A Multiple Attribute Decision Making Method Based on Uncertain Linguistic Heronian Mean

Abstract

The Heronian mean is a useful aggregation operator which can capture the interrelationship of the input arguments. In this paper, we develop some Heronian means based on uncertain linguistic variables, such as the generalized uncertain linguistic Heronian mean (GULHM) and uncertain linguistic geometric Heronian mean (ULGHM), and some of their desirable properties are also investigated. Considering the different importance of the input arguments, we define the generalized uncertain linguistic weighted Heronian mean (GULWHM) and uncertain linguistic weighted geometric Heronian mean (ULWGHM). Then, a method of multiple attribute decision making under uncertain linguistic environment is presented based on the GULWHM or the ULWGHM. In the end, an example is given to demonstrate the effectiveness and feasibility of the proposed method.

1. Introduction

Multiple attribute decision making exists here and there, and a multiple attribute decision making problem is to find the most desirable candidate from some feasible alternatives. In real life, decision-makers often provide their preferences on alternatives using linguistic term sets instead of numerical values owing to the fuzziness of human thinking process, and multiple attribute decision making under linguistic environment is a focus in recent years [1–12]. In the process of decision making, the input arguments need to be aggregated by some proper approaches so that the decision makers can select the most desirable alternative. Among these approaches, the operators are widely used. Yager [13] introduced the ordered weighted averaging (OWA) operator, which has only been used in situations in which the input arguments are the exact numerical values. But now, it has been extended to accommodate linguistic environment [2, 14–17], uncertain linguistic environment [18–22], and some other preference representation structures [23, 24]. Uncertain linguistic variable, as a generalization form of linguistic variable, is more powerful in dealing with uncertainty than linguistic variable since it is characterized by a linguistic interval rather than a linguistic value. Since its appearance, the uncertain linguistic variable has received much attention from researchers. Based on the weighted arithmetic averaging (WAA) operator [25] and the ordered weighted averaging (OWA) operator [13], Xu [18] introduced some uncertain linguistic aggregation operators called uncertain linguistic weighted averaging (ULWA) operator, uncertain linguistic ordered weighted averaging (ULOWA) operator, and uncertain linguistic hybrid aggregation (ULHA) operator. The ULWA operator only weights the uncertain linguistic arguments while the ULOWA operator only weights the ordered positions of the uncertain linguistic arguments. The ULHA operator combines the advantages of the ULWA and the ULOWA operator and weights not only the given arguments but also their ordered positions. From a geometric point of view, Xu [20] proposed some uncertain linguistic aggregation operators, such as the uncertain linguistic geometric mean (ULGM), uncertain linguistic weighted geometric mean (ULWGM), and uncertain linguistic ordered weighted geometric (ULOWG) operator. In order to solve the drawbacks of the ULWGM and the ULOWG operator, Wei [21] developed the uncertain linguistic hybrid geometric mean (ULHGM) operator and proposed an approach to multiple attribute group decision making with uncertain linguistic information based on the ULWGM and ULHGM operators. In [22], Park et al. proposed the uncertain linguistic weighted harmonic mean (ULWHM) operator, uncertain linguistic ordered weighted harmonic mean (ULOWHM) operator, and uncertain linguistic hybrid harmonic mean (ULHHM) operator, and an illustrative example about determining the air-conditioning system is also given to demonstrate the effectiveness and feasibility of the proposed method. Motivated by Yager and Filev [26], Xu [27] proposed some induced uncertain linguistic aggregation operators which can aggregate the decision making information in environments of mixing numeric and linguistic variables, such as the induced uncertain linguistic ordered weighted averaging (IULOWA) operator and the induced uncertain linguistic ordered weighted geometric (IULOWG) operator [20]. In [28], Xu generalized the IULOWA and the IULOWG operator and developed some generalized induced uncertain linguistic aggregation operators, including the generalized induced uncertain linguistic ordered weighted averaging (GIULOWA) operator and the generalized induced uncertain linguistic ordered weighted geometric (GIULOWG) operator.

However, the above uncertain linguistic aggregation approaches designed for solving multiple attribute decision making problems only consider the importance of the given arguments but ignore the correlation of them. Up to now, we are only aware of one paper on uncertain linguistic decision making that pays attention to the correlation of the input arguments [29]. In [29], Wei et al. utilized the uncertain linguistic Bonferroni mean (ULBM) operator and the uncertain linguistic geometric Bonferroni mean (ULGBM) operator which are an extension of the Bonferroni mean (BM) [30] to aggregate the uncertain linguistic arguments. The main advantage of the ULBM and ULGBM is that they can reflect the interrelationship of the input uncertain linguistic arguments. Nevertheless, these two means have their own disadvantages. For example, given a set of attributes 𝐶_𝑖(𝑖=1,2,…,𝑛), the BM can reflect the correlation between any pair of attributes 𝐶_𝑖 and 𝐶_𝑗(𝑖≠𝑗) but neglect the relationship between the attribute 𝐶_𝑖 and itself. Moreover, the BM considers the correlation between 𝐶_𝑖 and 𝐶_𝑗(𝑖≠𝑗) and the correlation between 𝐶_𝑗 and 𝐶_𝑖(𝑖≠𝑗) simultaneously, which results in potential redundancy. In order to solve these issues, we introduce the Heronian mean (HM) [31], the generalized Heronian mean (GHM₁) [32], and the geometric Heronian mean (GHM₂) [33] and extend them to accommodate uncertain linguistic environment.

