Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations

Information Sciences 180 (2010) 4877–4891

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Interval multiplicative transitivity for consistency, missing values and priority weights of interval fuzzy preference relations Serkan Genç a, Fatih Emre Boran a, Diyar Akay a,⇑, Zeshui Xu b a b

Department of Industrial Engineering, Gazi University, 06570 Ankara, Turkey Institute of Sciences, PLA University of Science and Technology, Nanjing, Jiangsu 210007, China

a r t i c l e

i n f o

Article history: Received 16 March 2009 Received in revised form 5 June 2010 Accepted 15 August 2010

Keywords: Interval multiplicative transitivity Interval fuzzy preference relation Consistency Missing values Priority vector

a b s t r a c t In this paper, the concept of multiplicative transitivity of a fuzzy preference relation, as defined by Tanino [T. Tanino, Fuzzy preference orderings in group decision-making, Fuzzy Sets and Systems 12 (1984) 117–131], is extended to discover whether an interval fuzzy preference relation is consistent or not, and to derive the priority vector of a consistent interval fuzzy preference relation. We achieve this by introducing the concept of interval multiplicative transitivity of an interval fuzzy preference relation and show that, by solving numerical examples, the test of consistency and the weights derived by the simple formulas based on the interval multiplicative transitivity produce the same results as those of linear programming models proposed by Xu and Chen [Z.S. Xu, J. Chen, Some models for deriving the priority weights from interval fuzzy preference relations, European Journal of Operational Research 184 (2008) 266–280]. In addition, by taking advantage of interval multiplicative transitivity of an interval fuzzy preference relation, we put forward two approaches to estimate missing value(s) of an incomplete interval fuzzy preference relation, and present numerical examples to illustrate these two approaches.  2010 Published by Elsevier Inc.

1. Introduction Decision-making, one of the most crucial and omnipresent human activities in the real world, is characterized as a process of choosing the best alternative(s) or course(s) of action, from a set of alternatives, to attain a goal (or goals). In the process of decision-making, a preference relation (or called pairwise comparison matrices, judgment matrices) is the most common representation format used by a decision-maker because it is very useful in expressing his/her information about alternatives. During the last few decades, various types of preference relations have been proposed, and among these preference relations, multiplicative preference relations [28] and fuzzy preference relations [25] have received much attention from a variety of research. Multiplicative preference relations have been applied in many fields [11,13,17,27,28,31,32,36,37,46]. However, in the real world, since a majority of decision-making problems take place in a fuzzy environment, much research has focused on fuzzy preference relations [1,3–5,8–10,12,15,16,19,21,22,25,26,29,30,35,41]. Fuzzy preference (reciprocal) relation was introduced by Orlovsky [25], whose element pij denotes the preference degree or intensity of the alternative i over j, and satisfies the following conditions pij 2 [0, 1], pij + pji = 1. A decision-maker may easily express his/her preference relations over alternatives with fuzzy preference relations. However, due to time limitation (or pressure) and sophisticated decision problems, these relations may not be consistent or complete. In such cases, inappropriate alternative(s) can be selected. Therefore, consistency and completeness, associated with the transitivity property, are important issues in preference relations. Various types of transitivity properties have been ⇑ Corresponding author. Tel.: +90 312 5823844; fax: +90 312 2308434. E-mail address: [email protected] (D. Akay). 0020-0255/$ - see front matter  2010 Published by Elsevier Inc. doi:10.1016/j.ins.2010.08.019

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introduced to verify consistency of fuzzy preference relations, and to estimate missing value(s) of fuzzy preference relations. The ones which are most widely cited are weak transitivity, max–min transitivity, max–max transitivity, restricted max–min transitivity, restricted max–max transitivity, additive transitivity and multiplicative transitivity [43]. Among these properties, additive transitivity and multiplicative transitivity only imply reciprocity. However, a conflict between the additive transitivity and the scale used for providing the preference values, i.e., the closed interval [0, 1], can appear. Chiclana et al. [4] recently proposed to model the cardinal consistency of reciprocal preference relations via a functional equation, and showed that when such a function is almost continuous and monotonic (increasing), then it must be a representable uninorm. It is then proved that consistency when represented by the conjunctive representable cross-ratio uninorm is equivalent to the multiplicative transitivity property. Therefore, multiplicative transitivity is the most suitable property to verify the consistency of a fuzzy preference relation [4,6]. In a fuzzy preference relation, it may be a tough task for a decision-maker to accurately express his/her preference information over alternatives with exact numerical values. In such cases, it is very suitable to use interval fuzzy preference relations. The interval fuzzy preference relation was introduced by Xu [33,34], whose element rij denotes the interval-valued h i preference degree or intensity of the alternative i over j and satisfies the following conditions r ij 2 r Lij ; r Uij ; r Uij P rLij P 0; rLij þ rUji ¼ rLji þ rUij ¼ 1, and rLij and rUij are the lower and upper limits of rij, respectively. Some authors have paid attention to interval fuzzy preference relations. Xu [33] represented a normalized rank aggregation method to determine the priority vector of an interval fuzzy preference relation using a possibility-degree formula. Xu [34] defined the concepts of compatibility degree and compatibility index of two interval fuzzy preference relations and introduced a compatibility theorem based on the acceptable compatibility for interval fuzzy preference relations. Herrera et al. [18] developed an aggregation process for managing non-homogenous information composed by different domains, including interval-valued preference relations. Xu [36] introduced the concept of an incomplete interval fuzzy preference relation and transformed them into incomplete fuzzy preference relations based on a continuous interval argument ordered weighted average [47]. Xu [42] utilized the continuous interval argument ordered weighted averaging operator to derive the priority vector of interval fuzzy preference relations. Jiang [20] gave an index to measure the similarity degree of two interval fuzzy preference relations and utilized the similarity index to check the consistency degree of group opinion. Alonso et al. [2] presented a procedure to estimate missing preference values in different types of incomplete preference relations: fuzzy, multiplicative, interval-valued and linguistic preference relations. Xu and Chen [44] introduced a multi-attribute group decision-making approach in which preference information on alternatives provided by decision-makers are represented with distinct uncertain preference structures (interval utility values, interval fuzzy preference relations, and interval multiplicative preference relations). The proposed approach ranks alternatives directly by solving some linear programming models using objective information depicted in the decision matrix and three distinct subjective preference information structures, without the need of unifying distinct preference structures or aggregating individual preferences into a collective one. Xu and Chen [40] defined an additive consistent interval fuzzy preference relation and a multiplicative consistent interval fuzzy preference relation, then established some linear programming models for deriving the priority weights from an interval fuzzy preference relation. The point of departure of this paper is the study of Xu and Chen [40]. Instead of checking the consistency and deriving the priority weights of an interval fuzzy preference relation by solving linear programming models, which are related to the concept of multiplicative consistent interval fuzzy preference relation, we prove that these can be identified exactly with the simple formulas derived through the introduced concept of interval multiplicative transitivity of an interval fuzzy preference, which is an extension of multiplicative transitivity of a fuzzy preference relation as defined by Tanino [29]. In addition, we introduce two approaches using the same concept to estimate missing value(s) of an interval fuzzy preference relation. To explain our achievements, the remainder of this paper is organized as follows: Section 2 comprises some preliminaries on the analysis of consistency, a short survey on the approaches to estimate the missing values in fuzzy preference relations, and the notion of an interval fuzzy preference relation. Section 3 defines the theory of interval multiplicative transitivity of an interval fuzzy preference relation. Then, with this concept in hand, it introduces the concept of a multiplicative consistent interval fuzzy preference relation, shows how to derive the priority vector of a multiplicative consistent interval fuzzy preference relation. Also, it defines two approaches to estimate missing value(s) of an incomplete interval fuzzy preference relation. Section 4 provides practical examples to illustrate our findings. Finally, we conclude the paper in Section 5.

