# Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables

^{*}

## Abstract

**:**

## 1. Introduction

_{0}(extremely poor), y

_{1}(very poor), y

_{2}(poor), y

_{3}(slightly poor), y

_{4}(moderate), y

_{5}(goodish), y

_{6}(good), y

_{7}(very good), y

_{8}(extremely good)}, the interval linguistic value [y

_{5}, y

_{7}] is given by two of the five experts and the HLS {y

_{4}, y

_{5}, y

_{6}} is given by three of the five experts under the situation of their uncertainty and hesitancy, and then the hybrid form of both [y

_{5}, y

_{7}] (the uncertain linguistic part) and {y

_{4}, y

_{5}, y

_{6}} (the hesitant linguistic part) cannot be expressed simultaneously using the aforementioned various linguistic concepts. Clearly, this expression problem requires us to solve the gap using a hybrid linguistic form. Hence, this study presents a new linguistic concept based on the combining form of both an LCV and a hesitant LV, which is called a linguistic cubic hesitant variable (LCHV), the operations and ranking method of LCHVs so as to solve MADM problems under the situation of decision makers’ uncertainty and hesitancy. In this study framework, Section 2 presents a LCHV concept to express the hybrid information of both an interval linguistic argument and a hesitant linguistic argument, and then the operational relations and linguistic score function of LCHVs. Section 3 presents the WAA and WGA operators of LCHVs based on the LCMN extension method, which contain the objectivity and suitability of the aggregation operations for LCHVs, and discusses their properties. Next, a MADM approach is proposed based on both the WAA and the WGA operators of LCHVs and the linguistic score function of LCHVs in Section 4. Section 5 applies the proposed MADM approach to the MADM problem regarding an illustrative example in a LCHV setting, and then its decision results show its applicability. Lastly, conclusions and the next study are contained in Section 6.

## 2. Linguistic Cubic Hesitant Variables (LCHVs)

**Definition**

**1.**

_{l}|l ∈ [0, q]}, where q is an even number. A LCHV z in Y is constructed using$z=({\tilde{y}}_{u},{\tilde{y}}_{h})$, where${\tilde{y}}_{u}=[{y}_{\alpha},{y}_{\beta}]$for β ≥ α and y

_{α}, y

_{β}∈ Y is an interval/uncertain LV and${\tilde{y}}_{h}=\left\{{y}_{{\lambda}_{k}}^{}|{y}_{{\lambda}_{k}}^{}\in Y,k=1,2,\dots ,r\right\}$ is a set of r possible LVs (i.e., a hesitant linguistic variable (HLV)) ranked in an ascending order.

**Definition**

**2.**

_{α}, y

_{β}∈Y and${\tilde{y}}_{h}=\left\{{y}_{{\lambda}_{k}}^{}|{y}_{{\lambda}_{k}}^{}\in Y,k=1,2,\dots ,r\right\}$. Then, we call

- (i)
- $z=({\tilde{y}}_{u},{\tilde{y}}_{h})=([{y}_{\alpha},{y}_{\beta}],\{{y}_{{\lambda}_{1}},{y}_{{\lambda}_{2}},\dots ,{y}_{{\lambda}_{r}}\})$an internal LCHV if every λ
_{k}∈ [α, β] (k = 1, 2, …, r) for α, β ∈ [0, q]; - (ii)
- $z=({\tilde{y}}_{u},{\tilde{y}}_{h})=([{y}_{\alpha},{y}_{\beta}],\{{y}_{{\lambda}_{1}},{y}_{{\lambda}_{2}},\dots ,{y}_{{\lambda}_{r}}\})$an external LCHV if every λ
_{k}∉ [α, β] (k = 1, 2, …, r) for α, β, λ_{k}∈ [0, q].

_{1}, r

_{1}, …, r

_{n}) for ${\tilde{y}}_{hj}$ (j = 1, 2, …, n) is c. Then, one can extend them to the same number of LVs based on the following extension forms:

**Example**

**1.**

_{1}= ([y

_{4}, y

_{6}], {y

_{4}, y

_{5}}) and z

_{2}= ([y

_{5}, y

_{6}], {y

_{4}, y

_{5}, y

_{6}}) are two linguistic cubic hesitant numbers (LCHNs) in the LTS Y = {y

_{0}, y

_{1}, y

_{2}, …, y

_{8}}.

