Operations on Hesitant Linguistic terms sets Induced By Archimedean Triangular Norms And Conorms

Zhaoyan Li; Chenfang Zhao; Zheng Pei

doi:10.2991/ijcis.11.1.38

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Volume 11, Issue 1, 2018, Pages 514 - 524

Operations on Hesitant Linguistic terms sets Induced By Archimedean Triangular Norms And Conorms

Authors

Zhaoyan Li, Chenfang Zhao, Zheng Peipqyz@263.net

School of Computer and Software Engineering, Xihua University, Chengdu 610039, Sichuan, China.

Received 23 August 2017, Accepted 16 January 2018, Available Online 31 January 2018.

DOI: 10.2991/ijcis.11.1.38 How to use a DOI?
Keywords: Triangular norms and conorms; 2-tuple linguistic representation; Hesitant fuzzy linguistic terms set; Linguistic aggregation operator; Linguistic decision making
Abstract: The aim of the paper is to discuss some new operations on hesitant fuzzy linguistic terms sets based on Archimedean t-norm and t-conorm. The advantage is that the operations on hesitant fuzzy linguistic terms sets are closed, by studying propositions of the operations on hesitant fuzzy linguistic terms sets, scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators are proposed. An example is presented to illustrate the practicality of the four well-known scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators, which are also compared with the symbolic aggregation-based method in the example, results show that scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators can be applied to fuse hesitant fuzzy linguistic terms sets.
Copyright: © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Group multi-criteria decision making (GMCDM) is to select a satisfying alternative from a group of possible alternatives with respect to multi-criteria. Because various types of uncertainties are in decision making process and the huge amounts of decision information and alternatives are continuously growing⁵, GMCDM is more and more complexity and difficulties in big data. Up to now, many different decision making methods have been proposed to solve various decision making problems¹⁻⁵, in which, because fuzzy linguistic variables provide a more direct way to effectively represent qualitative information in decision making process, linguistic decision makings based on fuzzy linguistic approach have become an important kind of decision makings, intuitively, linguistic decision makings are closest to human being’s cognitive processes that occurs in real life and have attracted many scholars to propose linguistic decision making methods⁶⁻¹⁴.

In linguistic decision makings, two common methods to represent linguistic assessments⁸ are: 1) 2-tuple linguistic model⁷, which is composed by linguistic phrases and numerical values in [−0.5, 0.5), i.e., let S_p = {s₀, …, s_p} be a initial linguistic term set, for β ∈ [0, p], a 2-tuple linguistic value corresponding to β is Δ : [0, p] → S_p × [−0.5, 0.5), Δ(β) = s_β = (s_i, α), where, i = round(β) and α = β− i ∈ [−0.5, 0.5), round(·) is the usual round operation and the linguistic term s_i is mostly close to β. Conversely, Δ⁻¹: S_p × [−0.5, 0.5) → [0, p] transforms 2-tuple linguistic value (s_i, α) as β = i + α ∈ [0, p]. Denote all 2-tuple linguistic values on S_p as H(S_p) = {s_α|0 ⩽ α ⩽ p}, and for any s_{β_i} = (s_i, α_i) and s_{β_j} = (s_j, α_j), then s_{β_i} ⩽ s_{β_j} if and only if β_i ⩽ β_j. Based on 2-tuple linguistic model, many linguistic aggregation operators have been proposed to fuse 2-tuple linguistic assessments of decision makers, such as 2-tuple linguistic weighted or ordered weighted aggregation operators and the probabilistic linguistic terms aggregation operators ^6,5,15,46; 2) The context-free grammar method⁸, a context-free grammar including different kinds of terminal symbols can be used to generate linguistic term set, i.e., the primary terms such as {low, medium, high}, hedges such as {not, little, much, very}, the relations such as {lower than, between, higher than}, conjunctions such as {and, but} and disjunctions such as {or}, for example, “higher than medium” generates a linguistic term set {medium, high}. By considering decision makers hesitate among different linguistic terms, Rodriguez, et al⁸ proposed hesitant fuzzy linguistic term set (HFLTS) by utilizing context-free grammars to serves as the basis of increasing the flexibility of the elicitation of linguistic information, it provides us different linguistic expressions to represent decision makers’ knowledge/preferences in decision making. Formally, a HFLTS on a linguistic term set S = {s₀, s₁, ⋯, s_p} is described as: H_S is an ordered finite subset of the consecutive linguistic terms of S⁸. Intuitively, a HFLTS is also H_S = {s ∈ S|s_i ⩽ s ⩽ s_j} for some i, j ∈ {0, ⋯, p} with i ⩽ j, here the non-empty HFLTS H_S = {s ∈ S|s_i ⩽ s ⩽ s_j} is denoted by H_S = [s_i, s_j], in which, if i = j, then H_S = [s_i, s_j] is the singleton {s_i}²², all HFLTS on S is denoted by HS = {[s_i, s_j]|i, j ∈ {0, ⋯, p} and i ⩽ j}. Basic operations on HFLTS are as follows⁸: 1) Lower bound: H_S⁻ = min(s_i) = s_j, s_i ∈ H_S and s_i ⩾ s_j ∀i; 2) Upper bound: H_S⁺ = max(s_i) = s_j, s_i ∈ H_S and s_i ⩽ s_j∀i; 3) Complement: HSc=S−HS={si|si∈S and si⊈HS} ; 4) Union: HS1∪HS2={si|si∈HS1 or si∈HS2} ; 5) Intersection: HS1∩HS2={si|si∈HS1 and si∈HS2} ; 6) Envelope: env(H_S) = [H_S⁻, H_S⁺]. After then, many hesitant fuzzy linguistic terms aggregation operators have been proposed for hesitant fuzzy linguistic decision makings^{9,10,11,12,13,16−24}.

