Consistency and consensus modeling of linear uncertain preference relations

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

European Journal of Operational Research xxx (xxxx) xxx

Contents lists available at ScienceDirect

European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

Consistency and consensus modeling of linear uncertain preference relations Zaiwu Gong a,∗, Weiwei Guo a, Enrique Herrera-Viedma b, Zejun Gong c, Guo Wei d a

School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China Department of Computer Science and Artificial Intelligence, University of Granada, Granada 18071, Spain c Information Computer Science, Xi’an Jiaotong-Liverpool University, Suzhou 215123, China d The University of North Carolina at Pembroke, Pembroke, North Carolina 28372, USA b

a r t i c l e

i n f o

Article history: Received 28 January 2019 Accepted 24 October 2019 Available online xxx Keywords: Group decisions and negotiations Interval fuzzy preference relations Linear uncertain preference relations Consensus Consistency

a b s t r a c t Interval operations, as currently defined, suffer from the problem of not satisfying the conditions of global complementarity and consistency of interval fuzzy preference relations (IFPRs). In this paper, we resolve this difficulty by constructing linear uncertain preference relations (LUPRs). By considering all the information and the uncertain distribution of an interval, we propose the concept of uncertain preference relations (UPRs) for the first time. Then we apply uncertainty distributions to characterize interval judgments that are considered as a whole to participate in the uncertain operation to achieve the desired conditions of global complementarity and consistency. Based on this, we prove that IFPRs and the definitions of their additive consistency are special cases of those of LUPRs. Moreover, we investigate two types of consensus models developed based on LUPRs between the minimum deviation and belief degree. We prove that the minimum deviation is a linear, increasing function of the belief degree, and then establish sufficient and necessary conditions for the consensus model to satisfy additive consistency. Finally, the LUPRs models presented in this paper is applied, incorporating with expert assistance in decisionmaking, to the sensitivity assessment of the meteorological industry in a region of China, and the LUPRs models can be utilized to obtain results with smaller deviations. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In group decision making (GDM), the preference relation is a general tool used by decision makers (DMs) to express preference information for a range of alternatives. The DMs compare alternatives according to their experience, construct a judgment matrix (preference relation), and then determine the priorities of alternatives through optimization modeling (Dong & Cooper, 2016; Liu, Zhang, & Wang, 2012; Ma, 2016; Meng & Chen, 2015). In the case in which DMs are unfamiliar with decision making problems, incomplete preferences regarding the alternatives could appear (Capuano, Chiclana, Fujita, Herrera-Viedma, & Loia, 2018; Ureña, Chiclana, Morente-Molinera, & Herrera-Viedma, 2015), and DMs are often in a state of hesitation and uncertainty when making judgments. In dealing with uncertainty, several theories and methods have been developed in addition to the probability theory, such as fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, etc. ∗

Corresponding author. E-mail addresses: [email protected] (Z. Gong), [email protected] (W. Guo), [email protected] (E. Herrera-Viedma), [email protected] (Z. Gong), [email protected] (G. Wei).

Bustince et al. (2015) reviewed the definition and basic properties of the different types of fuzzy sets and also analyzed the relationships between them. Barrenechea, Fernandez, Pagola, Chiclana, and Bustince (2014) put forward a method to construct an intervalvalued fuzzy set, when they define the membership values of the elements to that fuzzy set, generalizing the traditional fuzzy set concept by using an interval representation of the lack of knowledge or ignorance that experts are subject. Pal et al. (2013) proposed a new axiomatic framework to measure uncertainties of an intuitionistic fuzzy set. Interval judgment, intuitionistic fuzzy judgment, and natural linguistic judgment are three common forms of uncertain preferences adopted by DMs (Cabrerizo et al., 2017; Chen, 20 0 0; Gong et al., 2015; Liu, Shen, Zhang, Chen, & Wang, 2015; Orlovsky, 1978; Qin, Liu, & Pedrycz, 2017; Rodriguez, Martinez, & Herrera, 2012). Interval judgment uses the range of judgments to express pairwise comparisons of the alternatives (Tang, Meng, & Zhang, 2018; Wu, Chiclana, & Liao, 2018). By contrast, intuitionistic fuzzy judgment expresses inaccurate information about the judgment in terms of a membership degree, non-membership degree, and hesitation degree. This solves the problem that interval judgments cannot explain the hesitation phenomenon in the DMs’ judgment process (Atanassov, 1986; Ouyang & Pedrycz, 2016).

https://doi.org/10.1016/j.ejor.2019.10.035 0377-2217/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

JID: EOR 2

ARTICLE IN PRESS

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

Natural linguistic judgment (linguistic preference relation) is the most intuitive and concise form in which DMs can express their uncertain preferences. To resolve decision making issues, DMs use predefined basic language variables (e.g., good, medium, and poor) to evaluate the degree of preference for an element in a set of terms (Ben-Arieh & Chen, 2006; Li, Dong, Herrera, Herrera-Viedma, & Martínez, 2017; Wu, Chiclana, & Herrera-Viedma, 2015; Wu & Liao, 2019; Yan, Ma, & Huynh, 2017). In fact, intuitionistic fuzzy judgment and natural linguistic judgment are essentially interval preference relations, and can collectively be referred to as interval class judgments: intuitionistic fuzzy sets and interval fuzzy sets are equivalent in a mathematical sense; and linguistic preference and basic language sets, fuzzy sets, and interval fuzzy sets also have an equivalent mapping relationship. In the above studies, the interval preference relations are all based on the operation system of interval analysis developed by Moore (1966), Zadeh et al. (1965), etc. However, such an operation system has the following limitations: •

•

•

Only the endpoints of the interval data are involved in operations, that is, the inner characteristics of the interval are not considered, which makes the data processing discrete. There is no holistic complementarity between the elements of the symmetric position, thus destroying the essence of the complementary definition of fuzzy preference relations. The distribution characteristics of the intervals (e.g., normal distribution and uniform distribution) are not considered, so the interval is not involved in the operation as a whole.

This often leads to the loss or distortion of effective decision information. Consistency is not only the premise for evaluating the logic of DMs’ judgments, but also forms the basis of effective GDM. The earliest consistency study originated from the 1–9 scale of reciprocal judgment and was developed by Saaty (1978). Based on a 0–1 fuzzy scale, Tanino (1984) proposed the concept of additive consistency and multiplicative consistency in fuzzy preference relations, and these form the basis of consistent interval pref´ erence relations. Switalski (1999, 20 01, 20 03) studied the different types of transitivity and acyclicity conditions for fuzzy reciprocal relations and introduced the FG-transitivity. De Baets and De Meyer (2005) analyzed the advantages and disadvantages of FG-transitivity and compared cycle-transitivity with FG-transitivity, and they concluded that, under the reciprocal relations, the concept of cycle-transitivity provides a framework that validating for more types of transitivity than the FG-transitivity; They constructed the transitivity framework of reciprocal relations, laying a theoretical foundation for the consistency and transitivity of uncertain preference relations in this paper. De Schuymer, De Meyer, and De Baets (2005) proposed dice-transitivity, and concluded that the probabilistic relation generated by a collection of arbitrary independent random variables remains dice-transitive, and this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Chiclana, Herrera-Viedma, Alonso, and Herrera (2009) concluded that multiplicative transitivity is the most appropriate property for modeling the cardinal consistency of reciprocal preference relations. Xu (2011) developed two methods for constructing additively consistent interval fuzzy preference relations (IFPRs) and multiplicatively consistent IFPRs. Wan, Wang, and Dong (2017, 2018a, 2018b) studied the interval-valued intuitionistic fuzzy preference relation based on the additive and multiplicative consistent interval values of Atanassov and the interval-valued fuzzy preference relation based on geometric consistency. Liu, Peng, Yu, and Zhao (2018) proposed the concept of additive approximation consistency in interval additive reciprocal matrices, and Li, Rodríguez, Martínez, Dong, and Herrera (2018) proposed an interval consistency index to estimate the consistency range of hesitant fuzzy linguistic preference relations.

Meng and Tan (2017) and Meng, Tan, and Chen (2017) defined a new concept of consistency for the case of extended brittleness, and proposed an IFPR GDM method. Liu, Zhang, and Zhang (2014) proposed a new consistency definition for triangular fuzzy reciprocal preference relations, and Wang (2018) used a Lagrange multiplier method to analytically determine the interval weight of approximate solutions from inconsistent IFPRs. The common features of the above consistency analyses are that the upper and lower bounds of interval values are discretized from an operational point of view, and the intervals are not considered as a whole. Therefore, to ensure the integrity of the interval, guarantee the complementarity and consistency of the entire interval, and make full use of the judgment information to prevent any distortion of the decision making process, a more reasonable tool is needed to construct a new interval judgment and the corresponding operations. This tool should effectively overcome the above shortcomings, and not only fully represent the uncertainty of DMs, but also consider interval judgments as a whole. Liu’s uncertainty system theory (“uncertainty theory” for short) provides a new idea for the study of such problems: by introducing a belief degree and regarding the interval preference as an uncertain distribution, and the logical problem of consistency judgment can be better solved. In the judgment of interval class preferences, the decision value given by DMs has no reliable sample and it is not random; it relies only on subjective experience. However, although the specific value of the judgment interval cannot be determined, the approximate probability distribution of DMs’ judgments in the interval can always be given. For example, the possibilities of choosing any value in an interval are equal, but the closer a possibility is to the middle of the interval, the more likely that it will be selected. The characteristics of these interval judgments are consistent with the linear uncertain distribution or normal uncertain distribution in uncertainty theory. In particular, when judging the complementarity and consistency relationship for an interval, the linear uncertainty distribution is used to describe the interval judgment, which can achieve the overall complementarity and consistency of the interval, and ensure that the entire interval participates in the uncertain operation. •

•

•

In this paper, a new interval preference relation is proposed based on uncertainty theory, which can describe the global complementarity between its symmetric position elements. Linear uncertain distribution is used to characterize interval judgments, and a new definition of consistency is constructed to realize the global complementarity and consistency of intervals. This definition is also used as a constraint condition for optimal consensus modeling with uncertain preference relations (UPRs). In this paper, some conclusions regarding interval preference relations in the existing literature become special cases of UPRs, and the investigation of UPRs and its consistency is a theoretical extension of the study of interval preference relations.

The structure of this paper is as follows: In Section 2, the concepts of uncertainty theory, the linear uncertainty distribution, and its inverse are introduced. In Section 3, the concept of fuzzy preference relations and their definition of consistency are elaborated. In Section 4, a preference relation is constructed based on uncertain variables, and definitions of the additive consistency, general transitivity, and satisfactory transitivity of UPRs are presented. In Section 5, the optimal consensus model is built with the minimum deviation under the constraints of the linear uncertain distribution and belief degree, and the correlation between the minimum deviation and belief degree is explored. In Section 6, definitions and models of UPR, LUPR, FPR and IFPR are compared, and their connection, pros and cons are carefully investigated or compared. In Section 7, The LUPRs models proposed in this paper is applied to

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

3

the industry meteorological sensitivity assessment for a region in China, based on the assessment of experts specialized in decisionmaking, and the sensitivity ranking of four selected industries to meteorological conditions was achieved, which are also compared with the existing results obtained by other methods. Finally, in Section 8 our conclusions are summarized and ideas for future research are discussed. 2. Uncertainty theory Uncertainty theory mainly studies events for which the distribution function cannot fit the frequency or the belief degree problem of each event has to be evaluated by domain experts. The idea of uncertainty theory comes from probability theory, and it is complementary to probability theory. Uncertainty theory system is based on a rigorous mathematical reasoning system, and its system framework is complete. Uncertain systems research content includes: uncertain programming, uncertain risk analysis, uncertain reliability analysis, uncertain logic, uncertain inference, uncertain process, uncertain calculus, uncertain differential equation and uncertain financial (Chen, 2011; Liu & Liu, 2009; Zeng, Kang, Wen, & Zio, 2018). In practical applications, Uncertainty theory has achieved great success in applications such as vehicle scheduling, combinatorial optimization, and risk analysis (Gao & Ralescu, 2018; Liu, 2007; 2014; Memon, Lee, & Mari, 2015).

Fig. 1. Linear uncertainty distribution.

Theorem 1 (Liu, 2014). Let ξ1 , ξ2 , . . . , ξn be independent uncertain variables with regular uncertainty distributions 1 , 2 , . . . , n , respectively. If f (ξ1 , ξ2 , . . . , ξn ) is strictly increasing with respect to ξ1 , ξ2 , . . . , ξm and strictly decreasing with respect to ξm+1 , ξm+2 , . . . , ξn , then

ξ = f ( ξ1 , ξ2 , · · · , ξn )

2.1. Uncertain measure

has inverse uncertainty distribution Let  be a nonempty set (sometimes called a universal set). Collection L that consists of subsets of  is called an algebra over  if the following three conditions hold: (a)  ∈L; (b) if  ∈ L, then c ∈ L; and (c) if 1 , 2 , . . . , n ∈ L, then ni=1 i ∈ L. Collection L is called a σ -algebra over  if condition (c) is replaced by closure under the countable union, that is, when 1 , 2 , . . . ∈ L,  we have ∞ i=1 i ∈ L (Liu, 2014). To consider belief degrees rationally, Liu (2007, 2014) suggested the following axioms: Axiom 1 (Normality Axiom). M{ } = 1 for universal set  .