2. Preliminaries In this section, we briefly present definitions and short surveys related to the concept of the consistency, transitivity and missing values in fuzzy preference relations, and then represent the concept of interval fuzzy preference relation. 2.1. Consistency and transitivity in fuzzy preference relations When comparing alternatives in preference relations, a decision-maker can provide inconsistent preference information due to the complexity of a decision-making problem, time pressure, or lack of knowledge about problem domain. Consistency is of great importance in preference relations because, in decision-making, inconsistent information given by a deci-

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sion-maker can cause to select inappropriate alternative(s). In order to avoid undesirable conclusions, it is essential to test whether his/her preference relation is consistent or not. Transitivity property is the most important concept for consistency issues. Many properties have been proposed to model transitivity of fuzzy preference relations. Some of these properties can be given as follows [7,16]: (1) Triangle condition [23]: pik + pkj P pij, for all i, j, k = 1, 2, . . . , n. This condition can be geometrically explained. Let the alternatives xi, xk and xj be considered as the vertices of a triangle with length sides pik, pkj and pij, then the length corresponding to the vertices xi, xj should not exceed the sum of the lengths corresponding to the vertices xi, xk and xk, xj. (2) Weak transitivity [30]: If pik P 0.5 and pkj P 0.5, then pij P 0.5, for all i, j, k = 1, 2, . . . , n. This transitivity can be interpreted as follows: If the alternative xi is preferred to xk, and xk is preferred to xj, then xi should be preferred to xj. (3) Max–min transitivity [12,48]: pij P min(pik, pkj), for all i, j, k = 1, 2, . . . , n. The max–min transitivity is that the fuzzy preference value between the alternatives xi and xj should be equal to or greater than the minimum partial values between the alternatives xi, xk and xk, xj. (4) Max–max transitivity [12,48]: pij P max(pik, pkj) for all i, j, k = 1, 2, . . . , n. The max–min transitivity can be described as follows: the fuzzy preference value between the alternatives xi and xj should be equal to or greater than the maximum partial values between the alternatives xi, xk and xk, xj. (5) Restricted max–min transitivity [30]: if pik P 0.5 and pkj P 0.5, then pij P min(pik, pkj), for all i, j, k = 1, 2, . . . , n. In what follows, we explain the restricted max–min transitivity property: if the alternative xi is preferred to xk with a value pik, and xk is preferred to xj with a value pkj, then xi should be preferred to xj with at least a preference value pij equal to the minimum of the above values. The equality holds the condition under which there is indifference between at least two of the three alternatives. (6) Restricted max–max transitivity [30]: if pik P 0.5 and pkj P 0.5, then pij P max(pik, pkj) for all i, j, k = 1, 2, . . . , n. The restricted max–max transitivity property can be interpreted in the following way: when the alternative xi is preferred to xk with a value pik, and xk is preferred to xj with a value pkj, then xi should be preferred to xj with at least a preference value pij equal to the maximum of the above values. The equality holds only when there is indifference between at least two of the three alternatives. (7) Additive transitivity [29,30]: (pij  0.5) + (pjk  0.5) = (pik  0.5) for all i, j, k = 1, 2, . . . , n or equivalently pij þ pjk þ pki ¼ 32 for all i, j, k = 1, 2, . . . , n. The additive transitivity property can be interpreted as follows: the intensity of preference of the alternative xi over xk should be equal to the sum of the intensities of preferences of xi over xj and xj over xk when pij  0.5 is defined as an intensity of preference of alternative xi over xj. p p (8) Multiplicative transitivity [30]: pji  pkj ¼ ppki for all i; j; k ¼ 1; 2; . . . ; n. ij

jk

ik

Tanino [30] introduced the multiplicative transitivity property in the case where pij > 0 and pij/pji indicates a ratio of the preference intensity for the alternative xi to that of xj, that is, xi is pij/pji times as good as xj, for all i, j = 1, 2, . . . , n. The weak transitivity is the usual transitivity property that a logical and consistent person should use if he/she does not want to express inconsistent opinions; however, it is the minimum requirement condition to find out whether a fuzzy preference relation is consistent or not [16]. It is obvious that the max–max transitivity is better than the max–min transitivity. On the other hand, the max–max transitivity cannot be verified under reciprocity [4]. Also, neither the restricted max–min transitivity nor the restricted max–max transitivity implies reciprocity. Both the additive transitivity and the multiplicative transitivity imply reciprocity. However, a conflict between the additive transitivity and the scale used for providing the preference values, i.e., the closed interval [0, 1], can appear. Therefore, as already mentioned earlier, the multiplicative transitivity should be taken as the concept to verify the consistency of a fuzzy preference relation [7]. A fuzzy preference relation is a multiplicative consistent fuzzy preference relation if it satisfies the multiplicative transitivity [16,29,37]. A multiplicative consistent fuzzy preference relation can be interpreted as follows: the fuzzy preference degree or intensity pij of the alternative xi over xj should be equal to the multiplication of the intensities of preferences when using an intermediate alternative xk. If this holds for all i, j, k = 1, 2, . . . , n, then the fuzzy preference relation P is fully consistent. Several studies are reported in the literature related to the consistency of a fuzzy preference relation. Chiclana et al. [10] studied the internal consistency of various preference presentations, including preference orderings, utility functions and fuzzy preference relations. Xu and Da [39] developed an approach to improve consistency of a fuzzy preference relation and gave a practical iterative algorithm to derive a modified fuzzy preference matrices with acceptable consistency. Herrera-Viedma et al. [16] gave a characterization of the fuzzy consistency according to additive transitivity property and proposed a method for constructing consistent fuzzy preference relations from a set of n  1 preference values. Ma et al. [24] presented an analysis method to identify the inconsistency and weak transitivity of a fuzzy preference relation and to repair its inconsistency to reach weak transitivity. Alonso et al. [2] proposed a procedure guided by the additive consistency property to maintain experts’ consistency levels in different types of preference relations: multiplicative, fuzzy, interval-valued, and linguistic preference relations. Chiclana et al. [4,6] put forward a set of conditions for a fuzzy preference relation to be considered as fully consistent and showed that consistency of fuzzy preference relations can be characterized by representable uninorms, an example of which is Tanino’s multiplicative transitivity.

