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

Abstract

The Bonferroni mean (BM) operator is an important aggregation technique which reflects the correlations of aggregated arguments. Based on the BM and harmonic mean operators, H. Sun and M. Sun (2012) developed the fuzzy Bonferroni harmonic mean (FBHM) and fuzzy ordered Bonferroni harmonic mean (FOBHM) operators. In this paper, we study desirable properties of these operators and 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. Then based on the GFWBHM and GFOWBHM operators, we present an approach to multiple attribute group decision making and illustrate it with a practical example.

1. 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 [3–5], 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 [16–20].

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.

2. Generalized Fuzzy Bonferroni Harmonic Mean Operators

The Bonferroni mean operator was initially proposed by Bonferroni [12] and was also investigated intensively by Yager [13].

Definition 1. Let 𝑝,𝑞≥0 and let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If BM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1𝑖≠𝑗}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (1) then BM^𝑝,𝑞 is called the Bonferroni mean (BM) operator.

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 𝑝,𝑞,𝑟≥0 and let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If GBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)(𝑛−2)^𝑛_{𝑖,𝑗,𝑘=1𝑖≠𝑗≠𝑘}𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (2) then GBM^{𝑝,𝑞,𝑟} is called the generalized Bonferroni mean (GBM) operator.

In particular, if 𝑟=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 𝑖=𝑗 or 𝑗=𝑘 or 𝑖=𝑘, 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 𝑝,𝑞,𝑟≥0 and let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 such that 𝑤_𝑖>0, 𝑖=1,2,…,𝑛 and ∑^𝑛_𝑖=1𝑤_𝑖=1. If GWBM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (3) then GWBM^{𝑝,𝑞,𝑟} 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 𝑟=0, then the GWBM operator reduces to the following: GWBM^𝑝,𝑞,0𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=^𝑛_𝑖=1𝑤_𝑖𝑤_𝑗𝑎^𝑝_𝑖𝑎^𝑞_𝑗^𝑛_𝑘=1𝑤_𝑘^{1/(𝑝+𝑞)}=^𝑛_𝑖=1𝑤_𝑖𝑤_𝑗𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (4) which is the weighted Bonferroni mean (WBM) operator.

(2) If 𝑞=0 and 𝑟=0, then the GWBM operator reduces to the following: GWBM^𝑝,0,0𝑎₁,𝑎₂,…,𝑎_𝑛=^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘𝑎^𝑝_𝑖^1/𝑝=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖^𝑛_𝑗=1𝑤_𝑗^𝑛_𝑘=1𝑤_𝑘^1/𝑝=^𝑛_𝑖=1𝑤_𝑖𝑎^𝑝_𝑖^1/𝑝, (5) which is the generalized weighted averaging operator. Furthermore, in this case, let us look at the GWBM operator for some special cases of 𝑝.(1)If 𝑝=1, the GWBM operator reduces to the weighted averaging (WA) operator.(2)If 𝑝→0, then the GWBM operator reduces to the weighted geometric (WG) operator.(3)If 𝑝→+∞, then the GWBM operator reduces to the max operator.

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 𝑎 can be defined by a triplet [𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]. The membership function 𝜇𝑎(𝑥) is defined as 𝜇𝑎(𝑥)=0,𝑥<𝑎^𝐿;𝑥−𝑎^𝐿𝑎^𝑀−𝑎^𝐿,𝑎^𝐿≤𝑥≤𝑎^𝑀;𝑥−𝑎^𝑈𝑎^𝑀−𝑎^𝑈,𝑎^𝑀≤𝑥≤𝑎^𝑈;0,𝑥>𝑎^𝑈, (6) where 𝑎^𝑈≥𝑎^𝑀≥𝑎^𝐿≥0, 𝑎^𝐿, and 𝑎^𝑈 stand for the lower and upper values of 𝑎, respectively, and 𝑎^𝑀 stands for the modal value [24]. In particular, if any two of 𝑎^𝐿,𝑎^𝑀, and 𝑎^𝑈 are equal, then 𝑎 reduces to an interval number; if all 𝑎^𝐿,𝑎^𝑀, and 𝑎^𝑈 are equal, then 𝑎 reduces to a real number. For convenience, we let Ω be the set of all triangular fuzzy numbers.

Let 𝑎=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈] and 𝑏̂=[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈] be two triangular fuzzy numbers, then some operational laws defined as follows [24]:(1)𝑎+𝑏̂=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]+[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈]=[𝑎^𝐿+𝑏^𝐿,𝑎^𝑀+𝑏^𝑀,𝑎^𝑈+𝑏^𝑈];(2)𝜆𝑎=𝜆[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]=[𝜆𝑎^𝐿,𝜆𝑎^𝑀,𝜆𝑎^𝑈];(3)𝑎×𝑏̂=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]×[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈]=[𝑎^𝐿𝑏^𝐿,𝑎^𝑀𝑏^𝑀,𝑎^𝑈𝑏^𝑈];(4)1/𝑎=1/[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]=[1/𝑎^𝑈,1/𝑎^𝑀,1/𝑎^𝐿].

In order to compare two triangular fuzzy numbers, Xu [21] provided the following definition.

Definition 5. Let 𝑎=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈] and let 𝑏̂=[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈] be two triangular fuzzy numbers; then the degree of possibility of 𝑎≥𝑏̂ is defined as follows: 𝑝𝑏̂𝑏𝑎≥=𝛿max1−max^𝑀−𝑎^𝐿𝑎^𝑀−𝑎^𝐿+𝑏^𝑀−𝑏^𝐿+()𝑏,0,01−𝛿max1−max^𝑈−𝑎^𝑀𝑎^𝑈−𝑎^𝑀+𝑏^𝑈−𝑏^𝑀,[],,0,0𝛿∈0,1 (7) which satisfies the following properties: 𝑏̂(𝑎)𝑝𝑏̂̂𝑎0≤𝑝𝑎≥≤1,𝑝𝑎≥=0.5,𝑎≥+𝑝𝑏≥=1. (8)

Here, 𝛿 reflects the decision maker's risk-bearing attitude. If 𝛿>0.5, then the decision maker is risk lover; if 𝛿=0.5, then the decision maker is neutral to risk; if 𝛿<0.5, then the decision maker is risk avertor.