Axiom 3 (Subadditivity Axiom). For every countable sequence of events 1 , 2 , . . . , we have

M

∞ 



i

≤

∞ 

i=1

M{i }.

(1)

i=1

Axiom 4 (Product Axiom). Let (k , Lk , Mk ) be uncertainty spaces for k = 1, 2, . . . , where product uncertain measure M is an uncertain measure that satisfies



M

∞  k=1



k

=

∞ 

Mk {k },

(2)

k=1

2.2. Linear uncertain distribution and its inverse distribution Theorem 2 (Liu, 2014). Uncertain variable ξ is said to be linear if it has linear uncertainty distribution

0, x−a (x ) = , ⎪ ⎩b − a 1,

ifx ≤ a ifa ≤ x ≤ b

(5)

i f x ≥ b,

and is denoted by L[a, b], where a and b are real numbers, with a < b (see Fig. 1). Theorem 3 (Liu, 2014). The inverse uncertainty distribution of linear uncertain variable L[a, b] (see Fig. 2) is

−1 (α ) = (1 − α )a + α b.

(6)

3. Fuzzy preference relation 3.1. Fuzzy preference relation

where k are arbitrarily chosen events from Lk for k = 1, 2, . . .. Definition 1 (Liu, 2014). An uncertain variable is function ξ from uncertainty space ( , L, M) to the set of real numbers such that {ξ ∈ B} is an event for any Borel set B of real numbers. Uncertainty distribution  of uncertain variable ξ is defined as

(x ) = M{ξ ≤ x}.

(4)

⎧ ⎪ ⎨

Axiom 2 (Duality Axiom). M{} + M{c } = 1 for any event .



−1 −1 −1  −1 (α ) = f (−1 1 (α ), . . . , m (α ), m+1 (1 − α ), . . . , n (1 − α )).

(3)

For any real number x, where M{ξ ≤ x} is called the belief degree of event {ξ ≤ x}, which is measured by α (0 ≤ α ≤ 1), that is, α = (x ) = M{ξ ≤ x}. According to Axiom 2, M{ξ > x} = 1 − (x ) = 1 − α . When uncertainty distribution  is a continuous function, we also have M{ξ < x} = M{ξ ≤ x} = (x ), M{ξ ≥ x} = 1 − (x ).

Definition 2 (Kacprzyk, 1986; Nurmi, 1981; Orlovsky, 1978; Tanino, 1984). Nonnegative matrix R = (ri j )n×n is called a fuzzy preference relation if rii = 0.5, ri j + r ji = 1, i, j ∈ N. Element rij in fuzzy preference relation R expresses the membership degree of alternative xi over alternative xj . ri j = 0.5 indicates that there is no difference between xi and xj ; if rij > 0.5, then xi is superior to xj , and if rij < 0.5, then xj is superior to xi . Definition 3 (Tanino, 1984). Fuzzy preference relation R = (ri j )n×n is said to have additive consistency if

ri j + r jk = rik + 0.5, i, j, k ∈ N.

(7)

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 4

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

Fig. 2. Inverse linear uncertainty distribution.

(1) If r¯i j ≥ 0.5, r¯ jk ≥ 0.5, then r¯ik ≥ 0.5; and (2) if r¯i j ≤ 0.5, r¯ jk ≤ 0.5, then r¯ik ≤ 0.5.

3.2. Interval fuzzy preference relation Definition 4 (Xu, 2004). Nonnegative matrix R¯ = (r¯i j )n×n = ([riLj , rUi j ] )n×n is called an IFPR if r¯ii = [0.5, 0.5], riLj + rUji = rUi j + rLji = 1, i, j ∈ N. Element r¯i j = [riLj , rU ] in IFPR R¯ expresses the membership deij gree of alternative xi over alternative xj . r¯i j = [0.5, 0.5] indicates that there is no difference between xi and xj ; if r¯i j > [0.5, 0.5], then xi is superior to xj , and if r¯i j < [0.5, 0.5], then xj is superior to xi . If riLj = rU , i, j ∈ N, then the interval can be transformed into a crisp ij number. Therefore, the fuzzy preference relation is a special case of the IFPR. Definition 5 (Xu, Li, & Wang, 2014). IFPR R¯ = (r¯i j )n×n is said to have additive consistency if L riLj + r Ljk = rik + 0.5, i < j < k, i, j, k ∈ N,

(8)

rUi j + rUjk = rUik + 0.5, i < j < k, i, j, k ∈ N.

(9)

Eqs. (8) and (9) can be unified as follows:

r¯i j  r¯ jk = r¯ik + [0.5, 0.5], i < j < k, i, j, k ∈ N, where  denotes the addition of intervals: r¯i j  r¯ jk =

(10) [riLj

+ r Ljk , rU ij

+

rUjk ]. 3.3. General transitivity and satisfactory transitivity of fuzzy preference relations ´ Definition 6 (Jiang & Fan, 2008; Switalski, 2003). Let R¯ = (r¯i j )n×n be an IFPR for any i, j, k ∈ N, i = j = k. (1) When 0.5 ≤ λ ≤ 1, if r¯i j ≥ λ, r¯ jk ≥ λ, then r¯ik ≥ λ; and (2) when 0 ≤ λ ≤ 0.5, if r¯i j ≤ λ, r¯ jk ≤ λ, then r¯ik ≤ λ. Then, R¯ is said to have general transitivity. Theorem 4 (Gong, Lin, & Yao, 2012). Additively consistent IFPRs have general transitivity. Definition 7 (Jiang & Fan, 2008). Let R¯ = (r¯i j )n×n be an IFPR for any i, j, k ∈ N, i = j = k.

Then, R¯ is said to have satisfactory transitivity. Corollary 1 (Gong et al., 2012). Additively consistent IFPRs have satisfactory transitivity. Corollary 2 (Gong et al., 2012). Additively consistent fuzzy preference relations have the property of general transitivity and satisfactory transitivity. If an IFPR has satisfactory consistency, then the corresponding preference relation of the decision alternatives X = {x1 , x2 , . . . , xn } has the transitivity property; that is, there exists a priority chain xu1 ≥ xu2 ≥ · · · ≥ xun in X = {x1 , x2 , . . . , xn }, where xui denotes the ith decision alternative in the priority chain and xui ≥ xu j , “ ≥” indicates that xui is preferred (superior) to xu j . It can be seen that satisfactory consistency is a minimal logical requirement and a fundamental principle of fuzzy preference relations, which reflects a fundamental characteristic of human thought (Gong et al., 2012). The IFPR is a further extension of the fuzzy preference relation, which reflects the uncertainty of DMs in making judgments. However, it also has the following drawbacks: (1) As the addition and subtraction of intervals are irreversible, the complementarity of IFPR has to be defined in discrete form, for example, riLj + rUji = rU + r Lji = 1, i, j ∈ N. This defij inition only considers the value of the interval endpoint, and does not consider the inner information of the interval, which destroys the definition of global complementarity (ri j + r ji = 1, i, j ∈ N) of symmetric location elements in the fuzzy preference relation. Meng and Tan (2017) and Meng et al. (2017) proposed the concept of quasi-IFPRs by interchanging the upper and lower limits of the interval (e.g., [rU , r L ]) and solved the above problems, to some extent. ij ij However, the upper limit of the quasi-interval is less than or equal to the lower limit, which is contrary to mathematical logic. (2) The definition of additive consistency in IFPRs separates the upper and lower limits of the interval, and additive consistency of the upper and lower bounds of the interval is

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

5

defined separately, which improves the processing complexity of consistency and violates the connotation of global additive consistency in the original definition of fuzzy preference relations. (3) Although the DMs’ judgment is an interval, this value represents the comprehensive judgment from pairwise comparisons of alternatives, and should also be treated as a whole. Considering only the upper and lower limits of the interval loses a large amount of interval information and distorts the decision result.

ity requirement, which overcomes the shortcomings of Definition (4) for IFPRs, which only considers the upper and lower limits of the interval and divides them into different definitions. Simultaneously, this construction conforms to the connotation of the global complementarity of ri j + r ji = 1 in the original definition of fuzzy preference relations. Judgment element rij in the uncertainty preference relation indicates the degree to which alternative xi is superior to alternative xj . −1 (α ) = 0.5 means there is no difference between xi and xj ; if ij

Therefore, a new IFPR and associated consistency definitions are required.

xj is superior to xi . The definition of UPRs overcomes the shortcomings of the definition of IFPRs, and has the following advantages:

−1 (α ) > 0.5, then xi is superior to xj , and if −1 (α ) < 0.5, then ij ij

4. Preference relations based on uncertainty theory

(1) Using an uncertain distribution to describe interval judgments better simulates the real state of DMs and is more suitable for considering practical problems. (2) The elements of intervals are considered as a whole rather than discrete segments so that the information is completely retained, thereby ensuring the correctness of the final decision. (3) The definition of UPRs conforms to the connotation of the complementarity definition for fuzzy preference relations.

In real decision making, differences in people’s environment, education, completeness of information, and other factors mean that the judgment made is often uncertain, and there is no sample from which to estimate the distribution function of an individual’s specific decision making opinions. Simultaneously, the distribution function is unlike that in probability theory, which can be approximated in terms of frequency. Uncertainty theory provides a mathematical tool for studying this type of problem.

The additive consistency of UPRs is defined as follows: 4.1. UPRs and their consistency definitions Similar to the fuzzy preference relation, the definition of the UPR is based on uncertainty theory.

Definition 9. UPRs R = (ri j )n×n are consistency if, for any α , 0 ≤ α ≤ 1,

said

to

have

additive

−1 (α ) + −1 (α ) = −1 (α ) + 0.5, i < j < k, i, j, k ∈ N. ij jk ik

(11)

Definition 8. Let R = (ri j )n×n be a nonnegative matrix, where rij is an uncertain variable; ij is its uncertainty distribution function, that is, rij ∼ ij ; and −1 is the corresponding inverse unij certainty distribution function. For any belief degree α , 0 ≤ α ≤ 1, we have −1 (α ) = 0.5, −1 (α ) + −1 (1 − α ) = 1, i, j ∈ N (global ii ij ji complementarity). Then, R is called the UPR. If rij obeys a linear uncertain distribution, the matrix R = (ri j )n×n is called a linear UPR (LUPR).

For example, if r12 ∼ L[0.6, 0.8], r24 ∼ L[0.3, 0.7], then (6) and (11) imply that −1 (α ) = −1 (α ) + −1 ( α ) − 0.5 = 0.6 ( 1 − α ) + 14 12 24 0.8α + 0.3 ( 1 − α ) + 0.7α − 0.5 = 0.4 ( 1 − α ) + α . Clearly, r14 ∼ L[0.4, 1]. This result is consistent with the results obtained by (8) and (9), and (11) regards interval judgment information (linear uncertain distribution variables) as a whole, which conforms to the connotation of the definition of additive consistency for fuzzy preference relations and is more reasonable from the perspective of computational logic.

Belief degree is the core of uncertainty theory research. Introducing the belief degree into the UPRs has the following benefits:

Definition 10. Let R = (ri j )n×n be UPRs for any i, j, k ∈ N, i = j = k, 0 ≤ α ≤ 1.

•

•

•

The uncertainty decision based on belief degree can more reasonably reflect the confidence of DMs based on their own uncertainty experience judgment, which is especially suitable for the subjective decision-making environment. The UPRs based on belief degree is an extension of the traditional interval preference relations, and UPRs can be used to analyze whether the uncertain variables meet the definition of complementarity and consistency as a whole. It is easier to obtain the analytic solution by using the uncertainty decision built upon the belief degree. Similar to the belief degree, chance-constrained optimization (Cohen, Keller, Mirrokni, & Zadimoghaddam, 2019; Lejeune & Margot, 2016) is widely used in the confidence characterization under the condition of a certain degree of confidence, the possibility of random variables satisfy certain constraints. However, some stochastic optimization requires the use of intelligent algorithms to obtain approximate solutions, while uncertain optimization using belief degree is easier, to some extent, in obtaining analytic solutions than stochastic optimization.

In Definition (8), if r13 , r31 are uncertain variables with linear uncertain distributions r13 ∼ L[0.3, 0.6], r31 ∼ L[0.4, 0.7], then (6) and the definition of UPRs implies that −1 (α ) + −1 (1 − α ) = 13 31 0.3(1 − α ) + 0.6α + 0.4α + 0.7(1 − α ) = 1. By adjusting the value of α , any value within the interval can meet the complementar-

•

•

When 0.5 ≤ λ ≤ 1, M{rik ≥ λ} ≥ α ; and when 0 ≤ λ ≤ 0.5, M{rik ≤ λ} ≥ α .

if

M{rij ≥ λ} ≥ α ,

M{rjk ≥ λ} ≥ α ,

then

if

M{rij ≤ λ} ≥ α ,

M{rjk ≤ λ} ≥ α ,

then

Then, R is said to have general transitivity, where M{rij ≥ λ} ≥ α indicates that the DM’s uncertain judgment value rij ≥ λ has a belief degree of at least α . Definition 11. Let R = (ri j )n×n be UPRs for any i, j, k ∈ N, i = j = k, 0 ≤ α ≤ 1. • •

If M{rij ≥ 0.5} ≥ α , M{rjk ≥ 0.5} ≥ α , then M{rik ≥ 0.5} ≥ α ; and if M{rij ≤ 0.5} ≥ α , M{rjk ≤ 0.5} ≥ α , then M{rik ≤ 0.5} ≥ α .

