Intuitionistic fuzzy linguistic soft sets and their application in multi-attribute decision-making

Abstract

This paper introduced a new measurement method called intuitionistic fuzzy linguistic soft set (IFLSS) to represent individual points of view for decision-makers to evaluate attributes of different alternatives. IFLSS combines an intuitionistic fuzzy set and linguistic soft set to help users utilize a set of linguistic variables to perceive an aspect from different points of view. This method utilizes intuitionistic fuzzy numbers to measure the matching grades among the affirming, objection, and hesitation of linguistic variables which are more diversified and pragmatic than the traditional intuitionistic fuzzy sets and two-dimensional uncertainty linguistic variables. A generalized weighted aggregation operator for IFLSS was also developed, and two methods were proposed to settle relevant decision-making problems. At last, an example in numbers was given for illustrating the application of IFLSS in decision-making, and the practicality and effectiveness of the proposed methods were verified.

Keywords

Intuitionistic fuzzy set intuitionistic fuzzy linguistic soft set evaluation method multiple attribute decision making

1 Introduction

Multi-attribute decision-making (MADM) provides a quantitative approach to assist decision-makers in rationally considering all attributes for each alternative [1]. Many MADM methods have been developed to solve real-life issues [2 , 14]. During the MADM process, decision-makers frequently use qualitative and/or quantitative measures to evaluate the performance of each attribute of each alternative and consider the relative importance of each attribute in terms of the overall goal. In order to represent the uncertainty of human thinking, Zadeh proposed the fuzzy set (FS) theory [15]. Since then, many researchers have used FSs for multiple times in many different fields to help with uncertainty in decision making situations [16 , 19]. However, FSs are unable to express the objections or hesitant opinions of users. Therefore, Atanassov extended Zadeh’s traditional FS theory, and proposed the intuitionistic fuzzy sets theory (IFS) [12]. IFSs simultaneously represent the membership degree, non-membership degree, and degree of hesitation. They have shown their definite advantages over traditional FSs theory in handling with uncertainty and fuzziness. Since their development, IFSs have progressed to deal with valued intuitionistic fuzzy sets (IVIFSs) [13], and hesitant fuzzy sets (HFSs) [24] among others, and are widely used in decision-making situations.

In some cases, linguistic evaluations are preferred by those decision-makers in weighing attributes as in extremely bad, bad, ordinary, well and extremely well etc. By which, Zadeh (1975) proposed linguistic variables in fuzzy reasoning [17]. Based on the definition of linguistic variable, Zhu et al. proposed a method of two-dimensional linguistic (2DL) evaluation to assist with linguistic fuzzy decision problems [25]. In a two-dimensional linguistic evaluation method, users can apply one linguistic variable to evaluate an attribute, and apply another to evaluate the subjective measurement of their given results in the reliability. Now, 2DL has progressed into a two-dimensional intuitionistic uncertain linguistic set (2DIULS), which combines interval linguistic evaluation and intuitionistic fuzzy-valued subjective reliability. 2DIULSs have been widely used in multi-attribute decision-making problems [4 , 26]. Both the 2DL and 2DIULS evaluation methods help individuals simultaneously express their opinions and represent the reliability of their opinions.

When a decision-maker measures an attribute of an alternative, he may provide multiple measurement results from many points of view and it is hard to decipher the final result using only one dimension of linguistic evaluation (the other dimension represents the reliability of the 2DL method). For instance, when measuring the position attribute of one of variable by the view of public traffic, we will have the good answer; however, by viewing the number of consumers, we will have the bad answer. Obviously, at this time only one dimension of linguistic evaluation is not sufficient. Hence, a linguistic soft set was proposed to overcome such difficulties. Molodtsov proposed the soft set theory in 1999 [5]. By the soft set theory, users could choose different forms and parameters by different needs which are more useful in evaluating the imperfection and flexibility from a more objective view. Many studies have conducted multiple useful works by compiling soft set theory with the views from fuzzy sets, rough sets, and intuitionistic fuzzy sets together in which soft sets are represented in the attributes of the object [6 , 10, 21].

In this paper, an intuitionistic fuzzy linguistic soft set (IFLSS) by both linguistic soft set and intuitionistic fuzzy set was proposed in which the parameters of linguistic soft set were composed of assessment linguistic variables. Decision-makers could choose some of the linguistic variables to measure the attributes of the alternative which are introduced from multiple points of view as needed and the membership degree or non-membership degree of different linguistic measurements. IFLSS extends the functions of two-dimensional linguistic or two-dimensional intuitionistic uncertain linguistic methods both in usefulness and precision which can precisely show the evaluations of decision-makers regarding objects.

The major attributes in the proposed method are as follow:

An IFLSS form of evaluation was developed, which integrated a linguistic variable as an measurement, intuitionistic fuzzy measurement as the membership of a matching linguistic variable, and soft set structure which allows users could directly measure an object from multiple points of view if necessary. Consequently, it is much more useful and could be viewed as a measurement model.

