Linguistic hesitant intuitionistic fuzzy linear assignment method based on Choquet integral

Abstract

This study develops a new multiple attribute decision making method based on linear assignment method with linguistic hesitant intuitionistic fuzzy information considering correlation. First, we develop some linguistic hesitant intuitionistic fuzzy correlated aggregation operators by using Choquet integral. Then we propose a new linguistic hesitant intuitionistic fuzzy linear assignment method considering correlation of attributes. A numerical example has been presented to illustrate feasibility and practical advantages of the new method. Comparisons of new method with other methods, including the linguistic hesitant intuitionistic fuzzy correlated TOPSIS method, the method based on the generalized linguistic hesitant intuitionistic fuzzy correlated averaging (GLHIFCA) operator, have been conducted.

Keywords

Multiple attribute decision making hesitant fuzzy set linear assignment method Choquet integral aggregation operator

Hesitant fuzzy set (HFS) [1, 2] is the extension of fuzzy set, which permits the membership degree of an element to a set represented by several possible values. Torra and Narukawa [1] studied the relationship between the HFS and intuitionistic fuzzy set and showed that the envelope of HFS is an intuitionistic fuzzy set. Recently, the hesitant fuzzy set has received broad attention and has been studied and applied extensively [3 –45]. Hesitant fuzzy set has been generalized to accommodate interval-values [7 –9] triangular fuzzy values [10], type-2 fuzzy value [11], intuitionistic fuzzy value [12], interval-valued intuitionistic fuzzy value [13], linguistic term [14 –21], interval linguistic term [22], etc. Some aggregation operators have been developed [23 –29]. A series of hesitant fuzzy weighted aggregation operators and ordered weighted aggregation operators have been developed by Xia and Xu [24]. Wei [25] proposed some hesitant fuzzy prioritized aggregation operators and developed some models based on them, in which the attributes are in different priority level. Zhu et al. [26] studied the hesitant fuzzy geometric Bonferroni mean operator and the hesitant fuzzy Choquet geometric Bonferroni mean operator. Tan et al. [27] presented a family of hesitant fuzzy Hamacher operators based on the Hamacher t-norm and t-conorm. Zhang et al. [28] proposed a series of induced generalized aggregation operators for hesitant fuzzy or interval-valued hesitant fuzzy information. Some classic multiple attribute decision making methods have been generalized to accommodate hesitant fuzzy information [30 –37]. Beg and Rashid [31] extended the TOPSIS method to hesitant fuzzy linguistic term sets. Zhang and Wei [32] developed the extended VIKOR method and TOPSIS method to solve the multiple attribute decision making problems with hesitant fuzzy information. Chen and Xu [34] proposed hesitant fuzzy ELECTRE II approach by combining the idea of HFSs with the ELECTRE II method. Zhang and Xu [35] extended the TODIM to hesitant fuzzy setting to capture the decision maker’s psychological behavior based on prospect theory. Some entropy and cross-entropy measures, correlated coefficient, distance measures have been presented [38 –45]. Xu and Xia [38] proposed some hesitant fuzzy entropy and cross-entropy measures. Liao and Xu [40] developed a family of cosine distance and similarity measures for hesitant fuzzy linguistic term set. An entropy measure for interval-valued hesitant fuzzy sets based on three different functions, where each one represents a different measure: fuzziness, lack of knowledge and hesitance has been proposed by Quirós et al. [41]. Farhadinia [45] defined some distance and similarity measures for higher order hesitant fuzzy sets. From the above analysis we can see that, the hesitant fuzzy set is much useful in decision making process. In decision making process, decision makers would like to evaluate with linguistic terms. But they may express some extent of hesitation, which can be modeled by using intuitionistic fuzzy values effectively. For complex decision problems, experts from different fields are needed to evaluate alternatives. For example, in evaluating the construction alternatives of subway, it is need to invite experts from different fields including architects, structure engineer and geological engineer, etc. If expert is familiar with the attribute, he/she can give proper evaluation values and if he/she is not familiar with the attribute, he/she can refuse to give any evaluation values to assure the accuracy and reasonability of decision results. If two decision makers evaluate an alternative with respect to some attribute with the same linguistic terms and different intuitionistic fuzzy values, the intuitionistic fuzzy values are merged and we can get the linguistic hesitant intuitionistic fuzzy values [12]. We use linguistic terms and intuitionistic fuzzy values to evaluate and linguistic hesitant intuitionistic fuzzy information can be got.

The linear assignment method is developed by Bernardo and Blin [46]. Based on a set of criterion-wise rankings and a set of criterion weights, Bernardo and Blin made consumer choice among multi-attributed brands. Lin and Wen [47] developed a fuzzy assignment problem based on the labeling method. Chen [48] generalized the linear assignment method to accommodate interval type-2 trapezoidal fuzzy numbers and applied the new method to the selection of a landfill site. The linear assignment method in interval-valued intuitionistic fuzzy setting has been developed by Chen [49] to solve the investment company selection problem. An interactive fuzzy linear assignment method has been developed to select optimum maintenance strategy [50]. Bashiri and Badrithe [51] developed an interactive method to rank alternatives by using fuzzy linear assignment method. Liu and Wang [52] used a fuzzy linear assignment approach in the final selection of third-party logistics provider. Jahan et al. [53] used the linear assignment method to select the optimal material. Though linear assignment method is a useful multiple attribute decision making method, little research has been conducted on the problems with linguistic hesitant intuitionistic fuzzy information. Therefore, motivated by the idea of the linear assignment methodology, we develop an extended linear assignment method with linguistic hesitant intuitionistic fuzzy information based on the signed distance considering correlation of attributes.

In order to do so, the rest of the paper is organized as follows. Section 2 briefly reviews some concepts on linguistic hesitant intuitionistic fuzzy set. Some linguistic hesitant intuitionistic fuzzy correlation aggregation operators have been developed in Section 3. We develop the linguistic hesitant intuitionistic fuzzy correlated averaging (LHIFCA) operator, the linguistic hesitant intuitionistic fuzzy correlated geometric averaging (LHIFCGA) operator and the generalized linguistic hesitant intuitionistic fuzzy correlated averaging (GLHIFCA) operator. Some special cases of the new aggregation operators have been studied. In section 4, we present a new linguistic hesitant intuitionistic fuzzy linear assignment method based on Choquet integral. In Section 5, a numerical example has been presented and comparisons have been conducted. Conclusions have been presented in the last section.

1 Preliminaries

Torra and Narukawa [1, 2] generalized fuzzy set to make each membership include several values and developed the hesitant fuzzy set (HFS).

Definition 1. Let X be a reference set, an HFS A on X is a function h that returns a subset of values in [0, 1] when it is applied to X: $A = {< x, h_{A} (x) > ∣ x \in X},$

where h_A (x) is a set of some different values in [0, 1], representing the possible membership degrees of the element x ∈ X to A. h_A (x) is called a hesitant fuzzy element (HFE).

Suppose that S = {s_i| i = 1,. . . , g} is a finite and totally ordered discrete term set, where s_i represents a possible value for a linguistic variable. A set of nine terms S [54] can be expressed in the following S = {s₁ = extremelypoor, s₂ = verypoor, s₃ = poor, s₄ = slightly poor, s₅ = fair, s₆ = slightly good, s₇ = good, s₈ = verygood, s₉ = extremelygood} .