In the following, we will give a simple procedure for ranking of the triangular fuzzy numbers 𝑎_𝑖(𝑖=1,2,…,𝑛). First, by using (7), we compare each 𝑎_𝑖 with all the 𝑎_𝑗(𝑗=1,2,…,𝑛); for simplicity, let 𝑝_𝑖𝑗=𝑝(𝑎_𝑖≥𝑎_𝑗), and then we develop a possibility matrix [25, 26] as 𝑝𝑃=¹¹𝑝₁₂⋯𝑝_1𝑛𝑝₂₁𝑝₂₂⋯𝑝_2𝑛⋮𝑝_𝑛1𝑝_𝑛2⋯𝑝_𝑛𝑛, (9) where 𝑝_𝑖𝑗≥0, 𝑝_𝑖𝑗+𝑝_𝑗𝑖=1, 𝑝_𝑖𝑖=1/2, 𝑖,𝑗=1,2,…,𝑛.

Summing all elements in each line of matrix 𝑃, we have 𝑝_𝑖=∑^𝑛_𝑗=1𝑝_𝑖𝑗, 𝑖=1,2,…,𝑛. Then, in accordance with the values of 𝑝_𝑖(𝑖=1,2,…,𝑛), we rank the 𝑎_𝑖(𝑖=1,2,…,𝑛) in descending order.

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.

Definition 6 (see [23]). Let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers, let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight vector of 𝑎_𝑖(𝑖=1,2,…,𝑛), where 𝑤_𝑖 indicates the importance degree of 𝑎_𝑖, satisfying 𝑤_𝑖>0, 𝑖=1,2,…,𝑛 and ∑^𝑛_𝑖=1𝑤_𝑖=1. If FWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑀_𝑖^𝑝𝑎^𝑀_𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑈_𝑖^𝑝𝑎^𝑈_𝑗^{𝑞1/(𝑝+𝑞)}, (10) where 𝑝,𝑞≥0, then FWBHM^𝑝,𝑞 is called the fuzzy weighted Bonferroni harmonic mean (FWBHM) operator.

In particular, if 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the FWBHM operator reduces to the following: FBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (11) which we call the fuzzy Bonferroni harmonic mean (FBHM) operator.

In addition, a special case can obtained as the change of parameter. If 𝑞=0, then the FWBHM operator reduces to the following: FWBHM^𝑝,0𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝑖∑^𝑛_𝑗=1𝑤_𝑗^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝑖^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝐿_𝑖^𝑝1/𝑝,1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑀_𝑖^𝑝1/𝑝,1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑈_𝑖^𝑝1/𝑝, (12) 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.

Theorem 7. Let 𝑝,𝑞≥0, and let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers, and the following are valid.
(1) Idempotency. If all 𝑎_𝑖(𝑖=1,2,…,𝑛) are equal, that is, 𝑎_𝑖=𝑎, for all 𝑖, then FWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=𝑎. (13) (2) Boundedness. 𝑎⁻≤FWBHM^𝑝,𝑞(𝑎₁,𝑎₂,…,𝑎_𝑛)≤𝑎⁺, where 𝑎⁻=[min_𝑖{𝑎^𝐿_𝑖},min_𝑖{𝑎^𝑀_𝑖},min_𝑖{𝑎^𝑈_𝑖}] and 𝑎⁺=[max_𝑖{𝑎^𝐿_𝑖},max_𝑖{𝑎^𝑀_𝑖},max_𝑖{𝑎^𝑈_𝑖}].
(3) Commutativity. Let 𝑎^′_𝑖=[𝑎_𝑖^′𝐿,𝑎_𝑖^′𝑀,𝑎_𝑖^′𝑈](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers, and then FWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=FWBHM^𝑝,𝑞𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛, (14) where (𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛) is any permutation of (𝑎₁,𝑎₂,…,𝑎_𝑛).

Proof. Since (2) can be proven easily, we prove (1) and (3) as follows.
(1) Since 𝑎_𝑖=𝑎, we have FWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝𝑎^{𝑞1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^{𝑝+𝑞1/(𝑝+𝑞)}=𝑎∑^𝑛_𝑖=1𝑤_𝑖∑^𝑛_𝑗=1𝑤_𝑗^{1/(𝑝+𝑞)}=𝑎. (15)
(3) Since (𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛) is any permutation of (𝑎₁,𝑎₂,…,𝑎_𝑛), then FWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^′_𝑖^𝑝𝑎^′_𝑗^{𝑞1/(𝑝+𝑞)}=FWBHM^𝑝,𝑞𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛. (16)
In particular, if the triangular fuzzy numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) reduce to the interval numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛), then the FWBHM operator (10) reduces to the uncertain weighted Bonferroni harmonic mean (UWBHM) operator as follows: UWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑈_𝑖^𝑝(𝑎^𝑈_𝑗)^{𝑞1/(𝑝+𝑞)}. (17)
If 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the UWBHM operator reduces to the uncertain Bonferroni harmonic mean (UBHM) operator as follows: UBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^{𝑞1/(𝑝+𝑞)},11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝑈_𝑖^𝑝𝑎^𝑈_𝑗^{𝑞1/(𝑝+𝑞)}. (18)
If 𝑎^𝐿_𝑖=𝑎^𝑈_𝑖=𝑎_𝑖, for all 𝑖, then the UWBHM operator (17) reduces to the weighted Bonferroni harmonic mean (WBHM) operator as follows: WBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}. (19) In this case, if 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the WBHM operator reduces to the Bonferroni harmonic mean (BHM) operator: BHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}. (20)

Example 8. Given a collection of triangular fuzzy numbers: 𝑎₁=[2,3,4], 𝑎₂=[1,2,4], 𝑎₃=[2,4,6], 𝑎₄=[1,3,5], let 𝑤=(0.3,0.1,0.2,0.4)^𝑇 be the weight vector of 𝑎_𝑖(𝑖=1,2,3,4); then, by FWBHM operator (10) (let 𝑝=𝑞=2), we have FWBHM^2,2𝑎₁,𝑎₂,𝑎₃,𝑎₄=1∑⁴_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝐿_𝑖²𝑎^𝐿_𝑗^21/4,1∑⁴_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑀_𝑖²𝑎^𝑀_𝑗^21/4,1∑⁴_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑈_𝑖²𝑎^𝑈_𝑗^21/4=[1.27,2.95,4.64]. (21)

Based on the OWA and FWBHM operators and Definition 5, we define fuzzy ordered weighted Bonferroni harmonic mean (FOWBHM) operator as follows.

