Uncertain Linguistic Aggregation Distance Measures and Their Application to Group Decision Making

Abstract

We introduce a method based on distance measures for group decision making under uncertain linguistic environment. We develop some uncertain linguistic aggregation distance measures called the uncertain linguistic weighted distance (ULWD) measure, the uncertain linguistic ordered weighted distance (ULOWD) measure, and the uncertain linguistic hybrid weighted distance (ULHWD) measure. We study some of their characteristic, and we prove that the ULWD and the ULOWD are special cases of the ULHWD measure. Finally, we develop an application of the ULHWD measure in a group decision making problem concerning the evaluation of university faculty for tenure and promotion with uncertain linguistic information.

1. Introduction

As a common tool for measuring the deviations of different arguments, distance measures are fundamentally important in a variety of scientific fields such as decision making, pattern recognition, machine learning, and market prediction. A variety of distance measures have been introduced and investigated in the past several decades [1–11]. Most existing distance measures are the weighted distance measures, including some well-known distance measures such as the weighted Hamming distance and the weighted Euclidean distance. One problem of these distance measures is that they take the importance of the given individual distances into consideration, and then aggregate the difference elements together with their weights. Recently, motivated by the idea of the ordered weighted averaging (OWA) operator [12], Xu and Chen [13] introduced the ordered weighted distance (OWD) measure, which emphasizes the importance of the ordered position of the given individual distances instead of weighting arguments themselves. The prominent characteristic of the OWD is that it can relieve (or intensify) the influence of unduly large or unduly small deviations on the aggregation results by assigning them low (or high) weights. This desirable characteristic makes the OWD very useful in many actual fields such as group decision making, medical diagnosis, data mining, and pattern recognition. Since it was introduced, the OWD has been studied by many authors. Yager [14] generalized the OWD and provided a variety of ordered weighted averaging norms, based on which he proposed several similarity measures. Merigó and Gil-Lafuente [15] introduced an ordered weighted averaging distance (OWAD) operator and gave its application in the selection of financial products. The OWD measures are generally used to deal with situations where the input data are expressed in exact numerical values. Zeng and Su [16] extended the OWD to uncertain situation with intuitionistic fuzzy information [17] and developed an intuitionistic fuzzy OWD (IFOWD) operator. Zeng [18] developed some intuitionistic fuzzy aggregation distance measures, such as the intuitionistic fuzzy ordered weighted distance (IFOWD) measure, interval-valued intuitionistic fuzzy ordered weighted distance (IVIFOWD) measure, intuitionistic fuzzy hybrid weighted distance (IFHWD) measure, and interval-valued intuitionistic fuzzy hybrid weighted distance (IVIFHWD) measure and applied them to group decision making. Xu [19] developed some fuzzy ordered distance measures, including linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure. For further research on the use of the OWA operator in distance measures, see, for example, [10, 20–23].

In many complicated practical situations, the decision makers are willing or able to provide only uncertain linguistic information because of time pressure, lack of knowledge, or data, and their limited expertise related to the problem domain [24–28]. Therefore, it is necessary to pay attention to this issue. In this paper, we will develop some uncertain linguistic aggregation distance measures including the uncertain linguistic weighted distance (ULWD), the uncertain linguistic ordered weighted distance (ULOWD) measure, and uncertain linguistic hybrid weighted distance (ULHWD) measure. The fundamental aspect of the ULWD measure is that it only takes the importance of the given individual distances into consideration, and then aggregates these difference elements together with their weights. The ULOWD and ULHWD measures are extensions of the OWD with uncertain linguistic variables. The main advantage of the ULOWD and ULHWD is that they can alleviate the influence of unduly large (or small) deviations on the aggregation results by assigning them low (or high) weights. We have proved that the ULWD and the ULOWD are specials of the ULHWD measure. Moreover, we will apply the ULHWD measure to group decision making with uncertain linguistic information.

This paper is organized as follows. In Section 2, we briefly describe some basic aggregation operators and distance measures. In Section 3, we introduce the uncertain linguistic variables and the ULWD measure. In Section 4, we present the ULOWD and ULHWD measure in Section 5. In Section 6, we briefly describe the decision making process based on the ULHWD measure and we give a numerical example in Section 7. Section 8 summarizes the main conclusions of the paper.

2. Preliminaries

In this section we briefly review some basic distance measures, the OWA operator and the OWD measure.

Among the existing weighted distance measures, the weighted Hamming distance and the weighted Euclidean distance are the two most widely used ones, which can be described as follows.