From the algebraic operational laws point of view, aggregation operators are mainly based on triangular norm and conorm (briefly t-norm and t-conorm for short)²⁵, i.e., binary operations [0, 1] × [0, 1] → [0, 1] are such that commutativity, associativity, monotonicity and boundary condition, which serve as a natural generalization of the classical conjunction or disjunction in many valued reasoning systems²⁶, due to their interesting algebraic and logical properties, various extended forms of t-norm and t-conorm and applications in fuzzy logics and many practical problems have been studied in ²⁷⁻³⁵. The aggregation operators derived from the t-norms and t-conorms show great advantages in fusing numerical information, such as aggregation operators on intuitionistic fuzzy set based on Archimedean t-norm and t-conorm³⁶, new aggregation operators derived from Hamacher family of t-norms³⁷, and a family of hesitant fuzzy Hamacher operators for fusing hesitant fuzzy sets ^21,44.

In this paper, we investigate the linguistic hesitant fuzzy aggregation operators derived from Archimedean t-norms and t-conorms. To do so, we firstly review Archimedean t-norms and s-norms. Then we introduce the linguistic hesitant fuzzy Archimedean t-norms and s-norms and discuss their properties. Finally, we propose hesitant fuzzy linguistic terms weighted mean and geometric mean operators to fuse hesitant fuzzy linguistic terms in linguistic decision making. The rest of this paper is structured as follows: In Section 2, basic concepts of Archimedean t-norms and t-conorms are reviewed briefly; In Section 3, some new operational laws for HFLTSs based on the four Archimedean t-norms and t-conorms are proposed and their properties are analyzed, then hesitant fuzzy linguistic terms weighted mean and geometric mean operators induced by the new operational laws for HFLTSs are provided; In Section 4, we present an example to illustrate the practicality of hesitant fuzzy linguistic terms weighted mean and geometric mean operators, and compare with the symbolic aggregation-based method; Section 5 concludes the paper.

2. Preliminaries

In this section, we briefly review basic concepts of Archimedean t-norms and t-conorms, and their applications in aggregation operators.

Formally, t-norm is a binary operation T: [0, 1] × [0, 1] → [0, 1] such that commutative, associative, monotone and has 1 as neutral element, i.e., for any x, y, z ∈ [0, 1], 1) T(x, y) = T(y, x); 2) T(x, T(y, z)) = T(T(x, y), z); 3) T(x, y) ⩽ T(x, z), if y ⩽ z; 4) T(x, 1) = x. t-conorm is a binary operation S: [0, 1] × [0, 1] → [0, 1] such that commutative, associative, monotone and has 0 as neutral element, i.e., for any x, y, z ∈ [0, 1], S satisfies 1)-3) and 4′) S (x, 0) = x. Dual property of t-norm and t-conorm is that for any t-norm T, it’s t-conorm is S(x, y) = 1 − T (1 − x, 1 − y) and vice-versa. A t-norm is strict Archimedean and continuous if and only if it is obtained from a continuous additive function φ: [0, 1] → [0, ∞) that is strictly decreasing with φ(1) = 0, i.e.,

T(x,y)=ϕ−1(ϕ(x)+ϕ(y)),

where φ⁻¹ is the inverse function of φ and φ⁻¹(x) = sup{z ∈ [0, 1] |φ(z) > x}. Similarly, t-conorm is

S(x,y)=ψ−1(ψ(x)+ψ(y)),

where ψ(x) = φ(1 − x). The four well-known Archimedean t-norms and t-conorms are shown in Table 1⁴³, many interesting and important results about Archimedean t-norms and s-norms have been studied in²⁵⁻⁴¹. Here, we focus on two important applications in aggregation operators based on Archimedean t-norms and t-conorms.

Types	Notations	Formulas	Functions
Algebra	T^A(x, y) S^A(x, y)	xy x + y − xy	φ(z) = −log(z) ψ(z) = −log(1 − z)
Einstein	T^E(x, y) S^E(x, y)	xy(1+(1−x)(1−y)) x+y1+xy	ϕ(z)=log(2−zz) ψ(z)=log(2−(1−z)1−z)
Hamacher	TγH(x,y)(γ>0) SγH(x,y)(γ>0)	xy(r+(1−r)(x+y−xy)) (x+y−xy−(1−γ)xy)1−(1−γ)xy	ϕ(z)=log(γ+(1−γ)zz) ψ(z)=log(1−(1−γ)z1−z)
Frank	TγF(x,y)(γ>1) SγF(x,y)(γ>1)	logγ(1+(γx−1)(γy−1)γ−1) 1−logγ(1+(γ1−x−1)(γ1−y−1)r−1)	ϕ(z)=log(γ−1γz−1) ψ(z)=log(γ−1γ1−z−1)

Table 1.

The four Archimedean t-norms and t-conorms.

