Uncertain Linguistic Power Geometric Operators and Their Use in Multiattribute  Group Decision Making

Zhang, Huimin

doi:https://doi.org/10.1155/2015/948380

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 948380 | https://doi.org/10.1155/2015/948380

Uncertain Linguistic Power Geometric Operators and Their Use in Multiattribute Group Decision Making

Huimin Zhang¹

Academic Editor: Cheng C. Wang

Received16 Aug 2014

Revised02 Dec 2014

Accepted23 Dec 2014

Published24 Feb 2015

Abstract

Based on the extended triangular norm, several new operational laws for linguistic variables and uncertain linguistic variables (ULVs) are defined. To avoid the limitations of existing linguistic aggregation operators, a series of extended uncertain linguistic (UL) geometric aggregation operators are proposed on the basis of the extended triangular norm. In addition, a multiattribute group decision making (MAGDM) method dealing with UL information is developed based on the extended UL geometric aggregation operators. Finally, an example is presented to show the efficiency of the developed approach in solving MAGDM problems.

1. Introduction

Due to the increasing complexity of the decision problem and our limited ability, in the multiattribute group decision making (MAGDM) process, decision makers sometimes have difficulty in providing their opinions with crisp numbers. Alternatively, by means of linguistic information, they think they can express their opinions more naturally and straightforward. The studies on decision making with linguistic information have been widely performed and many methods have been developed to solve the decision making problem with linguistic information [1–12].

Sometimes, decision makers may feel that they cannot fully or accurately express their opinions with only one linguistic term. Xu [13] proposed the concept of ULV which can be regarded as one linguistic interval and is convenient for decision makers to better express their opinions. In view of the key role that aggregation operators always play in the decision making process, many UL aggregation operators have been put forward. For instance, Xu [13] introduced the UL ordered weighted averaging (ULOWA) operator and the UL hybrid aggregation (ULHA) operator. Xu [14] developed induced uncertain linguistic OWA (IULOWA) operators, where the second components are ULVs. Xu [15] defined the uncertain multiplicative linguistic preference relation and proposed several UL geometric operators. Wei [16] proposed an UL hybrid geometric mean (ULHGM) operator. Peng et al. [17] proposed an uncertain pure linguistic hybrid harmonic averaging (UPLHHA) operator. Based on the extended triangular conorm, Lan et al. [18] defined some new operational laws for linguistic variables and put forward several extended UL aggregation operators. Motivated by Bonferroni mean, Wei et al. [19] proposed several UL Bonferroni aggregation operators. Peng et al. proposed [20] some multigranular UL prioritized aggregation operators based on the prioritized aggregation operator.

Yager [21] introduced the power-average (PA) operator and the power OWA (POWA) operator which allow arguments to support each other in the aggregation process. On this basis, Xu and Yager [22] developed the power geometric (PG) operator, weighted PG operator, and POWG operator. Due to special properties of power aggregation operators, many linguistic power aggregation operators have been proposed. Zhou and Chen [23] proposed the generalized linguistic PA operator, the weighted generalized linguistic PA operator, and the generalized linguistic POWA operator. Xu and Wang [24] developed 2-tuple linguistic PA operator, 2-tuple linguistic weighted PA operator, and 2-tuple linguistic POWA operator. Xu et al. [25] defined the linguistic PA operator, the linguistic weighted PA operator, the linguistic POWA operator, the uncertain linguistic PA operator, and the uncertain linguistic POWA operator.

However, most of the abovementioned linguistic and UL aggregation operators are developed on the basis of the basic operational laws, by which the computing results derived sometimes may be beyond the discourse domain of the original linguistic variable and lead to a question on how to define the semantics for it. In addition, up to now, there is no PG operator developed for dealing with UL decision making problems. To overcome such issues, motivated by Lan et al. [18], we introduce the extended triangular norm and propose several uncertain linguistic PG operators based on the extended triangular norm. This paper is structured as below. Section 2 briefly introduces the basic knowledge that will be used in the following sections. Section 3 proposes several UL geometric operators based on extended triangular norm. In Section 4, a MAGDM method dealing with UL information is developed. In Section 5, an example is presented to show the efficiency of the developed approach in solving MAGDM problems. Section 6 gives the conclusions.