Then, R is said to have satisfactory transitivity. If rij , rji are linear uncertain variables, and ri j ∼ L[riLj , rU ], r ji ∼ L[r Lji , rUji ], then Theorem 3 implies that −1 (α ) = ij ij

(1 − α )riLj + α rUi j , −1 (1 − α ) = α rLji + (1 − α )rUji . When α = 0, ji −1 −1 i j (α ) +  ji (1 − α ) = 1 is the same as riLj + rUji = 1; when α = 1, −1 (α ) + −1 (1 − α ) = 1 is the same as rUi j + rLji = 1; and ij ji −1 (α ) = 0.5 is equivalent to rii = [0.5, 0.5]. If rij , rjk , rik are ii linear uncertain variables and ri j ∼ L[riLj , rU ], r jk ∼ L[r Ljk , rUjk ], rik ∼ ij

L , rU ], when α = 0, in (11) we recover (8), when α = 1, (11) is L[rik ik

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 6

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

the same as (9). Therefore, the definition of IFPRs and their additive consistency are special cases of the definition of UPRs and their additive consistency. Based on the definition of complementarity and additive consistency of UPRs, fuzzy preference relations and their additive consistency are also special cases of UPRs. In Definition 10, let α = 1. Then, M{rij ≥ λ} ≥ 1, and we know from Axiom 1 that M{rij ≥ λ} ≤ 1 always holds. Therefore, M{ri j ≥ λ} = 1; that is, the belief degree of event rij ≥ λ is 1, which is a deterministic event. Consequently, M{ri j ≥ λ} = 1 is equivalent to rij ≥ λ, and Definition 10 is the same as Definition 6. In the same way, when α = 1, Definition 11 is the same as Definition 7. This shows that Definitions 6 and 7 are special cases of Definitions 10 and 11, and Definitions 10 and 11 are general forms of Definitions 6 and 7. 4.2. General transitivity and satisfactory transitivity of additive consistent UPRs In this section, we prove that the additive consistent UPRs R =

(ri j )n×n also have general transitivity and satisfactory transitivity.

Theorem 5. Additive consistent UPRs R = (ri j )n×n has general transitivity.

In the transitivity frameworks of reciprocal relations, for any i, j, k ∈ N, the classical stochastic transitivity was introduced by ´ (Switalski, 2001): •

•

•

•

•

The relationship between these five types of transitivity is ´ shown in Fig. 3 (Switalski, 2001). To summarize these types of transitivity, De Baets and De Meyer (2005) proposed the g-stochastic transitivity and gisostochastic transitivity. Definition 12 (De Baets & De Meyer, 2005). Let g be an increasing [ 21 , 1]2 → [0, 1] mapping such that g( 21 , 12 ) ≤ 12 . A reciprocal relation Q on A is called g-stochastic transitive if for any (a, b, c) ∈ A3 it holds that

Proof.

M{ri j ≥ λ} ≥ α ⇒ 1 − M{ri j ≤ λ} ≥ α ⇒ 1 − i j (λ ) ≥ α ⇒ i j (λ ) ≤ 1 − α ⇒ λ ≤ −1 (1 − α ) ⇒ λ ≤ 1 − −1 (α ) ij ji

Strong stochastic transitivity (S-transitivity): min(ri j , r jk ) ≥ 12 , ⇒ rik ≥max(rij , rjk ). Moderate stochastic transitivity (M-transitivity): min(ri j , r jk ) ≥ 1 2 , ⇒ rik ≥min(rij , rjk ). Weak stochastic transitivity (W-transitivity): min(ri j , r jk ) ≥ 1 1 2 , ⇒ rik ≥ 2 . α -transitivity (α ∈ [0, 1]): min(ri j , r jk ) ≥ 12 , ⇒ rik ≥ α max(ri j , r jk ) + (1 − α )min(rij , rjk ). G transitivity: rik ≥ ri j + r jk − 1.

Q (a, b) ≥

1 1 ∧ Q (b, c ) ≥ 2 2



⇒ Q (a, c ) ≥ g(Q (a, b), Q (b, c ))

When g=max, min, 12 , α max+(1 − α )min, the g-stochastic transitivity corresponds to S-transitivity, M-transitivity, W-transitivity and α -transitivity, respectively.

⇒ −1 (α ) ≤ 1 − λ. ji Similarly,

Definition 13 (De Baets & De Meyer, 2005). Let g be an increasing [ 12 , 1]2 → [0, 1] mapping such that g( 21 , 12 ) = 12 and g( 12 , 1 ) = g(1, 12 ) = 1. A reciprocal relation Q on A is called g-isostochastic transitive if for any (a, b, c) ∈ A3 it holds that

M{r jk ≥ λ} ≥ α ⇒ −1 (α ) ≤ 1 − λ. kj Adding these two expressions, we obtain

 (α ) + −1 (α ) ≤ 2 − 2λ. kj −1 ji

As the UPRs R is additively consistent,

−1 (α ) + −1 (α ) = −1 ( α ) + 0.5. ji kj ki Because −1 (α ) + −1 (1 − α ) = 1 and λ ≥ 0.5 ⇒ 2λ − 0.5 ≥ λ, we ki ik have

−1 (α ) + 0.5 ≤ 2 − 2λ ⇒ −1 ( α ) ≤ 1.5 − 2λ ki ki ⇒ 1 − −1 (1 − α ) ≤ 1.5 − 2λ ⇒ −1 ( 1 − α ) ≥ 2λ − 0.5 ik ik

Q (a, b) ≥

1 1 ∧ Q (b, c ) ≥ 2 2



⇒ Q (a, c ) = g(Q (a, b), Q (b, c ))

Theorem 6. The additive consistency of FPR, IFPR, LUPR and UPR is a S-transitivity. Proof. When min(−1 (α ), −1 (α )) ≥ ij jk

−1 (α ) jk

≥

1 2,

⇒ 1 − ik (λ ) ≥ α ⇒ 1 − M{aik ≤ λ} ≥ α ⇒ M{aik ≥ λ} ≥ α .  In Theorem 5, setting λ = 0.5 gives the following: Corollary 3. Additively consistent UPRs R = (ri j )n×n have satisfactory transitivity. 4.3. A detailed description about transitivity Regarding the transitivity framework, De Baets and De Meyer (2005), De Schuymer et al. (2005), Freson, De Baets, and De Meyer ´ (2014), Switalski (2001, 2003), De Baets, De Loof, and De Meyer (2015) proposed and extended the cycle-transitivity, FG-transitivity, stochastic transitivity and fuzzy transitivity, etc., laying a foundation for the transitivity of reciprocal relations. Based on their studies, transitivity is used to solve many practical problems, such as machine learning, prediction, statistics and decision-making, and plays an increasingly important role (Cheng, Rademaker, De Baets, & Hüllermeier, 2010; Groves & Branke, 2019; Shah, Balakrishnan, Guntuboyina, & Wainwright, 2016).

that is −1 (α ) ≥ ij

1 2

and

from (11) we can get:

−1 (α ) = −1 (α ) + −1 (α ) − ij ik jk

⇒ −1 (1 − α ) ≥ λ ⇒ 1 − α ≥ ik (λ ) ik

1 2,

1 ≥ max(−1 (α ), −1 (α )) ij jk 2

Therefore, the additive consistency of UPR and LUPR is a Stransitivity. Since UPR and its additive consistency are general forms of FPR, IFPR and hence the additive consistency of FPR and IFPR is implied to be S-transitivity.  As can be seen from Fig. 3, S-transitivity can directly deduce the other four types of transitivity, and the additive consistency of UPR can directly deduce S-transitivity, which is also inspired by Definition 13. According to Definition 13, we, by letting g(x, y ) = x + y − 0.5, the additive consistency of UPR can be obtained. Theorem 7. The general transitivity and satisfactory transitivity of IFPR and LUPR are W-transitivity (which can be directly obtained by Definitions 6, 7, 10, 11). 4.4. The adaptability of uncertainty theory compared with other theories In terms of dealing with uncertain events, there are other theories, such as Probability Theory, Possibility Theory (Dubois & Prade, 2012) and Dempster-Shafer Evidence Theory (Sentz, Ferson et al., 2002), etc. This section compares the uncertainty theory with

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

7

Table 1 The applicability and characteristics: comparing four theories. Four theories

Uncertainty Theory

Probability Theory

Possibility theory

Dempster-Shafer Evidence Theory

Scope of application

Low frequency events or small sample events that cannot fit probabilities with frequency

High frequency and large sample events that can be fitted with probability

It is the generalization of fuzzy sets and fuzzy logic: suitable for uncertain situations with fuzzy nature

Core characteristics

Product measure: Product measure: Pr{1 × 2 } = Pr{1 } × Pr{2 } M { 1 × 2 } = M { 1 } ∧ M { 2 } Independent events: Independent events: M{1 ∩ 2 }=M{1 } ∧ M{2 } Pr{1 ∩ 2 } = Pr{1 } × Pr{2 } Theorem: 1 and 2 are Pr represents probability measure independent iff M { 1 ∪ 2 } = M { 1 } ∨ M { 2 } M represents uncertainty measure

Pos{1 ∪ 2 } = Pos{1 } ∨ Pos{2 } for any events 1 and 2 , no matter if they are independent or not, Pos represents possibility measure

An extension of subjective probability bayesian theory: a mathematical approach to uncertain inference problems based on evidence and combination  Bel (A ) = B⊆A m(B ), Supermodular: Bel (A ∪ B ) + Bel (A ∩ B ) ≥ Bel (A ) + Bel (B ) Bel(A) is the belief function of focal  element A, Pl (A ) = B∩A=∅ m(B ), Submodular: Pl (A ∪ B ) + Pl (A ∩ B ) ≤ Pl (A ) + Pl (B ) Pl(A) is the plausibility function of focal element A

Reliability interpretation

Belief degree is a subjective measure of the occurrence of uncertain events

Probability is represented as an objective measure of the occurrence of events

Possibility represents the membership of an event to a set

Advantages

It is easy to obtain the measure of the simultaneous occurrence of multiple uncertain independent events

It can represent the joint Whether an event is independent It satisfies the weaker condition than Bayes probability theory and has the distribution function of multiple or not has no effect on the ability to express uncertainty directly events measure of the event union

Difficulties in application

The distribution function of subjective data is difficult to obtain and can only be approximated

Computational complexity exists in joint distribution function

It is difficult to construct membership function scientifically

Belief function and plausibility function appear simultaneously, indicates the lower and upper levels of supporting evidence

The evidence must be independent and there is a potential combinatorial explosion problem in the calculation

Fig. 3. The relationship between the five transitivity.

these theories, by identifying the similarities and differences (refer to Table 1), and further explains why the uncertainty theory is more suitable for interval preference relationship research than other theories. Since this paper studies the problem of uncertain subjective judgment, such events have the following characteristics: •

•

•

•

Due to the small sample size and insufficient data, the probability of event occurrence cannot be obtained. The occurrence of events is highly uncertain, leading to a larger belief degree variance than the traditional probability theory. The experience of experts in the field of uncertain events is of great value, and subjective judgment based on their experience and some objective data is often used in the practice. When adopting other theories, it is difficult to build models for the events and the calculation process is usually complex, and in such circumstances the uncertainty theory can be more appropriate in solving the problems.

Consequently, the uncertainty theory is adopted in this paper, including the introduction of belief degree, finding the analytic solution through the uncertain chance-constrained modeling, data discretization, reduction in calculation complexity and avoiding distortion problems when using IFPR, which also provides a further extension of the study on IFPR.

