Research Article  Open Access
DingHong Peng, TieDan Wang, ChangYuan Gao, Hua Wang, "Multigranular Uncertain Linguistic Prioritized Aggregation Operators and Their Application to Multiple Criteria Group Decision Making", Journal of Applied Mathematics, vol. 2013, Article ID 857916, 13 pages, 2013. https://doi.org/10.1155/2013/857916
Multigranular Uncertain Linguistic Prioritized Aggregation Operators and Their Application to Multiple Criteria Group Decision Making
Abstract
We investigate multiple criteria group decisionmaking problems in which there are priority relationships between the decision elements (criteria and experts), and decision information provided by decision makers takes the form of multigranular uncertain linguistic information. Firstly, some operational laws and possibility degree of multigranular uncertain linguistic variables are introduced. Then, some new linguistic aggregation operators based on the prioritized aggregation operator, such as the multigranular uncertain linguistic prioritized weighted average (MULPWA) operator and the multigranular uncertain linguistic prioritized ordered weighted average (MULPOWA) operator, are developed and their desirable properties are studied. The prominent characteristics of these proposed operators are that they can aggregate directly the uncertain linguistic variables whose values form the linguistic term sets with different granularities and convey the prioritization phenomenon among the aggregated arguments. Furthermore, based on the MULPWA and MULPOWA operators, an approach to deal with multiple criteria group decisionmaking problems under multigranular uncertain linguistic environments is developed. Finally, a practical example is provided to illustrate the multiple criteria group decisionmaking process.
1. Introduction
Due to the fact that experiences and judgments of humans are usually represented by words in their natural language, decision making with linguistic information is becoming a hot research topic and has received many excellent results in recent years [1â€“35]. When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CWW). In the specialized literature, the main linguistic computational models can be divided into four kinds: the approximate model based on extension principle [1]; the ordered language model [2]; the 2tuple model [8], and the virtual linguistic variables model [9]. Compared with the former two models, the latter two models can avoid losing any linguistic information moreover, Dong et al. [11, 12] proved that the latter two models can be mutually retranslated, and they are equivalent.
However, in many situations, the decision information is expressed in the form of uncertain linguistic variables which can be considered as intervals of linguistic terms because of time pressure, lack of knowledge or data, and limited expertise related to the problem domain. Therefore, Xu [13â€“16] presented the concept of uncertain linguistic variables, and then various uncertain linguistic aggregation operators have been proposed, such as the uncertain linguistic averaging (ULA) operator, uncertain linguistic weighted averaging (ULWA) operator, uncertain linguistic ordered weighted averaging (ULOWA) operator, uncertain linguistic hybrid aggregation (ULHA) operator, uncertain linguistic geometric mean (ULGM) operator, uncertain linguistic weighted geometric mean (ULWGM) operator, uncertain linguistic ordered weighted geometric (ULOWG) operator, uncertain linguistic hybrid geometric mean (ULHG) operator [17], induced uncertain linguistic OWA (IULOWA), induced uncertain linguistic ordered weighted geometric(IULOWG) operator, uncertain linguistic correlated averaging (ULCA) operator, uncertain linguistic correlated geometric (ULCG) operator, and uncertain linguistic harmonic mean (ULHM) operators [18]. More recently, Zhang [19] proposed the intervalvalued 2tuple linguistic variable and introduced some aggregation operators of intervalvalued 2tuples. Chen and Lee [20] proposed interval linguistic labels ordered weighted average (ILLOWA) operator for autocratic decision making. Chen et al. [21] presented an adaptive consensus support model for group decision making systems based on intervals of linguistic 2tuples.
On the other hand, due to the difference in culture, cognition, experience, habits, backgrounds, and so on, decision makers may express their evaluation using different linguistic domains with different granularities, that is, multigranular linguistic term sets, in practical decision making problems. Up to now, many approaches have been proposed to deal with GDM problems with multigranular linguistic information [22â€“32]. In these approaches, the multigranularity linguistic terms are first unified into a basic linguistic term set (BLTS) by transform functions; usually the BLTS is linguistic term set with the greatest granularity, before the linguistic information aggregation. Zhang [19] pointed out that the transform processes are tedious and may produce the loss of information and proposed intervalvalued 2tuple linguistic variable to directly aggregate intervalvalued 2tuple linguistic information.