_{1}= 2 and r

_{2}= 3 in z

_{1}and z

_{2}is c = 6. Through Equation (1), the two LCHNs z

_{1}and z

_{2}can be extended to the following forms:

**Definition**

**3.**

_{l}|l ∈ [0, q]}. Then, we define

- (i)
- z
_{1}= z_{2}⇔${\tilde{y}}_{u1}={\tilde{y}}_{u2}$and${\tilde{y}}_{h1}={\tilde{y}}_{h2}$, i.e.,${y}_{{\alpha}_{1}}={y}_{{\alpha}_{2}}$,${y}_{{\beta}_{1}}={y}_{{\beta}_{2}}$, and${y}_{{\lambda}_{1k}}={y}_{{\lambda}_{2k}}$for k = 1, 2, …, r; - (ii)
- z
_{1}⊆ z_{2}⇔${\tilde{y}}_{u1}\subseteq {\tilde{y}}_{u2}$and${\tilde{y}}_{h1}\subseteq {\tilde{y}}_{h2}$, i.e.,${y}_{{\alpha}_{1}}\le {y}_{{\alpha}_{2}}$,${y}_{{\beta}_{1}}\le {y}_{{\beta}_{2}}$, and${y}_{{\lambda}_{1k}}\le {y}_{{\lambda}_{2k}}$for k = 1, 2, …, r.

**Definition**

**4.**

_{l}|l ∈ [0, q]}. Then, its linguistic score function is defined as the following:

**Definition**

**5.**

_{l}|l ∈ [0, q]}. Then, their linguistic scores are${y}_{L({z}_{1})}$and${y}_{L({z}_{2})}$. Thus, the ranking relations are defined as follows:

- (a)
- If${y}_{L({z}_{1})}$>${y}_{L({z}_{2})}$, then z
_{1}> z_{2}; - (b)
- If${y}_{L({z}_{1})}$<${y}_{L({z}_{2})}$, then z
_{1}< z_{2}; - (c)
- If${y}_{L({z}_{1})}$=${y}_{L({z}_{2})}$, then z
_{1}= z_{2}.

**Example**

**2.**

_{1}= ([y

_{5}, y

_{6}], {y

_{4}, y

_{6}}) and z

_{2}= ([y

_{4}, y

_{6}], {y

_{3}, y

_{5}, y

_{6}}) are two LCHNs in the LTS Y = {y

_{0}, y

_{1}, y

_{2}, …, y

_{8}}.

_{1}> z

_{2}.

**Definition**

**6.**

_{l}|l ∈ [0, q]}. Then, their operational relations are defined as below:

- (1)
- $\begin{array}{l}{z}_{1}+{z}_{2}=([{y}_{{\alpha}_{1}},{y}_{{\beta}_{1}}],\{{y}_{{\lambda}_{11}},{y}_{{\lambda}_{12}},\dots ,{y}_{{\lambda}_{1r}}\})+([{y}_{{\alpha}_{2}},{y}_{{\beta}_{2}}],\{{y}_{{\lambda}_{21}},{y}_{{\lambda}_{22}},\dots ,{y}_{{\lambda}_{2r}}\})\\ =\left(\left[{y}_{{\alpha}_{1}+{\alpha}_{2}-\frac{{\alpha}_{1}{\alpha}_{2}}{q}},{y}_{{\beta}_{1}+{\beta}_{2}-\frac{{\beta}_{1}{\beta}_{2}}{q}}\right],\left\{{y}_{{\lambda}_{11}+{\lambda}_{21}-\frac{{\lambda}_{11}{\lambda}_{21}}{q}},{y}_{{\lambda}_{12}+{\lambda}_{22}-\frac{{\lambda}_{12}{\lambda}_{22}}{q}},\dots ,{y}_{{\lambda}_{1r}+{\lambda}_{2r}-\frac{{\lambda}_{1r}{\lambda}_{2r}}{q}}\right\}\right)\end{array}$;
- (2)
- $\begin{array}{l}{z}_{1}\times {z}_{2}=([{y}_{{\alpha}_{1}},{y}_{{\beta}_{1}}],\{{y}_{{\lambda}_{11}},{y}_{{\lambda}_{12}},\dots ,{y}_{{\lambda}_{1r}}\})\times ([{y}_{{\alpha}_{2}},{y}_{{\beta}_{2}}],\{{y}_{{\lambda}_{21}},{y}_{{\lambda}_{22}},\dots ,{y}_{{\lambda}_{2r}}\})\\ =\left(\left[{y}_{\frac{{\alpha}_{1}{\alpha}_{2}}{q}},{y}_{\frac{{\beta}_{1}{\beta}_{2}}{q}}\right],\left\{{y}_{\frac{{\lambda}_{11}{\lambda}_{21}}{q}},{y}_{\frac{{\lambda}_{12}{\lambda}_{22}}{q}},\dots ,{y}_{\frac{{\lambda}_{1r}{\lambda}_{2r}}{q}}\right\}\right)\end{array}$;
- (3)
- $\begin{array}{l}\delta {z}_{1}=\delta \left([{y}_{{\alpha}_{1}},{y}_{{\beta}_{1}}],\left\{{y}_{{\lambda}_{11}},{y}_{{\lambda}_{12}},\dots ,{y}_{{\lambda}_{1r}}\right\}\right)\\ =\left(\left[{y}_{q-q{\left(1-\frac{{\alpha}_{1}}{q}\right)}^{\delta}},{y}_{q-q{\left(1-\frac{{\beta}_{1}}{q}\right)}^{\delta}}\right],\left\{{y}_{q-q{\left(1-\frac{{\lambda}_{11}}{q}\right)}^{\delta}},{y}_{q-q{\left(1-\frac{{\lambda}_{12}}{q}\right)}^{\delta}},\dots ,{y}_{q-q{\left(1-\frac{{\lambda}_{1r}}{q}\right)}^{\delta}}\right\}\right),\delta >0\end{array}$;
- (4)
- ${z}_{1}^{\delta}={\left([{y}_{{\alpha}_{1}}^{},{y}_{{\beta}_{1}}^{}],\left\{{y}_{{\lambda}_{11}}^{},{y}_{{\lambda}_{12}}^{},\dots ,{y}_{{\lambda}_{1r}}^{}\right\}\right)}^{\delta}=\left(\left[{y}_{q{\left(\frac{{\alpha}_{1}}{q}\right)}^{\delta}},{y}_{q{\left(\frac{{\beta}_{1}}{q}\right)}^{\delta}}\right],\left\{{y}_{q{\left(\frac{{\lambda}_{11}}{q}\right)}^{\delta}},{y}_{q{\left(\frac{{\lambda}_{12}}{q}\right)}^{\delta}},\dots ,{y}_{q{\left(\frac{{\lambda}_{1r}}{q}\right)}^{\delta}}\right\}\right),\delta >0$

## 3. Weighted Aggregation Operators of LCHVs

#### 3.1. Weighted Arithmetic Averaging (WAA) Operator of LCHVs

**Definition**

**7.**

_{l}|l∈ [0, q]}, along with its weight${\omega}_{j}\in [0,1]$for${\sum}_{j=1}^{n}{\omega}_{j}=1$. The corresponding WAA operator of the LCHVs is expressed using

**Theorem**

**1.**

_{l}|l ∈ [0, q]}, along with its weight ${\omega}_{j}\in [0,1]$ for ${\sum}_{j=1}^{n}{\omega}_{j}=1$, then the aggregation result of Equation (3) is still a LCHV, which is calculated using the following aggregation operation:

**Proof.**

- (1)
- Set n = 2, based on the operational relation (3) in Definition 6, we can get$${\omega}_{1}{z}_{1}=\left(\left[{y}_{q-q{(1-\frac{{\alpha}_{1}}{q})}^{{\omega}_{1}}},{y}_{q-q{(1-\frac{{\beta}_{1}}{q})}^{{\omega}_{1}}}\right],\left\{{y}_{q-q{(1-\frac{{\lambda}_{11}}{q})}^{{\omega}_{1}}},{y}_{q-q{(1-\frac{{\lambda}_{12}}{q})}^{{\omega}_{1}}},\dots ,{y}_{q-q{(1-\frac{{\lambda}_{1r}}{q})}^{{\omega}_{1}}}\right\}\right)$$$${\omega}_{2}{z}_{2}=\left(\left[{y}_{q-q{(1-\frac{{\alpha}_{2}}{q})}^{{\omega}_{2}}},{y}_{q-q{(1-\frac{{\beta}_{2}}{q})}^{{\omega}_{2}}}\right],\left\{{y}_{q-q{(1-\frac{{\lambda}_{21}}{q})}^{{\omega}_{2}}},{y}_{q-q{(1-\frac{{\lambda}_{22}}{q})}^{{\omega}_{2}}},\dots ,{y}_{q-q{(1-\frac{{\lambda}_{2r}}{q})}^{{\omega}_{2}}}\right\}\right)$$

- (2)
- Set n = k, the aggregation result of LCHVs based on Equation (4) can be expressed as$$\begin{array}{l}{F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{k})={\displaystyle \sum _{j=1}^{k}{\omega}_{j}{z}_{j}}\\ =\left(\left[{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\alpha}_{j}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\beta}_{j}}{q})}^{{\omega}_{j}}}}\right],\left\{{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{j1}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{j2}}{q})}^{{\omega}_{j}}}},\dots ,{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{jr}}{q})}^{{\omega}_{j}}}}\right\}\right)\end{array}$$
- (3)
- Set n = k + 1, based on Equations (7) and (8), the aggregation result of the LCHVs is given by$$\begin{array}{l}{F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{k})={\displaystyle \sum _{j=1}^{k}{\omega}_{j}{z}_{j}}+{\omega}_{k+1}{z}_{k+1}\\ =\left(\left[{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\alpha}_{j}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\beta}_{j}}{q})}^{{\omega}_{j}}}}\right],\left\{{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{j1}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{j2}}{q})}^{{\omega}_{j}}}},\dots ,{y}_{q-q{\displaystyle {\prod}_{j=1}^{k}{(1-\frac{{\lambda}_{jr}}{q})}^{{\omega}_{j}}}}\right\}\right)+{\omega}_{k+1}{z}_{k+1}\\ =\left(\left[{y}_{q-q{\displaystyle {\prod}_{j=1}^{k+1}{(1-\frac{{\alpha}_{j}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k+1}{(1-\frac{{\beta}_{j}}{q})}^{{\omega}_{j}}}}\right],\left\{{y}_{q-q{\displaystyle {\prod}_{j=1}^{k+1}{(1-\frac{{\lambda}_{j1}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle {\prod}_{j=1}^{k+1}{(1-\frac{{\lambda}_{j2}}{q})}^{{\omega}_{j}}}},\dots ,{y}_{q-q{\displaystyle {\prod}_{j=1}^{k+1}{(1-\frac{{\lambda}_{jr}}{q})}^{{\omega}_{j}}}}\right\}\right).\end{array}$$

**Example**

**3.**

_{1}= ([y

_{5}, y

_{6}], {y

_{4}, y

_{6}}), z

_{2}= ([y

_{4}, y

_{6}], {y

_{3}, y

_{5}, y

_{6}}), and z

_{3}= ([y

_{3}, y

_{5}], {y

_{3}, y

_{4}, y

_{5}}) as three LCHNs in the LTS Y = {y

_{0}, y

_{1}, y

_{2}, …, y

_{8}}. Then, their weight vector is given as ω = (0.4, 0.25, 0.35).

_{4}, y

_{6}}, {y

_{3}, y

_{5}, y

_{6}} and {y

_{3}, y

_{4}, y

_{5}} into the same components by using the LCMN extension method for them.

_{4}, y

_{6}}, {y

_{3}, y

_{5}, y

_{6}} and {y

_{3}, y

_{4}, y

_{5}} is obtained as c = 6. Thus, their extension forms are expressed as${z}_{1}^{e}$= ([y

_{5}, y

_{6}], {y

_{4}, y

_{4}, y

_{4}, y

_{6}, y

_{6}, y

_{6}}),${z}_{2}^{e}$= ([y

_{4}, y

_{6}], {y

_{3}, y

_{3}, y

_{5}, y

_{5}, y

_{6}, y

_{6}}), and${z}_{3}^{e}$= ([y

_{3}, y

_{5}], {y

_{3}, y

_{3}, y

_{4}, y

_{4}, y

_{5}, y

_{5}}).