To do so, the remainder of this paper is organized as follows. In Section 2, we briefly review some basic concepts, such as the uncertain linguistic variable, HM, GHM₁, and GHM₂. In Section 3, we extend these means to accommodate the situation in which the input arguments are uncertain linguistic variables and develop some uncertain linguistic Heronian means, such as generalized uncertain linguistic Heronian mean (GULHM), generalized uncertain linguistic weighted Heronian mean (GULWHM), uncertain linguistic geometric Heronian mean (ULGHM), and uncertain linguistic weighted geometric Heronian mean (ULWGHM). In Section 4, we propose a method for multiple attribute decision making with uncertain linguistic information based on GULWHM or ULWGHM. In Section 5, an example is given to verify the effectiveness and feasibility of the proposed method. Section 6 ends the paper with some concluding remarks.

2. Uncertain Linguistic Variables and Heronian Mean

2.1. Uncertain Linguistic Variables

Let 𝑆={𝑠_𝑖∣𝑖=1,2,…,𝑡} be a linguistic term set with odd cardinality, where 𝑠_𝑖 represents a possible value for a linguistic variable. For example, a set of seven terms, 𝑆, could be defined as follows: 𝑆=𝑠₁=extremelypoor,𝑠₂=verypoor,𝑠₃𝑠=poor,₄=medium,𝑠₅=good,𝑠₆𝑠=verygood,₇=extremelygood. (1)

It is usually required that there exist the following [7, 17, 21].(1)The set is ordered as 𝑠_𝑖≥𝑠_𝑗 if 𝑖≥𝑗.(2)There is the negation operator neg(𝑠_𝑖)=𝑠_𝑗 such that 𝑖+𝑗=𝑡+1.(3)Max operator max(𝑠_𝑖,𝑠_𝑗)=𝑠_𝑖, if 𝑠_𝑖≥𝑠_𝑗.(4)Min operator min(𝑠_𝑖,𝑠_𝑗)=𝑠_𝑖, if 𝑠_𝑖≤𝑠_𝑗.

To preserve all the given information, the discrete term set 𝑆 should be extended to a continuous term set 𝑆={𝑠_𝛼∣𝑠₁≤𝑠_𝛼≤𝑠_𝑞,𝛼∈[1,𝑞]}, where 𝑞 is a sufficiently large positive integer; if 𝑠_𝛼∈𝑆, then we call 𝑠_𝛼 the original term; otherwise, we call 𝑠_𝛼 the virtual term [17, 21]. The decision maker, in general, uses the original linguistic terms to evaluate alternatives, and the virtual linguistic terms can only appear in operations.

Definition 1 (see [18–22, 27, 28]). Let 𝑠̃=[𝑠_𝛼,𝑠_𝛽], where 𝑠_𝛼,𝑠_𝛽∈𝑆, 𝑠_𝛼, and 𝑠_𝛽 are the lower and the upper limits, respectively, and then we call 𝑠̃ the uncertain linguistic variable. Suppose that 𝑆 is the set of all uncertain linguistic variables.

If 𝑠_𝛼=𝑠_𝛽, then the uncertain linguistic variable 𝑠̃ is reduced to a linguistic value. Consider any three uncertain linguistic variables 𝑠̃=[𝑠_𝛼,𝑠_𝛽], 𝑠₁̃=[𝑠_𝛼1,𝑠_𝛽1], 𝑠₂̃=[𝑠_𝛼2,𝑠_𝛽2], and let 𝜆∈[0,1]; then their operational laws are defined as follows [18–21, 27, 28]:(1)  𝑠₁̃⊕𝑠₂̃=[𝑠_𝛼1,𝑠_𝛽1]⊕[𝑠_𝛼2,𝑠_𝛽2]=[𝑠_𝛼1⊕𝑠_𝛼2,𝑠_𝛽1⊕𝑠_𝛽2]=[𝑠_𝛼1+𝛼2,𝑠_𝛽1+𝛽2];(2)  𝜆𝑠̃=𝜆[𝑠_𝛼,𝑠_𝛽]=[𝜆𝑠_𝛼,𝜆𝑠_𝛽]=[𝑠_𝜆𝛼,𝑠_𝜆𝛽];(3)  𝑠₁̃⊗𝑠₂̃=[𝑠_𝛼1,𝑠_𝛽1]⊗[𝑠_𝛼2,𝑠_𝛽2]=[𝑠_𝛼1⊗𝑠_𝛼2,𝑠_𝛽1⊗𝑠_𝛽2]=[𝑠_𝛼1𝛼2,𝑠_𝛽1𝛽2];(4)  𝑠^𝜆̃=([𝑠_𝛼,𝑠_𝛽])^𝜆=[(𝑠_𝛼)^𝜆,(𝑠_𝛽)^𝜆]=[𝑠_𝛼𝜆,𝑠_𝛽^𝜆].

Moreover, the following relationship can be easily proved:(5)  𝑠₁̃⊕𝑠₂̃=𝑠₂̃⊕𝑠₁̃;(6)  𝑠₁̃⊗𝑠₂̃=𝑠₂̃⊗𝑠₁̃;(7)  𝜆(𝑠₁̃⊕𝑠₂̃)=𝜆𝑠₁̃⊕𝜆𝑠₂̃;(8)  (𝑠₁̃⊗𝑠₂̃)^𝜆=𝑠^𝜆₁̃⊗𝑠^𝜆₂̃;(9)  𝜆₁𝑠̃⊕𝜆₂𝑠̃=(𝜆₁+𝜆₂)𝑠̃;(10)  𝑠̃^𝜆1⊗𝑠̃^𝜆2=𝑠̃^𝜆1+𝜆2.

In order to compare the uncertain linguistic variables, we give the following definition.

