Abstract
Multiple attribute decision making (MADM) problems widely exist in real decision making, and MADM methods with linguistic information have achieved great success. However, as the complexity of decision making problems is increasing in the real world, it is of great necessity to further develop new expression of evaluation information and aggregation technologies that can reflect the correlation among multi-attributes under uncertain decision-making environment. In response, this paper originally presents q-rung orthopair fuzzy uncertain linguistic set (q-ROULS) by combining q-rung orthopair fuzzy set (q-ROFS) and uncertain linguistic set (ULS). Then operational laws, expected functions and accuracy functions of q-rung orthopair uncertain linguistic variables (q-ROULVs) are also defined. Considering the correlation between q-ROULVs, we propose a family of q-rung orthopair fuzzy uncertain linguistic Choquet integral operators to aggregate q-rung orthopair uncertain linguistic information. Further, a novel MADM technique is presented based on the proposed q-rung orthopair fuzzy uncertain linguistic Choquet integral operators. The developed MADM method with q-rung orthopair fuzzy uncertain linguistic information enriches fuzzy decision-making theory and provides a new way for decision makers (DMs) under q-rung orthopair fuzzy uncertain linguistic environment.
Keywords
Introduction
Multiple attribute decision making (MADM) is the process of ranking the alternatives and selecting an optimal one among a set of alternatives assessed on multiple attributes, which has been widely used in economy, management and other fields in recent years. For example, Roy et al. [1] proposed a rough strength relational decision making and trial evaluation laboratory model to analyze the internal strength and external impacts of key success factors in hospital service quality. Tang et al. [2] proposed an algorithm for MADM and applied it to flood disaster risk evaluation. Chatterjee et al. [3] developed a novel multi-attributive border approximation area comparison methods for non-traditional machining process selection problems in manufacturing domain. Mukhametzyanov and Pamučar [4] provided a model for the sensitivity analysis of MADM and selection of the optimal one. Vasiljević et al. [5] developed a rough group analytic hierarchy process approach to the evaluation supplier criteria in the company for producing metal washers for the automotive industry. Liu et al. [6] presented a multicriteria model for evaluating and selecting a transport service provider based on a single valued neutrosophic number. As MADM is becoming more and more complicated, we have to make decisions in different fuzzy environments. Therefore, one of the biggest difficulties in decision making is how to express decision makers’ assessments over alternatives. In general, the fuzzy information can be expressed by two forms: quantitative description and qualitative description. To appropiroretly describe the quantitative fuzzy information, quite a few tools, such as intuitionistic fuzzy set (IFS) [7], neutrosophic set [8], hesitant interval-valued fuzzy set [9], Pythagorean fuzzy set (PFS) [10–13] have been proposed. More recently, Yager [14] proposed a new concept of q-ROFS whose prominent feature is that the qth power of the membership degree and the qth power of the degree of non-membership is equal to or less than 1. Evidently, q-ROFS relaxes the constraints of both IFS and PFS, which makes it more powerful to express quantitative fuzzy information. Because of this advantage, q-ROFS has received more and more attention in recent years [15–19].
In real MADM problems, DMs prefer to make qualitative decisions instead of quantitative decisions due to time shortage and a lack of priori expterise. For example, people can give some linguistic terms like “excellent”, “medium” or “poor” to represent the evaluation values. To effectively address these circumtances, Zhadeh [20] provided a novle tool, called linguistic variables (LVs), to express fuzzy information. Later, Xu [21] proposed the concpt of uncertain linguistic variables (ULVs). After that, the researches on MADM under uncertain linguistic context received wide attention and several different linguistic sets have been developed, such as intuitionistic uncertain linguistic set (IULS) [22], interval intuitionistic uncertain linguistic set [23], hesitant fuzzy uncertain linguistic set [24], Pythagorean uncertain linguistic set (PULS) [25], interval-valued Pythagorean uncertain linguistic set [26], single value neutrosophic uncertain linguistic set (SVNULS) [27], interval neutrosophic uncertain linguistic set (INULS) [28]. An important research topic for these uncertain linguistic set is aggregation operator theory that can aggregate a collection of individual evaluated values into one. To aggregate intuitionistic uncertain linguistic information, Liu P.D. et al. [29], Liu P.D. et al. [30], Liu Z.M. et al. [31] successively extended some existing operators, such as weighted geometric average operator, ordered weighted geometric operator, Heronian mean, Bonferroni OWA and partitioned Bonferroni mean to the IULS. Liu Z.M. et al. [32] proposed intuitionistic uncertain linguistic Hamy mean operators with linguistic scale functions and apply them to health-care waste treatment technology selection. In addition, by combining interval neutrosophic uncertain linguistic variables with the Choquet integral, Liu and Tang [33] presented an interval neutrosophic uncertain linguistic Choquet integral method for MAGDM. Liu and Shi [34] introduced some neutrosophic uncertain linguistic Heronian mean operators for MAGDM. As PULS can describes fuzzy information better than IULS, Liu et al. [35] developed Pythagorean fuzzy uncertain linguistic prioritized weighted averaging aggregation operators and applied them to MADM with completely unknown weight of information. At the same time, they also extended the Maclaurin symmetric mean to PULS and introduced a family of Pythagorean fuzzy uncertain linguistic Maclaurin symmetric mean operator. Considering the partition interrelationship between attributes being aggregated, Z.M. Liu et al. [36] divided attributes into several different groups based on their correlation characteristics, and proposed a Pythagorean uncertain linguistic partitioned Bonferroni mean operator and its weighted form.