One is aggregation operators on intuitionistic fuzzy sets⁴⁵ proposed by Xia in ⁴², in which, Xia, et al used Archimedean t-norms and t-conorms to define new operations on two intuitionistic fuzzy sets, i.e., let α_i = (μ_{α_i}, ν_{α_i})(i = 1, 2, 3) be three intuitionistic fuzzy sets, where for any x ∈ X, 0 ⩽ μ_{α_i} (x) + ν_{α_i} (x) ⩽ 1, then we have: 1) α₁ ⊕ α₁ = (S(μ_α₁, μ_α₂), T(ν_α₁, ν_α₂)) = (ψ⁻¹(ψ(μ_α₁) + ψ(μ_α₂)), φ⁻¹(φ(ν_α₁) + φ(ν_α₂))); 2) α₁ ⊗ α₂ = (T(μ_α₁, μ_α₂), S(ν_α₁, ν_α₂)) = (φ⁻¹(φ(μ_α₁) + φ(μ_α₂)), ψ⁻¹(ψ(ν_α₁) + ψ(ν_α₂))); 3) λ α₃ = (ψ⁻¹(λ ψ(ν_α₃)), φ⁻¹(λ φ(ν_α₃)))(λ > 0); 4) α3λ=(ϕ−1(λϕ(να3)) , ψ⁻¹(λψ(ν_α₃)))(λ > 0). When φ and ψ are selected as four functions in Table 1, we can obtain Algebra, Einstein, Hamacher and Frank operations between two intuitionistic fuzzy sets. As pointed out in⁴², these operations have many interesting properties and are a uniform expressions of many existed operations on intuitionistic fuzzy sets. Accordingly, Xia, et al further proposed two kinds of intuitionistic fuzzy aggregation operators, i.e., let α_i = (μ_{α_i}, ν_{α_i})(i = 1, …, n) be n intuitionistic fuzzy sets and w = (w₁, …, w_n) the weight vector of α_i(i = 1, …, n), we have

ATS−IFWA(α1,⋯,αn)=⊕i=1nwiαi=⊕i=1n(ψ−1(∑i=1nwiψ(μαi)),ϕ−1(∑i=1nwiϕ(ναi))),ATS−IFWG(α1,⋯,αn)=⊗i=1nαiwi=⊗i=1n(ϕ−1(∑i=1nwiϕ(μαi)),ψ−1(∑i=1nwiψ(ναi))).

Formally, there are many many interesting properties for aggregation operators ATS − IFWA and ATS − IFWG⁴².

The other is aggregation operators on 2-tuple linguistic information proposed by Tao in⁴³, in which, Tao, et al used Archimedean t-norms and t-conorms to define new operations on two 2-tuple linguistic representations, i.e., let S_p = {s₀, …, s_p} be a initial linguistic term set, for any s_α₁, s_α₂, s_α₃ ∈ H(S_p), we have: 1) Additive operation sα1⊕sα2=Δ(p×ψ−1(ψ(α1p)+ψ(α2p))) ; 2) Multiplication sα1⊗sα2=Δ(p×ϕ−1(ϕ(α1p)+ϕ(α2p))) ; 3) Scalar-multiplication λsα3=Δ(p×ψ−1(λψ(α3p)))(λ>0) ; 4) Power operation sα3λ=Δ(p×ϕ−1(λϕ(α3p)))(λ>0) . Similarly, when φ and ψ are selected as four functions in Table 1, we can obtain Algebra, Einstein, Hamacher and Frank operations between two 2-tuple linguistic representations. Tao, et al discussed many interesting properties of these operations on H(S_p) and proposed aggregation operators on 2-tuple linguistic information, i.e., let s_{α_i} ∈ H(S_p)(i = 1, …, n) be n 2-tuple linguistic values and w = (w₁, …, w_n) the weight vector of s_{α_i} (i = 1, …, n), then a successive 2-tuple linguistic weighted arithmetic mean (S2T LW AM) is

S2TLWAM(sα1,…,sαn)=⊕i=1n(wisαi)=Δ(p×ψ−1(∑i=1n(wiψ(αip)))).

A successive 2-tuple linguistic weighted geometric mean (S2T LGM) is

S2TLGM(sα1,…,sαn)=⊗i=1nsαiwi=Δ(p×ϕ−1(∑i=1n(wiϕ(α3p)))).

Formally, S2T LWAM and S2T LGM of 2-tuple linguistic values are extensions of many existed aggregation operators of 2-tuple linguistic values⁴³.

Inspired by Xia and Tao’s works, in the follows, we discuss new operations for HFLTSs via Archimedean t-norms and t-conorms, formally, compared HFLTSs with intuitionistic fuzzy sets, it can be noticed that the constraint condition is different, i.e., 0 ⩽ μ_{α_i} (x) + ν_{α_i} (x) ⩽ 1 is in intuitionistic fuzzy set, however, i ⩽ j is in HFLTS H_S = [s_i, s_j]. In addition, HFLTS H_S = [s_i, s_j] is a discrete and consecutive linguistic terms set on S_p, for intuitionistic fuzzy set α_i = (μ_{α_i}, ν_{α_i}), μ_{α_i} and ν_{α_i} are generally continuous membership and non-membership function on X. Intuitively, 2-tuple linguistic value s_α ∈ H(S_p) can be understood as a special case of HFLTS H_S = [s_i, s_j], i.e., the singleton {s_α} when i = j. These differences lead us to obtain new operations for HFLTSs, which can provide more choices for the decision makers in hesitant fuzzy linguistic environment.

3. Operations for HFLTSs induced by Archimedean t-norms and t-conorms

In this section, we induce new operations on HFLTSs according to Archimedean t-norms and t-conorms and discuss properties of new operations.

3.1. Operations for HFLTSs

According to continuous additive function φ: [0, 1] → [0, ∞) and ψ: [0, 1] → [0, ∞) (ψ(x) = φ(1−x)) of Archimedean t-norms and t-conorms, we have the following operations on HFLTSs.

Definition 1.