Let 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛) and 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛) be two collections of real numbers, and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) be the weighting vector of the absolute difference |𝛼_𝑗−𝛽_𝑗|(𝑗=1,2,…,𝑛), where 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1, then(1)the weighted Hamming distance measure is WHD𝛼,𝛽=^𝑛_𝑗=1𝑤_𝑖𝛼_𝑗−𝛽_𝑗; (1)

(2) the weighted Euclidean distance is WED𝛼,𝛽=^𝑛_𝑗=1𝑤_𝑖𝛼_𝑗−𝛽_𝑗². (2)

Obviously, the above two weighted distance measures only take the importance of each difference element of data into consideration and then aggregate the difference elements together with their weights.

The ordered weighted averaging (OWA) operator introduced by Yager [12] provides a parameterized family of aggregation operators that include the maximum, the minimum, and the average criteria. The fundamental aspect of the OWA operator is the reordering step: it first reorders all the given arguments in descending order and then weights these ordered arguments and finally aggregates all these ordered weighted arguments into a collective one. Since its appearance, the OWA operator has been studied in a wide range of studies [29–44]. It can be defined as follows.

Definition 1. An OWA operator of dimension 𝑛 is a mapping OWA: 𝑅^𝑛→𝑅, which has an associated weighting 𝑊 with 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1, such that: OWA𝑎₁,…,𝑎_𝑛=^𝑛_𝑗=1𝑤_𝑗𝑏_𝑗, (3) where 𝑏_𝑗 is the 𝑗th largest of the 𝑎_𝑖.

Recently, motivated by the idea of the OWA operator, Xu and Chen [13] developed an ordered weighted distance measure (OWD).

Definition 2. Let 𝛼=(𝛼₁,𝛼₂,…,𝛼_𝑛) and 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛) be two collections of real numbers, and let 𝑑(𝛼_𝑗,𝛽_𝑗)=|𝛼_𝑗−𝛽_𝑗| be the distance between 𝛼_𝑗 and 𝛽_𝑗, then OWD𝛼,𝛽=^𝑛_𝑗=1𝑤_𝑗𝑑𝛼_𝜎(𝑗),𝛽_𝜎(𝑗)^𝜆1/𝜆 (4) is called an ordered weighted distance (OWD) between 𝛼 and 𝛽, where 𝜆>0, (𝜎(1),𝜎(2),…,𝜎(𝑛)) is any permutation of (1,2,…,𝑛), such that 𝑑𝛼_{𝜎(𝑗−1)},𝛽_{𝜎(𝑗−1)}≥𝑑𝛼_𝜎(𝑗),𝛽_𝜎(𝑗),𝑗=1,2,…,𝑛 (5) and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) be the weight vector of the absolute difference |𝛼_𝑗−𝛽_𝑗|(𝑗=1,2,…,𝑛), where 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1.
Especially, if 𝜆=1, then the OWD measure is called an ordered weighted Hamming distance (OWHD) measure: OWHD𝛼,𝛽=^𝑛_𝑗=1𝑤_𝑗𝑑𝛼_𝜎(𝑗),𝛽_𝜎(𝑗). (6)
In the case of 𝜆=2, then the OWD measure is reduced to the ordered weighted Euclidean distance (OWED) measure: OWED𝛼,𝛽=^𝑛_𝑗=1𝑤_𝑗𝑑𝛼_𝜎(𝑗),𝛽_𝜎(𝑗)^21/2. (7)
The OWD measure is very suitable to be used in many actual fields, including group decision making, medical diagnosis, data mining, and pattern recognition [13, 18]. However, the OWD measure is mainly used to aggregate or measure the data taking the form of exact numerical; in what follows, we extend the OWD to accommodate the situation in which the input data is provided with uncertain linguistic information.

3. Uncertain Linguistic Variables and Uncertain Linguistic Weighted Distance (ULWD) Measure

The linguistic approach is an approximate technique which represents qualitative aspects as linguistic values by means of linguistic variables [8, 24, 45, 46]. Suppose that 𝑆={𝑠_𝛼∣𝛼=−𝑡,…,0,1,…,𝑡} is a finite and totally ordered discrete term set, where 𝑠_𝛼 represents a possible value for a linguistic variable. For example, a set of nine terms 𝑆 could be define as 𝑆=𝑠₋₄=extremelypoor,𝑠₋₃𝑠=verypoor,₋₂=poor,𝑠₋₁𝑠=slightlypoor,₀=fair,𝑠₁=slightlygood,𝑠₂𝑠=good,₃=verygood,𝑠₄=extremelygood. (8)

To preserve all the given information, Xu [8] extended the discrete term set 𝑆 to a continuous term set 𝑆={𝑠_𝛼∣𝛼∈[−𝑡,𝑡]}, whose elements also meet all the characteristics above, and where, if 𝑠_𝛼∈𝑆, then we call 𝑠_𝛼 the original term, otherwise, we call 𝑠_𝛼 the virtual term; In general, the decision maker uses the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in operation [8, 26].

