2-tuple linguistic intuitionistic preference relation and its application in sustainable location planning voting system

Abstract

Sustainability has increasingly become a major consideration in location planning, directing urban distribution, and regional development. This scenario can be regarded as a group decision making (GDM) voting system case. Taking into account artificial thinking and the characteristics of the case, this paper introduces the 2-tuple linguistic intuitionistic preference relation (LIPR) to resolve the GDM voting system circumstance. Firstly, the Archimedean t-norm and t-conorm based operational laws and the 2-tuple linguistic intuitionistic fuzzy weighted geometric (ATS-2TLIFWG) operator are developed to aggregate and rank linguistic intuitionistic fuzzy numbers (LIFNs). Secondly, various concepts related to the 2-tuple LIPR and geometric consistency are extended for further analysis, including the completely geometrically consistent 2-tuple LIPR, the acceptably geometrically consistent 2-tuple LIPR, and the consistency index. Thirdly, motivated by the adverse impact of contradictory individual comparisons, we investigate a consistency modification algorithm to approximate the acceptable geometric consistency. Fourthly, we consider a consensus reaching algorithm as a unanimous elimination technique in the GDM procedure. A GDM algorithm, involving a consistency modification process as well as a consensus reaching process, and overall value aggregation process, is embedded into the GDM voting system. Finally, we conduct an illustrative example concerning smart city location planning, and employ the proposed algorithms in a comparison analysis to illustrate their feasibility and applicability.

Keywords

Group decision making 2-tuple linguistic intuitionistic preference relation geometric consistency group consensus sustainable location planning voting system

1 Introduction

As socioeconomic decision making has become increasingly complicated, it has become more difficult for a single decision maker (DM) to synthesize all indispensable factors when attempting to select the most appropriate alternative. For this reason, group decision making (GDM) has been extensively applied in real-life contexts such as recommendations, logistics outsourcing provider selection, green product development, and voting system [1 –4]. Voting systems, initially advocated by Hare [5], were designed as a GDM tool to elect representatives among a set of candidates. Traditionally, the aim of a voting system was to vote for a single candidate by means of Borda, DEA, and game theory [6 –8]. Considering the advantages of the preference matrix in voting systems, an improved AHP method [4] was developed to get over the cycling problems and rank the alternatives in order of preference. In response to the broadly rising awareness of the sustainable development, voting systems have been progressively generalized into sustainable location planning [9]. As a better description technique of this situation, this paper explores preference relations in sustainable location planning applications.

Preference relation tends to be a popular implement with which to model a DM’s preference information related to identifying the most desirable alternative [10]. On the basis of the fact that DMs cannot precisely provide their preference comparisons with crisp numbers over a set of alternatives, a variety of preference relations have emerged, briefly involving the fuzzy preference relation [11, 12], interval fuzzy preference relation [13], intuitionistic fuzzy preference relation (IFPR) [14 –17], multiplicative preference relation [18], interval multiplicative preference relation [19], intuitionistic multiplicative preference relation [20], linguistic preference relation (LPR) [21, 22], and uncertain linguistic preference relation [23, 24].

An intuitionistic fuzzy number (IFN) can exclusively depict DMs’ evaluation values from “preferred”, “not preferred”, and “indeterminate” perspectives [14, 25]. Owing to their flexibility in managing fuzziness and hesitancy, IFNs have been broadly extended to multiple criteria decision making [26 –28]. However, on account of the complexity of the evaluation context, it is preferable to represent human thinking by qualitative information rather than quantitative information. Therefore, combining the advantages of linguistic term sets with IFNs, linguistic intuitionistic fuzzy numbers (LIFNs) were introduced for application in GDM problems [29]. Subsequently, some operational rules and aggregation operators related to LIFNs were investigated [29, 30]. LIFNs were further applied into evaluating coal mine safety [31]. When it comes to dealing with artificial languages, models for computing with words (CWW) can be mainly classified into three categories: the semantic model [32], the symbolic model [33], and the 2-tuple linguistic model [34]. The voting system example is such a situation in which “yes”, “no”, and “abstain” votes are of significance possible to be delivered in the linguistic term set rather than in crisp numbers. Supporting votes, dissenting votes, and abstention votes, which can be regarded as the membership, non-membership, and hesitancy degrees of the LIFNs, are decimal fractions obtained through statistically averaging by a group of experts in the actual phenomenon. Because the 2-tuple linguistic model permits a continuous linguistic semantic representation and effectively addresses the issue of information loss [34], this paper adopts the 2-tuple model to dispose of linguistic information, and extends the concept of the Archimedean t-norm and t-conorm to the operational laws and aggregation operators of 2-tuple LIFNs in order to distinguish their strengths in reflecting human thinking logically and ensuring boundaries [35].

Based on the theoretical and practical foundations described above, it is advisable to introduce LIFNs into preference relations and apply them into voting system. A great deal of research concerning preference relations in GDM problems has focused on the consistency modification and consensus reaching approaches [17, 25]. Consistency is aimed at avoiding contradictory assessments of alternatives for each individual and ensuring an approximately reliable ranking result [36]. To optimize the unacceptable consistent LPR, two addictive consistency based automatic iterative algorithms were developed to reach the acceptable level [22]. It is of vital importance to come into agreement in GDM, thus, group consensus has been heavily emphasized. For more proper description of GDM problems, addictive consistency was taken into account when establishing consistency and consensus models, which not only detect conflicting individual evaluations but also non-cooperative group behavior [37]. However, it was recently suggested that addictive consistency could distort initial preference information in the conversion process [38]. Another fundamental type of consistency, multiplicative consistency, can effectively overcome this limitation of addictive consistency [38]. The concepts of multiplicative consistency, perfect multiplicative consistency, and acceptable multiplicative consistency have been used in elevating the inconsistency level [39]. For the sake of satisfying the acceptable consistency threshold, two optimization methods, which guarantee the DMs behave logically by combining with multiplicative consistency, were constructed to improve the consistency level [40]. Multiplicative consistency based estimation approaches were applied to create a decision support system for producing a completely consistent preference relation [41]. Considering the necessity of consistency and consensus in GDM problems, these algorithms, which minimize the interactions among DMs as less as possible, were designed to improve their level [42]. However, it should be noted that multiplicative consistency remains lacking in robustness, and it highly depends on alternative labels in calculation [43]. Derived from multiplicative consistency, geometric consistency was introduced to address that deficiency [44]. Taking advantage of its theoretical basis, it has been employed to feasibly model interval fuzzy preference relations and IFPRs [44].

Even so, very little research has focused on geometric consistency modification and consensus achieving methods. Based on aforementioned analysis, combined the strengths of 2-tuple LIFNs with preference relations in managing voting system cases, it is apparently appropriate to define the novel 2-tuple linguistic intuitionistic preference relation (LIPR). Furthermore, a GDM algorithm is applied to a sustainable location planning voting system incorporating the individual consistency process, group consensus reaching process, and overall value aggregation process.

The remainder of the paper is structured as follows. Section 2 briefly reviews the 2-tuple linguistic model as well as the concept of IFPR and its corresponding consistent properties. In Section 3, novel Archimedean t-norm and t-conorm based operational laws and an ATS-2TLIFWG operator are put forward. Based on the new definitions of 2-tuple LIPR, completely geometrically consistent 2-tuple LIPR, acceptably geometrically consistent 2-tuple LIPR, and consistency index, we propose a consistency modification algorithm to repair individual inconsistency. Subsequently, in Section 4, a consensus reaching algorithm is taken into consideration. A GDM algorithm, involving the consistency modification process as well as the consensus reaching process, overall value aggregation process, is embedded to the voting system. In Section 5, an illustrative example and a comparison analysis in the context of smart city location planning are provided in order to demonstrate the feasibility and applicability of the proposed algorithms. Finally, conclusions are presented in Section 6.

2 Preliminaries

In this section, some basic concepts related to the 2-tuple LIPR are reviewed.

2.1 2-tuple linguistic model

As an approximation technique, a linguistic approach uses linguistic terms as elements and indicates qualitative information using linguistic variables. Assume S ={ s₀, s₁, …, s_g } is a linguistic term set with an even cardinality g + 1. The element s_i indicates ith linguistic term for a linguistic variable, and the positive integer g is the upper limit of the linguistic term set S. Moreover, for any s_i, s_j ∈ S, there exists the following characteristics [45, 46]:

s_i > s_j, if and only if i > j.

neg (s_i) = s_j, such that j = g - i.

Definition 1. [34] Let S ={ s₀, s₁, …, s_g } be a linguistic term set and β ∈ [0, g] be a value representing the result of a symbolic aggregation operation. Then the 2-tuple which expresses the equivalent information to β can be obtained with the following formulae $Δ : [0, g] \to S \times [- 0.5, 0.5), Δ (β) = (s_{i}, α),$ (1) in which α = β - i, the round (·) is the usual round operation, s_i has the closet index label to β and α is the value of the symbolic translation. Additionally, the function Δ^-1, which can convert a 2-tuple into its equivalent value β ∈ [0, g], is shown as follows:

$\begin{matrix} Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, g], \\ Δ^{- 1} (s_{i}, α) = i + α = β . \end{matrix}$ (2)

Definition 2. [47] Let a t-norm function T (x, y) be a strict Archimedean t-norm, if it is a continuous function generating from f : [0, 1] → [0, ∞], which is strictly decreasing with f (1) = 0. Then, the function can be shown as $T (x, y) = f^{- 1} (f (x) + f (y)) .$ (3)

Likewise, employing it to its dual t-conorm, we have $S (x, y) = h^{- 1} (h (x) + h (y))$ (4) with h (x) = f (1 - x).

