Research Article  Open Access
Dominance Degree Multiple Attribute Decision Making Based on ZNumber Cognitive Information
Abstract
Cognitive information can be described by Znumber fully and effectively. However, many problems of Znumber need to be further studied. In this paper, two hidden probability models for calculating Znumber are established to provide more intuitive and abundant information. Next, the dominance degree relationship of Znumber is developed and subdivided. Furthermore, combined with the hidden probability of calculation, three different measurements of dominance degree are defined from three levels of geometry, algebra, and cross entropy based on the outranking relationship. The influencing factors are analyzed for different combinations of two probability models and three dominance degree measures. A multiattribute decision model is established on the basis of new grey association analysis and QUALIFLEX method. Finally, a decision example is given to verify the effectiveness and feasibility of the method. And sensitivity analysis is made to determine the impact of parameters and hidden probability on the decision model.
1. Introduction
Cognitive information for realworld decision making often has an element of uncertainty and is imprecise and only partially reliable. In 2011, Zadeh [1] put forward the Znumber theory to combine objective information with subjective understanding of cognitive information and enhance the understanding of natural language. A Znumber is expressed by a pair of ordered arrays A and B, where A represents the real value function of the uncertain variable X and B is the measure of reliability of A.
The current research on Znumbers can be mainly divided into three categories. The first is theoretical basis study [2]. The concepts of Zvaluation [3], [4] and Zinformation [5] are closely related to the concept of Znumbers. The proposed four concepts are shown in Table 1, where is the membership function of A and is the probability distribution of X. The second aspect is languagetype Znumber calculation and related extension [6, 7], for example, Zarrin and Azadeh [8] combined Znumber with fuzzy cognitive map (FCM) and proposed a novel approach named Znumber cognitive map. They evaluated and analyzed the impacts of resilience engineering (RE) principles on integrated health, safety, environment, and ergonomics (HSEE) management system. The main advantages of the proposed approach are determination of the weighted causality relations (for employing FCM) as well as handling uncertainty (for considering Znumber concept). Kant [9] extended the analysis of cognitive work as some requirements that gathered the framework of sociotechnical systems. It is helpful for the study of knowing and acting in technological contexts in the human. The third aspect is establishing a decision model based on Znumber [10–13]. For example, Li et al. [14] proposed a method to convert Znumbers into fuzzy numbers. Aliev et al. [4] proposed some algorithms about Znumbers. They considered two approaches for decision making with Zinformation. The first approach is based on converting the Znumbers to crisp numbers to determine the priority weight of each alternative, which would decrease some uncertain information during processing. The second approach is based on expected utility theory by using Znumbers. This method of selecting expected utility is an uncertain factor, and it influences the effect of using Znumber. Kang et al. [15] proposed a utility function of Znumbers. These decision methods, utility functions, or conversion methods may lose some information during the operation, and these shortcomings should be further studied.

At present, the motivation for using discrete Znumber in the study of Znumbers is mainly divided into three levels [16]. Firstly, Znumbers are used to describe relevant decision information in many decision problems by discrete language terms which are usually expressed as linguistic information. Secondly, the computational complexity of continuous fuzzy numbers [17, 18] and density functions is significantly higher than that of discrete fuzzy numbers [19–21] and discrete probability distributions. Thirdly, the universality of the multiattribute decision model is established according to the uncertain information. When the decision information is expressed as discrete Znumbers, one does not need to assume a probability distribution to limit the model. When the probability of natural state and surrogate results are described by Znumbers, it need transform numbers or sets according to some operators. Aliev and Zeinalova [22] performed decision analysis from two stages. Above all, the Znumber is converted to a fuzzy number; next, the value of the utility function is calculated and sorted, and then the scheme with the largest value of the utility function is selected as the optimal selection scheme. Zinformation [16] belongs to the category of probability limits; thus, the probability distribution can be regarded as the probability limit. Li et al. [14] proposed a fuzzy expectation based on the Znumber that is a fuzzy set as well. In the fuzzy expectation, B is decomposed into a detailed numerical value α. The fuzzy number obtained the first element A; multiplying by α is considered as a transformation of the original Znumber. Chen [23] combined the method in the literature [24] to establish a multicriteria decisionmaking method based on Znumber. This method used the standard weight and standard value as Znumber. Obviously, the Znumber of this method is a set of real numbers. Its means that a lot of uncertain information contained in Znumber is neglected and lost, that is, decision makers cannot obtain an accurate optimal decisionmaking method. Farhadinia [25] proposed a new measure of information entropy, which is closely related to probability and comprehensively reflected more uncertain information.
