VIKOR method for effect evaluation of ancient village landscape planning based on the heritage historical context under 2-tuple linguistic enviroment

Abstract

In this paper, we proposed the 2-tuple linguistic VIKOR method based on the fundamental theories of 2-tuple linguistic information and origin VIKOR model. Firstly, we introduce the concepts, operation formulas and the distance calculating method of 2-tuple linguistic information. Then we review some aggregation operator of 2-tuple linguistic number, thereafter, the calculating steps of the VIKOR model for 2-tuple linguistic MCGDM problems are simply presented, in our proposed method; it’s more scientific and reasonable for considering the conflicting attributes. Moreover, a numerical example for effect evaluation of ancient village landscape planning based on the heritage historical context has been proposed to illustrate the new method and some comparisons are also conducted to further illustrate advantages of the new method.

Keywords

MCGDM problems 2-tuple linguistic sets (2TLSs)VIKOR model 2-tuple linguistic VIKOR method ancient village landscape planning

1. Introduction

The VIKOR (VIseKriterijumska Optimizacija I KOmpromisno Resenje) model, which firstly defined by Opricovic [1], is a useful tool to investigate the multiple criteria group decision-making (MCGDM) problems and has been broadly used to industrial, commercial economy, and science of management. In previous literature, some traditional MCGDM model had been investigated such as the ELECTRE model [2], the PROMETHEE model [3], the TOPSIS model [4, 5], the GRA model [6 –8], and the MULTIMOORA model [9, 10]. Comparing with these existed methods, the VIKOR model has the advantage of considering the conflicting criteria with respect to the objectivity of decision maker and the complexity of decision environment to get more scientific and reasonable evaluation results. In practical decision problems, it’s difficult to present the criteria values with real values for the complexity and fuzziness of the alternatives, so it can be more useful and effective to express the criteria values by fuzzy numbers. The fuzzy set theory which initially introduced by Zadeh [11] has been proved as a feasible mean in the application of MCGDM [12, 13]. The decision information in some practical MAGDM situations may be unquantifiable due to its nature, or cannot be precisely assessed in a quantitative form, but may be assessed in a qualitative one. Thus, it may take the form of linguistic variables [14, 15], such as “poor”, “fair”, and “very good”. To utilize linguistic variables, a pre-defined linguistic assessment set is needed. Unfortunately, the traditional linguistic assessment set is discrete. So in many cases, the decision information provided by DMs may not match any of the original linguistic phrases in the linguistic assessment sets, resulting in loss of information. To overcome these limitations, Herrera and Martinez [16] introduced the 2-tuple linguistic representation model of which the significant advantage is to be continuous in its domain. Therefore, it can express any counting of information in the universe of the discourse.

In light of the fact that information aggregation always plays an important role in decision-making processes, many 2-tuple aggregation operators have been proposed to aggregate information. The ordered weighted averaging (OWA) operator is one of the most common aggregation methods [17 –20]. It provides a parameterized family of aggregation operators that include as special cases the maximum, the minimum and the average [21]. Motivated by the idea of the OWA operator, Xu and Wang [22] developed the 2-tuple linguistic power ordered weighted averaging (2TLPOWA) operator, which can take all the decision arguments and their relationships into account. Jiang and Fan [23] proposed the 2-tuple ordered weighted geometric (TOWG) operator on the basis of the 2-tuple OWA operator. Li et al. [21] developed the 2-tuple linguistic induced generalized ordered weighted averaging distance (2LIGOWAD) operator. Zeng et al. [24] developed the 2-tuple linguistic generalized ordered weighted averaging distance (2LGOWAD) operator, which is an extension of the OWA operator that utilizes generalized means, distance measures and uncertain information represented as 2-tuple linguistic variables. Wang and Hao [25] introduced the quantifier-guided OWA aggregation operator and anchoring value-based OWA aggregation operator for 2-tuples. However, it needs to point out that these above operators only take into account the importance degrees of relative position and fail to consider the individual importance. On the other hand, some operators just consider individual significance, but neglect the importance of ordered position. For instance, Liu et al. [26] developed a dependent interval 2-tuple weighted averaging (DITWA) operator and a dependent interval 2-tuple weighted geometric (DITWG) operator.