For any H₁ = [s_α₁, s_β₁], H₂ = [s_α₂, s_β₂] ∈ HS and a scalar λ > 0, operations based on Archimedean t-norms and t-conorms for HFLTSs are defined as:

1.
Additive operation: H1⊕H2=[Δ(p×ϕ−1(ϕ(α1p)+ϕ(α2p))),Δ(p×ψ−1(ψ(β1p)+ψ(β2p)))] ;
2.
Multiplication: H1⊗H2=[Δ(p×ϕ−1(ϕ(α1p)+ϕ(β1p)+ϕ(α2p)+ϕ(β2p))),Δ(p×ψ−1(ψ(ϕ−1(ϕ(β1p)+ϕ(α2p)))+ψ(ϕ−1(ϕ(α1p)+ϕ(β2p)))))] ;
3.
Scalar-multiplication: λ⊙H1=[Δ(p×ψ−1(λψ(α1p))),Δ(p×ψ−1(λψ(β1p)))] ;
4.
Power operation: H1λ=[Δ(p×ϕ−1(λϕ(α1p))),Δ(p×ϕ−1(λϕ(β1p)))] .

Compared Definition 1 with Xia and Tao’s works, operations on two 2-tuple linguistic representations are adopted in Definition 1. Because the constraint condition α_i ⩽ β_i of [s_{α_i}, s_{β_i}] must be satisfied, in additive operation on H₁ = [s_α₁, s_β₁] and H₂ = [s_α₂, s_β₂], we respectively use multiplication on s_α₁ and s_α₂ and additive operation on s_β₁ and s_β₂ to obtain H₁ ⊕ H₂, this is different to Xia’s additive operation on two intuitionistic fuzzy sets; In multiplication on H₁ = [s_α₁, s_β₁] and H₂ = [s_α₂, s_β₂], we first obtain multiplication on s_β₁ and s_α₂ and multiplication on s_α₁ and s_β₂, respectively, then we use multiplication and additive operation on them to obtain H₁ ⊗ H₂, this is different to Xia’s multiplication on two intuitionistic fuzzy sets; We use scalar-multiplication on s_α₁ and s_β₁ to obtain λ ⊙ H₁, and power operation on s_α₁ and s_β₁ to obtain H1λ , these are different to Xia’s scalar-multiplication and power operation on two intuitionistic fuzzy sets.

In Definition 1, if φ and ψ are selected as Algebra, Einstein, Hamacher and Frank t-norms and t-conorms, we have the following distinct representations of operations on HFLTSs, i.e., let H₁ = [s_i, s_j] and H₂ = [s_k, s_l] be two HFLTSs on S = {s₀, ···, s_p} and λ > 0 a scalar. Then

Case 1: Algebra t-norm and t-conorm based operations on HFLTSs are

1.
Algebra additive operation: H1⊕AH2=[Δ(p×TA(ip,kp)),Δ(p×SA(jp,lp))] ,
2.
Algebra Multiplication: H1⊗AH2=[Δ(p×TA(TA(jp,kp),TA(ip,lp))),Δ(p×SA(TA(jp,kp),TA(ip,lp)))] ,
3.
Algebra scalar-multiplication: λ⊙AH1=[Δ(p×(1−(1−ip)λ)),Δ(p×(1−(1−jp)λ))] ,
4.
Algebra power operation: H1λ=[Δ(p×(ip)λ),Δ(p×(jp)λ)] .

Case 2: Einstein t-norm and t-conorm based operations on HFLTSs are

1.
Einstein additive operation: H1⊕EH2=[Δ(p×TE(ip,kp)),Δ(p×SE(jp,lp))] ,
2.
Einstein Multiplication: H1⊗EH2=[Δ(p×TE(TE(jp,kp),TE(ip,lp))),Δ(p×SE(TE(jp,kp),TE(ip,lp)))] ,
3.
Einstein scalar-multiplication: λ⊙EH1=[Δ(p×(p+i)λ−(p−i)λ(p+i)λ+(p−i)λ),Δ(p×(p+j)λ−(p−j)λ(p+j)λ+(p−j)λ)] ,
4.
Einstein power operation: H1λ=[Δ(p×2iλ(2p−i)λ+iλ),Δ(p×2jλ(2p−j)λ+jλ)] .

Case 3: Hammer t-norm and t-conorm based operations on HFLTSs are

1.
Hammer additive operation: H1⊕HH2=[Δ(p×TH(ip,kp)),Δ(p×SH(jp,lp))] ,
2.
Hammer Multiplication: H1⊗HH2=[Δ(p×TH(TH(jp,kp),TH(ip,lp))),Δ(p×SH(TH(jp,kp),TH(ip,lp)))] ,
3.
Hammer scalar-multiplication: λ⊙HH1=[Δ(p×(p+(γ−1)i)λ−(p−i)λ(p+(γ−1)i)λ+(γ−1)(p−i)λ),Δ(p×(p+(γ−1)j)λ−(p−j)λ(p+(γ−1)j)λ+(γ−1)(p−j)λ)] ,
4.
Hammer power operation: H1λ=[Δ(p×γiλ(p+(γ−1)(p−i))λ+(γ−1)iλ),Δ(p×γjλ(p+(γ−1)(p−j))λ+(γ−1)jλ)] .

Case 4: Frank t-norm and t-conorm based operations on HFLTSs are

1.
Frank additive operation: H1⊕FH2=[Δ (p×TF(ip,kp)),Δ(p×SF(jp,lp))] ,
2.
Frank Multiplication: H1⊗FH2=[Δ(p×TF(TF(jp,kp),TF(ip,lp))),Δ(p×SF(TF(jp,kp),TF(ip,lp)))] ,
3.
Frank scalar-multiplication: λ⊙FH1=[Δ(p×(1−logγ(1+(γ1−ip−1)λ(γ−1)λ−1))),Δ(p×(1−logγ(1+(γ1−jp−1)−1(γ−1)λ−1)))] ,
4.
Frank power operation: H1λ=[Δ(p×logγ(1+(γ1−jp−1)λ(γ−1)λ−1)),Δ(p×logγ(1+(γ1−jp−1)λ(γ−1)λ−1))] .

Example 1.

