Generalized Fuzzy Bonferroni Harmonic Mean Operators and Their Applications in Group Decision Making

The Bonferroni mean (BM)operator is an important aggregation technique which reflects the correlations of aggregated arguments.


Introduction
Multiple attribute group decision making (MAGDM) is the common phenomenon in modern life, which is to select the optimal alternative(s) from several alternatives or to get their ranking by aggregating the performances of each alternative under several attributes, in which the aggregation techniques play an important role.Considering the relationships among the aggregated arguments, we can classify the aggregation techniques into two categories: the ones which consider the aggregated arguments dependently and the others which consider the aggregated arguments independently.For the first category, the well-known ordered weighted averaging (OWA) operator [1,2] is the representative, on the basis of which, a lot of generalizations have been developed, such as the ordered weighted geometric (OWG) operator [3][4][5], the ordered weighted harmonic mean (OWHM) operator [6], the continuous ordered weighted averaging (C-OWA) operator [7], the continuous ordered weighted geometric (C-OWG) operator [8].The second category can reduce to two subcategories: the first subcategory focuses on changing the weight vector of the aggregation operators, such as the Choquet integral-based aggregation operators [9], in which the correlations of the aggregated arguments are measured subjectively by the decision makers, and the representatives of another subcategory are the power averaging (PA) operator [10] and the power geometric (PG) operator [11], both of which allow the aggregated arguments to support each other in aggregation process, on the basis of which the weighted vector is determined.The second subcategory focuses on the aggregated arguments such as the Bonferroni mean (BM) operator [12].Yager [13] provided an interpretation of BM operator as involving a product of each argument with the average of the other arguments, a combined averaging and "anding" operator.Beliakov et al. [14] presented a composed aggregation technique called the generalized Bonferroni mean (GBM) operator, which models the average of the conjunctive expressions and the average of remaining.In fact, they extended the BM operator by considering the correlations of any three aggregated arguments instead of any two.However, both BM operator and the GBM operator ignore some aggregation information and the weight vector of the aggregated arguments.To overcome this drawback, Xia et al. [15] developed the generalized weighted Bonferroni mean (GWBM) operator as the weighted version of the GBM operator.Based on the GBM operator and geometric mean operator, they also developed the generalized Bonferoni geometric mean (GWBGM) operator.The fundamental characteristic of the GWBM operator is that it focuses on the group opinions, while the GWBGM operator gives more importance to the individual opinions.Because of the usefulness of the aggregation techniques, which reflect the correlations of arguments, most of them have been extended to fuzzy, intuitionistic fuzzy, or hesitant fuzzy environment [16][17][18][19][20].
Harmonic mean is the reciprocal of arithmetic mean of reciprocal, which is a conservative average to be used to provide for aggregation lying between the max and min operators, and is widely used as a tool to aggregate central tendency data [21].In the existing literature, the harmonic mean is generally considered as a fusion technique of numerical data information.However, in many situations, the input arguments take the form of triangular fuzzy numbers because of time pressure, lack of knowledge, and people's limited expertise related with problem domain.Therefore, "how to aggregate fuzzy data by using the harmonic mean?" is an interesting research topic and is worth paying attention to.So Xu [21] developed the fuzzy harmonic mean operators such as fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator and fuzzy hybrid harmonic mean (FHHM) operator, and applied them to MAGDM.Wei [22] developed fuzzy induced ordered weighted harmonic mean (FIOWHM) operator and then, based on the FWHM and FIOWHM operators, presented the approach to MAGDM.H. Sun and M. Sun [23] further applied the BM operator to fuzzy environment, introduced the fuzzy Bonferroni harmonic mean (FBHM) operator and the fuzzy ordered Bonferroni harmonic mean (FOBHM) operator, and applied the FOBHM operator to multiple attribute decision making.In this paper, we will develop some new harmonic aggregation operators, including the generalized fuzzy weighted Bonferroni harmonic mean (GFWBHM) operator and generalized fuzzy ordered weighted Bonferroni harmonic mean (GFOWBHM) operator, and apply them to MAGDM.
In order to do this, the remainder of this paper is arranged in following sections.Section 2 first reviews some aggregation operators, including the BM, GBM, and GWBM operators.Then, some basic concepts related to triangular fuzzy numbers and some operational laws of triangular fuzzy numbers are introduced.The desirable properties of the FBHM and FOBHM operators are discussed.We extend them, by considering the correlations of any three aggregated arguments instead of any two, to develop generalized fuzzy weighted Bonferroni harmonic mean (GFWBHM) operator and generalized fuzzy ordered weighted Bonferroni harmonic mean (GFOWBHM) operator.In particular, all these operators can be reduced to aggregate interval or real numbers.Section 3 presents an approach to MAGDM based on the GFWBHM and GFOWBHM operators.Section 4 illustrates the presented approach with a practical example, verifies and shows the advantages of the presented approach, and makes a comparative study to the existing approaches.Section 5 ends the paper with some concluding remarks.