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2.2. Estimation of missing values in fuzzy preference relations A complete preference relation needs n(n  1)/2 judgments in its entire top triangular portion [43]. However, when a decision-maker is unable to compare two alternatives, then this situation should not be reflected in the preference relation as an indifference situation, but with a missing entry for that particular pair of alternatives. In other words, a missing value in a preference relation is not equivalent to a lack of preference of one alternative over another. A missing value might also be the result of the incapacity of an expert to quantify the degree of preference of one alternative over another, in which case he/she may decide not to guess to maintain the consistency of the values already provided [2,5]. Estimation of missing value is one of the important issues in preference relations. In the literature, some procedures have been reported for computing the missing values in a preference relation. Alonso et al. [1] suggested a procedure to find out the missing values of an incomplete fuzzy preference relation using the known values, and defined an expert consistency measure according to additive consistency property. Xu [38] developed a system of equations, and then, based on this system of equations, proposed a method for decision-making with an incomplete fuzzy preference relation and estimating missing values. Herrera et al. [14] proposed the iterative procedure for estimating missing values in fuzzy preference relations and sufficient conditions to guarantee the successful estimation of all the missing values. The proposed method attempts to estimate the missing information in an expert’s incomplete fuzzy preference relation using only the preference values provided by that particular expert. By doing this, it is assured that the reconstruction of the incomplete fuzzy preference relation is compatible with the rest of the information provided by that expert. Alonso et al. [2] presented a procedure to estimate missing preference values when dealing with pairwise comparison and heterogeneous information. The method can be applied to estimate missing values for incomplete fuzzy, multiplicative, interval-valued, and linguistic preference relations. Chiclana et al. [5] summarized the problem of estimating missing values in decision-making and presented a new estimation method for incomplete fuzzy preference relations which is founded on the modeling of consistency of preferences via a self-dual almost continuous uninorm operator. Alonso et al. [3] introduced situations where an expert does not provide any information on a particular alternative, which are called situations of total ignorance. In the paper, a method proposed in Herrera et al. [14] is used to estimate missing values in preference relation. 2.3. Interval fuzzy preference relation A decision-maker may not estimate his/her preference relations over alternatives with exact numerical values because of the vague information about the preference degree between any two alternatives. In such cases, it is very suitable to express preference relations with interval values. The concept of interval fuzzy preference relation is as follows: Definition 1 ([33,34]). An interval fuzzy preference relation R on a finite set of alternatives X = {x1, x2, . . . , xn} is represented h i by an interval fuzzy preference matrix R = (rij)nxn  X  X, where r ij 2 rLij ; rUij ; rUij P rLij P 0; r Lij þ r Uji ¼ rLji þ rUij ¼ 1; rLii ¼ r Uii ¼ 0:5, for all i, j, and rij denotes the interval-valued preference degree or intensity of the alternative xi over xj, rLij and rUij are the lower and upper limits of rij, respectively. Especially, rij = [0.5, 0.5] indicates indifference between xi and xj, rij P [0.5, 0.5] indicates that xi is preferred to xj, rij 6 [0.5, 0.5] indicates that xj is preferred to xi, rij = [1, 1] indicates that xi is absolutely preferred to xj, and rij = [0, 0] indicates that xj is absolutely preferred to xi. Using the weighted arithmetic averaging operator, Xu [43] gave an approach to transform an interval fuzzy preference relation R = (rij)nxn into a fuzzy preference relation P = (pij)nn:

pij ¼ ar Lij þ ð1  aÞr Uij ;

06a61

ð1Þ

where a is an index that reflects the decision-maker’s risk-bearing attitude, and pij 2 [0, 1], pij + pji = 1, pii = 0.5 and pij denotes the fuzzy preference degree or intensity of the alternative xi over xj, for all i, j. Especially, a < 0.5 indicates that the decisionmaker is a risk lover, a > 0.5 indicates that the decision-maker is a risk averter, and a = 0.5 indicates that the decision-maker is neutral to the risk. Sometimes, a decision-maker may not consistently provide his/her preference information. Therefore, it is necessary to test the provided preference information. A multiplicative consistent preference relation is defined by Xu and Chen [40] as follows: Definition 2 [40]. Let R = (rij)nn be an interval fuzzy preference relation, if there exists a priority vector w = (w1, w2, . . . , wn)T, such that

rLij 6

wi 6 rUij wi þ wj

ð2Þ

Pn where wi > 0; i¼1 wi ¼ 1, for all i, j = 1, 2, . . . , n, then R is called a multiplicative consistent interval fuzzy preference relation. Obviously, Eq. (2) is equivalent to the following two equations:

(

wi P r Lij ðwi þ wj Þ wi 6 rUij ðwi þ wj Þ

ð3Þ

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for all i, j = 1, 2, . . . , n. Xu and Chen [40] introduced some linear programming models to obtain the priority vector w. If R = (rij)nn is a multiplicative consistent interval fuzzy preference relation, then the weights are calculated with the following linear programming models:

ðM-1Þ wi ¼ min wi =wþi ¼ max wi s:t: r Uji wi  r Lij wj P 0; r Uij wj  rLji wi P 0; wi > 0;

i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n

i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n

i ¼ 1; 2; . . . ; n;

n X

wi ¼ 1

i¼1 

þ

If R = (rij)nn is a multiplicative inconsistent interval fuzzy preference relation, and the deviation variables dij and dij are used to relax Eq. (3); then the following model is solved:

ðM-2Þ J ¼ min

n1 X n  X



þ

dij þ dij



i¼1 j¼iþ1 

s:t: r Uji wi  r Lij wj þ dij P 0; r Lji wi  r Uij wj 

þ dij

i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n

i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n n X i ¼ 1; 2; . . . ; n; wi ¼ 1

wi > 0;

6 0;

i¼1 

þ

dij ; dij P 0;

i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n

€ and d €þ are obtained by solving the model (M-2), and then the following linear programThe optimal deviation variables d ij ij ming models based on the optimal deviation variables are constructed:

ðM-3Þ wi ¼ min wi =wþi ¼ max wi € P 0; i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n s:t: r Uji wi  r Lij wj þ d ij L U þ € r w  r w  d 6 0; i ¼ 1; 2; . . . ; n  1; j ¼ i þ 1; . . . ; n ji

i

j

ij

wi > 0;

ij

i ¼ 1; 2; . . . ; n;

n X

wi ¼ 1

i¼1

  þ Solving the model (M-3), the priority vector w = (w1, w2, . . . , wn)T is obtained, where wi 2 w i ; wi . In addition, Xu and Chen [40] define the concept of consistent interval fuzzy preference relation based on the model (M-2): Definition 3 [40]. R = (rij)nn is a multiplicative consistent interval fuzzy preference relation if and only if J = 0. 3. Interval multiplicative transitivity, consistency and estimating missing values for interval fuzzy preference relations Initially, the concept of interval multiplicative transitivity of an interval fuzzy preference relation is introduced in this section. With this concept in hand, the concept of a multiplicative consistent interval fuzzy preference relation is defined, and how to derive the priority vector of a multiplicative consistent interval fuzzy preference relation is shown. Finally, two approaches are put forward to estimate missing value(s) of an incomplete interval fuzzy preference relation. 3.1. Interval multiplicative transitivity of an interval fuzzy preference relation Tanino [29] investigate a fuzzy preference relation P = (pij)nn and introduced the multiplicative transitivity:

pji pij

!

pkj  pjk

!