**Theorem**

**2.**

_{j}(j = 1, 2, …, n) as a group of LCHVs. Then, the WAA operator of${F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})$indicates the following properties:

- (1)
- Idempotency: If z
_{j}= z (j = 1, 2, …, n), then there exists${F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})=z$. - (2)
- Boundedness: Set${z}^{+}=\left(\left[\underset{j}{\mathrm{max}}({y}_{{\alpha}_{j}}),\underset{j}{\mathrm{max}}({y}_{{\beta}_{j}})\right],\left\{\underset{j}{\mathrm{max}}({y}_{{\lambda}_{j1}}),\underset{j}{\mathrm{max}}({y}_{{\lambda}_{j2}}),\dots ,\underset{j}{\mathrm{max}}({y}_{{\lambda}_{jr}})\right\}\right)$and${z}^{-}=\left(\left[\underset{j}{\mathrm{min}}({y}_{{\alpha}_{j}}),\underset{j}{\mathrm{min}}({y}_{{\beta}_{j}})\right],\left\{\underset{j}{\mathrm{min}}({y}_{{\lambda}_{j1}}),\underset{j}{\mathrm{min}}({y}_{{\lambda}_{j2}}),\dots ,\underset{j}{\mathrm{min}}({y}_{{\lambda}_{jr}})\right\}\right)$(j = 1, 2, …, n) as the maximum LCHV and the minimum LCHV, respectively. Then,${z}^{-}\le {F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le {z}^{+}$can hold.
- (3)
- Monotonicity: If${z}_{j}\le {z}_{j}^{*}$(j = 1, 2, …, n), then there exists${F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le {F}_{LCHV}({z}_{1}^{*},{z}_{2}^{*},\cdots ,{z}_{n}^{*})$.

**Proof.**

- (1)
- Since z
_{j}= z (j = 1, 2, …, n), the WAA aggregation result of LCHVs can be calculated using$$\begin{array}{l}{F}_{LCHV}({z}_{1},{z}_{2},\cdots {z}_{n})={\displaystyle \sum _{j=1}^{n}{\omega}_{j}{z}_{j}}=\left(\left[{y}_{q-q{\displaystyle \prod _{j=1}^{n}{(1-\frac{{\alpha}_{j}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle \prod _{j=1}^{n}{(1-\frac{{\beta}_{j}}{q})}^{{\omega}_{j}}}}\right],\left\{{y}_{q-q{\displaystyle \prod _{j=1}^{n}{(1-\frac{{\lambda}_{j1}}{q})}^{{\omega}_{j}}}},{y}_{q-q{\displaystyle \prod _{j=1}^{n}{(1-\frac{{\lambda}_{j2}}{q})}^{{\omega}_{j}}}},\dots ,{y}_{q-q{\displaystyle \prod _{j=1}^{n}{(1-\frac{{\lambda}_{jr}}{q})}^{{\omega}_{j}}}}\right\}\right)\\ \text{\hspace{1em}\hspace{1em}}=\left(\left[{y}_{q-q{(1-\frac{\alpha}{q})}^{{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}}},{y}_{q-q{(1-\frac{\beta}{q})}^{{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}}}\right],\left\{{y}_{q-q{(1-\frac{{\lambda}_{1}}{q})}^{{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}}},{y}_{q-q{(1-\frac{{\lambda}_{2}}{q})}^{{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}}},\dots ,{y}_{q-q{(1-\frac{{\lambda}_{r}}{q})}^{{\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}}}}\right\}\right)\\ \text{\hspace{1em}\hspace{1em}}=\left(\left[{y}_{q-q(1-\frac{\alpha}{q})},{y}_{q-q(1-\frac{\beta}{q})}\right],\left\{{y}_{q-q(1-\frac{{\lambda}_{1}}{q})},{y}_{q-q(1-\frac{{\lambda}_{2}}{q})},\dots ,{y}_{q-q(1-\frac{{\lambda}_{r}}{q})}\right\}\right)\\ \text{\hspace{1em}\hspace{1em}}=\left(\left[{y}_{\alpha},{y}_{\beta}\right],\left\{{y}_{{\lambda}_{1}},{y}_{{\lambda}_{2}},\dots ,{y}_{{\lambda}_{r}}\right\}\right)=z.\end{array}$$ - (2)
- Since ${z}^{-}$ and ${z}^{+}$ are the minimum LCHV and the maximum LCHV, respectively, there exists ${z}^{-}\le {z}_{j}\le {z}^{+}$. Thus, there is $\sum}_{j=1}^{n}{\omega}_{j}{z}^{-}}\le {\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}{z}_{j}}\le {\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}{z}^{+$. Corresponding to the property (1), there are ${\sum}_{j=1}^{n}{\omega}_{j}{z}^{-}}={z}^{-$ and ${\sum}_{j=1}^{n}{\omega}_{j}{z}^{+}}={z}^{+$. Hence, ${z}^{-}\le {F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le {z}^{+}$.
- (3)
- Since ${z}_{j}\le {z}_{j}^{*}$ (j = 1, 2, …, n), there exists $\sum}_{j=1}^{n}{\omega}_{j}{h}_{j}}\le {\displaystyle {\sum}_{j=1}^{n}{\omega}_{j}{z}_{j}^{*$. Hence, ${F}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le $ ${F}_{LCHV}({z}_{1}^{*},{z}_{2}^{*},\cdots ,{z}_{n}^{*})$.