Opricovic [1] used the VIKOR model to investigate some MCGDM problems with conflicting criteria [27, 28]. Bausys and Zavadskas [29] established INS VIKOR model. Liu and Park et al. [30] studied the VIKOR model under interval-valued intuitionistic fuzzy sets (IVIFSs). Selvakumari et al. [31] proposed the extended VIKOR model by constructing octagonal neutrosophic soft matrix. Wan et al. [32] proposed the VIKOR model with triangular intuitionistic fuzzy number (TIFN), Liu et al. [33] provided the linguistic VIKOR model, Qin et al. [34] developed interval type-2 fuzzy VIKOR model. Liu &Zhang [35] used VIKOR method to studied MADM problems based neutrosophic hesitant fuzzy environment. Liao et al. [36] explored VIKOR method with the hesitant fuzzy linguistic varies. Ren et al. [37] provided the dual hesitant fuzzy VIKOR model. Li et al. [38] provided the VIKOR model with linguistic intuitionistic fuzzy number, and Pouresmaeil et al. [39] established the SVNNs VIKOR model. Huang et al. [40] extended the VIKOR method to INN sets. Zhang and Wei [41] extended VIKOR method to hesitant fuzzy environment.

But, there has not yet been study about the VIKOR model for MCGDM problems with 2-tuple linguistic information. So, it’s of necessity to take 2-tuple linguistic VIKOR model into account. The goal of our article is to combine the origin VIKOR model with 2-tuple linguistic information to study MCGDM problems. The structure of our paper is organized as follows. Section 2 introduces the concepts, operation formulas, distance calculating method and some aggregation operators of 2-tuple linguistic information. Section 3 extends the origin VIKOR model to 2-tuple linguistic environment and introduce the calculating steps of 2-tuple linguistic VIKOR method. Section 5 provides a numerical example and introduces the comparison between our proposed methods with the existed method. Section 6 gives some summaries of our article.

2. Preliminaries

2.1. The linguistic 2-tuple representation model

Let S ={ s_i |i = 0, 1, ⋯ , t } be a linguistic term set with odd cardinality. Any label, s_i represents a possible value for a linguistic variable, and it should satisfy the following characteristics [38, 39]:

(1) The set is ordered: s_i > s_j, if i > j; (2) Max operator: max(s_i, s_j) = s_i, if s_i ⩾ s_j; (3) Min operator: min(s_i, s_j) = s_i, if s_i ⩽ s_j. For example, S can be defined as $\begin{matrix} S = {s_{0} = extremely poor (EP), s_{1} = very poor (VP), \\ s_{2} = poor (P), s_{3} = medium (M), s_{4} = good (G), \\ s_{5} = very good (VG), s_{6} = extremely good (EG)} \end{matrix}$

Herrera and Martinez [38, 39] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple (s_i, α_i), where s_i is a linguistic label from predefined linguistic term set S and α_i is the value of symbolic translation, and α_i ∈ [- 0.5, 0.5) .

Definition 1. Let β be the result of an aggregation of the indices of a set of labels assessed in a linguistic term set S, i.e., the result of a symbolic aggregation operation, β ∈ [0, t], being t the cardinality of S. Let i = round (β) and α = β - i be two values, such that, i ∈ [0, t] and α ∈ [- 0.5, 0.5) then α is called a symbolic translation [38, 39].

Definition 2. Let S ={ s₀, s₁, ⋯ , s_t } be a linguistic term set and β ∈ [0, t] is a number value representing the aggregation result of linguistic symbolic. Then the function Δ used to obtain the 2-tuple linguistic information equivalent to β is defined as: $Δ : [0, t] \to S \times [- 0.5, 0.5)$ (1) $Δ (β) = {\begin{matrix} s_{i}, i = round (β) \\ α = β - i, α \in [- 0.5, 0.5) \end{matrix}$ (2) where round(.) is the usual round operation, s_i has the closest index label to β and α is the value of the symbolic translation [38, 39].

Definition 3. Let S ={ s₀, s₁, ⋯ , s_t } be a linguistic term set and (s_i, α_i) be a 2-tuple. There is always a function Δ^-1 can be defined, such that, from a 2-tuple (s_i, α_i) it return its equivalent numerical value β ∈ [0, t] ⊂ R, which is [38, 39]. $Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, t]$ (3) $Δ^{- 1} (s_{i}, α) = i + α = β$ (4)

Definition 4. Let (s_k, a_k) and (s_l, a_l) be two 2-tuple, they should have the following properties [38, 39].