 ¼

pki pik

 ð4Þ

where pij/pji indicates a ratio of the preference intensity for the criterion xi to that for xj, that is, xi is pij/pji times as good as xj, and pij 2 [0, 1], for all i, j, k = 1, 2, . . . , n. Multiplicative transitivity is equivalent to the following form [16,39], for all i, j, k = 1, 2, . . . , n:

pij pjk pki ¼ pji pkj pik

ð5Þ

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which can also be expressed in the following form:

pij pjk pki ¼ ð1  pij Þpkj pik pij pjk pki ¼ pkj pik  pij pkj pik pij pjk pki þ pij pkj pik ¼ pkj pik pij ðpjk pki þ pkj pik Þ ¼ pkj pik pik  pkj pij ¼ pik  pkj þ pki pjk pik  pkj pij ¼ pik  pkj þ ð1  pik Þð1  pkj Þ

ð6Þ

h i Definition 4. Let r ij ¼ r Lij ; r Uij be an interval fuzzy preference value on a finite set of alternatives. X = {x1, x2, . . . , xn}, then the þ pessimistic case r  ij and the optimistic case r ij for the interval fuzzy preference value rij are defined using Eq. (1) as follows, respectively:

rij ¼ ar Lij þ ð1  aÞr Uij ¼ 1ðr Lij Þ þ ð1  1Þr Uij ¼ rLij

ð7Þ

rþij ¼ ar Lij þ ð1  aÞr Uij ¼ 0ðr Lij Þ þ ð1  0Þr Uij ¼ rUij

ð8Þ

and

þ It is obvious that r ij and r ij are two interval fuzzy preference values, and can be shown as follows:

h i rij ¼ rLij ; r Lij

ð9Þ

h i rþij ¼ rUij ; r Uij

ð10Þ

and

Based on the multiplicative transitivity of fuzzy preference relations, we introduce the concept of the interval multiplicative transitivity using the concept of the pessimistic case and the optimistic case. h i h i   Definition 5. Let r ij ¼ r Lij ; r Uij ; rik ¼ rLik ; rUik and rkj ¼ rLkj ; r Ukj be interval fuzzy preference values on a finite set of alternatives, X = {x1, x2, . . . , xn}, then the interval fuzzy multiplicative transitivity among these interval fuzzy preference relations is defined as follows:

" rij

¼

r ik r kj

r ik r kj

r þik r þkj

r þik r þkj

r ik rkj

r þik r þkj

# ð11Þ

; r ik r kj þ ð1  r ik Þ  ð1  r kj Þ rik r kj þ ð1  r ik Þ  ð1  r kj Þ

and

" rþij

¼

# ð12Þ

; r þik r þkj þ ð1  r þik Þ  ð1  r þkj Þ rþik r þkj þ ð1  r þik Þ  ð1  r þkj Þ

therefore,

" rij ¼

# ð13Þ

; r ik r kj þ ð1  r ik Þ  ð1  r kj Þ r þik r þkj þ ð1  r þik Þ  ð1  r þkj Þ

h i h i     þ þ þ þ þ þ    where r  ik ¼ r ik ; r ik and r kj ¼ r kj ; r kj are the pessimistic cases of rik and rkj, respectively, r ik ¼ r ik ; r ik and r kj ¼ r kj ; r kj are the optimistic cases of rik and rkj, respectively, for all i, j, k = 1, 2, . . . , n. It is obvious that Eq. (13) is equivalent to the following equation:

" rij ¼

r Lik rLkj

r Uik r Ukj

;

# ð14Þ

r Lik r Lkj þ ð1  r Lik Þ  ð1  r Lkj Þ r Uik r Ukj þ ð1  r Uik Þ  ð1  r Ukj Þ

for all i, j, k = 1, 2, . . . , n. In addition, rij is an interval fuzzy preference value, i.e.,

06

rLik r Lkj r Lik rLkj

þ ð1 

r Lik Þ

 ð1 

r Lkj Þ

6

r Uik r Ukj r Uik r Ukj

þ ð1  r Uik Þ  ð1  r Ukj Þ

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4883

Proof. Since

rLik 6 r Uik ;

r Lkj 6 r Ukj ;

r Lik rLkj 6 r Uik r Ukj

ð15Þ

and

1  r Lik P 1  r Uik ;

1  rLkj P 1  r Ukj ;

ð1  rLik Þ  ð1  rLkj Þ P ð1  r Uik Þ  ð1  r Ukj Þ

ð16Þ

then

ð1  r Lik Þ  ð1  r Lkj Þ rLik rLkj 1þ

P

ð1  r Uik Þ  ð1  r Ukj Þ

ð1  r Lik Þ  ð1  r Lkj Þ rLik rLkj

r Uik r Ukj P1þ

r Lik r Lkj þ ð1  r Lik Þ  ð1  r Lkj Þ rLik rLkj r Lik r Lkj þ ð1

rLik rLkj  r Lik Þ

 ð1  r Lkj Þ

P 6

ð1  r Uik Þ  ð1  r Ukj Þ rUik rUkj r Uik r Ukj þ ð1  r Uik Þ  ð1  r Ukj Þ r Uik r Ukj r Uik r Ukj

rUik r Ukj þ ð1  r Uik Þ  ð1  r Ukj Þ

ð17Þ

The above transitivity only holds for i – j. If i = j, we can not directly use the interval multiplicative transitivity because rik and rki (or rjk and rkj) are complement. h Theorem 1. Let rik and rki be two interval fuzzy preference values, then the estimated interval fuzzy preference value rii = rik  rki = [0.5, 0.5]. þ Proof. It is obvious that when the pessimistic case r  ik occurs, then the optimistic case r ki occurs, or when the optimistic case þ  r ik occurs, then the pessimistic case r ki occurs since rik and rki are complement. Therefore, Eqs. (11) and (12) are redefined as follows:

"

# r ik r þki r ik r þki     ; rik r þki þ 1  r ik  1  r þki r ik r þki þ 1  r ik  1  r þki " # r Lik r Uki r Lik r Uki     ¼ L U ; r ik r ki þ 1  r Lik  1  r Uki rLik r Uki þ 1  r Lik  1  r Uki " #   1  rUki rUki 1  r Uki r Uki    ;    ¼  1  r Uki r Uki þ 1  1  rUki  1  r Uki 1  r Uki r Uki þ 1  1  r Uki  1  rUki " #   1  rUki rUki 1  r Uki r Uki   ¼  ; ¼ ½0:5; 0:5 1  r Uki r Uki þ r Uki  1  r Uki ð1  rUki ÞrUki þ r Uki  1  r Uki

rii ¼ r ik  r þki ¼

ð18Þ

and

"

# r þik r ki r þik r ki     ; rþik r ki þ 1  r þik  1  r ki r þik r ki þ 1  r þik  1  r ki " # r Uik r Lki r Uik r Lki     ¼ U L ; r ik r ki þ 1  r Uik  1  r Lki rUik r Lki þ 1  r Uik  1  r Lki " #   1  rLki rLki 1  r Lki r Lki    ;    ¼  1  r Lki r Lki þ 1  1  rLki  1  r Lki 1  r Lki r Lki þ 1  1  r Lki  1  rLki " #   1  rLki rLki 1  r Lki r Lki    ¼ ½0:5; 0:5 ¼  ; 1  r Lki r Lki þ r Lki  1  r Lki 1  r Lki r Lki þ rLki  1  r Lki

rþii ¼ r þik  r ki ¼

ð19Þ

Therefore, the estimated interval fuzzy preference value is rii = [0.5, 0.5]. After this point, everywhere i = j, rii (or rjj) will be estimated as rii = [0.5, 0.5] (or rjj = [0.5, 0.5]) without any calculation since we cannot directly use the interval multiplicative transitivity for complements. h

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3.2. Consistent interval fuzzy preference relation From Definition 3, it is known that an interval fuzzy preference relation R = (rij)nn is consistent if and only if J = 0 in (M-2). In the following, we introduce some approaches founded on the interval multiplicative transitivity to test whether an interval fuzzy preference relation is consistent or not, and to find the priority vector of a consistent interval fuzzy preference relation without solving any linear programming models. But first, we need to define how to integrate different interval fuzzy preference relations estimated from each k = 1, 2, . . . , n, for all i, j = 1, 2, . . . , n:   Theorem 2. Let R0 ¼ r0ij

be an integrated interval fuzzy preference relation estimated from R = (rij)nn as follows:

nxn

" r0ij

(

¼ max

)

rLik r Lkj

r Lik rLkj þ ð1  r Lik Þ  ð1  rLkj Þ

k

(

; min k

h i where r 0ij ¼ r 0Lij ; r0U ij , for all i, j = 1, 2, . . . , n(i – j).