Combined with Einstein operations, the formulae of the Einstein Archimedean t-norm and t-conorm can be offered on the basis of the functions $f (z) = log \frac{2 - z}{z}$ and $h (z) = log \frac{2 - (1 - z)}{1 - z}$ , respectively, which are listed as follows: $T (x, y) = \frac{xy}{1 + (1 - x) (1 - y)}, S (x, y) = \frac{x + y}{1 + xy} .$ (5)

2.2 Consistency properties and concept of IFPR

This subsection introduces some basic definitions and consistent properties with regard to the IFPR, which will be applied to furtheranalysis.

Definition 3. [15] Let R = (r_ij) _n×n ⊂ X × X be a fuzzy preference relation matrix, and letX ={ x₁, …, x_n } be a finite alternatives set. If for any i, j = 1, 2, …, n, R satisfies $μ_{ij} = υ_{ji}, μ_{ji} = υ_{ij}, μ_{ii} = υ_{ii}, 0 \leq μ_{ij} + υ_{ij} \leq 1,$ (6) then it is said to be an IFPR matrix, in which r_ij is an IFN composed of the certainty degree μ_ij and υ_ij, which represent the preferred and non-preferred degree of the alternative x_i over x_j,respectively.

Definition 4. [44] Let R = (r_ij) _n×n be an IFPR matrix. If for any i, j = 1, 2, …, n, it satisfies the condition $\sqrt{\frac{μ_{ij} (1 - υ_{ij})}{(1 - μ_{ij}) υ_{ij}}} = \sqrt{\frac{μ_{ik} (1 - υ_{ik})}{(1 - μ_{ik}) υ_{ik}}} \sqrt{\frac{μ_{kj} (1 - υ_{kj})}{(1 - μ_{kj}) υ_{kj}}},$ (7) then the IFPR R is called geometrically consistent.

3 Geometric consistency of 2-tuple LIPRs

This section introduces some operational laws and aggregation operators related to LIFNs. Then, in order to facilitate a consistency modification algorithm, we present the new concept of a 2-tuple LIPR, its corresponding geometric consistency, acceptable geometric consistency, and consistency index. Then, based on the acceptable geometric consistency of a LIPR, an algorithm is utilized to revise individual inconsistency.

3.1 Operational laws and aggregation operators of 2-tuple LIFNs

Based on the concept of the Archimedean t-norm and t-conorm, this subsection puts forward some operational laws and aggregation operators concerning LIFNs.

Definition 5. Let $\bar{S} = {(s_{q}, α_{q}) | s_{q} \in S, α_{q} \in [- 0.5, 0.5)}$ be a linguistic 2-tuples set derived from the linguistic term set S ={ s₀, s₁, …, s_g }. Assuming that $(s_{μ}, α_{μ}), (s_{υ}, α_{υ}) \in \bar{S}$ , if u + υ ≤ g, then r = [(s_μ, α_μ) , (s_υ, α_υ)] is said to be a 2-tuple LIFN on $\bar{S}$ , in which (s_μ, α_μ) and (s_υ, α_υ) denote the linguistic membership degree and non-membership degree, respectively. In addition, if $(s_{μ}, α_{μ}), (s_{υ}, α_{υ}) \in \bar{S}$ , r is said to be an original 2-tuple LIFN, otherwise, r is said to be a virtual 2-tuple LIFN.

Definition 6. Let r₁ = [(s_μ1, α_μ1) , (s_υ1, α_υ1)] and r₂ = [(s_μ2, α_μ2) , (s_υ2, α_υ2)] be two 2-tuple LIFNs. Then, the Einstein Archimedean t-norm and t-conorm based 2-tuple LIFNs operational laws are defined as follows: $\begin{matrix} (1) \begin{matrix} r_{1} + r_{2} = \\ (Δ (\frac{g^{2} (Δ^{- 1} (s_{μ 1}, α_{μ 1}) + Δ^{- 1} (s_{μ 2}, α_{μ 2}))}{g^{2} + Δ^{- 1} (s_{μ 1}, α_{μ 1}) Δ^{- 1} (s_{μ 2}, α_{μ 2})}), \\ Δ (\frac{g Δ^{- 1} (s_{υ 1}, α_{υ 1}) Δ^{- 1} (s_{υ 2}, α_{υ 2})}{g^{2} + (g - Δ^{- 1} (s_{υ 1}, α_{υ 1})) (g - Δ^{- 1} (s_{υ 2}, α_{υ 2}))})), \end{matrix} \\ (2) \begin{matrix} r_{1} r_{2} = \\ (Δ (\frac{g Δ^{- 1} (s_{μ 1}, α_{μ 1}) Δ^{- 1} (s_{μ 2}, α_{μ 2})}{g^{2} + (g - Δ^{- 1} (s_{μ 1}, α_{μ 1})) (g - Δ^{- 1} (s_{μ 2}, α_{μ 2}))}), \\ Δ (\frac{g^{2} (Δ^{- 1} (s_{υ 1}, α_{υ 1}) + Δ^{- 1} (s_{υ 2}, α_{υ 2}))}{g^{2} + Δ^{- 1} (s_{υ 1}, α_{υ 1}) Δ^{- 1} (s_{υ 2}, α_{υ 2})})), \end{matrix} \\ (3) \begin{matrix} θ r_{1} = \\ (Δ (g \frac{(g + Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ} - (g - Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ}}{(g + Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ} + (g - Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ}}), \\ Δ (\frac{2 g (Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ}}{(2 g - Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ} + (Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ}})), \end{matrix} \\ (4) \begin{matrix} r_{1}^{θ} = (Δ (\frac{2 g (Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ}}{(2 g - Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ} + (Δ^{- 1} (s_{μ 1}, α_{μ 1}))^{θ}}, \\ Δ (g \frac{(g + Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ} - (g - Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ}}{(g + Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ} + (g - Δ^{- 1} (s_{υ 1}, α_{υ 1}))^{θ}})) . \end{matrix} \end{matrix}$ in which θ > 0 and g is the upper limit of the linguistic term set.

Definition 7. Let {r_i = [(s_μi, α_μi) , (s_υi, α_υi)] , i = 1, 2, …, n} be a collection of 2-tuple LIFNs with the associated weight vector ω = (ω₁, ω₂, …, ω_n) satisfying 0 ≤ ω_i ≤ 1, $\sum_{i = 1}^{n} ω_{i} = 1$ . Then, the ATS-2TLIFWG operator is

$\begin{matrix} ATS - 2 TLIFWG (r_{1}, r_{2}, \dots, r_{n}) \\ = r_{1}^{ω_{1}} \cdot r_{2}^{ω_{2}} \cdot \dots \cdot r_{n}^{ω_{n}} . \end{matrix}$ (8)

Theorem 1. Let {r_i = [(s_μi, α_μi) , (s_υi, α_υi)] , i = 1, 2, …, n} be a collection of 2-tuple LIFNs on the linguistic 2-tuples set $\bar{S} = {(s_{q}, α_{q}) | s_{q} \in S, α_{q} \in [- 0.5, 0.5)}$ , where the associated weight vector ω = (ω₁, ω₂, …, ω_n) satisfies 0 ≤ ω_i ≤ 1, $\sum_{i = 1}^{n} ω_{i} = 1$ . Then, the aggregation value, incorporating the Archimedean t-norm and t-conorm based 2-tuple LIFNs operational laws, is also a 2-tuple LIFN by employing LIFWG operator. $\begin{matrix} ATS - 2 TLIFWG (r_{1}, r_{2}, \dots, r_{n}) \\ = (Δ (\frac{2 g \prod_{i = 1}^{n} (Δ^{- 1} (s_{μ i}, α_{μ i}))^{ω_{i}}}{\prod_{i = 1}^{n} (2 g - Δ^{- 1} (s_{μ i}, α_{μ i}))^{ω_{i}} + \prod_{i = 1}^{n} (Δ^{- 1} (s_{μ i}, α_{μ i}))^{ω_{i}}}), Δ (\frac{g (\prod_{i = 1}^{n} (1 + Δ^{- 1} (s_{υ i}, α_{υ i}))^{ω_{i}} - \prod_{i = 1}^{n} (1 - Δ^{- 1} (s_{υ i}, α_{υ i}))^{ω_{i}}))}{\prod_{i = 1}^{n} (1 + Δ^{- 1} (s_{υ i}, α_{υ i}))^{ω_{i}} + \prod_{i = 1}^{n} (1 - Δ^{- 1} (s_{υ i}, α_{υ i}))^{ω_{i}}})) . \end{matrix}$ (9)

3.2 Definition and properties of geometric consistent 2-tuple LIPR

This subsection introduces LIFNs in the context of preference relations to improve the depiction of

DMs’ preferences among alternatives. The geometric consistency and relatively geometric consistent properties of the 2-tuple LIPR are explored for further analysis in GDM problems.