5. Optimal consensus matrix modeling for UPRs Based on the consistency constraint, in this section, we first construct two types of optimal consensus matrix models based on UPRs. 5.1. Consensus matrix model of UPRs without considering consistency constraints In a decision event, the UPRs given by DM dt , t = 1, 2, . . . , m can be represented as At = (ati j )n×n , i, j ∈ N, where ati j are uncertain variables with distribution function ijt and inverse distribution function −1 . For any belief degree β , 0 ≤ β ≤ 1, −1 (β ) + i jt i jt

−1 (1 − β ) = 1, −1 (β ) = 0.5 always hold. The consensus matrix jit iit is Ac = (aci j )n×n , i, j ∈ N, where aci j is a crisp number and aci j + acji = 1, acii = 0.5 always hold; when aci j is an uncertain variable, its distribution function is  ij and the inverse distribution function is i−1 . j

Then, i−1 (β ) +  −1 (1 − β ) = 1, ii−1 (β ) = 0.5 always hold. Let j ji m T ω = (ω1 , ω2 , . . . , ωm ) be the weight vector of the DMs, t=1 ωt = 1. We seek to minimize the deviation between the weighted agm gregation element t=1 ωt ati j of all UPRs and the consensus matrix m element aci j , that is, minimize ε ij under the condition | t=1 ωt ati j − aci j | ≤ εi j . As ati j are uncertain variables, without considering the

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 8

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

additive consistency of the UPRs, uncertainty theory implies that m εij is minimized when the belief degree of | t=1 ωt ati j − aci j | ≤ εi j is no less than α . The optimal consensus matrix model for UPRs without considering consistency constraints is as follows:

•

•



εi j  ⎧i, j∈N m  ⎪ t c ⎪ M | ωt ai j − ai j | ≤ εi j ≥ α , i, j ∈ N ⎪ ⎪ ⎨ t=1 s.t. 0 ≤ α ≤ 1, εi j ≥ 0, i, j ∈ N ⎪ ⎪ m ⎪  ⎪ ⎩ ω = 1, 0 ≤ ω ≤ 1. Min

t

t

t=1

α ) − i−1 ( α ) ≥ −ε i j . j

(12 − 1 ) (12 − 2 )

(12)

Theorem 8. The deterministic equivalent models of (12) are as follows: •

(12 − 3 )

m 

M

s.t.



m 

Simultaneously,



ω

t t ai j

t=1



⇒1−M

•

ωt −1 (α ) − aci j ≤ εi j . i jt

−



aci j

≥ −εi j

≥α



m 

ω

t t ai j

−

aci j

≤ −εi j

≥α

s.t.

ωt  (1 − α ) − −1 i jt

aci j

≥ −εi j .



ω

t t ai j

−

aci j

 ≤ εi j

≥ α ⇒ ϒ (εi j ) ≥ α ⇒ εi j ≥ ϒ −1 (α )

t=1 m 

⇒

εi j ⎧ ⎪ ωt −1 (α ) − i−1 (1 − α ) ⎪ i jt j ⎪ ⎪ t=1 ⎪ ⎪ ⎪ ⎪ ≤ εi j , i, j ∈ N ⎪ ⎪ ⎪ m ⎪ −1 −1 ⎪ ⎪ ⎨t=1 ωt i jt (1 − α ) − i j (α ) ≥ −εi j , i, j ∈ N (β ) +  −1 ( 1 − β ) = 1, ⎪i−1 j ji ⎪ ⎪ −1 ⎪  ( β ) = 0 . 5 , i, j ∈ N ⎪ ii ⎪ ⎪ ⎪ ⎪0 ≤ α ≤ 1, 0 ≤ β ≤ 1, εi j ≥ 0, i, j ∈ N ⎪ ⎪ ⎪ m ⎪  ⎪ ⎩ ω = 1, 0 ≤ ω ≤ 1. t

When aci j is an uncertain distribution, assume that the distribution function of aci j is  ij and the inverse distribution funcm tion is i−1 . Let ϒ be the joint distribution of t=1 ωt ati j − aci j , j  m −1 so that Theorem 1 implies that ϒ (α ) = t=1 ωt −1 (α ) − i jt

i−1 (1 − α ). Then, j

(13 − 3 ) (13 − 4 ) (13 − 5 )



t

t=1

t=1

M

c

i, j∈N m 

⇒ 1 − ϒ (−εi j ) ≥ α ⇒ ϒ (−εi j ) ≤ 1 − α ⇒ −εi j ≤ ϒ −1 (1 − α )

m 

c

(13 − 2 )

When aci j is an uncertain distribution,

Min

t=1

⇒

c

(13 − 1 )

(13)

t=1

m 

ai j + a ji = 1, aii = 0.5, i, j ∈ N ⎪ ⎪ ⎪ ⎪ ⎪0 ≤ α ≤ 1, εi j ≥ 0, i, j ∈ N ⎪ ⎪ ⎪ ⎪ m  ⎪ ⎪ ⎩ ωt = 1, 0 ≤ ωt ≤ 1.

ωt ati j − aci j ≤ εi j ≥ α ⇒ ϒ (εi j ) ≥ α

⇒ εi j ≥ ϒ −1 (α ) ⇒

M

t=1

t=1

t=1

m 



εi j ⎧i,mj∈N  ⎪ ⎪ ωt −1 (α ) − aci j ≤ εi j , i, j ∈ N ⎪ i jt ⎪ t=1 ⎪ ⎪ ⎪ m ⎪  ⎪ −1 c ⎪ ⎪ ⎨ ωt i jt (1 − α ) − ai j ≥ −εi j , i, j ∈ N

When aci j is a crisp number, let ϒ be the joint distribution m t c −1 (α ) = of t=1 ωt ai j − ai j . From Theorem 1, we then have ϒ m −1 c t=1 ωt i jt (α ) − ai j . Hence,



•

When aci j is a crisp number,

Min

According to Axiom (4), m m M{| t=1 ωt ati j − aci j | ≤ εi j } ≥ α ⇔ M{ t=1 ωt ati j − aci j ≤ εi j } ≥ m t c α , M{ t=1 ωt ai j − ai j ≥ −εi j } ≥ α . The following proves that the uncertain constraint (12–1) is equivalent to the deterministic nonlinear constraint when aci j is a crisp number or an uncertain distribution: •

When aci j is a crisp number, constraint (12–1) is equivalent to m m −1 −1 c c t=1 ωt i jt (α ) − ai j ≤ εi j ; t=1 ωt i jt (1 − α ) − ai j ≥ −εi j . c When ai j is an uncertain distribution, constraint (12–1) is equivm m −1 −1 −1 alent to t=1 ωt i jt (α ) − i j (1 − α ) ≤ εi j ; t=1 ωt i jt (1 −

ωt −1 (α ) − i−1 ( 1 − α ) ≤ εi j . i jt j

(14 − 1 ) (14 − 2 ) (14 − 3 ) (14 − 4 ) (14 − 5 ) (14)

Equations (13-1) and (13–2) are the range constraints of aci j , so m that the belief degree of | t=1 ωt ati j − aci j | ≤ εi j is no less than α . Therefore, ε ij also has the constraint that ε ij ≥ 0. Eq. (13-3) constrains the consensus matrix, (13–4) is the range constraint of α and ε ij , and (13–5) is the range constraint of ωt . Model (14) has a similar interpretation to (13), where (14–3) is the definition constraint of the UPRs. Under these models, when α , ωt , and ati j are known, the minimum value of ε ij and the value or distribution of aci j can be determined.

t=1

Simultaneously,



M

m 



5.2. Consensus matrix model of additive consistent UPRs

ωt ati j − aci j ≥ −εi j ≥ α

In this section, we derive additive consistency constraints based on the results in the previous section. The optimal consensus matrix model of additive consistent UPRs is constructed as follows:

t=1

⇒

m 

ωt −1 (1 − α ) − i−1 (α ) ≥ −εi j . i jt j

•

t=1

Lemma 1. The equivalent forms of constraint m M{| t=1 ωt ati j − aci j | ≤ εi j } ≥ α , are as follows:

(12–1),

When aci j is a crisp number, (7) shows that satisfying the additive consistency constraint implies that aci j + acjk = acik +

0.5, i, j, k ∈ N. Similar to (13), the consensus matrix model of

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

Theorem 9. Assuming that ξ1 , ξ2 , . . . , ξn are independent linear unn certain variables, ξi ∼ L[ui , vi ], i ∈ N, then ωi ξi is still a linear i=1  n n uncertain variable and i=1 ωi ξi ∼ L[ i=1 ωi ui , ni=1 ωi vi ].

additive consistent UPRs can be obtained as follows:



εi j ⎧ ⎪ ⎪ ωt −1 (α ) − aci j ≤ εi j , i, j ∈ N ⎪ i jt ⎪ t=1 ⎪ ⎪ ⎪ m ⎪  ⎪ −1 ⎪ ⎪ ωt i jt (1 − α ) − aci j ≥ −εi j , i, j ∈ N ⎪ ⎪ t=1 ⎨

Min

i, j∈N m 

c c c s.t. ai j + a ji = 1, aii = 0.5, i, j ∈ N

⎪ ⎪ aci j + acjk = acik + 0.5, i, j, k ∈ N ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ α ≤ 1, εi j ≥ 0, i, j ∈ N ⎪ ⎪ ⎪ ⎪ m  ⎪ ⎪ ⎩ ωt = 1, 0 ≤ ωt ≤ 1. t=1

(15 − 1 )

Proof. Assuming that the distribution function of linear uncertain variable ξ i is i , Theorem 3 implies that

(15 − 2 )

−1 (α ) = (1 − α )ui j + αvi j . i

(15 − 3 ) (15 − 4 ) (15 − 5 )

According to Theorem 1, the inverse uncertain distribution ϒ −1 (α )  of ni=1 ωi ξi is

(15 − 6 ) (15)

•

9

When aci j is an uncertain distribution, assume that the distribution function of aci j is  ij and the inverse distribution function

ϒ −1 (α ) =

n 

ωi −1 (α ) = (1 − α ) i

i=1

n 

ω i ξi ∼ L

i=1

ωi ui + α

i=1

where ϒ is the joint distribution of



n 

n 

ωi ui ,

i=1

n 

n



i=1

n 

ωi vi ,

i=1

ωi ξi . Therefore,

ωi vi .

(17)

i=1

is i−1 . If the consensus matrix is additively consistent, then j



(11) implies that the constraint i−1 (β ) +  −1 (β ) = ik−1 (β ) + j jk

0.5, i < j < k, i, j, k ∈ N must be satisfied. Similar to (14), the consensus matrix model of additive consistent UPRs can be obtained as follows:



εi j ⎧ ⎪ ⎪ ωt −1 (α ) − i−1 (1 − α ) ≤ εi j , i, j ∈ N ⎪ i jt j ⎪ t=1 ⎪ ⎪ ⎪ m  ⎪ ⎪ ωt −1 (1 − α ) − i−1 (α ) ≥ −εi j , i, j ∈ N ⎪ i jt j ⎪ ⎪ t=1 ⎪ ⎪ ⎪ ⎪ −1 (β ) +  −1 (1 − β ) = 1, ⎪ ⎨ ij ji −1 s.t. ii (β ) = 0.5, i, j ∈ N ⎪ ⎪ ⎪ i−1 (β ) +  −1 (β ) = ik−1 (β ) + 0.5, ⎪ j jk ⎪ ⎪ ⎪ i < j < k, i, j, k ∈ N ⎪ ⎪ ⎪ ⎪ ⎪ 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, εi j ≥ 0, i, j ∈ N ⎪ ⎪ ⎪ m  ⎪ ⎪ ⎩ ωt = 1, 0 ≤ ωt ≤ 1. Min

i, j∈N m 

t=1

(16 − 1 ) (16 − 2 ) (16 − 3 ) (16 − 4 )

Theorem 10. When aci j is a crisp number, there is a linear relationship between Min ε ij and α , where Min ε ij is a monotonic increasing  function of α . Objective function Min i, j∈N εi j , α , and elements aci j in consensus matrix Ac satisfy



Min

(16 − 6 )

In (15) and (16), the objective functions are the sum of the deviation ε ij between the consensus matrix and the weighted UPRs. Equations (15-1) and (15–2) are constraints of ε ij derived from (13–1) and (13–2). Eq. (15-3) constrains the consensus matrix, (15–4) is the additive consistency constraint of the consensus matrix, and (15–5) and (15–6) are derived from (13–4) and (13–5). The interpretation of (16) is similar to that of (14), where (16–3) is the definition constraint of UPRs and (16–4) is the additive consistency constraint of UPRs. In these models, when α , ωt , and ati j are known, the minimum value of ε ij and the value or distribution of aci j in the consensus matrix that satisfies additive consistency can be determined. 5.3. Further studies on the relationships between α and ε ij In this section, we provide a more in-depth study of α and ε ij without considering the additive consistency constraint. In particular, we explore the intrinsic relationship between α and ε ij . Consider the LUPR At = (ati j )n×n , i, j ∈ N, where ati j are uncertain variables, ati j ∼ L[uti j , vti j ], with distribution function ijt and inverse distribution function −1 . For any β , 0 ≤ β ≤ 1, −1 (β ) + i jt i jt

εi j =

Min

m 

   1 ωt vti j − uti j α − , 2

i, j∈N t=1

i, j∈N

εi j = Min ε ji , m  1  t ωt ui j + vti j , 2

aci j =

t=1

(16 − 5 )

(16)

−1 (1 − β ) = 1, −1 (β ) = 0.5 always hold. jit iit

5.3.1. Relationship between α and ε ij when aci j is a crisp number

1 2

α≥ . Proof. According to Theorem 9, m 



ω

t t ai j

∼L

t=1 m 

m 

ω

t t ui j ,

t=1

m 

ωv

t t ij

 ,

(18)

t=1

ωt −1 (α ) = (1 − α ) i jt

m 

t=1

ωt uti j + α

t=1

m 

ωt vti j .

(19)

t=1

According to constraints (13-1) and (13-2),

⎧m  ⎪ ⎪ (α ) − aci j ≤ εi j , i, j ∈ N ⎪ ωt −1 i jt ⎨ t=1 m 

⎪ ⎪ ⎪ ⎩

(20 − 1 ) (20)

ωt  (1 − α ) − −1 i jt

aci j

≥ −εi j , i, j ∈ N.