Peng and Wang [26] defined a kind of outranking relationship between two Znumbers, but it directly calculates the dominance degree of the first element A and the second element B in a Znumber by a simple comparison and accumulation, which makes no sense in some practical problems. For example, suppose that and ; then, the “high” and “low” can be directly compared. It is obvious that “high” is better than “low” in evaluating the grade of a commodity, but “absolutely” is better than “uncertainly” according to intuitive literal sense. It is worth noting that “uncertainly” and “absolutely” are measurements of the possibility of “high” and “low,” respectively. It is meaningless to compare them separately. That is, A and B in should be tied together to evaluate the target.
Form the work reviewed, it can be concluded that ranking of Znumber cognitive information is a necessary operation, and the hidden probability and dominance degree of Znumbers is a challenging practical issue.
The multiattribute decisionmaking problem is to give some alternative schemes, and each scheme needs to be comprehensively evaluated according to several attributes. The purpose of decision making is to find a scheme that makes the decision maker feel most satisfied from the given alternative schemes through the comprehensive evaluation sequence. Mao et al. [27] reported the intervalvalued intuitionistic fuzzy entropy which reflects intuitionism and fuzziness of intervalvalued intuitionistic fuzzy set (IvIFS) based on intervalvalued intuitionistic fuzzy cross entropy. According to the compositive entropy use for multiple attributes decision making, they adopted the weighted correlation coefficient between IvIFSs and pattern recognition by a similarity measure transformed from the compositive entropy. However, in practical problems, there are often some individual subjective factors of decision makers, objective factors of the attributes, unpreventable error factors, and other uncertain factors, which are one of the characteristics of uncertain information itself. How to make better use of dubious, inaccurate, and uncertain information in multiattribute decision making is the main research problem in the field of multiattribute decision making [28]. Yang and Wang [29] established a linear programming model to solve Znumber probability, and made a multicriteria decision based on Znumber probability. Yang and Wang [29] combined the concept of reliability to judge the decision maker and established the multicriteria decision aiding model based on stochastic multicriteria acceptability analysis (SMAA).
Li et al. [30] developed a linear programming methodology for solving multiattribute group decisionmaking problems using intuitionistic fuzzy (IF) sets. Wan and Li [31] extended the linear programming technique for multidimensional analysis of preference (LINMAP) for solving heterogeneous MADM problems which involve intuitionistic fuzzy (IF) sets (IFSs), trapezoidal fuzzy numbers (TrFNs), intervals, and real numbers. They presented decision maker’s preference given through pairwise comparisons of alternatives with hesitation degrees which are represented as IFSs. They constructed a new fuzzy mathematical programming model, obtained FIS and the attribute weights, and calculated the distances of all alternatives to the FIS, which are used to determine the ranking order of the alternatives. Yu et al. [32] developed a compromisetyped variable weight decision method for solving hybrid multiattribute decisionmaking problems with multiple types of attribute values, and variable weight synthesis and orness measures based on the coefficients of absolute risk aversion are analyzed in variable weight decision making. The comprehensive values of alternatives based on the compromisetyped variable weight decision method are calculated, and the decisionmaking results are determined according to the comprehensive values. Yu et al. [33] developed a novel method for HMAGDM problems based on the orness measures by analyzing the relative closeness of alternatives and preference deviation degrees of each decision maker (DM). The weights of the attributes of each alternative and weights preference of DMs are obtained using two linear programming models, and a ranking of alternative is determined according to the decision making preference of alternatives.