In order to preserve all information, the discrete linguistic term set S can be extended to a continuous one $\bar{S} = {s_{α} ∣ s_{0} \leq s_{α} \leq s_{g}, α \in [0, g]}$ .

Definition 2. [12]. Let X = {x₁, x₂,. . . , x_n} be a reference set. A linguistic hesitant intuitionistic fuzzy term set (LHIFS) $\overset{ˇ}{A}$ on X is defined as $\overset{ˇ}{A} = {< x_{i}, {\overset{ˇ}{h}}_{\overset{ˇ}{A}} (x_{i}) > ∣ x_{i} \in X},$ (1) where $\bar{S} = {s_{α} ∣ s_{0} \leq s_{α} \leq s_{g}}$ is a linguistic term set and H is the set of linguistic hesitant intuitionisic fuzzy elements. ${\overset{ˇ}{h}}_{\overset{ˇ}{A}} (x_{i}) : X \to H$ denotes all possible linguistic intuitionistic fuzzy evaluation values of element x_i ∈ X. For convenience, we call ${\overset{ˇ}{h}}_{\overset{ˇ}{A}} (x_{i})$ a linguistic hesitant intuitionistic fuzzy element (LHIFE), which can be represented as ${\overset{ˇ}{h}}_{\overset{ˇ}{A}} (x_{i}) = {(s_{θ_{i}}, lh (s_{θ_{i}})) ∣ x_{i} \in X},$ (2) where $s_{θ_{i}} \in \bar{S}$ is a linguistic argument and lh (s_{θ
_i}) = {(α_i, β_i)} is the set of intuitionistic fuzzy membership values that s_{θ
_i} satisfies x_i. (s_{θ
_i}, lh (s_{θ
_i})) is the linguistic intuitionistic fuzzy element (LIFE).

Definition 3. Let $\overset{ˇ}{h}$ , ${\overset{ˇ}{h}}_{1}$ and ${\overset{ˇ}{h}}_{2}$ be LHIFEs, λ > 0. $a_{k} = (s_{θ (k)}, lh (s_{θ (k)})) \in \overset{ˇ}{h}$ , $a_{i} = (s_{θ (i)}, lh (s_{θ (i)})) \in {\overset{ˇ}{h}}_{1}$ , $a_{j} = (s_{θ (j)}, lh (s_{θ (j)})) \in {\overset{ˇ}{h}}_{2}$ . (μ_k, ν_k) ∈ lh (s_θ(k)) , (μ_i, ν_i) ∈ lh (s_θ(i)) , (μ_j, ν_j) ∈ lh (s_θ(j)). Some operation laws can be defined as follows

$(1) {\overset{ˇ}{h}}_{1} \oplus {\overset{ˇ}{h}}_{2} = ⋃_{a_{i} \in {\overset{ˇ}{h}}_{1}, a_{j} \in {\overset{ˇ}{h}}_{2}} {(s_{θ (i) + θ (j)},$ ⋃ {(μ_i + μ_j - μ_iμ_j, ν_iν_j)})},

$(2) {\overset{ˇ}{h}}_{1} \otimes {\overset{ˇ}{h}}_{2} = ⋃_{a_{i} \in {\overset{ˇ}{h}}_{1}, a_{j} \in {\overset{ˇ}{h}}_{2}} {(s_{θ (i) θ (j)}, ⋃ {(μ_{i} μ_{j}, ν_{i}$ +ν_j - ν_iν_j)})},

$(3) λ \overset{ˇ}{h} = ⋃_{a_{k} \in \overset{ˇ}{h}} {(s_{λ θ (k)}, ⋃_{(μ_{k}, ν_{k}) \in lh (s_{θ (k)})} {(1 - (1 - μ_{k})^{λ}, ν_{k}^{λ})})}$ ,

$(4) (\overset{ˇ}{h})^{λ} = ⋃_{a_{k} \in \overset{ˇ}{h}} {(s_{θ (k)^{λ}}, ⋃_{(μ_{k}, ν_{k}) \in lh (s_{θ (k)})} {(μ_{k}^{λ},$ 1 - (1 - ν_k) ^λ)})}.

Definition 4. Let a_i = (s_{θ
_i}, lh (s_{θ
_i})) be a LIFE, then the score function s (a_i) of a_i can be defined as $s (a_{i}) = \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} (α_{i} - β_{i})),$ (3) the accuracy function h (a_i) of a_i can be defined as $h (a_{i}) = \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} (α_{i} + β_{i})),$ (4) the deviation function d (a_i) of a_i can be defined as

$\begin{matrix} d (a_{i}) & = & \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} \\ (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} ((α_{i} + β_{i}) - s (a_{i}))^{2}), \end{matrix}$ (5) where g is the number of linguistic arguments in linguistic term set S and ∣lh (s_{θ
_i})∣ is the number of intuitionistic fuzzy memberships in lh (s_{θ
_i}). If lh (s_{θ
_i}) = φ, let ∣lh (s_{θ
_i}) ∣ =1.

The following ranking method for LIFEs can be presented based on the score function s (a_i), the accuracy function h (a_i) and the deviation function d (a_i). Let a_i = (s_{θ
_i}, lh (s_{θ
_i})) and a_j = (s_{θ
_j}, lh (s_{θ
_j})) be two LIFEs, then

If s (a_i) < s (a_i), then a_i is smaller than a_j, denoted by a_i < a_j;

If s (a_i) = s (a_i), then

If h (a_i) < h (a_j), then a_i is smaller than a_j, denoted by a_i < a_j;

If h (a_i) = h (a_j), then

if d (a_i) > d (a_j), then a_i is smaller than a_j, denoted by a_i < a_j;

if d (a_i) = d (a_j), then a_i and a_j represent the same information, denoted thena_i ∼ a_j.

Definition 5. Let $\overset{ˇ}{h} = {(s_{θ_{i}},$ lh (s_{θ
_i})) ∣ (s_{θ
_i}, $lh (s_{θ_{i}})) \in \overset{ˇ}{h}}$ be LHIFE, the score function $S (\overset{ˇ}{h})$ can be defined as

$\begin{matrix} S (\overset{ˇ}{h}) \\ = \frac{1}{l_{\overset{ˇ}{h}}} (\sum \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} (α_{i} - β_{i}))), \end{matrix}$ (6) the accuracy function $A (\overset{ˇ}{h})$ can be defined as

$\begin{matrix} A (\overset{ˇ}{h}) \\ = \frac{1}{l_{\overset{ˇ}{h}}} (\sum \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} (α_{i} + β_{i}))), \end{matrix}$ (7) the deviation function $D (\overset{ˇ}{h})$ can be defined as

$\begin{matrix} D (\overset{ˇ}{h}) = \frac{1}{l_{\overset{ˇ}{h}}} (\sum \frac{θ_{i}}{g ∣ lh (s_{θ_{i}}) ∣} \\ (\sum_{(α_{i}, β_{i}) \in lh (s_{θ_{i}})} {(α_{i} + β_{i} - S (\overset{ˇ}{h}))}^{2})), \end{matrix}$ (8)

where $l_{\overset{ˇ}{h}}$ is the number of LIFEs in $\overset{ˇ}{h}$ and g is the number of linguistic arguments in linguistic term set S, ∣lh (s_{θ
_i})∣ is the count of intuitionistic fuzzy memberships in lh (s_{θ
_i}).

