷ the inverse operation of �deÞned by

a ÷ b = a � b(−1).

PROPOSITION 1. It results:

i) a = b � c if and only if c = b(−1) � a;ii) a = b � c if and only if a(−1) = b(−1) � c(−1);iii) (a ÷ c)(−1) = c ÷ a.

Proof. Because of the associativity and the cancellation property of a commutativegroup operation, we have: a = b � c ⇔ b−1 � a = b−1 � (b � c) ⇔ b−1 � a =c. Moreover, a = b � c ⇔ a(−1) = (b � c)(−1) ⇔ a(−1) = b(−1) � c(−1). Finally, byapplying (ii), we get (a ÷ c)(−1) = (a � c(−1))(−1) = a(−1) � c = c ÷ a. �

For a positive integer n, the (n)-power x(n) of x ∈ G is deÞned as follows:

x(1) = x

x(n) =n−1⊙i=1

xi � xn =n⊙

i=1xi, xi = x i = 1, . . . , n, n ≥ 2.

If b(n) = a, then we say that b is the (n)-root of a and write b = a(1/n).G is divisible if and only if for each positive integer n and each a ∈ G there

exists the (n)-root of a.



DEFINITION 1. Let G = (G, �, ≤) be a divisible alo-group. Then, the �-meanm�(a1, a2, . . . , an) of the elements a1, a2, . . . , an of G is deÞned by

m�(a1, a2, . . . , an) ={a1 n = 1,(⊙n

i=1 ai

)(1/n)n ≥ 2.

DEFINITION 2. An isomorphism between two alo-groups G = (G, �, ≤) and G ′ =(G′, ◦, ≤) is a bijection h : G → G′ that is both a lattice isomorphism and a groupisomorphism, that is,

x < y ⇔ h(x) < h(y)

h(x � y) = h(x) ◦ h(y).

PROPOSITION 2.6 Let h : G → G′ be an isomorphism between the alo-groups G =(G, �, ≤) and G ′ = (G′, ◦, ≤). Then G is divisible if and only if G ′ is divisible and,under the assumption of divisibility

m◦(y1, y2, . . . , yn) = h(m�(h−1(y1), h−1(y2), . . . , h−1(yn))

).

2.1. The Notion of Distance

The norm of an element a ∈ G is deÞned by setting

||a|| = a ∨ a(−1). (2)

Let us consider the operation

dG : (a, b) ∈ G2 → ||a ÷ b|| ∈ G. (3)

In Ref. 6, we prove that dG veriÞes the conditions:

1. dG(a, b) ≥ e;2. dG(a, b) = e ⇔ a = b;3. dG(a, b) = dG(b, a);4. dG(a, b) ≤ dG(a, c)� dG(b, c),

so, we provide the following deÞnition.

DEFINITION 3. The operation dG in (3) is a G-metric or G-distance.International Journal of Intelligent Systems DOI 10.1002/int


PROPOSITION 3.6 Let h : G → G′ be an isomorphism between the alo-groups G =(G, �, ≤) and G ′ = (G′, ◦, ≤). Then,

dG′(a′, b′) = h(dG(h−1(a′), h−1(b′))).

2.2. Real Alo-groups

An alo-group G = (G, �, ≤) is a real alo-group if and only if G is a subset ofthe real line R and ≤ is the total order on G inherited from the usual order on R

Under the above assumption for G and ≤, examples of real divisible andcontinuous alo-groups are the following:

Multiplicative alo-group. ]0, +∞[ = (]0, +∞[, ·, ≤), where · is the usualmultiplication on R. Then, e = 1, x(−1) = 1/x, x(n) = xn and x ÷ y = x

y. So

d]0,+∞[(a, b) = a

b∨ b

a

and

m·(a1, . . . , an) =(

n∏i=1

ai

) 1n

is the geometric mean.Additive alo-group.R = (R, +, ≤), where+ is the usual addition on R. Then,

e = 0, x(−1) = −x, x(n) = nx, x ÷ y = x − y. So

dR(a, b) = |a − b| = (a − b) ∨ (b − a)

and

m+(a1, . . . , an) =∑

i ai

n

is the arithmetic mean.Fuzzy alo-group. ]0,1[ = (]0, 1[, ⊗, ≤), where⊗ :]0, 1[2→]0, 1[ is the oper-

ation deÞned by

x ⊗ y = xy

xy + (1− x)(1− y).