Users have to extinguish the degree of different attributes of each alternatives in traditional multi-attribute decision-making cases. By using an IFLSS form, users are not needed to answer all different questions. So, not only is it quick for multiple perspective measurements, but also for numerous decision-makings with insufficient information.

An IFLSSGWA operator was developed to add multiple IFLSSs, and its fastness was validated. The IFLSSGWA user keeps most of the attributes of each IFLSS.

The following is a reminder of this paper in organization. In Section 2, the idea of IFLSS is introduced, basic operational rules are defined, characteristics of IFLSS are discussed, and a fast adding operator is explained on the basement of a traditional generalized weighted adding method and the operational rules from IFLSS. In Section 3, the operators are applied to deal with multiple-attribute decision-making problems in which the attributes were taken the form of IFLSSs, and the specific steps are detailed. In Section 4, the decision-making flow paths are illustrated on the basement of the proposed methods. Section 5 completes this paper with concluding remarks.

2 Intuitionistic fuzzy linguistic soft set

2.1 Definition of intuitionistic fuzzy linguistic soft set

Given that S = {s₀, s₁, . . . , s_l-1} is a finite and fully ordered discrete term set and l is an odd number and is equal to 3, 5, 7, 9, etc. For example, if l = 7, it can be represented as follows [22]:

S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆} = {extremely little, little, below ordinary, ordinary, above ordinary, great, extremely great}.

The element s_i and its subscript i in a linguistic set strictly monotonically increase [7]. To preserve all of the given information, the discrete linguistic label S = {s₀, s₁, . . . , s_l-1} is extended to a continuous linguistic label $\bar{S} = {s_{α} | α \in R}$ where s_α and its subscript α also strictly monotonically increase [27].

Definition 1. U is the initial universe set and S = {s₀, s₁, . . . , s_l-1} is a linguistic assessment set. IFV(U) denotes the set of all intuitionistic fuzzy subsets of U. If there exists a nonempty set H, H ⊆ S. Then (W,H) is a pair called an intuitionistic fuzzy linguistic soft set (IFLSS) over U, where W is a mapping given by W : H → IFV (U). That is, for ∀s_a ∈ H:

$\begin{matrix} W (s_{a}) & = & {〈 x, μ_{W (s_{a})} (x), v_{W (s_{a})} (x) 〉 | x \in U} \\ \in IFV (U) \end{matrix}$ (1) where μ_{W(s_a)} (x) ∈ [0, 1] and v_{W(s_a)} (x) ∈ [0, 1] represent the membership and non-membership degree of x (∈ U), respectively, belonging to linguistic assessment s_a (∈ H), μ_{W(s_a)} (x) + v_{W(s_a)} (x) ≤1.

For instance, when l = 7, there is S = {s₀, s₁, s₂, s₃, s₄, s₅, s₆}, as aforementioned. Then, A = {s₂, s₅} represents the linguistic evaluation results of “below fair” and “good” from two different perspectives for an object, and K (s₂) =〈 0.6, 0.1 〉 and K (s₅) =〈 0.8, 0.1 〉 are the corresponding intuitionistic fuzzy valued evaluations for the reliabilities of the two linguistic evaluations.

2.2 Definitions and theorems of IFLSS

Definition 2. If (P, H) and (Q, G) are two IFLSSs over U, their intersection and union rules can be defined as follow:

(P, H) ∩ (Q, G) = (X, F), where F = H ∩ G, and for ∀s_c ∈ F,

$\begin{matrix} X (s_{c}) & = & P (s_{c}) \cap Q (s_{c}) \\ = & 〈 μ_{P (s_{c})} μ_{Q (s_{c})}, v_{P (s_{c})} + v_{Q (s_{c})} \\ - v_{P (s_{c})} v_{Q (s_{c})} 〉 \end{matrix}$ (2)

(P, H) ∪ (Q, G) = (Y, F), where F = H ∪ G, and for ∀s_c ∈ F,

$Y (s_{c}) = {\begin{cases} P (s_{c}) = 〈 u_{P (s_{c})}, v_{P (s_{c})} 〉, \begin{matrix} \end{matrix} s_{c} \in P - Q \\ P (s_{c}) \cup Q (s_{c}) = 〈 u_{P (s_{c})} + μ_{Q (s_{c})} - μ_{P (s_{c})} μ_{Q (s_{c})}, v_{P (s_{c})} v_{Q (s_{c})} 〉 \begin{matrix} , & s_{c} \in P \cap Q \end{matrix} \\ Q (s_{c}) = 〈 u_{Q (s_{c})}, v_{Q (s_{c})} 〉, \begin{matrix} \end{matrix} s_{c} \in Q - P \end{cases}$ (3)

Definition 3. If (P, H) is an IFLSS over U, (P, H) ^C = (P, H^C) can be regarded as the complement of (P, H) where for ∀s_a ∈ H and $s_{a}^{c} = s_{l - 1 - a}$ , l is the number of linguistic variables, as aforementioned. For $\forall s_{a}^{C} \in H^{C}$ there is the following: $μ_{H (s_{a}^{C})} = μ_{H (s_{a})}, v_{H (s_{a}^{C})} = v_{H (s_{a})}$ (4)