Definition 9. Let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers. For 𝑝,𝑞≥0, a fuzzy ordered weighted Bonferroni harmonic mean (FOWBHM) operator of dimension 𝑛 is a mapping FOWBHM^𝑝,𝑞∶Ω^𝑛→Ω, that has an associated vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that 𝜔_𝑖≥0 and ∑^𝑛_𝑖=1𝜔_𝑖=1. Furthermore, FOWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝜎(𝑖)𝑎^𝑞_𝜎𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝐿_𝜎(𝑖)^𝑝𝑎^𝐿_𝜎𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑀_𝜎(𝑖)^𝑝𝑎^𝑀_𝜎𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑈_𝜎(𝑖)^𝑝𝑎^𝑈_𝜎𝑗^{𝑞1/(𝑝+𝑞)}, (22) where 𝑎_𝜎(𝑖)=[𝑎^𝐿_𝜎(𝑖),𝑎^𝑀_𝜎(𝑖),𝑎^𝑈_𝜎(𝑖)](𝑖=1,2,…,𝑛), and (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛) such that 𝑎_{𝜎(𝑖−1)}≥𝑎_𝜎(𝑖) for all 𝑖.

However, if there is a tie between 𝑎_𝑖 and 𝑎_𝑗, then we replace each of 𝑎_𝑖 and 𝑎_𝑗 by their average (𝑎_𝑖+𝑎_𝑗)/2 in process of aggregation. If 𝑘 items are tied, then we replace these by 𝑘 replicas of their average. The weighting vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 can be determined by using some weights determining methods like the normal distribution based method; see [27–29] for more details.

If 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the FOWBHM operator reduces to the FBHM operator; in addition, if 𝑞=0, then the FOWBHM operator reduces to the following: FOWBHM^𝑝,0𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝜎(𝑖)^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝜎(𝑖)∑^𝑛_𝑗=1𝑤_𝑗^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝜎(𝑖)^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝐿_𝜎(𝑖)^𝑝1/𝑝,1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑀_𝜎(𝑖)^𝑝1/𝑝,1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑈_𝜎(𝑖)^𝑝1/𝑝, (23) which we call the fuzzy ordered weighted generalized harmonic mean (FOWGHM) operator.

In particular, if the triangular fuzzy numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) reduce to the interval numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛), then the FOWBHM operator reduces to the uncertain ordered weighted Bonferroni harmonic mean (UOWBHM) operator as follows: UOWBHM^𝑝,𝑞𝑎_𝑖,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝜔_𝑖𝜔_𝑗/𝑎^𝑝_𝜎(𝑖)𝑎^𝑞_𝜎𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝐿_𝜎(𝑖)^𝑝𝑎^𝐿_𝜎𝑗^{𝑞1/(𝑝+𝑞)},1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑈_𝜎(𝑖)^𝑝𝑎^𝑈_𝜎𝑗^{𝑞1/(𝑝+𝑞)}, (24) where 𝑎_𝜎(𝑖)=[𝑎^𝐿_𝜎(𝑖),𝑎^𝑈_𝜎(𝑖)], and (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛) such that 𝑎_{𝜎(𝑖−1)}≥𝑎_𝜎(𝑖) for all 𝑖. If there is a tie between 𝑎_𝑖 and 𝑎_𝑗, then we replace each of 𝑎_𝑖 and 𝑎_𝑗 by their average (𝑎_𝑖+𝑎_𝑗)/2 in process of aggregation. If 𝑘 items are tied, then we replace these by 𝑘 replicas of their average.

If 𝑎^𝐿_𝑖=𝑎^𝑈_𝑖=𝑎_𝑖, for all 𝑖, then the UOWBHM operator reduces to the ordered weighted Bonferroni harmonic mean (OWBHM) operator as follows: OWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_𝑖,𝑗=1𝜔_𝑖𝜔_𝑗/𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{1/(𝑝+𝑞)}, (25) where 𝑏_𝑖 is the 𝑖th largest of 𝑎_𝑖(𝑖=1,2,…,𝑛). The OWBHM operator (25) has some special cases.(1)If 𝜔=(1,0,…,0)^𝑇, then OWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=max𝑎_𝑖=𝑏₁. (26)(2)If 𝜔=(0,0,…,1)^𝑇, then OWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=min𝑎_𝑖=𝑏_𝑛. (27)(3)If 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then OWBHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑏^𝑝_𝑖𝑏^𝑞_𝑗^{1/(𝑝+𝑞)}=11/𝑛²∑^𝑛_𝑖,𝑗=11/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=BHM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛. (28)

Example 10. Let 𝑎₁=[3,4,6], 𝑎₂=[1,2,4], 𝑎₃=[2,4,5], 𝑎₄=[3,5,6], and 𝑎₅=[2,5,7] be a collection of triangular fuzzy numbers. To rank these triangular fuzzy numbers, we first compare each triangular fuzzy number 𝑎_𝑖 with all triangular fuzzy numbers 𝑎_𝑗(𝑗=1,2,3,4,5) by using (7) (without loss of generality, set 𝛿=0.5); let 𝑝_𝑖𝑗=𝑝(𝑎_𝑖≥𝑎_𝑗)(𝑖,𝑗=1,2,3,4,5), then we utilize these possibility degrees to construct the following matrix 𝑃=(𝑝_𝑖𝑗)_5×5: 𝑃=0.50010.6670.3330.37500.5000000.33310.5000.125.0.2000.66710.8750.5000.4670.62510.8000.5330.500. (29) Summing all elements in each line of matrix 𝑃, we have 𝑝₁=2.875,𝑝₂=0.500,𝑝₃𝑝=2.158,₄=3.509,𝑝₅=3.458. (30) and then we rank the triangular fuzzy numbers 𝑎_𝑖(𝑖=1,2,3,4,5) in descending order in accordance with the values of 𝑝_𝑖(𝑖=1,2,3,4,5) as follows: 𝑎_𝜎(1)=𝑎₄,𝑎_𝜎(2)=𝑎₅,𝑎_𝜎(3)=𝑎₁,𝑎_𝜎(4)=𝑎₃,𝑎_𝜎(5)=𝑎₂. (31) Suppose that the weighting vector of the FOWBHM operator is 𝜔=(0.1117,0.2365,0.3036,0.2365,0.1117)^𝑇 (derived by the normal distribution based method [27]), and then by (22) (let 𝑝=𝑞=2), we get FOWBHM^2,2𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅=1∑⁵_𝑖,𝑗=1𝜔_𝑖𝜔_𝑗/𝑎^𝐿_𝜎(𝑖)²𝑎^𝐿_𝜎𝑗^21/4,1∑⁵_𝑖,𝑗=1𝜔_𝑖𝜔_𝑗/𝑎^𝑀_𝜎(𝑖)²𝑎^𝑀_𝜎𝑗^21/4,1∑⁵_𝑖,𝑗=1𝜔_𝑖𝜔_𝑗/𝑎^𝑈_𝜎(𝑖)²𝑎^𝑈_𝜎𝑗^21/4.=[1.901,3.632,5.509]. (32)

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.

Definition 11. Let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight vector of 𝑎_𝑖(𝑖=1,2,…,𝑛), where 𝑤_𝑖 indicates the importance degree of 𝑎_𝑖, satisfying 𝑤_𝑖>0, 𝑖=1,2,…,𝑛 and ∑^𝑛_𝑖=1𝑤_𝑖=1. For 𝑝,𝑞,𝑟≥0, if GFWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^𝑞𝑎^𝐿_𝑘^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑀_𝑖^𝑝𝑎^𝑀_𝑗^𝑞𝑎^𝑀_𝑘^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑈_𝑖^𝑝𝑎^𝑈_𝑗^𝑞𝑎^𝑈_𝑘^{𝑟1/(𝑝+𝑞+𝑟)}, (33) then GFWBHM^{𝑝,𝑞,𝑟} is called generalized fuzzy weighted Bonferroni harmonic mean (GFWBHM) operator.

In particular, if 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the GFWBHM operator reduces to the following: GFBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (34) which we call the generalized fuzzy Bonferroni harmonic mean (GFBHM) operator.

In addition, some special cases can be obtained as the change of parameters.

(1) If 𝑟=0, then the GFWBHM operator reduces to GFWBHM^𝑝,𝑞,0𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑘=1𝑤_𝑘∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=1∑^𝑛_𝑖,𝑗=1𝑤_𝑖𝑤_𝑗/𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (35) which is the FWBHM operator.

(2) If 𝑞=0 and 𝑟=0, then the GFWBHM operator reduces to GFWBHM^𝑝,0,0𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑝_𝑖^1/𝑝=1(∑^𝑛_𝑗=1𝑤_𝑗)∑^𝑛_𝑘=1𝑤_𝑘∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝑖^1/𝑝=1∑^𝑛_𝑖=1𝑤_𝑖/𝑎^𝑝_𝑖^1/𝑝, (36) which is FWGHM operator. In this case, if 𝑝=1, then FWGHM operator reduces to FWHM operator.

Similar to the FWBHM operator, the GFWBHM operator has the following properties.

Theorem 12. Let 𝑝,𝑞,𝑟≥0, and let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers, and the following are valid.
(1)  Idempotency. If all 𝑎_𝑖(𝑖=1,2,…,𝑛) are equal, that is, 𝑎_𝑖=𝑎, for all 𝑖, then GFWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=𝑎. (37)(2)  Boundedness.  𝑎⁻≤GFWBHM^{𝑝,𝑞,𝑟}(𝑎₁,𝑎₂,…,𝑎_𝑛)≤𝑎⁺, where 𝑎⁻=[min_𝑖{𝑎^𝐿_𝑖},min_𝑖{𝑎^𝑀_𝑖},min_𝑖{𝑎^𝑈_𝑖}] and 𝑎⁺=[max_𝑖{𝑎^𝐿_𝑖},max_𝑖{𝑎^𝑀_𝑖},max_𝑖{𝑎^𝑈_𝑖}]. 𝑎⁺=max_𝑖𝑎^𝐿_𝑖,max_𝑖𝑎^𝑀_𝑖,max_𝑖𝑎^𝑈_𝑖. (38)(3)  Commutativity. Let 𝑎^′_𝑖=[𝑎_𝑖^′𝐿,𝑎_𝑖^′𝑀,𝑎_𝑖^′𝑈](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers, and then GFWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=GFWBHM^{𝑝,𝑞,𝑟}𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛, (39) where (𝑎^′₁,𝑎^′₂,…,𝑎^′_𝑛) is any permutation of (𝑎₁,𝑎₂,…,𝑎_𝑛).