Then, e = 0.5, x(−1) = 1− x, x ÷ y = x(1−y)x(1−y)+ (1−x)y

. So

d]0,1[(a, b) = a(1− b)

a(1− b)+ (1− a)b∨ b(1− a)

b(1− a)+ (1− b)a.



We will compute the fuzzy mean m⊗(a1, . . . , an) in the sequel of this paper.

Remark 1. Our choice of the operation structuring the ordered interval ]0, 1[ as analo-group wants to obey the requests: 0,5 is the identity element and 1− x is thesymmetric of x. In this way, the condition of reciprocity for a PCM over a fuzzyalo-group is given again by aji = 1− aij , as deÞned in Section 1.

By setting G =]0, 1[ and

ψ : t ∈]0, +∞[→ t

t + 1∈]0, 1[, (4)

that is a continuous and strictly increasing function between ]0, +∞[ and ]0, 1[, weget

x ⊗ y = ψ(ψ−1(x) · ψ−1(y)).

Thus (G, ⊗, ≤) is a continuous alo-group as a consequence of the following resultof Ref. 6:

THEOREM 1. Let G be a proper open interval of R and ≤ the total order on G

inherited from the usual order on R, then the following assertions are equivalent:

1. G = (G, �, ≤) is a continuous alo-group;2. there exists a continuous and strictly increasing function ψ :]0, +∞[→ G verifying theequality

x � y = ψ(ψ−1(x) · ψ−1(y)).

Moreover,⊗ veriÞes our requests about the identity and the symmetric of an elementof the group.

Remark 2. The operation⊗ is the restriction to ]0, 1[2 of the uninorm (see Ref. 10):

U (x, y) ={0, (x, y) ∈ {(0, 1), (1, 0)};

xy

xy+(1−x)(1−y), otherwise.

The multiplicative, the additive and the fuzzy alo-groups are isomorphic; infact the bijection

h : x ∈]0, +∞[→ log x ∈ R

is an isomorphism between ]0,+∞[ andR andψ in (4) is an isomorphism between]0,+∞[ and ]0,1[. So, by Proposition 2, the mean m⊗(a1, . . . , an) related to thefuzzy alo-group can be computed, by means of the function in (4), as follows:



m⊗(a1, . . . , an) = ψ

⎛⎝(

n∏i=1

ψ−1(ai)

) 1n

⎞⎠ .

2.3. PCMs over an Alo-group

Let X = {x1, x2, . . . xn} a set of alternatives, A = (aij ) in (1) the related PCM.We claim thatA = (aij ) is a PCM over the alo-group G if and only aij ∈ G, ∀i, j ∈{1, . . . , n}. We will denote by

1. a1, a2, . . . , a n the rows of A;

2. a1, a2, . . . , a n the columns of A.

IfG = (G, �, ≤) is a divisible alo-group, then we consider themean vectorwm�(A),associated to A, that is deÞned as follows:

wm�(A) = (m�(a1), m�(a1), . . . , m�(an)). (5)

From now on, we assume that A = (aij ) is reciprocal with respect to �, that is,

aji = a(−1)ij ∀ i, j = 1, . . . , n, (6)

so

aii = e and aij � aji = e ∀i, j ∈ {1, 2, . . . , n}. (7)

DEFINITION 4.6 A = (aij ) is a consistent PCM with respect to �, if and only if

aik = aij � ajk ∀i, j, k.

In Ref. 6, we provide the following characterization.

PROPOSITION 4. A = (aij ) is a consistent PCM with respect to �, if and only if:

aik = aij � ajk ∀ i, j, k : i < j < k. (8)

2.4. A Consistency Index for a PCM

Let T be the set {(aij , ajk, aik), i < j < k} and nT = |T |. From Proposition2.3., we derive the following.



PROPOSITION 5. A = (aij ) is a consistent matrix with respect to �, if and only if

dG(aik, aij � ajk) = e ∀ i, j, k : i < j < k.

So, byProposition 5,A = (aij ) is inconsistent if and only if dG(aik, aij � ajk) >

e for some triple (aij , ajk, aik) ∈ T . Thus, in Ref. 6, we provide the followingdeÞnition of consistency index.

DEFINITION 5. The consistency index of A is given by

IG(A) ={dG(a13, a12 � a23) n = 3,

(⊙

T dG(aik, aij � ajk))( 1

nT)

n > 3.

with

nT = n(n − 2)(n − 1)

6.