Definition 2 (see [34]). Let 𝑠₁̃=[𝑠_𝛼1,𝑠_𝛽1] and 𝑠₂̃=[𝑠_𝛼2,𝑠_𝛽2] be two uncertain linguistic variables, and let len(𝑠₁̃)=𝛽₁−𝛼₁ and len(𝑠₂̃)=𝛽₂−𝛼₂; then the degree of possibility of  𝑠₁̃≥𝑠₂̃  is defined as 𝑝𝑠₁̃≥𝑠₂̃=max0,𝛽₁−𝛼₂−max0,𝛼₁−𝛽₂len𝑠₁̃+len𝑠₂̃. (2)

From Definition 2, we can easily get the following results:(1)0≤𝑝(𝑠₁̃≥𝑠₂̃)≤1, 0≤𝑝(𝑠₂̃≥𝑠₁̃)≤1; (2)𝑝(𝑠₁̃≥𝑠₂̃)+𝑝(𝑠₂̃≥𝑠₁̃)=1. Especially, 𝑝(𝑠₁̃≥𝑠₁̃)=𝑝(𝑠₂̃≥𝑠₂̃)=1/2.

2.2. Heronian Mean

Heronian mean (HM), which is one of the aggregation methods, has the desirable characteristic that it can reflect the interrelationship of the input arguments. The definition of HM is as follows.

Definition 3 (see [31]). Let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If HM𝑎₁,𝑎₂,…,𝑎_𝑛=2𝑛(𝑛+1)^𝑛𝑛_{𝑖=1𝑗=𝑖}𝑎_𝑖𝑎_𝑗, (3) then HM is called the Heronian mean (HM).

Based on Definition 3, Yu and Wu [32, 33] proposed the generalized Heronian mean (GHM₁) and the geometric Heronian mean (GHM₂).

Definition 4 (see [32]). Let  𝑝,𝑞≥0 and 𝑝,𝑞 do not take the value 0 simultaneously. Let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If GHM₁^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=2𝑛(𝑛+1)^𝑛𝑛_{𝑖=1𝑗=𝑖}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}, (4) then GHM₁ is called the generalized Heronian mean (GHM₁). If 𝑝=𝑞=1/2 especially, then the GHM₁ is reduced to HM.

It is noted that the GHM₁ has the following properties:(1)GHM₁^𝑝,𝑞(0,0,…,0)=0; (2)GHM₁^𝑝,𝑞(𝑎,𝑎,…,𝑎)=𝑎, if 𝑎_𝑖=𝑎, for all  𝑖;(3)GHM₁^𝑝,𝑞(𝑎₁,𝑎₂,…,𝑎_𝑛)≥GHM₁^𝑝,𝑞(𝑏₁,𝑏₂,…,𝑏_𝑛), that is, GHM₁^𝑝,𝑞  is monotonic, if  𝑎_𝑖≥𝑏_𝑖, for all  𝑖;(4)min_𝑖{𝑎_𝑖}≤GHM₁^𝑝,𝑞(𝑎₁,𝑎₂,…,𝑎_𝑛)≤max_𝑖{𝑎_𝑖}.

Example 5. Let 𝑎₁,𝑎₂,𝑎₃ be three nonnegative numbers and  𝑝=𝑞=2; then GHM₁^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=2𝑛(𝑛+1)^𝑛𝑛_{𝑖=1𝑗=𝑖}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=16𝑎²₁𝑎²₁+𝑎²₁𝑎²₂+𝑎²₁𝑎²₃+𝑎²₂𝑎²₂+𝑎²₂𝑎²₃+𝑎²₃𝑎²₃^1/4. (5)

If we use Bonferroni mean (BM) [30] to aggregate the above three nonnegative numbers, then BM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1𝑖≠𝑗}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}=16𝑎²₁𝑎²₂+𝑎²₁𝑎²₃+𝑎²₂𝑎²₁+𝑎²₂𝑎²₃+𝑎²₃𝑎²₁+𝑎²₃𝑎²₂^1/4. (6)

From the above analysis, we can find that the BM computes 𝑎²₁𝑎²₂, 𝑎²₂𝑎²₁, 𝑎²₁𝑎²₃, 𝑎²₃𝑎²₁, 𝑎²₂𝑎²₃, and 𝑎²₃𝑎²₂ separately. However, 𝑎²₁𝑎²₂ is equal to  𝑎²₂𝑎²₁, 𝑎²₁𝑎²₃ is equal to  𝑎²₃𝑎²₁, and 𝑎²₂𝑎²₃ is equal to  𝑎²₃𝑎²₂. Hence, it results in potential redundancy. Moreover, the BM has not paid attention to  𝑎²₁𝑎²₁, 𝑎²₂𝑎²₂, and  𝑎²₃𝑎²₃. Nevertheless, the GHM₁ can solve the two problems effectively.

Definition 6 (see [33]). Let 𝑝,𝑞≥0 and 𝑝,𝑞  do not take the value 0 simultaneously. Let  𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. If GHM₂^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑎_𝑖+𝑞𝑎_𝑗^{2/𝑛(𝑛+1)}, (7) then GHM₂ is called the geometric Heronian mean (GHM₂).

It is noted that the GHM₂ has the following properties:(1)GHM₂^𝑝,𝑞(0,0,…,0)=0; (2)GHM₂^𝑝,𝑞(𝑎,𝑎,…,𝑎)=𝑎, if  𝑎_𝑖=𝑎, for all  𝑖;(3)GHM₂^𝑝,𝑞(𝑎₁,𝑎₂,…,𝑎_𝑛)≥GHM₂^𝑝,𝑞(𝑏₁,𝑏₂,…,𝑏_𝑛) that is, GHM₂^𝑝,𝑞  is monotonic, if  𝑎_𝑖≥𝑏_𝑖, for all  𝑖;(4)min_𝑖{𝑎_𝑖}≤GHM₂^𝑝,𝑞(𝑎₁,𝑎₂,…,𝑎_𝑛)≤max_𝑖{𝑎_𝑖}.