(20 − 2 )

t=1

Substituting (19) into (20), we have

⎧ m m   ⎪ t ⎪ ( 1 − α ) ω u + α ωt vti j − aci j ≤ εi j , i, j ∈ N ⎪ t ij ⎨ t=1

t=1 m 

m  ⎪ ⎪ ⎪ − α ωt uti j − (1 − α ) ⎩ t=1

ωt vti j + aci j ≤ εi j , i, j ∈ N.

(21 − 1 ) (21 − 2 )

t=1

(21)

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 10







m m m (1 − α ) t=1 ωt uti j + α t=1 ωt vti j − aci j ≤ −α t=1 m 1 m t t c c ωt ui j − (1 − α ) t=1 ωt vi j + ai j , that is, ai j ≥ 2 t=1 ωt (uti j + vti j ), the common solution of (21) is

(1) When

εi j ≥ −α

m 

ωt uti j − (1 − α )

m 

t=1

≥ −α

m 

=

m  t=1

ωt vti j + aci j

t=1

ωt uti j − (1 − α )

t=1

m 

ωt vti j +

t=1

m 1 ωt (uti j + vti j ) 2 t=1



1 ωt (vti j − uti j ) α − . 2 m

m

εi j ≥ ( 1 − α )

m 

ωt uti j + α

t=1

≥ (1 − α ) =

m

t=1

m

t=1

ωt vti j − aci j

m 

ω

+α

t t ui j

m 

ωv

t t ij

t=1

ωt (vti j − uti j ) α −

t=1

 1

2

εi j ≥

ωt (vti j − uti j ) α −

t=1

 1 2

only

if

aci j =

1 2

.

(24)

m

t=1

Minεi j =

 1 ωt (vti j − uti j ) α − ,

(25)

2

t=1 m 1 aci j = ωt (uti j + vti j ). 2

(26)



m 

εi j =

 1 ωt (vti j − uti j ) α − , 2

i, j∈N t=1

εi j = Min ε ji , m 1 ωt (uti j + vti j ), 2 t=1

1 2

α≥ . Proof. According to constraints (14–1) and (14–2),

⎧ m −1 −1 ⎪ ⎨ ωt i jt (α ) − i j (1 − α ) ≤ εi j , i, j ∈ N

(30 − 1 )

m ⎪ ⎩  ωt −1 (1 − α ) −  −1 (α ) ≥ −εi j , i, j ∈ N.

(30 − 2 )

t=1

t=1

i jt

ij

(30) Substituting (19) and (29) into (30), we obtain

⎧ m m   ⎪ (1 − α ) ωt uti j + α ωt vti j − α uci j ⎪ ⎪ t=1 t=1 ⎪ ⎪ ⎨ − (1 − α )vc ≤ εi j , i, j ∈ N ij

(31 − 1 )

m m   ⎪ ⎪ −α ωt uti j − (1 − α ) ωt vti j + (1 − α )uci j ⎪ ⎪ ⎪ t=1 ⎩ t=1 c + αvi j ≤ εi j , i, j ∈ N.

(31 − 2 ) (31)

t=1

(1)

Simultaneously, we have Minε ji =

m 

 

 m 1 1 ωt (vtji − utji ) α − = ωt (1 − uti j − (1 − vti j )) α − 2

t=1

=

m 

ωt (vti j − uti j ) α −

t=1

 1 2

2

t=1

= Min εi j .

uti j ), and Minεi j = Minε ji always holds. Therefore, the objective function and the consensus matrix without considering the additive consistency constraint are as follows:

Min

 i, j∈N

εi j =

m  i, j∈N t=1

ωt (v − t ij

uti j

m

 1 ) α− , 2

(27)

m

t t=1 ωt ui j

When (1 − α ) + α t=1 ωt vti j − α uci j − (1 − m m c t α )vi j ≤ −α t=1 ωt ui j − (1 − α ) t=1 ωt vti j + (1 − α )uci j + m αvci j , that is, uci j + vci j ≥ t=1 ωt (uti j + vti j ), the common solution of (31) is m 

εi j ≥ −α

These show that there is a linear relationship between Min ε ij and α , that is, Min ε ij is a monotonically increasing function of m α . When α = 12 , Minεi j = 0; when α = 1, Minεi j = 12 t=1 ωt (vti j −

(29)

Theorem 11. When aci j is an uncertain distribution, there is a linear relationship between Min ε ij and α , that is, Min ε ij is a monotonic  increasing function of α . Objective function Min i, j∈N εi j , α , and elc c ements ai j in consensus matrix A satisfy

uci j = vci j =

decision can only be accepted when the belief degree of the event is at least 12 , so we have m 

i−1 (α ) = (1 − α )uci j + αvci j . j

(23)

ωt (uti j + vti j ), (24) becomes an m equality relation. Then, aci j + acji = 12 t=1 ωt (uti j + vti j ) +   m m 1 1 t t t t t t c t=1 ωt (u ji + v ji ) = 2 t=1 ωt (ui j + v ji + u ji + vi j ) = 1, aii = 2 1 m t t t=1 ωt (uii + vii ) = 0.5, which satisfy constraint (13–3). 2 As vti j ≥ uti j and the deviation ε ij should be positive, that is, εij ≥ 0, α ≥ 12 is reasonable. This is consistent with the fact that the and

5.3.2. Relationship between α and ε ij when aci j is an uncertain distribution Assuming that aci j also obeys the linear uncertain distribution, that is, aci j ∼ L[uci j , vci j ], its distribution function is  ij and the in-

Min

t=1

From (22) and (23), we can see that when aci j is a crisp number, regardless of its value, m 



i, j∈N

m 1 − ωt (uti j + vti j ) 2

.

(28)

t=1

Min

t=1

t=1 m 

m 

m 1 ωt (uti j + vti j ). 2

aci j =

verse distribution function is i−1 . Then, Theorem 3 implies that j

(22)

(1 − α ) t=1 ωt uti j + α t=1 ωt vti j − aci j ≥ −α m t ωt ui j − (1 − α ) t=1 ωt vti j + aci j , that is, aci j ≤ 12 t t ωt (ui j + vi j ), the common solution of (21) is

(2) When

If

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

 =α

t=1 c ij

v − 

≥α

ωt uti j − (1 − α ) m 

ω

m 

ωt vti j + (1 − α )uci j + αvci j

t=1





+ (1 − α )

t t ui j

t=1

m 



uci j

−



ωt vti j − uci j + (1 − α ) uci j −

t=1

= ( 2α − 1 )



m 

 ωv − t t ij

uci j

.

m 

 ωv

t t ij

t=1 m 

 ωt vti j

t=1

(32)

t=1

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

m m (1 − α ) t=1 ωt uti j + α t=1 ωt vti j − α uci j − (1 −   m m c t t α )vi j ≥ −α t=1 ωt ui j − (1 − α ) t=1 ωt vi j + (1 − α )uci j + m αvci j , that is, uci j + vci j ≤ t=1 ωt (uti j + vti j ), the common solution of (31) is

(2) When

m 

m 

εi j ≥ (1 − α ) ωt uti j + α ωt vti j − α uci j − (1 − α )vci j t=1 t=1     m m   =α ωt vti j − uci j + (1 − α ) ωt uti j − vci j t=1 t=1     (33) m m   ≥ α vci j − ωt uti j + (1 − α ) ωt uti j − vci j t=1 t=1   m  = (2α − 1 ) vci j − ωt uti j . t=1

m If and only if uci j + vci j = t=1 ωt (uti j + vti j ), both (32) and (33) retain the “ ≥ ” operator. Then,



m 

εi j ≥ ( 2α − 1 )



ωv − t t ij



= ( 2α − 1 )

uci j

c ij

v −

t=1



m 

2εi j ≥ ( 2α − 1 )

ωv − t t ij

uci j

c ij

+v −

t=1

m 

 ω

t t ui j

m 

5.3.3. Additive consistency based on the relationship between α and ε ij In the previous section, Theorem 10 was derived from (13) and Theorem 11 was derived from (14). Neither of these theorems takes into account the additive consistency constraint of consensus matrix aci j , but this constraint exists in (15) and (16). Therefore, it is necessary to study the overall relationship between them to solve these models. In this section, we study the necessary conditions when Theorems 10 and 11 hold and aci j satisfies additive consistency. (1) When aci j is a crisp number, according to Theorem 10, m we can see that when aci j = 12 t=1 ωt (uti j + vti j ), Minεi j = m 1 t t c t=1 ωt (vi j − ui j )(α − 2 ) can be obtained. If ai j must satisfy the additive consistency constraint aci j + acjk = acik + 0.5 and Theorem 10 holds, then



ω

t t ui j

m m 1 1 ωt (uti j + vti j ) + ωt (utjk + vtjk ) 2 2

,

t=1

t=1

.

t=1

that is,

From the definition of the linear uncertain distribution, vci j ≥ uci j and the belief degree α of an event should be no less than be generally acceptable. Thus, 2α − 1 ≥ 0. Therefore, m 

2εi j ≥ ( 2α − 1 )

ωt vti j − uci j + uci j −

t=1

εi j ≥

m 

m 

1 2

m 

to



(2)

t=1

m t t If and only if = = t=1 ωt (ui j + vi j ), this inequality becomes an equality relation. In this case, the variables aci j that obey the linear uncertain distribution, that is, aci j ∼ L[uci j , vci j ], have uci j = vci j , and aci j become crisp numbers. Similarly, Ac = (aci j )n×n satisfies constraint (14–3). As vti j ≥ uti j and the deviation ε ij should also be positive, that is, 1 2

vci j

of the event must be at least have

Min

εi j =

m 

uci j

=v

c ij

1 = 2

m 

t=1

ω(

+ v ). t ij

(35)

i, j∈N

uci j

=v

c ij

m 

εi j =

t=1

m 

ωt (uti j + vti j ) +

t=1

ωt (v − t ij

i, j∈N t=1

m 1 = ωt (uti j + vti j ). 2

uti j



1 ) α− , 2

(36)

t=1



m 

ωt (utjk + vtjk ) =

t=1

m 

ωt (utik + vtik ) + 1.

t=1

(39) From (38) and (39), we have the following. Theorem 12. When aci j is a crisp number and Theorem 10 holds, or when aci j is a linear uncertain distribution and Theorem 11 holds, if aci j satisfies the additive consistency constraint, then UPRs At =

(ati j )n×n , i, j ∈ N must satisfy

m 

(37)

t=1

that is,

t=1

Hence, there is a linear relationship between Min ε ij and α , that is, Min ε ij is a monotonically increasing function of α . When m α = 12 , Minεi j = 0; when α = 1, Minεi j = 12 t=1 ωt (vti j − uti j ), and Minεi j = Minε ji always holds. Therefore, the objective function and the consensus matrix without considering the additive consistency constraint are as follows:

Min

t=1

m m 1 1 = (1 − β ) · ωt (utik + vtik ) + β · ωt (utik + vtik ) + 0.5, 2 2

εi j = Min ε ji .



m m 1 1 ωt (uti j + vti j ) + β · ωt (uti j + vti j ) 2 2

t=1

t t ui j

ωt (utik + vtik ) + 1.

m m 1 1 + (1 − β ) · ωt (utjk + vtjk ) + β · ωt (utjk + vtjk ) 2 2

Because (34) is the same as (25), we can write

Min

m  t=1

t=1

(34)

2

t=1

t=1

(1 − β ) ·

to be accepted by people, so we

 1 ωt (vti j − uti j ) α − ,

ωt (utjk + vtjk ) =

When aci j obeys the linear uncertain distribution, that is, aci j ∼ L[uci j , vci j ], its distribution function is  ij and the inverse distribution function is i−1 . According to Theorem 11, j m c we can see that when ui j = vci j = 12 t=1 ωt (uti j + vti j ), m 1 t t Minεi j = t=1 ωt (vi j − ui j )(α − 2 ) can be obtained. If aci j must satisfy the additive consistency constraint i−1 (β ) + j −1  −1 ( β ) =  ( β ) + 0 . 5 and Theorem 11 holds, then jk ik

εij ≥ 0, α ≥ 12 , this is consistent with the fact that the belief degree 1 2

m 

(38)

ωt uti j ,

2

uci j

ωt (uti j + vti j ) +

t=1

 1 ωt (vti j − uti j ) α − .

t=1

t=1

m 1 = ωt (utik + vtik ) + 0.5, 2

t=1



11

ωt (uti j + vti j ) +

t=1

Because

m 

ωt (utjk + vtjk ) =

t=1

m

t=1

ωt = 1 and

m 

ωt (utik + vtik ) + 1.

t=1

utj j

=

vtj j

= 0.5,

m

t=1

ωt (utj j + vtj j ) = 1.