In particular, if the triangular fuzzy numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) reduce to the interval numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛), then the GFWBHM operator (24) reduces to the generalized uncertain weighted Bonferroni harmonic mean (GUWBHM) operator as follows: GUWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^𝑞𝑎^𝐿_𝑘^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑈_𝑖^𝑝𝑎^𝑈_𝑗^𝑞𝑎^𝑈_𝑘^{𝑟1/(𝑝+𝑞+𝑟)}. (40)

If 𝑤=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the GUWBHM operator reduces to the generalized uncertain Bonferroni harmonic mean (GUBHM): GUBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}=11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑎^𝐿_𝑖^𝑝𝑎^𝐿_𝑗^𝑞𝑎^𝐿_𝑘^{𝑟1/(𝑝+𝑞+𝑟)},11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑎^𝑈_𝑖^𝑝𝑎^𝑈_𝑗^𝑞𝑎^𝑈_𝑘^{𝑟1/(𝑝+𝑞+𝑟)}. (41)

Furthermore, if 𝑎^𝐿_𝑖=𝑎^𝑈_𝑖=𝑎_𝑖, for all 𝑖, then the GUWBHM operator reduces to the generalized weighted Bonferroni harmonic mean (GWBHM) operator: GWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}. (42) In this case, if 𝑝=1 and 𝑞=𝑟=0, the GWBHM operator reduces to the weighted harmonic mean (WHM) operator.

Example 13. Consider the four triangular fuzzy numbers 𝑎_𝑖 and their weight vector 𝑤 given in Example 8. Then by the GFWBHM operator (33) (without of generalization, let 𝑝=𝑞=𝑟=3), we have GFWBHM^3,3,3𝑎₁,𝑎₂,𝑎₃,𝑎₄=1∑⁴_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝐿_𝑖³𝑎^𝐿_𝑗³𝑎^𝐿_𝑘^31/9,1∑⁴_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑀_𝑖³𝑎^𝑀_𝑗³𝑎^𝑀_𝑘^31/9,1∑⁴_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑈_𝑖³𝑎^𝑈_𝑗³𝑎^𝑈_𝑘^31/9=[1.21,2.89,4.59]. (43)

Definition 14. Let 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) be a collection of triangular fuzzy numbers. For 𝑝,𝑞,𝑟≥0, a generalized fuzzy ordered weighted Bonferroni harmonic mean (GFOWBHM) operator of dimension 𝑛 is a mapping GFOWBHM^{𝑝,𝑞,𝑟}∶Ω^𝑛→Ω, that has an associated vector 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 such that 𝜔_𝑖≥0 and ∑^𝑛_𝑖=1𝜔_𝑖=1. Furthermore, GFOWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝑝_𝜎(𝑖)𝑎^𝑞_𝜎𝑗𝑎^𝑟_𝜎(𝑘)^{1/(𝑝+𝑞+𝑟)}=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝐿_𝜎(𝑖)^𝑝𝑎^𝐿_𝜎𝑗^𝑞𝑎^𝐿_𝜎(𝑘)^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝑀_𝜎(𝑖)^𝑝𝑎^𝑀_𝜎𝑗^𝑞𝑎^𝑀_𝜎(𝑘)^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝑈_𝜎(𝑖)^𝑝𝑎^𝑈_𝜎𝑗^𝑞𝑎^𝑈_𝜎(𝑘)^{𝑟1/(𝑝+𝑞+𝑟)}, (44) where 𝑎_𝜎(𝑖)=[𝑎^𝐿_𝜎(𝑖),𝑎^𝑀_𝜎(𝑖),𝑎^𝑈_𝜎(𝑖)](𝑖=1,2,…,𝑛), and (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛) such that 𝑎_{𝜎(𝑖−1)}≥𝑎_𝜎(𝑖) for all 𝑖.

However, if there is a tie between 𝑎_𝑖 and 𝑎_𝑗, then we replace each of 𝑎_𝑖 and 𝑎_𝑗 by their average (𝑎_𝑖+𝑎_𝑗)/2 in process of aggregation. If 𝑘 items are tied, then we replace these by 𝑘 replicas of their average.

If 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then the GFOWBHM operator reduces to the GFBHM operator. Moreover, some special cases can be obtained as the change of parameters. If 𝑟=0, then the GFOWBHM operator reduces to FOWBHM operator; if 𝑟=0 and 𝑞=0, then GFOWBHM operator reduces to FOWGHM operator. In particular, if the triangular fuzzy numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑀_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛) reduce to the interval numbers 𝑎_𝑖=[𝑎^𝐿_𝑖,𝑎^𝑈_𝑖](𝑖=1,2,…,𝑛), then the GFOWBHM operator reduces to the generalized uncertain ordered weighted Bonferroni harmonic mean (GUOWBHM) operator: GUOWBHM^{𝑝,𝑞,𝑟}𝑎_𝑖,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝑝_𝜎(𝑖)𝑎^𝑞_𝜎𝑗𝑎^𝑟_𝜎(𝑘)^{1/(𝑝+𝑞+𝑟)}=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝐿_𝜎(𝑖)^𝑝𝑎^𝐿_𝜎𝑗^𝑞𝑎^𝐿_𝜎(𝑘)^{𝑟1/(𝑝+𝑞+𝑟)},1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑎^𝑈_𝜎(𝑖)^𝑝𝑎^𝑈_𝜎𝑗^𝑞𝑎^𝑈_𝜎(𝑘)^{𝑟1/(𝑝+𝑞+𝑟)}, (45) where 𝑎_𝜎(𝑖)=[𝑎^𝐿_𝜎(𝑖),𝑎^𝑈_𝜎(𝑖)], and (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛) such that 𝑎_{𝜎(𝑖−1)}≥𝑎_𝜎(𝑖) for all 𝑖.

If 𝑎^𝐿_𝑖=𝑎^𝑈_𝑖=𝑎_𝑖, for all 𝑖=1,2,…,𝑛, then the GUOWBHM operator reduces to the generalized ordered weighted Bonferroni harmonic mean (GOWBHM) operator: GOWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=1∑^𝑛_{𝑖,𝑗,𝑘=1}𝜔_𝑖𝜔_𝑗𝜔_𝑘/𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (46) where 𝑏_𝑖 is the 𝑖th largest of 𝑎_𝑖(𝑖=1,2,…,𝑛). In this case, if 𝑝=1 and 𝑞=𝑟=0, then the GOWBHM operator reduces to the ordered weighted harmonic mean (OWHM) operator.

The GOWBHM operator (46) has some special cases.