In many situations, the decision information is expressed in the form of uncertain linguistic variables [24–28] because of time pressure, lack of knowledge or data, and their limited expertise related to the problem domain. Let 𝑠̃=[𝑠_𝛼,𝑠_𝛽], where 𝑠_𝛼,𝑠_𝛽∈𝑆, 𝑠_𝛼, and 𝑠_𝛽 are the lower and the upper limits, respectively, then we call 𝑠̃ the uncertain linguistic variable [26, 27]. Let 𝑆 be the set of all the uncertain linguistic variables.

For any three uncertain linguistic variables 𝑠̃=[𝑠_𝛼,𝑠_𝛽], 𝑠₁̃=[𝑠_𝛼1,𝑠_𝛽1], 𝑠₂̃=[𝑠_𝛼2,𝑠_𝛽2], the following two operational laws are valid:(1)𝑠₁̃⊕𝑠₂̃=[𝑠_𝛼1,𝑠_𝛽1]⊕[𝑠_𝛼2,𝑠_𝛽2]=[𝑠_𝛼1⊕𝑠_𝛼2,𝑠_𝛽1⊕𝑠_𝛽2]=[𝑠_𝛼1+𝛼2,𝑠_𝛽1+𝛽2];(2)𝜆𝑠̃=𝜆[𝑠_𝛼,𝑠_𝛽]=[𝜆𝑠_𝛼,𝜆𝑠_𝛽]=[𝑠_𝜆𝛼,𝑠_𝜆𝛽], where 𝜆∈[0,1].

Xu [9] defined an uncertain linguistic distance between 𝑠₁̃ and 𝑠₂̃ as follows.

Definition 3. Let 𝑠₁̃=[𝑠_𝛼1,𝑠_𝛽1], 𝑠₂̃=[𝑠_𝛼2,𝑠_𝛽2] be two uncertain linguistic labels, such that 𝑑𝑠₁̃,𝑠₂̃=12𝛼₁−𝛼₂+𝛽₁−𝛽₂ (9) is called an uncertain linguistic distance (ULD) between 𝑠₁̃ and 𝑠₂̃.
Based on the above information, now we can define the following uncertain linguistic weighted distance.

Definition 4. Let 𝐴=(𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃) and 𝐵=(𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃) be two collections of uncertain linguistic labels, and let 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃) be the distance between 𝑠_𝑗̃=[𝑠_𝛼𝑗,𝑠_𝛽𝑗] and let 𝑠^′_𝑗̃=[𝑠^′_𝛼𝑗,𝑠^′_𝛽𝑗], then ULWHD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠_𝑗̃,𝑠^′_𝑗̃ (10) is called an uncertain linguistic weighted Hamming distance (ULWHD) between 𝐴 and 𝐵, where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weighting vector associated with the 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃), such that 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1.

Definition 5. Let 𝐴=(𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃) and 𝐵=(𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃) be two collections of uncertain linguistic labels, and let 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃) be the distance between 𝑠_𝑗̃=[𝑠_𝛼𝑗,𝑠_𝛽𝑗] and 𝑠^′_𝑗̃=[𝑠^′_𝛼𝑗,𝑠^′_𝛽𝑗], then ULWED(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠_𝑗̃,𝑠̃^′_𝑗² (11) is called an uncertain linguistic weighted Euclidean distance (ULWED) between 𝐴 and 𝐵, where 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weighting vector associated with the 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃), such that 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1.
Now we generalized both (10) and (11) to the following form: ULWD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠_𝑗̃,𝑠̃^′_𝑗^𝜆1/𝜆 (12) which is called an uncertain linguistic weighted distance (ULWD) between 𝐴 and 𝐵. Specially, if 𝜆=1, then the ULWD measure is reduced to the ULWHD measure (10). If 𝜆=2, then the ULWD measure is reduced to the ULWED measure (11).

Example 6. Let 𝐴=(𝑠₁̃,𝑠₂̃,𝑠₃̃,𝑠₄̃)=([𝑠₀,𝑠₁],[𝑠₋₁,𝑠₁],[𝑠₁,𝑠₂],[𝑠₀,𝑠₃]), 𝐵=(𝑠^′₁̃,𝑠^′₂̃,𝑠^′₃̃,𝑠^′₄̃)=([𝑠₋₁,𝑠₀],[𝑠₁,𝑠₂],[𝑠₋₁,𝑠₂],[𝑠₁,𝑠₃]) be two collections of uncertain linguistic labels, then 𝑑𝑠₁̃,𝑠^′₁̃=12(|0−(−1)|+|1−0|)=1. (13) Similarly, we can get 𝑑𝑠₂̃,𝑠^′₂̃=1.5,𝑑𝑠₃̃,𝑠^′₃̃=1,𝑑𝑠₄̃,𝑠^′₄̃=0.5. (14) Suppose that 𝑤=(0.2,0.3,0.3,0.2), and without loss of generality, let 𝜆=2, then by (12), we can get the weighted distance between 𝐴 and 𝐵 as follows ()=ULWD𝐴,𝐵0.2×1²+0.3×1.5²+0.3×1²+0.2×0.5^21/2=1.225. (15)
The fundamental aspect of the ULWD measure is that it only takes the importance of the given individual distances into consideration, and then aggregates these difference elements together with their weights under the parameter 𝜆.