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 12

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx Table 2 Definition comparison of 4 preference relations. Preference relations

Element

The relations of symmetric elements

Diagonal elements

Additive consistency

UPR

rij ∼ ij

−1 (α ) + −1 (1 − α ) = 1 ij ji

−1 ( α ) = 0.5 ii

−1 (α ) + −1 (α ) = −1 ( α ) + 0.5, i < j < k ij jk ik

 (α ) +  (1 − α ) = 1

 ( α ) = 0.5

−1 (α ) + −1 (α ) = −1 ( α ) + 0.5, i < j < k ij jk ik

ri j ∼

LUPR

−1 ij

L[riLj , rUi j ]

−1 ji

−1 ii

FPR

rij

ri j + r ji = 1

rii = 0.5

ri j + r jk = rik + 0.5

IFPR

ri j = [riLj , rUi j ]

riLj + rUji = rUi j + r Lji = 1

riiL = rUii = 0.5

L riLj + r Ljk = rik + 0.5, rUi j + rUjk = rUik + 0.5, i < j < k

6. The Relationship between UPR, LUPR, FPR and IFPR 6.1. Definitions comparison Comparisons between definitions of UPR, LUPR, FPR and IFPR, and comparisons between their additive consistency are in Table 2. According to Table 2, the following can be concluded: •

•

•

LUPR is a special form of UPR, which is obtained by letting the uncertain variable obey a linear uncertainty distribution. Moreover, UPR can handle other uncertain distributions and traverse all the values of the uncertain variable by adjusting the value of α to satisfy the global complementarity. The definitions of LUPR and IFPR are essentially the same, but LUPR, along with UPR, solves the drawbacks in the definition of IFPR: IFPR only considers the global complementarity of interval endpoint values, but the global complementarity of the corresponding values in the interval is not taken care. In fact, when the global complementarity is defined by the endpoints of the interval, only when the interval is linearly distributed the complementarity of the interval as a whole can be ensured. In summary, the definitions of LUPR and IFPR are the same, but in LUPR the conditions that IFPR should satisfy are identified and included in the definition. FPR is a special form of IFPR, and when the upper and lower limits of the interval in IFPR (respectively, LUPR) are equal, IFPR (respectively, LUPR) is transformed into FPR.

Theorem 13. If IFPR satisfies the global complementarity, the interval must be linearly distributed, and, under the same condition, the definitions of IFPR and LUPR as well as their additive consistency are equivalent. When the upper and lower limits of the interval in IFPR (respectively, in LUPR) are equal, IFPR (respectively, LUPR) is transformed into FPR. Proof. When ri j ∼ j ∈ N, if −1 (α ) + −1 (1 − α ) = ij ji −1 1, ii (α ) = 0.5 hold for any α ∈ [0, 1], that is: (1 − α )riLj + α rUi j + α rLji + (1 − α )rUji = 1, (1 − α )riiL + α rUii = 0.5 always hold for L[riLj , rU ], i, ij

any α ∈ [0, 1], then:

⎧L ri j + rUji = 1 ⎪ ⎪ ⎪ ⎨−rL + rU + rL − rU = 0 ij ji ij ji L ⎪ r = 0 . 5 ⎪ ii ⎪ ⎩ L U −rii + rii = 0

⇒

⎧L ri j + rUji = 1 ⎪ ⎪ ⎪ ⎨rU + rL = 1 ij

ji

⎪ riiL = 0.5 ⎪ ⎪ ⎩U

 ⇒

L rUi j − riLj + rUjk − r Ljk + rik − rUik = 0 L riLj + r Ljk − rik − 0.5 = 0

 ⇒

L riLj + r Ljk = rik + 0.5

rUi j + rUjk = rUik + 0.5

,

this is consistent with the definition of additive consistency of IFPR. When riLj = rU , that is ri j = riLj = rU , then: ij ij

ri j + r jk = rik + 0.5, this is consistent with the definition of additive consistency of FPR.  In conclusion, the inclusion relation and transformation method of UPR, LUPR, FPR and IFPR are shown in Figs. 4 and 5. The advantages and disadvantages of UPR, LUPR, FPR, and IFPR are shown in Table 3: 6.2. Models comparison

,

rii = 0.5

this is consistent with the definition of IFPR. When riLj = rU , that is ri j = riLj = rU , then: ij ij

ri j + r ji = 1,

Fig. 4. The inclusion relation.

rii = 0.5,

this is consistent with the definition of FPR. If −1 (α ) + −1 (α ) = −1 (α ) + 0.5 holds for any α ∈ [0, 1], ij jk ik that is:

(1 − α )riLj + α rUi j + (1 − α )rLjk + α rUjk = (1 − α )rikL + α rUik + 0.5 L L ⇒ α (rUi j − riLj + rUjk − r Ljk + rik − rUik ) + riLj + r Ljk − rik − 0.5 = 0

If a¯ = [a− , a+ ] and b¯ = [b− , b+ ] are both nonnegative intervals, from Minkowski operations on intervals, we can derive the following: (i) (ii) (iii) (iv) (v)

a¯  b¯ = [a− + b− , a+ + b+ ]; a¯  b¯ = [a− − b+ , a+ − b− ]; a¯  b¯ = [a− b− , a+ b+ ]; a¯ /b¯ = [a− /b+ , a+ /b− ]; λa¯ = [λa− , λa+ ], λ ≥ 0.

If in GDM, the IFPRs given by DM dt , t = 1, 2, . . . , m can be represented as Rt = (rit j )n×n , i, j ∈ N, where rit j are interval values,

rit j = [rit− , rit+ ]. The consensus IFPR is Rc = (ricj )n×n , i, j ∈ N, where j j ricj are interval values, ricj = [ric− , ric+ ] and ric− + r c+ = 1, ric+ + r c− = j j ji j ji j

1, riic− = riic+ = 0.5 always hold. Let ω = (ω1 , ω2 , . . . , ωm )T be the m weight vector of the DMs, t=1 ωt = 1. In order to minimize the

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

13

Table 3 Advantages and disadvantages of the four preference relations Preference relations

Is the interval case considered? √ √

UPR LUPR FPR IFPR √

Is the interval distribution considered? √ √

× √

− ×

Is to be used as a whole in the operation? √ √ √

Can it achieve the global complementarity? √ √ √

×

×

Is it effective when the interval is linearly distributed? √ √ − √

Can it be adapted to all distributions? √ √ × ×

: represents yes or can; × : represents no or can’t; − : represents not involved

Fig. 5. The transformation method.

gap between DMs’ opinions and consensus opinions, and maximize the degree of consensus, the sum of deviations between the matrix formed by weighted aggregation of DMs’ IFPRs and the corresponding position interval values in consensus IFPR is required to be minimized.

Theorem 14. Let the matrix generated by the weighted aggregation be R˜, R˜ = (r˜i j )n×n , r˜i j = [r˜i−j , r˜i+j ], i, j ∈ N, then R˜ is an IFPR.

considering consistency constraints is as follows:

Min



di j

i, j∈N

⎧ di j = |r˜i−j − ric− | + |r˜i+j − ric+j |, i, j ∈ N ⎪ j ⎪ ⎪ ⎪ m m   ⎪ − t− + t+ ⎪ ⎪ ⎪r˜i j = ωt ri j , r˜i j = ωt ri j , i, j ∈ N ⎪ t=1 t=1 ⎨

s.t. ric− + r c+ = 1, ric+ + r c− = 1, j ji ji j

⎪ ⎪ ⎪ r c− = riic+ = 0.5, i, j ∈ N ⎪ ⎪ ii ⎪ ⎪ m ⎪  ⎪ ⎩ ωt = 1, 0 ≤ ωt ≤ 1

Proof. According to the algorithm of the interval number, we m m derive r˜i−j = t=1 ωt rit−j , r˜i+j = t=1 ωt rit+j . Because Rt , t = 1, 2, . . . , m

t=1

are IFPRs, rit− + rt+ = 1, rit+ + rt− = 1, riit− = riit+ = 0.5, then: ji j j ji

r˜i−j + r˜+ji = r˜i+j + r˜−ji =

m 

ωt rt+ = ji

m 

t=1

t=1

t=1

m 

m 

m 

t=1

r˜ii− =

ωt rit−j +

m 

m 

ωt rit+j +

ωt rt− = ji

t=1

ωt riit− = 0.5, r˜ii+ =

t=1

ωt (rit−j + rt+ ) = 1, ji

Min

ωt riit+ = 0.5.

t=1

In GDM, to maximize the degree of consensus, the matrix obtained after weighted aggregation should be modified to a minimum degree. Therefore, the optimal consensus IFPR without



(40 − 4 )

di j

⎧i, j∈N − c− di j = |r˜i j − ri j | + |r˜i+j − ric+ |, i, j ∈ N ⎪ j ⎪ ⎪ ⎪ m m   ⎪ ⎪ ⎪r˜i−j = ωt rit−j , r˜i+j = ωt rit+j , i, j ∈ N ⎪ ⎪ t=1 t=1 ⎪ ⎪ ⎪ c+ c− ⎪ = 1 , r + r = 1, riic− = riic+ = 0.5, ⎨ric−j + rc+ ji ji ij

ωt (rit+j + rt− ) = 1, ji



(40 − 3 )

The optimal consensus IFPR considering consistency constraints is as follows:

s.t. Therefore, R˜ is an IFPR.

(40 − 2 )

(40)

t=1 m 

(40 − 1 )

i, j ∈ N

⎪ ⎪ c− c+ ⎪ ric− + r c− = rik + 0.5, ric+ + r c+ = rik + 0.5, ⎪ j j jk jk ⎪ ⎪ ⎪ i < j < k, i, j, k ∈ N ⎪ ⎪ ⎪ m  ⎪ ⎪ ⎪ ⎩ ωt = 1, 0 ≤ ωt ≤ 1. t=1

(41 − 1 ) (41 − 2 ) (41 − 3 ) (41 − 4 ) (41 − 5 ) (41)

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 14

According to (40) and (41), the optimal consensus IFPR satisfying additive consistency and without additive consistency can be obtained respectively. In Section 7, the optimal consensus IFPR and LUPR are obtained by using IFPR and LUPR methods respectively, and the results are compared to explore the advantages and disadvantages of the two methods. 7. Industry meteorological sensitivity analysis 7.1. Problem description Industry meteorological sensitivity is a representation of the degree of change in the national economy affected by weatherrelated factors. Scientific analysis on the correlation and sensitivity between meteorological conditions and industrial economic development is helpful for improving the pertinence and efficiency of meteorological services. This work is widely valued by national meteorological departments in China and the United States. Since the 1960s, the United States has carried out industry meteorological economic and social benefit assessment for national decisionmaking input. The study of industry meteorological sensitivity evaluation also reflects one of the basic works of industry meteorological service benefit evaluation for the China Meteorological Administration. However, industry sensitivity evaluation is frequently limited by the available data. In addition to conventional quantitative analysis methods, field experts are assisted to qualitatively give, using their experience, the influence degree on an industry by meteorological factors. The China Meteorological Administration wants to know more accurately the impact of short-term weather changes on the industry through the evaluation of meteorological benefits, so as to guide the local governments to scientifically adjust the industrial structure and optimize the resource pattern (Guo, Yao, & Lv, 2015). In a meteorological sensitivity assessment of the industry, due to the difficulties in the objective data processing of four industries, transportation, wholesale and retail catering, electricity and gas, and public facilities management (respectively labeled x1 , x2 , x3 , x4 ), the quantitative assessment results cannot be reasonably provided. The local meteorological bureau invited some meteorologists, industry experts and economists to form an expert group to comprehensively analyze the sensitivity of these four industries by constructing LUPRs through pairwise comparison of industries. The industry meteorological sensitivity ranking can be modeled as a hierarchy, as shown in Fig. 6.