PROPOSITION 6. IG(A) ≥ e and A is consistent if and only if IG(A) = e.

3. THE ABELIAN GROUP GN = (GN, �×) AND THE NOTIONOF �-PROPORTIONALITY

Let G = (G, �, ≤) be an alo-group and

Gn = G × G × . . . × G︸︷︷︸n

, (9)

the cartesian product where elements are the vectors w = (w1, w2, . . . wn), wi ∈G, ∀i = 1, . . . , n.

We consider Gn embodied with the binary operation

�× : (w, v) → w �× v = (w1 � v1, w2 � v2, . . . wn � vn). (10)

The operation�× is associative and commutative, has identity element equal to e =(e, e, . . . , e), and, for each w = (w1, w2, . . . wn) ∈ Gn, w(−1) = (w(−1)

1 , w(−1)2 , . . . ,

w(−1)n ) is the symmetric element of w. Thus, Gn = (Gn, �×) is an abelian group.

Let ÷× be the inverse operation of �×, deÞned as follows: w ÷× v = w �×v(−1). Then



w ÷× v = (w1 ÷ v1, w2 ÷ v2, . . . wn ÷ vn). (11)

In the following, for simplicity of notation, we will use � for �× and ÷ for ÷×.

PROPOSITION 7. It results:

i) w = t � v if and only if v = t (−1) � w;ii) w = t � v if and only if w(−1) = t (−1) � v(−1);iii) (w ÷ v)(−1) = v ÷ w.

Proof. From Proposition 1. �

DEFINITION 6. Let c ∈ G and w = (w1, w2, . . . wn) ∈ Gn. Then the �-compositionof c and w is the vector c � w, where c = (c, c, . . . , c) ∈ Gn; that is,

c � w = (c � w1, c � w2, . . . , c � wn).

Then, c ÷ w denotes the composition c � w(−1); so

c ÷ w = (c ÷ w1, . . . , c ÷ wn) = (c � w(−1)1 , . . . , c � w(−1)

n ).

Similarly, w ÷ c denotes the vector

w ÷ c = (w1 ÷ c, . . . , wn ÷ c) = (w1 � c(−1), . . . , wn � c(−1)).

From Proposition 7, we derive

w = c � v ⇔ v = c(−1) � w. (12)

Hence we provide the following deÞnition.

DEFINITION 7. The vectors w and v are �-proportional if and only if there existsc ∈ G such that w = c � v.

PROPOSITION 8. The vectors w and v are �-proportional if and only the vectorsw(−1) and v(−1) are also �-proportional.

Proof. From Proposition 7, w = c � v if and only if w(−1) = c(−1) � v(−1). �

PROPOSITION 9. The vectors w and v are �-proportional if and only ifInternational Journal of Intelligent Systems DOI 10.1002/int


wi ÷ wj = vi ÷ vj ∀i, j = 1, . . . n. (13)

Proof. Let w and v be �-proportional vectors. Then there exists c ∈ G such thatwi = c � vi∀i = 1, . . . , n. Thus, wi ÷ wj = (c � vi)� (c � vj )

−1 = (c � vi)�(c−1 � v−1

j ) = vi ÷ vj for each i, j = 1, . . . , n.Let (13) be veriÞed, then there exists c ∈ G such that wj ÷ vj = c ∀j = 1, . . . , n.So w = c � v. �

Example 1. Let us consider the multiplicative alo-group ]0,+∞[.The vectors w = (1, 2, 3) and v = ( 1

2, 1, 3

2) are ·-proportional because v = 1

2· w.

Then

w1

w2= v1

v2= 1

2,

w1

w3= v1

v3= 1

3,

w2

w3= v2

v3= 2

3.

Example 2. Let us consider the additive alo-groupR.The vectors w = (−2, 0, 1) and v = (0, 2, 3) are +-proportional because v =2+ w.Then

w1 − w2 = v1 − v2 = −2, w1 − w3 = v1 − v3 = −3,

w2 − w3 = v2 − v3 = −1.