#### 3.2. Weighted Geometric Averaging (WGA) Operator of LCHVs

**Definition**

**8.**

_{l}|l ∈ [0, q]}, along with its weight${\omega}_{j}\in [0,1]$for${\sum}_{j=1}^{n}{\omega}_{j}=1$. The corresponding WGA operator of LCHVs is defined as

**Theorem**

**3.**

_{l}|l ∈ [0, q]}, along with its weight ${\omega}_{j}\in [0,1]$ for ${\sum}_{j=1}^{n}{\omega}_{j}=1$, then the aggregation result of Equation (11) is still a LCHV, which is computed using the following aggregation operation:

**Example**

**4.**

**Theorem**

**4.**

_{j}(j = 1, 2, …, n) as a group of LCHVs. Then, the WGA operator of${G}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})$indicates the following properties:

- (i)
- Idempotency: If z
_{j}= z (j = 1, 2, …, n), then there exists${G}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})=z$. - (ii)
- Boundedness: Set${z}^{+}=\left(\left[\underset{j}{\mathrm{max}}({y}_{{\alpha}_{j}}),\underset{j}{\mathrm{max}}({y}_{{\beta}_{j}})\right],\left\{\underset{j}{\mathrm{max}}({y}_{{\lambda}_{j1}}),\underset{j}{\mathrm{max}}({y}_{{\lambda}_{j2}}),\dots ,\underset{j}{\mathrm{max}}({y}_{{\lambda}_{jr}})\right\}\right)$and${z}^{-}=\left(\left[\underset{j}{\mathrm{min}}({y}_{{\alpha}_{j}}),\underset{j}{\mathrm{min}}({y}_{{\beta}_{j}})\right],\left\{\underset{j}{\mathrm{min}}({y}_{{\lambda}_{j1}}),\underset{j}{\mathrm{min}}({y}_{{\lambda}_{j2}}),\dots ,\underset{j}{\mathrm{min}}({y}_{{\lambda}_{jr}})\right\}\right)$(j = 1, 2, …, n) as the maximum LCHV and the minimum LCHV, respectively. Then${z}^{-}\le {G}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le {z}^{+}$can hold.
- (iii)
- Monotonicity: If${z}_{j}\le {z}_{j}^{*}$(j = 1, 2, …, n), then there exists${G}_{LCHV}({z}_{1},{z}_{2},\cdots ,{z}_{n})\le {G}_{LCHV}({z}_{1}^{*},{z}_{2}^{*},\cdots ,{z}_{n}^{*})$.