Based on the score function and accuracy function, we present the following method to compare LHIFEs. Let ${\overset{ˇ}{h}}_{i}$ and ${\overset{ˇ}{h}}_{j}$ be two LHIFEs,

If $S ({\overset{ˇ}{h}}_{i}) < S ({\overset{ˇ}{h}}_{j}),$ then ${\overset{ˇ}{h}}_{i}$ is smaller than ${\overset{ˇ}{h}}_{j}$ , denoted by ${\overset{ˇ}{h}}_{i} < {\overset{ˇ}{h}}_{j}$ ;

If $S ({\overset{ˇ}{h}}_{i}) = S ({\overset{ˇ}{h}}_{j}),$ then

If $A ({\overset{ˇ}{h}}_{i}) < A ({\overset{ˇ}{h}}_{j})$ , then ${\overset{ˇ}{h}}_{i}$ is smaller than ${\overset{ˇ}{h}}_{j}$ , denoted by ${\overset{ˇ}{h}}_{i} < {\overset{ˇ}{h}}_{j}$ ;

If $A ({\overset{ˇ}{h}}_{i}) = A ({\overset{ˇ}{h}}_{j})$ , then

If $D ({\overset{ˇ}{h}}_{i}) > D ({\overset{ˇ}{h}}_{j})$ , then ${\overset{ˇ}{h}}_{i}$ is smaller than ${\overset{ˇ}{h}}_{j}$ , denoted by ${\overset{ˇ}{h}}_{i} < {\overset{ˇ}{h}}_{j}$ ;

If $D ({\overset{ˇ}{h}}_{i}) = D ({\overset{ˇ}{h}}_{j})$ , then ${\overset{ˇ}{h}}_{i}$ and ${\overset{ˇ}{h}}_{j}$ represent the same information, denoted then ${\overset{ˇ}{h}}_{i} \sim {\overset{ˇ}{h}}_{j}$ .

Different LHIFEs may have different number of LIFEs and different LIFEs may have different number of intuitionistic fuzzy memberships in most cases. In order to define the distance between two LHIFEs ${\overset{ˇ}{h}}_{1},$ ${\overset{ˇ}{h}}_{2}$ more accurately, they should be extended according to the risk attitude of the decision maker until they have the same number of LIFEs and each IFLE should have the same number of intuitionistic fuzzy memberships. If decision maker is risk-seeking, the largest LIFE and the largest intuitionistic fuzzy membership can be added; if decision maker is risk-averse, the smallest LIFE and the smallest intuitionistic fuzzy membership can be added; if decision maker is risk-neutral, the average LIFE and average intuitionistic fuzzy membership can be added.

Definition 6. Let ${\overset{ˇ}{h}}_{1} = {a_{i}^{(1)}} = {(s_{θ_{i}^{(1)}}, lh (s_{θ_{i}^{(1)}}))},$ ${\overset{ˇ}{h}}_{2} = {a_{i}^{(2)}} = {(s_{θ_{i}^{(2)}}, lh (s_{θ_{i}^{(2)}}))} .$ $a_{σ (i)}^{(k)} \geq a_{σ (j)}^{(k)}$ for i ≥ j, k = 1, 2. $(α_{i}^{k (t)}, β_{i}^{k (t)}) \in lh (s_{θ_{i}^{(t)}})$ . l is the number of LIFEs in LHIFE, g is the number of linguistic variables in linguistic term set and ∣lh (s_{θ
_i})∣ is the number of intuitionistic fuzzy memberships in lh (s_{θ
_i}). Based on the operation laws of LHIFEs and the extension principle, we can define some distance measures between LHIFEs.

The hesitant Euclidean distance $d ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2})$ for LHIFEs ${\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}$ can be defined as:

$\begin{matrix} d ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}) \\ = (\frac{1}{3 l} \sum_{i = 1}^{l} ({(θ_{(σ (i))}^{(1)} - θ_{(σ (i))}^{(2)})}^{2} / g^{2} + \frac{1}{∣ lh (s_{θ_{i}}) ∣} \\ {\sum_{k = 1}^{∣ lh (s_{θ_{i}}) ∣} ({(α_{σ (i)}^{k (1)} - α_{σ (i)}^{k (2)})}^{2} + {(β_{σ (i)}^{k (1)} - β_{σ (i)}^{k (2)})}^{2})))}^{1 / 2} . \end{matrix}$ (9)

The linguistic hesitant intuitionistic fuzzy Hamming distance can be defined as follows:

$\begin{matrix} d ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}) \\ = \frac{1}{3 l} \sum_{i = 1}^{l} (| θ_{(σ (i))}^{(1)} - θ_{(σ (i))}^{(2)} | / g + \frac{1}{∣ lh (s_{θ_{i}}) ∣} \\ \sum_{k = 1}^{∣ lh (s_{θ_{i}}) ∣} (| α_{σ (i)}^{k (1)} - α_{σ (i)}^{k (2)} | + | β_{σ (i)}^{k (1)} - β_{σ (i)}^{k (2)} |)) . \end{matrix}$ (10)

2 Some linguistic hesitant intuitionistic fuzzy Choquet aggregation operators

In this section, we consider some linguistic hesitant intuitionistic fuzzy correlated aggregation operators based on Choquet integral.

Definition 7. [55]. A fuzzy measure μ on the set X is a set function m : P (X) →[0, 1] satisfying the following axioms:

m (φ) =0, m (X) =1 ;

B ⊆ C implies m (B) ≤ m (C), for all B, C ⊆ X;

m (B ∪ C) = m (B) + m (C) + ρm (B) m (C) for all B, C ⊆ X and B ∩ C = φ, where ρ ∈ (-1, + ∞) .

In the above definition, if ρ = 0, then the third condition reduces to the axiom of the additive measure:

$\begin{matrix} m (B \cup C) & = & m (B) + m (C) for all B, \\ C \subseteq X and B \cap C = φ . \end{matrix}$ (11)

If the elements of B in X are independent, we have $m (B) = \sum_{x_{i} \in B} m (x_{i}), for all B \subseteq X .$ (12)

Let X = {x₁, x₂,. . . , x_n} be a finite set. Sugeno [56] gave the following equation to determine fuzzy measure on X avoiding the computational complexity. $m (X) = {\begin{matrix} \frac{1}{ρ} (Π_{i = 1}^{n} (1 + ρ m (x_{i})) - 1), & ρ \neq 0, \\ \sum_{i = 1}^{n} m (x_{i}), & ρ = 0, \end{matrix}$ (13) the value ρ can be uniquely determined from m (X) =1 by the following equation $ρ = Π_{i = 1}^{n} (1 + ρ m (x_{i})) - 1 .$ (14)

Especially, for every subset A ⊂ X, we have $m (X) = {\begin{matrix} \frac{1}{ρ} (Π_{x_{i} \in A} (1 + ρ m (x_{i})) - 1), & ρ \neq 0, \\ \sum_{x_{i} \in A} m (x_{i}), & ρ = 0 . \end{matrix}$ (15)

We define linguistic hesitant intuitionistic fuzzy correlated averaging (LHIFCA) operator based on the Choquet integral as follows.

Definition 8. Let X = {x₁, x₂,. . . , x_n} be a reference set, m be the fuzzy measure [55] on X and ${{\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}}$ be a collection of LHIFEs on X. The LHIFCA operator can be defined as follows:

$\begin{matrix} LHIFCA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \oplus_{j = 1}^{n} (m (A_{σ (j)}) - m (A_{σ (j - 1)})) {\overset{ˇ}{h}}_{σ (j)}, \end{matrix}$ (16) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ .

Theorem 1. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x_n} and ${\overset{ˇ}{h}}_{j} (j = 1, 2, . . ., n)$ be a collection of LHIFEs on X. The aggregated value using LHIFCA operator is also LHIFE and

$\begin{matrix} LHIFCA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) = \oplus_{j = 1}^{n} w_{j} {\overset{ˇ}{h}}_{σ (j)} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{n} (m (A_{σ (j)}) - m (A_{σ (j - 1)})) θ_{σ (j)}^{it}}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} (1 - \prod_{j = 1}^{n} \\ (1 - μ_{σ (j) k}^{it})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}, \\ \prod_{j = 1}^{n} (ν_{σ (j) k}^{it})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}))}, \end{matrix}$ where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ . ${\overset{ˇ}{h}}_{j} = {a_{j}^{i} | a_{j}^{i} \in {\overset{ˇ}{h}}_{j}, i = 1, 2, . . ., l_{j}} (j = 1, 2, . . ., n)$ , $a_{j}^{i} = {(s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) |$ $(s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) \in a_{j}^{i}}, t = 1, 2, . . ., l_{ij}$ , $(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})), k = 1, 2, . . ., l_{ijt}$ .

Proof. We can prove the theorem by using mathematical induction method. Let w_i = m (A_σ(j)) - m (A_σ(j-1)). Then if n = 2, $\begin{matrix} w_{1} {\overset{ˇ}{h}}_{σ (1)} \oplus w_{2} {\overset{ˇ}{h}}_{σ (2)} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{2} w_{j} θ_{σ (j)}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{k}^{it}})} \\ {(1 - \prod_{j = 1}^{2} (1 - μ_{σ (j) k}^{it})^{w_{j}}, \prod_{j = 1}^{2} (ν_{σ (j) k}^{it})^{w_{j}})})} . \end{matrix}$ If $\begin{matrix} \oplus_{j = 1}^{m} w_{j} {\overset{ˇ}{h}}_{σ (j)} = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{m} w_{j} (θ_{σ (j)}^{it})}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{k}^{it}})} {((1 - \prod_{j = 1}^{m} (1 - μ_{σ (j) k}^{it})^{w_{j}}, \\ \prod_{j = 1}^{m} (ν_{σ (j) k}^{it})^{w_{j}})})} \end{matrix}$ holds for n = m, then for n = m + 1, we have $\begin{matrix} \oplus_{j = 1}^{m + 1} w_{j} {\overset{ˇ}{h}}_{σ (j)} = \oplus_{j = 1}^{m} w_{j} {\overset{ˇ}{h}}_{σ (j)} + w_{m + 1} {\overset{ˇ}{h}}_{σ (m + 1)} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{m} w_{j} θ_{σ (j)}^{it}}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{k}^{it}})} {(1 - \prod_{j = 1}^{m} (1 - μ_{σ (j) k}^{it})^{w_{j}}, \\ \prod_{j = 1}^{m} ν_{σ (j) k}^{it})^{w_{j}})})} \\ \oplus ⋃_{a_{σ (m + 1)}^{i} \in {\overset{ˇ}{h}}_{σ (m + 1)}} {(s_{w_{m + 1} θ_{σ (m + 1)}^{it}}, \\ ⋃_{(μ_{σ (m + 1) j}^{it}, ν_{σ (m + 1) j}^{it}) \in lh (s_{θ_{σ (m + 1)}^{it}})} \\ {(1 - (1 - μ_{σ (m + 1) k}^{it})^{w_{m + 1}}, (ν_{σ (m + 1) k}^{it})^{w_{m + 1}})})} \\ = ⋃_{a_{σ (i)} \in {\overset{ˇ}{h}}_{σ (i)}} {(s_{\sum_{j = 1}^{m + 1} w_{j} θ_{σ (j)}^{it}}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{k}^{it}})} {(1 - \prod_{j = 1}^{m + 1} \\ (1 - (μ_{σ (j) k}^{it})^{λ})^{w_{j}}, \prod_{j = 1}^{m + 1} (1 - (1 - ν_{σ (j) k}^{it})^{λ})^{w_{j}})})}, \end{matrix}$ then $\begin{matrix} \oplus_{j = 1}^{m + 1} w_{j} {\overset{ˇ}{h}}_{σ (j)} = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{m + 1} w_{j} θ_{σ (j)}^{it}}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{k}^{it}})} {(1 - \prod_{j = 1}^{m + 1} (1 - μ_{σ (j) k}^{it})^{w_{j}}, \\ \prod_{j = 1}^{m + 1} (ν_{σ (j) k}^{it})^{w_{j}})})} . \end{matrix}$ Hence, the value of LHIFCA operator is a LHIFE.

Definition 9. Let X = {x₁, x₂,. . . , x_n} be a reference set, m be the fuzzy measure on X and ${{\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}}$ be a collection of LHIFEs on X. The linguistic hesitant intuitionistic fuzzy correlated geometric averaging (LHIFCGA) operator can be defined as follows:

$\begin{matrix} LHIFCGA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{σ (j)})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}, \end{matrix}$ (17) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ .

Theorem 2. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x_n} and ${\overset{ˇ}{h}}_{j} (j = 1, 2, . . ., n)$ be a collection of LHIFEs on X. The aggregated value using LHIFCGA operator is also LHIFE and $\begin{matrix} LHIFCGA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{σ (j)})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\prod_{j = 1}^{n} (θ_{σ (j)}^{it})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}}, \\ ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} (\prod_{j = 1}^{n} (μ_{σ (j) k}^{it})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}, \\ 1 - \prod_{j = 1}^{n} (1 - ν_{σ (j) k}^{it})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))}))}, \end{matrix}$ where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ . ${\overset{ˇ}{h}}_{j} = {a_{j}^{i} | a_{j}^{i} \in {\overset{ˇ}{h}}_{j}, i = 1, 2, . . ., l_{j}} (j = 1, 2, . . ., n)$ , $a_{j}^{i} = {(s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) | (s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) \in a_{j}^{i}}, t = 1, 2, . . ., l_{ij}, (μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}}), k = 1, 2, . . ., l_{ijt}$ .

Definition 10. Let X = {x₁, x₂,. . . , x_n} be a reference set, m be the fuzzy measure on X and ${{\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}}$ be a collection of LHIFEs on X. The generalized linguistic hesitant intuitionistic fuzzy correlated averaging (GLHIFCA) operator can be defined as follows:

$\begin{matrix} GLHIFCA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = (\oplus_{j = 1}^{n} (m (A_{σ (j)}) - m (A_{σ (j - 1)})) ({\overset{ˇ}{h}}_{σ (j)})^{λ})^{1 / λ}, \end{matrix}$ (18) where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ .

Theorem 3. Let m be a fuzzy measure on X = {x₁, x₂,. . . , x_n} and ${\overset{ˇ}{h}}_{j} (j = 1, 2, . . ., n)$ be a collection of LHIFEs on X. The aggregated value using GLHIFCA operator is also LHIFE and

$\begin{matrix} GLHIFCA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{(\sum_{j = 1}^{n} (m (A_{σ (j)}) - m (A_{σ (j - 1)})) (θ_{σ (j)}^{it})^{λ})^{1 / λ}}, \\ ⋃_{(μ_{kj}^{it}, ν_{kj}^{it}) \in lh (s_{θ_{k}^{it}})} {((1 - \prod_{j = 1}^{n} (1 - (μ_{σ (j) k}^{it})^{λ})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))})^{1 / λ}, \\ 1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{σ (j) k}^{it})^{λ})^{(m (A_{σ (j)}) - m (A_{σ (j - 1)}))})^{1 / λ})})} . \end{matrix}$

where (σ (1) , σ (2) ,. . . , σ (n)) is a permutation of (1, 2, . . . , n) such that ${\overset{ˇ}{h}}_{σ (1)} \geq {\overset{ˇ}{h}}_{σ (2)} \geq . . . \geq {\overset{ˇ}{h}}_{σ (n)}$ . ${\overset{ˇ}{h}}_{j} = {a_{j}^{i} | a_{j}^{i} \in {\overset{ˇ}{h}}_{j}, i = 1, 2, . . ., l_{j}} (j = 1, 2, . . ., n)$ , $a_{j}^{i} = {(s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) |$ $(s_{θ_{j}^{it}}, lh (s_{θ_{j}^{it}})) \in a_{j}^{i}}, t = 1, 2, . . ., l_{ij}$ , $(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}}), k = 1, 2, . . ., l_{ijt}$ .

We consider some special cases of the new aggregation operators in the following.

Remark 1. For any A, B ∈ P (X) such that ∣A ∣ = ∣B∣, where ∣A∣ is the number of elements in the set A. If m (A) = m (B) and $m (A_{σ (i)}) = \frac{i}{n}, 1 \leq i \leq n$ . Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy averaging (LHIFA) operator as follows $\begin{matrix} LHIFA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \oplus_{j = 1}^{n} \frac{1}{n} ({\overset{ˇ}{h}}_{σ (j)}) = \oplus_{j = 1}^{n} \frac{1}{n} ({\overset{ˇ}{h}}_{j}) \\ = ⋃_{a_{j}^{i} \in {\overset{ˇ}{h}}_{j}} {(s_{\sum_{j = 1}^{n} \frac{1}{n} θ_{j}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (1 - \prod_{j = 1}^{n} (1 - μ_{jk}^{it})^{\frac{1}{n}}, \prod_{j = 1}^{n} ν_{jk}^{it})^{\frac{1}{n}}))} . \end{matrix}$

The LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy geometric averaging (LHIFGA) operator as follows $\begin{matrix} LHIFGA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{σ (j)})^{\frac{1}{n}} = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{j})^{\frac{1}{n}} \\ = ⋃_{a_{j}^{i} \in {\overset{ˇ}{h}}_{j}} {(s_{\prod_{j = 1}^{n} \frac{1}{n} θ_{σ (j)}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (\prod_{j = 1}^{n} (μ_{jk}^{it})^{\frac{1}{n}}, 1 - \prod_{j = 1}^{n} (1 - ν_{jk}^{it})^{\frac{1}{n}}))} . \end{matrix}$

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy averaging (GLHIFA) operator as follows $\begin{matrix} GLHIFA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = (\oplus_{j = 1}^{n} \frac{1}{n} ({\overset{ˇ}{h}}_{σ (j)})^{λ})^{1 / λ} = (\oplus_{j = 1}^{n} \frac{1}{n} ({\overset{ˇ}{h}}_{j})^{λ})^{1 / λ} \\ = ⋃_{a_{j}^{i} \in {\overset{ˇ}{h}}_{j}} {(s_{(\sum_{j = 1}^{n} \frac{1}{n} (θ_{j}^{it})^{λ})^{1 / λ}}, ⋃_{(μ_{kj}^{it}, ν_{kj}^{it}) \in lh (s_{θ_{k}^{it}})} \\ {((1 - \prod_{j = 1}^{n} (1 - (μ_{jk}^{it})^{λ})^{\frac{1}{n}})^{1 / λ}, \\ 1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{jk}^{it})^{λ})^{\frac{1}{n}})^{1 / λ})})} . \end{matrix}$

Remark 2. If m (A) = ∑_{x_i∈A}m ({x_i}) holds for all A ⊆ X with ∑_{x_i∈A}m ({x_i}) =1, then m ({x_σ(i)}) = m (A_σ(i)) - m (A_σ(i-1)) , i = 1, 2,. . . , n . Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy weighted averaging (LHIFWA) operator as follows $\begin{matrix} LHIFWA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \oplus_{j = 1}^{n} m (x_{σ (j)}) {\overset{ˇ}{h}}_{σ (j)} = \oplus_{j = 1}^{n} m (x_{j}) {\overset{ˇ}{h}}_{j} \\ = ⋃_{α_{j}^{k} \in {\overset{ˇ}{h}}_{j}} {(s_{\sum_{j = 1}^{n} m (x_{j}) θ_{j}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (1 - \prod_{j = 1}^{n} (1 - μ_{jk}^{it})^{m (x_{j})}, \prod_{j = 1}^{n} (ν_{jk}^{it})^{m (x_{j})}))} . \end{matrix}$

Then the LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy geometric weighted averaging (LHIFGWA) operator as follows $\begin{matrix} LHIFGWA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{σ (j)})^{m (x_{σ (j)})} = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{j})^{m (x_{j})} \\ = ⋃_{α_{j}^{k} \in {\overset{ˇ}{h}}_{j}} {(s_{\prod_{j = 1}^{n} m (x_{j}) θ_{j}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (\prod_{j = 1}^{n} (μ_{jk}^{it})^{m (x_{j})}, 1 - \prod_{j = 1}^{n} (1 - ν_{jk}^{it})^{m (x_{j})}))} . \end{matrix}$

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy weighted averaging (GLHIFWA) operator as follows $\begin{matrix} GLHIFWA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = (\oplus_{j = 1}^{n} m (x_{σ (j)}) ({\overset{ˇ}{h}}_{σ (j)})^{λ})^{1 / λ} \\ = (\oplus_{j = 1}^{n} m (x_{j}) ({\overset{ˇ}{h}}_{j})^{λ})^{1 / λ} \\ = ⋃_{a_{j}^{i} \in {\overset{ˇ}{h}}_{j}} {(s_{(\sum_{j = 1}^{n} m (x_{j}) (θ_{j}^{it})^{λ})^{1 / λ}}, \\ ⋃_{(μ_{kj}^{it}, ν_{kj}^{it}) \in lh (s_{θ_{k}^{it}})} {((1 - \prod_{j = 1}^{n} (1 - (μ_{jk}^{it})^{λ})^{m (x_{j})})^{1 / λ}, \\ 1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{jk}^{it})^{λ})^{m (x_{j})})^{1 / λ})})} . \end{matrix}$

Remark 3. If $m ({x_{i}}) = \sum_{i = 1}^{| A |} w_{i}$ holds for all A ⊆ X with ∑_{x_i∈A}m ({x_i}) =1, then w_i = m (A_σ(i)) - m (A_σ(i-1)) , i = 1, 2,. . . , n, and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the LHIFCA operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted averaging (LHIFOWA) operator $\begin{matrix} LHIFOWA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) = \oplus_{j = 1}^{n} w_{j} {\overset{ˇ}{h}}_{σ (j)} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\sum_{j = 1}^{n} w_{j} θ_{σ (j)}^{it}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (1 - \prod_{j = 1}^{n} (1 - μ_{σ (j) k}^{it})^{w_{j}}, \prod_{j = 1}^{n} (ν_{σ (j) k}^{it})^{w_{j}}))} . \end{matrix}$

