Decision Support
Consistency and consensus modeling of linear uncertain preference relations

https://doi.org/10.1016/j.ejor.2019.10.035Get rights and content

Highlights

  • Uncertain preference relation and its additive consistency definitions are proposed.

  • The interval preference relation is a special case of uncertain preference relation.

  • Optimal consensus models for uncertain preference relations are constructed.

  • The minimum deviation is a linear increasing function of the belief degree.

Abstract

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 decision-making, 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.

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. 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 interval-valued 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, Al-Hmouz, Morfeq, Balamash, Martínez, Herrera-Viedma, 2017, Chen, 2000, Gong, Xu, Zhang, Ozturk, Herrera-Viedma, Xu, 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). 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 preference relations. Świtalski, 1999, Świtalski, 2001, Świtalski, 2003 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, Dong, 2017, Wan, Wang, Dong, 2018a, Wan, Wang, Dong, 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 the industry meteorological sensitivity assessment for a region in China, based on the assessment of experts specialized in decision-making, 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.

Section snippets

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

Fuzzy preference relation

Definition 2

Kacprzyk, 1986, Nurmi, 1981, Orlovsky, 1978, Tanino, 1984

Nonnegative matrix R=(rij)n×n is called a fuzzy preference relation if rii=0.5, rij+rji=1,i,jN.

Element rij in fuzzy preference relation R expresses the membership degree of alternative xi over alternative xj. rij=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=(rij)n×n is said to have additive consistency ifrij+rjk=rik+0.5,i,j,kN.

Interval fuzzy preference relation

Definition 4

Xu, 2004

Nonnegative matrix R¯=(r¯ij)n×n=([rijL,r

Preference relations based on uncertainty theory

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

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.

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

Problem description

Industry meteorological sensitivity is a representation of the degree of change in the national economy affected by weather-related 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

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

Acknowledgments

This research is partially supported by the National Natural Science Foundation of China (71971121, 71571104), NUIST-UoR 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).

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