From the above analysis, we can find that researches on MADM methods with linguistic information have achieved great success. However, as the complexity of decision making problems is increasing in the real world, we may encounter the following MADM issues simultaneously, and the existing linguistic decision-making methods cannot express and deal with these situations:
There exist some situations where the square sum of the membership degree and the non-membership degree belonging to the uncertain linguistic variable is bigger than 1. For example: performance of a car is perhaps felt to be lower than “good” (s5) but higher than “fair” (s3). The membership degree to [s3, s5] is 0.9, and nonmembership degree is 0.7. The evaluation result can be denoted as < [s3, s5] , (0.9, 0.7)>. Due to the fact that 0.9 + 0.7 > 1 and 0.92 + 0.72 > 1, the evaluation value < [s3, s5] , (0.9, 0.7)> cannot be expressed by the existing IULS and PULS. Thus, it is of great necessity to further develop an extension of PFULS and IULS to better express linguistic information. In some situations, there exist correlations among attributes. For example, to evaluate patients based on the following symptoms of lung diseases: (vital signs, body temperature, cough and hemoptysis), we want to place more emphasis on hemoptysis than on body temperature. However, on the other hand, we also want to pay more attention to patients who have severe hemoptysis and high body temperature, because hemoptysis and hyperthermia are two classical symptoms of pneumonia. However, traditional aggregation operators, such as arithmetic and geometric operators are based on the assumption that the attributes are independent of one another. Therefore, it is meaningful to investigate aggregation technologies that can account the situation.
Motivated by q-ROFS and ULVs, we propose a new concept of uncertain linguistic set named q-ROFULS that can integrate the advantages of q-ROFS and ULVs. Compared to the existing PFULS and IULS, the constraint of new q-ROULS is that the sum of the qth power of membership degree and the qth power of non-membership degree belonging to the uncertain linguistic variable is less than or equal to one. The proposed q-ROULS are more general than PFULS and IULS due to the fact that PFULS and IULS are the special cases of q-ROULS when q = 1 and q = 2, respectively. Obviously, the larger the rung q, the more orthopairs satisfy the bounding constraint and thus the larger fuzzy information space can be expressed by q-ROULS. This feature makes q-ROULS more powerful and useful than PFULS and IULS in aspect of expressing the vagueness and fuzzy information. For the second issue, we note that Choquet integral operator originally developed by Choquet [37] can model correlation among attributes by using a fuzzy measure [38–42]. The fuzzy measure can be used to define a weight on each combination of criteria. So far, there is no research on MADM methods with q-rung orthopair uncertain linguistic information based on Choquet integral operator. Since it is better for q-ROULS to depict the actual situation, Choquet integral operator can capture the correlation among attributes, it is of great meaning to study the Choquet integral operator under q-rung orthopair uncertain linguistic environment for MCDM problems.
The above analysis describes the motivation behind proposing a comprehensive approach for tackling MADM with q-rung orthopair uncertain fuzzy linguistic information. Based on the above comprehensive analysis, the goal of this paper are: (1) to introduce a new linguistic fuzzy set called q-ROULS; (2) to introduce operational laws, expected functions and accuracy functions of an q-ROFULV; (3) to develop some novel Choquet integral aggregation operators to deal with q-rung orthopair uncertain linguistic information. Compared with existing aggregation operators, the proposed aggregation operators not only consider the correlations among attributes, but also demonstrate high generality than exiting uncertain linguistic aggregation operators; (4) to provide a new train of thought for MADM with q-rung orthopair fuzzy uncertain linguistic information based on the proposed operators.