Let a linguistic terms set be S = {nothing (s₀), low (s₁), medium(s₂), high (s₃), perfect (s₄)}, two HFLTSs H₁ = {s₁, s₂, s₃} = [s₁, s₃] and H₂ = {s₂, s₃, s₄} = [s₂, s₄]. Select φ(z) = − log(z), ψ(z) = −log(1 − z) and λ = 2, then

H1⊕H2=[Δ(4×ϕ−1(ϕ(14)+ϕ(24)),Δ(4×ψ−1(ψ(34)+ψ(44))]=[s0.5,s4]={s0.5,s1,s2,s3,s4},

H1⊗H2=[Δ(4×ϕ−1(ϕ(14)+ϕ(34)+ϕ(24)+ϕ(44)),Δ(4×ψ−1(ψ(ϕ−1(ϕ(34)+ϕ(24)))+ψ(ϕ−1(ϕ(14)+ϕ(44))))]=[s0.375,s2.125]={s0.375,s1,s2,s2.125},

λ⊙H1=[Δ(4×ψ−1(2×ψ(14))),Δ(4×ψ−1(2×ψ(34)))]=[s1.75,s3.75]={s1.75,s2,s3,s3.75},

H1λ=[Δ(4×ϕ−1(2×ϕ(14))),Δ(4×ϕ−1(2×ϕ(34)))]=[s0.25,s2.25]={s0.25,s1,s2,s2.25}.

3.2. Properties of operations on HFLTSs

In this subsection, we discuss several properties of operations on HFLTSs defined in Definition 1.

Proposition 1.

Let a linguistic term set S = {s₀, ⋯, s_p}. For any HFLTSs H₁ = [s_i, s_j] and H₂ = [s_k, s_l] in HS and λ > 0, H₁ ⊕ H₂, H₁ ⊗ H₂, λ ⊙ H₁ and H1λ are in HS.

Proof. According to t-norm and t-conorm, for any x, y ∈ [0, 1], we have T(x, y) ⩽ T(x, 1) = x and T(y, x) ⩽ T(y, 1) = y, i.e., T(x, y) ⩽ min{x, y}, and S(x, y) ⩾ S(x, 0) = x and S(y, x) ⩾ S(y, 0) = y, i.e., S(x, y) ⩾ max{x, y}, these meant that for any x, y ∈ [0, 1], t-norm and t-conorm, we have T(x, y) ⩽ min{x, y} ⩽ max{x, y} ⩽ S(x, y). Hence, for any Archimedean t-norms and t-conorms, we have φ⁻¹(φ(x) + φ(y)) ⩽ min{x, y} ⩽ max{x, y} ⩽ ψ⁻¹(ψ(x) + ψ(y)).

For any HFLTSs H₁ = [s_α₁, s_β₁], H₂ = [s_α₂, s_β₂] ∈ HS, due to α1p≤β1p and α2p≤β2p , we have ϕ−1(ϕ(α1p)+ϕ(α2p))≤ϕ−1(ϕ(β1p)+ϕ(β2p))≤ψ−1(ψ(β1p)+ψ(β2p)) , hence, H₁ ⊕ H₂ is such that Δ(p×ϕ−1(ϕ(α1p)+ϕ(α2p))≤Δ(p×ψ−1(ψ(β1p)+ψ(β2p)) , i.e., H₁ ⊕ H₂ is in HS.

In H₁ ⊗ H₂, due to ϕ−1( ϕ(α1p)+ϕ(β1p)+ϕ(α2p)+ϕ(β2p))=ϕ−1(ϕ(ϕ−1(ϕ(α1p)+ϕ(β1p)))+ϕ(ϕ−1(ϕ(α2p)+ϕ(β2p))))≤ψ−1(ψ(ϕ−1(ϕ(β1p)+ϕ(α2p)))+ψ(ϕ−1(α1p)+ϕ(β2p))) , hence, Δ(p×ϕ−1(ϕ(α1p)+ϕ(β1p)+ϕ(α2p)+ϕ(β2p))≤Δ(p×ψ−1(ψ(ϕ−1(ϕ(β1p)+ϕ(α2p)))+ψ(ϕ−1(ϕ(α1p)+ϕ(β2p)))) , i.e., H₁ ⊗ H₂ is in HS.

Due to for any λ > 0, ψ−1(λψ(α1p))≤ψ−1(λψ(β1p)) and φ−1(λφ(α1p))≤ϕ−1(λϕ(β1p)) , hence, Δ(p×ψ−1(λψ(α1p)))≤Δ(p×ψ−1(λψ(β1p))) and Δ(p×ϕ−1(λϕ(α1p)))≤Δ(p×ϕ−1(λϕ(β1p))) , i.e., λ ⊙ H₁ and H1λ are in HS.

The proposition means that Additive operation, Multiplication, scalar-multiplication and power operation on HFLTSs induced By Archimedean t-norms and t-conorms are closed.

Proposition 2.

For any HFLTSs H₁ = [s_α₁, s_β₁], H₂ = [s_α₂, s_β₂] and H₃ = [s_α₃, s_β₃] in HS, the following operational laws are held: 1) H₁ ⊕ H₂ = H₂ ⊕ H₁; 2) (H₁ ⊕ H₂) ⊕ H₃ = H₁ ⊕ (H₂ ⊕ H₃); 3) H₁ ⊗ H₂ = H₂ ⊗ H₁.

According to commutativity and associativity of functions φ and ψ, 1), 2) and 3) can be easily proved.

Example 2.

