# Consistency Checking and Improving for Interval-Valued Hesitant Preference Relations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Some specific methods of checking and repairing consistency for IVFPR and HFPR have been presented based on the multiplicative transitivity ${r}_{ij}{r}_{jk}{r}_{ki}={r}_{ji}{r}_{kj}{r}_{ik}$. [22,23], which is explicitly interpreted in Definition 9. This formula is the necessary condition of multiplicative transitivity ${r}_{ik}{r}_{kj}={r}_{ij}$, which is described in detail in Definition 8, but not the sufficient condition of it. That is to say, ${r}_{ij}{r}_{jk}{r}_{ki}={r}_{ji}{r}_{kj}{r}_{ik}$ holds when ${r}_{ik}{r}_{kj}={r}_{ij}$ is satisfied. However, ${r}_{ik}{r}_{kj}={r}_{ij}$ may be not true when ${r}_{ij}{r}_{jk}{r}_{ki}={r}_{ji}{r}_{kj}{r}_{ik}$. Therefore, it is important to find a way which is sufficient and necessary when ${r}_{ik}{r}_{kj}={r}_{ij}$

- Develop the concepts of expectation additive consistency for IVHFPR, IVHMPR, and the multiplicative consistency for IVHMPR. Construct a connection between the IVHFPR with expectation additive consistency and the IVHMPR with multiplicative consistency
- Give a checking and improving method for the multiplicative consistency of IVHMPR that is directly deduced from the original preference data and is also applied to IVHFPR

## 2. Preliminaries

#### 2.1. The Concepts of IVHFS and IVHFPR

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. Interval Number and IVMPR

**Definition**

**3.**

**Definition**

**4.**

- Addition operation:${I}_{1}+{I}_{2}=\left[{u}_{l},{u}_{r}\right]+\left[{\upsilon}_{l},{\upsilon}_{r}\right]=\left[{u}_{l}+{\upsilon}_{l},{u}_{r}+{\upsilon}_{r}\right]$;
- Multiplication operation:${I}_{1}\times {I}_{2}=\left[{u}_{l},{u}_{r}\right]\times \left[{\upsilon}_{l},{\upsilon}_{r}\right]=\left[{u}_{l}\times {\upsilon}_{l},{u}_{r}\times {\upsilon}_{r}\right]$;
- Logarithm operation:${\mathrm{log}}_{a}{I}_{1}=\left[{\mathrm{log}}_{a}{u}_{l},{\mathrm{log}}_{a}{u}_{r}\right]$.

**Definition**

**5.**

**Definition**

**6.**

#### 2.3. The Related Concepts on Consistency for Preference Relation

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

## 3. The IVHMPR and Its Connection with IVHFPR

#### 3.1. The IVHMPR

**Definition**

**11.**

**Definition**

**12.**

**Lemma**

**1.**

**Proof.**

**Definition**

**13.**

- (1)
- The first kind of multiplication operation:$\underset{i=1}{\overset{N}{{\displaystyle \odot}}}{m}_{i}=\left[{\left({\displaystyle \prod _{\sigma =1}^{L}{\underset{\u02dc}{m}}_{i}^{\sigma}}\right)}^{\frac{1}{L}},{\left({\displaystyle \prod _{\sigma =1}^{L}{\tilde{m}}_{i}^{\sigma}}\right)}^{\frac{1}{L}}\right]$
- (2)
- The second kind of multiplication operation:$\underset{i=1}{\overset{N}{{\displaystyle \otimes}}}{m}_{i}=\left[{\displaystyle \prod _{\sigma =1}^{L}{\underset{\u02dc}{m}}_{i}^{\sigma}},\text{\hspace{0.17em}}{\displaystyle \prod _{\sigma =1}^{L}{\tilde{m}}_{i}^{\sigma}}\right]$
- (3)
- The mean value of$M$
**:**$E\left(M\right)={\left({\displaystyle \prod _{i=1}^{L\left(M\right)}E\left({m}_{i}\left(x\right)\right)}\right)}^{\frac{1}{L\left(M\right)}}$ - (4)
- The variance value of$M$
**:**$Var(M)=E{\left(M-E\left(M\right)\right)}^{2}$

**Definition**

**14.**

- (1)
- ${m}_{ij}=\left\{\underset{N}{\underset{\u23b5}{\left[1,1\right],\cdots ,\left[1,1\right]}}\right\}$,$N\in {Z}^{+}$, if$i=j$;
- (2)
- $L\left({m}_{ij}\right)=L\left({m}_{ji}\right)$,$\forall i,j=1,2,\cdots ,n$;
- (3)
- All the elements in${m}_{ij}$are in an ascending order when$i<j$(the criterion of comparing any two elements in an IVHMN follows Definition 5);
- (4)
- ${\underset{\u02dc}{m}}_{ij}^{\sigma}\times {\tilde{m}}_{ji}^{L\left({m}_{ij}\right)-\sigma +1}=1$,$\forall i,j=1,2,\cdots ,n$.