If k < l then (s_k, a_k) is smaller than (s_l, a_l)

If k = l then

if a_k = a_l£¬then (s_k, a_k), (s_l, a_l) represents the same information

if a_k < a_l then (s_k, a_k) is smaller than (s_l, a_l); if a_k > a_l then (s_k, a_k) is bigger than (s_l, a_l)

2.2. The normalized Hamming distance

Definition 5. Let δ₁ = (r₁, a₁) and δ₂ = (r₂, a₂) be two 2-tuple, then we can get the normalized Hamming distance: $d (δ_{1}, δ_{2}) = \frac{1}{t} | Δ^{- 1} (r_{1}, a_{1}) - Δ^{- 1} (r_{2}, a_{2}) |$ (5)

Theorem 1. Let δ₁ = (r₁, a₁) , δ₂ = (r₂, a₂) and δ₃ = (r₃, a₃) be three 2-tuple, the Hamming distance d (δ₁, δ₂) has the following properties: $\begin{matrix} (P 1) if d (δ_{1}, δ_{2}) = 0, then δ_{1} = δ_{2}; \\ (P 2) d (δ_{1}, δ_{2}) = d (δ_{2}, δ_{1}); \\ (P 3) d (δ_{1}, δ_{2}) + d (δ_{2}, δ_{3}) ⩾ d (δ_{1}, δ_{3}); \\ (P 4) 0 ⩽ d (δ_{1}, δ_{2}) ⩽ 1 . \end{matrix}$

2.3. The aggregation operators of 2-tuple linguistic information

Definition 6 [16 –23]. Let δ_j = (r_j, a_j) (j = 1, 2, …, n) be a group of 2-tuple, then the TWA and TWG operators can be defined as follows. $\begin{matrix} TWA {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} = \sum_{j = 1}^{n} ω_{j} δ_{j} \\ = (ω_{1} Δ^{- 1} (r_{1}, a_{1}) + ω_{2} Δ^{- 1} (r_{2}, a_{2}) + \dots + ω_{n} Δ^{- 1} (r_{n}, a_{n})) \end{matrix}$ (6) and $\begin{matrix} TWG {(r_{1}, a_{1}), (r_{2}, a_{2}), \dots, (r_{n}, a_{n})} = \prod_{j = 1}^{n} {(δ_{j})}^{ω_{j}} \\ = {(Δ^{- 1} (r_{1}, a_{1}))}^{ω_{1}} \times {(Δ^{- 1} (r_{2}, a_{2}))}^{ω_{2}} \times \dots \times {(Δ^{- 1} (r_{n}, a_{n}))}^{ω_{n}} \end{matrix}$ (7)

3. The 2-tuple linguistic VIKOR model

Assume that {η₁, η₂, … η_m } be a group of alternatives, {d₁, d₂, … d_λ } be a list of experts with weighting vector be {v₁, v₂, … v_t }, and {c₁, c₂, … c_n } be a list of criteria with weighting vector be{ω₁, ω₂, … ω_n }, thereby satisfying ω_i ∈ [0, 1] , v_i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1, \sum_{i = 1}^{t} v_{i} = 1$ . Construct the evaluation matrixes $η^{λ} = {[η_{ij}^{λ}]}_{m \times n}, i = 1, 2, \dots, m, j = 1, 2, \dots, n$ where $δ_{ij}^{λ} = (r_{ij}^{λ}, a_{ij}^{λ})$ means the estimate results of the alternative η_i (i = 1, 2, …, m) based on the criterion c_j (j = 1, 2, …, n) by expert d^λ.

Consider both the 2-tuple linguistic information and the traditional VIKOR model; we try to propose a 2-tuple linguistic VIKOR model to study MAGDM problems effectively. The model can be depicted as follows:

Step 1. Construct the decision matrixes $η^{λ} = {[η_{ij}^{λ}]}_{m \times n}$ , and utilize overall values of $η^{λ} = {[η_{ij}^{λ}]}_{m \times n}$ to η = [η_ij] _m×n by using equal (6) or (7);

Step 2. Compute the positive ideal solution (PIS) η⁺ and the negative ideal solution (NIS)η^-; $η^{+} = {(Δ^{- 1} (r_{j}, a_{j}))}^{+} (j = 1, 2, \dots, n)$ (8) $η^{-} = {(Δ^{- 1} (r_{j}, a_{j}))}^{+} (j = 1, 2, \dots, n)$ (9)