2. Preliminaries

Suppose that is a linguistic term set (LTS), where the odd expresses the cardinality and denotes a possible linguistic value. For instance, a LTS with seven terms is expressed as below.

[6].

The LTS has the properties as below [6]:(1), if and only if ,(2)negation operator: such that .

Xu [26] proposed a continuous LTS , where is an original linguistic term if ; otherwise, is a virtual linguistic term.

Suppose that , and ; Xu [26] gave the following operational laws:

Besides, Xu [13] defined the ULV with the expression , where , and denote the upper and lower bounds, respectively. In particular, the ULV degenerates to one linguistic variable if . To rank ULVs, Xu [15] gave the concept of possibility degree.

Definition 1. Let and be two ULVs, and let and ; the possibility degree of is defined as
If , the order between and is denoted by .
Obviously, the possibility degree has the following properties:(1);(2). In particular, .
If a series of ULVs should be ranked, a likelihood matrix is constructed as below: where .
Thus, the order of the ULVs is obtained in accordance with the values of [27]:

Definition 2 (see [28, 29]). A -norm is a mapping satisfying, for all ,(1)(boundary condition) ;(2)(commutativity) ;(3)(associativity) ;(4)(monotonicity) whenever , .
Motivated by Lan et al. [18], in what follows, we extend the -norm to interval .

Definition 3. An extended -norm is a mapping satisfying, for all ,(1)(boundary condition) ;(2)(commutativity) ;(3)(associativity) ;(4)(monotonicity) whenever , .
Let be a strictly monotone decreasing and continuous function with and .
Lan et al. [18] proposed an interactive method to build the function of ; that is, given the utility values of by decision makers, the nonlinear curve-fit method is performed to obtain the approximate function of . For reason of simplicity, unless otherwise stated, we suppose in the following text. Obviously, has the properties that and . Furthermore, we can easily derive the inverse function of ; that is, .

Definition 4. Suppose that is a binary operator; for any , is defined as where is the inverse function of .

Theorem 5. is an extended -norm in .

Proof. Let .(1)Since , we have .(2)Consider .(3)Consider = .(4)Since is a strictly monotone decreasing and continuous function, then is also a strictly monotone decreasing and continuous function.
If and , we have , , and so .
Thus, we obtain ; that is, .

With the above analysis, the operational laws based on extended -norm can be derived as below.

Definition 6. Suppose that linguistic terms , and ; the operational laws are defined as below: where is a strictly monotone decreasing and continuous function with and .

Theorem 7. Suppose that linguistic terms , and ; then the properties of the operational laws are presented as below:(1);(2);(3);(4), if , ;(5);(6);(7), if and .

Proof. (1) Consider .
(2) Consider .
(3) Consider = .
(4) Since and , we have , and then .
Thus, we obtain .
(5) By (7), we have
(6) By (7), we have
(7) By (7), we have
Having the first four properties above, the operator can be regarded as an extended -norm in .

3. Some Uncertain Linguistic Geometric Operators Based on Extended -Norm

The operational laws discussed in Section 2 can be extended to UL environment.

Definition 8. Suppose that and are two ULVs and ; the operational laws are expressed as below: where is a strictly monotone decreasing and continuous function with and .
Similarly, the operational laws in Definition 8 have the properties as below.

Theorem 9. Suppose that , , , and are ULVs derived from the continuous LTS , and ; then some properties of the operational laws are presented as below:(1);(2);(3);(4), if , , , and ;(5);(6);(7), if and .

Proof. We only prove (4), (5), and (6).(4)By (11), we have
Since both and are strictly monotone decreasing functions, we have
Let , , , and ; then we have and .
If , then we have . Thus, by (3), we derive .
If , we can easily get , so we have .
Consequently, we obtain .(5)By (11) and (12), we have (6)By (11) and (12), we have

Example 10. Let , , , , , and ; then .
By (11) and (12), we have
It is easy to see that and .
However, if we follow the operational laws defined by (2), then we have
Obviously, the computing results are beyond the discourse domain of the linguistic variable. Alternatively, if we follow the method of Xu [14], then we have That is, , which seem to be counterintuitive and may not be easily accepted.
On the basis of the operational laws of ULVs, in what follows, we propose some extended UL geometric operators.