Example 7. Let 𝑎₁,𝑎₂,𝑎₃ be three nonnegative numbers and  𝑝=𝑞=1; then GHM₂^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑎_𝑖+𝑞𝑎_𝑗^{2/𝑛(𝑛+1)}=12𝑎₁+𝑎₁𝑎₁+𝑎₂𝑎₁+𝑎₃×𝑎₂+𝑎₂𝑎₂+𝑎₃𝑎₃+𝑎₃^1/6. (8) If we use geometric Bonferroni mean (GBM) proposed by Xia et al. [35] to aggregate the above three nonnegative numbers, then GBM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1𝑖≠𝑗}𝑝𝑎_𝑖+𝑞𝑎_𝑗^{1/𝑛(𝑛−1)}=12𝑎₁+𝑎₂𝑎₁+𝑎₃𝑎₂+𝑎₁×𝑎₂+𝑎₃𝑎₃+𝑎₁𝑎₃+𝑎₂^1/6. (9)
Similar to BM, the GBM also results in potential redundancy. Furthermore, it has not paid attention to (𝑎₁+𝑎₁),  (𝑎₂+𝑎₂),   and  (𝑎₃+𝑎₃). However, the GHM₂ can solve the two problems effectively.

3. Uncertain Linguistic Heronian Means

3.1. The GULHM and the GULWHM

The GHM₁ has the desirable characteristic capturing the interrelationship of the input arguments. However, the arguments suitable to be aggregated by the GHM₁ usually take the forms of nonnegative real numbers. In this section, we will extend the GHM₁ to accommodate the situations in which the input arguments are uncertain linguistic variables. Based on the operational rules on uncertain linguistic variables and Definition 4, we give the generalized uncertain linguistic Heronian mean (GULWHM) in the following.

Definition 8. Let 𝑝,𝑞≥0  and 𝑝,𝑞  do not take the value 0 simultaneously. Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛)   be a collection of uncertain linguistic variables. If GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝑖̃⊗𝑠̃^𝑞_𝑗^{1/(𝑝+𝑞)}=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛼𝑖⊗𝑠^𝑞_𝛼𝑗^{1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛽𝑖⊗𝑠^𝑞_𝛽𝑗^{1/(𝑝+𝑞)}, (10) then the GULHM is called the generalized uncertain linguistic Heronian mean (GULHM). If  𝑝=𝑞=1/2; then the GULHM reduces to GULHM^1/2,1/2𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠̃_𝑖^1/2⊗𝑠̃_𝑗^1/2=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛼^1/2_𝑖⊗𝑠_𝛼^1/2_𝑗,2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛽^1/2_𝑖⊗𝑠_𝛽^1/2_𝑗, (11) which we call the uncertain linguistic Heronian mean (ULHM).

In the following, we investigate the desirable properties of the GULHM.

Theorem 9 (idempotency). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛) be a collection of uncertain linguistic variables. If all  𝑠_𝑖̃  are equal, that is,  𝑠_𝑖̃=𝑠̃=[𝑠_𝛼,𝑠_𝛽]   for all  𝑖, then 𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=𝑠.̃ (12)

Proof. Consider the following: GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=GULHM^𝑝,𝑞(̃̃𝑠)̃=2𝑠,𝑠,…,𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝̃⊗𝑠̃^{𝑞1/(𝑝+𝑞)}=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛼⊗𝑠^𝑞_𝛼,2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛽⊗𝑠^𝑞_𝛽^{1/(𝑝+𝑞)}=𝑠_𝛼,𝑠_𝛽. (13)

Theorem 10 (permutation). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖]   and  𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛) be two collections of uncertain linguistic variables; then 𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃, (14) where 𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)   is any permutation of  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛).

Proof. Since 𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)   is any permutation of  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛), then GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝑖̃⊗𝑠̃^𝑞_𝑗^{1/(𝑝+𝑞)}=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠̃_𝑖^′𝑝⊗𝑠̃_𝑗^{′𝑞1/(𝑝+𝑞)}=GULHM^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃. (15)

Theorem 11 (monotonicity). Let 𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖]   and  𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)   be two collections of uncertain linguistic variables. If  𝑠_𝛼𝑖≤𝑠^′_𝛼𝑖,𝑠_𝛽𝑖≤𝑠^′_𝛽𝑖  for all  𝑖, then 𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃≤𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃. (16)

Proof. Since  𝑠_𝛼𝑖≤𝑠^′_𝛼𝑖, 𝑠_𝛽𝑖≤𝑠^′_𝛽𝑖  for all  𝑖, then 2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛼𝑖⊗𝑠^𝑞_𝛼𝑗^{1/(𝑝+𝑞)}≤2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛼^′𝑝_𝑖⊗𝑠_𝛼^′𝑞_𝑗^{1/(𝑝+𝑞)}2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛽𝑖⊗𝑠^𝑞_𝛽𝑗^{1/(𝑝+𝑞)}≤2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛽^′𝑝_𝑖⊗𝑠_𝛽^′𝑞_𝑗^{1/(𝑝+𝑞)}. (17) By Definition 2, we get that 2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛼𝑖⊗𝑠^𝑞_𝛼𝑗^{1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛽𝑖⊗𝑠^𝑞_𝛽𝑗^{1/(𝑝+𝑞)}≤2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛼^′𝑝_𝑖⊗𝑠_𝛼^′𝑞_𝑗^{1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛽^′𝑝_𝑖⊗𝑠_𝛽^′𝑞_𝑗^{1/(𝑝+𝑞)}. (18) Thus, GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝑖̃⊗𝑠̃^𝑞_𝑗^{1/(𝑝+𝑞)}≤2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠̃_𝑖^′𝑝⊗𝑠̃_𝑗^{′𝑞1/(𝑝+𝑞)}=GULHM^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃. (19)

Theorem 12 (boundedness). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛) be a collection of uncertain linguistic variables, and 𝑠⁻̃=min_𝑖𝑠_𝑖̃=min_𝑖𝑠_𝛼𝑖,min_𝑖𝑠_𝛽𝑖,𝑠⁺̃=max_𝑖𝑠_𝑖̃=max_𝑖𝑠_𝛼𝑖,max_𝑖𝑠_𝛽𝑖. (20) Then, 𝑠⁻̃≤𝐺𝑈𝐿𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃≤𝑠⁺̃. (21)

Proof. Consider the following:GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝑖̃⊗𝑠̃^𝑞_𝑗^{1/(𝑝+𝑞)}=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛼𝑖⊗𝑠^𝑞_𝛼𝑗^{1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝛽𝑖⊗𝑠^𝑞_𝛽𝑗^{1/(𝑝+𝑞)}≤2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}max_𝑖𝑠_𝛼𝑖^𝑝⊗max_𝑖𝑠_𝛼𝑖^{𝑞1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}max_𝑖𝑠_𝛽𝑖^𝑝⊗max_𝑖𝑠_𝛽𝑖^{𝑞1/(𝑝+𝑞)}=max_𝑖𝑠_𝛼𝑖,max_𝑖𝑠_𝛽𝑖=𝑠⁺̃. (22) Similarly, we can prove GULHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑝_𝑖̃⊗𝑠̃^𝑞_𝑗^{1/(𝑝+𝑞)}≥min_𝑖𝑠_𝛼𝑖,min_𝑖𝑠_𝛽𝑖=𝑠⁻̃, (23) which completes the proof of Theorem 12.

In most cases, the input arguments have their own importance. Each argument should be assigned a weight. Hence, it is necessary to consider the weighted form of the GULHM. In the following, we define the generalized uncertain linguistic weighted Heronian mean (GULWHM).

Definition 13. Let  𝑝,𝑞≥0 and 𝑝,𝑞  do not take the value 0 simultaneously. Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛) be a collection of uncertain linguistic variables. And  𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇  is the weight vector of  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛), where  𝑤_𝑖  indicates the importance degree of  𝑠_𝑖̃, satisfying  𝑤_𝑖≥0, and  ∑^𝑛_𝑖=1𝑤_𝑖=1.   If GULWHM_𝑤^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠̃_𝑖^𝑝⊗𝑤_𝑗𝑠̃_𝑗^{𝑞1/(𝑝+𝑞)}=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠_𝛼𝑖^𝑝⊗𝑤_𝑗𝑠_𝛼𝑗^{𝑞1/(𝑝+𝑞)},2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠_𝛽𝑖^𝑝⊗𝑤_𝑗𝑠_𝛽𝑗^{𝑞1/(𝑝+𝑞)}, (24) then GULWHM is called the generalized uncertain linguistic weighted Heronian mean (GULWHM). If 𝑝=𝑞=1/2; then the GULWHM reduces to GULWHM_𝑤^1/2,1/2𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠̃_𝑖^1/2⊗𝑤_𝑗𝑠̃_𝑗^1/2=2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠_𝛼𝑖^1/2⊗𝑤_𝑗𝑠_𝛼𝑗^1/2,2𝑛(𝑛+1)^𝑛_{𝑖=1,𝑗=𝑖}𝑤_𝑖𝑠_𝛽𝑖^1/2⊗𝑤_𝑗𝑠_𝛽𝑗^1/2, (25) which we call the uncertain linguistic weighted Heronian mean (ULWHM).

3.2. The ULGHM and the ULWGHM

The geometric Heronian mean (GHM₂) proposed by Yu [33] has the capability to capture the interrelationship among the input arguments. In this section, we will extend the GHM₂ to accommodate the situations in which the input arguments are uncertain linguistic variables. Based on the operational rules on uncertain linguistic variables and Definition 6, we give the uncertain linguistic geometric Heronian mean (ULGHM) as follows.

Definition 14. Let  𝑝,𝑞≥0  and 𝑝,𝑞  do not take the value 0 simultaneously. Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛)   be a collection of uncertain linguistic variables. If ULGHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠_𝑖̃⊕𝑞𝑠̃_𝑗^{2/𝑛(𝑛+1)}=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠_𝛼𝑖⊕𝑞𝑠_𝛼𝑗^{2/𝑛(𝑛+1)},1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠_𝛽𝑖⊕𝑞𝑠_𝛽𝑗^{2/𝑛(𝑛+1)}, (26) then the ULGHM is called the uncertain linguistic geometric Heronian mean (ULGHM). If  𝑝=𝑞, then the ULGHM reduces to ULGHM^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=12^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝑖̃⊕𝑠̃_𝑗^{2/𝑛(𝑛+1)}=12^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛼𝑖⊕𝑠_𝛼𝑗^{2/𝑛(𝑛+1)},^𝑛_{𝑖=1,𝑗=𝑖}𝑠_𝛽𝑖⊕𝑠_𝛽𝑗^{2/𝑛(𝑛+1)} (27) which we call the uncertain linguistic evolution Heronian mean (ULEHM).

In the following, we investigate the desirable properties of the ULGHM, and they can be derived easily.