Definition 8. Let R = (r_ij) _n×n be a preference relation matrix expressed by 2-tuple LIFNs r_ij = [(s_μij, α_μij) , (s_υij, α_υij)] on a linguistic 2-tuple term set $\bar{S} = {(s_{q}, α_{q}) | s_{q} \in S, α_{q} \in [- 0.5, 0.5)}$ with even cardinality. If for any i, j = 1, 2, …, n it satisfies $\begin{matrix} Δ^{- 1} (s_{μ ij}, α_{μ ij}) = Δ^{- 1} (s_{υ ji}, α_{υ ji}), \\ Δ^{- 1} (s_{μ ii}, α_{μ ii}) = Δ^{- 1} (s_{υ ii}, α_{υ ii}) = g / - 2, \\ 0 \leq Δ^{- 1} (s_{μ ij}, α_{μ ij}) + Δ^{- 1} (s_{υ ij}, α_{υ ij}) \leq g, \\ \frac{Δ^{- 1} (s_{μ ij}, α_{μ ij})}{g}, \frac{Δ^{- 1} (s_{υ ij}, α_{υ ij})}{g} \in [0, 1], \end{matrix}$ (10)

R is called a LIPR matrix. (s_μij, α_μij) and (s_υij, α_υij) are two 2-tuple linguistic variables, which represent the superior and inferior certainty degree of alternative x_i to x_j, respectively.

Definition 9. Let R = (r_ij) _n×n be a 2-tuple LIPR matrix expressed by r_ij = [(s_μij, α_μij) , (s_υij, α_υij)]. If for any i, j, k = 1, 2, …, n, it satisfies $\sqrt{\frac{Δ^{- 1} (s_{μ ij}, α_{μ ij}) (g - Δ^{- 1} (s_{μ ij}, α_{μ ij}))}{(g - Δ^{- 1} (s_{μ ij}, α_{μ ij})) Δ^{- 1} (s_{μ ij}, α_{μ ij})}} = \sqrt{\frac{Δ^{- 1} (s_{μ ik}, α_{μ ik}) (g - Δ^{- 1} (s_{μ ik}, α_{μ ik}))}{(g - Δ^{- 1} (s_{μ ik}, α_{μ ik})) Δ^{- 1} (s_{μ ik}, α_{μ ik})}} \sqrt{\frac{Δ^{- 1} (s_{μ kj}, α_{μ kj}) (g - Δ^{- 1} (s_{μ kj}, α_{μ kj}))}{(g - Δ^{- 1} (s_{μ kj}, α_{μ kj})) Δ^{- 1} (s_{μ kj}, α_{μ kj})}}, (11)$ (11)

then R is called a geometric consistent 2-tuple LIPR matrix.

Theorem 2. Suppose there exists an arbitrary 2- tuple LIPR matrix R = (r_ij) _n×n = [(s_μij, α_μij) , (s_υij, α_υij)], and it is geometrically consistent, when the following equations hold for any i, j, k = 1, 2, …, n: $Δ^{- 1} (s_{μ ij}, α_{μ ij}) = \frac{g Δ^{- 1} (s_{μ ik}, α_{μ ik}) Δ^{- 1} (s_{μ kj}, α_{μ kj})}{Δ^{- 1} (s_{μ ik}, α_{μ ik}) Δ^{- 1} (s_{μ kj}, α_{μ kj}) + (g - Δ^{- 1} (s_{μ ik}, α_{μ ik})) (g - Δ^{- 1} (s_{μ kj}, α_{μ kj}))},$ (12) $Δ^{- 1} (s_{υ ij}, α_{υ ij}) = \frac{g Δ^{- 1} (s_{υ ik}, α_{υ ik}) Δ^{- 1} (s_{υ kj}, α_{υ kj})}{Δ^{- 1} (s_{υ ik}, α_{υ ik}) Δ^{- 1} (s_{υ kj}, α_{υ kj}) + (g - Δ^{- 1} (s_{υ ik}, α_{υ ik})) (g - Δ^{- 1} (s_{υ kj}, α_{υ kj}))} .$ (13)

3.3 Acceptable geometric consistency and consistency modification algorithm of 2-tuple LIPR

This subsection focuses primarily on establishing a consistency modification algorithm to eliminate the effect of inconsistency on the final result. It should be noted that strict geometric consistency is too difficult to satisfy, therefore, acceptable geometric consistency is investigated. Before that, a distance formula and a consistency index are proposed.

Theorem 3. Let R = (r_ij) _n×n = [(s_μij, α_μij) , (s_υij, α_υij)] be a 2-tuple LIPR matrix, and let $\bar{R} = {({\bar{r}}_{ij})}_{n \times n} = [({\bar{s}}_{μ ij}, {\bar{α}}_{μ ij}), ({\bar{s}}_{υ ij}, {\bar{α}}_{υ ij})]$ be its corresponding geometrically consistent 2-tuple LIPR matrix. Then, R is completely geometrically consistent if and only if $R = \bar{R}$ .

Definition 10. Let R₁ = (r_ij,1) _n×n = [(s_μij,1, α_μij,1) , (s_υij,1, α_υij,1)] and R₂ = (r_ij,2) _n×n = [(s_μij,2, α_μij,2) , (s_υij,2, α_υij,2)] be two 2-tuple LIPRs, and the elements in those preference relations are on the same linguistic 2-tuples set $\bar{S} = {(s_{i}, α_{i}) | s_{i} \in S, α_{i} \in [- 0.5, 0.5)}$ . Then, the distance between two 2-tuple LIPRs is defined as

$\begin{matrix} d (R_{1}, R_{2}) \\ = \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (ln Δ^{- 1} (s_{μ ij, 1}, α_{μ ij, 1}) \\ - ln Δ^{- 1} (s_{υ ij, 1}, α_{υ ij, 1}) - ln Δ^{- 1} (s_{μ ij, 2}, α_{μ ij, 2}) \\ + {ln Δ^{- 1} (s_{υ ij, 2}, α_{υ ij, 2}))}^{2} \\ for any i, j, k = 1, 2, \dots, n . \end{matrix}$ (14)

Definition 11. Let R = (r_ij) _n×n be an arbitrary 2-tuple LIPR matrix expressed by R = [(s_μij, α_μij) , (s_υij, α_υij)], and let $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ be a consistent 2-tuple LIPR matrix characterized by $\bar{R} = [({\bar{s}}_{μ ij}, {\bar{α}}_{μ ij}), ({\bar{s}}_{υ ij}, {\bar{α}}_{υ ij})]$ . Then, the consistency index (CI) of the 2-tuple LIPR is defined as follows:

$\begin{matrix} CI (R) & = & 1 - d (R, \bar{R}) = 1 - \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} \\ (ln Δ^{- 1} (s_{μ ij}, α_{μ ij}) - ln Δ^{- 1} (s_{υ ij}, α_{υ ij}) \\ {- ln Δ^{- 1} ({\bar{s}}_{μ ij}, {\bar{α}}_{μ ij}) + ln Δ^{- 1} ({\bar{s}}_{υ ij}, {\bar{α}}_{υ ij}))}^{2} \\ for any i, j, k = 1, 2, \dots, n . \end{matrix}$ (15)

Remark 1. The consistency index CI (R) is adopted to measure the similarity between R and $\bar{R}$ . If and only if CI (R) =1, the 2-tuple LIPR can perfectly satisfy geometric consistency. We conclude that the higher the consistency index, the more consistent the 2-tuple LIPR.

Definition 12. Let R = (r_ij) _n×n be a 2-tuple LIPR matrix expressed by R = [(s_μij, α_μij) , (s_υij, α_υij)]. Then, R is called an acceptably geometrically consistent 2-tuple LIPR matrix, if the consistency index satisfies $CI (R) \geq \bar{CI}$ , where $\bar{CI}$ is the acceptable consistency threshold value.

Remark 2. It should be noted that the value of the consistency threshold is determined by practical circumstances and the DMs’ preferences. The existing literature provides no specific method to calculate it, and further study is required in the future. Generally, it can be set as $\bar{CI} = 0.9$ , as suggested by Ref. [38].

It is hardly realistic to expect an entirely geometrically consistent 2-tuple LIPR because of the complexity and uncertainty that exist during practical alternatives comparison. In this case, we can establish a consistency modification algorithm to ensure a reasonable ranking result.

Algorithm 1.

Step 1. Give a 2-tuple LIPR matrix R = (r_ij) _n×n.

Let an expert provide a 2-tuple LIPR matrix R = (r_ij) _n×n. $R^{(0)} = {(r_{ij}^{(0)})}_{n \times n}$ and $R^{(t)} = {(r_{ij}^{(t)})}_{n \times n}$ are the original and tth iterative 2-tuple LIPR matrices (i, j, l = 1, 2, …, n), respectively.

Step 2. Compute the completely geometrically consistent 2-tuple LIPR matrix ${\bar{R}}^{(t)} = {({\bar{r}}_{ij}^{(t)})}_{n \times n} = [({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}), ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})]$ .