Definition 11. Suppose that are a series of ULVs; then the extended UL geometric mean (EULGM) operator is defined as
The EULGM operator has the following properties.

Theorem 12. Suppose that are a series of ULVs; then one has(1)idempotency: if , , then (2)monotonicity: suppose that are a series of ULVs. If and , then (3)boundary: (4)commutativity: suppose that is any permutation of ; then

Definition 13. Suppose that are a series of ULVs, and (satisfying and ) is the weight vector regarding ULVs; then the extended UL weighted geometric mean (EULWGM) operator is defined as
If , , then the EULGWM operator degenerates to the EULGM operator. The EULWGM operator has the properties as below.

Theorem 14. Suppose that are a series of ULVs, and (satisfying and ) is the weight vector regarding ULVs; then one derives(1)idempotency: if , , then (2)monotonicity: suppose that are a series of ULVs. If and , then (3)boundary:

Definition 15. Suppose that are a series of ULVs; then the extended UL ordered weighted geometric (EULOGW) operator is defined as where is the th largest and (satisfying and ) is the aggregation-associated vector.

Theorem 16. Suppose that are a series of ULVs; then one derives(1)idempotency: if , , then (2)monotonicity: suppose that are a series of ULVs. If and , then (3)boundary: (4)commutativity: suppose that is any permutation of ; then

The above UL aggregation operators do not take the relationship between the ULVs being aggregated into account. To avoid such limitation, Yager [21] proposed the PA operator which allows argument values to support each other in the aggregation process. In addition, Xu and Yager [22] developed the PG operator, weighted PG operator, and POWG operator. On the basis of the operational laws of ULVs, in what follows, we introduce some PG operators for ULVs.

Definition 17. Suppose that are a series of ULVs; then the extended UL PG (EULPG) operator is defined as where and is the support for from , which meets the following properties [22]:(1);(2);(3), if , where is a similarity measure between ULVs.
In this paper, we defined as . Without loss of generality, the similarity measure between and is determined based on Hamming distance; that is, where and are derived from the continuous LTS .
Obviously, , where if and only if and , and if and only if ; that is, and (or and ), which means there is no support between and in the geometric aggregation process. In addition, if all the are equal, then the EULPG operator degenerates to the EULGM operator.

Theorem 18. Suppose that are a series of ULVs; then one derives(1)idempotency: if , then (2)boundary: (3)commutativity: suppose that is any permutation of ; then
If the weights of in the EULPG operator are taken into account, the extended UL weighted PG (EULWPG) operator is derived.

Definition 19. Suppose that are a series of ULVs; then the extended UL weighted PG (EULWPG) operator is defined as where and is the weight of (satisfying and ).
Obviously, the EULWPG operator is not commutative but idempotent and bounded.
Motivated by POWG operator [22], we propose an extended UL power ordered weighted geometric (EULPOWG) operator as below.

Definition 20. Suppose that are a series of ULVs; then the extended UL power ordered weighted geometric (EULPOWG) operator is defined as where is the th largest of , , , , , , is the support for from , and is a basic unit-interval monotonic function [22] with the following properties: (1) ; (2) ; (3) , if .
In particular, if , then the EULPOWG operator degenerates to the EULPG operator. By (40), we derive
The EULPOWG operator has the properties as below.

Theorem 21. Suppose that are a series of ULVs; then one derives(1)idempotency: if , , then (2)boundary: (3)commutativity: suppose that is any permutation of ; then

4. MAGDM Method Based on the Extended Uncertain Linguistic Aggregation Operators

In what follows, a MAGDM method presents the application of the aggregation operators proposed in Section 3.

Let , , and be the sets of alternatives, attributes, and decision makers, respectively. The attribute weights are known and satisfy and . Let be the decision matrix given by , where expresses the performance value for with respect to and takes the form of ULV with .