Definition 4. If (P, H), (Q, G) are two IFLSSs over U, when λ > 0, their operational rules can be represented as follow: $(1) (P, H) \oplus (Q, G) = (P, H) \cup (Q, G)$ (5) $(2) (P, H) \otimes (Q, G) = (P, H) \cap (Q, G)$ (6)

Suppose λ (P, H) = (X, F), then F = H, and for

$\begin{matrix} \forall s_{c} \in F, X (s_{c}) = λ P (s_{c}) \\ = 〈 1 - {(1 - μ_{P (s_{c})})}^{λ}, {(v_{P (s_{c})})}^{λ} 〉 \end{matrix}$ (7)

Suppose (P, H) ^λ = (X, F), then F = H, and for

$\begin{matrix} \forall s_{c} \in F, X (s_{c}) = {(P (s_{c}))}^{λ} \\ = 〈 {(u_{P (s_{c})})}^{λ}, 1 - {(1 - v_{P (s_{c})})}^{λ} 〉 \end{matrix}$ (8)

Theorem 1.For any two IFLSSs (P, H), (Q, G) over U, the following calculating results can be proved: $(1) (P, H) \oplus (Q, G) = (Q, G) \oplus (P, H)$ (9) $(2) (P, H) \otimes (Q, G) = (Q, G) \otimes (P, H)$ (10) $\begin{matrix} (3) & λ ((P, H) \oplus (Q, G)) \\ = λ (P, H) \oplus λ (Q, G), λ > 0 \end{matrix}$ (11) $\begin{matrix} (4) & λ_{1} (P, H) \oplus λ_{2} (P, H) \\ = (λ_{1} + λ_{2}) (P, H), λ_{1}, λ_{2} > 0 \end{matrix}$ (12) $\begin{matrix} (5) & (P, H)^{λ_{1}} \otimes (P, H)^{λ_{2}} \\ = (P, H)^{λ_{1} + λ_{2}}, λ_{1}, λ_{2} > 0 \end{matrix}$ (13) $\begin{matrix} (6) & (P, H)^{λ} \otimes (Q, G)^{λ} \\ = {((P, H) \otimes (Q, G))}^{λ}, λ > 0 \end{matrix}$ (14)

Proof.

(1) From Formulas (2) and (3), we can see that Rules (1) and (2) are obvious.

(2) Given Rule (3), if (P, H) ⊕ (Q, G) = (X, F), then F = H ∪ G for ∀s_c ∈ F, and the left of Rule (3) is as follows: $λ X (s_{c}) = {\begin{matrix} λ P (s_{c}), & s_{c} \in H - G \\ λ (P (s_{c}) \cup Q (s_{c})), & s_{c} \in H \cap G \\ λ Q (s_{c}), & s_{c} \in G - H \end{matrix}$

If (λ (P, H)) ⊕ (λ (Q, G)) = (Y, D), according to Formula (3), D = H ∪ G = F, so (Y, D) can be expressed as (Y, F). For ∀s_c ∈ F, the right of Rule (3) is as follows: $Y (s_{c}) = {\begin{matrix} λ P (s_{c}), & s_{c} \in H - G \\ (λ P (s_{c})) \cup (λ Q (s_{c})), & s_{c} \in H \cap G \\ λ Q (s_{c}), & s_{c} \in G - H \end{matrix}$ $\begin{matrix} ∵ λ (P (s_{c}) \cup Q (s_{c})) & = & λ 〈 1 - (1 - μ_{P (s_{c})}) (1 - μ_{Q (s_{c})}), v_{P (s_{c})} v_{Q (s_{c})} 〉 \\ = & 〈 1 - {(1 - (1 - μ_{P (s_{c})}) (1 - μ_{Q (s_{c})}))}^{λ}, {(v_{P (s_{c})} v_{Q (s_{c})})}^{λ} 〉 \\ = & 〈 1 - {(1 - μ_{P (s_{c})})}^{λ} {(1 - μ_{Q (s_{c})})}^{λ}, {(v_{P (s_{c})} v_{Q (s_{c})})}^{λ} 〉 \\ ∵ (λ P (s_{c})) \cup (λ Q (s_{c})) & = & 〈 1 - {(1 - μ_{P (s_{c})})}^{λ}, {(v_{P (s_{c})})}^{λ} 〉 \cup 〈 1 - {(1 - μ_{Q (s_{c})})}^{λ}, {(v_{G (s_{c})})}^{λ} 〉 \\ = & 〈 1 - (1 - {(1 - μ_{P (s_{c})})}^{λ}) (1 - {(1 - μ_{Q (s_{c})})}^{λ}), {(v_{P (s_{c})})}^{λ} {(v_{G (s_{c})})}^{λ} 〉 \\ = & 〈 1 - {(1 - μ_{P (s_{c})})}^{λ} {(1 - μ_{Q (s_{c})})}^{λ}, {(v_{P (s_{c})} v_{Q (s_{c})})}^{λ} 〉 \\ ∴ λ X (s_{c}) = Y (s_{c}) \end{matrix}$

Therefore, Rule (3) was proven true.

