On multiplicative consistency of interval and fuzzy reciprocal preference relations

Accepted Manuscript On multiplicative consistency of interval and fuzzy reciprocal preference relations Jana Krejč í PII: DOI: Reference:

S0360-8352(17)30292-9 http://dx.doi.org/10.1016/j.cie.2017.07.002 CAIE 4809

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Computers & Industrial Engineering

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2 November 2016 3 July 2017

Please cite this article as: Krejč í, J., On multiplicative consistency of interval and fuzzy reciprocal preference relations, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/j.cie.2017.07.002

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On multiplicative consistency of interval and fuzzy reciprocal preference relations Jana Krejˇc´ı1 Department of Industrial Engineering, University of Trento, Povo 38123, Italy Faculty of Law, Business and Economics, University of Bayreuth, Bayreuth D-95440, Germany

Abstract Extension of Saaty’s definition of consistency to interval and fuzzy reciprocal preference relations is studied in the paper. The extensions of the definition to interval and triangular reciprocal preference relations proposed by Wang (2005), Liu (2009), Liu et al. (2014) and Wang (2015) are reviewed and some shortcomings in the definitions are pointed out. Particularly, as was already shown by Wang (2015), the definitions of consistency proposed by Liu (2009) and Liu et al. (2014) are not invariant under permutation of compared objects. Wang’s (2015) definitions rectify this drawbacks. However, as is pointed out in this paper, Wang’s definitions of consistent interval and triangular reciprocal preference relations do not keep the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. In this paper, definitions of consistent interval, triangular and trapezoidal reciprocal preference relations invariant under permutation of compared objects and preserving the reciprocity of pairwise comparisons are proposed. Useful tools for verifying the consistency are proposed and some properties of consistent interval and fuzzy reciprocal preference relations are derived. Furthermore, the new definition of consistency for interval reciprocal preference relations is compared with the definition of consistency proposed by Wang et al. (2005), and numerical examples are provided to illustrate the difference between the consistency definitions. Keywords: Multiplicative consistency, triangular reciprocal preference relation, interval reciprocal preference relation, constrained fuzzy arithmetic, AHP

Email address: [email protected] (Jana Krejˇ c´ı)

Preprint submitted to Computers & Industrial Engineering

November 2, 2016

On multiplicative consistency of interval and fuzzy reciprocal preference relations

Abstract Extension of Saaty’s definition of consistency to interval and fuzzy reciprocal preference relations is studied in the paper. The extensions of the definition to interval and triangular reciprocal preference relations proposed by Wang (2005), Liu (2009), Liu et al. (2014) and Wang (2015) are reviewed and some shortcomings in the definitions are pointed out. Particularly, as was already shown by Wang (2015), the definitions of consistency proposed by Liu (2009) and Liu et al. (2014) are not invariant under permutation of compared objects. Wang’s (2015) definitions rectify this drawbacks. However, as is pointed out in this paper, Wang’s definitions of consistent interval and triangular reciprocal preference relations do not keep the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. In this paper, definitions of consistent interval, triangular and trapezoidal reciprocal preference relations invariant under permutation of compared objects and preserving the reciprocity of pairwise comparisons are proposed. Useful tools for verifying the consistency are proposed and some properties of consistent interval and fuzzy reciprocal preference relations are derived. Furthermore, the new definition of consistency for interval reciprocal preference relations is compared with the definition of consistency proposed by Wang et al. (2005), and numerical examples are provided to illustrate the difference between the consistency definitions. Keywords: Multiplicative consistency, triangular reciprocal preference relation, interval reciprocal preference relation, constrained fuzzy arithmetic, AHP

1. Introduction Reciprocal preference relations play a significant role in multi-criteria decisionmaking methods based on pairwise comparisons of objects. The most known method using reciprocal preference relations is Analytic Hierarchy Process (AHP in the following) developed by Saaty (1977, 1980). In this method, Saaty defined a scale of integers from 1 to 9 with assigned linguistic terms expressing the intensity of preference of one compared object over another one. Using this scale, reciprocal preference relations of alternatives and criteria are constructed. By applying various methods, priorities of alternatives and criteria

Preprint submitted to Computers & Industrial Engineering

July 5, 2017

are then elicited from the reciprocal preference relations and finally aggregated into overall priorities of alternatives. In practical applications of reciprocal preference relations, concept of consistency plays an important role. In particular, in order to guarantee that the priorities of objects derived from reciprocal preference relations are reasonable, consistency of the preference information provided in the reciprocal preference relations should be verified. For that, Saaty (1977) provided a definition of consistent reciprocal preference relations based on the multiplicative-transitivity property. Later, also other definitions of consistency were proposed; see, e.g., Basile and D’Apuzzo (2002); Stoklasa et al. (2013). Because not always reciprocal preference relations are consistent, various indices for measuring acceptable inconsistency of reciprocal preference relations have also been proposed. However, reciprocal preference relations are not able to handle the imprecision of information in real decision-making problems. Further, crisp numbers representing linguistic terms expressing the intensity of preference of one compared object over another one cannot handle the vagueness in their meaning. For these reasons, the extension of the AHP method and reciprocal preference relations to intervals and fuzzy numbers has been studied; see, e.g., Laarhoven and Pedrycz (1983); Buckley (1985); Cheng and Mon (1994); Chang (1996); Xu (2000); Buckley et al. (2001); Csutora and Buckley (2001); Enea and Piazza (2004); Krejˇc´ı et al. (2017); Krejˇc´ı (2016, 2017a,b,c). It should be mentioned here that harsh critics of fuzzy extension of AHP appeared recently, and fallacy of all well-known fuzzy AHP methods was claimed by Zh¨ u (2014). However, shortly afterwards, Fedrizzi and Krejˇc´ı (2015) demonstrated that these critics are not well-founded as they are based on arguments contradicting commonly accepted results of fuzzy set theory. Furthermore, Fedrizzi and Krejˇc´ı (2015) showed that it is possible to extend the AHP methods to fuzzy numbers properly by applying the constrained fuzzy arithmetic and preserving the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. Also in interval and fuzzy reciprocal preference relations consistency of preference information plays a very important role since the inconsistency can lead to wrong decisions. That is the reason why consistency of interval and fuzzy reciprocal preference relations and measures of inconsistency have been studied extensively; see, e.g., Buckley (1985); Wang et al. (2005); Liu (2009); Liu et al. (2014); Krejˇc´ı and Stoklasa (2016); Krejˇc´ı (2015, 2017a,b,c); Zheng et al. (2012); Gavalec et al. (2014); Li et al. (2016); Jandov´a et al. (2016). Definitions of consistency and inconsistency indices for interval and fuzzy reciprocal preference relations should preserve two basic properties - invariance under permutation of objects and reciprocity of pairwise comparisons. According to Brunelli and Fedrizzi (2015), invariance under permutation of objects is a desirable property. In fact, they introduced this property as one of the axioms characterizing inconsistency indices. Moreover, the lack of invariance under permutation of objects of some definitions of consistency for interval fuzzy preference relations was already pointed out and criticized by Wang (2014); Wang and Chen (2014); Krejˇc´ı (2017b) and Krejˇc´ı (2017c). Therefore, 2