4. Uncertain Linguistic OWD (ULOWD) Measure

Based on the (7) and (12), we define an uncertain linguistic ordered weighted distance measure as follows.

Definition 7. Let 𝐴=(𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃) and 𝐵=(𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃) be two collections of uncertain linguistic labels, and 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃) be the distance between 𝑠_𝑗̃=[𝑠_𝛼𝑗,𝑠_𝛽𝑗] and 𝑠^′_𝑗̃=[𝑠̃^′_𝛼𝑗,𝑠̃^′_𝛽𝑗], then ULOWD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎𝑗^𝜆1/𝜆 (16) is called an uncertain linguistic ordered weighted distance (ULOWD) between 𝐴 and 𝐵, where (𝜎(1),𝜎(2),…,𝜎(𝑛)) is any permutation of (1,2,…,𝑛), such that 𝑑𝑠̃_{𝜎(𝑗−1)},𝑠̃^′_𝜎𝑗−1≥𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎𝑗,𝑗=1,2,…,𝑛 (17)𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weighting vector associated with the ULOWD measure such that 𝑤_𝑗∈[0,1] and ∑^𝑛_𝑗=1𝑤_𝑗=1. Specially, if 𝜆=1, then the ULOWD measure is called an uncertain linguistic ordered weighted Hamming distance (ULOWHD) measure: ULOWHD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎𝑗. (18)
If 𝜆=2, then the ULOWD measure is reduced to the uncertain linguistic ordered weighted Euclidean distance (ULOWED) measure: ULOWED(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎𝑗^21/2. (19)

Example 8. Let 𝐴=(𝑠₁̃,𝑠₂̃,𝑠₃̃,𝑠₄̃)=([𝑠₀,𝑠₁],[𝑠₋₁,𝑠₁],[𝑠₁,𝑠₂],[𝑠₀,𝑠₃]), 𝐵=(𝑠^′₁̃,𝑠^′₂̃,𝑠^′₃̃,𝑠^′₄̃)=([𝑠₋₁,𝑠₀],[𝑠₁,𝑠₂],[𝑠₋₁,𝑠₂],[𝑠₁,𝑠₃]) be two collections of uncertain linguistic labels, then 𝑑𝑠₁̃,𝑠^′₁̃=12(|0−(−1)|+|1−0|)=1. (20) Similarly, we can get 𝑑𝑠₂̃,𝑠^′₂̃=1.5,𝑑𝑠₃̃,𝑠^′₃̃=1,𝑑𝑠₄̃,𝑠^′₄̃=0.5. (21) Reordering the above individual distances in descending order, then we get 𝑑𝑠̃_𝜎(1),𝑠̃^′_𝜎(1)=1.5,𝑑𝑠̃_𝜎(2),𝑠̃^′_𝜎(2)=𝑑𝑠̃_𝜎(3),𝑠̃^′_𝜎(3)𝑑𝑠̃=1,_𝜎(4),𝑠̃^′_𝜎(4)=0.5. (22)
Suppose that 𝑤=(0.2,0.3,0.3,0.2), and without loss of generality, let 𝜆=2, then by (16), we can get the ordered weighted distance between 𝐴 and 𝐵 as follows ()=ULOWD𝐴,𝐵0.2×1.5²+0.3×1²+0.3×1²+0.2×0.5^21/2=1.1. (23)
From the above definitions, we know that the ULWD measure takes the importance of given individual distances into consideration, while the ULOWD measure only emphasizes the importance of the ordered position of the given individual distances, it weights the ordered position of the given individual distances instead of weighting arguments themselves. Therefore, weights represent different aspects in both the ULWD and ULOWD measures. However, both the ULWD and ULOWD operators consider only one of them. To solve this drawback, in the following, we will propose an uncertain linguistic hybrid weighted distance (ULHWD) measure.

