Multiple Attribute Group Decision-Making Methods Under Hesitant Fuzzy Linguistic Environment

1 Introduction

As an extension of fuzzy set [51], hesitant fuzzy set proposed by Torra [25] and Torra and Narukawa [26] has become a very useful tool in managing the situation in which decision makers hesitate among several values to assess an alternative, variable, etc. Suppose some decision makers discuss the membership degree of an element to a given set: some people assign 0.5, some 0.7, while others 0.8. The decision makers cannot reach an agreement. For such cases, a hesitant fuzzy set {0.5, 0.7, 0.8} that incorporates all the decision makers’ opinions can represent the membership degree. Obviously, the hesitant fuzzy set, which permits the membership degree represented by several possible values, is different from the interval-valued fuzzy number [0.5, 0.8]. Torra [25] and Torra and Narukawa [26] proposed some basic operations for hesitant fuzzy sets and proved that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set [2]. Moreover, they discussed the relationship between hesitant fuzzy set and some other generalizations of fuzzy set, such as type 2 fuzzy set [7, 19], type n fuzzy set [7], and fuzzy multiset [18, 48]. To aggregate the attribute values, some aggregation operators for hesitant fuzzy information were developed [33, 36, 37, 57]. Some distance, similarity, and correlation measures for hesitant fuzzy sets have been presented [5, 44, 45]. Chen et al. [5] proposed several correlation coefficient formulas for hesitant fuzzy sets and applied them to do clustering analysis. Zhang and Wei [53] extended the VIKOR method to handle the multiple-attribute decision-making problems with hesitant fuzzy information. Xu and Zhang [46] developed an approach to deal with the hesitant fuzzy multiple-attribute decision-making problem with incomplete weight information based on the TOPSIS method.

However, in the real-life world, owing to the fuzziness of human thinking process, the decision-making information may not be assessed in a quantitative form but often in a qualitative one [11, 40, 52]. It allows us to represent the information more flexibly when we are unable to express it precisely. Usually, decision makers hesitate among several values to assess an alternative or indicator. Hesitant fuzzy sets are suitable for modeling the quantitative situation. Nevertheless, how to do so in qualitative situation seems to be an interesting and challenging problem. To solve this issue and increase the richness of linguistic elicitation, Rodríguez et al. [23] proposed hesitant fuzzy linguistic term set (HFLTS), which permits the membership degree of an element to a given set to be several consecutive linguistic values. Since its introduction, the HFLTS has attracted more and more attention from researchers in recent years [6, 14, 15, 16, 20, 24, 28, 29, 30, 31, 50, 54, 55, 56].

The decision-making information needs to be aggregated by proper methods, and the aggregation operators are a powerful tool and are commonly used. The linguistic aggregation methods [9, 10, 12, 17, 32, 39, 42, 47] cannot be applied directly to the situation where the input arguments take the form of HFLTS. To this end, Rodríguez et al. [23] presented a hesitant fuzzy linguistic decision-making model based on two symbolic aggregation operators, the min_upper operator and the max_lower operator. The min_upper operator selects the worse of the superior values, whereas the max_lower operator is just the opposite. Then, a linguistic interval is built, and each alternative can be ranked in accordance with the linguistic interval [27]. However, the linguistic interval obtained by the two operators is unable to incorporate all the decision-making information and the proposed method cannot be extended to deal with group decision-making problems. Wei et al. [34] defined hesitant fuzzy linguistic weighted averaging (HFLWA) operator and hesitant fuzzy linguistic ordered weighted averaging (HFLOWA) operator. Lee and Chen [13] also proposed some hesitant fuzzy linguistic aggregation operators to overcome the disadvantages of Rodríguez et al.’s [23] and Wei et al.’s [34] approach for MADM, such as HFLWA operator, HFLOWA, hesitant fuzzy linguistic weighted geometric (HFLWG) operator, and hesitant fuzzy linguistic ordered weighted geometric (HFLOWG) operator. However, the HFLWA and HFLWG operators weight the hesitant fuzzy linguistic arguments, whereas the HFLWOWA and HFLOWG operators weight the ordered position of the hesitant fuzzy linguistic arguments instead of weighting the arguments themselves. The operators consider only one of them. To solve these issues, we develop some hesitant fuzzy linguistic hybrid aggregation operators, which reflect the importance degree of the given arguments and their ordered position, and apply them to group decision making.