the definition of consistency for interval and fuzzy reciprocal preference relations, similarly as Saaty’s definition of consistency for reciprocal preference relations, should not depend on permutation of objects compared in the preference relation. Reciprocity of pairwise comparisons is an inherent property of reciprocal preference relations which needs to be extended properly also to interval and fuzzy reciprocal preference relations. This does not concern only the simple reciprocity of corresponding intervals and fuzzy numbers in the interval and fuzzy reciprocal preference relations, respectively. The constrained fuzzy arithmetic introduced by Klir and Pan (1998) needs to be employed in the extension of reciprocal preference relations to fuzzy numbers and intervals in order to handle properly the reciprocity property. With extension of reciprocal preference relations to intervals and fuzzy numbers, also another interesting issue emerges. Pairwise comparisons provided by decision makers in interval and fuzzy reciprocal preference relations can be highly indeterminate - the corresponding intervals or fuzzy numbers can be very vague. This may lead to highly indeterminate results (interval or fuzzy priorities of objects) with just a little information useful for making a decision. Therefore, it might be useful to measure the indeterminacy of interval and fuzzy reciprocal preference relations. An interesting approach for measuring indeterminacy of interval reciprocal preference relations was introduced by Li et al. (2016). In this paper, the extension of the original definition of consistency for reciprocal preference relations given by Saaty (1977) (that is the consistency based on the multiplicative-transitivity property) is focused on. The extensions of the definition proposed in the literature are reviewed and drawbacks of some of them are pointed out. The extension of the definition to interval reciprocal preference relations proposed by Liu (2009) and the fuzzy extension to triangular and trapezoidal reciprocal preference relations proposed by Liu et al. (2014) are reviewed briefly. As was already pointed out by Wang (2015b,a), the definitions proposed by Liu (2009) and Liu et al. (2014) are dependent on the labeling of objects. Wang (2015b,a) proposed other definitions of consistency for interval and triangular reciprocal preference relations. These definitions of consistency are based on the extension of a property equivalent to Saaty’s definition of consistency of reciprocal preference relations. However, it is shown in this paper that the extension of this property is not done properly as it is based on the standard interval fuzzy arithmetic and thus it violates the reciprocity of pairwise comparisons in interval and fuzzy reciprocal preference relations. Afterwards, a proper fuzzy extension of the definition of consistency given by Saaty (1977) is proposed. Properties of consistent interval and fuzzy reciprocal preference relations are studied, and the definition of consistent interval reciprocal preference relations is compared with the definition proposed by Wang et al. (2005). The definition of consistency proposed in this paper can be easily modified in order to be applied to fuzzy reciprocal preference relations with an arbitrary type of fuzzy numbers described uniquely by their α−cuts. The paper is organized as follows. In Section 2, basic notions of reciprocal preference relations, triangular and trapezoidal fuzzy numbers, standard 3

and constrained fuzzy arithmetic, and interval and fuzzy reciprocal preference relations are given. In Section 3, the definitions of consistency proposed by Wang et al. (2005), Liu (2009), Liu et al. (2014) and Wang (2015b,a) are reviewed. Further, the shortcomings of the definitions proposed by Liu (2009), Liu et al. (2014), Wang (2015b), and Wang (2015a) are discussed. In Section 4, new definitions of consistency for interval, triangular and trapezoidal reciprocal preference relations are provided. In Section 5, the properties of consistent interval and fuzzy reciprocal preference relations are studied, and the comparison with the definition of consistent interval reciprocal preference relations proposed by Wang et al. (2005) is done. Finally, in Section 6, illustrative examples are provided, and the conclusion is done in Section 7. 2. Preliminaries In this section, basic notions of reciprocal preference relations, triangular and trapezoidal fuzzy numbers, standard and constrained fuzzy arithmetic, and interval and fuzzy reciprocal preference relations are given. 2.1. Reciprocal preference relations Reciprocal preference relations have their origins in AHP (Saaty (1977, 1980)). A reciprocal preference relation on a finite set of n objects o1 , . . . , on n is represented by a square matrix A = {aij }i,j=1 . Element aij of the matrix represents the intensity of preference of object oi over object oj by means of the ratio of their priorities. In AHP, Saaty’s scale of integer numbers 1 − 9 together with their reciprocals is usually used for expressing the intensities of preference on pairs of compared objects. To each element of the scale a linguistic term expressing the intensity of preference of one compared object over another one n is assigned, see Table 1. A reciprocal preference relation A = {aij }i,j=1 is required to be (multiplicatively) reciprocal, that is aji = a1ij , i, j = 1, . . . , n, and n aii = 1, i = 1, . . . , n. Moreover, a reciprocal preference relation A = {aij }i,j=1 is said to be consistent if aij = aik akj ,

i, j, k = 1, . . . , n.

(1) n

Theorem 1. For a reciprocal preference relation A = {aij }i,j=1 , the following statements are equivalent: (i) aij = aik akj , i, j, k = 1, . . . , n, (ii) aij ajk aki = 1, i, j, k = 1, . . . , n, (iii) aij ajk aki = aik akj aji , i, j, k = 1, . . . , n. Therefore, consistency of a reciprocal preference relation can be checked by verifying any of the statements (i)–(iii) of Theorem 1.

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Table 1: Saaty’s scale.

Intensity of preference 1 3 5 7 9 2, 4, 6, 8

Linguistic term equal preference moderate preference strong preference very strong preference extreme preference intermediate values between the two adjacent judgments connected by word ”between”

2.2. Interval and fuzzy arithmetic The fuzzy set theory was initiated by Zadeh (1965, 1975). Let U be a nonempty set. A fuzzy set Se on the set U nis characterized by o its membership e e e function S : U → [0, 1]. The set Core S := u ∈ U ; S (u) = 1 denotes the core n o e and the set Supp Se := u ∈ U ; Se (u) > 0 denotes the support of S. e of S, According to Laarhoven and Pedrycz (1983), a triangular fuzzy number e c is a fuzzy set on R whose membership function is given uniquely by a triple of its representing values (cL , cM , cU ) as  x−cL , cL < x < cM ,  cM −cL      1, x = cM , e c (x) = (2)  cU −x M U  c < x < c , U −cM ,  c    0, otherwise, where cL and cU are called the lower and upper boundary values of the triangular fuzzy number e c, and cM is called the middle value of the triangular fuzzy number e c. Every triangular fuzzy number can be uniquely
described by a triple of these representing values; the notation e c = cL , cM
, cU is used in the paper hereafter. L A triangular fuzzy number e c = cL , cM , cU is said to be
positive if c > 0. L M U The core of a triangular fuzzy number e c = c ,c ,c is the singleton set Core e c = cM , and the support is an open interval Supp e c = ] cL , cU [ . Similarly, a trapezoidal fuzzy number ze is a fuzzy set on R whose membership function is given uniquely by a quadruple of its representing values (z α , z β , z γ , z δ ) as  x−zα , zα < x < zβ ,  z β −z α      1, zβ ≤ x ≤ zγ , ze (x) = (3) z δ −x   zγ < x < zδ , δ −z γ ,  z    0, otherwise. 5

Every trapezoidal fuzzy number can be uniquely described by a quadruple of
its representing values; the notation ze = z α , z β , z γ , z δ is used in the
paper hereafter. The core of a trapezoidal fuzzy number ze = z α , z β , z γ , z δ is a  β γ closed interval Core ze = z , z , and the support is an open interval Supp ze = ] zα, zδ [ . In order to handle the imprecision of information in real decision-making problems and the vagueness in meaning of linguistic terms used for expressing the intensities of preference on pairs of compared objects, the extension of reciprocal preference relations to intervals and fuzzy numbers has been introduced. Intervals are used in decision-making situations where the decision maker is able or prefers to provide only intervals of possible intensities of preference of one object over another one without further specifying which intensities of preference from the given intervals are more possible than others; all intensities of preference of one object over another one from the given interval are equally possible. Fuzzy numbers, most commonly triangular and trapezoidal fuzzy numbers, are applied in decision-making situations where the decision maker is able or wishes to provide more detailed information regarding the intensities of preference on the pairs of compared objects by specifying which intensities of preference are more possible than others. Particularly, in case of triangular fuzzy numbers, the decision maker is asked to provide the most possible intensity of preference and the lowest and highest possible intensities of preference for each pairwise comparison. In fuzzy multi-criteria decision making, and namely in fuzzy extension of AHP, simplified fuzzy arithmetic is usually applied; see, e.g., Laarhoven and Pedrycz (1983); Buckley (1985); Cheng and Mon (1994); Chang (1996); Xu (2000); Buckley et al. (2001); Csutora and Buckley (2001); Enea and Piazza (2004); Wang et al. (2005); Liu (2009); Krejˇc´ı (2015); Krejˇc´ı et al. (2017); Krejˇc´ı (2017a); Krejˇc´ı and Stoklasa (2016); Zheng et al. (2012); Liu et al. (2014); Gavalec et al. (2014). In fact, in most theoretical and application papers, simplified fuzzy arithmetic is applied without even mentioning this fact. In case of triangular and trapezoidal fuzzy numbers, the arithmetic operations are performed only on the representing values of these fuzzy numbers which are then connected by lines. This means that the representing values of the resulting fuzzy number are computed correctly and the functions connecting these representative values are approximated by linear functions. Thus, we obtain the results of the arithmetic operations again in the form of a triangular or trapezoidal fuzzy number, respectively.