In Example 1, for HFLTSs H₁ = [s₁, s₃], H₂ = [s₂, s₄] and H₃ = [s₃, s₄], we can easily check that H₁ ⊕ H₂ = H₂ ⊕ H₁ and H₁ ⊗ H₂ = H₂ ⊗ H₁, in addition, (H₁ ⊕ H₂) ⊕ H₃ = [s_0.5, s₄] ⊕ [s₃, s₄] = [s_0.375, s₄] and H₁ ⊕ (H₂ ⊕ H₃) = [s₁, s₃] ⊕ [s_1.5, s₄] = [s_0.375, s₄], i.e., (H₁ ⊕ H₂) ⊕ H₃ = H₁ ⊕ (H₂ ⊕ H₃). However, (H₁ ⊗ H₂) ⊗ H₃=[s_0.375, s_2.125] ⊗ [s₃, s₄] = [s_0.149, s_1.82] and H₁ ⊗ (H₂ ⊗ H₃) = [s₁, s₃] ⊗ [s_1.5, s_3.5] = [s_0.246, s_3.754], i.e., (H₁ ⊗ H₂) ⊗ H₃ ≠ H₁ ⊗ (H₂ ⊗ H₃). This means that multiplication on HFLTSs is not associative.

Denote * ∈ {A, E, H, F} and λ > 0, according to Propositions 1 and 2, for any HFLTSs H₁ = [s_α₁, s_β₁], H₂ = [s_α₂, s_β₂] and H₃ = [s_α₃, s_β₃] in HS and λ > 0, we have the following results: 1) H₁ ⊕_* H₂, H₁ ⊗_* H₂, λ ⊙_* H₁ and H1λ are in HS; 2) H₁ ⊕_* H₂ = H₂ ⊕_* H₁, (H₁ ⊕_* H₂) ⊕_* H₃ = H₁ ⊕_* (H₂ ⊕_* H₃) and H₁ ⊗_* H₂ = H₂ ⊗_* H₁.

3.3. Aggregation operators on HFLTSs

Based on operations on HFLTSs induced by Archimedean t-norm and t-conorm, we can propose two kinds of hesitant fuzzy linguistic aggregation operators to fuse hesitant fuzzy linguistic terms assessments provided by decision makers in linguistic decision makings, formally, two kinds of hesitant fuzzy linguistic aggregation operators are described as follows:

1.
Scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator: For any HFLTSs H₁ = [s_α₁, s_β₁], ⋯, H_n = [s_{α_n}, s_{β _n}] on linguistic term set S = {s₀, ⋯, s_p} and ∑i=1nλi=1(∀λi∈[0,1]) , scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator is
⊕i=1n(λi⊙Hi)=(λ1⊙H1)⊕⋯⊕(λn⊙Hn).
2.
Power multiplication hesitant fuzzy linguistic terms aggregation operator: For any HFLTSs H₁ = [s_α₁, s_β₁], ⋯, H_n = [s_{α_n}, s_{β_n}] on linguistic term set S = {s₀, ⋯, s_p} and ∑i=1nλi=1(∀λi∈[0,1]) , Power multiplication hesitant fuzzy linguistic terms aggregation operator is
⊗i=1nH1λ1⊗⋯⊗Hnλn.

In real world practices, when ⊕ ∈ {⊕_A, ⊕_E, ⊕_H, ⊕_F}, ⊙ ∈ {⊙ _A, ⊙_E, ⊙_H, ⊙_F} and ⊗ ∈ {⊗_A, ⊗_E, ⊗_H, ⊗_F}, we can obtain four scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operators and four power multiplication hesitant fuzzy linguistic terms aggregation operator, which can provide more choices for the decision makers to fuse hesitant fuzzy linguistic assessments in linguistic decision makings.

4. Applications

In this section, we provide an example to show scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator used in hesitant fuzzy linguistic decision making problem, the example was in⁸ to carry out hesitant fuzzy linguistic decision making by using the symbolic aggregation-based method. Here, we use scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator to deal with the example and compare their results with the symbolic aggregation-based method.

Let X = {x₁, x₂, x₃} be a set of alternatives, C = {c₁, c₂, c₃} be a set of criteria defined for each alternative, and S = {s₀: nothing, s₁: very low, s₂: low, s₃: medium, s₄: high, s₅: very high, s₆: perfect} be the linguistic term set that is used by the context-free grammar GH to generate the linguistic expressions. The HFLTS assessments that are provided in such a problem are shown in Table 2. Due to weights of criteria are not used in the symbolic aggregation-based method⁸, here we select weights ( 13 , 13 , 13 ) of criteria to compare scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator with the symbolic aggregation-based method.

	H_ij	Criteria

		c₁	c₂	c₃
Alternatives	x₁	[s₁, s₃]	[s₄, s₅]	[s₄, s₄]
	x₂	[s₂, s₃]	[s₃, s₃]	[s₀, s₂]
	x₃	[s₄, s₆]	[s₁, s₂]	[s₄, s₆]

Table 2.

The HFLTS assessments that are provided for the decision problem.

Using scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator to carry out the example, we fix ⊕ = ⊕_A, ⊙ = ⊙_A and ⊗ = ⊗_A, then the assessment of each alternative is

EA(x1)=(13⊙A[s1,s3])⊕A(13⊙A[s4,s5])⊕A(13⊙A[s4,s4])≐[s0.033,s4.84],CA(x1)=[s1,s3]13⊕A[s4,s5]13⊕A[s4,s4]13≐[s1.4,s4.737].

