Research Article | Open Access
Dominance Degree Multiple Attribute Decision Making Based on Z-Number Cognitive Information
Cognitive information can be described by Z-number fully and effectively. However, many problems of Z-number need to be further studied. In this paper, two hidden probability models for calculating Z-number are established to provide more intuitive and abundant information. Next, the dominance degree relationship of Z-number 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.
Cognitive information for real-world decision making often has an element of uncertainty and is imprecise and only partially reliable. In 2011, Zadeh  put forward the Z-number theory to combine objective information with subjective understanding of cognitive information and enhance the understanding of natural language. A Z-number 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 Z-numbers can be mainly divided into three categories. The first is theoretical basis study . The concepts of Z-valuation ,  and Z-information  are closely related to the concept of Z-numbers. 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 language-type Z-number calculation and related extension [6, 7], for example, Zarrin and Azadeh  combined Z-number with fuzzy cognitive map (FCM) and proposed a novel approach named Z-number 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 Z-number concept). Kant  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 Z-number [10–13]. For example, Li et al.  proposed a method to convert Z-numbers into fuzzy numbers. Aliev et al.  proposed some algorithms about Z-numbers. They considered two approaches for decision making with Z-information. The first approach is based on converting the Z-numbers 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 Z-numbers. This method of selecting expected utility is an uncertain factor, and it influences the effect of using Z-number. Kang et al.  proposed a utility function of Z-numbers. 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 Z-number in the study of Z-numbers is mainly divided into three levels . Firstly, Z-numbers 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 Z-numbers, 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 Z-numbers, it need transform numbers or sets according to some operators. Aliev and Zeinalova  performed decision analysis from two stages. Above all, the Z-number 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. Z-information  belongs to the category of probability limits; thus, the probability distribution can be regarded as the probability limit. Li et al.  proposed a fuzzy expectation based on the Z-number 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 Z-number. Chen  combined the method in the literature  to establish a multicriteria decision-making method based on Z-number. This method used the standard weight and standard value as Z-number. Obviously, the Z-number of this method is a set of real numbers. Its means that a lot of uncertain information contained in Z-number is neglected and lost, that is, decision makers cannot obtain an accurate optimal decision-making method. Farhadinia  proposed a new measure of information entropy, which is closely related to probability and comprehensively reflected more uncertain information.
Peng and Wang  defined a kind of outranking relationship between two Z-numbers, but it directly calculates the dominance degree of the first element A and the second element B in a Z-number 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 Z-number cognitive information is a necessary operation, and the hidden probability and dominance degree of Z-numbers is a challenging practical issue.
The multiattribute decision-making 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.  reported the interval-valued intuitionistic fuzzy entropy which reflects intuitionism and fuzziness of interval-valued intuitionistic fuzzy set (IvIFS) based on interval-valued 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 . Yang and Wang  established a linear programming model to solve Z-number probability, and made a multicriteria decision based on Z-number probability. Yang and Wang  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.  developed a linear programming methodology for solving multiattribute group decision-making problems using intuitionistic fuzzy (IF) sets. Wan and Li  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.  developed a compromise-typed variable weight decision method for solving hybrid multiattribute decision-making 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 compromise-typed variable weight decision method are calculated, and the decision-making results are determined according to the comprehensive values. Yu et al.  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  studied grey relational analysis of the series similarity and approximation. Liu et al.  defined the multiattribute and multistage decision-making 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  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 , supplier evaluation , and product design selection .
Based on the reviewed literature, the authors conclude that the little attention has been paid to the important issue of ranking Z-number and measuring uncertainty. And now many researchers do not consider the hidden probability and reliability of Z-number in the Z-number 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 Z-number 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 Z-number is introduced to limit the reliability.(2)The dominance degree accumulates A and B, respectively, in Z-number, which is unreasonable in practical problems. This paper distinguishes this from the geometric, algebraic, and cross-entropy levels to define the superiority of the three metrics. A multiattribute decision model with unknown attribute weights is established.(3)Z-number is an important tool with which humans communicate with a computer. With effective use of Z-number to express natural language, humans will not spend a lot of time learning their own computer language. Natural language and Z-number 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 Z-numbers. 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.
In this section, we introduce some basic knowledge about fuzzy numbers, discrete Z-number, 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 (Z-number) . A Z-number is an ordered pair of fuzzy number, . It is associated with a real-valued 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 Z-number) . 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 Z-number 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 ) . 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 Z-numbers, and R plays the role of the probability distribution.
3. Two Probability Models of Discrete Z-Numbers
The uncertainty of some information includes fuzziness and randomness. In order to obtain more uncertain information, we considered the hidden probability distribution of Z-numbers, comprehensively considered the relation between A and B in Z-numbers, and established two linear programming models to determine the hidden probability of discrete Z-numbers.
A discrete Z-number described imperfect information on real-valued random variable X values and satisfies the following conditions:(1) (the slash symbol denotes division)(2) and
3.1. Probability Model I of Discrete Z-Numbers
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 Z-number is calculated by MATLAB programming language, denoted as
Use P matrix to calculate the comprehensive probability of :
3.2. Probability Model II of Discrete Z-Numbers
On the basis of probability of Z-number, 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 Z-number 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 Z-numbers, 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 Z-number. In , first we need to figure out the hidden probability matrix and of and and then obtain the comprehensive probabilities and of Z-number from .Then, the same logic applies toUsing , the hidden probability of Z-number 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 Z-Number
This section redefines a more detailed Z-number outranking relation for the outranking relation in article . Combining the three aspects of geometry, algebra, and information entropy with Z-number and the hidden probability information of Z-number itself, three different computing methods of dominance degree are defined. And a model for calculating the hidden probability P of discrete Z-number 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 Z-number). Let two Z-numbers characterized by language variables, denoted as and , where and and , be two LTSs. Then, the outranking relationship of two Z-numbers 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 Z-numbers. In this paper, we define the property of extremely strong dominant relation of Z-number 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 Z-number). Let and be two random variables, and be the discrete Z-numbers of and , and and be the membership function of and ; is similar to . Firstly, Z-number is normalized:normalized:where , , , and ; is similar to . After Z-number standardization, dimensional relationship between data is eliminated, making data comparable.
Then, three measurements of dominance degree for discrete Z-number as follows.
Definition 7 (the geometric measurements of dominance degree for discrete Z-number). The geometric measurements of dominance degree for discrete Z-number is Z-number after the number of standardized data one by one to quantify. The angle is measured as the degree of dominance between the two Z-numbers. 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 X-axis, 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 Z-number 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.
Definition 8 (the algebra measurements of dominance degree for discrete Z-number). Similar to the geometric measurements of dominance degree for discrete Z-number, the algebra measurements of dominance degree for discrete Z-number are defined as follows after Z-number standardization:where + + , and ; substituting λ into Z-numbers as the weight, the hidden information of Z-number 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 cross-entropy measurements of dominance degree for discrete Z-number). Similar to the geometric and algebra measurements of dominance degree for discrete Z-number, the cross-entropy measurements of dominance degree for discrete Z-number are defined as follows after Z-number standardization. The cross-entropy measurement of dominance degree for discrete Z-number was defined as follows:
However, is not symmetric, so in analogy with article , we proceed to the following definition of the Z-number symmetric cross-entropy measure uncertain information:and then we define the cross-entropy measurements of dominance degree for discrete Z-number as follows:where n is the number of discrete fuzzy number A in discrete Z-number.
Property 4. Let be the Z-number 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