By utilizing $\begin{matrix} Δ (Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)})) \\ = Δ (\sqrt[n]{\prod_{k = 1}^{n} \frac{g Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)}) Δ^{- 1} (s_{μ kj}^{(t)}, α_{μ kj}^{(t)})}{Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)}) Δ^{- 1} (s_{μ kj}^{(t)}, α_{μ kj}^{(t)}) + (g - Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)})) (g - Δ^{- 1} (s_{μ kj}^{(t)}, α_{μ kj}^{(t)}))}}), \end{matrix}$ (16) $\begin{matrix} Δ (Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})) \\ = Δ (\sqrt[n]{\prod_{k = 1}^{n} \frac{g Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)}) Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)})}{Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)}) Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)}) + (g - Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)})) (g - Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)}))}}) . \end{matrix}$ (17)

The given 2-tuple LIPR matrix $R^{(t)} = {(r_{ij}^{(t)})}_{n \times n}$ can be transformed into its corresponding completely consistent 2-tuple LIPR matrix $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ .

Step 3. Calculate the consistency index of the 2-tuple LIPR matrix R^(t).

Compute the consistency index of the 2-tuple LIPR matrix R^(t), referring to Equation (15). Determine whether or not the 2-tuple LIPR matrix R^(t) can satisfy the acceptable consistency. If the consistency index $CI (R^{(t)}) \geq \bar{CI}$ , then go directly to Step 5, otherwise, proceed to Step 4.

Step 4. Construct an improved consistent 2-tuple LIPR matrix $R^{(t + 1)} = {(r_{ij}^{(t + 1)})}_{n \times n} = [(s_{μ ij}^{(t + 1)}, α_{μ ij}^{(t + 1)}), (s_{υ ij}^{(t + 1)}, α_{υ ij}^{(t + 1)})]$ .

Build the improved consistent 2-tuple LIPR matrix R^(t+1) using $\begin{matrix} Δ (Δ^{- 1} (s_{μ ij}^{(t + 1)}, α_{μ ij}^{(t + 1)})) \\ = Δ (\frac{g Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)})^{λ}}{Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)})^{λ} + (g - Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)}))^{1 - λ} (g - Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}))^{λ}}), \end{matrix}$ (18) $\begin{matrix} Δ (Δ^{- 1} (s_{υ ij}^{(t + 1)}, α_{υ ij}^{(t + 1)})) \\ = Δ (\frac{g Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})^{λ}}{Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})^{λ} + (g - Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)}))^{1 - λ} (g - Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}))^{λ}}) . \end{matrix}$ (19)

Let t = t + 1 and return to Step 2.

Step 5. Output the improved 2-tuple LIPR matrix $R^{(t)} = {(r_{ij}^{(t)})}_{n \times n}$ , its consistency index CI (R^(t)) and the iteration number t.

Let $\overset{⌢}{R} = R^{(t)}$ and $CI (\overset{⌢}{R}) = CI (R^{(t)})$ .

Theorem 4. Let R be an unacceptable geometrically consistent 2-tuple LIPR matrix, and let $\bar{CI}$ be its consistency threshold. Assume that $R^{(t)} = {(r_{ij}^{(t)})}_{n \times n}$ is a 2-tuple LIPR matrix conducted by Algorithm 1, and that CI (R^(t)) is its consistency index. Then, for each t, CI (R^(t+1)) > CI (R^(t)).

Proof.

Because $\begin{matrix} Δ^{- 1} (s_{μ ij}^{(t + 1)}, α_{μ ij}^{(t + 1)}) \\ = \frac{g Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{μ kj}^{(t)}, {\bar{α}}_{μ kj}^{(t)})^{λ}}{Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)})^{1 - λ} Δ^{- 1} ({\bar{s}}_{μ kj}^{(t)}, {\bar{α}}_{μ kj}^{(t)})^{λ} + (g - Δ^{- 1} (s_{μ ik}^{(t)}, α_{μ ik}^{(t)}))^{1 - λ} (g - Δ^{- 1} ({\bar{s}}_{μ kj}^{(t)}, {\bar{α}}_{μ kj}^{(t)}))^{λ}}, \\ Δ^{- 1} (s_{υ ij}^{(t + 1)}, α_{υ ij}^{(t + 1)}) \\ = \frac{g Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)})^{1 - λ} Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)})^{λ}}{Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)})^{1 - λ} Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)})^{λ} + (g - Δ^{- 1} (s_{υ ik}^{(t)}, α_{υ ik}^{(t)}))^{1 - λ} (g - Δ^{- 1} (s_{υ kj}^{(t)}, α_{υ kj}^{(t)}))^{λ}}, \end{matrix}$ consider Theorem 3, it can be inferred that

$\begin{matrix} CI (R^{(t + 1)}) \\ = 1 - d (R^{(t + 1)}, {\bar{R}}^{(t + 1)}) \geq 1 - d (R^{(t + 1)}, {\bar{R}}^{(t)}) \\ = 1 - \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (ln Δ^{- 1} (s_{μ ij}^{(t + 1)}, α_{μ ij}^{(t + 1)}) - ln Δ^{- 1} (s_{υ ij}^{(t + 1)}, α_{υ ij}^{(t + 1)}) - ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) \\ + ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})) \\ = 1 - \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (ln (Δ^{- 1} {(s_{μ ij}^{(t)}, α_{μ ij}^{(t)})}^{1 - λ} Δ^{- 1} {({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)})}^{λ}) \\ - ln (Δ^{- 1} {(s_{υ ij}^{(t)}, α_{υ ij}^{(t)})}^{1 - λ} Δ^{- 1} {({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)})}^{λ}) {- ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) + ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}))}^{2} \\ = 1 - \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} ((1 - λ) ln Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)}) + λ ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) - (1 - λ) ln Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)}) \\ {- λ ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}) - ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) + ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}))}^{2} \\ = 1 - \frac{1}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} ((1 - λ) ln Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)}) - (1 - λ) ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) \\ {- (1 - λ) ln Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)}) + (1 - λ) ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}))}^{2} \\ = 1 - \frac{1 {(1 - λ)}^{2}}{2 n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (ln Δ^{- 1} (s_{μ ij}^{(t)}, α_{μ ij}^{(t)}) - ln Δ^{- 1} (s_{υ ij}^{(t)}, α_{υ ij}^{(t)}) {- ln Δ^{- 1} ({\bar{s}}_{μ ij}^{(t)}, {\bar{α}}_{μ ij}^{(t)}) + ln Δ^{- 1} ({\bar{s}}_{υ ij}^{(t)}, {\bar{α}}_{υ ij}^{(t)}))}^{2} \\ = 1 - {(1 - λ)}^{2} d (R^{(t)}, {\bar{R}}^{(t)}) \geq CI (R^{(t)}), \end{matrix}$

which completes the proof.

4 Consensus measure for GDM with 2-tuple LIPRs

In this subsection, we summarize a consensus reaching algorithm with a group consensus index.

Subsequently, we demonstrate the specific procedure of the GDM algorithm.

4.1 Group consensus index

Definition 13. Let R = (r_ij) _n×n be an arbitrary 2-tuple LIPR matrix, and let R_c = (r_ij,c) _n×n be a collective 2-tuple LIPR matrix characterized by R_c = [(s_μij,c, α_μij,c) , (s_υij,c, α_υij,c)]. The group consensus index (GCI) of the 2-tuple LIPR is defined as follows: $\begin{matrix} GCI (R) = 1 - d (R, R_{c}) \\ = 1 - \frac{2}{n (n - 1) g} \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} (ln Δ^{- 1} (s_{μ ij}, α_{μ ij}) - ln Δ^{- 1} (s_{υ ij}, α_{υ ij}) {- ln Δ^{- 1} (s_{μ ij, c}, α_{μ ij, c}) + ln Δ^{- 1} (s_{υ ij, c}, α_{υ ij, c}))}^{2} . \end{matrix}$ (20)

4.2 Consensus reaching algorithm

Inspired by the ideal of the automatic iteration, this subsection puts forward an algorithm to help reach a consensus goal.

Algorithm 2.

Step 1. Give individual 2-tuple LIPR matrices R = (r_ij,l) _n×n.

Let each expert provide his 2-tuple LIPR matrix R. Then, $R_{l}^{^{(0)}} = {(r_{i j, l}^{(0)})}_{n \times n}$ and $R_{l}^{^{(p)}} = {(r_{i j, l}^{(p)})}_{n \times n}$ are the original and pth iterative 2-tuple LIPR matrices (i, j, l = 1, 2, …, n), respectively.

Step 2. Compute the collective 2-tuple LIPR matrix $R_{c}^{(0)} = {(r_{ij, c}^{(0)})}_{n \times n}$ .

Utilize Equation (9) to calculate the collective 2-tuple LIPR matrix $R_{c}^{(0)}$ .

Step 3. Calculate the group consensus index of the 2-tuple LIPR matrix ${\overset{⌢}{R}}_{l}^{(p)} = {({\overset{⌢}{r}}_{ij, l}^{(p)})}_{n \times n}$ .