Similarly, we can obtain E_A(x₂) ≐ [s₀, s_2.7], C_A(x₂) ≐ [s₀, s_3.47], E_A(x₃) ≐ [s_0.033, s₆] and C_A(x₃) ≐ [s_1.44, s_4.87]. By using the score function and the variance function for HFLTS²⁴, we can order assessments E_A(x₁), E_A(x₂) and E_A(x₃) (or C_A(x₁), C_A(x₂) and C_A(x₃)) to select the best one alternative, i.e., the number of linguistic terms in HFLTSs E_A(x₁), E_A(x₂) and E_A(x₃) (or C_A(x₁), C_A(x₂) and C_A(x₃)) are ♯E_A(x₁) = ♯{s_0.033, s₁, s₂, s₃, s₄, s_4.84} = 6, ♯E_A(x₂) = 4 and ♯E_A(x₃) = 7 (or ♯C_A(x₁) = 5, ♯C_A(x₂) = 5 and ♯C_A(x₃) = 5), then the score functions of E_A(x₁), E_A(x₂) and E_A(x₃) (or C_A(x₁), C_A(x₂) and C_A(x₃)) are

ρ(EA(x1))=s1♯EA(x1)∑sα∈EA(x1)α=s0.033+1+2+3+4+4.846≐s2.479,ρ(EA(x2))≐s1.425,ρ(EA(x3))≐s3.519.ρ(CA(x1))≐s3.027,ρ(CA(x2))≐s1.894,ρ(CA(x3))≐s3.062.

The variance functions of E_A(x₁), E_A(x₂) and E_A(x₃) (or C_A(x₁), C_A(x₂) and C_A(x₃)) are

σ(EA(x1))=s1♯EA(x1)∑sα,sβ∈EA(x1)(α−β)2≐s1.607,σ(EA(x2))≐s0.68,σ(EA(x3))≐s1.961.σ(CA(x1))≐s1.231,σ(CA(x2))≐s1.273,σ(CA(x3))≐s1.258.

Due to ρ(E_A(x₃)) > ρ(E_A(x₁)) > ρ(E_A(x₂)) (or ρ(C_A(x₃)) > ρ(C_A(x₁)) > ρ(C_A(x₂))), we obtain the ordering on alternatives, x₃ ⥼ x₁ ⥼ x₂ and the best alternative is x₃.

Similarly, we can calculate assessments of alternatives when ⊕ ∈ {⊕_E, ⊕_H, ⊕_F}, ⊙ ∈ {⊙_E, ⊙_H, ⊙_F} and ⊗ ∈ {⊗_E, ⊗_H, ⊗_F}, then compute the score functions and the variance functions for HFLTS assessments of alternatives and their ordering, accordingly, we can obtain the best one alternative, all these are shown in Table 3.

Types	Addition (E_*)	(ρ(), σ())	Ordering	Multiplication (C_*)	(ρ(), σ())	Ordering
Algebra	x₁ : [s_0.033, s_4.84]	(s_2.479, s_1.607)	x₃ ⥼ x₁ ⥼ x₂	x₁ : [s_1.4, s_4.737]	(s_3.027, s_1.231)	x₃ ⥼ x₁ ⥼ x₂
	x₂ : [s₀, s_2.7]	(s_1.425, s_1.021)		x₂ : [s₀, s_3.47]	(s_1.894, s_1.273)
	x₃ : [s_0.033, s₆]	(s_3.004, s_1.961)		x₃ : [s_1.44, s_4.87]	(s_3.062, s_1.258)

Einstein	x₁ : [s_0.008, s_4.15]	(s_2.360, s_1.583)	x₃ ⥼ x₁ ⥼ x₂	x₁ : [s_1.251, s_5.316]	(s_3.428, s_1.49)	x₃⥼ x₁ ⥼ x₂
	x₂ : [s₀, s_1.395]	(s_0.798, s_0.587)		x₂ : [s₀, s_2.3]	(s_1.325, s_0.904)
	x₃ : [s_0.008, s₆]	(s_3.001, s_1.967)		x₃ : [s_2.072, s_5.402]	(s_3.895, s_1.235)

Hamacher γ = 0.5	x₁ : [s_0.071, s_4.225]	(s_2.383, s_1.516)	x₃ ⥼ x₁⥼ x₂	x₁ : [s_1.481, s_4.622]	(s_3.021, s_0.982)	x₁ ⥼ x₃ ⥼ x₂
	x₂ : [s₀, s_2.71]	(s_1.425, s_1.023)		x₂ : [s₀, s_2.278]	(s_1.32, s_0.825)
	x₃: [s_0.071, s₆]	(s_3.01, s_1.965)		x₃: [s_1.439, s_4.462]	(s_2.980, s_1.146)

Frank γ = 2	x₁: [s_0.017, s_4.164]	(s_2.364, s_1.518)	x₃ ⥼ x₁ ⥼ x₂	x₁: [s_0.569, s_4.004]	(s_2.429, s_1.352)	x₂ ⥼ x₃ ⥼ x₁
	x₂: [s₀, s_2.692]	(s_1.423, s_1.018)		x₂: [s_2.557, s_5.6]	(s_4.031, s_1.152)
	x₃: [s_0.017, s₆]	(s_3.002, s_1.981)		x₃: [s_2.538, s_5.438]	(s_4.00, s_1.114)

Table 3.

The four scalar-multiplication addition and power multiplication hesitant fuzzy linguistic terms aggregation operators.

In⁸, Rodríguez used the symbolic aggregation-based method to carry out the example, more detail, the min-upper and max-lower operators are adopted to obtain the core information of each alternative, such as for alternative x₁, HFLTSs assessments of x₁ is H(x₁) = {{s₁, s₂, s₃}, {s₄, s₅}, {s₄}}, then the upper bound of H(x₁) is H⁺(x₁) = {max{s₁, s₂, s₃}, max{s₄, s₅}, max{s₄}} = {s₃, s₅, s₄}, the lower bound of H(x₁) is H− (x₁) = {min{s₁, s₂, s₃}, min{s₄, s₅}, min{s₄}} = {s₁, s₄}, Hmin+(x1)=min H+(x1)=min{s3,s5,s4}=s3 and Hmax−(x1)=maxH−1(x1)=max{s1,s4}=s4 , the core information of x₁ is the linguistic interval [min{Hmax−(x1),Hmin+(x1)},max{Hmax−(x1),Hmin+(x1)}]=[s3,s4] , others are shown in Table 4.