Decision making with uncertain linguistic information and with multigranular linguistic information has significantly advanced decision analysis with linguistic information. However, in these studies, the multigranular uncertain linguistic information is seldom addressed [30]. Xu (2009) [28] developed some transformation functions to unify the unbalanced linguistic labels with different granularities and then utilized the uncertain linguistic weighted averaging (ULWA) operator to aggregate the unified unbalanced linguistic information. Fan and Liu (2010) [30] proposed a method with multigranular uncertain linguistic information, in which multigranular uncertain linguistic values are first transformed into trapezoidal fuzzy numbers and then an extended TOPSIS methodology is used to solve the decisionmaking problem. Gao and Peng (2011) [31] defined the transformation function of uncertain linguistic preferences with different granularities. Zhang and Guo (2012) [32] proposed a method for multigranular uncertain linguistic group decision making with incomplete weight information. Meanwhile, current linguistic decision making methods are under the assumption that the criteria are at the same priority level, and they are characterized by the ability to trade off between criteria. However, in lots of real and practical multiple criteria decisionmaking problems, the criteria have different priority levels commonly. Such as in the case of buying a car based upon the criteria of safety and cost, usually we may not allow compensation between cost and safety, and in this case of organizational decision making, superiors generally have a higher priority than those of their subordinates. Yager [36] paid attention to this issue and presented the prioritized aggregation operators by modeling the prioritization of attributes with respect to the weights associated with the attributes dependent upon the satisfaction of the higher priority attributes. Afterwards, some authors devoted their attentions to this issue [37â€“41]. However, to our knowledge, no study has reported an appropriate method for solving the prioritization phenomenon among the criteria using linguistic information, letting alone the problems with multigranular uncertain linguistic information. Therefore, it is necessary to pay attention to this issue.
To do this, the remainder of this paper is organized as follows. Section 2 introduces the operational laws of multigranular uncertain linguistic variables and briefly reviews the prioritized aggregation operators. Section 3 proposes the multigranular uncertain linguistic prioritized weighted average (MULPWA) operator and multigranular uncertain linguistic prioritized ordered weighted average (MULPOWA) operator to aggregate the multigranular uncertain linguistic variables, whose desirable properties are also studied in this section. In Section 4, we develop an approach for multicriteria group decision making based on the proposed operators under multigranular uncertain linguistic environment. In Section 5, a practical example is provided to verify the practicality and effectiveness of the developed approach. Section 6 concludes the paper.
2. Preliminaries
2.1. Uncertain Linguistic Variables
Let be a linguistic term set with odd cardinality, where represents a possible value for a linguistic variable and denote the granularity of the term set, and it usually has the following characteristics [1â€“5]: (1) the set is ordered: if ; (2) there is the negation operator: such that ; (3) max operator: if ; (4) min operator: if . For example, a linguistic term set with 9 terms can be defined as .
Usually, the granularity of must be small enough so as not to impose useless precision on the experts and it must be rich enough in order to allow a discrimination of the performances of each object in a limited number of grades; the limit of cardinality is 11 or not more than 13 [3, 4].
To preserve all the given information, Xu [9, 10] extended the discrete term set to a continuous term set , , whose elements also meet all the characteristics above, and if , then we call the original term; otherwise, we call the virtual term.
Definition 1 (see [13, 14]). Let , where , and and are the lower and the upper limits, respectively; we then call the uncertain linguistic variable.
In the existing literature, the operational laws of uncertain linguistic variables are all assumed that the values of uncertain linguistic variables are from the same linguistic term set. So they may be not fit to the uncertain linguistic information from multigranular linguistic term sets, that is, the multigranular uncertain linguistic information. In order to process the multigranular uncertain linguistic information, we give the following definitions.
Definition 2. For any three uncertain linguistic variables , , and and their linguistic values from linguistic term sets with granularities , , and , respectively, then their operational laws can be defined as follows: where is the granularity of basic term set.
From the operational laws (3) and (4), we can get the following equations: where is the granularity of basic term set.
To compare uncertain linguistic variables, Xu (2004) [13] proposed a possibility degree formula to calculate an uncertain linguistic term that is greater than the other one. In the following, we further extend the possibility degree formula to compare the multigranular uncertain linguistic variables.
Definition 3. Let and be uncertain linguistic variables and their linguistic values from linguistic terms sets with granularities and , respectively, and then the possibility degree dominate is defined as
If and , then the possibility degree dominate is defined as
From Definition 3, we can easily get the following results:(1), Especially, ,(2),â€‰especially, ,(3), where .
