# Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality

^{*}

## Abstract

**:**

## 1. Introduction

_{7}, h

_{3}, h

_{4}> and <h

_{5}, h

_{3}, h

_{1}>, from the given LT set H = {h

_{s}|s $\in $ [0, 8]} regarding an evaluated object. However, it is difficult to express the hesitation information and the LNN information of the DMs simultaneously by a unique LNN or a unique HFS. Therefore, for the purposes of satisfying the demand of hesitant decision-making with LNNs and ensuring the objectivity of the measure calculation, this paper aims to (i) define the concept of HLNNs by combining HFSs with LNNs, (ii) present the LCMC-based generalized distance and similarity measures of HLNNs for more objective measure calculation of HLNN information, and (iii) to propose a novel multiple-attribute decision-making (MADM) method based on the proposed LCMC-based similarity measure in the HLNN setting.

## 2. Linguistic Neutrosophic Numbers (LNNs)

**Definition**

**1**

**.**Let H = {h

_{0}, h

_{1}, ..., h

_{τ}} be a LT set, where τ + 1 is an odd cardinality. A LNN can be defined as ϑ = < h

_{T}, h

_{U}, h

_{F}> for h

_{T}, h

_{U}, h

_{F}$\in $ H and T, U, F $\in $ [0, τ], where h

_{T}, h

_{U}, h

_{F}represent the degrees of truth, indeterminacy, and falsity, respectively.

**Definition**

**2**

**.**Let ϑ = <h

_{T}, h

_{U}, h

_{F}> be a LNN in H. Then its score function can be given by:

**Definition**

**3**

**.**Let ${\vartheta}_{\alpha}=<{h}_{{T}_{\alpha}},{h}_{{U}_{\alpha}},{h}_{{F}_{\alpha}}>$ and ${\vartheta}_{\beta}=<{h}_{{T}_{\beta}},{h}_{{U}_{\beta}},{h}_{{F}_{\beta}}>$ be two LNNs in H. There exist the following relations:

- (1)
- If S(ϑ
_{α}) < S(ϑ_{β}), then ϑ_{α}< ϑ_{β}; - (2)
- If S(ϑ
_{α}) > S(ϑ_{β}), then ϑ_{α}> ϑ_{β}; - (3)
- If S(ϑ
_{α}) = S(ϑ_{β}) and V(ϑ_{α}) < V(ϑ_{β}), then ϑ_{α}< ϑ_{β}; - (4)
- If S(ϑ
_{α}) = S(ϑ_{β}) and V(ϑ_{α}) > V(ϑ_{β}), then ϑ_{α}> ϑ_{β}; - (5)
- If S(ϑ
_{α}) = S(ϑ_{β}) and V(ϑ_{α}) = V(ϑ_{β}), then ϑ_{α}= ϑ_{β}.

## 3. Hesitant Linguistic Neutrosophic Numbers (HLNNs) and HLNN Set

**Definition**

**4**

**Definition**

**5.**

_{1}, s

_{2}, …, s

_{q}} and a finite LT set H = {h

_{0}, h

_{1}, …, h

_{τ}}, and then a HLNN set N

_{l}on S can be expressed as

_{l}(s

_{j}) is a set of m

_{j}LNNs, denoted by a HLNN${E}_{l}({s}_{j})=\{<{h}_{{T}_{j}^{k}},{h}_{{U}_{j}^{k}},{h}_{{F}_{j}^{k}}>{h}_{{T}_{j}^{k}}\in H,{h}_{{U}_{j}^{k}}\in H,{h}_{{F}_{j}^{k}}\in H,k=1,2,\cdots ,{m}_{j}\}$for s

_{j}$\in $S.

## 4. LCMC-Based Distance and Similarity Measures of HLNNs

_{1}, s

_{2}, …, s

_{q}} are ${E}_{{l}_{1}}({s}_{j}),{E}_{{l}_{2}}({s}_{j}),\cdots ,{E}_{{l}_{p}}({s}_{j})$ for s

_{j}$\in $ S (j = 1, 2, ..., q). Then, the HLNNs ${E}_{{l}_{i}}({s}_{j})$ for i = 1, 2, …, p can be given by

_{ij}is the cardinal number of ${E}_{{l}_{i}}({s}_{j})$ (i = 1, 2, …, p and j = 1, 2, …, q).