Generalized Fuzzy Bonferroni Harmonic Mean Operators
The Bonferroni mean operator was initially proposed by Bonferroni [12] and was also investigated intensively by Yager [13].
Beliakov et al. [14] further extended the BM operator by considering the correlations of any three aggregated arguments instead of any two.Definition 2. Let , ,  ≥ 0 and let   ( = 1, 2, . . ., ) be a collection of nonnegative numbers.If then GBM ,, is called the generalized Bonferroni mean (GBM) operator.
In particular, if  = 0, then the GBM operator reduces to the BM operator.However, it is noted that both BM operator and the GBM operator do not consider the situation that  =  or  =  or  = , and the weight vector of the aggregated arguments is not also considered.To overcome this drawback, Xia et al. [15] defined the weighted version of the GBM operator.Definition 3. Let , ,  ≥ 0 and let   ( = 1, 2, . . ., ) be a collection of nonnegative numbers with the weight vector  = ( 1 ,  2 , . . .,   )  such that   > 0,  = 1, 2, . . .,  and then GWBM ,, is called the generalized weighted Bonferroni mean (GWBM) operator.
Some special cases can be obtained as the change of the parameters as follows.
(1) If  = 0, then the GWBM operator reduces to the following: which is the weighted Bonferroni mean (WBM) operator.
(2) If  = 0 and  = 0, then the GWBM operator reduces to the following: which is the generalized weighted averaging operator.Furthermore, in this case, let us look at the GWBM operator for some special cases of .
The previous aggregation techniques can only deal with the situation that the arguments are represented by the exact numerical values, but are invalid if the aggregation information is given in other forms, such as triangular fuzzy number [24], which is a widely used tool to deal with uncertainty and fuzziness, described as follows.
Definition 4 (see [24]).A triangular fuzzy number â can be defined by a triplet [  ,   ,   ].The membership function  â() is defined as where   ≥   ≥   ≥ 0,   , and   stand for the lower and upper values of â, respectively, and   stands for the modal value [24].In particular, if any two of   ,   , and   are equal, then â reduces to an interval number; if all   ,   , and   are equal, then â reduces to a real number.
For convenience, we let Ω be the set of all triangular fuzzy numbers.
To aggregate the triangular fuzzy correlated information, based on the BM and weighted harmonic mean operators, H. Sun and M. Sun [23] developed the fuzzy Bonferroni harmonic mean operator.Because this operator considers the weight vector of the aggregated arguments, we redefine this operator as fuzzy weighted Bonferroni harmonic mean operator.
In addition, a special case can obtained as the change of parameter.If  = 0, then the FWBHM operator reduces to the following: which we call the fuzzy weighted generalized harmonic mean (FWGHM) operator.
On the basis of the operational laws of triangular fuzzy numbers, the FWBHM operator has the following properties.
However, if there is a tie between â and â , then we replace each of â and â by their average (â  + â )/2 in process of aggregation.If  items are tied, then we replace these by  replicas of their average.The weighting vector  = ( 1 ,  2 , . . .,   )  can be determined by using some weights determining methods like the normal distribution based method; see [27][28][29] for more details.
If there is a tie between ã and ã , then we replace each of ã and ã by their average (ã  + ã )/2 in process of aggregation.If  items are tied, then we replace these by  replicas of their average.
Suppose that the weighting vector of the FOWBHM operator is  = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117)  (derived by the normal distribution based method [27]), and then by (22) (let  =  = 2), we get )) )) )) Both FWBHM and FOWBHM operators, however, can only deal with the situation in which there are correlations between any two aggregated arguments, but not the situation in which there exist connections among any three aggregated arguments.To solve this issue, motivated by Definition 3, we define the following.
In addition, some special cases can be obtained as the change of parameters.
However, if there is a tie between â and â , then we replace each of â and â by their average (â  + â )/2 in process of aggregation.If  items are tied, then we replace these by  replicas of their average.
The GOWBHM operator (46) has some special cases.
In the following, we apply the GFWBHM and GFOWBHM operators to group decision making with triangular fuzzy information.
Step 1. Normalize each attribute value â() in the matrix  () into a corresponding element in the matrix  () = (r ()  ) × (r ()   = [ ()  ,  ()  ,  ()  ]) using the following formulas: (51) Step 2. Utilize the GFWBHM operator (33) as follows: to aggregate all the elements in the th column of  ()  and get the overall attribute value r()  of the alternative   corresponding to the decision maker   .