For two positive triangular fuzzy numbers e a = aL , aM , aU and eb = bL , bM , bU , the arithmetic operations in simplified standard fuzzy arithmetic are defined in

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the following way:
e a + eb = aL + bL , aM + bM , aU + bU ,
e a − eb = aL − bU , aM − bM , aU − bL ,
e a ⊗ eb = aL bL , aM bM , aU bU ,  L M U e a a a a = , , , eb bU bM bL   1 1 1 1 = , , . e a aU aM aL

(4) (5) (6) (7) (8)



For two positive trapezoidal fuzzy numbers ye = y α , y β , y γ , y δ and ze = z α , z β , z γ , z δ , the arithmetic operations in simplified standard fuzzy arithmetic are defined in the following way:
ye + ze = y α + z α , y β + z β , y γ + z γ , y δ + z δ ,
ye − ze = y α − z δ , y β − z γ , y γ − z β , y δ − z α ,
ye ⊗ ze = y α z α , y β z β , y γ z γ , y δ z δ ,  α β γ δ ye y y y y = , γ, β, α , δ ze z z z z  1 1 1 1 1 = , , , . ye yδ yγ yβ yα The interval standard arithmetic is defined in an analogous way; the arithmetic operations are defined as (4)–(8) except for the middle values that are now missing. 2.3. Constrained fuzzy arithmetic As it is pointed out by Klir and Pan (1998), the concept of standard fuzzy arithmetic can be applied only if there are no interactions between the fuzzy numbers. In case of any interactions between the fuzzy numbers, the concept of constrained fuzzy arithmetic should be considered instead. Let f be a continuous
M U function, f : Rn → R, and let triangular fuzzy numbers e ai = aL i , ai , ai , i = 1, . . . , n, express uncertain values of an n-tuple of variables. Let g(a1 , . . . , an ) = 0 represent a constraint imposed on the n-tuple
of variables. Then f (e a1 , . . . , e an ) is a triangular fuzzy number e c = cL , cM , cU whose representing values are given as follows:    U cL = min f (a1 , . . .
, an ) ; ai ∈ aL i , ai , i = 1, . . . , n, g(a1 , . . . , an ) = 0 , M cM = f aM 1 , . . . , an ,   U U c = max f (a1 , . . . , an ) ; ai ∈ aL i , ai , i = 1, . . . , n, g(a1 , . . . , an ) = 0 . Let us illustrate the use of the constrained fuzzy arithmetic on a very simple example. Let the triangular fuzzy number e a = (3, 5, 6) represent a certain 7

linguistic variable. According to the standard fuzzy arithmetic, the difference e a−e a would be computed as e a−e a = (3, 5, 6) − (3, 5, 6) = (−3, 0, 3) . However, this is not precise since both operands express the same state of one variable. Therefore, the constraint a1 = a2 , a1 ∈ [3, 6] , a2 ∈ [3, 6] should be imposed on the variables. Then, according to the constrained fuzzy arithmetic, the difference e c = (cL , cM , cU ) = e a−e a should be computed correctly as follows: cL cM cU

= = =

min {a1 − a2 ; a1 ∈ [3, 6] , a2 ∈ [3, 6] , a1 = a2 } = min {a1 − a1 ; a1 ∈ [3, 6]} = 0, aM − aM = 5 − 5 = 0, max {a1 − a2 ; a1 ∈ [3, 6] , a2 ∈ [3, 6] , a1 = a2 } = max {a1 − a1 ; a1 ∈ [3, 6]} = 0.

For more examples on applying constrained fuzzy arithmetic and a more detailed description of the problematic, see Klir and Pan (1998). Analogously to the constrained fuzzy arithmetic, the constrained interval arithmetic for computing with intervals is defined; the procedure for computing the lower and upper boundary values is the same as in the constrained fuzzy arithmetic and the middle values are missing. The constrained interval and fuzzy arithmetic will be applied later in the paper in order to define properly the consistency of interval and fuzzy reciprocal preference relations based on the multiplicative-transitivity property. 2.4. Interval and fuzzy reciprocal preference relations By using simplified fuzzy interval arithmetic, reciprocal preference relations are extended to triangular and trapezoidal fuzzy numbers and to intervals. A triangular reciprocal preference relation on a finite set of n objects o1 , . . . ,
on n M U e = {e is represented by a square fuzzy matrix A aij }i,j=1 , e aij = aL ij , aij , aij , where element e aij represents the intensity of preference of object oi over object oj . Similarly as for crisp reciprocal preference relations, a triangular ren e = {e ciprocal preference relation A aij }i,j=1 is required to be reciprocal, that is 1 e aji = eaij , i, j = 1, . . . , n, according to (8) and e aii = 1, i = 1, . . . , n. n

Analogously, interval reciprocal preference relation A = {aij }i,j=1 , aij =  L U n e = {e a ,a and trapezoidal reciprocal preference relation A aij }i,j=1 , e aij =  ij ij  β γ α δ aij , aij , aij , aij , are defined. 3. Definition of consistency In this section, the definitions of consistency for interval, triangular and trapezoidal reciprocal preference relations introduced by Wang et al. (2005), Liu (2009), Liu et al. (2014), Wang (2015b), and Wang (2015a) will be reviewed. Further, the drawbacks in the definitions proposed by Liu (2009), Liu et al. (2014), Wang (2015b), and Wang (2015a) will be pointed out. In particular, violation of invariance under permutation of objects and violation of reciprocity of pairwise comparisons, which is the key property of reciprocal preference relations, will be discussed.

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Wang et al. (2005) defined the consistency of interval reciprocal preference relations as follows:   n U Definition 1. (Wang et al. (2005)) Let A = {aij }i,j=1 , aij = aL ij , aij , be an relation. If the convex feasible region S = n interval reciprocal preference o Pn wi L U w1 , . . . , wn : aij ≤ wj ≤ aij , i=1 wi = 1, wi > 0, i = 1, . . . , n is nonempty, then A is said to be consistent. Otherwise, A is said to be inconsistent. Further, they proved the following theorem which provides a useful tool for verifying the consistency according to Definition 1.   n U Theorem 2. Interval reciprocal preference relation A = {aij }i,j=1 , aij = aL ij , aij , is consistent if and only if it satisfies the following constraints:   U U L max aL aik akj , ∀i, j = 1, . . . , n. (9) ik akj ≤ min k=1,...,n

k=1,...,n

In the following theorem, it will be shown that it is sufficient to verify the inequality (9) just for i, j = 1, . . . , n, i < j, thus saving more than half of the computations.   n U Theorem 3. Interval reciprocal preference relation A = {aij }i,j=1 , aij = aL ij , aij , is consistent if and only if it satisfies the following constraints:   U U L max aL aik akj , ∀i, j = 1, . . . , n, i < j. (10) ik akj ≤ min k=1,...,n

k=1,...,n

Proof. It is sufficient to show that validity of inequalities (10) for i, j = 1, . . . , n, i < j implies automatically the validity for all i, j = 1, . . . , n, i.e. that (9) is valid. The validity of inequalities (9) for i = j is trivial from the definition of interval reciprocal preference relations since  L  U   aik aik L U max aL a = max ≤ 1 ≤ min ≤ min aU ik ki ik aki . U L k=1,...,n k=1,...,n a k=1,...,n a k=1,...,n ik ik Further, for i > j, by using (10) and the reciprocity properties, we get ) (  L L 1 1 1 n o≤ max aik akj = max = U U k=1,...,n k=1,...,n aki ajk min aU aU k=1,...,n

1 n o = min k=1,...,n L max aL jk aki

(

1 1 L aL jk aki

jk ki

) =

min

k=1,...,n

 U U aik akj .

k=1,...,n

Another definition of consistency for interval reciprocal preference relations was given by Liu (2009).