Note that if , then the above results reduced to the ones of uncertain linguistic variables [13]. Based on the possibility degree and the idea of [42], the relative possibility degree of over all can be defined as follows:
2.2. Prioritized Aggregation Operators
The prioritized average (PA) operator is an aggregation operator in which there exists a prioritization relationship between the arguments [36]. The prioritizations of arguments are modeled by using importance weights in which the weights associated with the lower priority criteria are related to the satisfaction of the higher priority criteria, which were defined as follows.
Definition 4 (see [37]). Let be a collection of the criteria and that there is a prioritization between the criteria expressed by the linear ordering , indicating that criteria have a higher priority than , if . The value is the performance of any alternative under attribute and satisfies . If where , , , then PWA is called the prioritized weighted average (PWA) operator.
The PWA operator emphasizes the situation where lack of satisfaction of higher priority criteria cannot be compensated for by satisfaction of lower priority criteria. The PWA operator is monotonic, bounded, and idempotent [37].
Afterwards, Yager embedded priority weights into the importance weighted OWA operator and presented a prioritized ordered weighted average (POWA) operator.
Definition 5 (see [43]). An OWA operator of dimension is a mapping so that where is a permutation function such that , are ordered weights such that in which is a basic unitinterval monotonic (BUM) function having the following properties: (1) ,â€‰â€‰(2) ,â€‰â€‰(3) , if .
In [44], Yager considered the situation that the argument variable is associated with an importance and gave a policy to derive ordered weights of OWA operator based on the importance of argument variable: Substituting (9) into (7), we get
According to the importance weighted OWA operator, Yager takes the prioritized weights as a special importance weights and normally presented the prioritized OWA operator [38], which was defined as follows.
Definition 6. Let be a collection of criteria and that there is a prioritization between the criteria expressed by the linear ordering , indicating attribute has a higher priority than , if . The value is the performance of any alternative under attribute and satisfies . If where is a permutation function such that , , , . Then POWA is called the prioritized ordered weighting average (POWA) operator.
The POWA operator is also monotonic, bounded, and idempotent.
3. Multigranular Uncertain Linguistic Prioritized Operators
In this section, we will investigate the prioritized aggregation operators under multigranular uncertain linguistic environments.
3.1. Multigranular Uncertain Linguistic Prioritized WA Operator
Based on Definition 4 and the operational laws of multigranular uncertain linguistic variables, we give the definition of the multigranular uncertain linguistic prioritized weighted average (MULPWA) operator as follows.
Definition 7. Let , be a collection of multi granular uncertain linguistic variables (MULV), which are prioritized such that if , and then the MULPWA operator is defined as follows: where , , and is the possibility degree ofâ€‰ among the . In particular, if , then the MULPWA operator returns to an uncertain linguistic prioritized weighted average (ULPWA) operator; if for all , then the MULPWA operator returns to a multigranular linguistic prioritized weighted average (MLPWA) operator.
From (12), we know that: (1)since for , then a criterion can never have a bigger priority weight than a criterion that has a higher priority than it;(2)the MULPWA operator consists of the following three steps:(a)calculate the relative possibility degrees of the multigranular uncertain linguistic arguments variables by (3) and (5);(b)determine the prioritized levels and ;(c)derive the aggregated results according to .
Example 8. Given the collection of multigranular uncertain linguistic variables , , , and , the prioritized relations are .
By (3), we have
Similarly, we can derive all pair comparisons among the multigranular uncertain linguistic values and form a possibility degree matrix as shown in Table 1.

Utilize (5) to derive the relative possibility degrees over all the other alternatives:
Then determine the prioritized levels
, , , , and .
Thus,
Similar to the PWA operator, it can be easily proved that the MULPWA operator has the following properties.
Theorem 9 (idempotency). Let , â€‰ be a collection of MULV, where , , and is the possibility degree of â€‰â€‰. Ifâ€‰â€‰ for all , then
Proof. Since for all , we can get for all , then , .
Theorem 10 (boundedness). Letâ€‰â€‰, be a collection of MULV, , , and is the possibility degree of , and then
Proof. Since thus .