Shih and Chen [34] studied grey relational analysis of the series similarity and approximation. Liu et al. [35] defined the multiattribute and multistage decisionmaking problem, that is, the attribute weights and time weights in each decision stage are unknown and the attribute value is interval numbers; after dimensionless processing of attribute values, the grey correlation analysis method is used to determine the attribute weights of each attribute in different time periods to make decisions. Paelinck [36] proposed a simple and flexible outranking model. The QUALIFLEX method tested the binary relation of each possible ranking possibility and directed distance measure under different attributes and then calculated the comprehensive concordance/discordance index of each ranking to determine the optimal ranking. The cardinal and ordinal information can be correctly processed and the uncertain information can be fully considered by this method as well. The QUALIFLEX method has been studied and extended to various applications, such as investment risk assessment [37], supplier evaluation [38], and product design selection [39].
Based on the reviewed literature, the authors conclude that the little attention has been paid to the important issue of ranking Znumber and measuring uncertainty. And now many researchers do not consider the hidden probability and reliability of Znumber in the Znumber multiattribute decision problem. This paper will consider the important role of hidden probability and reliability in decision making.
Based on the previous discussion, the three primary motivations of this paper are as follows:(1)The structure of Znumber expresses the subjectivity and objectivity of natural language, and the expression of its reliability cannot completely limit the subjective uncertainty information. Therefore, the hidden probability of Znumber is introduced to limit the reliability.(2)The dominance degree accumulates A and B, respectively, in Znumber, which is unreasonable in practical problems. This paper distinguishes this from the geometric, algebraic, and crossentropy levels to define the superiority of the three metrics. A multiattribute decision model with unknown attribute weights is established.(3)Znumber is an important tool with which humans communicate with a computer. With effective use of Znumber to express natural language, humans will not spend a lot of time learning their own computer language. Natural language and Znumber combination can better reflect uncertain information.
This paper is divided into nine sections. Section 2 gives the basic definitions of , discrete , and . In Section 3, two probability models, and , are established. Section 4 is the main part of this paper: outranking relationships and three measurements of dominance degree of discrete Znumbers. Section 5 is about two models as well: one is to establish a model for solving attribute weight by new grey relational analysis and the other is a multiattribute decision model based on three measurements of dominance degree, mainly using the QUALIFLEX method. Section 6 is about a multiattribute decision case of a venture capital company. Section 7 is a sensitivity analysis based on the decision cases. In Section 8, comparison and summary of the decision method of Section 6 are given, and Section 9 concludes this paper.
2. Preliminaries
In this section, we introduce some basic knowledge about fuzzy numbers, discrete Znumber, and in detail. Let X be a universal set, and a fuzzy set A in X is represented aswhere indicates the membership degree of the element to A subset of X.
Definition 1 (discrete fuzzy number) [4, 26]. A fuzzy subset A of the real line R with membership function is a discrete fuzzy number if its support is finite, i.e., there exist with , such that and there exist natural numbers with satisfying the following conditions:(1) for any natural number i with (2) for each natural number with (3) for each natural number with
Definition 2 (Znumber) [1]. A Znumber is an ordered pair of fuzzy number, . It is associated with a realvalued uncertain variable, X. The first component, A, playing the role of a fuzzy restriction, , about the values which X can take, written as X is A, where A is a fuzzy set. The restrictionis referred to as a possibility restriction (constraint), where is the membership function of A and u is a generic value of X. The second component, B, is referred to as certainty. Closely related to certainty are the concepts of sureness, confidence, reliability, strength of belief, and probability. The underlying probability distribution, , is not known. What is known is a restriction on which may be expressed as
Definition 3 (discrete Znumber) [4]. Let X be a random variable, A and B be two discrete fuzzy numbers, and and . For the membership function of A and B, respectively, and . A discrete Znumber is defined as an ordered pair of discrete fuzzy numbers on X, where A is the fuzzy restriction of X and B is the fuzzy restriction of the probability measure of A.