5. Uncertain Linguistic Hybrid Weighted Distance (ULHWD) Measure

Definition 9. Let 𝐴=(𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃) and 𝐵=(𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃) be two collections of uncertain linguistic labels, and let 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃) be the distance between 𝑠_𝑗̃=[𝑠_𝛼𝑗,𝑠_𝛽𝑗] and 𝑠^′_𝑗̃=[𝑠̃^′_𝛼𝑗,𝑠̃^′_𝛽𝑗], then ULHWD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑̇𝑎_𝜎(𝑗),𝑎_𝜎(𝑗)^1/𝜆 (24) is called an uncertain linguistic hybrid weighted distance (ULHWD) between 𝐴 and 𝐵, where 𝑑(̇𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎(𝑗)) represents the 𝑗th largest of the weighted distance 𝑑(̇𝑠_𝑗̃,𝑠^′_𝑗̃) (here 𝑑(̇𝑠_𝑗̃,𝑠^′_𝑗̃)=𝑛𝜔_𝑗(𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃))^𝜆, 𝑗=1,2,…,𝑛), 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛) is the weighting vector associated with the ULHWD measure, 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 is the weighting vector of the 𝑑(𝑠_𝑗̃,𝑠^′_𝑗̃), with 𝜔_𝑖∈[0,1] and the the sum of the weights is 1. 𝑛 is the balancing coefficient, which plays a role of balance (in such a case, if the vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛) approaches (1/𝑛,1/𝑛,…,1/𝑛), then (𝑛𝜔₁|𝑠₁̃−𝑠^′₁̃|^𝜆,𝑛𝜔₂|𝑠₂̃−𝑠^′₂̃|^𝜆,…,𝑛𝜔_𝑛|𝑠_𝑛̃−𝑠^′_𝑛̃|^𝜆) approaches (|𝑠₁̃−𝑠^′₁̃|^𝜆,|𝑠₂̃−𝑠^′₂̃|^𝜆,…,|𝑠_𝑛̃−𝑠^′_𝑛̃|^𝜆).

Example 10. Let 𝐴=(𝑠₁̃,𝑠₂̃,𝑠₃̃,𝑠₄̃,𝑠₅̃)=([𝑠₁,𝑠₃],[𝑠₋₁,𝑠₂],[𝑠₀,𝑠₂],[𝑠₋₂,𝑠₀],[𝑠₋₁,𝑠₃]), 𝐵=(𝑠₁̃,𝑠₂̃,𝑠₃̃,𝑠₄̃,𝑠₅̃)=([𝑠₃,𝑠₄],[𝑠₀,𝑠₁],[𝑠₋₂,𝑠₂],[𝑠₋₁,𝑠₂],[𝑠₀,𝑠₃]) be two collections of uncertain linguistic labels, then 𝑑𝑠₁̃,𝑠^′₁̃=12(|1−3|+|3−4|)=1.5. (25) Similarly, we can get 𝑑𝑠₂̃,𝑠^′₂̃=1,𝑑𝑠₃̃,𝑠^′₃̃𝑑𝑠̃=1,₄,𝑠^′₄̃=1.5,𝑑𝑠₅̃,𝑠^′₅̃=0.5. (26) Suppose that 𝜔=(0.10,0.15,0.25,0.30,0.20), and without loss of generality, let 𝜆=2, then we can get 𝑑̇𝑠₁,𝑠^′₁=5×0.10×1.5²=1.125. (27) Similarly, we can have 𝑑̇𝑠₂,𝑠^′₂=0.75,𝑑̇𝑠₃,𝑠^′₃𝑑̇𝑠=1.25,₄,𝑠^′₄=3.375,𝑑̇𝑠₅,𝑠^′₅=0.25. (28) Reordering the above weighted distances in descending order, then we get 𝑑̇𝑠̃_𝜎(1),𝑠̃^′_𝜎(1)=3.375,𝑑̇𝑠̃_𝜎(2),𝑠̃^′_𝜎(2)𝑑̇𝑠̃=1.25,_𝜎(3),𝑠̃^′_𝜎(3)=1.125,𝑑̇𝑠̃_𝜎(4),𝑠̃^′_𝜎(4)𝑑̇𝑠̃=0.75,_𝜎(5),𝑠̃^′_𝜎(5)=0.25. (29) Let the weighting vector associating with the ULHWD measure be 𝑤=(0.11,0.24,0.30,0.24,0.11), which is derived by the normal distribution based method [37], then by (24), we can get the hybrid weighted distance between 𝐴 and 𝐵: ()=()ULHWD𝐴,𝐵0.11×3.375+0.24×1.25+0.30×1.125+0.24×0.75+0.11×0.25^1/2=1.1028. (30)

Theorem 11. The ULWD measure is a special case of the ULHWD measure.