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  n U Definition 2. (Liu (2009)) Let A = {aij }i,j=1 , aij = aL ij , aij , be an interval reciprocal preference relation. Further, let reciprocal preference relations L = n n {lij }i,j=1 , U = {uij }i,j=1 be constructed from the interval reciprocal preference relation A as  L  U  aij , i < j  aij , i < j 1, i=j 1, i=j . lij = uij = (11)  U  L aij , i > j aij , i > j If the matrices L, U are consistent according to (1), then A is said to be consistent. Otherwise, A is said to be inconsistent. Later, Liu et al. (2014) defined consistency of a triangular reciprocal preference relation in the following way:
n M U e = {e Definition 3. (Liu et al. (2014)) Let A aij }i,j=1 , e aij = aL ij , aij , aij , be a triangular reciprocal preference relation. Further, let reciprocal preference n n n relations L = {lij }i,j=1 , M = {mij }i,j=1 and U = {uij }i,j=1 be constructed e as from the triangular reciprocal preference relation A  L  M  U  aij , i < j  aij , i < j  aij , i < j 1, i=j 1, i=j 1, i=j . lij = mij = uij =  U  M  L aij , i > j aij , i > j aij , i > j (12) e is said to be If the matrices L, M and U are consistent according to (1), then A e is said to be inconsistent. consistent. Otherwise, A Similarly to Definition 3, the definition of consistency of trapezoidal reciprocal preference relations was given by Liu et al. (2014). Liu et al. (2014) state that Definition 3 of consistency naturally reflects the reciprocity property of triangular reciprocal preference relations since the preference relations L, M and U are clearly reciprocal. Further, the authors claim that the definition of consistency is closely related to the typical definition of consistent reciprocal preference relations given by Saaty (1980). However, as was demonstrated by Wang (2015a), Definition 3 highly depends on the ordering of objects compared in the triangular reciprocal preference relation, i.e. it has no invariance to permutation of the objects. Analogously, Wang (2015b) demonstrated the invalidity of Definition 2 for interval reciprocal preference relations. The drawback in Definition 3 is caused by the fact that the reciprocal preference relations L and U are dependent on the labeling of compared objects. There is no logical justification for choosing the lower boundary values of triangular fuzzy numbers above the main diagonal and the upper boundary values of the triangular fuzzy numbers below the main diagonal to construct the reciprocal preference relation L. A similar problem appears for the reciprocal preference relation U . This way of constructing reciprocal preference relations L and U from a triangular reciprocal preference relation does not reflect naturally the reciprocity property of triangular reciprocal preference relations; the 10

reciprocal preference relations L and U change completely with a permutation of compared objects. Notice that the reciprocal preference relation M does not change with any permutation of compared objects; the reciprocal preference relation M is permuted accordingly but no changes in the intensities of preference occur. Definition 2 of consistency for interval reciprocal preference relations suffers from the same drawbacks, and the same is valid also for the consistency of trapezoidal reciprocal preference relations defined by Liu et al. (2014). Clearly, Definitions 2 and 3 are not proper fuzzy extensions of the definition of consistency (1) proposed by Saaty (1980). Properly defined consistency of interval and triangular reciprocal preference relations has to be independent of alternatives labeling and, naturally, as was already pointed out by Liu et al. (2014), it has to preserve the reciprocity condition. Wang (2015b,a) proposed definitions of consistency for interval and triangular reciprocal preference relations independent of alternatives labeling as follows:   n U Definition 4. (Wang (2015b)) Let A = {aij }i,j=1 , aij = aL ij , aij , be an inn terval reciprocal preference relation. A = {aij }i,j=1 is said to be consistent if U L U L U aL ij aij = aik aik akj akj ,

∀i, j, k = 1, . . . , n.

(13)


n M U e = {e Definition 5. (Wang (2015a)) Let A aij }i,j=1 , e aij = aL ij , aij , aij , be a n e = {e triangular reciprocal preference relation. A aij }i,j=1 is said to be consistent if e aij ⊗ e ajk ⊗ e aki = e aik ⊗ e akj ⊗ e aji , ∀i, j, k = 1, . . . , n. (14) Further, according to Wang (2015a), the following theorem is valid. Theorem 4. (Wang (2015a)) For a triangular reciprocal preference relation
n M U e = {e A aij }i,j=1 , e aij = aL ij , aij , aij , the following statements are equivalent: (i) (ii) (iii) (iv)

e is consistent according to Definition 5, A M M L U L U L U aM ij = aik akj , aij aij = aik aik akj akj , i, j, k = 1, . . . , n, M M M M M M L L L L L aij ajk aki = aik akj aji , aL ij ajk aki = aik akj aji , i, j, k = 1, . . . , n, M M M M M U U U U U U aM ij ajk aki = aik akj aji , aij ajk aki = aik akj aji , i, j, k = 1, . . . , n.

A theorem similar to Theorem 4 can be formulated for interval reciprocal n preference relations A = {aij }i,j=1 by just removing the middle values aM ij , i, j = 1, . . . , n, and the associated formulas. Note 1. Notice that multiplication in the formula (14) is done according to the simplified standard fuzzy arithmetic as defined in Section 2. In fact, using the simplified standard fuzzy arithmetic, the formula (14) can be written as (iii) and (iv) of Theorem 4. Therefore, based on Theorem 4, it follows that Definitions 4 and 5 for interval and triangular reciprocal preference relations, respectively, are practically the same, even though consistency conditions (13) and (14) seem to be different.

11

Let us focus more in detail on the requirement of reciprocity of pairwise comparisons in crisp and triangular reciprocal preference relations and on its impact on the consistency condition. As is stated in Section 2, reciprocity of pairwise n comparisons is a key property of reciprocal preference relations A = {aij }i,j=1 . The reciprocity means that when, for example, the intensity of preference of object oi over object oj is aij = 3 (i.e. object oi is moderately preferred over object oj ), then the intensity of preference of object oj over object oi has to be aji = 13 (i.e. object oj has to be moderately less preferred over object oi ). Therefore, the reciprocity is a reasonable property that results from a very natural requirement on the pairwise comparisons of objects. Moreover, because of the reciprocity property, the pairwise comparisons aii , i = 1, . . . , n, are always equal to 1 standing for equal preference. Also this result is very natural since the pairwise comparison aii expresses the intensity of preference of object oi over itself (clearly, any object has to be equally preferred to itself). Because of the reciprocity of pairwise comparisons, the consistency condition (1) for reciprocal preference relations is equivalent to statements (ii) and (iii) in Theorem 1. Conception of the reciprocity becomes more complicated when extended to fuzzy numbers or intervals. For
a triangular reciprocal preference relation n M U e = {e A aij }i,j=1 , e aij = aL aji = ea1ij = ij , aij , aij , the reciprocity is defined as e   1 , 1 , 1 . According to this property, when e.g. the highest possible inaU aM aL ij

ij

ij

U tensity of preference aU ij of object oi over object oj is aij = 5 (i.e. object oi is at most strongly preferred over object oj ), this means that the lowest possible 1 L intensity of preference aL ji of object oj over object oi is automatically aji = 5 (i.e. object oj is at least strongly less preferred over object oi ). However, this is not all. It is important to realize that the reciprocity of pairwise comparisons extended to triangular reciprocal preference relations does not concern only the reciprocity of triangular fuzzy numbers above and below the main diagonal of e The requirement of reciprocity carries much more information. For example, A.
M U when we consider any particular value aij of e aij = aL ij , aij , aij as a possible intensity of preference of object oi over object oj , this intensity of preference is
M U associated with a corresponding intensity of preference aji of e aji = aL ji , aji , aji 1 such that aji = aij . Definition 1 of consistency for reciprocal preference relations should be extended appropriately to triangular reciprocal preference relations so that Theorem 1 can be extended to triangular reciprocal preference relations accordingly. This can be done by employing properly the reciprocity property. Wang (2015a) defined consistency of triangular reciprocal preference relations by fuzzifying the expression (iii) in Theorem 1 using the simplified standard fuzzy arithmetic; see Definition 5. As was emphasized by Wang (2015a), the definition is invariant to permutation of objects in the triangular reciprocal preference relation. This is an advantage over the definition formerly proposed by Liu et al. (2014). Wang (2015a) applied the simplified standard fuzzy arithmetic on computations with triangular fuzzy numbers. Therefore, the expression (14) is nothing

12

else but the statements (iii) and (iv) in Theorem 4. However, the expressions L L L L L U U U U U U aL ij ajk aki = aik akj aji and aij ajk aki = aik akj aji in the statements (iii) and (iv) do not preserve the reciprocity of pairwise comparisons. For example, in the first expression, the intensity of preference aL ij of oi over oj and the intensity of preference aL of o over o are used at the same time. This clearly j i ji 1 1 violates the reciprocity of pairwise comparisons since aL ji = aU 6= aL (unless ij