Theorem 15 (idempotency). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛) be a collection of uncertain linguistic variables. If all  𝑠_𝑖̃  are equal, that is,  𝑠_𝑖̃=𝑠̃=[𝑠_𝛼,𝑠_𝛽]   for all  𝑖, then 𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=𝑠.̃ (28)

Theorem 16 (permutation). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖]   and  𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)   be two collections of uncertain linguistic variables; then 𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃=𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃, (29) where 𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)   is any permutation of  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛).

Theorem 17 (monotonicity). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖]   and 𝑠^′_𝑖̃=[𝑠^′_𝛼𝑖,𝑠^′_𝛽𝑖](𝑖=1,2,…,𝑛)  be two collections of uncertain linguistic variables. If  𝑠_𝛼𝑖≤𝑠^′_𝛼𝑖, 𝑠_𝛽𝑖≤𝑠^′_𝛽𝑖  for all  𝑖, then 𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃≤𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠^′₁̃,𝑠^′₂̃,…,𝑠^′_𝑛̃. (30)

Theorem 18 (boundedness). Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛)  be a collection of uncertain linguistic variables, and 𝑠⁻̃=min_𝑖𝑠_𝑖̃=min_𝑖𝑠_𝛼𝑖,min_𝑖𝑠_𝛽𝑖,𝑠⁺̃=max_𝑖𝑠_𝑖̃=max_𝑖𝑠_𝛼𝑖,max_𝑖𝑠_𝛽𝑖, (31) then 𝑠⁻̃≤𝑈𝐿𝐺𝐻𝑀^𝑝,𝑞𝑠₁̃,𝑠₂̃,…,𝑠_𝑛̃≤𝑠⁺̃. (32)

It is noted that the uncertain linguistic geometric Heronian mean (ULGHM) does not consider the importance of each argument. In the following, we introduce the uncertain linguistic weighted geometric Heronian mean (ULWGHM).

Definition 19. Let  𝑝,𝑞≥0  and 𝑝,𝑞  do not take the value 0 simultaneously. Let  𝑠_𝑖̃=[𝑠_𝛼𝑖,𝑠_𝛽𝑖](𝑖=1,2,…,𝑛)   be a collection of uncertain linguistic variables. If ULWGHM_𝑤^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠^𝑤𝑖_𝑖̃⊕𝑞𝑠̃^𝑤𝑗_𝑗^{2/𝑛(𝑛+1)}=1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠^𝑤𝑖_𝛼𝑖⊕𝑞𝑠^𝑤𝑗_𝛼𝑗^{2/𝑛(𝑛+1)},1𝑝+𝑞^𝑛_{𝑖=1,𝑗=𝑖}𝑝𝑠^𝑤𝑖_𝛽𝑖⊕𝑞𝑠^𝑤𝑗_𝛽𝑗^{2/𝑛(𝑛+1)}, (33) then ULWGHM is called the uncertain linguistic weighted geometric Heronian mean (ULWGHM). If  𝑝=𝑞, then the ULWGHM reduces to ULWGHM_𝑤^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=12^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑤𝑖_𝑖̃⊕𝑠̃^𝑤𝑗_𝑗^{2/𝑛(𝑛+1)}=12^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑤𝑖_𝛼𝑖⊕𝑠^𝑤𝑗_𝛼𝑗^{2/𝑛(𝑛+1)},^𝑛_{𝑖=1,𝑗=𝑖}𝑠^𝑤𝑖_𝛽𝑖⊕𝑠^𝑤𝑗_𝛽𝑗^{2/𝑛(𝑛+1)} (34) which we call the uncertain linguistic weighted evolution Heronian mean (ULWGHM).

4. A Method for Multiple Attribute Decision Making Based on Heronian Means under Uncertain Linguistic Environment

In this section, we consider a multiple attribute decision making problem with uncertain linguistic information. The generalized uncertain linguistic weighted Heronian mean (GULWHM) or the uncertain linguistic weighted geometric Heronian mean (ULWGHM) proposed in Section 3 will be used to solve the multiple attribute decision making problem.

Let 𝐴={𝐴₁,𝐴₂,…,𝐴_𝑚} be the set of alternatives and 𝐶={𝐶₁,𝐶₂,…,𝐶_𝑛} the set of attributes, whose weight vector is 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇such that 𝑤_𝑗∈[0,1],∑^𝑛_𝑗=1𝑤_𝑗=1. The decision makers use the uncertain linguistic variable to provide the linguistic expression under the attribute 𝐶_𝑗 for the alternative 𝐴_𝑖 and construct the uncertain linguistic decision matrix 𝐷=(𝑑_𝑖𝑗)_𝑚×𝑛. In the following, based on the GULWHM or the ULWGHM, we develop an approach to multiple attribute decision making with uncertain linguistic information.

Step 1. Utilize the GULWHM as 𝑑_𝑖=𝑠_𝛼𝑖,𝑠_𝛽𝑖=GULWHM_𝑤^𝑝,𝑞𝑑_𝑖1,𝑑_𝑖2,…,𝑑_𝑖𝑛=2𝑛(𝑛+1)^𝑛_𝑘=1⊕_,𝑙=𝑘𝑤_𝑘𝑑_𝑖𝑘^𝑝⊗𝑤_𝑙𝑑_𝑖𝑙^{𝑞1/(𝑝+𝑞)}(𝑖=1,2,…,𝑚), (35) or the ULWGHM as 𝑑_𝑖=𝑠_𝛼𝑖,𝑠_𝛽𝑖=ULWGHM_𝑤^𝑝,𝑞𝑑_𝑖1,𝑑_𝑖2,…,𝑑_𝑖𝑛=1𝑝+𝑞^𝑛_𝑘=1⊗_,𝑙=𝑘𝑝𝑑^𝑤𝑘_𝑖𝑘⊕𝑞𝑑^𝑤𝑙_𝑖𝑙^{2/𝑛(𝑛+1)}(𝑖=1,2,…,𝑚) (36) to get the overall attribute value 𝑑_𝑖 of the alternative 𝐴_𝑖(𝑖=1,2,…,𝑚).