_{ij}(i = 1, 2, ..., p and j = 1, 2, ..., q) is c

_{j}(j = 1, 2, …, q), by increasing the number of LNNs $<{h}_{{T}_{ij}^{k}},{h}_{{U}_{ij}^{k}},{h}_{{F}_{ij}^{k}}>$ (k = 1, 2, ..., m

_{ij}) in ${E}_{{l}_{i}}({s}_{j})$ depending on c

_{j}(j = 1, 2, …, q), the extended HLNN ${E}_{{l}_{i}}^{o}({s}_{j})$ (i = 1, 2, …, p and j = 1, 2, …, q) will be obtained by the extension forms:

_{ij}is the number of LNNs $<{h}_{{T}_{ij}^{k}},{h}_{{U}_{ij}^{k}},{h}_{{F}_{ij}^{k}}>$ (k = 1, 2, ..., m

_{ij}) in ${E}_{{l}_{i}}^{o}({x}_{j})$ (i = 1, 2, …, p and j = 1, 2, …, q), calculated by:

_{j}) in ${E}_{{l}_{i}}^{o}({x}_{j})$ are arranged in an ascending order, denoted as ${E}_{{l}_{i}}^{o}({x}_{j})=\{{\vartheta}_{ij}^{\sigma (1)},{\vartheta}_{ij}^{\sigma (2)},\cdots ,{\vartheta}_{ij}^{\sigma ({c}_{j})}\}$ (i = 1, 2, …, p and j = 1, 2, …, q), where $\sigma :(1,2,\dots ,{c}_{j})\to (1,2,\dots ,{c}_{j})$ is a permutation satisfying ${\vartheta}_{ij}^{\sigma (k)}\le {\vartheta}_{ij}^{\sigma (k+1)}$ (k = 1, 2, …, c

_{j}).

**Definition**

**6.**

_{1}, s

_{2}, ..., s

_{q}}, where${E}_{{l}_{1}}({s}_{j})$and${E}_{{l}_{2}}({s}_{j})$(j = 1, 2, …, q) are HLNNsin a LT set H = {h

_{0}, h

_{1}, ..., h

_{τ}} for h

_{j}$\in $H. Let f(h

_{j}) = j/τ be a linguistic scale function. Then, the normalized generalized distance between${N}_{{l}_{1}}$and${N}_{{l}_{2}}$can be represented as:

**Proposition**

**1.**

- (HP1)
- $0\le d({N}_{{l}_{1}},{N}_{{l}_{2}})\le 1$;
- (HP2)
- $d({N}_{{l}_{1}},{N}_{{l}_{2}})=0$if and only if${N}_{{l}_{1}}={N}_{{l}_{2}}$;
- (HP3)
- $d({N}_{{l}_{1}},{N}_{{l}_{2}})=d({N}_{{l}_{2}},{N}_{{l}_{1}})$;
- (HP4)
- Let${N}_{{l}_{3}}=\{{E}_{{l}_{3}}({s}_{1}),{E}_{{l}_{3}}({s}_{2}),\cdots ,{E}_{{l}_{3}}({s}_{q})\}$be a HLNN set, then$d({N}_{{l}_{1}},{N}_{{l}_{2}})\le d({N}_{{l}_{1}},{N}_{{l}_{3}})$and$d({N}_{{l}_{2}},{N}_{{l}_{3}})\le d({N}_{{l}_{1}},{N}_{{l}_{3}})$if${N}_{{l}_{1}}\subseteq {N}_{{l}_{2}}\subseteq {N}_{{l}_{3}}$.

**Proof.**

_{j}$\in $ S (j = 1, 2, ..., q), which implies ${T}_{3j}^{\sigma (k)}\ge {T}_{2j}^{\sigma (k)}\ge {T}_{1j}^{\sigma (k)}$, ${U}_{3j}^{\sigma (k)}\le {U}_{2j}^{\sigma (k)}\le {U}_{1j}^{\sigma (k)}$, ${F}_{3j}^{\sigma (k)}\le {F}_{2j}^{\sigma (k)}\le {F}_{1j}^{\sigma (k)}$ for k = 1, 2, ..., c

_{j}. It follows that

_{j}of an element s

_{j}$\in $ S with w

_{j}$\in $ [0, 1] and ${\sum}_{j=1}^{q}{w}_{j}=1$, the generalized weighted distance between ${N}_{{l}_{1}}$ and ${N}_{{l}_{2}}$ is