Example 3. Let us consider the fuzzy alo-group ]0,1[.The vectors w = (0.5, 0.6, 0.4) and v = (0.4, 0.5, 0.3077) are ⊗-proportional be-cause v = 0.4⊗ w.Then

w1 ⊗ w(−1)2 = v1 ⊗ v

(−1)2 = 0.4, w2 ⊗ w

(−1)1 = v2 ⊗ v

(−1)1 = 0.6,

w1 ⊗ w(−1)3 = v1 ⊗ v

(−1)3 = 0.6, w3 ⊗ w

(−1)1 = v3 ⊗ v

(−1)1 = 0.4,

w2 ⊗ w(−1)3 = v2 ⊗ v

(−1)3 = 0.6923, w3 ⊗ w

(−1)2 = v

(−1)3 ⊗ v

(−1)2 = 0.3077.

PROPOSITION 10. Given w = (w1, w2, . . . , wn) and v = (v1, v2, . . . , vn) inGn, therelation ∼�, deÞned as



w ∼� v ⇔ ∃c ∈ G : w = c � v

is an equivalence relation.

Proof. As w = e � w then w ∼� w. So ∼� is a reßexive relation.Because of the equivalence (12), ∼� is a symmetric relation.If w = c1 � v and v = c2 � u, then, by associativity property, w = (c1 � c2)� u.Thus (w ∼� v, v ∼� u) ⇒ w ∼� u and ∼� is a transitive relation. �

We denote by [w] the �-equivalence class of w, that is,

[w] = {v ∈ Gn : w ∼� v}

and with Gn/ ∼� the quotient set of Gn by ∼�.

4. CONSISTENT PCMS: FIRST CHARACTERIZATIONS

In this section, G = (G, �, ≤) is a divisible alo-group and A = (aij ) in (1) is aPCM over G. By assumption, A is reciprocal with respect to �, so by (6) we get

ai = a(−1)i ∀ i = 1, . . . , n. (14)

By means of the following proposition we reformulate the condition of consis-tency given in DeÞnition 4.

PROPOSITION . A = (aij ) is a consistent PCM with respect to �, if and only if

ai = aij � aj ∀i, j = 1, 2, . . . , n, (15)

or, equivalently

ai = a(−1)ij � aj ∀i, j = 1, 2, . . . , n. (16)

Proof. The equalities (15) follow from DeÞnitions 4 and 6; the equalities (16)follow from the equalities (14), (15) and item (ii) of Proposition 7. �

By the reciprocity property, the consistency conditions (15) can be written asfollows:

ai = a(−1)ji � aj ∀i, j = 1, 2, . . . , n. (17)



The conditions (16) and (17) characterize the consistent PCMs and allows usto build a consistent PCM starting from a column or a row, respectively.

By applying (17), we provide Algorithm 1 for building a consistent PCMstarting from n − 1 comparisons of the row ai∗ ; this algorithm generalizes andimproves the algorithm proposed in Ref. 7.

Algorithm 1: BUILD A CONSISTENT PCM FROM THE SET {ai∗i : i �= i∗}ai∗i∗ = e

for i = 1 . . . n doif i �= i∗ then

ai = a(−1)i∗i � a i∗

end ifend for

Example 4. Let us consider the multiplicative alo-group ]0,+∞[.Let {x1, x2, x3} be a set of alternatives. We suppose that the DM expresses thefollowing preference ratios: a12 = 2 and a13 = 3. Thus, i∗ = 1 and the rows of therelated consistent PCM are the following:

a1 = (1, 2, 3),

and

a2 = 1

2· a1 =

(1

2, 1,

3

2

),

a3 = 1

3· a1 =

(1

3,2

3, 1

).

Example 5. Let us consider the additive alo-groupR.Let {x1, x2, x3} be a set of alternatives. We suppose that the DM expresses thefollowing preference differences: a21 = 2 and a23 = 3. Thus, i∗ = 2 and the rowsof the related consistent PCM are the following:

a2 = (2, 0, 3),

and

a1 = −2+ a2 = (0, −2, 1)

a3 = −3+ a2 = (−1, −3, 0).International Journal of Intelligent Systems DOI 10.1002/int


Example 6. Let us consider the fuzzy alo-group ]0,1[.Let {x1, x2, x3} be a set of alternatives. We suppose that the DM expresses thefollowing fuzzy preferences: a12 = 0.4 and a13 = 0.6. Thus, i∗ = 1 and the rows ofthe related consistent PCM are the following:

a1 = (0.5, 0.4, 0.6),

and

a2 = 0.6⊗ a1 = (0.6, 0.5, 0.69),

a3 = 0.4⊗ a1 = (0.4, 0.31, 0.5).

The following theorem gives a characterization of a consistent PCM in termsof �-proportional vectors. This characterization allows us to check in an easy waythe property of consistency.