The remainder of this paper is arranged as follows. In Section 2, the relationship between the HFLTS and uncertain linguistic variable is discussed. Section 3 develops some aggregation operators for hesitant fuzzy linguistic information. Section 4 presents a method for group decision making based on the developed operators. Section 5 illustrates the proposed method with a numerical example and compares the proposed method with other methods. Section 6 ends the paper with some concluding remarks.

2 Uncertain Linguistic Variable and HFLTS

Suppose S={s₁, s₂, L, s_t} is a linguistic term set by means of a finite and totally ordered discrete structure, where s_i represents a possible value for a linguistic variable [9, 12]. For example, a set of nine terms S could be given as follows:

S = {s1 = extremely poor, s2 = very poor, s3 = poor, s4 = slightly poor, s5 = fair, s6 = slightly good, s7 = good, s8 = very good, s9 = extremely good}.

It is usually required that there exist the following:

1. The set is ordered: 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. A maximization operator: Max(s_i, s_j)=s_i, if s_i≥s_j;

4. A minimization operator: Min(s_i, s_j)=s_i, if s_i≤s_j.

The decision-making problems with linguistic information have received attention from researchers, and many approaches have been adopted to deal with multiple-attribute decision making with linguistic information [6, 11, 14, 15, 16, 20, 24, 29, 56]. In this paper, we follow the ideas of Xu [39, 40]. Then, to preserve all the given information, the discrete term set S should be extended to a continuous term set S̅={s_α | s₀<s_α≤s_q, α∈(0, q]}, where q is a sufficiently large positive integer. If s_α∈S, we call s_α the original term; otherwise, we call s_α the virtual term. The decision maker, in general, uses the original linguistic terms to evaluate alternatives, whereas the virtual linguistic terms can only appear in operations. According to Rodríguez and Martínez [22], the virtual linguistic model has no fuzzy representation in that no semantics and proper linguistic syntax are provided for the virtual linguistic term. However, it can avoid loss of information and has been widely used to deal with multiple-attribute decision-making problems [21, 32, 39, 40, 47].

In different situations, the experts involved in complicated decision-making problems may hesitate among several values to assess an alternative. Hesitant fuzzy sets [25, 26] can be used to model the quantitative settings. However, similar situations could be found in qualitative settings. For such cases, Rodríguez et al. [23] proposed another generalization of fuzzy linguistic term sets.

Definition 1: ([23]) Let S={s₁, s₂, L, s_t} be a linguistic term set by means of a finite and totally ordered discrete structure. A HFLTS, H_s, is an ordered finite subset of the consecutive linguistic terms of S.

Example 1: Let S be the set of a linguistic term set,

S = {s1 = extremely poor, s2 = very poor, s3 = poor, s4 = slightly poor, s5 = fair,s6 = slightly good, s7 = good, s8 = very good, s9 = extremely good}.

A different HFLTS for a variable ϑ might be

HS(ϑ) = {s2, s3, s4}, HS(ϑ) = {s4, s5, s6, s7}.

The HFLTS can be directly used by the decision makers to evaluate an alternative or a linguistic variable. To increase the flexibility of hesitant fuzzy linguistic expressions and make them more similar to the ways in which human beings think and reason. Rodríguez et al. [23] proposed the comparative linguistic terms represented by HFLTSs and defined the following context-free grammar.