**Proposition**

**2.**

- (HP1)
- $0\le {S}_{w}({N}_{{l}_{1}},{N}_{{l}_{2}})\le 1$;
- (HP2)
- ${S}_{w}({N}_{{l}_{1}},{N}_{{l}_{2}})=1$if and only if${N}_{{l}_{1}}={N}_{{l}_{2}}$;
- (HP3)
- ${S}_{w}({N}_{{l}_{1}},{N}_{{l}_{2}})={S}_{w}({N}_{{l}_{2}},{N}_{{l}_{1}})$;
- (HP4)
- Let${N}_{{l}_{3}}$be a HLNN set, then there are${S}_{w}({N}_{{l}_{1}},{N}_{{l}_{2}})\ge {S}_{w}({N}_{{l}_{1}},{N}_{{l}_{3}})$and${S}_{w}({N}_{{l}_{2}},{N}_{{l}_{3}})\ge {S}_{w}({N}_{{l}_{1}},{N}_{{l}_{3}})$if${N}_{{l}_{1}}\subseteq {N}_{{l}_{2}}\subseteq {N}_{{l}_{3}}$.

**Proof.**

## 5. MADM Method Using the Similarity Measure of HLNNs

_{1}, g

_{2}, …, g

_{p}}) over q attributes (denoted by S = {s

_{1}, s

_{2}, …, s

_{q}}) from the LT set H = {h

_{0}, h

_{1}, …, h

_{τ}}. Then, a weight vector W = (ω

_{1}, ω

_{2}, …, ω

_{q}), which is on the conditions of 0 ≤ ω

_{j}≤ 1 (j = 1, 2, ..., q) and ${\sum}_{j=1}^{q}{\omega}_{j}=1$, represents the importance of the attributes in S. Thus, the HLNN decision matrix M can be expressed as:

_{j}$\in $ S, and m

_{ij}is the number of LNNs in ${E}_{{l}_{i}}({s}_{j})$ (i = 1, 2, …, p and j = 1, 2, …, q).

**Step 1**: For any HLNN ${E}_{{l}_{i}}({s}_{j})$ (j = 1, 2, …, q) in M, rank all elements ${\vartheta}_{ij}^{\sigma (k)}$ (k = 1, 2, …, m

_{ij}) in each HLNN ${E}_{{l}_{i}}({s}_{j})$ (j = 1, 2, …, q) in an ascending order according to their score and accuracy functions, then yield the corresponding extended HLNN ${E}_{{l}_{i}}^{o}({s}_{j})$ based on the LCMC c

_{j}and the occurrence number R

_{ij}of every LNN in ${E}_{{l}_{i}}({s}_{j})$ obtained by Equation (3). Hence, the extended decision matrix ${M}^{\xb0}$ is

_{j}).

**Step 2**: Specify an ideal HLNN set as ${g}^{*}=\{{E}_{l}^{o}({s}_{1}),{E}_{l}^{o}({s}_{2}),\dots ,{E}_{l}^{o}({s}_{q})\}=\{\{{\vartheta}_{1}^{\sigma (1)},{\vartheta}_{1}^{\sigma (2)},\dots ,{\vartheta}_{1}^{\sigma ({c}_{1})}\},\{{\vartheta}_{2}^{\sigma (1)},{\vartheta}_{2}^{\sigma (2)},\dots ,{\vartheta}_{2}^{\sigma ({c}_{2})}\},\dots ,\{{\vartheta}_{q}^{\sigma (1)},{\vartheta}_{q}^{\sigma (2)},\dots ,{\vartheta}_{q}^{\sigma ({c}_{q})}\}\}$ for all ${\vartheta}_{j}^{\sigma (k)}=<{h}_{\tau},{h}_{0},{h}_{0}>$ (k = 1, 2, ..., c

_{j}and j = 1, 2, ..., q).

_{i}(i = 1, 2, …, p) and g* can be calculated by

**Step 3:**According to the similarity measure results, rank the alternatives in G = {g

_{1}, g

_{2}, …, g

_{m}} in a descending order and choose the best one.