THEOREM 2. The following assertions related to A = (aij ) are equivalent:

1. A = (aij ) is a consistent PCM;2. for every choice of i, j = 1, 2, . . . , n the rows ai and aj are �-proportional vectors; sofor a Þxed index i∗, [ai] = [ai∗ ] for i = 1, 2, . . . , n;

3. for every choice of i, j = 1, 2, . . . , n the columns ai and aj are �-proportional vectors;so for a Þxed index i∗, [ai] = [ai∗ ] for i = 1, 2, . . . , n.

Proof. 1. ⇔ 2. If A = (aij ) is a consistent PCM the item 2 follows immediatelyfrom the condition of consistency as formulated in (17). Alternatively, if item 2is veriÞed, then, for every choice of i, j = 1, 2, . . . , n, there exists an elementcij ∈ G such that ai = cij � aj , so that aik = cij � ajk for each k = 1, 2, . . . , n.By choosing k = j , we get aij = cij � e = cij ; thus the condition (15) is veriÞedand A = (aij ) is a consistent PCM.2 ⇔ 3. From Proposition 8 or reasoning as above starting from equality (16). �

Remark 3. We stress that, because of the assumption of reciprocity, the �-proportionality of the rows (resp. of the columns) is a necessary and sufÞcientcondition for the consistency. Moreover, the condition of reciprocity allows us todetermine the constants of�-proportionality by means of one of the equalities (15),(16) or (17).

Example 7. Let



A =

⎛⎜⎜⎜⎜⎜⎜⎝

12

3

1

2

3

21

3

4

24

31

⎞⎟⎟⎟⎟⎟⎟⎠

be a PCMover themultiplicative alo-group ]0,+∞[.A is consistent because a2, a3 ∈[a1]. Indeed, a2 = 2

3� a1 and a3 = 1

2� a1.

The matrix

B =

⎛⎜⎜⎜⎜⎜⎝1

2

3

1

2

3

21

1

4

2 4 1

⎞⎟⎟⎟⎟⎟⎠

over ]0,+∞[ is not consistent because a21a31

= 34is not equal to a22

a32= 1

4, so for the

column vectors a1 and a2 the condition (13) in Proposition 9 is not veriÞed anda2 /∈ [a1].

COROLLARY 1. The following assertions related to A = (aij ) are equivalent:

1. A = (aij ) is a consistent PCM;2. for every choice of i < n, the rows a i and a i+1 are �-proportional vectors,3. for every choice of i < n, the columns a i and a i+1 are �-proportional vectors.

Proof. From Theorem 2 and the transitivity of the relation ∼�. �

Remark 4. As we stressed in Remark 3 the constants of�-proportionality betweentwo rows (resp. columns) are computed by means of (17) (resp. (16)), so

ai+1 = a(−1)i i+1 � ai ∀i < n;

ai+1 = a(−1)i+1 i � ai ∀i < n.

4.1. Consistent Vectors and Consistent Matrices

DEFINITION 8.6 A vectorw = (w1, . . . , wn), withwi ∈ G, is a consistent vector forA = (aij ) if and only if



wi ÷ wj = aij ∀ i, j = 1, 2, . . . , n.

PROPOSITION 12.6 A = (aij ) is a consistent PCM if and only if there exists aconsistent vector w = (w1, w2, . . . , wn), wi ∈ G.

PROPOSITION 13. Let A = (aij ) be a consistent PCM and w a consistent vector forA = (aij ). A vector v is consistent for A = (aij ) if and only if v ∈ [w].

Proof. From DeÞnition 8 and Proposition 9. �

We stress that the condition of consistency in DeÞnition 4 can be written as

aik ÷ ajk = aij ∀i, j, k,

thus, in a consistent PCM, each column is a consistent vector. In Ref. 6 the followingresult is provided.

PROPOSITION 14. The following assertions related to A = (aij ) are equivalent:

i) A = (aij ) is a consistent PCM;ii) each column ak is a consistent vector;iii) the mean vector wm� = (m�(a1), . . . , m�(an)) is a consistent vector.

4.2. Bijection between Gn/ ∼� and the Set of Consistent PCMs

THEOREM 3 . Let CM be the set of the consistent PCM, then the relation:

F : [v] ∈ Gn/ ∼�→ Av =

⎛⎜⎝

v1 ÷ v1 v1 ÷ v2 . . . v1 ÷ vn

v2 ÷ v1 v2 ÷ v2 . . . v2 ÷ vn

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

vn ÷ v1 vn ÷ v2 . . . vn ÷ vn

⎞⎟⎠ ∈ CM

is a bijective function.