U aL ij = aij =

1 aL ji

=

1 ). aU ji

ij

Therefore, even though Definitions 4 and 5 of consistent

interval and triangular reciprocal preference relations are invariant to permutation of objects in the reciprocal preference relations, their meaning is questionable since they violate the reciprocity property of pairwise comparisons. Further, when applying the simplified standard fuzzy arithmetic, as Wang (2015a) did, the condition (14) is not equivalent neither to e aij = e aik ⊗ e akj , i, j, k = 1, . . . , n, nor to e aij ⊗ e ajk ⊗ e aki = 1, i, j, k = 1, . . . , n. The same is true for Definition 4 of consistency for interval reciprocal preference relations. This means that, for the definitions of consistency proposed by Wang (2015b,a), Theorem 1 cannot be extended to interval and triangular reciprocal preference relations. Another serious drawback of applying the simplified standard fuzzy arithmetic on computation with pairwise comparisons of triangular reciprocal preference relations is the fact that ! L U
a a ij ij L M M U U e aij ⊗ e aji = aL , 1, L 6= 1, i, j = 1, . . . , n. ij aji , aij aji , aij aji = aU aij ij (15) From the reciprocity property e aij = ea1ji of triangular reciprocal preference relations, it should follow that e aij e aji = 1. Obviously, the expression (15) violates again the reciprocity property of pairwise comparisons. The constrained fuzzy arithmetic needs to be applied on computations with pairwise comparisons of triangular reciprocal preference relations in order to remedy this drawback, and also the definition of consistency for triangular reciprocal preference relations needs to be based on the constrained fuzzy arithmetic. In the following Section, a proper extension of the consistency condition (1) and Theorem 1 based on simplified constrained fuzzy arithmetic and constrained interval arithmetic will be proposed. 4. New definition of consistency In this section, consistency of triangular, trapezoidal and interval reciprocal preference relations will be defined, and useful tools for verifying consistency will be provided.
n M U e = {e Definition 6. Let A aij }i,j=1 , e aij = aL ij , aij , aij , be a triangular recipron e = {e cal preference relation. A aij } is said to be consistent if i,j=1

M M aM ij = aik akj

13

and

(16)

   L U  L U U ∀aij ∈ aL ij , aij ∃aik ∈ aik , aik ∧ ∃akj ∈ akj , akj : aij = aik akj

(17)

for each i, j, k ∈ {1, . . . , n} . Note 2. Definition 6 is a natural fuzzy extension of the definition of consistency (1) proposed by Saaty (1980). According to this definition, a triangular reciprocal preference relation is said to be consistent if the middle values of the triangular fuzzy numbers form a consistent reciprocal preference relation and if for any possible value aij from the closure of the support of the fuzzy pairwise comparison e aij , i, j ∈ {1, . . . , n} , there exist possible values aik and akj from the closures of the supports of e aik and e akj , k ∈ {1, . . . , n} , respectively, such that they preserve consistency according to (1). Condition (16) is equivalent to the requirement of consistency of reciprocal preference relation M in Definition 3. Definition 6 models the following decision-making situation under uncertainty. The preference system of the decision maker is in accordance with consistency definition (1); condition (16) means that the most possible intensities of preferences provided by the decision maker are consistent according to (1). At the same time, by using triangular fuzzy numbers to express the intensities of preference on pairs of compared objects, the decision maker acknowledges both the vagueness in meaning of linguistic terms used to express the intensities of preference and the imprecision of information in real decision-making problems. The decision maker admits that the intensities of his or her preferences can slightly vary from the middle values of the provided triangular fuzzy numbers within the range given by the lower and upper boundary values of the triangular fuzzy numbers. At the same time, however, the decision maker states that his or her preferences are always in accordance with consistency (1), which is captured by condition (17). Unlike the definition of consistent triangular reciprocal preference relations proposed by Liu et al. (2014), new Definition 6 is invariant under permutation of objects compared in triangular reciprocal preference relations. That is, no matter in which order the preference information is inserted in the matrix (as long as the intensities of preference do not change), the conclusion about the consistency/inconsistency based on Definition 6 is always the same. Further, unlike the definition of consistent triangular reciprocal preference relations proposed by Wang (2015a), Definition 6 does not violate the reciprocity property of pairwise comparisons. This is shown in the following proposition. Proposition 1. Definition 6 preserves the reciprocity of pairwise comparisons  U in sense that any fixed value aij ∈ aL , a ij ij , i, j ∈ {1, . . . , n} , representing the intensity of preference of object o over object oj is associated with the correi  U sponding value aji ∈ aL , a representing the intensity of preference of object ji ji 1 oj over object oi is such that aji = aij . Proof. To prove this proposition we just need to show that expression (17)   does U not violate the reciprocity property, i.e. for any fixed aij ∈ aL , a ij ij , i, j ∈ {1, . . . , n} , the intensity of preference of oj over oi is expressed by aji = a1ij . 14

For an ordered triplet i, j, k ∈ {1, . .. , n} , i 6= j 6= k, no reciprocals appear in U expression aij = aik akj for any aij ∈ aL ij , aij . For i = j = k, expression (17) reduces to ∀aii = 1 ∃a∗ii = 1 ∧ ∃a∗∗ ii = 1 : 1 = 1 · 1, which again does not violate the reciprocity property. Further, for i 6= j = k, expression (17) is as   ∗  L U U ∗ ∀aij ∈ aL ij , aij ∃aij ∈ aij , aij ∧ ∃ajj = 1 : aij = aij · 1. This means that a∗ij = aij and, therefore, the reciprocity property is not violated. For i = k 6= j the proof is analogous. Finally, for i = j 6= k, expression (17) is as    L U U ∗ ∗ ∀aii = 1 ∃aik ∈ aL ik , aik ∧ ∃aki ∈ aki , aki : 1 = aik aki . This means that a∗ki =

1 aik

and, therefore, the reciprocity is preserved.

By handling properly the reciprocity property of pairwise comparisons, Theorem 1 can be extended to triangular reciprocal preference relations as follows. n e = {e Theorem 5. For a triangular reciprocal preference relation A aij }i,j=1 , e aij =
L M U aij , aij , aij , the following statements are equivalent:

e is consistent according to Definition 6. (i) A (ii) For each i, j, k ∈ {1, . . . , n} :

M M aM ij ajk aki = 1

and

   L U  L U U ∀aij ∈ aL ij , aij ∃ajk ∈ ajk , ajk ∧ ∃aki ∈ aki , aki : aij ajk aki = 1. (18) (iii) For each i, j, k ∈ {1, . . . , n} :

M M M M M aM ij ajk aki = aik akj aji

and

   L U  L U U ∀aij ∈ aL ij , aij ∃ajk ∈ ajk , ajk ∧ ∃aki ∈ aki , aki : 1 1 1 aij ajk aki = aik akj aji , aji = , aki = , ajk = . (19) aij aik akj Proof. From the reciprocity property e aij =  L U  L U ∀aij ∈ aij , aij ∃aji ∈ aji , aji : aji = a1ij .

1 e aji , i, j

= 1, . . . , n, it follows that

(a) First, let us show that the statements (i) and (ii) are equivalent. Because of the reciprocity property, (17) can be equivalently written as    L U  L U 1 1 U ∀aij ∈ aL , ij , aij ∃aki ∈ aki , aki ∧ ∃ajk ∈ ajk , ajk : aij = ajk aki which is equivalent to (18).

15

(b) Now, let us show that the statements (ii) and (iii) are equivalent. Statement (19) can be equivalently written as    L U  L U 2 2 2 U ∀aij ∈ aL ij , aij ∃ajk ∈ ajk , ajk ∧ ∃aki ∈ aki , aki : aij ajk aki = 1. (20) Because all the fuzzy numbers in a triangular reciprocal preference relation are positive, the square roots can be removed from expression a2ij a2jk a2ki = 1, which means that (20) is equivalent to (19).

The following theorems give us useful tools for verifying consistency of triangular reciprocal preference relations.
n e = {e Theorem 6. Let A aij } ,e aij = aL , aM , aU , be a triangular reciprocal ij

i,j=1

ij

ij

e is consistent according to Definition 6 if and only preference relation. Then A if for each i, j, k ∈ {1, . . . , n} the following holds: M M aM ij = aik akj

aL ij aU ij

≥ ≤

(21)

L aL ik akj U aU ik akj

(22) (23)

e be a consistent triangular reciprocal preference relation acProof. First, let A cording to Definition 6. The equations (16) and (21) are identical. Further, let aij := aL ij . Then according to (17):    L U L U ∃aik ∈ aL ik , aik ∧ ∃akj ∈ akj , akj : aij = aik akj . L Since aik ≥ aL ik , akj ≥ akj , then clearly (22) is valid. Analogously, let aij := Then according to (17):    L U U U ∃aik ∈ aL ik , aik ∧ ∃akj ∈ akj , akj : aij = aik akj .

aU ij .