Definition 4 (discrete ) [4]. A discrete , denoted as , where A is the fuzzy restriction and R is the probability distribution of X, is expressed aswhere means that , is the possibility that . Similarly, is the probability that . And A plays the same role in as it does in Znumbers, and R plays the role of the probability distribution.
3. Two Probability Models of Discrete ZNumbers
The uncertainty of some information includes fuzziness and randomness. In order to obtain more uncertain information, we considered the hidden probability distribution of Znumbers, comprehensively considered the relation between A and B in Znumbers, and established two linear programming models to determine the hidden probability of discrete Znumbers.
A discrete Znumber described imperfect information on realvalued random variable X values and satisfies the following conditions:(1) (the slash symbol denotes division)(2) and
3.1. Probability Model I of Discrete ZNumbers
Condition (1) ensures that these distributions are compatible when the centroid of and is coincident. Condition (2) is the normalization and nonnegativity condition of the probability. We can define the mathematical model as follows:and then the hidden probability matrix P of Znumber is calculated by MATLAB programming language, denoted as
Use P matrix to calculate the comprehensive probability of :
3.2. Probability Model II of Discrete ZNumbers
On the basis of probability of Znumber, after simplification is shown as follows. Compared with , is relatively concise, but its response has less hidden uncertain information than .where ; to make the distinction, the probability calculated under the model is denoted as . The hidden probability of the problem containing Znumber is solved, that is to say, the problem is resolved, and the optimization problem is solved by MATLAB language. Subsequently, the probability model presented in this paper can be used to evaluate the basic probability distribution.
Example 1. Let and be two discrete Znumbers, whereThe hidden probabilities and of and are calculated, respectively, by using the above probabilistic and , which are called and , where and are the following discrete probability distributions:where n is the number of discrete fuzzy number A in discrete Znumber. In , first we need to figure out the hidden probability matrix and of and and then obtain the comprehensive probabilities and of Znumber from .Then, the same logic applies toUsing , the hidden probability of Znumber is calculated as follows:From the calculation results of Example 1, it can be seen that the calculation results of reflect more hidden information, and the results of are only part of the results of . However, which model the dominance degree proposed in the following part applies to depends on the results of our sensitivity analysis.
4. Outranking Relationship and Dominance Degree of Discrete ZNumber
This section redefines a more detailed Znumber outranking relation for the outranking relation in article [26]. Combining the three aspects of geometry, algebra, and information entropy with Znumber and the hidden probability information of Znumber itself, three different computing methods of dominance degree are defined. And a model for calculating the hidden probability P of discrete Znumber is established.
A linguistic term set (LTS) noted is a finite and completely ordered discrete LTS with odd cardinality, where represents the possible value for a linguistic variable, and satisfy if , and if for any .
Definition 5 (outranking relationship of discrete Znumber). Let two Znumbers characterized by language variables, denoted as and , where and and , be two LTSs. Then, the outranking relationship of two Znumbers can be defined as follows:(1)Extremely strong dominance: when , extremely and strongly dominates ; let us label this relationship as or (2)Strong dominance: when and , strongly dominates ; let us label this relationship as or (3)Weak dominance: when and , weakly dominates ; let us label this relationship as or (4)Equal dominance: when and , equally dominates ; let us label this relationship as or (5)Incomparable relation: if neither nor satisfies the above conditions, then and are called incomparable; let us label this relationship as or
Property 1. Let , and be three arbitrary Znumbers. In this paper, we define the property of extremely strong dominant relation of Znumber as follows:(1)Nonreflexivity: , where indicates nonextremely strong dominance(2)Asymmetry: (3)Transitivity: and
Proof. (1)The nonreflexivity property uses the method of proof by contradiction; if , then and . It is obviously contradictory.(2)The asymmetry property uses the method of proof by contradiction as well. Let us say that is true. If , then and . If , then and . These two conclusions are obviously contradictory, and the asymmetric property is proved.(3)If , then and ; if , then and . According to the transitivity of inequality, we can draw a conclusion and , that is, .