Proof. The function F is well deÞned because if w ∈ [v] then, by Proposition 9,wi ÷ wj = vi ÷ vj ∀i, j = 1, . . . , n and so Av = Aw.



Moreover,

� if [v] �= [w] then, by Proposition 9, ∃i, j : wi ÷ wj �= vi ÷ vj , as a consequenceAv �= Aw;so F is injective;

� if A is a consistent PCM, then, by Proposition 12, there exists a consistent vector v; byDeÞnition 8 and Proposition 13, F ([v]) = A. Thus F is surjective.

�

Theorem 3 allows us to build a consistent PCM starting form a vector; so weprovide Algorithm 2.

Algorithm 2: BUILD A CONSISTENT PCM FROM A VECTOR v = (v1, . . . vn)

for i = 1 . . . n doai = vi ÷ v

end for

Similarly, we can build the PCM for columns, by setting ai = v ÷ vi.

Example 8. Let us consider the multiplicative alo-group ]0,+∞[.Let us assume v = (2, 3, 4). Let us set

a1 = v1/v = 2/(2, 3, 4) =(1,2

3,1

2

)

a2 = v2/v = 3/(2, 3, 4) =(3

2, 1,

3

4

)

a3 = v3/v = 4/(2, 3, 4) =(2,4

3, 1

).

So v and each vector in [v] generates the consistent PCM

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

12

3

1

2

3

21

3

4

24

31

⎞⎟⎟⎟⎟⎟⎟⎟⎠

.

Example 9. Let us consider the additive alo-groupR.The vector v = (−2, 1, 2) and each vector in [v] generates the consistent PCM



A =⎛⎝0 −3 −43 0 −14 1 0

⎞⎠ .

Example 10. Let us consider the fuzzy alo-group ]0,1[.The vector v = (0.3, 0.4, 0.7) and each vector in [v] generates the consistent PCM

A =⎛⎝ 0.5 0.39 0.1550.61 0.5 0.220.845 0.78 0.5

⎞⎠ .

In order to give a simple way to check the consistency of a PCM we providethe following proposition.

PROPOSITION 15. A = (aij ) is a consistent PCM if and only if A = F [ai] for somei = 1, 2, . . . , n.

Proof. From Theorem 3, DeÞnition 8 and Proposition 14. �

For instance, the matrix A in Example 7 is consistent as A = F (a1).

5. FURTHER CHARACTERIZATIONS OF CONSISTENT PCMS

In Section 4, we have provided characterizations for building a consistent PCM,starting from comparisons contained in a row (resp. a column) of the built PCM orfrom a vector that represents a consistent vector for the built PCM, and for checkingthe consistency by checking each entry of the given PCM.

In this section, we provide new characterizations of the consistency that allowsus to build a consistent PCM from a different set of comparisons and to check theconsistency, by checking only a minimum number of entries of the given PCM.

5.1. Useful Characterizations for Building a Consistent PCM

PROPOSITION 16. The following assertions are equivalent:

1. A is a consistent PCM with respect to �;2. aik = ai i+1 � ai+1 k ∀i, k : i < k;3. aik = ai i+1 � ai+1 i+2 � . . . � ak−1 k ∀i, k : i < k.

Proof. 1 ⇒ 2. It is straightforward because of Proposition 4.



2 ⇒ 3. By 2:

aik = ai i+1 � ai+1 k

ai+1 k = ai+1 i+2 � ai+2 k

...

ak−2 k = ak−2 k−1 � ak−1 k

Thus, by associativity of �, 3 is achieved.3 ⇒ 1. By Proposition 4, it is enough to prove that 3 ⇒ (8).

Let i < j < k. By 3, we have that

aik = ai i+1 � . . . aj−1 j � aj j+1 . . . � ak−1 k;

so, by associativity of � and applying again 3, we have that

aik = (ai i+1 � ai+1 i+2 � . . . � aj−1 j )

�(aj j+1 � aj+1 j+2 � . . . � ak−1 k) = aij � ajk.

�

Proposition 16 generalizes to PCMs deÞned on alo-group a result provided inRef. 8 for the fuzzy case.