(3) From the results above we can see that Rule (3), Rules (4), (5), and (6) can also be proved.

Definition 5. If (P, H) is an IFLSS over U for ∀s_a ∈ H, then the expected value of 〈s_a, (μ_{P(s_a)}, v_{P(s_a)})〉 can be expressed as follows: $E_{〈 s_{a}, (μ_{P (s_{a})}, v_{P (s_{a})}) 〉} = \frac{a \times (μ_{P (s_{a})} + 1 - v_{P (s_{a})})}{2}$

Therefore, the expected value E_(P,H) of (P, H) can be defined as follows:

$\begin{matrix} E_{(P, H)} & = & \frac{\sum_{s_{a} \in H} E_{〈 s_{a}, (μ_{P (s_{a})}, v_{P (s_{a})}) 〉}}{| H |} \\ = & \frac{\sum_{s_{a} \in H} (a \times (μ_{P (s_{a})} + 1 - v_{P (s_{a})}))}{2 | H |} \end{matrix}$ (15)

Definition 6. If (P, H) is an IFLSS over U for ∀s_a ∈ H, then the accuracy function of 〈s_a, (μ_{P(s_a)}, v_{P(s_a)}) 〉 can be expressed as follows: $R_{〈 s_{a}, (μ_{F (s_{a})}, v_{F (s_{a})}) 〉} = \frac{a \times (μ_{P (s_{a})} + v_{P (s_{a})})}{2}$

Therefore, the accuracy value R_(P,H) of (P, H) can be defined as follows:

$\begin{matrix} R_{(P, H)} & = & \frac{\sum_{s_{a} \in H} R_{〈 s_{a}, (μ_{P (s_{a})}, v_{P (s_{a})}) 〉}}{| H |} \\ = & \frac{\sum_{s_{a} \in H} (a \times (μ_{P (s_{a})} + v_{P (s_{a})}))}{2 | H |} \end{matrix}$ (16)

Definition 7. For any two IFLSSs (P, H), (Q, G) over U, the comparison rules are defined as follow:

If E_(P,H) > E_(Q,G), then (P, H) ≻ (Q, G).

If E_(P,H) = E_(Q,G), then

If R_(P,H) > R_(Q,G), then (P, H) ≻ (Q, G).

If R_(P,H) = R_(Q,G), then (P, H) ∼ (Q, G).

For instance, if (P, H) = {〈 s₃, (0.8, 0.1) 〉 , 〈 s₄, (0.6, 0.4) 〉 , 〈 s₆, (0.5, 0.3) 〉}, and (Q, G) ={ 〈 s₄, (0.8, 0.2) 〉 , 〈 s₅, (0.7, 0.2) 〉 }, then the following is true, according to Formulas (15) and (16): $\begin{matrix} E_{(P, H)} \\ = \frac{3 (0.8 + 1 - 0.1) + 4 (0.6 + 1 - 0.4) + 6 (0.5 + 1 - 0.3)}{3 \times 2} \\ = 2.85 \end{matrix}$ $\begin{matrix} E_{(Q, G)} \\ = \frac{4 (0.8 + 1 - 0.2) + 5 (0.7 + 1 - 0.2)}{2 \times 2} \\ = 3.475 \\ R_{(P, H)} \\ = \frac{3 (0.8 + 0.1) + 4 (0.6 + 0.4) + 6 (0.5 + 0.3)}{3 \times 2} \\ \approx 1.92 \\ R_{(Q, G)} \\ = \frac{4 (0.8 + 0.2) + 5 (0.7 + 0.2)}{2 \times 2} \\ = 2.125 \end{matrix}$

In this case, according to above comparison rules, (Q, G) is superior to (P, H).

2.3 Generalized weighted summation operator based on IFLSS

Definition 8. From traditional generalized weighted summation methods [23], the generalized weighed summation operator for IFLSSs were defined. Let (F_i, A_i) (i = 1, 2, . . . , n) be an aggregation of IFLSSs, where F_i : A_i → IFV (U), and for ∀s_{a
_i} ∈ A_i, F_i (s_{a
_i}) ={ 〈 x, (μ_{F_i(s_{a
_i})} (x) , v_{F_i(s_{a
_i})} (x)) 〉 |x ∈ U }. Let W = (w₁, w₂, ⋯ , w_n) ^T be the weighting vector of (F_i, A_i) with w_i ≥ 0 and $\sum_{i = 1}^{n} w_{i} = 1$ . Their generalized weighted aggregation IFLSSGWA Ωⁿ → Ω can be defined as follows:

$\begin{matrix} (H, X) & = & IFLSS GWA ((F_{1}, A_{1}), (F_{2}, A_{2}), \\ . . ., (F_{n}, A_{n})) \end{matrix}$ (17) where $X = ⋃_{i = 1}^{n} A_{i}$ , and for ∀s_x ∈ X,