**Example**

**1.**

#### 3.2. The Connection between IVHFPR and IVHMPR

**Definition**

**15.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

- (a)
- $L\left(V\left({r}_{ij,1}\right)\right)=L\left(V\left({r}_{ij,2}\right)\right)$, $\forall i,j=1,2,\cdots ,n$;
- (b)
- $V\left({r}_{ij,1}\right)=V\left({r}_{ij,2}\right)$, $\forall i,j=1,2,\cdots ,n$.

- (c)
- $L\left({r}_{ij,1}\right)=L\left({r}_{ij,2}\right)$, $\forall i,j=1,2,\cdots ,n$;
- (d)
- ${r}_{ij,1}={r}_{ij,2}$, $\forall i,j=1,2,\cdots ,n$.

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

## 4. Analysis of the Consistency for IVHFPR and IVHMPR

#### 4.1. The Normalizations of IVHFPR and IVHMPR

**Definition**

**16.**

- (1)
- $\forall i,j=1,2,\cdots ,n$,$L\left(\#{r}_{ij}\right)={L}_{R}$, where${L}_{R}=\mathrm{max}\left(L\left({r}_{ij}\right)|i,j=1,2,\cdots ,n\right)$
- (2)
- If$i=j$, then$\#{r}_{ij}=\left\{\underset{{L}_{R}}{\underset{\u23b5}{\left[1,1\right],\cdots ,\left[1,1\right]}}\right\}$
- (3)
- If${L}_{R}>L\left({r}_{ij}\right)$, where$i<j$, then we should infill${L}_{R}-L\left({r}_{ij}\right)$number of$\left[\#{\underset{\u02dc}{r}}_{ij}^{\sigma},\#{\tilde{r}}_{ij}^{\sigma}\right]\notin {r}_{ij}$into${r}_{ij}$, where$\left[\#{\underset{\u02dc}{r}}_{ij}^{\sigma},\#{\tilde{r}}_{ij}^{\sigma}\right]=\left[\frac{{\displaystyle \sum _{\sigma =1}^{L\left({r}_{ij}\right)}{\underset{\u02dc}{r}}_{ij}^{\sigma}}}{L\left({r}_{ij}\right)},\text{\hspace{0.17em}}\frac{{\displaystyle \sum _{\sigma =1}^{L\left({r}_{ij}\right)}{\tilde{r}}_{ij}^{\sigma}}}{L\left({r}_{ij}\right)}\right]$. Furthermore, all the elements in$\#{m}_{ij}$($i<j$) are in a non-descending order
- (4)
- If$i>j$, then$\#{r}_{ij}=\left\{\left[\#{\underset{\u02dc}{r}}_{ij}^{\sigma},\#{\tilde{r}}_{ij}^{\sigma}\right]|\sigma =1,2,\cdots ,{L}_{R}\right\}$, where$\left[\#{\underset{\u02dc}{r}}_{ij}^{\sigma},\#{\tilde{r}}_{ij}^{\sigma}\right]=\left[1-\#{\tilde{r}}_{ji}^{L\left(\#{r}_{ij}\right)-\sigma +1},1-\#{\underset{\u02dc}{r}}_{ji}^{L\left(\#{r}_{ij}\right)-\sigma +1}\right]$

**Theorem**

**3.**

**Proof.**

- (1)
- If $i\le j$, then we calculate the mean value of each ${r}_{ij}$, and denote it as $E\left({r}_{ij}\right)=\left[\frac{{\displaystyle \sum _{\sigma =1}^{L\left({r}_{ij}\right)}{\underset{\u02dc}{r}}_{ij}^{\sigma}}}{L\left({r}_{ij}\right)},\frac{{\displaystyle \sum _{\sigma =1}^{L\left({r}_{ij}\right)}{\tilde{r}}_{ij}^{\sigma}}}{L\left({r}_{ij}\right)}\right]$.