By Proposition 16, the following corollary follows.

COROLLARY 2. A = (aij ) is a consistent PCM with respect to �, if and only if:

dG(aik, ai i+1 � ai+1 k) = e ∀ i, k : i < k.

We provide Algorithm 3 to build a consistent PCM starting from a12, a23, . . .

an−1 n.

Algorithm 3: BUILDING A CONSISTENT PCM STARTING FROM a12, a23, . . . an−1 n

for i = 1, . . . n − 2 doai i+2 = ai i+1 � ai+1 i+2temp = ai i+2for k = i + 3 . . . n do

ai k = temp � ak−1 k

temp = ai k

end forend for



for i = 2, . . . n doforj = 1, . . . i − 1 do

ai j = a(−1)j i

end forend forfor i = 1, . . . , n do

aii = e

end for

Example 11. Let {x1, x2, x3, x4, x5} be a set of alternatives.We suppose that theDMexpresses the following preference ratios (multiplicative case): a12 = 2, a23 = 2,a34 = 5

4and a45 = 6

5. By means of Algorithm 3, we obtain

a13 = a12 · a23 = 2 · 2 = 4,

a14 = a12 · a23 · a34 = a13 · a34 = 5,

a15 = a12 · a23 · a34 · a45 = a14 · a45 = 6,

a24 = a23 · a34 = 5

2,

a25 = a23 · a34 · a45 = a24 · a45 = 3,

a35 = a34 · a45 = 3

2.

Computing the symmetric elements, we have

a21 = 1

2,

a31 = 1

4, a32 = 1

2,

a41 = 1

5, a42 = 2

5, a43 = 4

5,

a51 = 1

6, a52 = 1

3, a53 = 2

3, a54 = 5

6.

Finally,

a11 = a22 = a33 = a44 = a55 = 1.

Thus,



A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 2 4 5 61

21 2

5

23

1

4

1

21

5

4

3

21

5

2

5

4

51

6

51

6

1

3

2

3

5

61

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

Example 12. Let {x1, x2, x3, x4, x5} be a set of alternatives. We suppose that theDM expresses the following preferences (fuzzy case): a12 = 0.6, a23 = 0.609, a34 =0.632 and a45 = 0.692. By means of Algorithm 3, we obtain

A =

⎛⎜⎜⎜⎝0.5 0.6 0.7 0.8 0.90.4 0.5 0.609 0.727 0.8570.3 0.391 0.5 0.632 0.7940.2 0.273 0.368 0.5 0.6920.1 0.143 0.206 0.308 0.5

⎞⎟⎟⎟⎠ .

5.2. Useful Characterizations for Checking the Consistency

In the following, if w = (w1, . . . , wn) ∈ Gn, then w[l : m], with 1 ≤ l ≤ m ≤n, will denote the vector (wl, . . . , wm) ∈ Gm−l+1.

PROPOSITION 17. The following assertions related to A = (aij ) are equivalent:

1. A = (aij ) is a consistent PCM;2. the sub-rowsai[i + 2 : n]andai+1[i + 2 : n]are�-proportional vectors for i = 1 . . . n −2, that is

ai+1 k = a(−1)i i+1 � aik ∀i = 1, . . . n − 2 ∀k = i + 2, . . . n;

3. the sub-columns ai[1 : i − 1] and ai+1[1 : i − 1] are �-proportional vectors for i =2, . . . n − 1, that is,

ak i = a(−1)i i+1 � ak i+1 ∀i = 2, . . . n − 1 ∀k = 1, . . . i − 1.

Proof. By assumption of reciprocity for A = (aij ), Corollary 1 and Remark 4. �



In order to check whether or not a PCM is consistent, we use item 2 ofProposition 17 and we provide Algorithm 4. In Algorithm 4, we assume that

� i is the index of the rows of A, it is initialized to i = 1;� k is the index of the columns of A, it is initialized to k = i + 2;� n is the order of A;� ConsistentMatrix is a boolean variable and the algorithm returns ConsistentMatrix =

true if and only if the matrix is consistent. It is initialized to true, butConsistentMatrix = f alse is immediately returned when an inconsistent triple(aik, ai i+1, ai+1 k) occurs.

Similar algorithm can be provided if we use item 3 of Proposition 17.