$\begin{matrix} H (s_{x}) & = & {(\sum_{i = 1, s_{x} \in S_{i}}^{n} \frac{w_{i}}{| S_{i} |} {(F_{i} (s_{x}))}^{λ})}^{1 / - λ} \\ = & {(\sum_{i = 1, s_{x} \in S_{i}}^{n} \frac{w_{i}}{| S_{i} |} (〈 (μ_{F_{i} (s_{x})})^{λ}, 1 - (1 - v_{F_{i} (s_{x})})^{λ} 〉))}^{1 / - λ} \\ = & {(\sum_{i = 1, s_{x} \in S_{i}}^{n} (〈 1 - {(1 - (μ_{F_{i} (s_{x})})^{λ})}^{\frac{w_{i}}{| S_{i} |}}, {(1 - (1 - v_{F_{i} (x)})^{λ})}^{\frac{w_{i}}{| S_{i} |}} 〉))}^{1 / - λ} \\ = & {(〈 1 - \prod_{i = 1, s_{x} \in S_{i}}^{n} {(1 - (μ_{F_{i} (s_{x})})^{λ})}^{\frac{w_{i}}{| S_{i} |}}, \prod_{i = 1, s_{x} \in S_{i}}^{n} {(1 - (1 - v_{F_{i} (s_{x})})^{λ})}^{\frac{w_{i}}{| S_{i} |}} 〉)}^{1 / - λ} \\ = & 〈 {(1 - \prod_{i = 1, s_{x} \in S_{i}}^{n} {(1 - (μ_{F_{i} (s_{x})})^{λ})}^{\frac{w_{i}}{| S_{i} |}})}^{1 / - λ}, 1 - {(1 - \prod_{i = 1, s_{x} \in S_{i}}^{n} {(1 - (1 - v_{F_{i} (s_{x})})^{λ})}^{\frac{w_{i}}{| S_{i} |}})}^{1 / - λ} 〉 \end{matrix}$ (18)

Theorem 2.The aggregation result utilizing operator $H (s_{x}) = {(\sum_{i = 1, s_{x} \in S_{i}}^{n} \frac{w_{i}}{| S_{i} |} {(F_{i} (s_{x}))}^{λ})}^{1 / - λ}$ can preserve the maximum information of original IFLSSs because the square sum of the difference between the aggregation value and the original values is the minimum under the level ofλpower.

Proof. The optimization function was constructed as follows: $min J = \sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |} {({(H (s_{x}))}^{λ} - {(F_{i} (s_{x}))}^{λ})}^{2})$

If $\frac{dJ}{d (H (s_{x}))} = 0$ , then the following is true: $\begin{matrix} \frac{dJ}{d (H (s_{x}))} \\ = 2 λ \sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |} ({(H (s_{x}))}^{λ} - {(F_{i} (s_{x}))}^{λ}) {(H (s_{x}))}^{λ - 1}) \\ = 0 \end{matrix}$

So, $H (s_{x}) = \sum_{i = 1, s_{x} \in S_{i}}^{n} {(\frac{w_{i}}{| S_{i} |} {(F_{i} (s_{x}))}^{λ})}^{1 / - λ}$ is the only solution to achieve the extreme value.

Meanwhile, when $\sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |}) = 1$ , the following is true:

$\begin{matrix} ∵ \frac{d^{2} J}{d^{2} (H (s_{x}))} = 2 {(H (s_{x}))}^{λ - 2} \sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |} ((2 λ^{2} - λ) {(H (s_{x}))}^{λ} - (λ^{2} - λ) {(F_{i} (s_{x}))}^{λ})) \\ = 2 {(H (s_{x}))}^{λ - 2} \sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |} ((2 λ^{2} - λ) ({(H (s_{x}))}^{λ} - {(F_{i} (s_{x}))}^{λ}) + λ^{2} {(F_{i} (s_{x}))}^{λ})) \\ = 2 {(H (s_{x}))}^{λ - 2} λ^{2} \sum_{i = 1, s_{x} \in S_{i}}^{n} (\frac{w_{i}}{| S_{i} |} {(F_{i} (s_{x}))}^{λ}) > 0 \end{matrix}$

So, the extreme value is the minimum value. Theorem 2 was proved.