Definition 2: ([23]). Let G_H be a context-free grammar and S={s₁, s₂, L, s_t} a linguistic term set. The elements of G_H=(V_N, V_T, I, P) are defined as follows:

VN = {〈primary term〉, 〈composite term〉, 〈unary relation〉, 〈binary relation〉, 〈conjunction〉},VT = {lower than, greater than, between, and, s1, s2, ⋯, st},I ∈ VN.

The production rules are defined in an extended Backus–Naur form, so that the brackets enclose optional elements and the symbol “|” indicates alternative elements [3]. For the context-free grammar G_H, the production rules are as follows:

P = {I :: = 〈primary term〉 | 〈composite term〉〈composite term〉 :: = 〈unary relation〉 〈primary term〉 | 〈binary relation〉 〈primary term〉 |〈conjunction〉 〈primary term〉〈primary term〉 :: = s1 | s2 | ⋯ | st〈unary relation〉 :: = lower than | greater than〈binary relation〉 :: = between〈conjunction〉 :: = and}.

Example 2: Let S be the set of a linguistic term set as in Example 1. Some linguistic expressions obtained by the context-free grammar G_H might be

II1 = poor, II2 = lower than slightly poor, II3 = greater than good, II4 = between fair and good.

The linguistic expressions provided by decision makers must be transformed into HFLTSs by a transformation function EGH to accomplish the aggregation processes [23]. Then, to obtain the HFLTSs from the comparative linguistic terms, a transformation function EGH is defined as follows.

Definition 3: ([23]). Let EGH be a function that transforms linguistic expression II, which are obtained by G_H, into HFLTS H_S, where S is the linguistic term set that is used by G_H:

Example 3: Use the linguistic expressions in Example 2, and the HFLTSs obtained by the transformation function can be

EGH(ΙΙ1) = {s3}, EGH(ΙΙ2) = {s1, s2, s3, s4}, EGH(ΙΙ3) = {s7, s8, s9}, EGH(ΙΙ4) = {s5, s6, s7}.

Given three HFLTSs represented by HS, HS1, and HS2, Rodríguez et al. [23] defined some operations on them, which can be described as

1. HSC = S − HS = {si | si ∈ S and si ∉ HS}.

2. HS1 ∪ HS2 = {si | si ∈ HS1 or si ∈ HS2}.

3. HS1 ∩ HS2 = {si | si ∈ HS1 and si ∈ HS2}.

4. HS+ = max(si) = sj, si ∈ HS and si ≤ sj ∀i.

5. HS− = min(si) = sj, si ∈ HS and si ≥ sj ∀i.

Rodríguez et al. [23] showed that the envelope of an HFLTS is a linguistic interval expressed in the following definition.

Definition 4: ([23]). The envelope of the HFLTS, env(H_S), is a linguistic interval whose limits are obtained by means of upper bound (max) and lower bound (min). Hence,

In fact, the linguistic interval [HS−, HS+] is an uncertain linguistic variable.

Wei et al. [35] presented the hesitancy index of an HFLTS to measure the decision maker’s hesitant degree and gave a novel score function for ranking two HFLTSs.

Definition 5: ([35]). Let S={s₁, s₂, L, s_t} be a linguistic term set and Hs = {sδl | l = 1, 2, ⋯ ∥Hs∥} be an HFLTS on S, where ‖H_s‖ is the number of linguistic terms in an HFLTS H_S. A score function F(H_S) of H_S is defined as follows,

where δ¯ = 1∥Hs∥∑l = 1∥Hs∥δl and var(t) = (1 − (t + 1)/2)2 + ⋯ + (t − (t + 1)/2)2t. For two HFLTSs, HS1 and HS2, if F(HS1) > F(HS2), then HS1 > HS2; if F(HS1) = F(HS2), then HS1 = HS2.

However, in some cases, the score function proposed by Wei et al. [35] does not work and it yields illogical results.