The LHIFCGA operator becomes the linguistic hesitant intuitionistic fuzzy ordered weighted geometric averaging (LHIFOWGA) operator as follows $\begin{matrix} LHIFOWGA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) = \otimes_{j = 1}^{n} ({\overset{ˇ}{h}}_{σ (j)})^{w_{j}} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{\prod_{j = 1}^{n} (θ_{σ (j)}^{it})^{w_{j}}}, ⋃_{(μ_{jk}^{it}, ν_{jk}^{it}) \in lh (s_{θ_{j}^{it}})} \\ (\prod_{j = 1}^{n} (μ_{σ (j) k}^{it})^{w_{j}}, 1 - \prod_{j = 1}^{n} (1 - ν_{σ (j) k}^{it})^{w_{j}}))} . \end{matrix}$

The GLHIFCA operator becomes the generalized linguistic hesitant intuitionistic fuzzy ordered weighted averaging (GLHIFOWA) operator as follows $\begin{matrix} GLHIFOWA ({\overset{ˇ}{h}}_{1}, {\overset{ˇ}{h}}_{2}, . . ., {\overset{ˇ}{h}}_{n}) \\ = (\oplus_{j = 1}^{n} w_{j} ({\overset{ˇ}{h}}_{σ (j)})^{λ})^{1 / λ} \\ = ⋃_{a_{σ (j)}^{i} \in {\overset{ˇ}{h}}_{σ (j)}} {(s_{(\sum_{j = 1}^{n} w_{j} (θ_{σ (j)}^{it})^{λ})^{1 / λ}}, \\ ⋃_{(μ_{kj}^{it}, ν_{kj}^{it}) \in lh (s_{θ_{k}^{it}})} {((1 - \prod_{j = 1}^{n} (1 - (μ_{σ (j) k}^{it})^{λ})^{w_{j}})^{1 / λ}, \\ 1 - (1 - \prod_{j = 1}^{n} (1 - (1 - ν_{σ (j) k}^{it})^{λ})^{w_{j}})^{1 / λ})})} . \end{matrix}$

3 Linguistic hesitant intuitionistic fuzzy linear assignment method

In this section, we develop linguistic hesitant intuitionistic fuzzy linear assignment method based on Choquet integral and distance measures.

Let A = {A₁, A₂,. . . , A_n} be the set of alternatives, C = {C₁, C₂,. . . , C_n} be the set of attributes. Multiple experts from different fields are invited to evaluate alternatives with respect to attributes with linguistic terms and intuitionistic fuzzy memberships. Hence, we can get linguistic hesitant intuitionistic fuzzy decision matrix $H = ({\overset{ˇ}{h}}_{ij})_{m \times n}$ , ${\overset{ˇ}{h}}_{ij} = {a_{ij} ∣ a_{ij} \in {\overset{ˇ}{h}}_{ij}} = {(s_{θ_{ij}}, lh (s_{θ_{ij}})) ∣ a_{ij} \in {\overset{ˇ}{h}}_{ij}}$ , lh (s_{θ
_ij}) = {(μ_ij, ν_ij)}. The concrete steps used in the proposed linear assignment algorithm are as follows.

Algorithm.

Step 1. Construct a linguistic hesitant intuitionistic fuzzy decision matrix $H = ({\overset{ˇ}{h}}_{ij})_{m \times n}$ as in Table 1, where ${\overset{ˇ}{h}}_{ij}$ is the linguistic hesitant intuitionistic fuzzy element given by experts in evaluating alternative A_i with respect to the attribute C_j. Extend decision matrix according to the risk attitude of the decision makers until all the LHIFEs have the same number of LIFEs and each IFLE should have the same number of intuitionistic fuzzy memberships.

Step 2. Determine ρ by using equation $ρ = Π_{i = 1}^{n} (1 + ρ m (x_{i})) - 1)$ and determine the fuzzy measure m on X by using Equation (15).

Step 3. Determine the linguistic hesitant intuitionistic fuzzy positive ideal solution (LHIFPIS) I⁺ and the linguistic hesitant intuitionistic fuzzy negative ideal solution (LHIFPIS) I^- as $\begin{matrix} I^{+} & = & ({\overset{ˇ}{h}}_{1}^{+}, {\overset{ˇ}{h}}_{2}^{+}, . . ., {\overset{ˇ}{h}}_{n}^{+}) \\ = & (max_{i} {\overset{ˇ}{h}}_{i 1}, max_{i} {\overset{ˇ}{h}}_{i 2}, . . ., max_{i} {\overset{ˇ}{h}}_{in}), \end{matrix}$ (19) $\begin{matrix} I^{-} & = & ({\overset{ˇ}{h}}_{1}^{-}, {\overset{ˇ}{h}}_{2}^{-}, . . ., {\overset{ˇ}{h}}_{n}^{-}) \\ = & (min_{i} {\overset{ˇ}{h}}_{i 1}, min_{i} {\overset{ˇ}{h}}_{i 2}, . . ., min_{i} {\overset{ˇ}{h}}_{in}) . \end{matrix}$ (20)

Calculate the signed measure degree s_ij as $s_{ij} = \frac{d ({\overset{ˇ}{h}}_{ij}, {\overset{ˇ}{h}}_{j}^{-})}{d ({\overset{ˇ}{h}}_{ij}, {\overset{ˇ}{h}}_{j}^{-}) + d ({\overset{ˇ}{h}}_{ij}, {\overset{ˇ}{h}}_{j}^{+})}$ (21)

Step 4. Rank the s_ij value of alternative for each attribute C_j.

Step 5. Determine the rank frequency matrix Π, where Π_ij represents the frequency of alternative A_i is ranked as jth attribute-wise ranking in accordance with decreasing order of s_ij. If ρ alternatives are tied with respect to a criterion, there are ρ ! equalized rankings list separately. $Π = [\begin{matrix} Π_{11} & Π_{12} & . . . & Π_{1 m} \\ Π_{21} & Π_{22} & . . . & Π_{2 m} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ Π_{m 1} & Π_{m 2} & . . . & Π_{mm} \end{matrix}] .$

Step 6. Compute weighted rank frequency matrix $Π^{'} = (Π_{ij}^{'})_{m \times m}$ , where $Π_{ij}^{'} = m (x_{k_{1}},$ x_{k
₂}, . . . , x_{k
_j}) is the contribution of A_i in the ranking if A_i is assigned to the jth overall rank. The larger $Π_{ij}^{'}$ , the greater concordance of assigning A_i to the jth overall rank.

Step 7. Define the permutation matrix P_m×m and set up the following linear assignment model (M-1) $\begin{matrix} (M - 1) & \max & \sum_{i = 1}^{m} \sum_{j = 1}^{m} Π_{ij}^{'} P_{ij} \\ s . t . & \sum_{j = 1}^{m} P_{ij} = 1, i = 1, 2, . . ., m, \\ \sum_{i = 1}^{m} P_{ij} = 1, j = 1, 2, . . ., m, \\ P_{ij} = 0 for each i and k . \end{matrix}$

Step 8. Solve the above linear programming model by using the Simplex method to get the optimal permutation matrix P′.

Step 9. Compute the optimal ranking of alternatives by AP′ and select the optimal alternative.

The new algorithm has the following advantages. First, the experts evaluate alternatives with respect to the attributes by linguistic terms and intuitionistic fuzzy memberships. The fuzzy nature of human thinking and hesitation existing in evaluation process has been properly modeled. Experts can refuse to give any evaluation values if they are not familiar with the attributes to assure the results are scientific and reasonable. Second, correlation of attributes is considered by using Choquet integral. Third, linear assignment method has been used to rank alternatives to avoid the subjective affect.