)#

rUik r Ukj

   T Proof. Let w0 ¼ w01 ; w02 ; . . . ; w0n be an interval priority vector for R0 ¼ r 0ij

r0Lij 6

w0i

ð20Þ

r Uik rUkj þ ð1  r Uik Þ  ð1  r Ukj Þ

  with w0i ¼ w0Li ; w0U such that i

nxn

w0i 6 r0U ij þ w0j

ð21Þ

for all i, j = 1, 2, . . . , n. Here, it is obvious that the different interval fuzzy preference relations estimated for each k = 1, 2, . . . , n, the maximum value of the lower membership degree and the minimum value of the upper membership degree satisfy the w0

i ratio w0 þw 0 but the other ones do not satisfy it, for all i, j = 1, 2, . . . , n. This can be interpreted in detail as follows: the minimum i

ratio

j

w0L i 0O1

w0L þwj i

and the maximum ratio

w0U i 0O2

w0U þwj i

0O1 2 of the priority vector w0i are always between w0Li and w0U and w0O are i , where wj j

the optimal weights minimizing and maximizing the ratio, respectively.

h

  Theorem 3. R = (rij)nn is a multiplicative consistent interval fuzzy preference relation if there exists a R ¼ r ij

"

(

(

rij ¼ max r Lij ; max k

))

rLik r Lkj

r Lik r Lkj

þ ð1 

r Lik Þ

 ð1 

r Lkj Þ

(

(

; min r Uij ; min k

r Uik r Ukj

rUik rUkj

))#

nn

, where

ð22Þ

þ ð1  rUik Þ  ð1  rUkj Þ

h i U U satisfies the condition 0 6 r L and r ij ¼ rL ij ; r ij ij 6 r ij 6 1 for any i, j = 1, 2, . . . , n. If Eq. (22) does not satisfy the condition U 0 6 r L ij 6 r ij 6 1 for any i, j = 1, 2, . . . , n, then we call R an inconsistent interval fuzzy preference relation. This can be easily proven

in a similar way to the proof of Theorem 2. If we calculate the minimum ratio

wLi

O1

wLi þwj

wOj 1

and the maximum ratio

wU i

O2

wU þwj i

of the priority

wOj 2

and are the optimal weights minimizing and vector wi for a consistent interval fuzzy preference relation R = (rij)nn where         , i.e., R ¼ rij is the consistent part of R = (rij)nn, for all maximizing the ratio, respectively, then we get R ¼ rij nn

nn

0U i, j = 1, 2, . . . , n. In addition, if Eq. (20) does not satisfy the condition 0 6 r0L ij 6 r ij 6 1, then R is an inconsistent interval fuzzy preference relation, and we do not need to use Eq. (22). However, if Eq. (20) holds, then it cannot be said whether R is a consistent interval fuzzy preference relation or not, and we need to use Eq. (22).

  If R is consistent, then there exists a unique interval priority vector w = (w1, w2, . . . , wn)T with wi ¼ wLi ; wUi of R, for all * i, j = 1, 2, . . . , n. It can be calculated with the consistent interval fuzzy preference relation R without solving any linear proh i U gramming models because R* is the consistent part of R, and the lower and upper values (the pair r L ) of R* exactly satij ; r ij isfy the minimum and maximum values of the ratio

wi , wi þwj

for all i, j = 1, 2, . . . , n, which can be defined as follows:

Theorem 4. If R = (rij)nn is a consistent interval fuzzy preference relation, then the priority vector w = (w1,w2, . . ., wn)T of R can be estimated by the following equation:

3

2

7 6 7 6 7 6 1 1 6 1 ; 17 wi ¼ 6 0 0 7 L U 7 6P 1r ij 1r ij Pn 5 4 n @ A @ A L U j¼1

for all i, j = 1, 2, . . . , n.

rij

j¼1

rij

ð23Þ

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Proof. It is obvious that the membership degree of wi is of minimum value, if we equalize the ratio

wLi

wLi þwU j

and the value r L ij ,

i.e., the membership degree wLi of wi can be estimated by this equality, and that the membership degree of wi is of maximum wU

U i value, if we equalize the ratio wU þw L and the value r ij . In addition, we know that the sum of the weights of the priority vector i

j

wi is equal to 1. Thus, we can prove the above theorem as follows, for all i = 1, 2, . . . , n:

wLi

rL ij ¼

wLi þ wOj 1   O1 L rL ¼ wLi ij wi þ wj   O1 L L rL ij wj ¼ 1  r ij wi   1  rL ij wOj 1 ¼ wLi r L ij 1 0 n n 1  r L X X ij @ AwL wOj 1 ¼ i rL ij j¼1 j¼1 Pn

1 0

j¼1

@

1r L ij

1 ¼ wLi

r L ij

ð24Þ

A

and

wUi

¼ r U ij wUi þ wOj 2   wUi ¼ r U wUi þ wOj 2 ij   O2 1  r U wUi ¼ r U ij ij wj   1  r U ij wUi ¼ wOj 2 rU ij 1 0 n n 1  r U X X ij @ AwU ¼ wOj 2 i U r ij j¼1 j¼1 wUi ¼ Pn

j¼1

1 0 @

1r U ij r U ij

1

ð25Þ

A

where wOj 1 and wOj 2 are the optimal weights minimizing and maximizing wi, respectively, for all j = 1, 2, . . . , n, and satisfy the P P condition nj¼1 wOj 1 ¼ 1 and nj¼1 wOj 2 ¼ 1. h Based on the interval multiplicative transitivity, the weights of the priority vector derived from Eq. (23) are in the interval numbers form. In this paper, a straightforward possibility-degree formula introduced by Xu and Da [45] is used to compare and rank interval weights: h i   Definition 6 [45]. Let wi ¼ wLi ; wUi be any two interval weights, where 0 6 wLi 6 wUi 6 1 and and wj ¼ wLj ; wUj 0 6 wLj 6 wUj 6 1, then the degree of possibility of wi P wj is defined as

( pðwi P wj Þ ¼ max 1  max

(

wUj  wLi

) )

;0 ;0 wUi  wLi þ wUj  wLj

that is, wi is superior to wj to degree of p(wi P wj), denoted by wi

ð26Þ pðwi Pwj Þ



wj .