- (2)
- If $i>j$, then we can prove that the theorem is true in the same way based on Axiom (4) of Definition 15. This completes the proof of Theorem 3. □

**Corollary**

**2.**

**Proof.**

**Example**

**3.**

**Definition**

**17.**

- (1)
- $\forall i,j=1,2,\cdots ,n$,$L\left(\#{m}_{ij}\right)={L}_{M}$, where${L}_{M}=\mathrm{max}\left(L\left({m}_{ij}\right),\text{\hspace{0.17em}}i,j=1,2,\cdots ,n\right)$
- (2)
- If$i=j$,$\#{m}_{ij}=\left\{\underset{{L}_{M}}{\underset{\u23b5}{\left[1,1\right],\cdots ,\left[1,1\right]}}\right\}$
- (3)
- If${L}_{M}>L\left({m}_{ij}\right)$, where$i<j$, then we should add${L}_{M}-L\left({m}_{ij}\right)$number of$\left[\#{\underset{\u02dc}{m}}_{ij}^{\sigma},\#{\tilde{m}}_{ij}^{\sigma}\right]\notin {m}_{ij}$into${m}_{ij}$, and calculate it by the formula$\left[\#{\underset{\u02dc}{m}}_{ij}^{\sigma},\#{\tilde{m}}_{ij}^{\sigma}\right]=\left[{a}^{\frac{{\displaystyle \sum _{\sigma =1}^{L\left({m}_{ij}\right)}{\mathrm{log}}_{a}{\underset{\u02dc}{m}}_{ij}^{\sigma}}}{L\left({m}_{ij}\right)}},{a}^{\frac{{\displaystyle \sum _{\sigma =1}^{L\left({m}_{ij}\right)}{\mathrm{log}}_{a}{\tilde{m}}_{ij}^{\sigma}}}{L\left({m}_{ij}\right)}}\right]$
- (4)
- All the elements in$\#{m}_{ij}$($i<j$) are in a non-descending order.
- (5)
- If$i>j$, then$\#{m}_{ij}=\left\{\left[\#{\underset{\u02dc}{m}}_{ij}^{\sigma},\#{\tilde{m}}_{ij}^{\sigma}\right]|\sigma =1,2,\cdots ,{L}_{M}\right\}$, where$\left[\#{\underset{\u02dc}{m}}_{ij}^{\sigma},\#{\tilde{m}}_{ij}^{\sigma}\right]=\left[\frac{1}{\#{\tilde{m}}_{ji}^{L\left(\#{m}_{ij}\right)-\sigma +1}},\frac{1}{\#{\underset{\u02dc}{m}}_{ji}^{L\left(\#{m}_{ij}\right)-\sigma +1}}\right]$.