U Since aik ≤ aU ik , akj ≤ akj , then clearly (23) is valid. Second, let (21)–(23) be valid for a triangular reciprocal preference relation e A. Again, (21) and (16) are identical. Further,from inequalities (22) and (23)  L U U L U we get aL ik akj ≤ aij ≤ aik akj for every aij ∈ aij , aij and, therefore, (17) is satisfied.

According to the second part of the proof of Theorem 6, the consistency of a triangular reciprocal preference relation can be checked also by using the following theorem.
n e = {e Theorem 7. Let A aij } ,e aij = aL , aM , aU , be a triangular reciprocal ij

i,j=1

ij

ij

e is consistent according to Definition 6 if and only preference relation. Then A if for each i, j, k ∈ {1, . . . , n} the following holds: M aM = aM ij ik akj h i  L U L U U aij , aij ⊆ aL a , a a ik kj ik kj

16

(24) (25)

Proof. The proof is a direct consequence of the second part of the proof of Theorem 6. The following theorem allows us to examine only the upper part of a triangular fuzzy pairwise comparison matrix in order to verify the consistency, thus saving a substantial part of the computations.
n M U e = {e Theorem 8. Let A aij }i,j=1 , e aij = aL ij , aij , aij , be a triangular reciprocal e is consistent according to Definition 6 if and only preference relation. Then A if aM ij

M = aM ik akj

aL ij

n o L ≥ max aL ik akj k=1,...,n n o U ≤ min aU ik akj

aU ij

k=1,...,n

i, j, k ∈ {1, . . . , n} , i < k < j

(26)

i, j ∈ {1, . . . , n} , i < j

(27)

i, j ∈ {1, . . . , n} , i < j

(28)

Proof. It is trivial to show that (26) is equivalent to (21) for each i, j, k ∈ {1, . . . , n} . Further, let us show that properties (27), (28) for i < j imply the validity of these properties also for i ≥ j. Using (27) and (28) and the reciprocity properties of a triangular reciprocal preference relation, for i > j, we get ( )  L L 1 1 1 1 n o ≤ U = aL max aik akj = max = ij U U k=1,...,n k=1,...,n aki ajk aji min aU aU k=1,...,n

jk ki

and ( min

k=1,...,n



U aU ik akj



=

min

k=1,...,n

1 1 L aL ki ajk

)

1 1 n o ≥ L = aU ij . L aji max aL a jk ki

=

k=1,...,n

For i = j, we get immediately max



min



k=1,...,n

L aL ik aki = max

k=1,...,n

and k=1,...,n



U aU ik aki =

 min

k=1,...,n

aL ik

1 aU ik



aU ik

1 aL ik



≤ 1 = aL ii

≥ 1 = aU ii .

To prove the theorem it is sufficient to show that (27) and (28) are equivalent to (22) and (23), respectively. Inequality (22) n o is valid for each i, j, k ∈ {1, . . . , n} L L if and only if aL ij ≥ maxk=1,...,n aik akj

for each i, j ∈ {1, . . . , n} which is

equivalent to (27). Similarly, inequality (23) n o is valid for each i, j, k ∈ {1, . . . , n} U U if and only if aU ≤ min a a for each i, j ∈ {1, . . . , n} which is k=1,...,n ij ik kj equivalent to (28).

17

Analogously to the definition of consistent triangular reciprocal preference relations, consistent interval reciprocal preference relations and consistent trapezoidal reciprocal preference relations are defined.   n U Definition 7. Let A = {aij }i,j=1 , aij = aL ij , aij , be an interval reciprocal n preference relation. A = {aij }i,j=1 is said to be consistent if    L U  L U U ∀aij ∈ aL (29) ij , aij ∃aik ∈ aik , aik ∧ ∃akj ∈ akj , akj : aij = aik akj for each i, j, k ∈ {1, . . . , n} .   n β γ δ e = {e Definition 8. Let A aij }i,j=1 , e aij = aα ij , aij , aij , aij , be a trapezoidal e is said to be consistent if reciprocal preference relation. A  α δ  α δ    δ ∀aij ∈ aij , aij ∃aik ∈ aik , aik ∧ ∃akj ∈ aα (30) kj , akj : aij = aik akj , h i h i h i ∀aij ∈ aβij , aγij ∃aik ∈ aβik , aγik ∧ ∃akj ∈ aβkj , aγkj : aij = aik akj (31) for each i, j, k ∈ {1, . . . , n} . Again, unlike the definition of consistent interval reciprocal preference relations proposed by Liu (2009) and the definition of consistent trapezoidal reciprocal preference relations proposed by Liu et al. (2014), new Definitions 7 and 8 are invariant to permutation of objects. Further, unlike the definition of consistent interval reciprocal preference relations proposed by Wang (2015a), Definitions 7 and 8 do not violate the reciprocity of pairwise comparisons. The following two theorems give us a useful tool for a consistency check for interval and trapezoidal reciprocal preference relations.   n U Theorem 9. Let A = {e aij }i,j=1 , aij = aL ij , aij , be an interval reciprocal preference relation. Then A is consistent according to Definition 7 if and only if n o L L aL ≥ max a a ∀i, j = 1, . . . , n, i < j (32) ij ik kj k=1,...,n n o U aU ≤ min aU ∀i, j = 1, . . . , n, i < j (33) ij ik akj k=1,...,n

  n β γ δ e = {e Theorem 10. Let A aij }i,j=1 , e aij = aα ij , aij , aij , aij , be a trapezoidal e is consistent according to Definition 8 if reciprocal preference relation. Then A and only if n o α α aα ≥ max a a ∀i, j = 1, . . . , n, i < j (34) ij ik kj k=1,...,n n o aβij ≥ max aβik aβkj ∀i, j = 1, . . . , n, i < j (35) k=1,...,n n o aγij ≤ min aγik aγkj ∀i, j = 1, . . . , n, i < j (36) k=1,...,n n o aδij ≤ min aδik aδkj ∀i, j = 1, . . . , n, i < j (37) k=1,...,n

18

Note 3. Similarly as Theorems 9 and 10 were derived based on Theorem 8, we could easily derive other theorems for interval and trapezoidal fuzzy reciprocal preference relations based on Theorems 6 and 7. 5. Properties of consistent triangular reciprocal preference relations In this section, properties of consistent triangular reciprocal preference relations given by Definition 6 will be examined. Any of these properties can be easily adapted to consistent interval and trapezoidal reciprocal preference relations. Therefore, for the sake of simplicity, the focus will be put only on consistent triangular reciprocal preference relations here. Further, the new definition of consistency for interval reciprocal preference relations will be compared with Wang et al.’s Definition 1. e be a consistent triangular reciprocal preference relation Theorem 11. Let A e∗ conaccording to Definition 6. A triangular reciprocal preference relation A e is structed by eliminating the l-th row and the l-th column, l ∈ {1, . . . , n} , of A again consistent. e (16) and (17) are valid for each i, j, k ∈ {1, . . . , n} . After eliminatProof. For A, e (16) and (17) are still valid for each ing the l-th row and the l-th column of A, remaining i, j, k ∈ {1, . . . , n} − {l} . Therefore, the new triangular reciprocal e∗ is consistent. preference relation A n on
n M U e = {e e = ebij Theorem 12. Let A aij }i,j=1 , e aij = aL , a , a , and B , ebij = ij ij ij i,j=1
M U bL ij , bij , bij , be consistent triangular reciprocal preference relations according n e = {e to Definition 6. Then C cij }i,j=1 , where
 L
δ L
1−δ
M
1−δ
U
1−δ  M U M δ U δ e cij = cL , c , c = a b , a b , a bij , δ ∈ [0, 1] , ij ij ij ij ij ij ij ij is a consistent triangular reciprocal preference relation. e is a triangular reciprocal preference relation. Proof. First, let us show that C For i = 1, . . . , n we get L δ L 1−δ cL = 1δ 11−δ = 1, ii = (aii ) (bii )

U δ U 1−δ cU = 1δ 11−δ = 1, ii = (aii ) (bii )

and thus, e cii = 1. Further, cL ij

cM ij

=

=

δ L 1−δ (aL ij ) (bij )

δ M 1−δ (aM ij ) (bij )

=

1 aU ji

=

1 aM ji

!δ

!δ

1 bU ji 1 bM ji

19

!1−δ = !1−δ =

1 δ U 1−δ (aU ji ) (bji )

=

1 δ M 1−δ (aM ji ) (bji )

1 , cU ji

=

1 , cM ji

U δ U 1−δ cU = ij = (aij ) (bij )