**Step 4:**End.

## 6. Actual Example

_{1}, g

_{2}, g

_{3}, g

_{4}}, for producing computers (g

_{1}), cars (g

_{2}), food (g

_{3}), and clothing (g

_{4}), respectively. The four alternatives must satisfy a set of three attributes, S = {s

_{1,}s

_{2}, s

_{3}}, including the risk (s

_{1}), the growth (s

_{2}), and the environmental impact (s

_{3}), with the importance given by the weight vector W = (0.35, 0.25, 0.4). Now, some DMs are assigned to assess the alternatives over the attributes by HLNN expressions from the given LT set H = {h

_{0}: none, h

_{1}: lowest, h

_{2}: lower, h

_{3}: low, h

_{4}: moderate, h

_{5}: high, h

_{6}: higher, h

_{7}: highest, h

_{8}: perfect}. Then, the assessment results regarding the four alternatives g

_{1}, g

_{2}, g

_{3}, and g

_{4}on the three attributes s

_{1}, s

_{2}, and s

_{3}can be constructed as

**Step 1**: According to the score and accuracy functions obtained by Equations (1) and (2), rank the LNNs ${\vartheta}_{ij}^{\sigma (k)}$ (k = 1, 2, …, m

_{ij}) in each HLNN ${E}_{{l}_{i}}({s}_{j})$ (i = 1, 2, 3, 4 and j = 1, 2, 3) in an ascending order, and obtain the following matrix:

_{j}= 6 (j = 1, 2, 3) and the number of occurrences of LNNs R

_{ij}of ${E}_{{l}_{i}}({s}_{j})$ (i = 1, 2, 3, 4 and j = 1, 2, 3) obtained by Equation (3), yield the following extended decision matrix ${M}^{\xb0}$:

**Step 2**: Obtain the similarity measures between the alternatives g

_{1}, g

_{2}, g

_{3}, and g

_{4}and the ideal solution g* = {{<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}, {<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}, {<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}} by Equation (7) for ρ = 1 and 2:

**Step 3**: Due to S

_{w}(g

_{4}, g*) > S

_{w}(g

_{2}, g*) > S

_{w}(g

_{3}, g*) > S

_{w}(g

_{1}, g*) for ρ = 1 and 2, the ranking of the four alternatives is g

_{4}> g

_{2}>g

_{3}> g

_{1}; thus, the best choice is g

_{4}.

_{4}> g

_{2}> g

_{3}> g

_{1}for ρ = 1 and 2, and then it becomes g

_{4}> g

_{3}> g

_{2}> g

_{1}for ρ > 2; while the best alternative is always g

_{4}.

## 7. Discussion and Analysis

#### 7.1. Resolution Analysis

#### 7.2. Sensitivity Analysis of Weights

_{4}> g

_{2}> g

_{3}> g

_{1}for W = (0.35, 0.25, 0.4) and g

_{4}> g

_{3}> g

_{2}> g

_{1}for W = (1/3, 1/3, 1/3) indicate a little difference. Then, the best alternatives are the same within the entire range of ρ. Hence, the ranking orders in this example imply a little sensitivity to the attribute weights.

## 8. Conclusions

- (1)
- The proposed HLNN provides a new effective way to express more decision information than existing LNNs by considering the hesitancy of DMs.
- (2)
- The proposed MADM method of HLNNs solves the MADM problems with HLNN information for the first time, as well as the gap of existing linguistic decision-making methods.
- (3)

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Similarity measure values of four alternatives for ρ$\in $ [1, 100]. (

**a**) W = (0.35, 0.25, 0.4) and (

**b**) W = (1/3, 1/3, 1/3).

**Table 1.**Decision results of the proposed multiple-attribute decision-making (MADM) method for ρ $\in $ [1, 100] and W = (0.35, 0.25, 0.4).

ρ^{1} | S_{w}(g_{1}, g*), S_{w}(g_{2}, g*), S_{w}(g_{3}, g*), S_{w}(g_{4}, g*) ^{2} | Ranking Order | AV ^{3} | SD ^{4} | Best Alternative |
---|---|---|---|---|---|

1 | 0.7354, 0.7493, 0.7406, 0.7747 | g_{4} > g_{2} > g_{3} > g_{1} | 0.7500 | 0.0151 | g_{4} |

2 | 0.7121, 0.7224, 0.7217, 0.7525 | g_{4} > g_{2} > g_{3} > g_{1} | 0.7272 | 0.0152 | g_{4} |

3 | 0.6905, 0.6985, 0.7037, 0.7335 | g_{4} > g_{3} > g_{2} > g_{1} | 0.7066 | 0.0163 | g_{4} |

4 | 0.6710, 0.6781, 0.6867, 0.7182 | g_{4} > g_{3} > g_{2} > g_{1} | 0.6885 | 0.018 | g_{4} |