Example 4: Let

S = {s1 = extremely poor, s2 = very poor, s3 = poor, s4 = slightly poor, s5 = fair,s6 = slightly good, s7 = good, s8 = very good, s9 = extremely good}

and HS1 = {s2, s6}, HS2 = {s1, s5, s9} be two HFLTSs. Clearly, HS1 ≠ HS2.

By Definition 5, we obtain F(HS1) = 4 − 0.6 = 3.4, and F(HS2) = 5−1.6 = 3.4. Then, F(HS1) = F(HS2), which is contradictory.

Taking the hesitant degree of an HFLTS into account, Wei et al. [35] defined a novel score function for ranking HFLTSs. They utilized the normalized variance of the subscripts of the HFLTS to measure the hesitant degree, but neglect the number of linguistic terms in an HFLTS. For any HFLTS, the number of linguistic terms in an HFLTS also reflects the hesitant degree of decision makers when they determine the membership degree of an element to a given set. The larger the number, the more hesitant the decision makers. For example, if there is only one linguistic term in an HFLTS, it indicates that there is no hesitancy for decision makers to determine the membership degree. However, if the number of the linguistic terms in an HFLTS intends to be infinite, it implies that the decision makers are hesitant completely. In the following, we present a modified score function for ranking HFLTSs.

Definition 6: Let S={s₁, s₂, L, s_t} be a linguistic term set, and Hs = {sδl | l = 1, 2, ⋯ ∥Hs∥} be an HFLTS on S. A score function F_M(H_S) of H_S is defined as follows,

where δ¯ = 1∥Hs∥∑l = 1∥Hs∥δl, var(t) = (1 − (t + 1)/2)2 + ⋯ + (t − (t + 1)/2)2t. For two HFLTSs, HS1 and HS2, if FM(HS1) > FM(HS2), then HS1 > HS2; if FM(HS1) = FM(HS2), then HS1 = HS2.

For Example 4, by Definition 6, we obtain FM(HS1) = 3.45 and FM(HS2) = 3.87. Thus, HS2 > HS1, which fits our intuition.

In comparison with the method proposed by Wei et al. [35], the modified score function for ranking HFLTSs reflects the importance degree of both the number of the given assessments and the variance of the subscripts of the HFLTS. It has the character of higher distinguishability.

Based on the relationship between the HFLTSs and the uncertain linguistic variable, we define some new operations on the HFLTSs HS, HS1 and HS2:

1. λHS = {λsi | si ∈ HS} = ∪si ∈ HS{λsi} = ∪si ∈ HS{sλi}, λ > 0.

2. (HS)λ = {(si)λ | si ∈ HS} = ∪si ∈ HS{siλ} = ∪si ∈ HS{siλ}, λ > 0.

3. HS1 ⊕ HS2 = {si ⊕ sj | si ∈ HS1, sj ∈ HS2} = ∪si ∈ HS1, sj ∈ HS2{si + j}.

4. HS1 ⊗ HS2 = {si ⊗ sj | si ∈ HS1, sj ∈ HS2} = ∪si ∈ HS1, sj ∈ HS2{si ⋅ j}.

5. 1HS = {1si | si ∈ HS} = ∪si ∈ HS{1si} = ∪si ∈ HS{s1/i}.

Theorem 1: Let HS, HS1, and HS2 be three HFLTSs, then

1. env(λH_S) =λenv(H_S).

2. env(HSλ) = (env(HS))λ.

3. env(HS1 ⊕ HS2) = env(HS1) ⊕ env(HS2).

4. env(HS1 ⊗ HS2) = env(HS1) ⊗ env(HS2).

5. env(1HS) = 1env(HS).

Proof: For any three HFLTSs, HS, HS1, and HS2, we have

1. env(λHS) = env(∪si ∈ HS{λsi}) = [min(λsi), max(λsi)] = λ[min(si), max(si)], = λ[Hs−, Hs+] = λenv(HS).

2. env(HSλ) = env(∪si ∈ HS{siλ}) = [min(siλ), max(siλ)] = [min(si), max(si)]λ = [Hs−, Hs+]λ = (env(HS))λ.