**Theorem**

**4.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Example**

**4.**

#### 4.2. Checking and Improving the Consistency of IVHFPR and IVHMPR

**Definition**

**18.**

**Definition**

**19.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Definition**

**20.**

Algorithm 1. |

Input: The IVHFPR $R$ or the IVHMPR $M$.Output: the acceptably expectation additively consistent IVHFPR $\overline{R}$ and the acceptably multiplicatively consistent IVHMPR $\overline{M}$.Step 1. If the preference information provided by the decision maker is expressed as an IVHFPR, then we should transform it into an IVHMPR and normalize it through the methods presented in Section 3 and Section 4.1. When the preferences are directly expressed as an IVHMPR, we only need to normalize it by the approach defined in Definition 17. Let ${N}_{M}\left(M\right)$ be the final normalized IVHMPR, go to Step 2.Step 2. Choose the value of the threshold $\lambda $. Based on Definitions 7 and 20, check whether the multiplicative consistency of ${N}_{M}\left(M\right)$ is up to level $\lambda $. Denote the consistency index of ${N}_{M}\left(M\right)$ by $C.I.\left({N}_{M}\left(M\right)\right)$. If $C.I.\left({N}_{M}\left(M\right)\right)<\lambda $, then go to Step 5. Otherwise, go to Step 3.Step 3. Construct an induced matrix $C={B}^{2}\left({N}_{M}\left(M\right)\right)-nB\left({N}_{M}\left(M\right)\right)$, where ${B}^{2}\left({N}_{M}\left(M\right)\right)=$ $B\left({N}_{M}\left(M\right)\right)$ $\otimes B\left({N}_{M}\left(M\right)\right)$.Step 4. Identify all the elements in $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$; there should be one of the following situations:Case 1. All of the elements in $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$ are zero elements $\left\{\left[0,0\right]\right\}$, which means that the present normalized IVHMPR is multiplicatively consistent. In this case, go to Step 5.Case 2. There are some zero elements in $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$. Thus, we should adjust the corresponding original element(s) in the normalized IVHMPR ${N}_{M}\left(M\right)$ according to the following sub-steps:**Sub-step 1.**Find the maximal value within $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$ and denote its location by $\left(\xi ,\eta \right)$. Go to the next sub-step.**Sub-step 2.**Denote the $\xi $th row in ${N}_{M}\left(M\right)$ by $\#{m}_{\xi \cdot}=\left(\#{m}_{\xi 1}\text{\hspace{0.17em}}\#{m}_{\xi 2}\text{\hspace{0.17em}}\cdots \#{m}_{\xi {L}_{{N}_{M}\left(M\right)}}\right)$ and denote the $\eta $th column in ${N}_{M}\left(M\right)$ by $\#{m}_{\cdot \eta}=\left(\#{m}_{1\eta}\text{\hspace{0.17em}}\#{m}_{2\eta}\text{\hspace{0.17em}}\cdots \#{m}_{{L}_{{N}_{M}\left(M\right)}\eta}\right)$. Compute $\#{m}_{\xi \cdot}\cdot \#{m}_{\cdot \eta}-(\text{\hspace{0.17em}}\underset{{L}_{{N}_{M}\left(M\right)}}{\underset{\u23b5}{\#{m}_{\xi \eta}\text{\hspace{0.17em}}\#{m}_{\xi \eta}\text{\hspace{0.17em}}\cdots \#{m}_{\xi \eta}}})$ $=(\#{m}_{\xi 1}\cdot \#{m}_{1\eta}\text{\hspace{0.17em}}-\#{m}_{\xi \eta}\#{m}_{\xi 2}\cdot \#{m}_{2\eta}-\#{m}_{\xi \eta}\cdots \#{m}_{\xi {L}_{{N}_{M}\left(M\right)}}\cdot \#{m}_{{L}_{{N}_{M}\left(M\right)}\eta}-\#{m}_{\xi \eta})$ and find the absolute minimal nonzero value within it, which is denoted by $\#{m}_{\xi k}\cdot \#{m}_{k\eta}-\#{m}_{\xi \eta}$. Go to the next sub-step.**Sub-step 3.**Adjust $\#{m}_{\xi \eta}$ as $\#{m}_{\xi k}$ and adjust $\#{m}_{\eta \xi}$ according to $\#{m}_{\xi \eta}$ based on Definition 17. Denote the adjusted matrix as a new IVHMPR $M$, go to Step 2.
Step 5. Denote the IVHMPR as $\overline{M}$ and compute the corresponding IVHFPR $\overline{R}$ using Equation (3). Output the results and end Algorithm 1. |

## 5. Illustrative Example

**Step 1.**Normalize $R$ and translate it into an IVHMPR as $\#M$ according to Theorem 5 with the base number $a=6$:

**Step 2.**Let $\lambda =0.02$ be the threshold of consistency index. According to Definitions 7 and 20, we know that the multiplicatively consistent indexes of $B\left(\#M\right)$ are 0.00, 0.03, 0.01, 0.05, 0.00, and 0.02. Therefore, the multiplicative consistency of $\#M$ should be improved.

**Step 3.**Construct an induced matrix $C={B}^{2}\left({N}_{M}\left(M\right)\right)-nB\left({N}_{M}\left(M\right)\right)$ as follows:

**Step 4.**Identify all the elements in $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$, where

**Substep 1.**From the above $\left|E\left(C\right)\right|+\left|Var\left(C\right)\right|$, we find the maximal value to be $26.93$ and denote its location as $\left(\xi ,\eta \right)=\left(1,3\right)$. Go to the next sub-step.