3.3. Incomplete interval fuzzy preference relations Sometimes, a decision-maker may not completely express his/her preference relation over other alternatives because of time limitation, lack of knowledge, or the complexity of decision problem, thus developing an incomplete preference relation. Xu [36] introduced the concept of incomplete preference relations as follows:

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e is called an incomplete interval fuzzy preference relation if some Definition 7 [36]. An interval fuzzy preference relation R of its elements cannot be given by the decision-maker. As it is known, the completeness of a preference relation is related with the transitivity property, and the multiplicative transitivity is the strongest concept among all of the transitivity properties. Based on the multiplicative transitivity, a multiplicative consistent incomplete interval fuzzy preference relation was defined by Xu [43]: e is called a multiplicative consistent incomplete Definition 8 [43]. An incomplete interval fuzzy preference relation R interval fuzzy preference relation if there exists a priority vector w = (w1, w2, . . . , wn)T, such that the condition e 0 6 wLi 6 wUi 6 1 holds for all the known elements of R. In the following, we introduce two approaches to estimate the missing value(s) of an incomplete interval fuzzy preference relation using the interval multiplicative transitivity. The first approach is defined as follows: e ¼ ð~rij Þ e Theorem 5. Let R nn be an incomplete interval fuzzy preference relation. If R is a multiplicative consistent interval fuzzy preference relation, then the missing value ~r ij of an incomplete interval fuzzy preference relation is estimated with the following equation:

"

(

~rij ¼ max k

)

rLik r Lkj

r Lik rLkj þ ð1  r Lik Þ  ð1  rLkj Þ

(

; min k

rUik r Ukj

)#

r Uik rUkj þ ð1  r Uik Þ  ð1  r Ukj Þ

ð27Þ

where k 2 K, k – i, k – j and ~r ij 2 U, K is a set of the known values and U is a set of the unknown values, for all i, j = 1, 2, . . . , n(i – j). This can be interpreted in detail as follows: the missing value(s) should be estimated without changing the priority vector w = (w1, w2, . . . , wn)T of an incomplete interval fuzzy preference relation. Proof. It is similar to the proof of Theorem 2. h It is obvious that Eq. (27) holds if and only if an incomplete interval fuzzy preference relation is of consistency. However, an interval fuzzy preference relation may not be consistent. In such cases, the missing value(s) could be estimated with the following approach. e ¼ ð~rij Þ ~ Theorem 6. Let R nn be an incomplete interval fuzzy preference relation, then the missing value r ij of an incomplete interval fuzzy preference relation is estimated by means of the interval multiplicative transitivity:

" ~rijðkÞ ¼

r Lik r Lkj

r Uik r Ukj

#

; rLik r Lkj þ ð1  r Lik Þ  ð1  r Lkj Þ r Uik rUkj þ ð1  rUik Þ  ð1  rUkj Þ

ð28Þ

where k 2 K, k – i, k – j and ~rij 2 U, K is a set of the known values, # K is the number of the elements in K, and U is a set of the unknown values, for all i, j = 1, 2, . . . , n(i – j). ðkÞ We utilize the weighted geometric operator to aggregate different estimated preference relations ~rij ðk 2 KÞ. The weighted geometric operator is as follows:

WGx ða1 ; a2 ; . . . ; an Þ ¼

n Y

xj

aj

ð29Þ

j¼1

for all ~r ij 2 U. Eq. (29) can be adapted for interval fuzzy preference relations:

Y

~rij ¼

!1=#K

2

¼4

~r ðkÞ ij

k2K

K Y

!1=#K ~rijLðkÞ

k¼1

;

K Y

~r ijUðkÞ

!1=#K 3 5

ð30Þ

k¼1

Proof. Since

~rijLðkÞ 6 ~r ijUðkÞ

ð31Þ

for all k 2 K, thus,

~rLik~rLkj 6 ~r Uik~rUkj K K Y Y ~r LðkÞ ~rijUðkÞ 6 ij k¼1 K Y

k¼1

!1=#K ~rijLðkÞ

k¼1

for all i, j = 1, 2, . . . , n.

6

K Y k¼1

!1=#K ~r UðkÞ ij

ð32Þ

S. Genç et al. / Information Sciences 180 (2010) 4877–4891

4887

Sometimes, the lower and upper limits of the missing value(s) estimated with the first approach can be very close. In such cases, the second approach may be also used to estimate the missing value(s) of a multiplicative consistent interval fuzzy preference relation. Because, it covers the values estimated with the first approach, i.e., the lower values estimated with the second approach are smaller than the ones estimated with the first approach, and the upper values estimated with the second approach are larger than the ones estimated with the first approach. h h i ðkÞ ðkÞL ðkÞU Proof. Let ~r ij ¼ ~r ij ; ~rij be an interval fuzzy preference relation estimated by using any k 2 K, then

~rijðkÞL 6 r~00 Lij K Y

 #K ~r ðkÞL 6 r~00 Lij ij

k¼1 K Y

!1=#K ~rijðkÞL

k¼1 K Y

6

  1=#K #K r~00 Lij

!1=#K ~rijðkÞL

6 r~00 Lij

ð33Þ

k¼1

and

~rijðkÞU P r~00 Uij K Y

 #K ~r ðkÞU P r~00 Uij ij

k¼1 K Y

!1=#K ~rijðkÞU

k¼1 K Y

P

  1=#K #K r~00 Uij

!1=#K ~rijðkÞU

P r~00 Uij

ð34Þ

k¼1 ðkÞL ðkÞU where r~00 Lij and r~00 Uij are the maximum and minimum values among all of ~rij and ~rij , respectively, i.e., they are the values estimated with the first approach. h

4. Illustrative examples In this section, we present some practical examples to illustrate the developed approaches. Example 1. This example is taken from Xu and Chen [40], and suppose that a decision-maker provides his/her preference information over a collection of alternatives x1, x2, x3, x4 with the following interval fuzzy preference relation:

2

½0:5; 0:5 ½0:3; 0:4 ½0:5; 0:7 ½0:4; 0:5

3

6 ½0:6; 0:7 ½0:5; 0:5 ½0:6; 0:8 ½0:2; 0:6 7 6 7 R ¼ ðrij Þ44 ¼ 6 7 4 ½0:3; 0:5 ½0:2; 0:4 ½0:5; 0:5 ½0:4; 0:8 5 ½0:5; 0:6 ½0:4; 0:8 ½0:2; 0:6 ½0:5; 0:5 Solving the model (M-2), we get J = 0. Thus, we know that R is a multiplicative consistent interval fuzzy preference relation. Then by the model (M-1), we get

w1 ¼ ½0:1739; 0:2400;

w2 ¼ ½0:3000; 0:3913;

w3 ¼ ½0:1600; 0:2222;

w4 ¼ ½0:2222; 0:3000

Secondly, we utilize the concept of interval multiplicative transitivity. We initially estimate the interval fuzzy preference   relation R ¼ r ij using Eq. (22): nxn

2   R ¼ r ij

44

½0:5; 0:5

½0:3077; 0:4

½0:5; 0:6

½0:4; 0:5

3

6 ½0:6; 0:6923 ½0:5; 0:5 ½0:6; 0:6923 ½0:5; 0:6 7 7 6 ¼6 7 4 ½0:4; 0:5 ½0:3077; 0:4 ½0:5; 0:5 ½0:4; 0:5 5 ½0:5; 0:6

  It is obvious that R ¼ r ij

nxn

½0:4; 0:5

½0:5; 0:6

½0:5; 0:5

U satisfies the condition 0 6 rL ij 6 r ij 6 1 for any i, j = 1, 2, . . . , 5. Thus, we know that the interval

fuzzy preference relation R is a consistent interval fuzzy preference relation, and then we get the weights of the priority vector of R using Eq. (23):

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w1 ¼ ½0:1739; 0:2400;

w2 ¼ ½0:3000; 0:3913;

w3 ¼ ½0:1600; 0:2222;

w4 ¼ ½0:2222; 0:3000

Obviously, the weights derived by the linear programming models are exactly the same as the priority weights derived by our proposed formula. Example 2. Suppose that a decision-maker provides his/her preference information over a collection of alternatives x1, x2, x3, x4, x5 with the following interval fuzzy preference relation:

2

R ¼ ðrij Þ55

½0:5; 0:5 6 6 ½0:4; 0:5 6 6 ¼6 6 ½0:5; 0:8 6 6 ½0:3; 0:7 4

½0:5; 0:6 ½0:2; 0:5 ½0:3; 0:7 ½0:3; 0:5

3

7 ½0:5; 0:5 ½0:3; 0:6 ½0:6; 0:8 ½0:2; 0:4 7 7 7 ½0:4; 0:7 ½0:5; 0:5 ½0:7; 0:8 ½0:4; 0:5 7 7 7 ½0:2; 0:4 ½0:2; 0:3 ½0:5; 0:5 ½0:1; 0:4 7 5

½0:5; 0:7 ½0:6; 0:8 ½0:5; 0:6 ½0:6; 0:9 ½0:5; 0:5 Solving the model (M-2), we get J = 0. Therefore, R is a consistent interval fuzzy preference relation, and then the weights of the priority vector are derived with the model (M-1):

wL1 ¼ 0:1364;

wU1 ¼ 0:2442;

wL2 ¼ 0:1111;

wU2 ¼ 0:2029

wL3 ¼ 0:2029;

wU3 ¼ 0:3218;

wL4 ¼ 0:0662;

wU4 ¼ 0:1154

wL5 ¼ 0:2442;

wU5 ¼ 0:3899

i.e.,

w1 ¼ ½0:1364; 0:2442;

w2 ¼ ½0:1111; 0:2029;

w4 ¼ ½0:0662; 0:1154;

w5 ¼ ½0:2442; 0:3899

w3 ¼ ½0:2029; 0:3218

Secondly, we applied the concept of interval multiplicative transitivity. We initially estimated the interval fuzzy prefer  ence relation R ¼ rij using Eq. (22): nxn

2

55

55

½0:5; 0:6

½0:3; 0:5

½0:6; 0:7

½0:5; 0:5

½0:3; 0:5

½0:6; 0:7

½0:3; 0:5

3

7 ½0:2222; 0:4 7 7 ½0:5; 0:7 ½0:5; 0:5 ½0:7; 0:8 ½0:4; 0:5 7 7 7 ½0:3; 0:4 ½0:2; 0:3 ½0:5; 0:5 ½0:1552; 0:3 5 ½0:5; 0:7 ½0:6; 0:7778 ½0:5; 0:6 ½0:7; 0:8448 ½0:5; 0:5

6 6 ½0:4; 0:5 6 ¼6 6 ½0:5; 0:7 6 4 ½0:3; 0:4

  R ¼ rij 

  where R ¼ rij

½0:5; 0:5

U satisfies the condition 0 6 r L ij 6 r ij 6 1 for any i,j = 1, 2, . . ., 5, which means that it is a consistent interval

fuzzy preference relation. The weights of priority vector of R are then calculated using Eq. (23):

wL1 ¼ 0:1364;

wU1 ¼ 0:2442;

wL2 ¼ 0:1111;

wU2 ¼ 0:2029

wL3 ¼ 0:2029;

wU3 ¼ 0:3218;

wL4 ¼ 0:0662;

wU4 ¼ 0:1154

wL5 ¼ 0:2442;

wU5 ¼ 0:3899

i.e.,

w1 ¼ ½0:1364; 0:2442;

w2 ¼ ½0:1111; 0:2029;

w4 ¼ ½0:0662; 0:1154;

w5 ¼ ½0:2442; 0:3899

w3 ¼ ½0:2029; 0:3218

Again, the weights derived by the linear programming models are exactly the same as the weights derived by the priority weights formula. After deriving the weights, the possibility-degree formula can be used to compare each wi with all wj (j = 1, 2, . . . , 5), and construct the following fuzzy preference relation:

2

0:5

6 6 0:3332 6 P¼6 6 0:8178 6 4 0 1

0:6668 0:1822 0:5

0

1 0:9695

1

0:5

1

0:0305

0

0:5

1

0:7067

1

0

3

7 7 7 0:2933 7 7: 7 0 5 0:5 0

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Summing all elements in each line of P, we get:

p1 ¼ 2:3490;

p2 ¼ 1:8027;

p3 ¼ 3:6111;

p4 ¼ 0:5305;

p5 ¼ 4:2067

then 0:7067

0:8178

0:6668

0:9695

w5  w3  w1  w2  w4 which indicates that w5 is superior to w3 to degree of 70.67%, w3 is superior to w1 to degree of 81.78%, w1 is superior to w2 to degree of 66.68%, and w2 is superior to w4 to degree of 96.95%. Therefore, the ranking of wj (j = 1, 2, . . . , 5) is as follows:

x5  x3  x1  x2  x4 where the notion ‘‘” indicates one alternative is preferred to another. Example 3. Suppose that a decision-maker provides his/her preference information over a collection of alternatives x1, x2, x3, x4, x5, x6 with the following incomplete interval fuzzy preference relation:

2

e ¼ ð~rij Þ R 66

½0:5; 0:5 ½0:2; 0:4 ½0:2; 0:4

x

½0:6; 0:9

x

3

7 6 6 ½0:6; 0:8 ½0:5; 0:5 ½0:4; 0:5 ½0:6; 0:8 x ½0:4; 0:6 7 7 6 7 6 7 6 ½0:6; 0:8 ½0:5; 0:6 ½0:5; 0:5 x ½0:8; 0:9 x 7 6 ¼6 7 7 6 x ½0:2; 0:4 x ½0:5; 0:5 ½0:5; 0:6 ½0:1; 0:4 7 6 7 6 7 6 ½0:1; 0:4 x ½0:1; 0:2 ½0:4; 0:5 ½0:5; 0:5 ½0:2; 0:4 5 4 x

½0:4; 0:6

x

½0:6; 0:9 ½0:6; 0:8 ½0:5; 0:5

where x denotes the unknown variable. We firstly use the interval multiplicative transitivity, for all k 2 K: ð2Þ ~r 14 ¼ r 12 r 24 ¼ ½0:2727; 0:7273; ð2Þ ~r 16 ¼ r 12 r 26 ¼ ½0:1429; 0:5;

ð5Þ ~r14 ¼ r 15 r 54 ¼ ½0:5; 0:9

ð5Þ ~r 16 ¼ r 15 r 56 ¼ ½0:2727; 0:8571

ð1Þ ~r 25 ¼ r 21 r 15 ¼ ½0:6923; 0:9730;

~r ð3Þ 25 ¼ r 23 r 35 ¼ ½0:7273; 0:9

ð4Þ ~r 25 ¼ r 24 r 45 ¼ ½0:6; 0:8571;

ð6Þ ~r 25 ¼ r 26 r 65 ¼ ½0:5; 0:8571

ð2Þ ~r 34 ¼ r 32 r 24 ¼ ½0:6; 0:8571;

ð5Þ ~r 34 ¼ r 35 r 54 ¼ ½0:7273; 0:9

ð2Þ ~r 36 ¼ r 32 r 26 ¼ ½0:4; 0:6923;

ð5Þ ~r 36 ¼ r 35 r 56 ¼ ½0:5; 0:8571

By Eq. (27), the missing values are estimated as follows:

~r 14 ¼ ½maxf0:2727; 0:5g; minf0:7273; 0:9g ¼ ½0:5; 0:7272 ~r 16 ¼ ½maxf0:1429; 0:2727g; minf0:5; 0:8571g ¼ ½0:2727; 0:5 ~r 25 ¼ ½maxf0:6923; 0:7273; 0:6; 0:5g; minf0:9730; 0:9; 0:8571; 0:8571g ¼ ½0:7273; 0:8571 ~r 34 ¼ ½maxf0:6; 0:7273g; minf0:8571; 0:9g ¼ ½0:7273; 0:8571 ~r 36 ¼ ½maxf0:4; 0:5g; minf0:6923; 0:8571g ¼ ½0:5; 0:6923 It is obvious that the missing values estimated with Eq. (27) satisfy the condition 0 6 rLij 6 rUij 6 1, i.e., the incomplete interval fuzzy preference relation is consistent. Therefore, we get the following complete interval fuzzy preference relation:

3 ½0:5; 0:5 ½0:2; 0:4 ½0:2; 0:4 ½0:5; 0:7272 ½0:6; 0:9 ½0:2727; 0:5 6 ½0:6; 0:8 7 ½0:5; 0:5 ½0:4; 0:5 ½0:6; 0:8 ½0:7273; 0:8571 ½0:4; 0:6 7 6 6 7 6 ½0:6; 0:8 ½0:5; 0:6 ½0:5; 0:5 ½0:7273; 0:8571 ½0:8; 0:9 ½0:5; 0:6923 7 e¼6 7 R 6 ½0:2728; 0:5 ½0:2; 0:4 ½0:1429; 0:2727 ½0:5; 0:5 ½0:5; 0:6 ½0:1; 0:4 7 6 7 6 7 4 ½0:1; 0:4 ½0:1429; 0:2727 ½0:1; 0:2 ½0:4; 0:5 ½0:5; 0:5 ½0:2; 0:4 5 2

½0:5; 0:7273

½0:4; 0:6

½0:3077; 0:5

½0:6; 0:9

½0:6; 0:8

½0:5; 0:5

The estimated values of the missing values ~r25 and ~r34 are in a small interval. In such cases, we know that Eq. (30) can be used to relax the values estimated with Eq. (27):

" ~r 25 ¼

ð0:6923  0:7273  0:6  0:5Þ1=4 ; ð0:9730  0:9  0:8571  0:8571Þ1=4

# ¼ ½0:6234; 0:8956

~r 34 ¼ ½ð0:6  0:7273Þ1=2 ; ð0:8571  0:9Þ1=2  ¼ ½0:6606; 0:8783

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Example 4. Suppose that a decision-maker provides his/her preference information over a collection of alternatives x1, x2, x3, x4, x5 with the following incomplete interval fuzzy preference relation:

3 ½0:5; 0:5 x ½0:7; 0:9 ½0:2; 0:4 ½0:4; 0:5 7 6 x ½0:5; 0:5 ½0:5; 0:7 x ½0:8; 0:9 7 6 7 6 7 ¼6 6 ½0:1; 0:3 ½0:3; 0:5 ½0:5; 0:5 ½0:6; 0:9 ½0:3; 0:4 7 7 6 x ½0:1; 0:4 ½0:5; 0:5 x 5 4 ½0:6; 0:8 ½0:5; 0:6 ½0:1; 0:2 ½0:6; 0:7 x ½0:5; 0:5 2

R ¼ ðrij Þ55

where x denotes the unknown variable. It can be easily proven that the missing values estimated by Eq. (27) do not satisfy the condition 0 6 rLij 6 rUij 6 1 (note that ~r12 and ~r 45 do not satisfy this condition). Thus, we utilize Eq. (28) to estimate the missing values:

~r ð3Þ 12 ¼ r 13 r 32 ¼ ½0:5; 0:9;

~r ð5Þ 12 ¼ r 15 r 52 ¼ ½0:0690; 0:2

~r ð3Þ 24 ¼ r 23 r 34 ¼ ½0:6; 0:9545; ~r ð1Þ 45 ¼ r 41 r 15 ¼ ½0:5; 0:8;

~r ð5Þ 24 ¼ r 25 r 54 ¼ ½0:5; 0:8571

~r ð3Þ 45 ¼ r 43 r 35 ¼ ½0:0454; 0:3077

Using Eq. (30), we estimate the missing values as follows:

~r 12 ¼ ½ð0:5Þ1=2 ð0:0690Þ1=2 ; ð0:9Þ1=2 ð0:2Þ1=2  ¼ ½0:3255; 0:5646 ~r 24 ¼ ½ð0:6Þ1=2 ð0:5Þ1=2 ; ð0:9545Þ1=2 ð0:8571Þ1=2  ¼ ½0:6694; 0:9353 ~r 45 ¼ ½ð0:5Þ1=2 ð0:0454Þ1=2 ; ð0:8Þ1=2 ð0:3077Þ1=2  ¼ ½0:2833; 0:6267 and then have the following complete interval fuzzy preference relation:

2

½0:5; 0:5 ½0:3255; 0:5646 6 ½0:5; 0:5 6 ½0:4354; 0:6745 6 e¼6 R ½0:1; 0:3 ½0:3; 0:5 6 6 ½0:6; 0:8 ½0:0647; 0:3306 4 ½0:5; 0:6

½0:1; 0:2

½0:7; 0:9

½0:2; 0:4

½0:4; 0:5

3

7 7 7 7 ½0:5; 0:5 ½0:6; 0:9 ½0:3; 0:4 7 7 ½0:1; 0:4 ½0:5; 0:5 ½0:2833; 0:6267 5 ½0:6; 0:7 ½0:3733; 0:7177 ½0:5; 0:5 ½0:5; 0:7 ½0:6694; 0:9353

½0:8; 0:9

5. Conclusions and future work In this paper, we have studied the issue of consistency, missing value(s) and derivation of the priority vector of interval fuzzy preference relations. The concept of interval multiplicative transitivity of an interval fuzzy preference relation has been introduced. On the basis of this concept, a straightforward approach has been proposed to check whether an interval fuzzy preference relation is consistent or not, and to derive a formula to determine the priority vector of a consistent interval fuzzy preference relation. It has been shown that the proposed approach can yield the same results without solving any linear programming models. Two examples have been presented to show the validity and the practicality of the proposed approach. Furthermore, we have developed two approaches to estimate the missing value(s) of an incomplete interval fuzzy preference relation. The first approach estimates the missing value(s) without changing the priority weights and only holds under the condition where an incomplete fuzzy preference relation is consistent. The second approach has been introduced for an inconsistent, incomplete interval fuzzy preference relation to estimate the missing values. Also, we have proven that the second approach relaxes the missing value(s) estimated with the first approach. Finally, two illustrative examples have been given to show the missing value(s) estimation approaches. This paper studies the concept of the interval multiplicative transitivity and shows some new approaches, which are the most important issues in an interval fuzzy preference relation. However, there are still some problems which need further study. First, the interval multiplicative transitivity holds in the case of being rij > 0 "i, j, i.e., the lower limit r  ij of each element of an interval fuzzy preference relation should be greater than zero. Secondly, the priority vector of an interval fuzzy preference relation is derived from a multiplicative consistent interval fuzzy preference relation but not an inconsistent one. These two issues require further research in the future. Acknowledgments The authors are very grateful to the Editor-in-Chief, Professor Witold Pedrycz, and the anonymous referees, for their constructive comments and suggestions that led to an improved version of this paper. The work was supported in part by the National Science Fund for Distinguished Young Scholars of China (No. 70625005).

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