4 An illustrate example

In this section, an example adapted from [43] to illustrate the feasibility and practical advantages of the new algorithm.

In most metropolis, more and more congestions appear since transportation demand increases rapidly due to the economic development and urbanization. Many methods have been presented, such as traffic ban, widen roads, encouraging public transport, etc. Like that in other metropolis, the subway has become an inevitable choice for Xi’an in construction of urban public transport since it is timely, comfortable. It can also avoid road congestion, parking difficulty, etc. There are various types of risk in the construction of subway. If accidents happen during construction, the project may suffer from serious losses and social effects. Suppose that there is project need to be constructed. Multiple experts from different fields have been invited to evaluate risks of the project. They mainly consider the following four types of risk in selecting alternative: C₁-environmental risk, C₂-policy risk, C₃-financing risk, C₄-technical risk. After pre-evaluation, five alternatives are left for further evaluation.

Step 1. Experts evaluate alternatives with respect to attributes with linguistic terms and intuitionistic fuzzy memberships and the linguistic hesitant intuitionistic fuzzy decision matrix $H = ({\overset{ˇ}{h}}_{ij})_{5 \times 4}$ is formed as in Table 2. Assume experts are risk-averse. The smallest intuitionistic fuzzy value (0.5, 0.4) and the smallest LIFE (s₂,(0.5,0.4)) are added until all the LHIFEs have the same number of LIFEs and all the LIFEs have the same number of intuitionistic fuzzy values. The extended decision matrix $H^{'} = ({\overset{ˇ}{h}}_{ij}^{'})_{5 \times 4}$ is shown in Table 3.

Step 2. Assume that m (C₁) =0.3, m (C₂) =0.25, m (C₃) =0.2, m (C₄) =0.35. By using the Equation (14), we can get ρ = -0.2330. Then wecan get m (C₁) =0.3, m (C₂) =0.25, m (C₃) =0.2, m (C₄) = 0.35, m ({C₁, C₂}) =0.5325, m ({C₁, C₃}) =0.4860, m ({C₁, C₄}) =0.6255, m ({C₂, C₃}) =0.4383, m ({C₂, C₄}) =0.5796, m ({C₃, C₄}) =0.5337, m ({C₁, C₂, C₃}) = 0.7077, m ({C₁, C₂, C₄}) =0.8391, m ({C₁, C₃, C₄}) =0.7964, m ({C₂, C₃, C₄}) =0.7526, m ({C₁, C₂, C₃, C₄}) =1.0.

Step 3. The LHIFPIS ${\overset{ˇ}{h}}^{+}$ and the LHIFNIS ${\overset{ˇ}{h}}^{-}$ can be determined as $\begin{matrix} {\overset{ˇ}{h}}^{+} & = & ({(s_{8}, (0.6, 0.2), (0.5, 0.4)), \\ (s_{7}, (0.7, 0.3), (0.5, 0.4))}, \\ {(s_{8}, (0.6, 0.1), (0.7, 0.2)), \\ (s_{2}, (0.5, 0.4), (0.5, 0.4))}, \\ {(s_{6}, (0.6, 0.2), (0.5, 0.4)), \\ (s_{5}, (0.7, 0.3), (0.5, 0.4))}, \\ {(s_{8}, (0.8, 0.2), (0.5, 0.4)), \\ (s_{7}, (0.6, 0.1), (0.5, 0.4))}) . \\ {\overset{ˇ}{h}}^{-} & = & ({(s_{2}, (0.5, 0.4), (0.5, 0.4)), \\ (s_{3}, (0.5, 0.2), (0.5, 0.4))}, \\ {(s_{4}, (0.5, 0.2), (0.5, 0.4)), \\ (s_{2}, (0.5, 0.4), (0.5, 0.4))}, \\ {(s_{2}, (0.6, 0.2), (0.5, 0.3)), \\ (s_{2}, (0.5, 0.4), (0.5, 0.4))}, \\ {(s_{5}, (0.7, 0.3), (0.5, 0.4)), \\ (s_{2}, (0.5, 0.4), (0.5, 0.4))}) . \end{matrix}$

Calculate the individual signed measure by using Equation (21) and the results are shown in Table 4.

Step 4. Rank the individual signed measure degree of each alternative in the same attribute to get

For C₁, we have s₃₁ > s₅₁ > s₄₁ > s₁₁ > s₂₁.

For C₂, we have s₂₂ > s₁₂ > s₃₂ > s₄₂ > s₅₂.

For C₃, we have s₅₃ > s₃₃ > s₂₃ > s₄₃ > s₁₃.

For C₄, we have s₄₄ > s₃₄ > s₅₄ > s₁₄ > s₂₄.

Step 5. The rank frequency matrix can be determined as in Table 5.

Step 6. Calculate the weighted individual signed measure and the results are shown in Table 6.

Step 7. The permutation matrix can be defined as P = (p_ij) _5×5 and the following linear programming model can be set up as following model (M-2). $\begin{matrix} (M - 2) & \max & 0.2500 p_{12} + 0.6255 p_{14} + 0.2200 p_{15} \\ + 0.2500 p_{21} + 0.2000 p_{23} + 0.6255 p_{25} \\ + 0.3000 p_{31} + 0.5337 p_{32} + 0.2500 p_{33} \\ + 0.3500 p_{41} + 0.3000 p_{43} + 0.4860 p_{44} \\ + 0.2000 p_{51} + 0.3000 p_{52} + 0.3500 p_{53} \\ + 0.2500 p_{55} \\ s . t . & p_{11} + p_{12} + p_{13} + p_{14} + p_{15} = 1, \\ p_{21} + p_{22} + p_{23} + p_{24} + p_{25} = 1, \\ p_{31} + p_{32} + p_{33} + p_{34} + p_{35} = 1, \\ p_{41} + p_{42} + p_{43} + p_{44} + p_{45} = 1, \\ p_{51} + p_{52} + p_{53} + p_{54} + p_{55} = 1, \\ p_{11} + p_{21} + p_{31} + p_{41} + p_{51} = 1, \\ p_{12} + p_{22} + p_{32} + p_{42} + p_{52} = 1, \\ p_{13} + p_{23} + p_{33} + p_{43} + p_{53} = 1, \\ p_{14} + p_{24} + p_{34} + p_{44} + p_{54} = 1, \\ p_{15} + p_{25} + p_{35} + p_{45} + p_{55} = 1, \\ p_{ij} = 0 or 1 for i, j = 1, 2, . . ., 5 . \end{matrix}$

Step 8. Use the Simplex method to solve the linear assignment model (M-2) to get the optimal permutation matrix P′ as follows $P^{'} = [\begin{matrix} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix}] .$

Step 9. Calculate the alternatives’ ranking by using the permutation matrix P′ ${AP}^{'} = (A_{1}, A_{2}, A_{3}, A_{4}, A_{5}) [\begin{matrix} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \end{matrix}] .$ By using matrix multiplication, we can get AP′ = (A₄, A₃, A₅, A₁, A₂) . The ranking of the alternatives can be get as A₄ > A₃ > A₅ > A₁ > A₂ and the optimal alternative is A₄.