Conduct Algorithm 1 and obtain individual ${\overset{⌢}{R}}_{1}^{(p)}$ with acceptable consistency. Compute the group consensus index of the individual 2-tuple LIPR matrix ${\overset{⌢}{R}}_{1}^{(p)} = {({\overset{⌢}{r}}_{ij, l}^{(p)})}_{n \times n}$ according to Equation (20). Judge whether or not the individual 2-tuple LIPR can satisfy the acceptable consensus. If the consensus index $GCI ({\overset{⌢}{R}}_{l}^{(p)}) \geq \bar{GCI}$ , then go directly to Step 5, otherwise, proceed to Step 4.

Step 4. Construct a modified 2-tuple LIPR matrix ${\overset{⌢}{R}}_{l}^{(p + 1)} = {({\overset{⌢}{r}}_{i j, l}^{(p + 1)})}_{n \times n} = [({\overset{⌢}{s}}_{μ i j, l}^{(p + 1)}, {\overset{⌢}{α}}_{μ i j, l}^{(p + 1)}), ({\overset{⌢}{s}}_{υ i j, l}^{(p + 1)}, {\overset{⌢}{α}}_{υ i j, l}^{(p + 1)})]$ .

Repair the 2-tuple LIPR matrix ${\overset{⌢}{R}}_{l}^{(p + 1)}$ by $\begin{array}{l} Δ (Δ^{- 1} ({\overset{⌢}{s}}_{μ i j, l}^{(p + 1)}, {\overset{$ ⌢ $}{α}}_{μ i j, l}^{(p + 1)})) \\ = Δ (Δ^{- 1} {({\overset{⌢}{s}}_{μ i j, l}^{(p)}, {\overset{⌢}{α}}_{μ i j, l}^{(p)})}^{1 - λ} \\ Δ^{- 1} {({\overset{⌢}{s}}_{μ i j, c}^{(p)}, {\overset{⌢}{α}}_{μ i j, c}^{(p)})}^{λ}), \end{array}$ (21) $\begin{array}{l} Δ (Δ^{- 1} ({\overset{⌢}{s}}_{υ i j, l}^{(p + 1)}, {\overset{⌢}{α}}_{υ i j, l}^{(p + 1)})) \\ = Δ (Δ^{- 1} {({\overset{⌢}{s}}_{υ i j, l}^{(p)}, {\overset{⌢}{α}}_{υ i j, l}^{(p)})}^{1 - λ} \\ Δ^{- 1} {({\overset{⌢}{s}}_{υ i j, c}^{(p)}, {\overset{⌢}{α}}_{υ i j, c}^{(p)})}^{λ}) . \end{array}$ (22)

Let p = p + 1 and return to Step 3.

Step 5. Output a set of modified 2-tuple LIPR matrices ${\overset{⌢}{R}}_{l}^{(p)} = {({\overset{⌢}{r}}_{ij, l}^{(p)})}_{n \times n}$ with consensus, their group consensus index $GCI ({\overset{⌢}{R}}_{l}^{(p)})$ and iteration number p.

Theorem 5. Let R be an unacceptable consensus 2-tuple LIPR matrix, and let $\bar{GCI}$ be its consensus threshold. Suppose that $R^{(p)} = {(r_{ij}^{(p)})}_{n \times n}$ is a 2-tuple LIPR matrix conducted by Algorithm 2 and CI (R^(t)) is its consensus index. Then, for each p, there exists GCI (R^(p+1)) > GCI (R^(p)).

Proof. The proof process of the convergence is similar to Theorem 4.

4.3 GDM algorithm with acceptably geometric consistent 2-tuple LIPR

Based on the individual consistency and the group consensus 2-tuple LIPR, this subsection presents a GDM algorithm to select the best solution from a series of alternatives while considering the overall value aggregation process.

Algorithm 3.

Step 1. Obtain consensus modified 2-tuple LIPRs $({\overset{⌢}{R}}_{l}^{(p)} = {({\overset{⌢}{r}}_{ij, l}^{(p)})}_{n \times n}$ .

Employ Algorithm 2 to obtain a set of modified 2-tuple LIPRs ${\overset{⌢}{R}}_{l}^{(p)}$ with acceptable geometric consistency.

Step 2. Compute the collective 2-tuple LIPR matrix ${\overset{⌢}{R}}_{c}^{(p)} = {({\overset{⌢}{r}}_{ij, c}^{(p)})}_{n \times n} .$

Utilize the ATS-2TLIFWG operator defined in Equation (9) to calculate the collective 2-tuple LIPR matrix ${\overset{⌢}{R}}_{c}^{p}$ .

Step 3. Aggregate the average preference degree of each alternative in ${\overset{⌢}{R}}_{c}^{(p)}$ .

Utilize the ATS-2TLIFWG operator defined in Equation (9) to fuse ith row. The overall value aggregation stands for the average preference degree of ith alternative over other alternatives for all i = 1, 2, …, n.

Step 4. Generate the ranking value of $P ({\overset{⌢}{r}}_{i, c} \geq {\overset{⌢}{r}}_{j, c})$ .

Motivated by the ideal of the likelihood and the comparison rules presented in Ref. [48], we derive the ranking values using the likelihood between two 2-tuple LIFNs as follows:

$\begin{matrix} P ({\overset{⌢}{r}}_{i, c} \geq {\overset{⌢}{r}}_{j, c}) \\ = max (1 - max (\frac{g - Δ^{- 1} (s_{υ j, c}, α_{υ j, c}) - Δ^{- 1} (s_{μ i, c}, α_{μ i, c})}{L ({\overset{⌢}{r}}_{i, c}) + L ({\overset{⌢}{r}}_{j, c})}, 0), 0), \end{matrix}$ (23) in which $\begin{matrix} L ({\overset{⌢}{r}}_{i, c}) & = & g - Δ^{- 1} (s_{υ i, c}, α_{υ j, c}) \\ - Δ^{- 1} (s_{μ i, c}, α_{μ i, c}), \\ L ({\overset{⌢}{r}}_{j, c}) & = & g - Δ^{- 1} (s_{υ j, c}, α_{υ j, c}) \\ - Δ^{- 1} (s_{μ j, c}, α_{μ j, c}) . \end{matrix}$

Step 5. Rank all the alternatives and select the optimal one(s).

According to the ranking value of each alternative, determine the ranking result of all alternatives and select the optimal one(s).

5 Application to sustainable location planning voting system

To highlight the applicability and feasibility of the proposed method, this section employs it to a practical sustainable location planning voting system. It is further validated through a comparison analysis.

5.1 Illustrative example

Assume that government departments plan to set up a smart city in a certain region. Taking consideration of sustainable development, this task

is a long-term problem that requires a synthetic consideration of the location planning problem with respect to voting systems. In order to select the most sustainable location for the smart city, experts in the fields of urban planning (e₁), circumstances (e₂), and laws (e₃) are invited to vote on four potential cities A_i (i = 1, 2, …, 4), synthesizing environmental factors, land utilization factors, and urban planning factors in making their decisions. The associated weight vector for each filed expert is subjectively given as w = (0.5, 0.3, 0.2). Since an expert cannot give a precise total ordering of the cities, a group of experts of equal importance in each field devote themselves to providing evaluations using linguistic information to express their degree of support, objection or abstention between each pair of cities. With the purpose of dealing with the support, objection, and abstention degrees, LIFNs (the linguistic information is derived from a discrete term set S ={ s₀ = slightly, s₁ = fair, s₂ = very, s₃ = extremely, s₄ = absolutely }) are applied to indicate the votes data. Three collective preference relations can be generated from the weighted average statistical data of experts in each field. In most cases, the statistical results are decimal values that cannot be expressed in linguistic term, then, we convert the LIFNs in the synthetic preference relation into 2-tuple LIFNs. The 2-tuple LIPRs of the three domains are $\begin{matrix} R_{1} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{2}, 0.7), (s_{1}, - 0.5)] & [(s_{3}, - 0.3), (s_{1}, - 0.2)] & [(s_{1}, 0.1), (s_{3}, - 0.4)] \\ [(s_{1}, - 0.5), (s_{2}, 0.7)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{2}, - 0.4), (s_{2}, 0.4)] & [(s_{1}, 0.7), (s_{1}, - 0.8)] \\ [(s_{1}, - 0.2), (s_{3}, - 0.3)] & [(s_{2}, 0.4), (s_{2}, - 0.4)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{4}, - 0.4), (s_{0}, 0.1)] \\ [(s_{3}, - 0.4), (s_{1}, 0.1)] & [(s_{1}, - 0.8), (s_{1}, 0.7)] & [(s_{0}, 0.1), (s_{4}, - 0.4)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}], \\ R_{2} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.3), (s_{3}, 0.2)] & [(s_{3}, 0.1), (s_{0}, 0.3)] & [(s_{2}, 0.1), (s_{1}, 0.4)] \\ [(s_{0}, 0.3), (s_{3}, 0.1)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{0}, 0.1), (s_{2}, 0.7)] & [(s_{2}, 0.7), (s_{1}, - 0.4)] \\ [(s_{3}, 0.2), (s_{1}, - 0.3)] & [(s_{2}, 0.7), (s_{0}, 0.1)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{4}, - 0.4), (s_{0}, 0.1)] \\ [(s_{1}, 0.4), (s_{2}, 0.1)] & [(s_{1}, - 0.4), (s_{2}, 0.7)] & [(s_{0}, 0.1), (s_{4}, - 0.4)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}], \\ R_{3} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{0}, 0.7), (s_{3}, - 0.4)] & [(s_{1}, 0.9), (s_{0}, 0.8)] & [(s_{3}, 0), (s_{1}, 0)] \\ [(s_{3}, - 0.4), (s_{0}, 0.7)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{0}, 0.5), (s_{2}, 0.3)] & [(s_{2}, - 0.4), (s_{1}, 0.3)] \\ [(s_{0}, 0.8), (s_{1}, 0.9)] & [(s_{2}, 0.3), (s_{0}, 0.5)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{4}, - 0.4), (s_{0}, 0.1)] \\ [(s_{1}, 0), (s_{3}, 0)] & [(s_{1}, 0.3), (s_{2}, - 0.4)] & [(s_{0}, 0.1), (s_{4}, - 0.4)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}] . \end{matrix}$

Algorithm 1.