Proof. Let 𝑤=(1/𝑛,1/𝑛,…,1/𝑛), then ULHWD(𝐴,𝐵)=^𝑛_𝑗=1𝑤_𝑗𝑑̇𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎𝑗^1/𝜆=1𝑛^𝑛_𝑗=1𝑑̇𝑠_𝑗̃,𝑠̃^′_𝑗^1/𝜆=1𝑛^𝑛_𝑗=1𝑛𝜔_𝑗𝑑𝑠_𝑗̃,𝑠̃^′_𝑗^𝜆1/𝜆=^𝑛_𝑗=1𝜔_𝑗𝑑𝑠_𝑗̃,𝑠̃^′_𝑗^𝜆1/𝜆=ULWD(𝐴,𝐵) (31) which completes the proof of Theorem 11.

Theorem 12. The ULOWD measure is a special case of the ULHWD measure.

Proof. Let 𝜔=(1/𝑛,1/𝑛,…,1/𝑛), then 𝑑̇𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎(𝑗)=𝑛𝜔_𝑗𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎(𝑗)^𝜆=𝑑𝑠̃_𝜎(𝑗),𝑠̃^′_𝜎(𝑗)^𝜆 (32) which completes the proof of Theorem 12.

From Definition 9 and the above theorems, we know that(1)the ULHWD measure first weights the given individual distances, and then reorders the weighted individual distances in descending order and weights these ordered individual distances by the ULHWD weights and finally aggregates these individual distances into a collective one under the parameter 𝜆;(2)the ULHWD measure generalizes both the ULWD and ULOWD measure and reflects the importance degrees of both the given individual distances and their ordered positions.

In addition, a prominent characteristic of the ULHWD measure is that it can relieve (or intensify) the influence of unduly large or unduly small difference elements on the aggregation results by assigning them low (or high) weights.

6. An Approach to Group Decision Making Based on the ULHWD Measure

For a group decision making problem, let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a finite set of alternatives, let and 𝐸={𝑒₁,𝑒₂,…,𝑒_𝑚} be the set of decision makers (whose weight vector is 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑚),𝑣_𝑘≥0,∑^𝑚_𝑘=1𝑣_𝑘=1). The decision makers 𝑒_𝑘(𝑘=1,2,…,𝑚) provide their preferences with uncertain linguistic labels 𝑎_𝑘𝑗(𝑘=1,2,…,𝑚;𝑗=1,2,…,𝑛) over all the alternatives 𝑥_𝑗(𝑗=1,2,…,𝑛)  in respect to a criterion. For convenience, we denote the preference vectors of all the decision makers 𝑒_𝑘(𝑘=1,2,…,𝑚) as 𝐴_𝑘=𝑎_𝑘1,𝑎_𝑘2,…,𝑎_𝑘𝑛,𝑘=1,2,…,𝑚. (33)

Based on the above decision information, we can utilize the ULHWD measure to develop an approach to reaching consensus of group opinions, which can be summarized as follows [18].

Step 1. Calculate the collective preference vector 𝐴₀=(𝑎₀₁,𝑎₀₂,…,𝑎_0𝑛) by using the uncertain linguistic weighted averaging (ULWA) operator [26], where 𝑎_0𝑗=𝑣₁𝑎_1𝑗⊕𝑣₂𝑎_2𝑗⊕⋯𝑣_𝑚𝑎_𝑚𝑗,𝑗=1,2,…𝑛. (34)

Step 2. By (24), we calculate ULHWD𝐴_𝑘,𝐴₀=^𝑛_𝑗=1𝑤_𝑘𝑗𝑑̇𝑎_{𝜎(𝑘𝑗)},𝑎_𝜎(0𝑗)^1/𝜆 (35) which is the distance between the preference vectors 𝐴_𝑘 and 𝐴₀, where 𝑑(̇𝑎_{𝜎(𝑘𝑗)},𝑎_𝜎(0𝑗)) is the 𝑗th largest of the 𝑑(̇𝑎_𝑘𝑗,𝑎_0𝑗) (here 𝑑(̇𝑎_𝑘𝑗,𝑎_0𝑗)=𝑛𝜔_𝑗𝑑(𝑎_𝑘𝑗,𝑎_0𝑗)), 𝑗=1,2,…,𝑛, the weighting vector associating with the ULHWD measure can be derived by using some determining methods like the normal distribution based method, see [37] for more details.

Step 3. If all ULHWD(𝐴_𝑘,𝐴₀)≤𝜂(𝑘=1,2,…,𝑚), then the group is of acceptable consensus, where 𝜂 is the threshold value of acceptable consensus, which can be determined by the group in practical applications. Otherwise, if there exists some 𝑘₀, such that ULHWD(𝐴_𝑘0,𝐴₀)>𝜂, then we will return 𝐴_𝑘0 (together with 𝐴₀ as a reference) to the decision maker 𝑒_𝑘 for revaluation and repeat this procedure until ULHWD(𝐴_𝑘0,𝐴₀)≤𝜂 or the process will stop as the repetition times reach the maximum number predefined by the group.