The main processes of the method proposed are described as below.

Step 1. Each decision maker is asked to provide the UL decision matrix .

Step 2. By the EULPOWG operator, the aggregated decision matrix can be derived; that is, .

Step 3. Aggregate to obtain the overall performance values for each alternative based on the EULWGM operator.

Step 4. Obtain the order of the alternatives by Definition 1.

5. Application Example

To demonstrate the application of the proposed model, in this section, we provide an example where the decision makers from a manufacturing company have to select a third-party logistics provider.

With the rapid change of competitive environment, more and more companies focus on their core competency and logistics outsourcing has become an important strategy to promote enterprise’s competitiveness. Within new strategies for purchasing and manufacturing, third-party logistics providers play a key role in achieving corporate competition. Particularly for manufacturing companies, the selection of appropriate third-party logistics provider is of high importance. Sustainable third-party logistics provider selection requires the evaluation of providers’ performance from several metrics. That is, in the first stage of third-party logistics provider selection, the manufacturing company should define the qualitative and quantitative attributes which are taken into account to evaluate and select a supplier. After full consideration of the long-term strategy, the manufacturing company constructs the following evaluation systems including four aspects (suppose the weight vector is ): is the innovation capability; is the sense of social responsibility and environment performance; is the ability of sustainability; is the quality of service. Then, the manufacturing company organizes a committee of experts who are responsible for providing their evaluations on each supplier with respect to each attribute. Suppose there are four third-party logistics providers (alternatives: , , , and ) to be evaluated. In addition, three experts take part in the decision making and give their evaluations with the LTS , = .

In the existing research on third-party logistics provider or supplier selection models, although the criteria used by most of them and performance values given by decision makers are not completely independent, the dependent and feedback effects are often neglected. In what follows, we applied the extended UL geometric aggregation operators to solve the above MAGDM problem.

Step 1. The decision makers provide their evaluation values and construct the UL decision matrix () as shown in Tables 1–3, respectively.

Step 2. Suppose that and ; utilize the EULPOWG operator to derive the aggregated decision matrix , which is shown in Table 4.

Taking as an example, we have , where , , and .

Then, by Definition 1, we can easily derive , where is the th largest of , .

According to Definition 20, we have and then,

Thus, we derive , , and .

Consequently, by (40), we obtain

Step 3. Aggregate () to yield the overall performance values for each alternative () based on the EULWGM operator with the weight vector , as shown in Table 5.

Step 4. By ranking () based on Definition 1, the order of the alternatives can be obtained, which is listed in Table 5. That is, is the best third-party logistics provider to be selected.

Just as Xu and Yager [22] pointed out, both the EULWPG and EULPOWG operators take the relationships between the arguments into account. The difference between such two operators is that the EULWPG operator stresses the importance of every ULV, while the EULPOWG operator emphasizes the importance of the ordered position of every ULV. In consequence, in the group decision making process, by using the EULWPG or EULPOWG operator, we can not only allow individual opinions to support each other in the aggregation process but also reduce the influence of excessive low or high evaluation values on the decision result by assigning them lower weights.

6. Conclusions

In this paper, the MAGDM problems with UL information are investigated. Motivated by the ideal of Lan et al. [18], the extended triangular norm is defined, based on which several UL geometric aggregation operators are proposed, such as the EULGM operator, EULWGM operator, EULOGW operator, EULPG operator, EULWPG operator, and EULPOWG operator, together with their properties. In the process of decision making, these UL aggregation operators can avoid the limitations of existing linguistic aggregation operators; that is, the results derived in the computing processes sometimes may be counterintuitive or beyond the discourse domain of the linguistic variable. As a result, more intuitive and acceptable results may be obtained by these aggregation operators proposed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant no. U1304701, the High-Level Talent Project of Henan University of Technology under Grant no. 2013BS015, and the Plan of Nature Science Fundamental Research in Henan University of Technology under Grant no. 2013JCYJ14.

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Copyright © 2015 Huimin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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