## 4. MADM Approach Using the WAA and WGA Operators of LCHVs

_{1}, Z

_{2}, …, Z

_{m}} and R = {R

_{1}, R

_{2}, …, R

_{n}} are two sets of m alternatives and n attributes, respectively. When decision makers are requested to assess the alternative Z

_{i}(i = 1, 2, …, m) over the attribute R

_{j}(j = 1, 2,…, n), they may assign an interval linguistic value to ${\tilde{y}}_{uij}$ and a set of several possible linguistic values to ${\tilde{y}}_{hij}$ due to their hesitancy and indeterminacy from the predefined LTS Y = {y

_{l}|l ∈ [0, q]}, where q is an even number. Thus, the assessed hybrid information of ${\tilde{y}}_{uij}$ and ${\tilde{y}}_{hij}$ corresponding to each attribute R

_{j}on each alternative Z

_{i}can be represented as a LCHV ${z}_{ij}=({\tilde{y}}_{uij},{\tilde{y}}_{hij})=([{y}_{{\alpha}_{ij}},{y}_{{\beta}_{ij}}],\{{y}_{{\lambda}_{ij(1)}},{y}_{{\lambda}_{ij(2)}},\dots ,{y}_{{\lambda}_{ij({r}_{ij})}}\})$ (j = 1, 2,…, n; i = 1, 2,…, m). Hence, a LCHV decision matrix M = (z

_{ij})

_{m}

_{×n}can be constructed based on all the assessed LCHVs. Then, the weight of each attribute R

_{j}is ω

_{j}∈ [0,1] and ${\sum}_{j=1}^{n}{\omega}_{j}=1$.

**Step 1.**The LCMNs of (r

_{i}

_{1}, r

_{i}

_{2}, …, r

_{in}) (i = 1, 2, …, m) in M = (z

_{ij})

_{m}

_{×n}can be obtained as c

_{i}, where r

_{ij}is the number of LVs in ${\tilde{y}}_{hij}$ for z

_{ij}. Based on the number of occurrences of c

_{i}/r

_{ij}in a LCHV z

_{ij}(i = 1, 2, …, m; j = 1, 2, …, n), z

_{ij}is extended to the following form:

**Step 2.**Based on Equations (4) or (12), the aggregation values of z

_{i}for Z

_{i}are calculated using the following formula:

**Step 3.**The linguistic score values of ${y}_{L({z}_{i})}$ for Z

_{i}are calculated using Equation (2).

**Step 4.**All the alternatives are ranked in a descending order of the linguistic score values and the best one is selected corresponding to the biggest linguistic score value.

## 5. Illustrative Example

_{1}, Z

_{2}, Z

_{3}, and Z

_{4}—from all the applicants, they need to be further assessed based on the requirements (attributes) of innovation capability (R

_{1}), work experience (R

_{2}), and self-confidence (R

_{3}). Then, five experts (decision makers) are invited to choose the most suitable candidate among them in the interview. Here, the importance of the three attributes is indicated by the weight vector

**ω**= (0.45, 0.35, 0.2). Thus, the five experts will assess each potential candidate Z

_{i}(i = 1, 2, 3, 4) over the three attributes R

_{j}(j = 1, 2, 3) using the hybrid information of uncertain and hesitant linguistic terms so as to express the assessment values of LCHVs from the predefined LTS Y = {y

_{0}(extremely poor), y

_{1}(very poor), y

_{2}(poor), y

_{3}(slightly poor), y

_{4}(moderate), y

_{5}(goodish), y

_{6}(good), y

_{7}(very good), and y

_{8}(extremely good)} with q = 8.

_{1}for Z

_{1}, the interval linguistic value [y

_{4}, y

_{6}] is assigned by two of the five experts corresponding to the two uncertain ranges [y

_{4}, y

_{5}] and [y

_{4}, y

_{6}], and then the HLS {y

_{4}, y

_{6}} is assigned by three of the five experts corresponding to the three linguistic evaluation values y

_{4}, y

_{4}, and y

_{6}, which can be expressed as the LCHN ([y

_{4}, y

_{6}], {y

_{4}, y

_{6}}). Using a similar evaluation method, all the LCHNs assessed by the five experts can be constructed as the following LCHN decision matrix:

**Step 1.**The LCMNs of (r

_{i}

_{1}, r

_{i}

_{2}, …, r

_{in}) (i = 1, 2, …, m) in M = (z

_{ij})

_{m}

_{×n}can be obtained as c

_{i}= 6 for i = 1, 2, 3, 4. After that, the decision matrix M can be extended into the extension decision matrix below:

**Step 2.**Based on Equation (13), the aggregated LCHN z

_{1}for Z

_{1}can be obtained as follows:

_{i}for Z

_{i}(i = 2, 3, 4):

_{2}= ([y

_{4.0011}, y

_{6.1167}], {y

_{4.8617}, y

_{4.8617}, y

_{5.2427}, y

_{5.9577}, y

_{6.2983}, y

_{6.2983}}), z

_{3}= ([y

_{5.3969}, y

_{7}], {y

_{4.8223}, y

_{4.8223}, y

_{5.2337}, y

_{6}, y

_{6.2589}, y

_{6.2589}}), and z

_{4}= ([y

_{5.3523}, y

_{6.8513}], {y

_{5.2337}, y

_{5.2337}, y

_{5.695}, y

_{6.6340}, y

_{7}, y

_{7}}).

**Step 3.**Through Equation (2), the linguistic score value of ${y}_{L({z}_{1})}$ is given below:

**Step 4.**The four candidates are ranked as Z

_{4}≻ Z

_{3}≻ Z

_{1}≻ Z

_{2}based on the linguistic score values. Thus, Z

_{4}is the best one among them.

**Step 1.**The same as Step 1.

**Step 2.**Through Equation (14), the aggregated LCHN z

_{1}for Z

_{1}is yielded as follows:

_{i}for Z

_{i}(i = 2, 3, 4):

_{2}= ([y

_{3.7998}, y

_{5.8338}], {y

_{4.6099}, y

_{4.6099}, y

_{5.0968}, y

_{5.6249}, y

_{6.1059}, y

_{6.1059}}), z

_{3}= ([y

_{5.3295}, y

_{7}], {y

_{4.7818}, y

_{4.7818}, y

_{5.1857}, y

_{6}, y

_{6.1879}, y

_{6.1879}}), and z

_{4}= ([y

_{5.1906}, y

_{6.7875}], {y

_{5.1857}, y

_{5.1857}, y

_{5.6291}, y

_{6.5309}, y

_{7}, y

_{7}}).

**Step 3.**Through Equation (2), the linguistic score value of ${y}_{L({z}_{1})}$ is given below:

**Step 4.**The four candidates are ranked as Z

_{4}≻ Z

_{3}≻ Z

_{1}≻ Z

_{2}. Thus, Z

_{4}is still the best one among them.

## 6. Conclusions

- (1)
- The LCHV concept extends the existing LCV concept to express the interval/uncertain linguistic and hesitant linguistic arguments simultaneously for the first time.
- (2)
- This paper proposed two weighted aggregation operators of LCHVs based on the LCMN extension method and the linguistic score function of LCHV for the first time.
- (3)
- The advantages of the WAA and WGA operators of LCHVs are that: (a) The two aggregation operators based on the LCMN show the objective extension operations without the subjective extension forms depending on decision makers’ preferences; (b) all the linguistic values in the aggregated LCHVs still belong to the predefined LTS; and (c) the WAA and WGA operators are the simplest and most common weighted aggregation operations used for MADM problems.
- (4)
- The developed MADM approach with LCHVs extends the existing MADM approaches with LCVs so as to carry out the MADM problems with the hybrid information of both interval linguistic arguments and hesitant linguistic arguments, which the existing ones cannot handle.

## Author Contributions

## Funding

## Conflicts of Interest

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Ye, J.; Cui, W.
Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables. *Algorithms* **2018**, *11*, 135.
https://doi.org/10.3390/a11090135

**AMA Style**

Ye J, Cui W.
Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables. *Algorithms*. 2018; 11(9):135.
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**Chicago/Turabian Style**

Ye, Jun, and Wenhua Cui.
2018. "Multiple Attribute Decision-Making Method Using Linguistic Cubic Hesitant Variables" *Algorithms* 11, no. 9: 135.
https://doi.org/10.3390/a11090135