The remainder of the paper is organized as follows. Section 2 recalls some basic notions, and further proposes a new concept of uncertain linguistic term named q-ROULS. Section 3 proposes a family of q-rung orthopair fuzzy uncertain linguistic Choquet integral operators, such as q-rung orthopair uncertain linguistic Choquet integral averaging (q-ROULCA) operator, q-rung orthopair uncertain linguistic Choquet integral geometric (q-ROULCG) operator, generalized q-rung orthopair uncertain linguistic Choquet integral averaging q-ROULGCA) operator and generalized q-rung orthopair uncertain linguistic Choquet integral geometric (q-ROULGCG) operator. Section 4 presents a novel approach to MADM based on proposed operators and Section 5 provides a numerical instance to show the validity of propose method.
Preliminaries
In this section, we briefly review concepts about q-ROFS and Choquet integral, and then we define a new notation of q-ROULS, some operational laws on q-ROULVs are also introduced.
q-Rung orthopair fuzzy set
The indeterminacy degree is defined as π A (x) = (u A (x) q + v A (x) q - u A (x) q v A (x) q ) 1/q. For convenience, (u A (x) , v A (x)) is called a q-rung orthopair fuzzy number (q-ROFN) by Liu and Wang [15], which can be denoted by A = (u A , v A ).
Liu and Wang [15] also proposed operational laws for q-ROFNs.
To compare two q-ROFNs, Liu and Wang [15] proposed a comparison method for q-ROFNs.
If If
if
if
Let S = {s
i
|i = 0, 1, . . . , l - 1} be a discrete linguistic term set with odd cardinality, where the term s
i
∈ S represents a possible value for a linguistic variable. For the linguistic set S, the following conditions should be satisfied: The set is ordered, namely, the s
i
> s
j
, if and only if i > j; Negation operator: Neg (s
i
) = s
j
, such that j = l - 1 - i; Maximum and minimum operators: max {s
i
, s
j
} = s
i
, if s
i
≥ s
j
; min {s
i
, s
j
} = s
i
, if s
i
≤ s
j
.
Take l = 7 for example, a possible set S can be: S= (s0, s1, s2, s3, s4, s5, s6)= very poor, poor, slightly poor, fair, slightly good, good, very good.
Based on the linguistic set, Xu [21] proposed uncertain linguistic set (ULS).
In addition, Xu [21] provided some operations for ULVs, which are shown in Definition 5.
q-Rung orthopair fuzzy uncertain linguistic set
By combing the ULS with q-ROFS, we propose the concept of q-ROULS.
For a q-ROULS A, the indeterminacy of x to the uncertain linguistic set [sθ(x), sτ(x)] is represented by π A (x) = (u A (x) q + v A (x) q -u A (x) q v A (x) q ) 1/q. For convenience, we call 〈 [sθ(x), sτ(x)] , (u A (x) , v A (x))〉 a q-ROULV, which can be denoted by 〈 [s θ , s τ ] , (u, v)〉.
Based on the operations for ULVs and for q-ROFNs, we provide the following operational laws for q-ROULVs.
a1 ⊕ a2 = a2 ⊕ a1; a1 ⊗ a2 = a2 ⊗ a1; λ (a1 ⊕ a2) = λa1 ⊕ λa2; λ1a1 ⊕ λ2a1 = (λ1 + λ2) a1;
To compare two q-ROULVs, we firstly provide the concepts of expected value and accuracy function of a q-ROULV.
Based on the two above concepts, a comparison law for q-ROULVs can be provided.
If E (a1) > E (a2), then a1 > a2; If E (a1) = E (a2), then
If H (a1) > H (a2), then a1 > a2;
If H (a1) = H (a2), then a1 = a2.
Sugeno [38] initiated the notation of fuzzy measure, which can be used to define a weight on each combination of criteria in the Choquet integral model. In this subsection, we introduce the definitions of fuzzy measure and Choquet integral.