Step 2. To rank these overall attribute values  𝑑_𝑖(𝑖=1,2,…,𝑚), we first compare each 𝑑_𝑖 with all the 𝑑_𝑗(𝑗=1,2,…,𝑚)   by using (2). Then a complementary matrix  𝑃=(𝑝_𝑖𝑗)_𝑚×𝑚  is developed, where 𝑝_𝑖𝑗=𝑝𝑑_𝑖≥𝑑_𝑗,𝑝_𝑖𝑗≥0,𝑝_𝑖𝑗+𝑝_𝑗𝑖𝑝=1,_𝑖𝑖=0.5,𝑖,𝑗=1,2,…,𝑚. (37)

Summing all the elements in each line of matrix  𝑃=(𝑝_𝑖𝑗)_𝑚×𝑚, we have 𝑝_𝑖=∑^𝑚_𝑗=1𝑝_𝑖𝑗, 𝑖=1,2,…,𝑚.   Then we rank the overall attribute values  𝑑_𝑖  in descending order according to the values of  𝑝_𝑖(𝑖=1,2,…,𝑚).

Step 3. Rank all the alternatives 𝐴_𝑖 and select the desirable one in accordance with the values of  𝑑_𝑖(𝑖=1,2,…,𝑚).

Step 4. End.

5. Example Illustration and Discussion

In this section, an example adapted from [29] is given to illustrate the application of the methods proposed in this paper.

5.1. Example Illustration

Example 20 (see [29]). Suppose an organization plans to implement ERP system. The first step is to form a project team that consists of CIO and two senior representatives from user departments. By collecting all possible information about ERP vendors and systems, project team chooses four potential ERP systems 𝐴_𝑖(𝑖=1,2,3,4)   as candidates. The company employs some external professional organizations (or experts) to aid this decision making. The project team selects four attributes to evaluate the alternatives: (1) function and technology  𝐶₁,  (2) strategic fitness  𝐶₂,  (3) vendor’s ability  𝐶₃, and (4) vendor’s reputation  𝐶₄. Decision makers use the uncertain linguistic variables to evaluate the four possible alternatives 𝐴_𝑖(𝑖=1,2,3,4)  under the above four attributes (whose weight vector is  𝑤=(0.2,0.1,0.3,0.4)^𝑇) and construct the uncertain linguistic decision matrix  𝐷=(𝑑_𝑖𝑗)_4×4  listed in Table 1.

Table 1 

Uncertain linguistic decision matrix 𝐷.

In the following, we use the proposed methods to get the most desirable system.

Step 1. Utilize the GULWHM as 𝑑_𝑖=𝑠_𝛼𝑖,𝑠_𝛽𝑖=GULWHM_𝑤^𝑝,𝑞𝑑_𝑖1,𝑑_𝑖2,…,𝑑_𝑖𝑛=2𝑛(𝑛+1)^𝑛_{𝑘=1,𝑙=𝑘}𝑤_𝑘𝑑_𝑖𝑘^𝑝⊗𝑤_𝑙𝑑_𝑖𝑙^{𝑞1/(𝑝+𝑞)}(𝑖=1,2,…,𝑚) (38) to obtain the overall attribute value 𝑑_𝑖 for the alternative  𝐴_𝑖(𝑖=1,2,3,4), and let  𝑝=𝑞=1. We have 𝑑₁=𝑠_0.74,𝑠_1.01,𝑑₂=𝑠_0.89,𝑠_1.22,𝑑₃=𝑠_1.32,𝑠_1.59,𝑑₄=𝑠_1.13,𝑠_1.43. (39)

Step 2. To rank these overall attribute values  𝑑_𝑖(𝑖=1,…,4), we first compare each 𝑑_𝑖  with all the  𝑑_𝑗(𝑗=1,…,4) by using (2). Then a complementary matrix 𝑃=(𝑝_𝑖𝑗)_4×4  is developed as 𝑃=𝑝_𝑖𝑗4×4=0.5000.2000.0000.0000.8000.5000.0000.1431.0001.0000.5000.8071.0000.8570.1930.500. (40)

Summing all the elements in each line of matrix  𝑃=(𝑝_𝑖𝑗)_4×4, we have 𝑝₁=0.700,𝑝₂𝑝=1.443,₃=3.307,𝑝₄=2.550. (41)

Then we rank the overall attribute values 𝑑_𝑖 in descending order according to the values of  𝑝_𝑖(𝑖=1,2,3,4) as 𝑑₃≻𝑑₄≻𝑑₂≻𝑑₁. (42)

Step 3. Rank all the alternatives 𝐴_𝑖 in accordance with the values of  𝑑_𝑖(𝑖=1,2,3,4) as 𝐴₃≻𝐴₄≻𝐴₂≻𝐴₁. (43) Thus, the most desirable system is  𝐴₃.