(1) If 𝜔=(1,0,…,0)^𝑇, then GOWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=max𝑎_𝑖=𝑏₁. (47)

(2) If 𝜔=(0,0,…,1)^𝑇, then GOWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=min𝑎_𝑖=𝑏_𝑛. (48)

(3) If 𝜔=(1/𝑛,1/𝑛,…,1/𝑛)^𝑇, then GOWBHM^{𝑝,𝑞,𝑟}𝑎₁,𝑎₂,…,𝑎_𝑛=11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑏^𝑝_𝑖𝑏^𝑞_𝑗𝑏^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}=11/𝑛³∑^𝑛_{𝑖,𝑗,𝑘=1}1/𝑎^𝑝_𝑖𝑎^𝑞_𝑗𝑎^𝑟_𝑘^{1/(𝑝+𝑞+𝑟)}, (49) which we call the generalized Bonferroni harmonic mean (GBHM) operator.

Example 15. Consider the four triangular fuzzy numbers 𝑎_𝑖 and their weight vector 𝑤 given in Example 10. Then by the GFOWBHM operator (44) (let 𝑝=𝑞=𝑟=3), we have GFOWBHM^3,3,3𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅=1∑⁵_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝐿_𝜎(𝑖)³𝑎^𝐿_𝜎𝑗³𝑎^𝐿_𝜎(𝑘)^31/9,1∑⁵_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑀_𝜎(𝑖)³𝑎^𝑀_𝜎𝑗³𝑎^𝑀_𝜎(𝑘)^31/9,1∑⁵_{𝑖,𝑗,𝑘=1}𝑤_𝑖𝑤_𝑗𝑤_𝑘/𝑎^𝑈_𝜎(𝑖)³𝑎^𝑈_𝜎𝑗³𝑎^𝑈_𝜎(𝑘)^31/9=[1.751,3.410,5.422]. (50) In the following section, we will apply the developed operators to multiple attribute group decision making.

3. An Approach to Multiple Attribute Group Decision Making with Triangular Fuzzy Information

For a group decision making with triangular fuzzy information, let 𝑋={𝑥₁,𝑥₂,…,𝑥_𝑛} be a discrete set of 𝑛 alternatives, let 𝐺={𝐺₁,𝐺₂,…,𝐺_𝑚} be the set of 𝑚 attributes, whose weight vector is 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑚)^𝑇 with 𝑤_𝑖≥0 and ∑^𝑚_𝑖=1𝑤_𝑖=1, and let 𝐷={𝑑₁,𝑑₂,…,𝑑_𝑠} be the set of decision makers, whose weight vector is 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑠)^𝑇, where 𝑣_𝑘≥0 and ∑^𝑠_𝑘=1𝑣_𝑘=1. Suppose that 𝐴^(𝑘)=(𝑎^(𝑘)_𝑖𝑗)_𝑚×𝑛 is the decision matrix, where 𝑎^(𝑘)_𝑖𝑗=[𝑎^𝐿(𝑘)_𝑖𝑗,𝑎^𝑀(𝑘)_𝑖𝑗,𝑎^𝑈(𝑘)_𝑖𝑗] is an attribute value, which takes the form of triangular fuzzy number, of the alternative 𝑥_𝑗∈𝑋 with respect to the attribute 𝐺_𝑖∈𝐺.

In the following, we apply the GFWBHM and GFOWBHM operators to group decision making with triangular fuzzy information.

Step 1. Normalize each attribute value 𝑎^(𝑘)_𝑖𝑗 in the matrix 𝐴^(𝑘) into a corresponding element in the matrix 𝑅^(𝑘)=(𝑟̂^(𝑘)_𝑖𝑗)_𝑚×𝑛 (𝑟̂^(𝑘)_𝑖𝑗=[𝑟^𝐿(𝑘)_𝑖𝑗,𝑟^𝑀(𝑘)_𝑖𝑗,𝑟^𝑈(𝑘)_𝑖𝑗]) using the following formulas: 𝑟̂^(𝑘)_𝑖𝑗=𝑎^(𝑘)_𝑖𝑗∑^𝑛_𝑗=1𝑎^(𝑘)_𝑖𝑗=𝑎^𝐿(𝑘)_𝑖𝑗∑^𝑛_𝑗=1𝑎^𝑈(𝑘)_𝑖𝑗,𝑎^𝑀(𝑘)_𝑖𝑗∑^𝑛_𝑗=1𝑎^𝑀(𝑘)_𝑖𝑗,𝑎^𝑈(𝑘)_𝑖𝑗∑^𝑛_𝑗=1𝑎^𝐿(𝑘)_𝑖𝑗,forbenefitattribute𝐺_𝑖𝑟̂,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑠,_{(𝑘)𝑖𝑗}=1/𝑎^(𝑘)_𝑖𝑗∑^𝑛_𝑗=11/𝑎^(𝑘)_𝑖𝑗=1/𝑎^𝑈(𝑘)_𝑖𝑗∑^𝑛_𝑗=11/𝑎^𝐿(𝑘)_𝑖𝑗,1/𝑎^𝑀(𝑘)_𝑖𝑗∑^𝑛_𝑗=11/𝑎^𝑀(𝑘)_𝑖𝑗,1/𝑎^𝐿(𝑘)_𝑖𝑗∑^𝑛_𝑗=11/𝑎^𝑈(𝑘)_𝑖𝑗,forcostattribute𝐺_𝑖,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑠. (51)

Step 2. Utilize the GFWBHM operator (33) as follows: 𝑟̂_𝑗^(𝑘)=GFWBHM^{𝑝,𝑞,𝑟}𝑟̂^(𝑘)_1𝑗,𝑟̂^(𝑘)_2𝑗,…,𝑟̂^(𝑘)_𝑚𝑗=1∑^𝑚_{𝑖,ℎ,𝑙=1}𝑤_𝑖𝑤_ℎ𝑤_𝑙/𝑟̂^(𝑘)_𝑖𝑗^𝑝𝑟̂^(𝑘)_ℎ𝑗^𝑞𝑟̂^(𝑘)_𝑙𝑗^{𝑟1/(𝑝+𝑞+𝑟)}=^𝑚_{𝑖,ℎ,𝑙=1}𝑤_𝑖𝑤_ℎ𝑤_𝑙𝑟^𝐿(𝑘)_𝑖𝑗^𝑝𝑟^𝐿(𝑘)_ℎ𝑗^𝑞𝑟^𝐿(𝑘)_𝑙𝑗^{𝑟−1/(𝑝+𝑞+𝑟)}_{𝑚𝑖,ℎ,𝑙=1},𝑤_𝑖𝑤_ℎ𝑤_𝑙𝑟^𝑀(𝑘)_𝑖𝑗^𝑝𝑟^𝑀(𝑘)_ℎ𝑗^𝑞𝑟^𝑀(𝑘)_𝑙𝑗^{𝑟−1/(𝑝+𝑞+𝑟)}_{𝑚𝑖,ℎ,𝑙=1},𝑤_𝑖𝑤_ℎ𝑤_𝑙𝑟^𝑈(𝑘)_𝑖𝑗^𝑝𝑟^𝑈(𝑘)_ℎ𝑗^𝑞𝑟^𝑈(𝑘)_𝑙𝑗^{𝑟−1/(𝑝+𝑞+𝑟)}, (52) to aggregate all the elements in the 𝑗th column of 𝑅^(𝑘) and get the overall attribute value 𝑟̂_𝑗^(𝑘) of the alternative 𝑥_𝑗 corresponding to the decision maker 𝑑_𝑘.

Step 3. Utilize the GFOWBHM operator (44): 𝑟_𝑗̂=GFOWBHM^{𝑝,𝑞,𝑟}𝑟̂_𝑗⁽¹⁾,𝑟̂_𝑗⁽²⁾,…,𝑟̂_𝑗^(𝑠)=1∑^𝑠_{𝑘,ℎ,𝑙=1}𝜔_𝑘𝜔_ℎ𝜔_𝑙/𝑟̇̂_𝑗^{𝜎(𝑘)𝑝}𝑟̇̂_𝑗^{𝜎(ℎ)𝑞}𝑟̇̂_𝑗^{𝜎(𝑙)𝑟1/(𝑝+𝑞+𝑟)}=^𝑠_{𝑘,ℎ,𝑙=1}𝜔_𝑘𝜔_ℎ𝜔_𝑙𝑟̇_𝑗^{𝐿(𝜎(𝑘))𝑝}𝑟̇_𝑗^{𝐿(𝜎(ℎ))𝑞}𝑟̇_𝑗^{𝐿(𝜎(𝑙))𝑟−1/(𝑝+𝑞+𝑟)}_{𝑠𝑘,ℎ,𝑙=1},𝜔_𝑘𝜔_ℎ𝜔_𝑙𝑟̇_𝑗^{𝑀(𝜎(𝑘))𝑝}𝑟̇_𝑗^{𝑀(𝜎(ℎ))𝑞}𝑟̇_𝑗^{𝑀(𝜎(𝑙))𝑟−1/(𝑝+𝑞+𝑟)}_{𝑠𝑘,ℎ,𝑙=1},𝜔_𝑘𝜔_ℎ𝜔_𝑙𝑟̇_𝑗^{𝑈(𝜎(𝑘))𝑝}𝑟̇_𝑗^{𝑈(𝜎(ℎ))𝑞}𝑟̇_𝑗^{𝑈(𝜎(𝑙))𝑟−1/(𝑝+𝑞+𝑟)}, (53) to aggregate the overall attribute values 𝑟̂_𝑗^(𝑘)(𝑘=1,2,…,𝑠) corresponding to the decision maker 𝑑_𝑘(𝑘=1,2,…,𝑠) and get the collective overall attribute value 𝑟_𝑗̂, where 𝑟̇̂_𝑗^(𝜎(𝑘))=[𝑟̇_𝑗^{𝐿(𝜎(𝑘))},𝑟̇_𝑗^{𝑀(𝜎(𝑘))},𝑟̇_𝑗^{𝑈(𝜎(𝑘))}] is the 𝑘th largest of the weighted data and 𝑟̇̂_𝑗^(𝑘)(𝑟̇̂_𝑗^(𝑘)=𝑠𝑣_𝑘𝑟̂_𝑗^(𝑘), 𝑘=1,2,…,𝑠), 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑠)^𝑇 is the weighting vector of the GFOWBHM operator, with 𝜔_𝑘≥0 and ∑^𝑠_𝑘=1𝜔_𝑘=1.

Step 4. Compare each 𝑟_𝑗̂ with all 𝑟_𝑖̂(𝑖=1,2,…,𝑛) by using (7), and let 𝑝_𝑖𝑗=𝑝(𝑟_𝑖̂≥𝑟_𝑗̂), and then construct the possibility matrix 𝑃=(𝑝_𝑖𝑗)_𝑛×𝑛, where 𝑝_𝑖𝑗≥0, 𝑝_𝑖𝑗+𝑝_𝑗𝑖=1, 𝑝_𝑖𝑖=0.5, 𝑖,𝑗=1,2,…,𝑛. Summing all elements in each line of matrix 𝑃, we have 𝑝_𝑖=∑^𝑛_𝑗=1𝑝_𝑖𝑗, 𝑖=1,2,…,𝑛, and then reorder 𝑟_𝑗̂(𝑗=1,2,…,𝑛) in descending order in accordance with the values of 𝑝_𝑗(𝑗=1,2,…,𝑛).

Step 5. Rank all alternatives 𝑥_𝑗(𝑗=1,2,…,𝑛) by the ranking of 𝑟_𝑗̂(𝑗=1,2,…,𝑛), and then select the most desirable one.

Step 6. End.

4. Example Illustrations

In this section, we use a multiple attribute group decision making problem of determining what kind of air-conditioning systems should be installed in a library (adopted from [21, 30]) to illustrate the proposed approach.