By Step 2, compute the completely geometrically consistent 2-tuple LIPR matrix. $\begin{matrix} {\bar{R}}_{1} & = & {({\bar{r}}_{ij, 1})}_{n \times n} \\ = & [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.15), (s_{1}, - 0.24)] & [(s_{1}, - 0.1), (s_{1}, 0.14)] & [(s_{2}, - 0.17), (s_{0}, 0.27)] \\ [(s_{1}, - 0.24), (s_{1}, 0.15)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.35), (s_{2}, - 0.22)] & [(s_{1}, 0.19), (s_{0}, 0.37)] \\ [(s_{1}, 0.14), (s_{1}, - 0.1)] & [(s_{2}, - 0.22), (s_{1}, - 0.35)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{2}, - 0.25), (s_{0}, 0.26)] \\ [(s_{0}, 0.27), (s_{2}, - 0.17)] & [(s_{0}, 0.37), (s_{1}, 0.19)] & [(s_{0}, 0.26), (s_{2}, - 0.25)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}] . \end{matrix}$

The consistency index of the 2-tuple LIPR matrix R⁽⁰⁾ is CI (R⁽⁰⁾) =0.68 < 0.9, and turn to the next step to improve the consistency.

Utilize Step 4, a 2-tuple LIPR matrix $R_{1}^{(1)} = {(r_{ij, 1}^{(1)})}_{n \times n}$ with consistency improved can be obtained. Let t = 1 and return to Step 2.

Repeat the above process. Because t = 2, the consistency index $CI (R^{(2)}) = 0.90 \leq \bar{CI}$ . Then, $CI ({\overset{⌢}{R}}_{1}) = CI (R_{1}^{(2)}) = 0.9$ and we can output the consistency improved 2-tuple LIPR. $\begin{matrix} {\overset{⌢}{R}}_{1} = R_{1}^{(2)} \\ = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.43), (s_{1}, - 0.41)] & [(s_{1}, 0.17), (s_{1}, 0)] & [(s_{2}, - 0.45), (s_{1}, - 0.47)] \\ [(s_{1}, - 0.41), (s_{1}, 0.43)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.28), (s_{2}, - 0.34)] & [(s_{1}, 0.27), (s_{0}, 0.3)] \\ [(s_{1}, 0), (s_{1}, 0.17)] & [(s_{2}, - 0.34), (s_{1}, - 0.28)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{2}, 0.32), ((s_{0}, 0.19))] \\ [(s_{1}, - 0.47), (s_{2}, - 0.45)] & [(s_{0}, 0.3), (s_{1}, 0.27)] & [(s_{0}, 0.19), (s_{2}, 0.32)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}] . \end{matrix}$

Conduct the identical procedures with R₂ and R₃, respectively. It can be interfered that R₂ and R₃ are of acceptable geometric consistency with $CI ({\overset{⌢}{R}}_{2}) = CI (R_{2}^{(2)}) = 0 . 94$ ,

$CI ({\overset{⌢}{R}}_{3}) = CI (R_{3}^{(1)}) = 0 . 94$ . Therefore, we output ${\overset{⌢}{R}}_{2}$ and ${\overset{⌢}{R}}_{3}$ in the same way.

Then, apply Algorithm 2 to reach group consensus for the 2-tuple LIPRs.

Algorithm 2.

Utilize the ${\overset{⌢}{R}}_{1}$ , ${\overset{⌢}{R}}_{2}$ , and ${\overset{⌢}{R}}_{3}$ obtained in Algorithm 1. Calculate the collective 2-tuple LIPR matrix $R_{c}^{(0)} = {(r_{ij, c}^{(0)})}_{n \times n}$ by step 2.

$R_{c}^{(0)} = {\overset{⌢}{R}}_{1}^{(2)} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.45), (s_{2}, - 0.01)] & [(s_{3}, - 0.36), (s_{1}, - 0.35)] & [(s_{2}, - 0.33), (s_{2}, 0)] \\ [(s_{2}, - 0.01), (s_{1}, 0.45)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.42), (s_{2}, 0.48)] & [(s_{2}, - 0.05), (s_{1}, - 0.45)] \\ [(s_{1}, - 0.35), (s_{3}, - 0.36)] & [(s_{2}, 0.48), (s_{1}, - 0.42)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{4}, - 0.4), (s_{0}, 0.1)] \\ [(s_{2}, 0), (s_{2}, - 0.33)] & [(s_{1}, - 0.45), (s_{2}, - 0.05)] & [(s_{0}, 0.1), (s_{4}, - 0.4)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}],$

Repeat the step 2–4, and output the modified 2-tuple LIPRs ${\overset{⌢}{R}}_{2}^{(2)}$ , ${\overset{⌢}{R}}_{2}^{(2)}$ , and ${\overset{⌢}{R}}_{3}^{(2)}$ . ${\overset{⌢}{R}}_{1}^{(2)} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.34), (s_{1}, 0.4)] & [(s_{2}, - 0.04), (s_{1}, - 0.3)] & [(s_{2}, - 0.39), (s_{1}, 0.25)] \\ [(s_{1}, 0.4), (s_{1}, 0.34)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.44), (s_{2}, 0.09)] & [(s_{2}, - 0.33), (s_{0}, 0.46)] \\ [(s_{1}, - 0.3), (s_{2}, - 0.04)] & [(s_{2}, 0.09), (s_{1}, - 0.44)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{3}, 0.05), (s_{0}, 0.11)] \\ [(s_{1}, 0.25), (s_{2}, - 0.39)] & [(s_{0}, 0.46), (s_{2}, - 0.33)] & [(s_{0}, 0.11), (s_{3}, 0.05)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}],$ $\begin{matrix} {\overset{⌢}{R}}_{2}^{(2)} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.25), (s_{1}, 0.5)] & [(s_{2}, - 0.33), (s_{1}, - 0.3)] & [(s_{2}, - 0.32), (s_{1}, 0.24)] \\ [(s_{1}, 0.5), (s_{1}, 0.25)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{0}, 0.43), (s_{2}, 0.13)] & [(s_{2}, - 0.21), (s_{0}, 0.48)] \\ [(s_{1}, - 0.3), (s_{2}, - 0.33)] & [(s_{2}, 0.13), (s_{0}, 0.43)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{3}, 0.26), (s_{0}, 0.1)] \\ [(s_{1}, 0.24), (s_{2}, - 0.32)] & [(s_{0}, 0.48), (s_{2}, - 0.21)] & [(s_{0}, 0.1), (s_{3}, 0.26)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}], \\ {\overset{⌢}{R}}_{3}^{(2)} = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.32), (s_{2}, - 0.25)] & [(s_{2}, - 0.07), (s_{1}, - 0.23)] & [(s_{2}, - 0.2), (s_{1}, 0.42)] \\ [(s_{1}, - 0.25), (s_{1}, 0.32)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.46), (s_{2}, 0.21)] & [(s_{2}, - 0.08), (s_{1}, - 0.45)] \\ [(s_{1}, - 0.23), (s_{2}, - 0.07)] & [(s_{2}, 0.21), (s_{1}, - 0.46)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{3}, 0.38), (s_{0}, 0.11)] \\ [(s_{1}, 0.42), (s_{2}, - 0.2)] & [(s_{1}, - 0.45), (s_{2}, - 0.08)] & [(s_{0}, 0.11), (s_{3}, 0.38)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}] . \end{matrix}$

Their corresponding group consensus indices are $GCI ({\overset{⌢}{R}}_{1}^{(2)}) = 0.96$ , $GCI ({\overset{⌢}{R}}_{2}^{(2)}) = 0.95$ , and $GCI ({\overset{⌢}{R}}_{3}^{(2)}) = 0.96$ , with the same iteration p = 2.

Algorithm 3.