If we use the ULWGHM to solve the above multiple attribute decision making problem and let  𝑝=𝑞, then the overall attribute values  𝑑_𝑖  of the alternative  𝐴_𝑖(𝑖=1,2,3,4)   can be obtained as follows: 𝑑₁=𝑠_1.30,𝑠_1.41,𝑑₂=𝑠_1.35,𝑠_1.47,𝑑₃=𝑠_1.51,𝑠_1.59,𝑑₄=𝑠_1.45,𝑠_1.55. (44)

To rank these overall attribute values  𝑑_𝑖(𝑖=1,2,3,4), we first compare each 𝑑_𝑖  with all the  𝑑_𝑗(𝑗=1,2,3,4)   by using (2). Then a complementary matrix  𝑃=(𝑝_𝑖𝑗)_4×4  is developed as 𝑃=𝑝_𝑖𝑗4×4=0.5000.2610.0000.0000.7390.5000.0000.0911.0001.0000.5000.7781.0000.9090.2220.500. (45)

Summing all the elements in each line of matrix  𝑃=(𝑝_𝑖𝑗)_4×4, we have 𝑝₁=0.761,𝑝₂𝑝=1.330,₃=3.278,𝑝₄=2.631. (46)

Then we rank the overall attribute values  𝑑_𝑖  in descending order according to the values of  𝑝_𝑖(𝑖=1,2,3,4) as 𝑑₃≻𝑑₄≻𝑑₂≻𝑑₁. (47)

Rank all the alternatives  𝐴_𝑖  in accordance with the values of  𝑑_𝑖(𝑖=1,2,3,4) as 𝐴₃≻𝐴₄≻𝐴₂≻𝐴₁. (48)

Thus, the most desirable system is 𝐴₃ and the ranking is the same as obtained by the GULWHM.

5.2. Discussion

If the parameter 𝑝 or 𝑞 takes the value of zero, then the GULWHM and ULWGHM cannot capture the interrelationship of the input arguments. Moreover, different overall attribute values  𝑑_𝑖  of the alternatives  𝐴_𝑖(𝑖=1,2,3,4)   can be obtained, and it needs much more calculation effort as the parameters 𝑝 and 𝑞 change. Here, we will list some of them. From Table 2, we can find that the overall attribute values obtained by the GULWHM become bigger as the parameters 𝑝 and 𝑞 increase simultaneously for the same aggregation arguments. If the parameter  𝑝  is fixed (without loss of generality, 𝑝 takes the value 1) and the parameter 𝑞 increases, the overall attribute values obtained by the GULWHM and shown in Table 3 become bigger for the same aggregation arguments. Similarly, if the parameter 𝑞 is fixed (𝑞=1), the aggregated results in Table 4 show that the overall attribute values obtained by the GULWHM for the same aggregation arguments firstly experience a decrease and then become bigger as the parameter  𝑝  increases. The different parameters play an important part in decision making. The decision makers who take a pessimistic view for prospect can choose the smaller values of the parameters  𝑝  and  𝑞, while the decision makers who take an optimistic view for prospect can choose the bigger values of the parameters  𝑝  or  𝑞.

Table 2 

Overall attribute values by the GULWHM_𝑤^𝑝,𝑝 and the rankings of the alternatives.

Table 3 

Overall attribute values by the GULWHM_𝑤^1,𝑞 and the rankings of the alternatives.

Table 4 

Overall attribute values by the GULWHM_𝑤^𝑝,1 and the rankings of the alternatives.

If we utilize the ULWGHM to aggregate the arguments, some different overall attribute values  𝑑_𝑖  of the alternatives 𝐴_𝑖(𝑖=1,2,3,4) are listed in Tables 5 and 6. If the parameter 𝑝 is fixed (𝑝=1), the overall attribute values obtained by the ULWGHM become bigger as the parameter  𝑞  increases for the same aggregation arguments. If the parameter 𝑞 is fixed (𝑞=1), the overall attribute values obtained by the ULWGHM become smaller as the parameter 𝑝 increases for the same aggregation arguments. Therefore, the decision makers who take a pessimistic view for prospect can choose the smaller values of the parameter 𝑞 or the bigger values of the parameter 𝑝, while the decision makers who take an optimistic view for prospect can choose the bigger values of the parameter  𝑞  or the smaller values of the parameter  𝑝. From Tables 2 to 6, we can find that the overall attribute values of each alternative derived by the GULWHM or ULWGHM depend on the choice of the parameters  𝑝  and  𝑞, but the ranking is kept unchanged.

Table 5 

Overall attribute values by the ULWGHM_𝑤^1,𝑞 and the rankings of the alternatives.

Table 6 

Overall attribute values by the ULWGHM_𝑤^𝑝,1 and the rankings of the alternatives.

6. Concluding Remarks

The Heronian mean can reflect the correlation of the aggregated arguments and is usually used to aggregate the information taken the form of numerical numbers. In this paper, we extend the Heronian mean to accommodate the situation where the input arguments are uncertain linguistic variables and develop some uncertain linguistic Heronian means such as the generalized uncertain linguistic Heronian mean (GULHM) and uncertain linguistic geometric Heronian mean (ULGHM). Some desirable properties of these means such as idempotency, permutation, monotonicity, and boundedness are also discussed. Moreover, to aggregate uncertain linguistic variables and embody different importance of the input arguments, we then define the generalized uncertain linguistic weighted Heronian mean (GULWHM) and uncertain linguistic weighted geometric Heronian mean (ULWGHM). The proposed means take the interrelationship of the input arguments into account, and it is a flexible multiple attribute decision making method in that the decision makers can choose different values of the parameters  𝑝  and  𝑞  according to their actual needs. To demonstrate the effectiveness and feasibility of the developed uncertain linguistic Heronian means, an example about ERP system is given. In future research, we will continue to study the Heronian mean, and some other types of Heronian mean will also be investigated.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 71171112), the Key Project of University Philosophy and Social Science in Jiangsu Province (no. 2012ZDIXM007 and 2012JDXM003), the Funding of Jiangsu Innovation Program for Graduate Education (no. CXLX13_171), the Fundamental Research Funds for the Central Universities, and the Young Teachers’ Science Foundation of Anhui University of Technology (no. QZ201321).