**Substep 2.**Denote the 1st row in $\#M$ as $\#{m}_{1\cdot}$ and denote the 3rd column in ${N}_{M}\left(M\right)$ as $\#{m}_{\cdot 3}$. Compute $\#{m}_{1\cdot}\cdot \#{m}_{\cdot 3}-\#{m}_{13}$=$\left(\left[2.34,46,21\right],\left[2.65,148.67\right],\left[1.83,167.67\right]\right)$ and find the absolute minimal nonzero value within it as 1.83, which is denoted by $\#{\underset{\u02dc}{m}}_{12}\cdot \#{\underset{\u02dc}{m}}_{23}-\#{\underset{\u02dc}{m}}_{13}$. Go to the next sub-step.

**Substep 3.**Adjust all elements in $\#{m}_{13}$ as $\#{\underset{\u02dc}{m}}_{13}$=1.71. Denote the adjusted matrix as a new IVHMPR $M$, go to Step 2.

**Step 5.**Output $\#{M}^{2}$ as the acceptably multiplicatively consistent IVHMPR $\overline{M}$ and output $\#{R}^{2}$ as the acceptably additively consistent IVHFPR $\overline{R}$. Obviously, $\overline{R}$ also is an acceptably expectation additively consistent IVHMPR. End Algorithm 1.