1 aL ji

!δ

1 bL ji

!1−δ =

1 1 = L, δ (bL )1−δ (aL ) c ji ji ji

1 e cji , ∀i, j

= 1, . . . , n. e is consistent. It is sufficient to prove inequalities Second, let us show that C (21)–(23). Since (21)–(23) are valid for triangular reciprocal preference relations e and B, e we obtain A and thus, e cij =

M cM ik ckj

= aM ik


δ

L cL ik ckj

= aL ik


δ

U cU ik ckj

= aU ik


δ


1−δ  M δ  M 1−δ M δ M M 1−δ δ M 1−δ bM akj bkj = (aM = (aM = cM ij ) (bij ) ij ik ik akj ) (bik bkj )     δ 1−δ
1−δ L δ L L 1−δ δ L 1−δ bL aL bL = (aL ≤ (aL = cL ij ) (bij ) ij ik kj kj ik akj ) (bik bkj )     δ 1−δ
1−δ U δ U U 1−δ δ U 1−δ bU aU bU = (aU ≥ (aU = cU ij ) (bij ) ij ik kj kj ik akj ) (bik bkj )

which proves the theorem. Theorem 12 can be further extended to the aggregation of n ≥ 2 consistent triangular reciprocal preference relations as follows.  τ n
M U fτ = e Theorem 13. Let A aij i,j=1 , e aτij = aL τ ij , aτ ij , aτ ij , τ = 1, . . . , m, be n e = {e consistent triangular reciprocal preference relations. Then A aij } , where i,j=1

e aij =

M U aL ij , aij , aij




=

m Y


δτ aL τ ij

τ =1

,

m Y


δτ aM τ ij

τ =1

,

m Y


δτ aU τ ij

! ,

τ =1

is a consistent P triangular reciprocal preference relation for any δτ ∈ [0, 1] , τ = m 1, . . . , m, with τ =1 δτ = 1. Proof. The proof is similar to the proof of Theorem 12. In the following theorem, the definition of consistency proposed in this paper will be compared with the definition of consistency proposed by Wang et al. (2005). Since Wang et al. defined the consistency only for interval reciprocal preference relations (Definition 1), we will compare their definition with Definition 7 for interval reciprocal preference relations.   n U Theorem 14. Let A = {aij }i,j=1 , aij = aL ij , aij , be an interval reciprocal n preference relation. If A = {aij }i,j=1 is consistent according to the new Definition 7, then it is consistent also according to Wang et al.’s Definition 1. Proof. To prove this theorem, Theorems 3 and 9 will be used. Because A is consistent according to Definition 7, then clearly inequalities (32) and (33) hold. From this we obtain   U U L L U max aL aik akj ∀i, j = 1, . . . , n, i < j, ik akj ≤ aij ≤ aij ≤ min k=1,...,n

k=1,...,n

which implies (10). 20

Note 4. According to Theorem 14, when an interval reciprocal preference relation is consistent according to Definition 7 then it is also automatically consistent according to Wang et al.’s Definition 1. However, when an interval reciprocal preference relation is consistent according to Wang et al.’s definition, it does not mean that it is consistent also according to the definition introduced in this paper. Clearly, Wang et al.’s definition of consistency is weaker then the new one; it only requires existence of one crisp consistent reciprocal preference relation obtainable by combining particular elements from the intervals in the interval reciprocal preference relation. Thus, the set of all interval reciprocal preference relations consistent according to Definition 7 is a proper subset of the set of all interval reciprocal preference relations consistent according to Wang et al.’s Definition 1. 6. Illustrative examples In In this section, three illustrative examples are given. In the first example, it is demonstrated that, beside violating the reciprocity of pairwise comparisons, Definitions 4 and 5 of consistency allow that unreasonable intensities of preference are present in consistent interval and fuzzy reciprocal preference relations. After, two illustrative examples are given in order to compare the definition of consistency proposed in this paper with the definition of consistency given by Wang et al. (2005). Example 1. Let us assume the interval reciprocal preference relation 3    1 2 , 2 [x, y]  3  1 2   1 2, 2  . A =  2, 3 i h   1 1 1 2 1 y, x 2, 3

(38)

For now, the interval pairwise comparisons a13 and a31 = a113 are unknown. Definition 4 proposed by Wang (2015b) and Definition 7 will be applied on the interval reciprocal preference relation A in order to find out what values of a13 are allowed in order to preserve the consistency. First, by applying Definition 4, for i = 1, k = 2, j = 3, we obtain xy =

3 3 · 2 · · 2 = 9. 2 2

Therefore, h ithe interval pairwise comparison a13 = [x, y] , x ≤ y, can be in the 9 form y , y , y ∈ [3, 9] , in order to keep consistency of the interval reciprocal preference relation A. This means that, for example, interval [1, 9] is allowed. However, let us have a closer look on the intensities of preferences in such interval reciprocal preference relation. Clearly, object o1 is preferred  over  object o2 and object o2 is preferred over object o3 since a12 = a23 = 32 , 2 , 32 > 1. Therefore, object o1 should be also prefered over object o3 . However, according 21

to a13 = [1, 9] , equal preference (aL 13 = 1) of objects o1 and o3 is admitted. Moreover, the intensity of preference of object o1 over object o2 is at most 2 (between equal and slight preference) and also the intensity of preference of object o2 over object o3 is at most 2. However, the intensity of preference of object o1 over object o3 can be up to 9 (absolute preference) which is much  Lhigher  than U 2 · 2 = 4. In fact, there are not intensities of preference a ∈ a , a 12 12 12 and  U a23 ∈ aL , a such that a a = 1 or a a = 9. 12 23 12 23 23 23 Now let us apply Definition 7 on the interval reciprocal preference relation (38). By using Theorem 8, we obtain the following: i = 1, j = 2 : i = 1, j = 3 : i = 2, j = 3 :

3 2

≥ x · 12 = x ≥ 23 · 32 = 3 1 2 ≥ 2 ·x=

x 2 9 4 x 2

⇒ x ≤ 3, ⇒ x ≥ 49 , ⇒ x ≤ 3,

2 ≤ y · 23 ⇒ y ≥ 3 y ≤2·2⇒y ≤4 2 ≤ 23 · y ⇒ y ≥ 3

Therefore, the interval reciprocal preference relation (38) is consistent according  to Definition 7 if a13 = [x, y] , x ≤ y, is such that x ∈ 94 , 3 , y ∈ [3, 4] . This means that the lowest possible intensity of preference of object o1 over object o3 is at least 94 > 1, i.e. object o1 is definitely preferred over object o3 . Moreover, the highest possible intensity of preference of object o1 overobject  o3 is 4, which  is reachable under the consistency condition for a12 = 2 ∈ 32 , 2 , a23 = 2 ∈ 32 , 2 . Example 2. Let us assume the interval reciprocal preference relation   1 [2, 5] [2, 4] [1, 3]  1 1    ,   5 2 1 [1, 3] [1, 2]   1 1  1   1  A=  ,   4 2 3, 1 1 2, 1    1  1   3 , 1 2 , 1 [1, 2] 1

(39)

examined by Liu (2009). A is not consistent according to the new Definition 7 since inequalities (32), (33) are violated; e.g.    L L 1 max a1k ak4 = max 1, 2, 2 , 1 = 2 ≥ 1 = aL 14 . k=1,...,4 2 However, A is consistent according to Wang et al.’s Definition 1 since inequalities (10) are satisfied:    U U L 2 = max 2, 2, 23 , 12 = max aL a1k ak2 = min {5, 5, 4, 3} = 3 1k ak2 ≤ min k=1,...,4 k=1,...,4 L L U 2 = max {2, 2, 2, 1} = max a1k ak3 ≤ min aU 1k ak3 = min {4, 15, 4, 6} = 4 k=1,...,4  k=1,...,4  U U L 2 = max {1, 2, 1, 1} = max aL a1k ak4 = min {3, 10, 4, 3} = 3 1k ak4 ≤ min 2 k=1,...,4  L L k=1,...,4  U U 1 = max 5 , 1, 1, 1 = max a2k ak3 ≤ min a2k ak3 = min {2, 3, 3, 4} = 2  k=1,...,4  k=1,...,4  U U  L 1 = max 15 , 1, 12 , 1 = max aL a2k ak4 = min 32 , 2, 3, 2 = 32 2k ak4 ≤ min  1 1 1 1 k=1,...,4  L L k=1,...,4  U U  1 = max a3k ak4 ≤ min a3k ak4 = min 32 , 2, 1, 1 = 1 2 4 , 3 , 2 , 2 = max k=1,...,4

k=1,...,4

22

Therefore, according to Definition there exists at least one consistent recipn o1, n wi wi U rocal preference relation R = wj , such that aL ij ≤ wj ≤ aij , wi = 1, i = i,j=1 Pn 1, . . . , n, i=1 wi = 1. For A, it is for example   1 2 2 2 1  1 1 1  2  R=  1 1 1 1 , 2 1 1 1 1 2 where w1 = 25 , w2 = w3 = w4 = 15 . Example 3. Now, let us examine triangular reciprocal preference relation
1 1 2
  2 1 2 1 5, 2, 3 5, 2, 3  3