A city is planning to build a municipal library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning systems should be installed in the library. The contractor offers five feasible alternatives, which might be adapted to the physical structure of the library. The alternatives 𝑥_𝑗(𝑗=1,2,3,4,5) are to be evaluated using triangular fuzzy numbers by the three decision makers 𝑑_𝑘(𝑘=1,2,3) (whose weight vector is 𝑣=(0.4,0.3,0.3)^𝑇) under three major impacts: economic, functional, and operational. Two monetary attributes and six nonmonetary attributes (i.e., 𝐺₁: owning cost ($/ft²), 𝐺₂: operating cost ($/ft²), 𝐺₃: performance (*), 𝐺₄: noise level (Db), 𝐺₅: maintainability (*), 𝐺₆: reliability (%), 𝐺₇: flexibility (*), 𝐺₈: safety (*), where * unit is from 0 to 1 scale, three attributes 𝐺₁, 𝐺₂, and 𝐺₄ are cost attributes, and the other five attributes are benefit attributes, and suppose that the weight vector of the attributes 𝐺_𝑖(𝑖=1,2,…,8) is 𝑤=(0.05,0.08,0.14,0.12,0.18,0.21,0.05,0.17)^𝑇) emerged from three impacts is Tables 1, 2, and 3.

Table 1 

Triangular fuzzy number decision matrix 𝐴⁽¹⁾.

Table 2 

Triangular fuzzy number decision matrix 𝐴⁽²⁾.

Table 3 

Triangular fuzzy number decision matrix 𝐴⁽³⁾.

In the following, we utilize the decision procedure to select the best air-conditioning system.

Step 1. By using (51), we normalize each attribute value 𝑎^(𝑘)_𝑖𝑗 in the matrices 𝐴^(𝑘)(𝑘=1,2,3) into the corresponding element in the matrices 𝑅^(𝑘)=(𝑟̂_𝑖𝑗)_8×5 (𝑘=1,2,3) (Tables 4, 5, and 6).

Table 4 

Normalized triangular fuzzy number decision matrix 𝑅⁽¹⁾.

Table 5 

Normalized triangular fuzzy number decision matrix 𝑅⁽²⁾.

Table 6 

Normalized triangular fuzzy number decision matrix 𝑅⁽³⁾.

Step 2. Utilize the GFWBHM operator (52) (let 𝑝=𝑞=𝑟=3) to aggregate all elements in the 𝑗th column 𝑅^(𝑘) and get the overall attribute value 𝑟̂_𝑗^(𝑘): 𝑟̂₁⁽¹⁾=[],𝑟̂0.1390,0.1753,0.2187₂₍₁₎=[],𝑟̂0.1347,0.1586,0.1927₃₍₁₎=[],𝑟̂0.1581,0.1852,0.2178₄₍₁₎=[],𝑟̂0.1900,0.2289,0.2651₅₍₁₎=[],𝑟̂0.1565,0.1911,0.2311₁₍₂₎=[],𝑟̂0.1480,0.1851,0.2248₂₍₂₎=[],𝑟̂0.1434,0.1706,0.1992₃₍₂₎=[],𝑟̂0.1561,0.1792,0.2057₄₍₂₎=[],𝑟̂0.1927,0.2228,0.2477₅₍₂₎=[],𝑟̂0.1499,0.1761,0.2098₁₍₃₎=[],𝑟̂0.1459,0.1811,0.2104₂₍₃₎=[],𝑟̂0.1370,0.1607,0.1938₃₍₃₎=[],𝑟̂0.1679,0.1921,0.2173₄₍₃₎=[],𝑟̂0.1883,0.2138,0.2395₅₍₃₎=[0.1678,0.1922,0.2215]. (54)Step 3. Utilize the GFOWBHM operator (53) (suppose that its weight vector is 𝜔=(0.243,0.514,0.243)^𝑇 determined by using the normal distribution based method [27] (let 𝛿=0.5 and 𝑝=𝑞=𝑟=3) to aggregate the overall attribute value 𝑟̂_𝑗^(𝑘)(𝑘=1,2,3), corresponding to the decision maker 𝑑_𝑘(𝑘=1,2,3), and get the collective overall attribute value 𝑟_𝑗̂: 𝑟₁̂=[],𝑟̂0.1459,0.1816,0.2195₂=[],𝑟̂0.1395,0.1649,0.1962₃=[],𝑟̂0.1592,0.1837,0.2111₄=[],𝑟̂0.1909,0.2218,0.2493₅=[0.1552,0.1830,0.2171]. (55)Step 4. Compare each 𝑟_𝑗̂ with all 𝑟_𝑖̂(𝑖=1,2,3,4,5) by using (7) (without loss of generality, set 𝛿=0.5), and let 𝑝_𝑖𝑗=𝑝(𝑟_𝑖̂≥𝑟_𝑗̂), and then construct a possibility matrix: 𝑃=0.50.73870.459800.46100.26130.50.163800.19210.54020.83620.500.50101110.510.53900.80790.499000.5 (56) Summing all elements in each line of matrix 𝑃, we have 𝑝₁=2.1595,𝑝₂=1.1171,𝑝₃𝑝=2.3774,₄=4.5,𝑝₅=2.3459, (57) and then reorder 𝑟_𝑗̂(𝑗=1,2,3,4,5) in descending order in accordance with the values of 𝑝_𝑗(𝑗=1,2,3,4,5): 𝑟₄̂>𝑟₃̂>𝑟₅̂>𝑟₁̂>𝑟₂̂. (58)Step 5. Rank all the alternatives 𝑥_𝑗(𝑗=1,2,3,4,5) by the ranking of 𝑟_𝑗̂(𝑗=1,2,3,4,5): 𝑥₄≻𝑥₃≻𝑥₅≻𝑥₁≻𝑥₂, (59) and thus the most desirable alternative is 𝑥₄.

From the previous analysis, the results obtained by the proposed approach are very similar to the ones obtained Xu's approach [21], but our approach is more flexible than that of Xu [21] because it can provide the decision makers more choices as parameters are assigned different values.

5. 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.

Table 7 

Comparison of the proposed approach with other approaches.

6. 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 interval-valued triangular fuzzy environment.

Acknowledgment

This work was supported by a Research Grant of Pukyong National University (2013).