When λ → 0, the summation is similar to weighted geometric summation operator (WG), as follows:

$\begin{matrix} H (s_{x}) \\ = \prod_{i = 1, s_{x} \in S_{i}}^{n} {(F_{i} (s_{x}))}^{\frac{w_{i}}{| S_{i} |}} \\ = \prod_{i = 1, s_{x} \in S_{i}}^{n} (〈 (μ_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}}, 1 - (1 - v_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}} 〉) \\ = 〈 \prod_{i = 1, s_{x} \in S_{i}}^{n} (μ_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}}, 1 - \prod_{i = 1}^{n} (1 - v_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}} 〉 \end{matrix}$ (19)

When λ = 1, the summation is similar to a weighted arithmetic summation operators (WA), as follows:

$\begin{matrix} H (s_{x}) & = & \sum_{i = 1, s_{x} \in S_{i}}^{n} \frac{w_{i}}{| S_{i} |} (F_{i} (s_{x})) \\ = & \sum_{i = 1, s_{x} \in S_{i}}^{n} (〈 1 - {(1 - (μ_{F_{i} (s_{x})}))}^{\frac{w_{i}}{| S_{i} |}}, (v_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}} 〉) \\ = & 〈 1 - \prod_{i = 1, s_{x} \in S_{i}}^{n} {(1 - (μ_{F_{i} (s_{x})}))}^{\frac{w_{i}}{| S_{i} |}}, \prod_{i = 1, s_{x} \in S_{i}}^{n} (v_{F_{i} (s_{x})})^{\frac{w_{i}}{| S_{i} |}} 〉 \end{matrix}$ (20)

3 Two approaches for multi-attribute decision-making method based on IFLSS

Consider a multiple-decision-making problem with intuitionistic fuzzy linguistic soft set information, and let A = {A₁, A₂, . . . , A_m} be the set of alternatives and C = {C₁, C₂, . . . , C_n} be the set of attributes; w = (w₁, w₂, ⋯ , w_n) ^T is the weighting vector of the attribute C = {C₁, C₂, . . . , C_n} where w_j ∈ [0, 1] (j = 1, 2, . . . n) and $\sum_{j = 1}^{n} w_{j} = 1$ . In this case, the characteristic of the attribute C_j of alternative A_i is represented by an IFLSS (F_ij, X_ij), and for ∀s_{x
_ij} ∈ X_ij, μ_{F_ij(s_{x
_ij})} ∈ [0, 1], v_{F_ij(s_{x
_ij})} ∈ [0, 1] and μ_{F_ij(s_{x
_ij})} + v_{F_ij(s_{x
_ij})} ≤ 1. Here, μ_{F_ij(s_{x
_ij})} and v_{F_ij(s_{x
_ij})} represent the degree to the alternative A_i satisfies or dissatisfies, respectively, the linguistic evaluation s_{x
_ij} for attribute C_j. Therefore, an IFLSS decision matrix R = [(F_ij, X_ij)] _m×n can be obtained. We applied the above operators to calculate multi-attribute decision-making issues on the basement of IFLSSinformation.

3.1 Method 1: Aggregate the attribute values first

The following steps are involved in this method:

Step 1. Convert evaluation results to standard decision matrix.

If the characteristic of attribute C_j is benefit, then (B_ij, Y_ij) = (F_ij, X_ij).

If the characteristic of attribute C_j is cost, then $(B_{ij}, Y_{ij}) = (F_{ij}, X_{ij}^{C})$ .

Then, the standard decision matrix B = [(B_ij, Y_ij)] _m×n is obtained.

Step 2. Apply IFLSSGWS operator to generate summary evaluation results.

Using Formulas (17) and (18), or (19) and (20), adding the IFLSS attribute values ((B_i1, Y_i1) , (B_i2, Y_i2) , . . . (B_in, Y_in)) of each alternative into an exclusive IFLSS attribute value (B_i, Y_i) by the weights of all the elements, as in the following: $\begin{matrix} (B_{i}, Y_{i}) & = & IFLSS GWA ((B_{i 1}, Y_{i 1}), (B_{i 2}, Y_{i 2}), \\ . . ., (B_{in}, Y_{in})) \end{matrix}$

Step 3. Find the expected and accuracy values for all alternatives.

Using Definitions 5 and 6, calculate the expected value E_{(B_i,Y_i)} and accuracy value R_{(B_i,Y_i)} (i = 1, 2, . . . , m) for each aggregated value (B_i, Y_i) (i = 1, 2, . . . , m).

Step 4. Rank all of the alternatives A_i (i = 1, 2, . . . , m).

Rank the alternatives in accordance with the comparison rules of Definition 7.

3.2 Method 2: Using GWA operator based on expected values

This method involves the following steps:

Step 1. Refer to Step 1 in Section 3.1.

Step 2. Calculate the expected value for each attribute of each alternative.

Using Formulas (5) and (6), calculate the expected value E_{(B_ij,Y_ij)} (i = 1, 2, . . . , m, j = 1, 2, . . . , n) for each attribute of each alternative, as follows: $\begin{matrix} E_{(B_{ij}, Y_{ij})} & = & \frac{\sum_{s_{a} \in Y_{ij}} E_{〈 s_{a}, (μ_{B_{ij} (s_{a})}, v_{B_{ij} (s_{a})}) 〉}}{| Y_{ij} |} \\ = & \frac{\sum_{s_{a} \in Y_{ij}} (a \times (μ_{B_{ij} (s_{a})} + 1 - v_{B_{ij} (s_{a})}))}{2 | Y_{ij} |} \end{matrix}$

Step 3. Utilize traditional GWS operator to aggregate the expected values for each alternative, as follows: $Z_{i} = {(\sum_{j = 1}^{n} w_{j} {(E_{(B_{ij}, {YB}_{ij})})}^{λ})}^{1 / - λ}$ (21)

Whenλ → 0, then refer to the following: $Z_{i} = \prod_{j = 1}^{n} {(E_{(B_{ij}, {YB}_{ij})})}^{w_{j}}$ (22)

When λ = 1, refer to the following: $Z_{i} = \sum_{j = 1}^{n} w_{j} (E_{(B_{ij}, {YB}_{ij})})$ (23)

Step 4. Rank all of the alternatives A_i (i = 1, 2, . . . , m).

