Extended IOWG Operator and its Use in Group Decision Making Based on Multiplicative Linguistic Preference Relations

: In [1], Xu and Da introduced the Induced Ordered Weighted Geometric (IOWG) operator, which takes as its argument pairs, called OWG pairs, in which one component is used to induce an ordering over the second components which are exact numerical values and then aggregated. In this study, we develop an extended IOWG (EIOWG) operator, in which the second components are linguistic variables. We study some desirable properties of the EIOWG operator, and then apply the EIOWG operator to group decision making based on multiplicative linguistic preference relations.


INTRODUCTION
The ordered weighted averaging (OWA) operator was developed by Yager [2]. The fundamental aspect of the OWA operator is a reordering step in which the input arguments are rearranged in descending order [2][3][4][5][6][7][8]. The ordered weighted geometric (OWG) operator is an aggregation operator that is based on the OWA operator and the geometric mean [1,[9][10][11][12][13][14][15]. Yager and Filev [16] introduced a more general type of OWA operator called induced ordered weighted averaging (IOWA) operator. The IOWA operator takes as its argument pairs, called OWA pairs, in which one component is used to induce an ordering over the second components which are exact numerical values and then aggregated. Recently, Xu and Da [1] developed an induced ordered weighted geometric (IOWG) operator that is based on the IOWA operator and the geometric mean, which can be used to aggregate multiplicative preference relations with exact numerical values in group decision making problems [17]. However, in many situations, the input arguments take the form of linguistic variables rather than numerical ones [13,. Therefore, it is necessary to pay attention to this issue. In this study, we shall develop an extended IOWG (EIOWG) operator, and study some desirable properties of the EIOWG operator. Then, we shall develop an approach, based on the EIOWG and the extended OWG (EOWG) operators, for ranking alternatives in group decision making with multiplicative linguistic preference relations. Finally, we shall apply the developed approach to the evaluation of investment alternatives of an investment company and draw our conclusions.
The ordered weighted geometric (OWG) operator is an aggregation operator that Chiclana et al. [9] defined and characterized to design multiplicative decision-making models [10,11,14]. It is based on the ordered weighted averaging (OWA) operator [2] and on the geometric mean. Xu and Da [12] presented some families of OWG operators.
The OWG operator has only been used in situations in which the input arguments are the exact numerical values.
However, judgements of people depend on personal psychological aspects such as experience, learning, situation, state of mind, and so forth. It is more suitable to provide their preferences by means of linguistic variables rather than numerical ones (for example when evaluating the comfort or design of a car, terms like good, fair, poor can be used).
In [37], Xu extended the OWG operator to accommodate the situations where the input arguments are linguistic variables.

Definition 2 [37]:
An extended ordered weighted geometric (EOWG) operator of dimension n is a mapping S S EOWG n → : , which has associated with it an exponential weighting vector     [16] introduced a more general type of OWA operator called induced ordered weighted averaging (IOWA) operator. The IOWA operator takes as its argument pairs, called OWA pairs, in which one component is used to induce an ordering over the second components which are exact numerical values and then aggregated. Xu and Da [1] developed an induced ordered weighted geometric (IOWG) operator that is based on the IOWA operator and the geometric mean, which can be used to aggregate multiplicative preference relations with exact numerical values in group decision-making problems.

Definition 3 [1]:
An IOWG operator is defined as follows: where T n w w w w ) ,..., , ( = is an exponential weighting vector, such that is referred to as the order inducing variable and i a as the argument variable, is the ordered position of the i a , then IOWG is the weighted geometric mean operator.
In the following, we shall extend the IOWG operator to accommodate the situations where the input arguments are linguistic variables.

Definition 4:
An extended IOWG (EIOWG) operator is defined as follows: However, if we replace the objects in Example 3 with 0 s with respect to order inducing variable. In this case, we can follow the policy presented by Yager and Filev [16], that is, to replace the arguments of the tied objects by the average of the arguments of the tied objects. Thus, for Example 3, we replace the argument component of each of In the following, we shall study some desirable properties of the EIOWG operator.    It is well known that the multiplicative preference relations to express the judgements are reciprocal, however, In [14], Herrera and Herrera-Viedma showed that reciprocity generally is not preserved when aggregating multiplicative preference relations using the OWG operator. In the section, we shall show that the reciprocal property can be maintained when aggregating multiplicative linguistic preference relations using the EIOWG operator, where the order inducing variable This completes the proof of Theorem 5. In the following, we shall apply the EIOWG and the EOWG operators to group decision making based on multiplicative linguistic preference relations.
Step 1: For a group decision making problem with linguistic preference information. The decision maker D d l ∈ compares these alternatives with respect to a single criterion by the multiplicative linguistic terms in S , and constructs the multiplicative linguistic preference relation Step 4: Rank all the alternatives and select the best one(s) in accordance with the values of ) ,..., 2 , 1 ( n i r i = .
Illustrative Example: Let us suppose an investment company, which wants to invest a sum of money in the best option (adapted from [32]). There is a panel with five possible alternatives in which to invest the money: x is a car industry; 2) 2 x is a food company; 3) 3 x is a computer company; 4) 4 x is an arms company;

5) 5
x is a TV company.
One main criterion used is growth analysis. There are three decision makers    To get the most desirable alternative(s), the following steps are involved: Step 1: Utilize the EIOWG operator (let its weighting vector be  Step 2: Utilize the EOWG operator (let its weighting vector be x x x x x and thus the most desirable alternative is 3 x .

CONCLUSION
In this study, we have developed an extended induced ordered weighted geometric (EIOWG) operator, which takes as its argument pairs, called OWG pairs, in which one component is used to induce an ordering over the second components which are linguistic variables. We have studied some desirable properties of the EIOWG operator, and then applied the EIOWG operator to group decision making based on multiplicative linguistic preference relations. In the future, we shall continue working in the application of the EIOWG operator to other domains.