5 | 0.6539, 0.6608, 0.6710, 0.7061 | g_{4} > g_{3} > g_{2} > g_{1} | 0.6730 | 0.0201 | g_{4} |

10 | 0.5972, 0.6047, 0.6133, 0.6722 | g_{4} > g_{3} > g_{2} > g_{1} | 0.6219 | 0.0296 | g_{4} |

15 | 0.5690, 0.5754, 0.5817, 0.6575 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5959 | 0.0358 | g_{4} |

20 | 0.5531, 0.5582, 0.5631, 0.6497 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5810 | 0.0398 | g_{4} |

30 | 0.5361, 0.5397, 0.5432, 0.6417 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5652 | 0.0443 | g_{4} |

40 | 0.5273, 0.5301, 0.5327, 0.6376 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5569 | 0.0466 | g_{4} |

50 | 0.5220, 0.5242, 0.5264, 0.6351 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5519 | 0.048 | g_{4} |

100 | 0.5111, 0.5123, 0.5134, 0.6301 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5417 | 0.051 | g_{4} |

^{1}ρ: parameter;

^{2}S

_{w}(g

_{i}, g*): the similarity measures between the alternatives g

_{i}(i = 1, 2, 3, 4) and the ideal solution g* = {{<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}, {<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}, {<8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>, <8,0,0>}};

^{3}AV: average value;

^{4}SD: standard deviation.

**Table 2.**Decision results of the proposed MADM method for ρ $\in $ [1, 100] and W = (1/3, 1/3, 1/3).

ρ | S_{w}(g_{1}, g*), S_{w}(g_{2}, g*), S_{w}(g_{3}, g*), S_{w}(g_{4}, g*) | Ranking Order | AV | SD | Best Alternative |
---|---|---|---|---|---|

1 | 0.7431, 0.7546, 0.7500, 0.7755 | g_{4} > g_{2} > g_{3} > g_{1} | 0.7558 | 0.0121 | g_{4} |

2 | 0.7189, 0.7278, 0.7300, 0.7529 | g_{4} > g_{3} > g_{2} > g_{1} | 0.7324 | 0.0125 | g_{4} |

3 | 0.6968, 0.7039, 0.7112, 0.7336 | g_{4} > g_{3} > g_{2} > g_{1} | 0.7114 | 0.0138 | g_{4} |

4 | 0.6769, 0.6833, 0.6938, 0.7180 | g_{4} > g_{3} > g_{2} > g_{1} | 0.693 | 0.0156 | g_{4} |

5 | 0.6596, 0.6659, 0.6779, 0.7057 | g_{4} > g_{3} > g_{2} > g_{1} | 0.6773 | 0.0177 | g_{4} |

10 | 0.6019, 0.6088, 0.6193, 0.6717 | g_{4} > g_{3} > g_{2} > g_{1} | 0.6254 | 0.0274 | g_{4} |

15 | 0.5727, 0.5786, 0.5865, 0.6572 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5988 | 0.0341 | g_{4} |

20 | 0.556, 0.5608, 0.5671, 0.6495 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5834 | 0.0384 | g_{4} |

30 | 0.5381, 0.5415, 0.5459, 0.6415 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5668 | 0.0432 | g_{4} |

40 | 0.5289, 0.5315, 0.5349, 0.6374 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5582 | 0.0458 | g_{4} |

50 | 0.5232, 0.5254, 0.5281, 0.6350 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5529 | 0.0474 | g_{4} |

100 | 0.5118, 0.5128, 0.5142, 0.6300 | g_{4} > g_{3} > g_{2} > g_{1} | 0.5422 | 0.0507 | g_{4} |

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**MDPI and ACS Style**

Cui, W.; Ye, J.
Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality. *Symmetry* **2018**, *10*, 330.
https://doi.org/10.3390/sym10080330

**AMA Style**

Cui W, Ye J.
Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality. *Symmetry*. 2018; 10(8):330.
https://doi.org/10.3390/sym10080330

**Chicago/Turabian Style**

Cui, Wenhua, and Jun Ye.
2018. "Multiple-Attribute Decision-Making Method Using Similarity Measures of Hesitant Linguistic Neutrosophic Numbers Regarding Least Common Multiple Cardinality" *Symmetry* 10, no. 8: 330.
https://doi.org/10.3390/sym10080330