Algorithm 4: CHECKING CONSISTENCY

i = 1;ConsistentMatrix = true;while i ≤ n − 2 and ConsistentMatrix do

k = i + 2;while k ≤ n and ConsistentMatrix doif ai+1 k �= a

(−1)i i+1 � aik then

ConsistentMatrix = false;end ifk = k + 1;

end whilei = i + 1;

end whilereturn ConsistentMatrix;

Example 13. Let {x1, x2, x3, x4, x5} be a set of alternatives and

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 2 4 5 61

21 2

5

33

1

4

1

21

5

4

3

21

5

3

5

4

51

6

51

6

1

3

2

3

5

61

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

the related multiplicative PCM. In order to check whenever A is consistent we haveto check the following equalities:

a2[3 : 5] = a−112 · a1[3 : 5] = 1

2· a1[3 : 5]

a3[4 : 5] = a−123 · a2[4 : 5] = 1

2· a2[4 : 5]



and

a4[5 : 5] = a−134 · a3[5 : 5] = 4

5· a3[5 : 5].

We stress that, for instance, a24 �= 12

· a14; so the PCM is not consistent.

6. A NEW CONSISTENCY INDEX

In Section 2.4, we have introduced a consistency index IG(A). At the light ofCorollary 2, it is reasonable to deÞne a new consistency index, considering onlythe distances dG(aik, ai i+1 � ai+1 k), with i < k − 1; of course, if i = k − 1 thendG(aik, ai i+1 � ai+1 k) = e. Let T ∗ be the set {(ai i+1, ai+1 k, aik), i < k − 1} andnT ∗ = |T ∗|, then we consider the following index:

I ∗G(A) = dG(a13, a12 � a23) n = 3,( ⊙

T ∗ dG(aik, ai i+1 � ai+1 k))( 1

nT ∗ ) n > 3.

with

nT ∗ = (n − 2)(n − 1)

2.

From Corollary 2, we have the following.

PROPOSITION 18. I ∗G(A) ≥ e and A is consistent if and only if I ∗

G(A) = e.

As for n > 3 it results nT ∗ < nT , the index I ∗G(A) is more easy to compute than

the consistency index IG(A), thus, we provide the following deÞnition.

DEFINITION 9. A consistency index of A is given by I ∗G(A).

PROPOSITION 19. Let G ′ = (G′, ◦, ≤) be a divisible alo-group isomorphic to G andA′ = (h(aij )) the transformed of A = (aij ) by means of the isomorphism h : G →G′. Then I ∗

G(A) = h−1(I ∗G′(A′)).

Example 14. Let us consider

A =

⎛⎜⎝0.5 0.3 0.4 0.40.7 0.5 0.1 0.20.6 0.9 0.5 0.80.6 0.8 0.2 0.5

⎞⎟⎠



that is a PCM over the fuzzy alo-group ]0,1[. By applying the function ψ−1, with ψ

in 4, to the entries of A, we get the matrix

A′ =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

13

7

2

3

2

37

31

1

9

1

43

29 1 4

3

24

1

41

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

A′ is a PCM over the multiplicative alo-group ]0,+∞[ and its consistency index is

I ∗]0,+∞[(A

′) = 3

√I ∗]0,+∞[(A

′123) · I ∗

]0,+∞[(A′124) · I ∗

]0,+∞[(A′234)

= 3

√14 · 56

9· 169

= 5.37.

Applying Proposition 19, we can compute the consistency index ofA by meansof the isomorphism ψ in 4:

I ∗]0,1[(A) = ψ(I ∗

]0,+∞[(A′)) = 5.37

6.37= 0.84.

7. CONCLUSION AND FUTUREWORK

We consider PCMs over an alo-group G = (G, �); in this framework, theseveral approaches to PCMs are uniÞed and the drawbacks linked to the possibilityof the DM to be consistent are removed. Moreover,

� we introduce the abelian group Gn = (Gn, �×) and the notion of �-proportionality;� we provide new characterizations of a consistent PCM;� we introduce an equivalence relation ∼� on Gn and a bijection between the quotient set

Gn/ ∼� and the set of the consistent PCMs;� we provide efÞcient algorithms to check the consistency and to build a consistent PCM;� we deÞne a new consistency index.

Following the results in Refs. 2, 4, 5, and 9 for themultiplicative case, our futurework will be directed to investigate, in the general context of the PCMs over alo-groups, conditions that allow us to obtain an evaluation vector w = (w1, . . . , wn) ∈Gn, and able to represent the actual ranking of the alternatives at different levels.



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Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups

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