For benefit attribute $η^{+} = {(Δ^{- 1} (r_{j}, a_{j}))}^{+} = max_{i} (Δ^{- 1} (r_{j}, a_{j}))$ (10) $η^{-} = {(Δ^{- 1} (r_{j}, a_{j}))}^{-} = min_{i} (Δ^{- 1} (r_{j}, a_{j}))$ (11)

For cost attribute $η^{+} = {(Δ^{- 1} (r_{j}, a_{j}))}^{+} = min_{i} (Δ^{- 1} (r_{j}, a_{j}))$ (12) $η^{-} = {(Δ^{- 1} (r_{j}, a_{j}))}^{-} = max_{i} (Δ^{- 1} (r_{j}, a_{j}))$ (13)

Step 3. Based on the Equation (5) and the attribute weighting vector ω_j, we can calculate the values of τ_i and ψ_i which express the average and the worst group scores of η_i. $τ_{i} = \sum_{j = 1}^{n} ω_{j} \frac{d ({(Δ^{- 1} (r_{j}, a_{j}))}^{+}, (Δ^{- 1} (r_{j}, a_{j})))}{d ({(Δ^{- 1} (r_{j}, a_{j}))}^{+}, {(Δ^{- 1} (r_{j}, a_{j}))}^{-})}$ (14) $ψ_{i} = max_{j} {ω_{j} \frac{d ({(Δ^{- 1} (r_{j}, a_{j}))}^{+}, (Δ^{- 1} (r_{j}, a_{j})))}{d ({(Δ^{- 1} (r_{j}, a_{j}))}^{+}, {(Δ^{- 1} (r_{j}, a_{j}))}^{-})}}$ (15) where d is the normalized Hamming distance and 0 ⩽ ω_j ⩽ 1 means the weight of attributes which satisfies $\sum_{i = 1}^{n} ω_{i} = 1$ .

Step 4. Compute the values of Ω_i based on the results of τ_i and ψ_i, the calculating formula is characteristic as follows. $Ω_{i} = ρ \frac{(τ_{i} - τ^{+})}{(τ^{-} - τ^{+})} + (1 - ρ) \frac{(ψ_{i} - ψ^{+})}{(ψ^{-} - ψ^{+})}$ (16) where $τ^{+} = min_{i} τ_{i}, τ^{-} = max_{i} τ_{i}$ (17) $ψ^{+} = min_{i} ψ_{i}, ψ^{-} = max_{i} ψ_{i}$ (18) where ρ means the coefficient of decision making strategic. ρ > 0.5 depicts “the maximum group utility”, ρ = 0.5 depicts equality and ρ < 0.5 depicts the minimum regret.

Step 5. To choose the best alternative by rank the values of Ω_i, the alternative with minimum value is the best choice.

4. The numerical example

4.1. Numerical for 2-tuple linguistic MAGDM problems

In this chapter, we provide a numerical example to select best ancient village landscape planning projects by using 2-tuple linguistic VIKOR method. Assume that five possible construction projects η_i (i = 1, 2, 3, 4, 5) to be selected and four criteria to assess these construction projects: 1G₁ is the human factors in construction projects; 2G₂ is the building materials and equipment factors; 3G₃ is the management factors; 4G₄ is the environmental factors. The five possible ancient village landscape planning projects η_i (i = 1, 2, 3, 4, 5) are to be evaluated with 2-tuple linguistic infomation with the four criteria by three experts d^λ (criteria weight ω = (0.15, 0.25, 0.15, 0.45), experts weight v = (0.5, 0.3, 0.2) .), which are given in Tables 1–3.