Obtain consensus modified 2-tuple LIPRs by Algorithm 3, and calculate the collective 2-tuple LIPR matrix by step 2. $\begin{matrix} {\overset{⌢}{R}}_{c}^{p} = {({\overset{⌢}{r}}_{ij, c}^{(p)})}_{n \times n} \\ = [\begin{matrix} [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, 0.31), (s_{2}, - 0.49)] & [(s_{2}, - 0.14), (s_{1}, - 0.29)] & [(s_{2}, - 0.33), (s_{1}, 0.28)] \\ [(s_{2}, - 0.49), (s_{1}, 0.31)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{1}, - 0.49), (s_{2}, 0.13)] & [(s_{2}, - 0.24), (s_{0}, 0.49)] \\ [(s_{1}, - 0.29), (s_{2}, - 0.14)] & [(s_{2}, 0.13), (s_{1}, - 0.49)] & [(s_{2}, 0), (s_{2}, 0)] & [(s_{3}, 0.17), (s_{0}, 0.11)] \\ [(s_{1}, 0.28), (s_{2}, - 0.33)] & [(s_{0}, 0.49), (s_{2}, - 0.24)] & [(s_{0}, 0.11), (s_{3}, 0.17)] & [(s_{2}, 0), (s_{2}, 0)] \end{matrix}] . \end{matrix}$

The overall value aggregation results are $\begin{matrix} {\overset{⌢}{r}}_{1, c} = [(s_{2}, - 0.28), (s_{1}, 0.4)], \\ {\overset{⌢}{r}}_{2, c} = [(s_{1}, 0.47), (s_{2}, - 0.48)] \\ {\overset{⌢}{r}}_{3, c} = [(s_{2}, 0.15), (s_{1}, 0.17)], \\ {\overset{⌢}{r}}_{4, c} = [(s_{1}, 0.01), (s_{2}, 0.25)] . \end{matrix}$

According to Step 4, the likelihood matrix is $P ({\overset{⌢}{r}}_{i, c} \geq {\overset{⌢}{r}}_{j, c}) = [\begin{matrix} 0.5 & 0.6 & 0.29 & 0.98 \\ 0.4 & 0.5 & 0.19 & 0.84 \\ 0.71 & 0.81 & 0.5 & 0 \\ 0.02 & 0.16 & 1 & 0.5 \end{matrix}] .$

The ranking values are $V ({\overset{⌢}{r}}_{1, c}) = 0.34$ , $V ({\overset{⌢}{r}}_{2, c}) = 0.23$ , $V ({\overset{⌢}{r}}_{3, c}) = 0.25$ , and

$V ({\overset{⌢}{r}}_{4, c}) = 0.17$ . The ranking result is A₁ > A₃ > A₂ > A₄, and the optimal choice is A₄.

5.2 Comparison analysis and discussion

In what follows, to validate the feasibility and applicability of the proposed algorithms, a comparison analysis is conducted by comparing with other methods provided in the existing literature. It is due

to that the preference information in Refs. [25, 49] is described with IFNs, the evaluation values distinguished by linguistic 2-tuples in this paper are transformed into linguistic labels for convenience of calculation. In addition, the linguistic labels must be further normalized to [0, 1] to match the approach proposed by Refs. [25, 49].

The critical steps in Ref. [49] are to directly derive the priority weight of each individual using the fractional programming model that considers the multiplicative consistency, and to generate a synthetic priority vector using the aggregation operator. The synthetic priority vector is identified as ([0.15, 0.55] , [0.08, 0.8] , [0.21, 0.51] , [0.01, 0.99]). According to the similarity function, we have L (ω₁) = 0.34, L (ω₂) = 0.18, L (ω₃) = 0.38, and L (ω₄) = 0.01. It yields the ranking result A₃ > A₁ > A₂ > A₄.

The framework of the GDM method in Ref. [49] can be roughly divided into three procedures: the inconsistency repair process, the consensus reaching process, and the selection process. With the assistance of the fractional programming models, the priority weight of the IFPRs can be generated directly. The inconsistency repair process is an iterative procedure due to the multiplicative consistency. The consensus reaching process operates on the basis of the rule that excluding the one who would not change his evaluation with the furthest distance between two experts but furnishing the one referring to others until reaching the consensus threshold. In the selection process, the overall priority weight can be aggregated by the IFWA operator and ranked by the similarity function. Conducting the aforementioned steps, we have L (ω₁) = 0.35, L (ω₂) = 0.18, L (ω₃) = 0.38, and L (ω₄) = 0.01. Therefore, the ranking result is A₂ > A₃ > A₁ > A₄.

Then, the final ranking results provided by the different methods are listed in Table 1.

Table 1
Ranking results from different methods

Methods Operators Ranking of alternatives

Method in Ref. [49] IFWG A₃ > A₁ > A₂ > A₄

Method in Ref. [25] IFWA A₂ > A₃ > A₁ > A₄

The proposed method ATS-2TLIFWG A₁ > A₃ > A₂ > A₄

Methods	Operators	Ranking of alternatives
Method in Ref. [49]	IFWG	A₃ > A₁ > A₂ > A₄
Method in Ref. [25]	IFWA	A₂ > A₃ > A₁ > A₄
The proposed method	ATS-2TLIFWG	A₁ > A₃ > A₂ > A₄

The illustrate example described above reveals that both of the methods developed by Refs. [25, 49] produce slightly different ranking results from our proposed method. Roughly speaking, as shown in Table 1, the aggregation operator may be one of the factors that influence differences in the rankings. The method in Ref. [49] generates different positions for A₃ and A₁. It seems quite simple because it derives the priority weight directly, and only need to do the aggregation process after that. However, in practical situations, consistency and consensus play an essential role in offering a more rational final ranking result for GDM. Therefore, with the consistency modification algorithm, our approach offers a relatively convincing order. Although the approach in Ref. [25] includes inconsistency repair and consensus reaching processes, it has some significant defects in comparison to our method, leading it to invert order of A₁ and A₂. Multiplicative consistency, which is introduced as the foundational concept of the GDM framework [25, 49], has demonstrated a lack of robustness in permutations of intuitionistic fuzzy comparisons, and it depends highly on alternative labels [44]. In contrast, the geometric consistency applied in this paper overcomes these drawbacks in robustness, and it is feasible in dealing with IFNs.

The strengths of the proposed method can be categorically drawn in the following.

As an extension of IFNs, 2-tuple LIFNs are well able to manage practical situations because preference intentions can be accurately conveyed by quantitative information in a linguistic form, reflecting the actual semantics flexibly and effectively.

According to the characteristics of the GDM voting system, the novel 2-tuple LIPR can be considered an efficient technique to handle this issue. It indicates that the 2-tuple LIPR is of applicability and effectiveness in reality.

The proposed method is quite convincing as it not only circumvents individual inconsistency but also considers the group consensus, thereby delivering a reasonable and acceptable ranking result identified by group DMs. Furthermore, the iterative procedure makes it more applicable to real situations.

It is feasible and accurate to apply geometric consistency in disposing of the LIFPR. Meanwhile, compared to other methods in the existing literature, the proposed method can effectively overcome the problems of non-robustness and information loss, and it reduces computation to some extent.

6 Conclusion

Sustainable location planning, which plays an essential role in enhancing regional competitiveness and urban development, is a long-term issue that can be addressed by GDM voting systems. Considering the strengths of artificial languages and preference relations in depicting the case, 2-tuple LIPR can be generalized to this situation. To optimize the GDM voting system, a GDM algorithm, which is focused on individual consistency modification, group consensus reaching, and overall value aggregation, is adapted to practical.

The main advantages of the proposed method lie not only in its capacity to effectively manage the voting system, but also in its ability to settle GDM problems by considering individual consistency and group consensus, yielding solutions that more closely accord with realistic contexts. The limitation of these algorithms is that they cannot be applied to situation in which expert weights are unknown. Moreover, as a new type of preference relation, some distinct characteristics of the 2-tuple LIPR concerning consistency need to be further investigated. Additionally, more attention should be paid to incomplete and multi-granularity information.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 71571193).

References

Chen

S.M.

, Cheng

S.H.

and Lin

T.E.

, Group decision making systems using group recommendations based on interval fuzzy preference relations and consistency matrices, Information Sciences 298 (2015), 555–567.

Wang

, Wang

J.Q.

and Zhang

H.Y.

, A likelihood-based TODIM approach based on multi-hesitant fuzzy linguistic information for evaluation in logistics outsourcing, Computers & Industrial Engineering 99 (2016), 287–299.

Tian

Z.P.

, Wang

J.Q.

and Zhang

H.Y.

, Simplified neutrosophic linguistic multi-criteria group decision-making approach to green product development, Group Decision and Negotiation (2016). DOI: 10.1007/s10726-016-9479-5

Pan

D.B.

, Liu

and Deng

, An improved AHP method in preferential voting system, pp, Computer Science & Education (ICCSE), 2016 11th International Conference on IEEE (2016), 82–85.

Hare

, The election of representatives, parliamentary and municipal: A treatise, London: Longman, Green, Longman, Roberts & Green, 1865.

Reilly

, The global spread of preferential voting: Australian institutional imperialism? Australian Journal of Political Science 39(2) (2004), 253–266.

Angiz

M.Z.

, Tajaddini

, Mustafa

and Kamali

M.J.

, Ranking alternatives in a preferential voting system using fuzzy concepts and data envelopment analysis, Computers & Industrial Engineering 63(4) (2012), 784–790.

Rivest

R.L.

and Shen

, An optimal single-winner preferential voting system based on game theory, Proc of 3rd International Workshop on Computational Social Choice, 2010, pp. 399–410.

Izadikhah

and Saen

R.F.

, A new preference voting method for sustainable location planning using geographic information system and data envelopment analysis, Journal of Cleaner Production 137 (2016), 1347–1367.