Rank the alternatives according to Z_i.

4 Numeric example

Consider a new supermarket that wants to choose a location. There are four alternatives, represented as A₁, A₂, A₃, and A₄. Decision is made according to four attributes: C₁ is the number of potential customers, C₂ is the traffic conditions, C₃ is the consumption capacity, C₄ is the number of competing stores.

The normalized weights of the four attributes are 0.31, 0.16, 0.24 and 0.29 respectively. The evaluations of the four possible locations (A₁, A₂, A₃, A₄) are measured under the above four aspects with the linguistic term set S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆), as aforementioned. C₁, C₂, C₃ are beneficial aspects, the representation of the linguistic set is extremely bad, bad, below ordinary, ordinary, over ordinary, well, or extremely well; C₄ is a cost aspect, and the representation of its linguistic set is extremely few, few, below ordinary, ordinary, over ordinary, many, or a great many. For each aspect of every variable, decision-makers can select some linguistic variables. A good example is when evaluating the attribute C₂ for the alternative A₃ from the point of view in public transport conditions, the result is “well,” that is s₅, and one’s membership and non-membership degrees are 0.8 and 0.1, respectively. However, from the point of view in parking lot, the result is “extremely well,” which is s₆, and its intuitionistic fuzzy value would be 〈0.7, 0.2〉. Next, the measurement for C₂ of A₃ can be represented as (B₃₂, Y₃₂), where Y₃₂ = {s₅, s₆}, B₃₂ (s₅) =〈 0.8, 0.1 〉, B₃₂ (s₆) =〈 0.7, 0.2 〉; In a word, the measurement can be viewed as {〈 s₅, (0.8, 0.1) 〉 , 〈 s₆, (0.7, 0.2) 〉 }.

Decision matrix R would be the following: $\begin{matrix} R = [\begin{matrix} {〈 s_{2}, (0 . 8, 0 . 1) 〉} & {〈 s_{5}, (0 . 9, 0 . 1) 〉} \\ {〈 s_{6}, (0 . 7, 0 . 2) 〉} & {〈 s_{3}, (0 . 8, 0 . 2) 〉} \\ {〈 s_{6}, (0 . 9, 0 . 1) 〉, & {〈 s_{5}, (0 . 8, 0 . 1) 〉, \\ 〈 s_{5}, (0 . 8, 0 . 1) 〉} & 〈 s_{6}, (0 . 7, 0 . 2) 〉} \\ {〈 s_{5}, (0 . 9, 0 . 1) 〉} & {〈 s_{5}, (0 . 7, 0 . 3) 〉} \end{matrix} \\ \to \begin{matrix} {〈 s_{2}, (0 . 8, 0 . 1) 〉} & {〈 s_{0}, (0 . 9, 0 . 1) 〉} \\ {〈 s_{5}, (0 . 6, 0 . 1) 〉, & {〈 s_{5}, (0 . 8, 0 . 2) 〉, \\ 〈 s_{3}, (0 . 3, 0 . 2) 〉} & 〈 s_{3}, (0 . 6, 0 . 2) 〉} \\ {〈 s_{5}, (0 . 8, 0 . 1) 〉} & {〈 s_{1}, (0 . 8, 0 . 1) 〉} \\ {〈 s_{6}, (0 . 8, 0 . 2) 〉} & {〈 s_{0}, (0 . 7, 0 . 1) 〉} \end{matrix}] \end{matrix}$

4.1 Using the method of aggregating the attribute values first

To obtain the best alternative(s), the method introduced in Section 3 can be used to generate the ranking orders of the alternatives.

Step 1. Convert evaluation results to standard decision matrix.