7. Illustrative Example

A group decision problem of evaluating university faculty for tenure and promotion (adapted from [13]) is used to illustrate the developed approach. One main criterion used is “teaching.” There are five decision makers 𝑒_𝑘(𝑘=1,2,…,5) (whose weighting vector is 𝑣=(0.20,0.15,0.25,0.30,0.10)) and there are seven faculty candidates (alternatives) 𝑥_𝑗(𝑗=1,2,…,7). By using linguistic label set (8), each decision maker 𝑒_𝑘 provides his/her preferences 𝑎_𝑘𝑗∈𝑆(𝑗=1,2,…,7) over all the faculty candidates 𝑥_𝑗(𝑗=1,2,…,7), shown in Table 1.

Table 1 

The decision makers' preferences.

For convenience, we denote the preferences of all the decision makers 𝑒_𝑘(𝑘=1,2,…,5) in the vector forms: 𝐴₁=𝑎₁₁,𝑎₁₂,…,𝑎₁₇=𝑠₁,𝑠₂,𝑠₀,𝑠₂,𝑠₁,𝑠₂,𝑠₂,𝑠₃,𝑠₂,𝑠₃,𝑠₀,𝑠₂,𝑠₂,𝑠₃,𝐴₂=𝑎₂₁,𝑎₂₂,…,𝑎₂₇=𝑠₀,𝑠₂,𝑠₁,𝑠₃,𝑠₂,𝑠₃,𝑠₀,𝑠₁,𝑠₁,𝑠₂,𝑠₁,𝑠₂,𝑠₁,𝑠₃,𝐴₃=𝑎₃₁,𝑎₃₂,…,𝑎₃₇=𝑠₁,𝑠₂,𝑠₀,𝑠₂,𝑠₁,𝑠₃,𝑠₁,𝑠₂,𝑠₂,𝑠₃,𝑠₁,𝑠₂,𝑠₀,𝑠₂,𝐴₄=𝑎₄₁,𝑎₄₂,…,𝑎₄₇=𝑠₁,𝑠₂,𝑠₂,𝑠₃,𝑠₀,𝑠₂,𝑠₀,𝑠₁,𝑠₂,𝑠₄,𝑠₁,𝑠₂,𝑠₂,𝑠₃,𝐴₅=𝑎₅₁,𝑎₅₂,…,𝑎₅₇=𝑠₀,𝑠₁,𝑠₁,𝑠₂,𝑠₁,𝑠₂,𝑠₀,𝑠₁,𝑠₀,𝑠₃,𝑠₂,𝑠₃,𝑠₁,𝑠₂. (36) Then, we can calculate the collective preference vector by using (34): 𝐴₀=𝑎₀₁,𝑎₀₂,…,𝑎₀₇=𝑠_0.75,𝑠_1.9,𝑠_0.85,𝑠_2.45,𝑠_0.85,𝑠_2.4,𝑠_0.65,𝑠_1.65,𝑠_1.65,𝑠_3.75,𝑠_0.90,𝑠_2.1,𝑠_1.25,𝑠_2.65. (37) Then, utilize (9) to calculate the distance 𝑑(𝑎_𝑘𝑗,𝑎_0𝑗) of the corresponding collective preference value 𝑎_0𝑗 and the corresponding preference value 𝑎_𝑘𝑗 provided by the decision maker 𝑒_𝑘: 𝑑𝑎₁₁,𝑎₀₁=0.175,𝑑𝑎₁₂,𝑎₀₂𝑑𝑎=0.75,₁₃,𝑎₀₃𝑑𝑎=0.275₁₄,𝑎₀₄=1.35,𝑑𝑎₁₅,𝑎₀₅𝑑𝑎=0.55,₁₆,𝑎₀₆𝑑𝑎=0.50₁₇,𝑎₀₇=0.55,𝑑𝑎₂₁,𝑎₀₁𝑑𝑎=0.425,₂₂,𝑎₀₂𝑑𝑎=0.35₂₃,𝑎₀₃=0.875,𝑑𝑎₂₄,𝑎₀₄𝑑𝑎=0.65,₂₅,𝑎₀₅𝑑𝑎=1.20₂₆,𝑎₀₆=0.10,𝑑𝑎₂₇,𝑎₀₇𝑑𝑎=0.30,₃₁,𝑎₀₁𝑑𝑎=0.175₃₂,𝑎₀₂=0.75,𝑑𝑎₃₃,𝑎₀₃𝑑𝑎=0.325,₃₄,𝑎₀₄𝑑𝑎=0.35₃₅,𝑎₀₅=0.55,𝑑𝑎₃₆,𝑎₀₆𝑑𝑎=0.10,₃₇,𝑎₀₇𝑑𝑎=0.95₄₁,𝑎₀₁=0.175,𝑑𝑎₄₂,𝑎₀₂𝑑𝑎=0.85,₄₃,𝑎₀₃𝑑𝑎=0.625₄₄,𝑎₀₄=0.65,𝑑𝑎₄₅,𝑎₀₅𝑑𝑎=0.30,₄₆,𝑎₀₆𝑑𝑎=0.10₄₇,𝑎₀₇=0.55,𝑑𝑎₅₁,𝑎₀₁𝑑𝑎=0.825,₅₂,𝑎₀₂𝑑𝑎=0.30₅₃,𝑎₀₃=0.275,𝑑𝑎₅₄,𝑎₀₄𝑑𝑎=0.65,₅₅,𝑎₀₅𝑑𝑎=1.20₅₆,𝑎₀₆=1.0,𝑑𝑎₅₇,𝑎₀₇=0.45. (38)

