Hesitant fuzzy linguistic ordered weighted distance operators for group decision making

Abstract

Since the concept of hesitant fuzzy sets was put forward, different types of extensions have been proposed to deal with actual problems. A hesitant fuzzy linguistic term set provides a linguistic and computational basis to increase the flexibility and richness of linguistic elicitation based on the fuzzy linguistic approach. In this paper, we consider the concept of distance operator and develop a hesitant fuzzy linguistic ordered weighted distance (HFLOWD) operator. The HFLOWD operator is very suitable to deal with the uncertain situations with linguistic information. Moreover, it is also a new aggregation operator that provides parameterized families of distance aggregation operators between the minimum and the maximum distance. Some of its main properties and different families of HFLOWD operators are investigated. Finally, an application of the new approach is offered and comparative analyses are also provided to show the advantages over existing methods.

Appendix

In this appendix, we introduce some notations in [9].

The upper bound $h_S^+$, and the lower bound $h_S^-$ of $h_S$ are defined as follows:

• (1) $h_S^+ =\max (s_i )=s_{\max } ,s_i \in h_S$ and $s_{\max } \ge s_i ,\forall i$.

• (2) $h_S^- =\min (s_i )=s_{\min } ,s_i \in h_S$ and $s_i \ge s_{\min } ,\forall i$.

The notions of the hesitant fuzzy linguistic positive ideal solution $x^{+}$ and the hesitant fuzzy linguistic negative ideal solution $x^{-}$ are as follows, respectively:

$$\begin{aligned} x^{+}&= \{h_1^+ ,h_2^+ ,\cdots ,h_n^+ \}\end{aligned}$$

(12)

$$\begin{aligned} x^{-}&= \{h_1^- ,h_2^- ,\cdots ,h_n^- \} \end{aligned}$$

(13)

where

$$\begin{aligned} h_j^+ \!=\!\left\{ {\begin{array}{l} \mathop {\max }\limits _{i=1,2,\cdots ,n} h_{ij}^+ = \mathop {\max }\limits _{\begin{array}{l} i=1,2,\cdots ,n\\ l=1,2,\cdots ,\# h_{ij} \\ \end{array}} \{x_{ij} \}\hbox { for benefit criterion G}_j \\ \mathop {\min }\limits _{i=1,2,\cdots ,n} h_{ij}^- =\mathop {\max }\limits _{\begin{array}{l} i=1,2,\cdots ,n \nonumber \\ l=1,2,\cdots ,\# h_{ij} \\ \end{array}} \{x_{ij} \}\hbox { for cost criterion G}_j \\ \end{array}} \right. ,\quad \hbox { for } j=1,2,\cdots ,n\\ \end{aligned}$$

(14)

and

$$\begin{aligned} h_j^- \!=\!\left\{ {\begin{array}{l} \mathop {\min }\limits _{i=1,2,\cdots ,n} h_{ij}^- =\mathop {\max }\limits _{\begin{array}{l} i=1,2,\cdots ,n \\ l=1,2,\cdots ,\# h_{ij} \\ \end{array}} \{x_{ij} \}\hbox { for benefit criterion G}_j \\ \mathop {\max }\limits _{i=1,2,\cdots ,n} h_{ij}^+ =\mathop {\max }\limits _{\begin{array}{l} i=1,2,\cdots ,n \nonumber \\ l=1,2,\cdots ,\# h_{ij} \\ \end{array}} \{x_{ij} \}\hbox { for cost criterion G}_j \\ \end{array}} \right. ,\quad \hbox {for} j=1,2,\cdots ,n\\ \end{aligned}$$

(15)

Let us define the hesitant fuzzy linguistic term set as $h_i =\cup _{s_{\delta _i } \in h_i } \{s_{\delta _i } \big | l=1,2, \cdots , \# h_i \}(\# h_i$ is the number of linguistic terms in $h_i )$.