If we don’t consider the correlation of attributes, we can set up the linear programming model by using rank frequency matrix in Table 5. The optimal permutation matrix P″ can be got as $P^{″} = [\begin{matrix} 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix}] .$ Then the ranking of the alternatives can be got as A₅ > A₃ > A₄ > A₁ > A₂ and the optimal alternative is A₅, which is different from the proposed method. Since the contribution of different attributes to the ranking of the alternatives is different, more accurate results can be got by considering weights of attributes.

In order to illustrate the practical advantages of the new algorithm, we compare it with some other methods. We first consider the correlated TOPSIS method by using Choquet integral, which is denoted as M₁. If the TOPSIS method considering correlation is used to rank alternatives, the first three steps are the same as the proposed algorithm. Then we calculate the distances of each alternative evaluation values to LHIFPIS ${\overset{ˇ}{h}}^{+}$ and the LHIFNIS ${\overset{ˇ}{h}}^{-}$ by using the distance measures as follows. $\begin{matrix} d_{i}^{+} & = & w_{i 1} d ({\overset{ˇ}{h}}_{i 1}^{'}, {\overset{ˇ}{h}}_{1}^{+}) + w_{i 2} d ({\overset{ˇ}{h}}_{i 2}^{'}, {\overset{ˇ}{h}}_{2}^{+}) + . . . \\ + w_{i 4} d ({\overset{ˇ}{h}}_{i 4}^{'}, {\overset{ˇ}{h}}_{4}^{+}) . \\ d_{i}^{-} & = & w_{i 1} d ({\overset{ˇ}{h}}_{i 1}^{'}, {\overset{ˇ}{h}}_{1}^{-}) + w_{i 2} d ({\overset{ˇ}{h}}_{i 2}^{'}, {\overset{ˇ}{h}}_{2}^{-}) + . . . \\ + w_{i 4} d ({\overset{ˇ}{h}}_{i 4}^{'}, {\overset{ˇ}{h}}_{4}^{-}) . \end{matrix}$ Here, we take Hamming distance. For example, in order to calculate $d_{1}^{+}$ , we need to determine attribute weights first. Since ${\overset{ˇ}{h}}_{12} > {\overset{ˇ}{h}}_{14} > {\overset{ˇ}{h}}_{11} > {\overset{ˇ}{h}}_{13}$ , we can calculate the weights as w₁₂ = m (C₂) =0.25, w₁₄ = m (C₂, C₄) - m (C₂) =0.5796 - 0.2500 = 0.3296, w₁₁ = m (C₁, C₂, C₄) - m (C₂, C₄) =0.2595, w₁₃ = m (C₁, C₂, C₄, C₄) - m (C₁, C₂, C₄) = 1 - 0.8391 = 0.1609 . The weight vector is got as w₁ = (0.2595, 0.2500, 0.1609, 0.3296). Other weights can be calculated similarly. $\begin{matrix} d ({\overset{ˇ}{h}}_{11}^{'}, {\overset{ˇ}{h}}_{1}^{+}) \\ = (\frac{1}{3} ((| (3 - 8) / 9 | + (| 0.6 - 0.7 | + | 0.2 - 0.2 | \\ + | 0.5 - 0.6 | + | 0.4 - 0.3 |) / 2) \\ + \frac{1}{3} ((| (7 - 2) / 9 | + | 0.7 - 0.5 | + | 0.3 - 0.4 | \\ + | 0.5 - 0.5 | + | 0.4 - 0.4 |) / 2)) / 2 \\ = 0.2269 . \end{matrix}$ Similarly, we can get $d ({\overset{ˇ}{h}}_{12}^{'}, {\overset{ˇ}{h}}_{2}^{+}) = 0.1593, d ({\overset{ˇ}{h}}_{13}^{'}, {\overset{ˇ}{h}}_{3}^{+})$ = 0.1630, $d ({\overset{ˇ}{h}}_{14}^{'}, {\overset{ˇ}{h}}_{4}^{+}) = 0.1713$ . We cancalculate $d_{1}^{+} = 0.2595 * 0.2269 + 0.2500 * 0.1593 + 0.1609 * 0.1630 + 0.3296 * 0.1713 = 0.1814 .$ Other distances can be calculated similarly and the results are shown in Table 7. Then we calculate the relative closeness by using the following equation ${CC}_{i} = \frac{d_{i}^{-}}{d_{i}^{-} + d_{i}^{+}} .$ Rank the alternatives according to the ranking of CC_i to get A₄ > A₅ > A₃ > A₁ > A₂ and select the optimal alternative as A₄. We also consider the method of TOPSIS, which is denoted as M₂. The weight vector (0.3000, 0.2036, 0.1709, 0.3255) of A₅ in the proposed algorithm is used in calculating the weighted distances. We can calculate the weighted distance measures and the relative coefficients and the results are shown in Table 7. The alternatives can be ranked as A₅ > A₄ > A₃ > A₁ > A₂ and the optimal alternative is A₅.

If we use aggregation operators introduced in Section 3, we use the decision matrix H and need not to extend the matrix. Step 2 is the same. Then we can determine the attribute weights for each alternative as the same as in M₁. Then we can calculate the collective evaluation values of A_i by using aggregation operators and the results are omitted here for space limit. Then we can calculate the scores of ${\overset{ˇ}{h}}_{i}$ by using Equation (6) to get $S ({\overset{ˇ}{h}}_{1}) = 0.3587, S ({\overset{ˇ}{h}}_{2}) = 0.3206, S ({\overset{ˇ}{h}}_{3}) = 0.3774, S ({\overset{ˇ}{h}}_{4}) = 0.4070, S ({\overset{ˇ}{h}}_{5}) = 0.3496 .$ According to the ranking of $S ({\overset{ˇ}{h}}_{i})$ , we can get ${\overset{ˇ}{h}}_{4} > {\overset{ˇ}{h}}_{3} > {\overset{ˇ}{h}}_{1} > {\overset{ˇ}{h}}_{5} > {\overset{ˇ}{h}}_{2}$ . Alternatives can be ranked as A₄ > A₃ > A₁ > A₅ > A₂. We also can use other aggregation operators and the results are shown in Table 8. From the results we can see that, A₄ is the optimal alternative in all the aggregation operators, which is the same as that in the proposed method. There is slightly difference in alternative ranking for different λ, which can be seen as the risk-attitude of the decision makers. With increasing of λ, the largest evaluation value plays more and more important role in aggregation process and decision makers are more risk-seeking.

5 Conclusions

Linear assignment method has the advantages of avoiding the effect of experts’ subjectivity. Meanwhile, the computation complexity can be significantly reduced, more reasonable and robust ranking results can be provided more importantly. In this paper, we develop linguistic hesitant intuitionistic fuzzy linear assignment considering correlation by using Choquet integral. The current researches are improved from perspectives such as: the description of fuzziness and uncertainty existing in decision process, the correlation of attributes, the objectivity and accuracy of the ranking results and the overall computation efficiency. The linguistic arguments with intuitionistic fuzzy memberships can properly model the fuzzy nature of human thinking. The proposed procedure is a useful approach to deal with multiple attribute decision making problems with correlated and hesitant fuzzy information.

Footnotes

Acknowledgments

The authors would like to express appreciation to the anonymous reviewers for their very helpful comments on improving the paper. This work is partly supported by National Natural Science Foundation of China (No. 11401457, 61403298), Postdoctoral Science Foundation of China (2015M582624), Shaanxi Province Natural Science Fund of China (Nos. 2014JQ1019, 2014JM1010), Shaanxi Provincial Education Department fund of China (No. 2013JK0565).

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