10.

Z.S.

, A survey of preference relations, International Journal of General Systems 36(2) (2007), 179–203.

11.

Meng

F.Y.

and Chen

X.H.

, A new method for triangular fuzzy compare wise judgment matrix process based on consistency analysis, International Journal of Fuzzy Systems 19 (2017) 27–46.

12.

Zhou

and Xu

Z.S.

, Asymmetric hesitant fuzzy sigmoid preference relations in the analytic hierarchy process, Information Sciences 358 (2016), 191–207.

13.

Z.S.

, On compatibility of interval fuzzy preference relations, Fuzzy Optimization and Decision Making 3(3) (2004), 217–225.

14.

Zhou

, Wang

, Li

X.E.

and Wang

J.Q.

, Intuitionistic hesitant linguistic sets and their application in multi-criteria decision-making problems, Operational Research 16(1) (2016), 131–160.

15.

Zhou

, Xu

Z.S.

and Chen

M.H.

, Preference relations based on hesitant-intuitionistic fuzzy information and their application in group decision making, Computers & Industrial Engineering 87 (2015), 163–175.

16.

Wang

C.H.

and Wang

J.Q.

, A multi-criteria decision-making method based on triangular intuitionistic fuzzy preference information, Intelligent Automation & Soft Computing 22(3) (2016), 473–482.

17.

Tong

X.Y.

and Wang

Z.J.

, A group decision framework with intuitionistic preference relations and its application to low carbon supplier selection, International Journal of Environmental Research and Public Health 13(9) (2016), 923.

18.

Saaty

T.L.

, The analytic hierarchy process, McGrew Hill, New York 1980.

19.

Wang

Y.M.

, Yang

J.B.

and Xu

D.L.

, A two-stage logarithmic goal programming method for generating weights from interval comparison matrices, Fuzzy Sets and Systems 152(3) (2005), 475–498.

20.

Jiang

, Xu

Z.S.

and Yu

X.H.

, Compatibility measures and consensus models for group decision making with intuitionistic multiplicative preference relations, Applied Soft Computing 13(4) (2013), 2075–2086.

21.

Meng

F.Y.

, Tang

, An

Q.X.

and Chen

X.H.

, Decision making with intuitionistic linguistic preference relations, International Transactions in Operational Research (2016). DOI: 10.1111/itor.12383

22.

Jin

F.F.

, Ni

Z.W.

, Chen

H.Y.

and Li

Y.P.

, Approaches to decision making with linguistic preference relations based on additive consistency, Applied Soft Computing 49 (2016), 71–80.

23.

Z.S.

, Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment, Information Sciences 168(1) (2004), 171–184.

24.

Zhang

and Guo

C.H.

, A method for multi-granularity uncertain linguistic group decision making with incomplete weight information, Knowledge-Based Systems 26 (2012), 111–119.

25.

Liao

H.C.

, Xu

Z.S.

, Zeng

X.J.

and Merigó

J.M.

, Framework of group decision making with intuitionistic fuzzy preference information, IEEE Transactions on Fuzzy Systems 23(4) (2015), 1211–1227.

26.

Wang

, Xu

X.H.

, Wang

J.Q.

and Cai

C.G.

, Some new operation rules and a new ranking method for interval-valued intuitionistic linguistic numbers, Journal of Intelligent & Fuzzy Systems 32(1) (2017), 1069–1078.

27.

Cao

Y.X.

, Zhou

and Wang

J.Q.

, An approach to interval-valued intuitionistic stochastic multi-criteria decision-making using set pair analysis, International Journal of Machine Learning and Cybernetics (2016). DOI: 10.1007/s13042-016-0589-9

28.

S.M.

, Wang

and Wang

J.Q.

, An extended TODIM approach with intuitionistic linguistic numbers, International Transactions in Operational Research (2016). DOI: 10.1111/itor.12363

29.

Chen

Z.C.

, Liu

P.H.

and Pei

, An approach to multiple attribute group decision making based on linguistic intuitionistic fuzzy numbers, International Journal of Computational Intelligence Systems 8(4) (2015), 747–760.

30.

Liu

P.D.

and Qin

X.Y.

, Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision making, Journal of Intelligent & Fuzzy Systems 32(1) (2017) 1029–1043.

31.

Peng

H.G.

, Wang

J.Q.

and Chen

P.F.

, A linguistic intuitionistic multi-criteria decision-making method based on the Frank Heronian mean operator and its application in evaluating coal mine safety, International Journal of Machine Learning and Cybernetics (2016). DOI: 10.1007/s13042-016-0630-z

32.

Herrera

, Alonso

, Chiclana

and Herrera-Viedma

, Computing with words in decision making: Foundations, trends and prospects, Fuzzy Optimization and Decision Making 8(4) (2009), 337–364.

33.

Martínez

, Ruan

and Herrera

, Computing with words in decision support systems: An overview on models and applications, International Journal of Computational Intelligence Systems 3(4) (2010), 382–395.

34.

Y.X.

, Wang

J.Q.

and Chen

X.H.

, 2-tuple linguistic aggregation operators based on subjective sensation and objective numerical scales for multi-criteria group decision-making problems, Scientia Iranica 23(3) (2016), 1399–1417.

35.

Tao

Z.F.

, Chen

H.Y.

, Zhou

L.G.

and Liu

J.P.

, On new operational laws of 2-tuple linguistic information using Archimedean t-norm and s-norm, Knowledge-Based Systems 66 (2014), 156–165.

36.

Z.S.

and Xia

M.M.

, Iterative algorithms for improving consistency of intuitionistic preference relations, Journal of the Operational Research Society 65(5) (2014), 708–722.

37.

Zhang

and Guo

C.H.

, Consistency and consensus models for group decision-making with uncertain 2-tuple linguistic preference relations, International Journal of Systems Science 47(11) (2016), 2572–2587.

38.

Zhang

Z.M.

and Wu

, On the use of multiplicative consistency in hesitant fuzzy linguistic preference relations, Knowledge-Based Systems 72 (2014), 13–27.

39.

Liao

H.C.

, Xu

Z.S.

and Xia

M.M.

, Multiplicative consistency of hesitant fuzzy preference relation and its application in group decision making, International Journal of Information Technology & Decision Making 13(01) (2014), 47–76.

40.

Zhu

and Xu

Z.S.

, Consistency measures for hesitant fuzzy linguistic preference relations, IEEE Transactions on Fuzzy Systems 22(1) (2014), 35–45.

41.

Xia

M.M.

, Xu

Z.S.

and Wang

, Multiplicative consistency-based decision support system for incomplete linguistic preference relations, International Journal of Systems Science 45(3) (2014), 625–636.

42.

Xia

M.M.

, Xu

Z.S.

and Chen

, Algorithms for improving consistency or consensus of reciprocal [0, 1]-valued preference relations, Fuzzy Sets and Systems 216 (2013), 108–133.

43.

Z.S.

, Cai

X.Q.

and Szmidt

, Algorithms for estimating missing elements of incomplete intuitionistic preference relations, International Journal of Intelligent Systems 26(9) (2011), 787–813.

44.

Wang

Z.J.

, Geometric consistency based interval weight elicitation from intuitionistic preference relations using logarithmic least square optimization, Fuzzy Optimization and Decision Making 14(3) (2015), 289–310.

45.

Zhou

and Xu

Z.S.

, Generalized asymmetric linguistic term set and its application to qualitative decision making involving risk appetites, European Journal of Operational Research 254(2) (2016), 610–621.

46.

Zhou

, Wang

J.Q.

and Zhang

H.Y.

, Multi-criteria decision-making approaches based on distance measures for linguistic hesitant fuzzy sets, Journal of the Operational Research Society (2016). DOI: 10.1057/jors.2016.41

47.

Xia

M.M.

, Xu

Z.S.

and Zhu

, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012), 78–88.

48.

Zhou

, Wang

J.Q.

and Zhang

H.Y.

, Grey stochastic multi-criteria decision-making approach based on prospect theory and distance measures, the Journal of Grey System 29(1) (2017), 15–33.

49.

Liao

H.C.

and Xu

Z.S.

, Some algorithms for group decision making with intuitionistic fuzzy preference information, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems 22(04) (2014), 505–529.

2-tuple linguistic intuitionistic preference relation and its application in sustainable location planning voting system

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 2-tuple linguistic model

3.1 Operational laws and aggregation operators of 2-tuple LIFNs

4.1 Group consensus index

5.1 Illustrative example

5.2 Comparison analysis and discussion

Table 1 Ranking results from different methods Methods Operators Ranking of alternatives Method in Ref. [49] IFWG A3 > A1 > A2 > A4 Method in Ref. [25] IFWA A2 > A3 > A1 > A4 The proposed method ATS-2TLIFWG A1 > A3 > A2 > A4

Footnotes

Acknowledgments

References

Table 1
Ranking results from different methods

Methods Operators Ranking of alternatives

Method in Ref. [49] IFWG A₃ > A₁ > A₂ > A₄

Method in Ref. [25] IFWA A₂ > A₃ > A₁ > A₄

The proposed method ATS-2TLIFWG A₁ > A₃ > A₂ > A₄