For two sets of HFLSs $\tilde{A}=\{h_{\alpha _1 } ,h_{\alpha _2 } ,\cdots ,h_{\alpha _n } \}$ and $\tilde{B}=\{h_{\beta _1 } ,h_{\beta _2 } ,\cdots ,h_{\beta _n } \}$, we define the generalized weighted distance measure between two HFLTSs as follows:

$$\begin{aligned} d_{gowd} (\tilde{A},\tilde{B})=\left( {\sum _{j=1}^n {w_j \left( {\frac{1}{l}\sum _{l=1}^{\# h_i } {\left( {\frac{\left| {\delta _1^{\sigma (j)} -\delta _2^{\sigma (j)} } \right| }{2t+1}} \right) } ^{\lambda }} \right) } } \right) ^{1/\lambda } \end{aligned}$$

(16)

where $w$ is the weighting vector of $h_i$, $w=(w_1 ,w_2 ,\cdots ,w_n )^{T}$ with $w_i \in [0,1]$ and $\sum _{i=1}^n {w_i } =1$, respectively. $\lambda >0$ and $\sigma (j):(1,2,\cdots ,n)\rightarrow (1,2,\cdots ,n)$ is a permutation such that

$$\begin{aligned} \frac{1}{l}\sum _{l=1}^{\# b} {\left( {\frac{\left| {\delta _1^{\sigma (j)} -\delta _2^{\sigma (j)} } \right| }{2t+1}} \right) } ^{\lambda }\ge \frac{1}{l}\sum _{l=1}^{\# b} {\left( {\frac{\left| {\delta _1^{\sigma (j+1)} -\delta _2^{\sigma (j+1)} } \right| }{2t+1}} \right) } ^{\lambda },\quad j=1,2,\cdots ,n \end{aligned}$$

In particular, if $\lambda =1$, then the generalized ordered weighted distance becomes the ordered weighted Hamming distance between $\tilde{A}$ and $\tilde{B}$:

$$\begin{aligned} d_{owhd} (\tilde{A},\tilde{B})=\sum _{j=1}^n {\frac{w_j }{l}\sum _{l=1}^{\# h_i } {\frac{\left| {\delta _1^{\sigma (j)} -\delta _2^{\sigma (j)} } \right| }{2t+1}} } \end{aligned}$$

(17)

If $\lambda =1$, then the generalized ordered weighted distance becomes the ordered weighted Euclidean distance between $\tilde{A}$ and $\tilde{B}$:

$$\begin{aligned} d_{owed} (\tilde{A},\tilde{B})=\left( {\sum _{j=1}^n {w_j \left( {\frac{1}{l}\sum _{l=1}^{\# b} {\left( {\frac{\left| {\delta _1^{\sigma (j)} -\delta _2^{\sigma (j)} } \right| }{2t+1}} \right) } ^{2}} \right) } } \right) ^{1/2} \end{aligned}$$

(18)

To take the distance between each alternative $Y_i$ and the hesitant fuzzy linguistic positive ideal solution $x^{+}$, and the distance between each alternative $Y_i$ and the hesitant fuzzy linguistic negative ideal solution $x^{-}$ into consideration simultaneously, Liao et al.’s defined the definition below:

A satisfaction degree of a given alternative $Y_i$ over the criteria $G_j (j=1,2,\cdots ,n)$ is defined as:

$$\begin{aligned} \eta (Y_i )=\frac{(1-\theta )d(Y_i ,x_i^- )}{\theta d(Y_i ,x_i^+ )+(1-\theta )d(Y_i ,x_i^- )} \end{aligned}$$

(19)

where the parameter $\theta$ denotes the risk preferences of the decision maker: $\theta >0.5$ means that the decision maker is pessimists; while $\theta <0.5$ means the opposite. The value of the parameter $\theta$ is provided by the decision maker in advance. The higher the satisfaction degree, the better the alternative is.