3. env(HS1 ⊕ HS2) = env(∪si ∈ HS1, sj ∈ HS2{si + j}) = [min(si + j), max(si + j)], env(HS1) ⊕ env(HS2) = [min(si), max(si)] ⊕ [min(sj), max(sj)].= [min(si) ⊕ min(sj), max(si) ⊕ max(sj)] = [min(si ⊕ sj), max(si ⊕ sj)] = [min(si + j), max(si + j)].

4. env(HS1 ⊗ HS2) = env(∪si ∈ HS1, sj ∈ HS2{si ⋅ j}) = [min(si ⋅ j), max(si ⋅ j)] env(HS1) ⊗ env(HS2) = [min(si), max(si)] ⊗ [min(sj), max(sj)]= [min(si) ⊗ min(sj), max(si) ⊗ max(sj)] = [min(si ⊗ sj), max(si ⊗ sj)]= [min(si ⋅ j), max(si ⋅ j)].

5. env(1HS) = env(∪si ∈ HS{1si}) = [min(1si), max(1si)], 1env(HS) = 1[min(si), max(si)] = [1max(si), 1min(si)] = [min(1si), max(1si)],

which complete the proof of the theorem.□

3 Some Aggregation Operators for Hesitant Fuzzy Linguistic Information

In the following, we introduce some well-known operators.

1. The weighted arithmetic averaging (WAA) operator [8]:

   (5)WAAw(a1, a2, ⋯, an) = ∑j = 1nwjaj,

   where w=(w₁, w₂, L, w_n)^T is the weight vector of real numbers a₁, a₂, L, a_n, with wj ∈ [0, 1], ∑j = 1nwj = 1.

2. The weighted geometric averaging (WGA) operator [1]:

   (6)WGAw(a1, a2, ⋯, an) = ∑j = 1najwj,

   where w=(w₁, w₂, L, w_n)^T is the weight vector of positive real numbers a₁, a₂, L, a_n, with w_j∈[0, 1], ∑j = 1nwj = 1.

3. The weighted harmonic averaging (WHA) operator [4]:

   (7)WHAw(a1, a2, ⋯, an) = 1∑j = 1nwjaj,

   where w=(w₁, w₂, L, w_n)^T is the weight vector of positive real numbers a₁, a₂, L, a_n, with w_j∈[0, 1], ∑j = 1nwj = 1.

4. The ordered weighted averaging (OWA) operator [49]:

   (8)OWAω(a1, a2, ⋯, an) = ∑j = 1nωjbj,

where b_j is the jth largest of the real numbers a_i(i=1, 2, L, n) and ω=(ω₁, ω₂, L, ω_n)^T is the aggregation-associated vector such that ωj ∈ [0, 1], ∑j = 1nωj = 1.

All the above operators have only been used in the situation where the input arguments are the exact numerical values. However, in many situations, decision makers would provide their preferences by means of linguistic values rather than numerical ones because of people’s limited expertise related to the problem domain and so on. Especially, decision makers are thinking of several linguistic values at the same time as an expression of their knowledge. In the following, based on the operations defined on HFLTSs, we will extend the above operators to accommodate the situation where the input arguments take the form of HFLTSs.

Definition 7: Let HSj (j = 1, 2, ⋯, n) be a collection of HFLTSs. A hesitant fuzzy linguistic weighted arithmetic averaging (HFLWAA) operator is a mapping Hⁿ→H such that

where w=(w₁, w₂, L, w_n)^T is the weight vector of HS1, HS2, ⋯, HSn, with wj ∈ [0, 1], ∑j = 1nwj = 1. In the case, if w=(1/n, 1/n, L, 1/n)^T, then the HFLWAA operator reduces to the HFLAA operator:

Definition 8: Let HSj (j = 1, 2, ⋯, n) be a collection of HFLTSs. A hesitant fuzzy linguistic weighted geometric averaging (HFLWGA) operator is a mapping Hⁿ→H such that

where w=(w₁, w₂, L, w_n)^T is the weight vector of HS1, HS2, ⋯, HSn, with wj ∈ [0, 1], ∑j = 1nwj = 1. In the case, if w=(1/n, 1/n, L, 1/n)^T, the HFLWGA operator reduces to the HFLGA operator:

Definition 9: Let HSj (j = 1, 2, ⋯, n) be a collection of HFLTSs. A hesitant fuzzy linguistic weighted harmonic averaging (HFLWHA) operator is a mapping Hⁿ→H such that

where w=(w₁, w₂, L, w_n)^T is the weight vector of HS1, HS2, ⋯, HSn, with wj ∈ [0, 1], ∑j = 1nwj = 1. In the case, if w=(1/n, 1/n, L, 1/n)^T, the HFLWGA operator reduces to the HFLHA operator:

Example 5: Assume w=(0.3, 0.3, 0.4)^T, HS1 = {s1, s2}, HS2 = {s3}, and HS3 = {s3, s4, s5}, then

HFLWAAw(HS1, HS2, HS3) = ⊕j = 13(wjHSj) = ∪sα1 ∈ HS1, sα2 ∈ HS2, sα3 ∈ HSn{s∑j = 13wjαj}= {s0.3 × 1 + 0.3 × 3 + 0.4 × 3, s0.3 × 2 + 0.3 × 3 + 0.4 × 3, s0.3 × 1 + 0.3 × 3 + 0.4 × 4,   s0.3 × 2 + 0.3 × 3 + 0.4 × 4, s0.3 × 1 + 0.3 × 3 + 0.4 × 5, s0.3 × 2 + 0.3 × 3 + 0.4 × 5}= {s2.400, s2.700, s2.800, s3.100, s3.200, s3.500},

HFLWGAw(HS1, HS2, HS3) = ⊗j = 13(HSj)wj = ∪sα1 ∈ HS1, sα2 ∈ HS2, sα3 ∈ HS3{s∏j = 13αjwj}= {s10.3⋅30.3⋅30.4, s10.3⋅30.3⋅40.4, s10.3⋅30.3⋅50.4, s20.3⋅30.3⋅30.4, s20.3⋅30.3⋅40.4, s20.3⋅30.3⋅50.4}= {s2.158, s2.421, s2.647, s2.656, s2.980, s3.259}

HFLWHAw(HS1, HS2, HS3) = 1⊕j = 13wjHSj = ∪sα1 ∈ HS1, sα2 ∈ HS2, sα3 ∈ HS3{s1∑j = 13wjαj}= {s10.31 + 0.33 + 0.43, s10.31 + 0.33 + 0.44, s10.31 + 0.33 + 0.45, s10.32 + 0.33 + 0.43, s10.32 + 0.33 + 0.44, s10.32 + 0.33 + 0.45}= {s1.875, s2.000, s2.083, s2.609, s2.857, s3.030}.

Lemma 1: Let x_j>0, λ_j>0 and ∑j = 1nλj = 1, then

with equality if and only if x₁=x₂=L=x_n.□

Proof: Xu [38] has proved that ∏j = 1nxjλj ≤ ∑j = 1nλjxj. If we replace x_j with 1xj, then

1∑j = 1nλjxj ≤ ∏j = 1nxjλj,

with equality if and only if x₁=x₂=L=x_n.□

Theorem 2: Let HSj (j = 1, 2, ⋯, n) be a collection of HFLTSs, w=(w₁, w₂, L, w_n)^T is the weight vector of HS1, HS2, ⋯, HSn, with wj ∈ [0, 1], ∑j = 1nwj = 1, then

Proof: By Lemma 1, for any sαj ∈ HSj (j = 1, 2, ⋯, n), we have

1∑j = 1nwjαj ≤ ∏j = 1nαjwj ≤ ∑j = 1nwjαj.