## 6. Comparisons

#### 6.1. Comparing Algorithm 1 to the Method with ${D}^{2}-nD=O$

#### 6.2. Comparing Algorithm 1 to the Method with 0–1 Programming Models

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chen, N.; Xu, Z.S.; Xia, M.M. Interval-valued hesitant preference relations and their applications to group decision making. Knowl.-Based Syst.
**2013**, 37, 528–540. [Google Scholar] [CrossRef] - Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Chen, N.; Xu, Z.S. Properties of interval-valued hesitant fuzzy sets. J. Intell. Fuzzy Syst.
**2014**, 27, 143–158. [Google Scholar] - Wei, G.W.; Lin, R.; Wang, H.J. Distance and similarity measures for hesitant interval-valued fuzzy sets. J. Intell. Fuzzy Syst.
**2014**, 27, 19–36. [Google Scholar] - Farhadinia, B. Study on division and subtraction operations for hesitant fuzzy sets, interval-valued hesitant fuzzy sets and typical dual hesitant fuzzy sets. J. Intell. Fuzzy Syst.
**2015**, 28, 1393–1402. [Google Scholar] - Farhadinia, B. Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets. Inf. Sci.
**2013**, 240, 129–144. [Google Scholar] [CrossRef] - Zhai, Y.L.; Xu, Z.S.; Liao, H.C. Measures of probabilistic interval-valued intuitionistic hesitant fuzzy sets and the application in reducing excessive medical examinations. IEEE Trans. Fuzzy Syst.
**2017**, 26, 1651–1670. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z.S.; Xu, J.P. Deriving a ranking from hesitant fuzzy preference relations under group decision making. IEEE Trans. Cybern.
**2014**, 44, 1328–1337. [Google Scholar] [CrossRef] - Li, L.G.; Peng, D.H. Interval-valued hesitant fuzzy Hamacher synergetic weighted aggregation operators and their application to shale gas areas selection. Math. Probl. Eng.
**2014**, 1, 759–765. [Google Scholar] [CrossRef] - Wei, G.W.; Zhao, X.F.; Lin, R. Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2013**, 46, 43–53. [Google Scholar] [CrossRef] - Jun, Y. Interval-valued hesitant fuzzy prioritized weighted aggregation operators for multiple attribute decision making. J. Algorithms Comput. Technol.
**2014**, 8, 179–192. [Google Scholar] - Meng, F.Y.; Wang, C.; Chen, X.H.; Zhang, Q. Correlation coefficients of interval-valued hesitant fuzzy sets and their application based on the Shapley function. Int. J. Intell. Syst.
**2016**, 31, 17–43. [Google Scholar] [CrossRef] - Meng, F.Y.; Chen, X.H. An approach to interval-valued hesitant fuzzy multi-attribute decision making with incomplete weight information based on hybrid Shapley operators. Informatica
**2014**, 25, 617–642. [Google Scholar] [CrossRef] - Tavakkoli-Moghaddam, R.; Gitinavard, H.; Mousavi, S.M.; Siadat, A. An interval-valued hesitant fuzzy TOPSIS method to determine the criteria weights. Lect. Notes Bus. Inf. Process.
**2015**, 218, 157–169. [Google Scholar] - Wei, G.W.; Zhao, X.F.; Lin, R.; Wang, H.J. Models for hesitant interval-valued fuzzy multiple attribute decision making based on the correlation coefficient with incomplete weight information. J. Intell. Fuzzy Syst.
**2014**, 26, 1631–1644. [Google Scholar] - Pérez-Fernández Raúl Alonso, P.; Bustince, H.; Díazd, I.; Montes, S. Applications of finite interval-valued hesitant fuzzy preference relations in group decision making. Inf. Sci.
**2016**, 326, 89–101. [Google Scholar] [CrossRef] - Khalid, A.; Beg, I. Incomplete interval-valued hesitant fuzzy preference relations in decision making. Iran. J. Fuzzy Syst.
**2018**, 15, 6. [Google Scholar] [CrossRef] - Zhang, Z.M. Multi-criteria decision-making using interval-valued hesitant fuzzy QUALIFLEX methods based on a likelihood-based comparison approach. Neural Comput. Appl.
**2017**, 28, 1835–18554. [Google Scholar] [CrossRef] - Tang, J.; Meng, F.Y. Ranking objects from group decision making with interval-valued hesitant fuzzy preference relations in view of additive consistency and consensus. Knowledge-Based Systems
**2018**, 162, 46–61. [Google Scholar] [CrossRef] - Zhang, Y.N.; Tang, J.; Meng, F.Y. Programming model-based method for ranking objects from group decision making with interval-valued hesitant fuzzy preference relations. Appl. Intell.
**2018**. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.S.; Xia, M.M. Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J. Intell. Fuzzy Syst.
**2014**, 27, 2969–2985. [Google Scholar] - Liao, H.C.; Xu, Z.S.; Xia, M.M. Multiplicative consistency of hesitant fuzzy preference relation and ITS application in group decision making. Int. J. Inf. Technol. Decis. Mak.
**2014**, 13, 47–76. [Google Scholar] [CrossRef] - Liu, H.F.; Xu, Z.S.; Liao, H.C. The multiplicative consistency index of hesitant fuzzy preference relation. IEEE Trans. Fuzzy Syst.
**2016**, 24, 82–93. [Google Scholar] [CrossRef] - Moore, R.E. Intervalanalysis; Prentiee-Hall INE.: Englewood Cliffs, NJ, USA, 1966. [Google Scholar]
- Xu, Z.S.; Da, Q.L. Research on method for ranking interval numbers. Syst. Eng.
**2001**, 19, 94–96. [Google Scholar] - Saaty, T.L.; Vargas, L.G. Uncertainty and rank order in the analytic hierarchy process. Eur. J. Oper. Res.
**1987**, 32, 107–117. [Google Scholar] [CrossRef] - Saaty, T.L. Modeling unstructured decision problems—The theory of analytical hierarchies. Math. Comput. Simul.
**1978**, 20, 147–158. [Google Scholar] [CrossRef] - Tanino, T. Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems. Fuzzy Sets Syst.
**1984**, 12, 117–131. [Google Scholar] [CrossRef] - Ergu, D.; Kou, G.; Peng, Y.; Shi, Y. A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP. Eur. J. Oper. Res.
**2011**, 213, 246–259. [Google Scholar] [CrossRef] - Jemal, A.; Bray, F.; Center, M.M.; Ferlay, J.; Ward, E.; Forman, D. Global cancer statistics. CA Cancer J. Clin.
**1999**, 49, 33–64. [Google Scholar] [CrossRef] - Chen, W.; Zheng, R.; Zhang, S.; Zhao, P.; Li, G.; Wu, L.; He, J. Report of incidence and mortality in China cancer registries 2009. Chin. J. Cancer Res.
**2013**, 24, 171–180. [Google Scholar] [CrossRef]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhai, Y.; Xu, Z.
Consistency Checking and Improving for Interval-Valued Hesitant Preference Relations. *Symmetry* **2019**, *11*, 466.
https://doi.org/10.3390/sym11040466

**AMA Style**

Zhai Y, Xu Z.
Consistency Checking and Improving for Interval-Valued Hesitant Preference Relations. *Symmetry*. 2019; 11(4):466.
https://doi.org/10.3390/sym11040466

**Chicago/Turabian Style**

Zhai, Yuling, and Zeshui Xu.
2019. "Consistency Checking and Improving for Interval-Valued Hesitant Preference Relations" *Symmetry* 11, no. 4: 466.
https://doi.org/10.3390/sym11040466