A1 = ([u1i j , v1i j ] )4×4

⎛

[0 . 5 , 0 . 5 ]

⎜ [0.15, 0.38] =⎝ [0.44, 0.58] [0.35, 0.39]

[0.62, 0.85] [0 . 5 , 0 . 5 ] [0.51, 0.72] [0.58, 0.64]

[0.42, 0.56] [0.28, 0.49] [0 . 5 , 0 . 5 ] [0.35, 0.62]

A2 = ([u2i j , v2i j ] )4×4

⎛

[0 . 5 , 0 . 5 ] ⎜[0.27, 0.46] =⎝ [0.42, 0.6] [0.3, 0.38]

[0.54, 0.73] [0.5, 0.5] [0.46, 0.62] [0.36, 0.8]

[0.4, 0.58] [0.38, 0.54] [0 . 5 , 0 . 5 ] [0.28, 0.46]

A3 = ([u3i j , v3i j ] )4×4

⎛

[0 . 5 , 0 . 5 ]

⎜[0.68, 0.82] =⎝ [0.22, 0.54] [0.46, 0.48]

At

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

(ati j )4×4 , t

[0.18, 0.32] [0 . 5 , 0 . 5 ] [0.55, 0.76] [0.66, 0.72]

[0.46, 0.78] [0.24, 0.45] [0 . 5 , 0 . 5 ] [0.38, 0.6] A1 ,

A2 ,

A3

⎞

[0.61, 0.65] [0.36, 0.42]⎟ , [0.38, 0.65]⎠ [0 . 5 , 0 . 5 ]

⎞

[0.62, 0.7] [0.2, 0.64] ⎟ , [0.54, 0.72]⎠ [0 . 5 , 0 . 5 ]

⎞

[0.52, 0.54] [0.28, 0.34]⎟ . [0.4, 0.62] ⎠ [0 . 5 , 0 . 5 ]

= = 1, 2, 3, i, j = 1, 2, 3, 4. are the LUPRs given by meteorologists, industry experts and economists respectively for transportation, wholesale and retail catering, electricity

and gas, and public facilities management. ati j indicates the degree to which the t-th expert considers alternative xi to be superior to alternative xj . ati j is an uncertain variable and obeys a linear uncertain distribution, that is, ati j ∼ L[uti j , vti j ], where uti j and vti j are the lower and upper limits of the distribution, respectively. uti j and vti j come from subjective judgments made by experts based on their experience and objective data. Due to the rich experience of industry experts in different industries, the China Meteorological Administration pays more attention to the opinions of industry experts, so the weight of these three types of experts is ω1 = 0.3, ω2 = 0.4, ω3 = 0.3. After the discussion of the participating experts and referring to other types of chance-constrained models, it is considered that the belief degree is no less than 0.8, that is α = 0.8 is more appropriate. To obtain consensus matrix Ac with the smallest deviation from the three experts’ LUPRs, we apply the models established in Section 5. 7.2. Analysis of results (1) When the consensus opinion aci j is a crisp number, (13) can be adopted to solve the problem, so the belief degree of m ω at − ac | ≤ ε |t=1 t ij i j is no less than α . Using Lingo 15.0 ij software to solve the problem, we obtained the following results:

⎛

Ac = (aci j )4×4

0.5495 0.5 0.597 0.622

0.529 0.403 0.5 0.4405

0.667 0.378 ⎟ , 0.5595⎠ 0.5

0 ⎜0.0561 =⎝ 0.063 0.048

0.0561 0 0.057 0.0636

0.063 0.057 0 0.0657

0.048 0.0636⎟ , 0.0657⎠ 0

⎛

Min (εi j )4×4

Min



⎞

0.5 ⎜0.4505 =⎝ 0.471 0.333

⎞

εi j = 0.7068.

i, j∈N

Clearly, Minεi j = Minε ji . Simultaneously, the values of the obtained consensus matrix aci j and Min ε ij are verified, and the following results are satisfied:

Min



εi j =

εi j =

 1 ωt (vti j − uti j ) α − ,

m 

 1 ωt (vti j − uti j ) α − , 2

t=1

aci j =

2

i, j∈N t=1

i, j∈N

Min

m 

m 1 ωt (uti j + vti j ). 2 t=1

This proves the correctness of Theorem 10, but the consenm sus matrix is not additively consistent because t=1 ωt (uti j +   m m t t t t t vi j ) + t=1 ωt (u jk + v jk ) = t=1 ωt (uik + vik ) + 1. This also proves the correctness of Theorem 12. If the consensus opinion aci j must have additive consistency, (15) is used to solve the problem. The following results are obtained:

⎛

Ac = (aci j )4×4

0.626 0.5 0.597 0.5375

0.529 0.403 0.5 0.4405

0.5885 0.4625⎟ , 0.5595⎠ 0.5

0 ⎜0.1326 =⎝ 0.063 0.0385

0.1326 0 0.057 0.1481

0.063 0.057 0 0.0657

0.0385 0.1481⎟ , 0.0657⎠ 0

⎛

Min (εi j )4×4

⎞

0.5 ⎜ 0.374 =⎝ 0.471 0.4115

⎞

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

15

Fig. 6. Industry meteorological sensitivity ranking hierarchy chart.

Min



εi j = 1.0098.

Min

i, j∈N

ωt (uti j + vti j ) +

t=1

m 

ωt (utjk + vtjk ) =

t=1

m 

uci j = vci j =

ωt (utik + vtik ) + 1.

m ω at − ac | ≤ ε is no problem, so the belief degree of |t=1 t ij ij ij less than α . Using Lingo 15.0 software to solve the problem, the following results can be obtained:

Ac = (aci j )4×4

⎛

Min(εi j )4×4

Min



[0.5495, 0.5495] [0.5, 0.5] [0.597, 0.597] [0.622, 0.622]

0.0561 0 0.057 0.0636

0 ⎜0.0561 =⎝ 0.063 0.048

0.063 0.057 0 0.0657

[0.529, 0.529] [0.403, 0.403] [0.5, 0.5] [0.4405, 0.4405]

⎞

0.048 0.0636⎟ , 0.0657⎠ 0

εi j = 0.7068.

i, j∈N

Again, Minεi j = Minε ji . Simultaneously, the values of the obtained consensus matrix aci j and Min ε ij are verified, and the following results are satisfied:

Min

 i, j∈N

εi j =

m  i, j∈N t=1

 1 ωt (vti j − uti j ) α − , 2

2

m 1 ωt (uti j + vti j ). 2

This proves the correctness of Theorem 11, but the consenm sus matrix is not additively consistent because t=1 ωt (uti j +   m m vti j ) + t=1 ωt (utjk + vtjk ) = t=1 ωt (utik + vtik ) + 1. This proves the correctness of Theorem 12. If the consensus opinion aci j must satisfy additive consistency, then (16) can be used to solve the problem. The result is uci j = vci j in aci j ∼ L[uci j , vci j ], which is consistent with the result that additive consistency is required when aci j is a crisp  number, and Min i, j∈N εi j is not affected by α changes.

In this case, consensus matrix Ac still pursues additive consistency, which results in (25) and (26) becoming inapplicable. However, the solution at this time is still meaningful, giving the optimal solution when the consensus matrix has  additive consistency and i, j∈N εi j is minimized. (2) When the consensus opinion aci j is an uncertain distribution, that is, aci j ∼ L[uci j , vci j ], (14) can be adopted to solve the

[0.5, 0.5] ⎜[0.4505, 0.4505] =⎝ [0.471, 0.471] [0.333, 0.333]

 1 ωt (vti j − uti j ) α − ,

t=1

t=1

⎛

m  t=1

The reason for the inconsistency with the previous result is that the UPRs A1 , A2 , A3 are not satisfied: m 

εi j =

⎞

[0.667, 0.667] [0.378, 0.378] ⎟ , [0.5595, 0.5595]⎠ [0.5, 0.5]

According to the consensus matrix results with additive consistency, the industry meteorological sensitivity ranking, from high to low, identified is transportation, electricity and gas, public facilities management and wholesale and retail catering. Based on this result, government departments formulate different policies according to the meteorological sensitivity of different industries, make reasonable adjustments to industries, and maximize the use of resources. The transportation industry, electricity and gas industry and other basic industries develop reasonable measures to reduce their exposure to meteorological disasters. As the value of α changes, the relationship between Min ε ij and α is shown in Fig. 7 for the case in which the consensus matrix is not required to satisfy additive consistency and when aci j

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

ARTICLE IN PRESS

JID: EOR 16

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

⎛

Min(di j )4×4

Min



0.146 0 0.021 0.01

0 ⎜0.146 =⎝ 0 0.213

⎞

0.213 0.01 ⎟ , 0.166⎠ 0

0 0.021 0 0.166

di j = 1.112.

i, j∈N

Fig. 7. Relationship between Min ε ij and α .

When the additive consistency is not considered in Rc , Rc is equal  to R˜, thus Min i, j∈N di j = 0. When the additive consistency is con sidered in Rc , from the results we know: Min i, j∈N di j = 1.112, and the industry meteorological sensitivity ranking is, from high to low, transportation, public facilities management, electricity and gas, and wholesale and retail catering. If LUPR is adopted and the ad ditive consistency is considered, then Min i, j∈N εi j = 1.0098 (see Section 7.2), and the industry meteorological sensitivity ranking is transportation, electricity and gas, public facilities management and wholesale and retail catering. The LUPR method can be used to obtain results with smaller deviations.

is a crisp number or an uncertain distribution. From Fig. 7, we can directly conclude that there is a linear relationship between Min ε ij and α , and that Min ε ij is a monotonic increasing function of α . When α ≥ 12 , Min ε ij is meaningful; when α = 1, Minεi j = 1 m t t t=1 ωt (vi j − ui j ). Fig. 7 also directly illustrates the correctness 2 of Theorems 10 and 11. As the LUPRs given by experts are often m m not satisfied in accordance with t=1 ωt (uti j + vti j ) + t=1 ωt (utjk + m t t t v jk ) = t=1 ωt (uik + vik ) + 1, when the consensus matrix does not require additive consistency, the problem can be directly solved using Theorems 10 and 11. If additive consistency is required, then the results will be slightly different; although, if such deviations are allowed in actual problems, Theorems 10 and 11 can still be m used. If the LUPRs given by the experts satisfy t=1 ωt (uti j + vti j ) + m  m t t t t t=1 ωt (u jk + v jk ) = t=1 ωt (uik + vik ) + 1, then Theorems 10 and 11 can be directly used, and the consensus matrix will also have additive consistency.

7.3.2. Comparison of interval distance In order to avoid the difference in results caused by the difference between models, we will employ the calculation formula of interval distance in fuzzy sets to compare LUPR and IFPR. In the theory of fuzzy sets, the measurement of interval distance has always been an important topic in decision-making. When the decision opinion or consensus opinion is an interval value, the calculation of the interval distance is particularly important. The smaller the distance between the decision and consensus intervals, obviously the higher the degree of consensus. Therefore, In GDM, the distance between decision interval and consensus interval should be minimized and the degree of consensus should be maximized. For intervals a¯ = [a− , a+ ] and b¯ = [b− , b+ ], Tran and Duckstein (2002) defined the following interval distance formula by considering the difference of each point in the two interval numbers: d (a¯ , b¯ )

7.3. Results comparison

  ! ! a− + a+ b− + b+ 2 1 a+ − a− 2  b+ − b− 2 " = − + + 2

7.3.1. Comparisons of the results from different models If A1 , A2 , A3 are IFPRs, under the same data, we obtain consensus IFPR with minimal deviation by using (40) and (41) respectively: (1) When the consensus IFPR Rc = (ricj )n×n does not consider additive consistency, the result is: Rc = ([ric− , ric+ ] )4×4 j j

⎛

[0.5, 0.5] ⎜[0.357, 0.544] =⎝ [0.366, 0.576] [0.363, 0.413]

Min



[0.456, 0.643] [0.5, 0.5] [0.502, 0.692] [0.516, 0.728]

[0.424, 0.634] [0.308, 0.498] [0.5, 0.5] [0.331, 0.55]

⎞

[0.587, 0.637] [0.272, 0.484]⎟ , [0.45, 0.669] ⎠ [0.5, 0.5]

di j = 0,

2

3

2

2

(42) According to (42), we have (1 ) d (a¯ , b¯ ) ≥ 0; (2 ) d (a¯ , b¯ ) = d (b¯ , a¯ ); (3) If d (a¯ , b¯ ) = 0, then a¯ = b¯ . Therefore, the interval distance measure is effective. The consensus matrices Ac and Rc which satisfy the additive consistency, obtained by LUPR (Model (16)) and IFPR (Model (41)) respectively, and the decision matrix R˜ obtained by weighted aggregation through IFPR method are used to measure the interval distance. The results are as follows:

d (Ac , R˜ ) = 0.81,

d (Rc , R˜ ) = 1.15.