Comparison of the Proposed Approach with Other Approaches
In this section, we compare the proposed approach with other approaches.The approaches to be compared here are the approaches proposed by Xu [21], Wei [22], and H. Sun and M. Sun [23], respectively.Each of methods has its advantages and disadvantages and none of them can always perform better than the others in any situations.It perfectly depends on how we look at things and not on how they are themselves.The differences in four approaches are the following.
(1) The H. Sun and M. Sun's approach is only suitable for solving multiple attribute decision making (MADM), while the proposed approach and Xu's and Wei's approaches are suitable for solving MAGDM because the approaches provide the aggregation stage in aggregation process.
(2) The Xu's and Wei's approaches have simple computation process than the proposed approach and H. Sun and M. Sun's approach, while the proposed approach and H. Sun and M. Sun's approach are more flexible than Xu's and Wei's approaches because these can provide the decision makers more choices as parameters are assigned different values.
(3) The Wei's approach uses the weights of decision makers as the order inducing variables in aggregation stage, while other approaches use the weights of decision makers to determine the order positions of the overall attribute values in exploitation stage.
Others of relative comparison with Xu's, Wei's, and H. Sun and M. Sun's approaches are shown in Table 7.

Conclusions
In this paper, we have extended the GWBM operator to the triangular fuzzy environment and developed the fuzzy harmonic aggregation operators including the FWBHM and GFWBHM operators.Based on the these operators and Yager's OWA operator, we have developed the FOWBHM operator and the GFOWBHM operator, respectively, and discussed their properties and special cases.It has been pointed out that if all the input fuzzy data reduce to the interval or numerical data, then the GFWBHM operator reduces to the GUWBHM operator and GWBHM operator, respectively; the GFOWBHM operator reduces to the GUOWBHM operator and GOWBHM operator, respectively.In these situations, the WHM (resp., OWHM) operator is the special case of the GWBHM (resp.GOWBHM) operator.Based on the GFWBHM and GFOWBHM operators, we have developed an approach to multiple attribute group decision making with triangular fuzzy information and have also applied the proposed approach to the problem of determining what kind of air-conditioning systems should be installed in the library.Furthermore, the comparison of the proposed approach with other existing approaches is presented.The merit of the proposed approach is that it is more flexible than the classical ones because it can provide the decision makers more choices as parameters are assigned different values.Apparently, the proposed aggregation techniques and decision making method can also extended to the intervalvalued triangular fuzzy environment.

Table 7 :
Comparison of the proposed approach with other approaches.