Table 1
2-tuple linguistic information decision matrix by d¹

G ₁ G ₂ G ₃ G ₄

η ₁ (s₄, 0) (s₂, 0) (s₁, 0) (s₃, 0)

η ₂ (s₂, 0) (s₃, 0) (s₁, 0) (s₂, 0)

η ₃ (s₅, 0) (s₂, 0) (s₁, 0) (s₄, 0)

η ₄ (s₃, 0) (s₄, 0) (s₅, 0) (s₃, 0)

η ₅ (s₂, 0) (s₄, 0) (s₁, 0) (s₂, 0)

	G ₁	G ₂	G ₃	G ₄
η ₁	(s₄, 0)	(s₂, 0)	(s₁, 0)	(s₃, 0)
η ₂	(s₂, 0)	(s₃, 0)	(s₁, 0)	(s₂, 0)
η ₃	(s₅, 0)	(s₂, 0)	(s₁, 0)	(s₄, 0)
η ₄	(s₃, 0)	(s₄, 0)	(s₅, 0)	(s₃, 0)
η ₅	(s₂, 0)	(s₄, 0)	(s₁, 0)	(s₂, 0)

Table 2

2-tuple linguistic information decision matrix by d²

	G ₁	G ₂	G ₃	G ₄
η ₁	(s₂, 0)	(s₄, 0)	(s₂, 0)	(s₃, 0)
η ₂	(s₂, 0)	(s₁, 0)	(s₃, 0)	(s₄, 0)
η ₃	(s₃, 0)	(s₁, 0)	(s₄, 0)	(s₂, 0)
η ₄	(s₄, 0)	(s₃, 0)	(s₂, 0)	(s₅, 0)
η ₅	(s₂, 0)	(s₁, 0)	(s₂, 0)	(s₄, 0)

Table 3

2-tuple linguistic information decision matrix by d³

	G ₁	G ₂	G ₃	G ₄
η ₁	(s₁, 0)	(s₂, 0)	(s₃, 0)	(s₂, 0)
η ₂	(s₂, 0)	(s₄, 0)	(s₂, 0)	(s₁, 0)
η ₃	(s₃, 0)	(s₄, 0)	(s₁, 0)	(s₁, 0)
η ₄	(s₂, 0)	(s₅, 0)	(s₄, 0)	(s₄, 0)
η ₅	(s₃, 0)	(s₃, 0)	(s₂, 0)	(s₁, 0)

Step 1. Utilize overall values of $η^{λ} = {[η_{ij}^{λ}]}_{m \times n}$ to η = [η_ij] _m×n by using TWA operator, the aggregation results are listed in Table 4 as follows.

Table 4

The aggregation values by TWA operator

	G ₁	G ₂	G ₃	G ₄
η ₁	(s₃, - 0.2)	(s₃, - 0.4)	(s₂, - 0.3)	(s₃, - 0.2)
η ₂	(s₂, 0.0)	(s₃, - 0.4)	(s₂, - 0.2)	(s₂, 0.4)
η ₃	$\begin{matrix} (s_{4}, 0.0) \end{matrix}$	(s₂, 0.1)	(s₂, - 0.1)	(s₃, - 0.2)
η ₄	(s₃, 0.1)	(s₄, - 0.1)	(s₄, - 0.1)	(s₄, - 0.2)
η ₅	$\begin{matrix} (s_{2}, 0.2) \end{matrix}$	(s₃, - 0.1)	(s₂, - 0.5)	(s₂, 0.4)

Step 2. Compute the values of η⁺ (PIS) and η^-(NIS), for all attributes are benefit and based on the formula (12) and (13), we can obtain the (PIS) η⁺ and (NIS)η^- as follows. $η^{+} = {(s_{4}, 0.0), (s_{4}, - 0.1), (s_{4}, - 0.2), (s_{4}, - 0.2)}$ $η^{-} = {(s_{2}, 0.0), (s_{2}, 0.1), (s_{2}, - 0.5), (s_{2}, 0.4)}$

Step 3. Based on the Equation (5) and the attribute weighting vector ω_j, calculate the values of τ_i and ψ_i. $\begin{matrix} τ_{1} = 0.7295, τ_{2} = 0.9118, τ_{3} = 0.6964 \\ τ_{4} = 0.0675, τ_{5} = 0.8739 \end{matrix}$ $\begin{matrix} ψ_{1} = 0.3214, ψ_{2} = 0.4500, ψ_{3} = 0.3214 \\ ψ_{4} = 0.0675, ψ_{5} = 0.4500 . \end{matrix}$

Step 4. Compute the values of Ω_i based on the results of τ_i and ψ_i, the calculating values are listed as follows. (Let ρ = 0.4) $\begin{matrix} Ω_{1} = 0 . 7119, Ω_{2} = 1 . 0000, Ω_{3} = 0 . 6963 \\ Ω_{4} = 0 . 0000, Ω_{5} = 0 . 9820 . \end{matrix}$

Step 5. To choose the best alternative by rank the values of Ω_i, the ranking of Ω_i is Ω₄ > Ω₃ > Ω₁ > Ω₅ > Ω₂, and the best choice is η₄.