 1 e =  , 2, 5  1 B (40) 2 2 , 1, 1  .  2
3 1 2 , 2, 5 (1, 1, 2) Notice that this interval reciprocal preference relation is consistent according to Definition 3 proposed by Liu et al. (2014). However, by changing the ordering of objects o1 , o2 , o3 to e.g. o1 , o3 , o2 or o2 , o1 , o3 , the permuted interval reciprocal preference relation is not consistent anymore according to Definition 3. To verify the consistency given by Definition 6, let us check the validity of the conditions (26)–(28): 1 2 2 5 2 3 1 5 2 3 1 2

M M = bM = 12 1 = 12 , 13 = b12 b23   2 2 1 2 L L = b12 ≥ max bL 1k bk2 = max 1 5 , 5 1, 5 1 = 5 , k=1,2,3   2 2 2 2 U = bU bU 12 ≤ min 1k bk2 = min 1 3 , 3 1, 3 2 = 3 , k=1,2,3   1 21 1 1 L = bL bL 13 ≥ max 1k bk3 = max 1 5 , 5 2 , 5 1 = 5 , k=1,2,3   2 2 2 2 U = bU bU 13 ≤ min 1k bk3 = min 1 3 , 3 1, 3 1 = 3 , k=1,2,3  3 1 1 1 1 L = bL bL 23 ≥ max 2k bk3 = max 2 5 , 1 2 , 2 1 = 2 , k=1,2,3  5 2 U 1 = bU bU 23 ≤ min 2k bk3 = min 2 3 , 1, 1 = 1. k=1,2,3

e satisfies the conditions (26)–(28), it is consistent according to DefiBecause B nition 6. Let us analyze the meaning of Definition 6 on triangular reciprocal preference e Condition (17) says that, for any triplet i, j, k of indices, for any relation B.    L U U element from the interval bL ij , bij there exist elements in the intervals bik , bik h i U and bL kj , bkj such that the consistency (1) is preserved. For example, for a par  ticular choice b23 = 1 (that is we fix one particular value from the interval 12 , 1 which expresses that object o2 is equally preferred to object o3 ), the preference   of object o1 over object o3 can be expressed by any number x ∈ 25 , 23 ⊂ 15 , 23 , and the preference of object o2 over object o1 can be expressed by any number  y ∈ 32 , 52 such that 1 = x · y, e.g. x = 25 and y = 52 , or x = 12 and y = 2. 23

7. Conclusion Extension of Saaty’s definition of consistent reciprocal preference relations based on the multiplicative-transitivity property was dealt with in this paper. Particularly, the extension of the definition of consistency to interval and fuzzy reciprocal preference relations was focused on, and the importance of reciprocity of pairwise comparisons and of invariance of definitions of consistency under permutation of objects was emphasized. Approaches to the extension of the definition of consistency proposed by other authors were reviewed first. As already pointed out by Wang (2015b,a), the definitions of consistent interval and triangular reciprocal preference relations proposed by Liu (2009) and Liu et al. (2014) are dependent on the labeling of compared objects, i.e. they are not invariant under permutation of objects. Wang (2015b,a) proposed definitions of consistency that are invariant under permutation of objects. These definitions are based on the extension of a property equivalent to Saaty’s definition of consistent reciprocal preference relations. However, after the extension to interval and triangular reciprocal preference relations, the equivalence does not hold anymore. Moreover, as it was demonstrated in this paper, Wang’s definitions of consistency violate the reciprocity of pairwise comparisons, which is the substance of reciprocal preference relations. These drawbacks are caused by an inappropriate extension of the formulas by applying standard interval and fuzzy arithmetic. As shown in the paper, based on the reviewed definitions of consistency, the conclusions about consistency and inconsistency of interval and fuzzy reciprocal preference relations are non unanimous. By using two different definitions of consistency we may come to two different conclusions. By using a definition of consistency that is not invariant under permutation of objects, the conclusions may vary even for the same interval/fuzzy reciprocal preference relation. This has a negative impact on practical applications. In particular, by using an inappropriate definition of consistency, we may wrongly judge an inconsistent interval/fuzzy reciprocal preference relation as consistent and, as a consequence, we may derive interval/fuzzy priorities of objects that are not reliable. On the other hand, by wrongly judging an interval/fuzzy reciprocal preference relation containing consistent preference information as inconsistent, the decision maker who provided this preference information could be wrongfully excluded from the decision-making process. Therefore, it is of high importance for practical applications to prevent these problems. This was done in this paper by properly reflecting all properties of interval/fuzzy reciprocal preference relations when extending Saaty’s definition of consistency. As emphasized in the paper, the constrained interval and fuzzy arithmetic has to be employed in the extension of the definition of consistency instead of the standard interval and fuzzy arithmetic in order to preserve the reciprocity of pairwise comparisons and the invariance of the definition under permutation. Given this, new definitions of consistency for interval, triangular, and trapezoidal reciprocal preference relations were proposed. The new definitions are obtained by a proper extension of Saaty’s definition of consistency based on the 24

multiplicative-transitivity property. Unlike the definitions given by Liu (2009) and Liu et al. (2014), the new definitions are invariant under permutation of compared objects. Moreover, unlike the definitions given by Wang (2015b,a), the new definitions reflect naturally the reciprocity property of pairwise comparisons. Useful tools for verifying the consistency of interval and fuzzy reciprocal preference relations were provided in the paper, and some properties of consistent interval and fuzzy reciprocal preference relations were examined. The definition of consistency proposed in this paper can be easily modified in order to be applied on fuzzy reciprocal preference relations with an arbitrary type of fuzzy numbers described uniquely by their α−cuts. Similarly, all the tools for verifying the consistency of fuzzy reciprocal preference relations proposed in this paper can be modified accordingly. The new definition of consistency for interval reciprocal preference relations was compared with the definition proposed by Wang et al. (2005) and tt was demonstrated that the new definition is stronger than the definition proposed by Wang et al. (2005). In particular, interval reciprocal preference relations consistent according to the new definition of consistency are also consistent according to Wang et al.’s definition of consistency. However, this does not hold true the other way around. The new definitions of consistency introduced in this paper are quite strong and often they might be difficult to fulfill (especially for large sets of compared objects). When these definitions of consistency are too strong or too difficult to fulfill for the decision maker, he or she may use Wang et al.’s Definition 1 that is much weaker. In such case, an extension of Definition 1 to triangular and trapezoidal reciprocal preference relations would be needed first. Since the definitions of consistency proposed in this paper as well as the definition of consistency proposed by Wang et al. (2005) are quite extreme (very strong or very weak), a compromise between these definitions might be useful in some practical applications. Therefore, searching for a definition of consistency that is obtained as a compromise between both types of consistency but still keeping the reciprocity of pairwise comparisons and the invariance under permutation might be an interesting topic for future research. Basile, L., D’Apuzzo, L., 2002. Weak consistency and quasi-linear means imply the actual ranking. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10(3), 227–239. Brunelli, M., Fedrizzi, M., 2015. Axiomatic properties of inconsistency indices for pairwise comparisons. Journal of Operational Research Society 66, 1–15. Buckley, J.J., 1985. Fuzzy hierarchical analysis. Fuzzy Sets and Systems 17, 233–247. Buckley, J.J., Feuring, T., Hayashi, Y., 2001. Fuzzy hierarchical analysis revised. European Journal of Operational Research 129, 48–64.

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Highlights -

Multiplicative consistency of interval and fuzzy reciprocal preference relations is studied Definitions provided by other authors are reviewed and drawbacks are pointed out New definition of multiplicative consistency is proposed The new definition is invariant under permutation of objects The new definition preserves the multiplicative reciprocity of pairwise comparisons