ρ (φ) = 0, ρ (x) = 1 (boundary conditions) A, B ∈ X, A ⊆ B, then ρ (A) ≤ ρ (B) (monotonicity)
However, we generally need to determine 2 n - 2 values for n criteria, which make it quite complex, and thus it is not easy to give such fuzzy measure according to Definition 11. Therefore, Sugeno [38] defined the following ρ- fuzzy measure:
Where A ∪ B = φ, and the parameter ∂ ∈ [- 1, + ∞) denotes interaction between the attributes. In Equation (5), If ∂=0, then ∂- fuzzy measure reduces to ρ (A ∪ B) = ρ (A) + ρ (B), which is defined as additive measure.
In this situation, if all the elements in X are independent, we get
If ∂≻0, then ∂- fuzzy measure reduces toρ (A ∪ B) ≻ ρ (A) + ρ (B), which is defined as super-additive measure. If -1 ≤ ∂≺0, then∂- fuzzy measure reduces toρ (A ∪ B) ≺ ρ (A) + ρ (B), which is defined as sub-additive measure.
When using a fuzzy measure to model the importance of decision criteria set S, a well-known aggregation function is the Choquet integral [37].
To capture correlations among attributes, in this section, we propose a family of q-rung orthopair uncertain linguistic Choquet integral operators, such as q-ROULCA operator, q-ROULCG, q-ROULGCA operator and q-ROULGCG operator. Moreover, desirable properties of them are discussed.
The q-ROULCA and q-ROULCG operators
Based on the operational laws for q-ROULVs, the following theorem can be obtained.
When n = 2, by the operational law (1) and (3) in Definition 6, we have
Then
Thus, result is true for n = 2. If Equation (8) holds for n = k, that is
Thus, Equation (10) holds for n = k + 1.
Therefore, Equation (10) holds for all n, which completes the proof. □
In the followings, we discuss some desirable properties of the q-ROULCA operator.
Theorems 3, 4 can be easily proven, and thus the proofs are omitted.
It is noted that there is a parameter q in the q-ROULCA operator, which makes the aggregation processes flexible and feasible. Thus, we discuss some special cases of the q-ROULCA operator.
Similar to the definition of the q-ROULCA operator, we define the following q-rung orthopair uncertain linguistic Choquet geometric mean (q-ROULCG) operator.
Based on the operational laws for q-ROULVs, the following theorem can be obtained.
Parallel to Theorems 3, 4, the q-ROULCG operator have properties such as idempotency and boundedness under same conditions.
By giving different values of the parameters λ, q, we get the following special cases.
Based on the operations for q-ROULVs, the following theorem can be obtained.
The q-ROULGCA operators have properties similar to q-ROULCA operators such as idempotency, and boundedness under some conditions, which are omitted in order to save space.
By giving different values of the parameters λ, q, we get the following special cases.
Similar to the definition of q-ROULGCA operator, we define the following q-ROULGCG operator.
Based on the operational laws for q-ROULVs, the following theorem can be obtained.
By giving different values of the parameters λ, q, we get the following special cases.
In this section, utilize the proposed operators to deal with q-rung orthopair fuzzy uncertain linguistic MADM problems.
A typical q-rung orthopair uncertain linguistic MADM problem can be described as: Let X ={ x1, x2, . . . , x n } be a collection of alternatives, G ={ G1, G2, . . . , G t } be a set of attributes. For attribute G j (j = 1, 2, . . . , t) of alternative x i (i = 1, 2, . . . , n), decision maker is required to utilize a q-ROULV to express his evaluation information, which can be denoted asa ij =〈 [s θ ij , s τ ij ] , (u ij , v ij ) 〉. Finally, q-rung orthopair uncertain linguistic decision matrix can be obtained, which can be denoted as A = (a ij ) n×t. In the following, based on the proposed q-rung orthopair uncertain linguistic Choquet integral aggregation operators, a novel approach to solve this problem is introduced.
Numerical experiment
q-Rung orthopair uncertain linguistic decision matrix A
q-Rung orthopair uncertain linguistic decision matrix A
The decision-making steps based on the q-ROULCA operators
Take the alternative x1 for an example, the weight of the alternative x1 is calculated as follows:
Similarly, the weight of other alternatives x i (i = 1, 2, 3, 4) are obtained. Thus the weight matrix is shown in Table 2.
The weight matrix
The weight matrix
Therefore, the rank of the overall values isa1 ≻ a2 ≻ a4 ≻ a3.
The decision-making steps based on the q-ROULGCA operators
Therefore, the rank of the overall values is a1 ≻ a2 ≻ a3 ≻ a4.
The prominent characteristic of proposed operators is that they can model practical decision making problems more flexible with additional parameter. To reflect the influences of parameters λ, q on result, we utilize different values of parameters q, λ to rank the alternatives by the proposed q-ROULGCA operator, and results are shown in Fig. 1 and Table 3.