Theorem 11 (monotonicity). Let = , , â€‰â€‰â€‰â€‰ be two collections of MULV, where , , , , and and are the possibility degrees ofâ€‰â€‰â€‰â€‰and , respectively. If for all , then
Proof. Since , and = and meanwhile, since for all , , , and .
Thus, .
Theorem 12. Letâ€‰â€‰, be a collection of MULV, where , , and is the possibility degree of . If , then
Proof. Since then , , and then
3.2. Multigranular Uncertain Linguistic Prioritized OWA Operator
Based on Definition 6 and the operational laws of multigranular uncertain linguistic variables, we give the definition of the multigranular uncertain linguistic prioritized ordered weighted average (MULPOWA) operator as follows.
Definition 13. Let , be a collection of multigranular uncertain linguistic variables (MULV), which are prioritized such that . If , then the MULPOWA operator is defined as follows: where is a permutation function such that , , , , and is the possibility degree of . In particular, if , then the MULPOWA operator returns to an uncertain linguistic prioritized ordered weighted average (ULPOWA) operator; if for all , then the MULPOWA operator returns to a multigranular linguistic prioritized weighted average (MLPWA) operator.
The aggregation process of the MULPOWA operator consists of the following four steps:(1) calculate the possibility degrees of the multigranular; uncertain linguistic arguments variables by (3) and (5);(2) determine the prioritized levels and ;(3) reorder the input arguments in descending order and associate with ;(4) derive the aggregated results of according to
Example 14. Given the collection of multigranular uncertain linguistic variables , , , and , the prioritized relations are .
Since , . , , , , and , . Consider
Similarly to Theorems 9â€“12 and (24), it can be easily proved that the MULPOWA operator has the following properties.
Theorem 15 (idempotency). Let, be a collection of MULV, where , , , and is the possibility degree ofâ€‰â€‰ . If â€‰â€‰ for all , then
Theorem 16 (boundedness). Let, be a collection of MULV, , , , and is the possibility degree ofâ€‰ , and then
Theorem 17 (monotonicity). Letâ€‰â€‰, , , be two collections of MULV, where , , , , , , and are the possibility degrees ofâ€‰â€‰ â€‰â€‰and , respectively. If â€‰â€‰ for all , then
Theorem 18. Let , be a collection of MULV, where , , , and is the possibility degree ofâ€‰â€‰ . If , then
4. An Approach to Multiple Criteria Group Decision Making with Prioritized Levels
Let be a set of alternatives, a collection of criteria, and the set of DMs. There is a prioritization between the criteria expressed by the linear ordering , indicating that attribute has a higher priority than , if and a prioritization between the experts expressed by the linear ordering , indicating that attribute has a higher priority than , if . Suppose that is the multigranular uncertain linguistic decision matrix provided by , where is the criteria value of with respect to alternative .
In the following, we apply the MULPWA and MULPOWA operators to multiple criteria group decision making based on multigranular uncertain linguistic information. The method involves the following steps.
Step 1. Select the basic linguistic term set (in general, the BLT can be chosen by the decision makers directly).
Step 2. Calculate the relative possibility degree of by utilizing (3) and (5):
Step 3. Based on the relative possibility degree, determine the values of with the following expression:
Step 4. Utilize the MULPWA operator (12) to aggregate the valuesâ€‰ in the row of and get the individual decision results corresponding to the alternative :
Step 5. Calculate the relative possibility degree of , by utilizing (3) and (5) again:
Step 6. Calculate the values of the values of :
Step 7. Employ the MULPOWA operator (24) to derive the collective decision results of alternative :
Step 8. Rank all the alternatives by the possibility degree again.
5. Numerical Example
In this section, an example from [40] is used to illustrate the use of the proposed method. To strengthen scientific research and promote the building of teaching body, the school of business in a Chinese university wants to introduce an outstanding professor. The work has been raised great attention from the school, university president , dean of management school , and human resource officer setting up the panel of decision makers which will take the whole responsibility for this introduction. They made strict evaluation for 5 candidates from four aspects, namely, morality , Tangibles , Convenience , and Reliability . This introduction will be in strict accordance with the principle of combine ability with political integrity. The prioritization relationship for the criteria is as follows, . Besides, the university president has the absolute priority for decision making; the dean of the management school comes next. Three decision makers evaluate the candidates with respect to the criteria with linguistic terms sets with different granularities, where provides of his judgments uses of linguistic term set with 9 granularities, , ,