Without loss of generality, let 𝜆=1 and 𝜔=(0.12,0.15,0.10,0.13,0.14,0.20,0.16), suppose that the weighting vector associating with the ULHWD measure is 𝑤=(0.07,0.13,0.19,0.22,0.19,0.13,0.07), which is derived by using the normal distribution based method [37], then we calculate distance ULHWD(𝐴_𝑘,𝐴₀) between the preference vectors 𝐴_𝑘 and 𝐴₀: ULHWD𝐴₁,𝐴₀=0.59,ULHWD𝐴₂,𝐴₀𝐴=0.48,ULHWD₃,𝐴₀=0.42,ULHWD𝐴₄,𝐴₀𝐴=0.44,ULHWD₅,𝐴₀=0.66. (39)

Let us suppose the threshold value of acceptable consensus is 𝜂=0.60, then ULHWD(𝐴_𝑘,𝐴₀)<0.60  (𝑘=1,2,3,4), ULHWD(𝐴₅,𝐴₀)>0.60, and thus, we need to return 𝐴₅ (together with 𝐴₀ as a reference) to the decision maker 𝑒₅ for revaluation. Suppose that the revaluated 𝐴₅ is 𝐴₅=𝑎₅₁,𝑎₅₂,…,𝑎₅₇=𝑠₀,𝑠₁,𝑠₁,𝑠₂,𝑠₁,𝑠₂,𝑠₀,𝑠₁,𝑠₀,𝑠₂,𝑠₁,𝑠₂,𝑠₁,𝑠₂. (40)

Then, we can calculate the collective preference vector by using (34): 𝐴₀=𝑎₀₁,𝑎₀₂,…,𝑎₀₇=𝑠_0.75,𝑠_1.9,𝑠_0.85,𝑠_2.45,𝑠_0.85,𝑠_2.4,𝑠_0.65,𝑠_1.65,𝑠_1.65,𝑠_3.05,𝑠_0.80,𝑠_2.0,𝑠_1.25,𝑠_2.65. (41) Respectively, then by (9) and (24) (let 𝜆=1), we get ULHWD𝐴₁,𝐴₀=0.50,ULHWD𝐴₂,𝐴₀𝐴=0.45,ULHWD₃,𝐴₀=0.35,ULHWD𝐴₄,𝐴₀𝐴=0.49,ULHWD₅,𝐴₀=0.51. (42)

Thus, all the distances ULHWD(𝐴_𝑘,𝐴₀)(𝑘=1,2,3,4,5) are 1ess than 0.60, which indicates that the group reaches consensus.

8. Conclusions

In this paper,we have suggested several extensions of the OWD measure when dealing with uncertain situations. The increasing complexity of the socioeconomic environment makes it more suitable for a decision maker to express his/her preferences over alternatives with uncertain linguistic information instead of exact numerical values. We have developed some uncertain linguistic aggregation distance measures. such as the uncertain linguistic weighted distance measure (ULWD), the uncertain linguistic ordered weighted distance measure (ULOWD), and the uncertain linguistic hybrid weighted distance measure (ULHWD). These developed distance measures are very suitable to deal with the situation where the input data is represented in uncertain linguistic information. We have analyzed that the ULHWD measure generalizes both the ULWD and ULOWD measure, and reflects the importance degrees of both the given difference element and their ordered positions. Finally, based on the ULHWD measure, we have proposed a consensus reaching process for group decision making with uncertain linguistic preference information.

Acknowledgments

The work was supported by the Projects in Science and Technique of Ningbo Municipal (no. 2012B82003), the MOE Project of Key Research Institute of Humanities and Social Sciences in Universities (no. 13JJD910002), and Humanities and Social Science Key Research Base (Statistics) of Zhejiang College.