Definition 6. (three measurements of dominance degree for discrete Znumber). Let and be two random variables, and be the discrete Znumbers of and , and and be the membership function of and ; is similar to . Firstly, Znumber is normalized:normalized:where , , , and ; is similar to . After Znumber standardization, dimensional relationship between data is eliminated, making data comparable.
Then, three measurements of dominance degree for discrete Znumber as follows.
Definition 7 (the geometric measurements of dominance degree for discrete Znumber). The geometric measurements of dominance degree for discrete Znumber is Znumber after the number of standardized data one by one to quantify. The angle is measured as the degree of dominance between the two Znumbers. The geometric measurements of dominance degree for discrete and are defined as follows:whereand θ and γ are the positive angles between and vectors and the Xaxis, respectively. represents the degree of preference of to calculated with the hidden probability solved by probability , .
Property 2. Let be the geometric measurements of dominance degree for over ; they have the following properties:(1) if and only if (2) or (3)
Example 2. Use and in Example 1 and to calculate the dominance degree of geometric measurements and .The first step is to calculate the hidden probability of Znumber with two models; then, we calculated the results as follows:And then, the dominance degree is calculated by equation (16), as shown in Table 2.
In the polar coordinate system, the geometric measurement dominance degree of , , and is shown in Figure 1. Figure 1 shows the dominance degree to which is partial to intuitively. In the figure, the green line represents the negative angle, the red line represents the positive angle, and a complete circle represents . The superposition of the angles gives the final dominance degree.
(a)
(b)
(c)
(d)
(e)
(f)
Definition 8 (the algebra measurements of dominance degree for discrete Znumber). Similar to the geometric measurements of dominance degree for discrete Znumber, the algebra measurements of dominance degree for discrete Znumber are defined as follows after Znumber standardization:where + + , and ; substituting λ into Znumbers as the weight, the hidden information of Znumber is fully applied in combination with the hidden probability .
Property 3. Let be the algebra measurements of dominance degree for over ; they have the following properties:(1) if and only if (2)(3)
Example 3. Use , , and in Example 2 to calculate the dominance degree and algebra measurements and with different λ values; then, we use equation (20) to calculate the results as shown in Table 3.
Since there is a big difference in the change of algebraic measurement dominance degree when and , we divide the algebraic measurement dominance degree into two cases and obtain Figures 2 and 3. When , and decrease with the increase of λ, while the monotonicity of is the opposite. This is because there is a large gap between the value of calculated by the probability model and the value of calculated by the probability model , resulting in a large gap between the value of . When , and decrease with the increase of λ, and the monotonicity of basically increases monotonically.
Definition 9 (the crossentropy measurements of dominance degree for discrete Znumber). Similar to the geometric and algebra measurements of dominance degree for discrete Znumber, the crossentropy measurements of dominance degree for discrete Znumber are defined as follows after Znumber standardization. The crossentropy measurement of dominance degree for discrete Znumber was defined as follows:
However, is not symmetric, so in analogy with article [40], we proceed to the following definition of the Znumber symmetric crossentropy measure uncertain information:and then we define the crossentropy measurements of dominance degree for discrete Znumber as follows:where n is the number of discrete fuzzy number A in discrete Znumber.

(a)
(b)
(c)
(d)
(e)
(f)
(a)
(b)
(c)
(d)
(e)
(f)
Property 4. Let be the Znumber symmetric cross entropy for and ; they have the following properties:(1)Symmetry: (2)Negative: (3)Normative:
Proof. (1)The proof is trivial(2)Because , , , , , and are all members of , obviously(3) when , can be obtained similarly:so