4.2. comparative analyses for 2-tuple linguistic MAGDM problems

In this section, we compare our proposed 2-tuple linguistic VIKOR model with the TWA and TWG operators [16]. Based on the values of Table 4 and attributes weighting vector ω = (0.15, 0.25, 0.45, 0.15) ^T, we can utilize overall η_ij toη_iby TWA and TWG operators.

Calculate results η_iby TWA operator: $\begin{matrix} η_{1} = (s_{3}, - 0.4150), η_{2} = (s_{2}, 0.3000) \\ η_{3} = (s_{3}, - 0.3300), η_{4} = (s_{4}, - 0.2650) \\ η_{5} = (s_{2}, 0.3600) \end{matrix}$

Calculate results η_i by TWG operator: $\begin{matrix} η_{1} = (s_{3}, - 0.4496), η_{2} = (s_{2}, 0.2818) \\ η_{3} = (s_{3}, - 0.4064), η_{4} = (s_{4}, - 0.2758) \\ η_{5} = (s_{2}, 0.3146) \end{matrix}$

The ranking of alternatives by TWA and TWG operators are listed in the Table 5.

Compare the values of our proposed 2-tuple linguistic VIKOR method with TWA and TWG operators, the results in ranking of alternatives and the best alternatives are same, however, the 2-tuple linguistic VIKOR method can consider the conflicting attributes and can be more reasonable and scientific in the application of MAGDM problems.

Table 5
Rank of Alternatives by TWA and TWG operators

order

TWA η₄ > η₃ > η₁ > η₅ > η₂

TWG η₄ > η₃ > η₁ > η₅ > η₂

2-tuple linguistic VIKOR η₄ > η₃ > η₁ > η₅ > η₂

	order
TWA	η₄ > η₃ > η₁ > η₅ > η₂
TWG	η₄ > η₃ > η₁ > η₅ > η₂
2-tuple linguistic VIKOR	η₄ > η₃ > η₁ > η₅ > η₂

5. Conclusion

In our article, we proposed the 2-tuple linguistic VIKOR method based on the fundamental theories of 2-tuple linguistic information and origin VIKOR model. Firstly, we introduce the concepts, operation formulas and the distance calculating method of 2-tuple linguistic information. Then we review some aggregation operator of 2-tuple linguistic number, thereafter, the calculating steps of the VIKOR model for 2-tuple linguistic MCGDM problems are simply presented, in our proposed method; it’s more scientific and reasonable for considering the conflicting attributes. Moreover, a numerical example for effect evaluation of ancient village landscape planning based on the heritage historical context has been proposed to illustrate the new method and some comparisons are also conducted to further illustrate advantages of the new method. In the future, our proposed 2-tuple linguistic VIKOR model can be applied to the risk analysis, the MCGDM problems and many other uncertain and fuzzy environments[42 –65].

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VIKOR method for effect evaluation of ancient village landscape planning based on the heritage historical context under 2-tuple linguistic enviroment

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1. The linguistic 2-tuple representation model

4.1. Numerical for 2-tuple linguistic MAGDM problems

Table 1 2-tuple linguistic information decision matrix by d1 G 1 G 2 G 3 G 4 η 1 (s4, 0) (s2, 0) (s1, 0) (s3, 0) η 2 (s2, 0) (s3, 0) (s1, 0) (s2, 0) η 3 (s5, 0) (s2, 0) (s1, 0) (s4, 0) η 4 (s3, 0) (s4, 0) (s5, 0) (s3, 0) η 5 (s2, 0) (s4, 0) (s1, 0) (s2, 0)

Table 5 Rank of Alternatives by TWA and TWG operators order TWA η4 > η3 > η1 > η5 > η2 TWG η4 > η3 > η1 > η5 > η2 2-tuple linguistic VIKOR η4 > η3 > η1 > η5 > η2

References

Table 5
Rank of Alternatives by TWA and TWG operators

order

TWA η₄ > η₃ > η₁ > η₅ > η₂

TWG η₄ > η₃ > η₁ > η₅ > η₂

2-tuple linguistic VIKOR η₄ > η₃ > η₁ > η₅ > η₂