	H⁺, Hmin+	H⁻, Hmax−	linguistic interval
x₁	{s₃, s₄, s₅}, s₃	{s₁, s₄}, s₄	[s₃, s₄]
x₂	{s₃, s₂}, s₂	{s₂, s₃, s₀}, s₃	[s₂, s₃]
x₃	{s₆, s₂}, s₂	{s₄, s₁}, s₄	[s₂, s₄]

Table 4.

The core information of each alternative.

Based on the core information of each alternative in Table 4, the binary preference relation between alternatives is built⁴⁷, i.e., let linguistic intervals I₁ = [x_jL, x_jR] over interval I₂ = [x_j′L, x_j′R], the binary preference relation is p_jj′ = P(I₁ > I₂), where

P(I1>I2)=max{xjRxj′L,0}−max{xjL−xj′R,0}(xjR−xjL)+(xj′R−xj′L),

such as for alternatives x₁ and x₂, p12=P(a1>a2)=max{4−2,0}−max{3−3,0}(4−3)+(3−2)=1 , based on Table 4, the binary preference relation between three alternatives is the following P

P=[pjj′]3×3=(−10.6670−0.3330.3330.667−),

and nondominance degrees of three alternatives are NDD1=min{1−p21c,1−p31c}=min{1−max{p21−p12,0} , 1 − max{p₃₁ − p₁₃}} = min{1 − max{0 – 1, 0}, 1 − max{0.333 − 0.667, 0}} = 1, NDD₂ = min{1 − max{p₁₂ − p₂₁, 0}, 1 − max{p₃₂ − p₂₃}} = min{1 − max{1 − 0, 0}, 1 − max{0.667 − 0.333, 0}} = 0 and NDD₃ = min{1 − max{p₁₃ − p₃₁, 0}, 1 − max{p₂₃ − p₃₂}} = min{1 − max{0.667 − 0.333, 0}, 1 − max{0.333 − 0.667, 0}} = 0.666. Accordingly, the ordering of three alternatives is x₁ ⥼ x₃ ⥼ x₂ and the alternative x₁ is selected due to NDD₁ = max{NDD₁, NDD₂, NDD₃}.

Compared Table 3 with Table 4, we notice the following results:

1)
The symbolic aggregation-based method does not consider weights of criteria, the core information of each alternative is obtained by using the min-upper and max-lower operators. Linguistic intervals of three alternatives in Table 3 are obtained by using scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator, in which, weights of criteria are considered in these operators.
2)
Based on the core information of each alternative, the symbolic aggregation-based method adopts the binary preference relation between three alternatives to order alternatives. However, the score functions and the variance functions for HFLTS are adopted in Table 3 to order alternatives. In fact, if we use the score functions and the variance functions for the core information of each alternative in 4 to order alternatives, we obtain the same ordering x₁ ⥼ x₃ ⥼ x₂.
3)
From the Archimedean t-norm and t-conorm point of view, power multiplication hesitant fuzzy linguistic terms aggregation operators are more similar to the min-upper and max-lower operators than scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator, intuitively, power multiplication hesitant fuzzy linguistic terms aggregation operators are used to obtain common information (or the core information) of assessments of each alternative, in fact, Hamacher power multiplication hesitant fuzzy linguistic terms aggregation operators (γ = 0.5) and the symbolic aggregation-based method obtain the same ordering x₁ ⥼ x₃ ⥼ x₂. Because Archimedean t-norm is less than min and Archimedean t-conorm is more than max, scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operators are used to obtain general information of assessments of each alternative, which can be seen from widths of linguistic intervals shown in Table 3.

5. Conclusion

In this paper, we have studied a further application of Archimedean t-norm and t-conorm under hesitant fuzzy linguistic environment, and proposed scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator and power multiplication hesitant fuzzy linguistic terms aggregation operator, especially, Algebra, Einstein, Hamacher and Frank scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operators and power multiplication hesitant fuzzy linguistic terms aggregation operators are used in the example to fuse hesitant fuzzy linguistic terms sets. By comparing with the symbolic aggregation-based method in the example, we notice that power multiplication hesitant fuzzy linguistic terms aggregation operator can be used to obtain the core information of assessments of each alternative, scalar-multiplication addition hesitant fuzzy linguistic terms aggregation operator can be used to obtain general information of assessments of each alternative, which provide more choices to fuse hesitant fuzzy linguistic assessments in linguistic decision makings.

Acknowledgments

We appreciate Dr. Rodríguez and the reviewers for their valuable suggestions in improving this paper. This work is partially supported by National Nature Science Foundation of China (Grant No. 61372187) and the fund of lab of Security Insurance of Cyberspace, Sichuan Province (SZJJ2016-038) and Sichuan Educational Committee(17ZA0360).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 11 - 1
Pages: 514 - 524
Publication Date: 2018/01/31
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
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TY  - JOUR
AU  - Zhaoyan Li
AU  - Chenfang Zhao
AU  - Zheng Pei
PY  - 2018
DA  - 2018/01/31
TI  - Operations on Hesitant Linguistic terms sets Induced By Archimedean Triangular Norms And Conorms
JO  - International Journal of Computational Intelligence Systems
SP  - 514
EP  - 524
VL  - 11
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.11.1.38
DO  - 10.2991/ijcis.11.1.38
ID  - Li2018
ER  -

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