The results also show that the LUPR method can be used to achieve consensus results with smaller deviation from decision-making opinions. Therefore, LUPR is not only better defined than IFPR, but also better in terms of results.

i, j∈N

(2) When the consensus IFPR Rc = (ricj )n×n involves additive consistency, the result is: Rc = ([ric− , ric+ ] )4×4 j j

⎛

[0.5, 0.5] ⎜[0.357, 0.398] =⎝ [0.366, 0.576] [0.363, 0.626]

[0.602, 0.643] [0.5, 0.5] [0.509, 0.678] [0.506, 0.728]

[0.424, 0.634] [0.322, 0.491] [0.5, 0.5] [0.497, 0.55]

⎞

[0.374, 0.637] [0.272, 0.494]⎟ , [0.45, 0.503] ⎠ [0.5, 0.5]

8. Conclusions Uncertainty theory uses uncertain variables based on a belief degree to fit a distribution function to decision-making judgment values, allowing the judgment elements to be considered as a whole in the uncertain operation. This method overcomes the shortcomings of the traditional IFPR, that is, it only considers the endpoints of intervals, neglecting the inner information, and it cannot ensure complementarity and consistency of entire intervals. Additionally, we proved that both the traditional consistency and

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

JID: EOR

ARTICLE IN PRESS Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

consensus models of IFPR are special cases of the model developed in this study. Based on the uncertain distribution of DMs’ judgments, we have defined the UPRs and their additive consistency, and also explored their main properties. We constructed the minimum deviation consensus matrix model for a linear uncertain distribution and derived the optimal conditions for consensus by analyzing the relationship between the minimum deviation and belief degree. In addition, the LUPR model proposed in this paper is applied to the evaluation of industrial meteorological sensitivity in a certain area of China where experts are assisted in decision-making, and it is concluded that the LUPRs model can ensure consistent results with smaller deviations. To summarize, the main contributions of this research are as follows: (1) A linear uncertain distribution was used to fit the DMs’ interval judgments, and the entire interval judgment value was considered to participate in the operations of uncertain variables. (2) Based on the belief degree of experts, the complementarity and consistency of UPRs were constructed, and their general transitivity and satisfactory transitivity were studied. (3) Based on the consistency constraints of the linear uncertain distribution, two types of optimal consensus matrix models for IFPRs with the minimum deviation were constructed. (4) We proved that the minimum deviation is a linear increasing function of the belief degree, and derived the optimality conditions for the optimal consensus matrix. In this study, we only discussed the case in which the DMs’ judgments obey the linear uncertain distribution. In future research, we will investigate the consistency of preference relations that obey the normal uncertain distribution or multiple uncertain distributions and the simulation of group uncertain preference distributions. At the same time, based on the transitivity frameworks ´ of reciprocal relations proposed by De Baets, Switalski, etc. we will further study the cycle-transitivity and FG-transitivity in UPR, expand and generalize the consistency and transitivity theory, such as the multiplicative consistency and geometric consistency of UPR, and use these properties to complement the incomplete UPR. Acknowledgments This research is partially supported by the National Natural Science Foundation of China (71971121, 71571104), NUISTUoR International Research Institute, the Major Project Plan of Philosophy and Social Sciences Research in Jiangsu Universities (2018SJZDA038), the 2019 Jiangsu Province Policy Guidance Program (Soft Science Research) (BR2019064), and the Spanish Ministry of Economy and Competitiveness with FEDER funds (Grant number TIN2016-75850-R). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ejor.2019.10.035. References Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and Systems, 20(1), 87–96. Barrenechea, E., Fernandez, J., Pagola, M., Chiclana, F., & Bustince, H. (2014). Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. application to decision making. Knowledge-Based Systems, 58, 33–44. Ben-Arieh, D., & Chen, Z. (2006). Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 36(3), 558–568. Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J., Xu, Z., Bedregal, B., . . . De Baets, B. (2015). A historical account of types of fuzzy sets and their relationships. IEEE Transactions on Fuzzy Systems, 24(1), 179–194.

[m5G;November 15, 2019;16:41] 17

Cabrerizo, F. J., Al-Hmouz, R., Morfeq, A., Balamash, A. S., Martínez, M., & Herrera-Viedma, E. (2017). Soft consensus measures in group decision making using unbalanced fuzzy linguistic information. Soft Computing, 21(11), 3037–3050. Capuano, N., Chiclana, F., Fujita, H., Herrera-Viedma, E., & Loia, V. (2018). Fuzzy group decision making with incomplete information guided by social influence. IEEE Transactions on Fuzzy Systems, 26(3), 1704–1718. Chen, C.-T. (20 0 0). Extensions of the topsis for group decision-making under fuzzy environment. Fuzzy sets and systems, 114(1), 1–9. Chen, X. (2011). American option pricing formula for uncertain financial market. International Journal of Operations Research, 8(2), 32–37. Cheng, W., Rademaker, M., De Baets, B., & Hüllermeier, E. (2010). Predicting partial orders: ranking with abstention. In Proceedings of the Joint European conference on machine learning and knowledge discovery in databases (pp. 215–230). Springer. Chiclana, F., Herrera-Viedma, E., Alonso, S., & Herrera, F. (2009). Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Transactions on Fuzzy Systems, 17(1), 14–23. Cohen, M. C., Keller, P. W., Mirrokni, V., & Zadimoghaddam, M. (2019). Overcommitment in cloud services: Bin packing with chance constraints. Management Science, 65(7), 2947–3448. De Baets, B., De Loof, K., & De Meyer, H. (2015). A frequentist view on cycle-transitivity of reciprocal relations. Fuzzy Sets and Systems, 281, 198–218. De Baets, B., & De Meyer, H. (2005). Transitivity frameworks for reciprocal relations: cycle-transitivity versus FG-transitivity. Fuzzy Sets and Systems, 152(2), 249–270. De Schuymer, B., De Meyer, H., & De Baets, B. (2005). Cycle-transitive comparison of independent random variables. Journal of Multivariate Analysis, 96(2), 352–373. Dong, Q., & Cooper, O. (2016). A peer-to-peer dynamic adaptive consensus reaching model for the group ahp decision making. European Journal of Operational Research, 250(2), 521–530. Dubois, D., & Prade, H. (2012). Possibility theory. Springer. Freson, S., De Baets, B., & De Meyer, H. (2014). Closing reciprocal relations wrt stochastic transitivity. Fuzzy Sets and Systems, 241, 2–26. Gao, R., & Ralescu, D. A. (2018). Convergence in distribution for uncertain random variables. IEEE Transactions on Fuzzy Systems, 26(3), 1427–1434. Gong, Z., Lin, Y., & Yao, T. (2012). Uncertain fuzzy preference relations and their applications: 281. Springer. Gong, Z., Xu, X., Zhang, H., Ozturk, U. A., Herrera-Viedma, E., & Xu, C. (2015). The consensus models with interval preference opinions and their economic interpretation. Omega, 55, 81–90. Groves, M., & Branke, J. (2019). Top-κ selection with pairwise comparisons. European Journal of Operational Research, 274(2), 615–626. Guo, J., Yao, X., & Lv, M. (2015). Research and operational advance on sensitivity industry evaluation of meteorological service benefits. Plateau and Mountain Meteorology Research, 35(2), 88–95. Jiang, Y., & Fan, Z. (2008). Theory and methods of decision making based on judgment matrices. Science Press, Beijing. Kacprzyk, J. (1986). Group decision making with a fuzzy linguistic majority. Fuzzy sets and systems, 18(2), 105–118. Lejeune, M. A., & Margot, F. (2016). Solving chance-constrained optimization problems with stochastic quadratic inequalities. Operations Research, 64(4), 939–957. Li, C.-C., Dong, Y., Herrera, F., Herrera-Viedma, E., & Martínez, L. (2017). Personalized individual semantics in computing with words for supporting linguistic group decision making. an application on consensus reaching. Information Fusion, 33, 29–40. Li, C.-C., Rodríguez, R. M., Martínez, L., Dong, Y., & Herrera, F. (2018). Consistency of hesitant fuzzy linguistic preference relations: An interval consistency index. Information Sciences, 432, 347–361. Liu, B. (2007). Uncertainty theory. Uncertainty theory (pp. 205–234). Springer. Liu, B. (2014). Uncertainty theory. Springer. Liu, B., & Liu, B. (2009). Theory and practice of uncertain programming: 239. Springer. Liu, B., Shen, Y., Zhang, W., Chen, X., & Wang, X. (2015). An interval-valued intuitionistic fuzzy principal component analysis model-based method for complex multi-attribute large-group decision-making. European Journal of Operational Research, 245(1), 209–225. Liu, F., Peng, Y.-N., Yu, Q., & Zhao, H. (2018). A decision-making model based on interval additive reciprocal matrices with additive approximation-consistency. Information Sciences, 422, 161–176. Liu, F., Zhang, W.-G., & Wang, Z.-X. (2012). A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making. European Journal of Operational Research, 218(3), 747–754. Liu, F., Zhang, W.-G., & Zhang, L.-H. (2014). Consistency analysis of triangular fuzzy reciprocal preference relations. European Journal of Operational Research, 235(3), 718–726. Ma, L.-C. (2016). A new group ranking approach for ordinal preferences based on group maximum consensus sequences. European Journal of Operational Research, 251(1), 171–181. Memon, M. S., Lee, Y. H., & Mari, S. I. (2015). Group multi-criteria supplier selection using combined grey systems theory and uncertainty theory. Expert Systems with Applications, 42(21), 7951–7959. Meng, F., & Chen, X. (2015). An approach to incomplete multiplicative preference relations and its application in group decision making. Information Sciences, 309, 119–137. Meng, F., & Tan, C. (2017). A new consistency concept for interval multiplicative preference relations. Applied Soft Computing, 52, 262–276. Meng, F., Tan, C., & Chen, X. (2017). Multiplicative consistency analysis for interval fuzzy preference relations: A comparative study. Omega, 68, 17–38.

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035

JID: EOR 18

ARTICLE IN PRESS

[m5G;November 15, 2019;16:41]

Z. Gong, W. Guo and E. Herrera-Viedma et al. / European Journal of Operational Research xxx (xxxx) xxx

Moore, R. E. (1966). Interval analysis: 4. Englewood Cliffs, NJ: Prentice-Hall. Nurmi, H. (1981). Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and systems, 6(3), 249–259. Orlovsky, S. (1978). Decision-making with a fuzzy preference relation. Fuzzy sets and systems, 1(3), 155–167. Ouyang, Y., & Pedrycz, W. (2016). A new model for intuitionistic fuzzy multi-attributes decision making. European Journal of Operational Research, 249(2), 677–682. Pal, N. R., Bustince, H., Pagola, M., Mukherjee, U., Goswami, D., & Beliakov, G. (2013). Uncertainties with Atanassov’s intuitionistic fuzzy sets: Fuzziness and lack of knowledge. Information Sciences, 228, 61–74. Qin, J., Liu, X., & Pedrycz, W. (2017). An extended todim multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment. European Journal of Operational Research, 258(2), 626–638. Rodriguez, R. M., Martinez, L., & Herrera, F. (2012). Hesitant fuzzy linguistic term sets for decision making. IEEE Transactions on Fuzzy Systems, 20(1), 109–119. Saaty, T. L. (1978). Modeling unstructured decision problemshe theory of analytical hierarchies. Mathematics and computers in simulation, 20(3), 147–158. Sentz, K., Ferson, S., et al. (2002). Combination of evidence in Dempster-Shafer theory: 4015. Citeseer. Shah, N. B., Balakrishnan, S., Guntuboyina, A., & Wainwright, M. J. (2016). Stochastically transitive models for pairwise comparisons: Statistical and computational issues. IEEE Transactions on Information Theory, 63(2), 934–959. ´ Switalski, Z. (1999). Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems, 107(2), 187–190. ´ Switalski, Z. (2001). Transitivity of fuzzy preference relations–an empirical study. Fuzzy Sets and Systems, 118(3), 503–508. ´ Switalski, Z. (2003). General transitivity conditions for fuzzy reciprocal preference matrices. Fuzzy Sets and Systems, 137(1), 85–100. Tang, J., Meng, F., & Zhang, Y. (2018). Decision making with interval-valued intuitionistic fuzzy preference relations based on additive consistency analysis. Information Sciences, 467, 115–134. Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy sets and systems, 12(2), 117–131. Tran, L., & Duckstein, L. (2002). Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy sets and Systems, 130(3), 331–341. Ureña, R., Chiclana, F., Morente-Molinera, J. A., & Herrera-Viedma, E. (2015). Managing incomplete preference relations in decision making: a review and future trends. Information Sciences, 302, 14–32.

Wan, S., Wang, F., & Dong, J. (2017). Additive consistent interval-valued atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. European Journal of Operational Research, 263(2), 571–582. Wan, S., Wang, F., & Dong, J. (2018a). A group decision-making method considering both the group consensus and multiplicative consistency of interval– valued intuitionistic fuzzy preference relations. Information Sciences, 466, 109– 128. Wan, S., Wang, F., & Dong, J. (2018b). A group decision making method with interval valued fuzzy preference relations based on the geometric consistency. Information Fusion, 40, 87–100. Wang, Z.-J. (2018). A goal programming approach to deriving interval weights in analytic form from interval fuzzy preference relations based on multiplicative consistency. Information Sciences, 462, 160–181. Wu, J., Chiclana, F., & Herrera-Viedma, E. (2015). Trust based consensus model for social network in an incomplete linguistic information context. Applied Soft Computing, 35, 827–839. Wu, J., Chiclana, F., & Liao, H. (2018). Isomorphic multiplicative transitivity for intuitionistic and interval-valued fuzzy preference relations and its application in deriving their priority vectors. IEEE Transactions on Fuzzy Systems, 26(1), 193–202. Wu, X., & Liao, H. (2019). A consensus-based probabilistic linguistic gained and lost dominance score method. European Journal of Operational Research, 272(3), 1017–1027. Xu, Y., Li, K. W., & Wang, H. (2014). Incomplete interval fuzzy preference relations and their applications. Computers and Industrial Engineering, 67, 93– 103. Xu, Z. (2004). On compatibility of interval fuzzy preference relations. Fuzzy Optimization and Decision Making, 3(3), 217–225. Xu, Z. (2011). Consistency of interval fuzzy preference relations in group decision making. Applied Soft Computing, 11(5), 3898–3909. Yan, H.-B., Ma, T., & Huynh, V.-N. (2017). On qualitative multi-attribute group decision making and its consensus measure: a probability based perspective. Omega, 70, 94–117. Zadeh, L. A., et al. (1965). Fuzzy sets. Information and control, 8(3), 338–353. Zeng, Z., Kang, R., Wen, M., & Zio, E. (2018). Uncertainty theory as a basis for belief reliability. Information Sciences, 429, 26–36.

Please cite this article as: Z. Gong, W. Guo and E. Herrera-Viedma et al., Consistency and consensus modeling of linear uncertain preference relations, European Journal of Operational Research, https://doi.org/10.1016/j.ejor.2019.10.035