We need to convert all attributes to benefit attributed before future processes. The standardized decision making matrix B would be the followings: $\begin{matrix} B & = & [\begin{matrix} {〈 s_{2}, (0 . 8, 0 . 1) 〉} & {〈 s_{5}, (0 . 9, 0 . 1) 〉} \\ {〈 s_{6}, (0 . 7, 0 . 2) 〉} & {〈 s_{3}, (0 . 8, 0 . 2) 〉} \\ {〈 s_{6}, (0 . 9, 0 . 1) 〉, & {〈 s_{5}, (0 . 8, 0 . 1) 〉, \\ 〈 s_{5}, (0 . 8, 0 . 1) 〉} & 〈 s_{6}, (0 . 7, 0 . 2) 〉} \\ {〈 s_{5}, (0 . 9, 0 . 1) 〉} & {〈 s_{5}, (0 . 7, 0 . 3) 〉} \end{matrix} \\ \to \begin{matrix} {〈 s_{2}, (0 . 8, 0 . 1) 〉} & {〈 s_{6}, (0 . 9, 0 . 1) 〉} \\ {〈 s_{5}, (0 . 6, 0 . 1) 〉, & {〈 s_{1}, (0 . 8, 0 . 2) 〉, \\ 〈 s_{3}, (0 . 3, 0 . 2) 〉} & 〈 s_{3}, (0 . 6, 0 . 2) 〉} \\ {〈 s_{5}, (0 . 8, 0 . 1) 〉} & {〈 s_{5}, (0 . 8, 0 . 1) 〉} \\ {〈 s_{6}, (0 . 8, 0 . 2) 〉} & {〈 s_{6}, (0 . 7, 0 . 1) 〉} \end{matrix}] \end{matrix}$

Step 2. Apply IFLSSGWS operator to generate summary evaluation results.

Using Formulas (17) and (18), (19) or (20), let λ → 0, λ = 1, or λ = 2. The summarized results are shown in Table 1.

Step 3. Find the expected value and accuracy value for all alternatives.

Using Definitions 5 and 6, the expected values and accuracy values of the alternatives were obtained, as listed in Table 2.

Step 4. Rank all of the alternatives A_i (i = 1, 2, . . . , m).

According to the comparison rules described in Definition 7, the ranking results can be obtained. $\begin{matrix} λ \to 0 : A_{3} ≻ A_{4} ≻ A_{1} ≻ A_{2} \\ λ = 1 : A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} \\ λ = 2 : A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} \end{matrix}$

It can be seen that the ranking orders have a slight change with the change of λ.

4.2 Using GWA operator based on expected values

Step 1. See Step 1 in Section 4.1.

Step 2. Find the expected values for all attributes of all alternatives.

Using Formulas (5) and (6), the expected values can be obtained, as listed in Table 3.

Step 3. Utilize traditional GWS operator to aggregate the expected values for each alternative.

Using Formulas (21), (22), or (23), let λ → 0, λ = 1, or λ = 2. The aggregation results are listed in Table 4.

Step 4. Rank all of the alternatives A_i (i = 1, 2, . . . , m). According to the aggregated values, the ranking results can be obtained, as follow: $\begin{matrix} λ \to 0 : A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} \\ λ = 1 : A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} \\ λ = 2 : A_{4} ≻ A_{3} ≻ A_{1} ≻ A_{2} \end{matrix}$

It can be seen that the ranking orders are consistent under different λs, and approximately similar to the ranking orders using the other method.

4.3 The influence of parameter λ

To discuss the influence of parameter λ, ranking orders were calculated using the aforementioned methods under different λs. The results are listed in Table 5.

From Table 5, it can be observed that the ranking orders are consistent under different λs using the GWA method. However, while using a traditional GWS method, the ranking orders are different under different λs. Therefore, the proposed GWA method was proved to be more stable and reliable.

5 Conclusion

We discussed a new evaluation means named intuitionistic fuzzy linguistic soft set (IFLSS) in this paper, which takes the measurement linguistic variables as the parameters of the soft set. Decision-makers could select some linguistic variables to measure an aspect of an alternative in multiple points of view. In this way, it can be more diversified and pragmatic when representing the complexity of decision-makers’ subjective judgment over traditional linguistic evaluation methods. An exclusive weighted summation operator designed for intuitionistic fuzzy linguistic soft sets was also developed, and its special forms and characteristics were discussed. Using expected values of IFLSSs and the proposed aggregation operator, two methods were proposed to deal with multi-attribute decision-making studies where aspect weights are taken in the means of real numbers and the aspect values are taken in the form of IFLSSs. Illustrated examples presented the procedure of the researched outcomes, and results verified their practicality and effectiveness.

The most significant advantage of this model is that it allows decision-makers to express their opinions from various perspectives, and assists them in more rationally considering all of the attributes for each alternative. Compared to using a general assessment for each attribute of an alternative as in traditional models, this model can more accurately reflect a realistic decision-making process. Employing this model, decision-makers no longer need to answer all questions if they aren’t sure. In fact, the IFLSS is more flexible when dealing with incomplete information. Unfortunately, along with improvements in flexibility, the complexity of the process was increased. In more complex applications, corresponding software based on the proposed model can help to improve its efficiency. This model is more suitable for group decision-making conditions. Each element of an evaluation can be assigned a weight according to its importance for improving the universality of the model. In future analysis, the operators and methods in this paper could be used in some else fuzzy information.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants (71471076), and by the Joint Research of the NSFC-NRF Scientific Cooperation Program under Grant (71411140250). This work was also sponsored by the Fund of Ministry of education of Humanities and Social Sciences (14YJAZH025), the Fund of China Nation Tourism Administration (15TACK003), and the Natural Science Foundation of Shandong Province (ZR2013GM003).

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