And then

s1∑j = 1nwjαj ≤ s∏j = 1nαjwj ≤ s∑j = 1nwjαj.

This implies that 1⊕j = 1nwjHSj ≤ ⊗j = 1n(HSj)wj ≤ ⊕j = 1n(wjHSj).□

In the decision-making process, the decision makers can select an appropriate operator to solve the multiple-attribute decision-making problems according to the decision maker’s risk preference. If the decision maker’s risk preference is risk-seeking, he or she can choose the HFLWAA operator to handle the multiple-attribute decision-making problems. If the decision maker’s risk preference is risk-averse, he or she can select the HFLWHA operator. If the decision maker’s risk preference is risk-neutral, he or she can select the HFLWGA operator.

The prominent feature of the OWA operator [49] is the reordering step in which the arguments are ordered by their values. Motivated by the OWA operator, we develop some operators for hesitant fuzzy linguistic information, such as the HFLOWAA operator, the HFLOWGA operator and the HFLOWHA operator.

Definition 10: Let HSj (j = 1, 2, ⋯, n) be a collection of HFLTSs, HSσ(j) be the jth largest of them, and the aggregation-associated vector is ω=(ω₁, ω₂, L, ω_n)^T such that ωj ∈ [0, 1], ∑j = 1nωj = 1, then

1. A hesitant fuzzy linguistic ordered weighted arithmetic averaging (HFLOWAA) operator of dimension n is a mapping:

   HFLOWAA: Hⁿ→H, where

   (17)HFLOWAAω(HS1, HS2, ⋯, HSn) = ⊕j = 1n(ωjHSσ(j)) = ∪sασ(1)∈HSσ(1), sασ(2)∈HSσ(2), ⋯ sασ(n)∈HSσ(n){s∑j = 1nωjασ(j)}.

2. A hesitant fuzzy linguistic ordered weighted geometric averaging (HFLOWGA) operator of dimension n is a mapping:

   HFLOWGA: Hⁿ→H, where

   (18)HFLOWGAω(HS1, HS2, ⋯, HSn) = ⊗j = 1n(HSσ(j))ωj = ∪sασ(1) ∈ HSσ(1), sασ(2) ∈ HSσ(2), ⋯ sασ(n) ∈ HSσ(n){s∏j = 1nασ(j)ωj}.

3. A hesitant fuzzy linguistic ordered weighted harmonic averaging (HFLOWHA) operator of dimension n is a mapping:

   HFLOWHA: Hⁿ→H, where

   (19)HFLOWHAω(HS1, HS2, ⋯, HSn) = 1⊕j = 1nωjHSσ(j) = ∪sασ(1) ∈ HSσ(1), sασ(2) ∈ HSσ(2), ⋯ sασ(n) ∈ HSσ(n){s1∑j = 1nωjασ(j)}.

Especially, if ω=(1/n, 1/n, L, 1/n)^T, then the HFLOWAA operator is reduced to the HFLAA operator, the HFLOWGA operator is reduced to the HFLGA operator, and the HFLOWHA operator is reduced to the HFLHA operator. The weighting vector ω=(ω₁, ω₂, L, ω_n)^T can be obtained using some weight-determining methods like linguistic quantifiers [49], the Gaussian distribution-based method [41], and so forth.

Example 6: Given the collection of HFLTSs HS1 = {s2, s3}, HS2 = {s3, s4, s5}, HS3 = {s4, s5}, and assume that the aggregation-associated vector is ω=(0.25, 0.4, 0.35)^T. To rank these arguments, we first compute the score values FM(HSj) ( j = 1, 2, 3) of HSj ( j = 1, 2, 3) by Definition 6,

FM(HS1) = 2.23, FM(HS2) = 3.62, FM(HS3) = 4.23.

Then, we rank the arguments HSj (j = 1, 2, 3) in descending order according to FM(HSj)(j = 1, 2, 3),