Expected values when λ = 2 and q ∈ (1, 10) by the q-ROULGCA operator. Rankings with different value of parameter
Figure 1 illustrates the expected values of the alternatives obtained by the q-ROULGCA operator if we take λ = 2, q ∈ (1, 10). From Fig. 1, we can easily find the ranking results may be slightly different for different parameters. For example, when we take λ = 2, q ∈ [1, 6] , then the ranking is a1 ≻ a2 ≻ a3 ≻ a4; when we take λ = 2, q ∈(6, 10] then the ranking isa1≻ a2 ≻ a4 ≻ a3 ; However, the best choice is always x1 . Moreover, as the values of q increase from 1 to 10, the score values become smaller and smaller.
Table 3 illustrates the expected values of the alternatives obtained by the q-ROULGCA operator as assigned different values λ. It is easily find that we get different scores in Table 3 when we change the parameter λ, which makes decision making more flexible and can meet the needs of different types of DMs.
The new uncertain linguistic fuzzy decision matrix
Comparison of rankings with different aggregation operators
From Table 4, we know the elements a12, a21, a32 and a44 are (0.8, 0.7), respectively. Given 0.8 + 0.7 > 1 and 0.82 + 0.72 > 1, the evaluation attribute value (0.8, 0.7) cannot be expressed by IFNs and PFNs. Thus, as shown in Table 5, IULWGA, IULOWG, PULPWG and PULPWGA operators cannot solve the above problem as the membership degree and non-membership degree do not satisfy the constraint conditions of IFNs and PFNs. However, the proposed approach based on the q-ROULCA, q-ROULCG q-ROULGCA and q-ROULGCA operators can still work as (0.8, 0.7) can be represented by q–ROFNs by adjusting the value of q. Therefore, the applicable range of our approach is wider than methods based on the IULWGA, IULOWG operators, PULPWG and PULPWGA operators. Further, according to Table 5, we find that the ranking results may be slightly different by our MADM method, but the best alternative is always x1.
By further analysis, we can draw the following conclusions. The proposed q-ROULS can more precisely ex press the uncertainty than the IFULS and PFULS. In today’s complex decision-making environment, the q-ROULS proposed in this paper are a very powerful tool in decision making. The approaches in this paper are the optimization of the existing methods. The IULCA and IULCG, PULCA and PULCG operators are the special cases of proposed operators in this paper when q = 1, 2, respectively. Furthermore, the proposed operators can include almost all of the geometric and arithmetic aggregation operators for IULVs, PULVs, and q-ROULVs according to different values of parameter. Our methods can efficiently take the correlations among the decision data into account. Furthermore, when we change the values of parameters, we get different scores, which makes decision making more flexible and can meet the needs of different types of DMs.
Based on the comparisons and analysis above, the methods proposed in this paper are more superior than other approaches.
Conclusions
In this paper, we have proposed q-ROULS by combining q-ROFS and ULS, which is a further generalization of the concepts of IFULS and PFULS. Then, based on q-rung orthopair fuzzy uncertain linguistic Choquet integral operators, we developed a framework to the MADM problem under q-rung orthopair fuzzy uncertain linguistic environment. The strengths of the proposed method have been discussed via comparative analysis.
The main contributions of this paper are four aspects. First, a new linguistic fuzzy set called q-ROULS are proposed. The proposed q-ROULS more powerful and useful than PFULS and IULS in aspect of dealing with the vagueness and fuzzy information. Second, some novel Choquet integral aggregation operators for q-ROULS are proposed. The proposed operators not only consider the correlations among attributes, but also demonstrate high generality than exiting aggregation operators. Third, a novel approach to MADM with q-rung orthopair uncertain linguistic information is proposed. Compared with the existing MADM methods, the novel method has wider and more flexible applicable scope for MADM problems, and can be further applied to other MADM problems, such as pilot hospitals selection, supplier selection and so on.
In the future study, we can extend the proposed operators to some other fuzzy sets [43–45], and further explore the generalisation of Choquet integral by combining with other aggregation operators, such as Heronian mean, and Frank operator.
Footnotes
Acknowledgments
This work was partially supported by a Key Project of National Natural Science Foundation of China (NSFC) with grant number 71532002, Fundamental Research Funds for the Central Universities (No.2019YJS055) and Beijing Logistics Informatics Research Base.
