# Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

## 3. Proposed Procedure

#### 3.1. Normalization Process

**Example**

**1.**

#### 3.2. Consistency Analysis

- ${T}_{L}$ consistency index (${T}_{L}CI$) for a pair of alternatives evaluated as:$${T}_{L}CI\left({h}_{ij}^{*q}\right)=1-\frac{1}{\left|{h}_{ij}^{*}\right|}\underset{\beta =1}{{\displaystyle \sum}^{\left|{h}_{ij}^{*}\right|}}d\left({h}_{ij}^{*q\beta},\tilde{{h}_{ij}^{*q\beta}}\right)$$
- ${T}_{L}CI$ for alternatives ${a}_{i}$, $1\le i\le n$, is determined as:$${T}_{L}CI\left({a}_{i}\right)=\frac{1}{2(n-1)}\underset{j=1,j\ne i}{{\displaystyle \sum}^{n}}({T}_{L}CI({h}_{ij}^{*q})+{T}_{L}CI({h}_{ji}^{*q}))$$
- At the end, ${T}_{L}CI$ against NHFPR ${H}^{*q}$ is evaluated using average operator:$${T}_{L}CI\left({H}^{*q}\right)=\frac{1}{n}\underset{i=1}{{\displaystyle \sum}^{n}}{T}_{L}CI\left({a}_{i}\right)$$

#### 3.3. Consensus Analysis

- At the first level, the consensus degree on a pair of alternatives $({a}_{i},{a}_{j})$, denoted by $c{d}_{ij}$ is defined to estimate the degree of consensus amongst all experts on that pair of alternatives:$$c{d}_{ij}={s}_{ij}$$
- At the second level, the consensus degree on alternatives ${a}_{i}$ denoted by $C{D}_{i}$, is defined to determine the consensus degree amongst all the experts on that alternative:$$C{D}_{i}=\frac{1}{2(n-1)}\underset{j=1,j\ne i}{{\displaystyle \sum}^{n}}({s}_{ij}+{s}_{ji})$$
- At the third level, the consensus degree on the relation denoted by $CR$, is defined to calculate the global degree of consensus amongst all DMs:$$CR=\frac{1}{n}\underset{i=1}{{\displaystyle \sum}^{n}}C{D}_{i}$$

#### 3.4. Enhancement Mechanism

#### 3.5. Rating of Decision Makers

#### 3.6. Aggregated NHFPR

#### 3.7. Ranking of Alternatives

## 4. Comparative Example

**Consistency measures:**

- (i).
- The consistency measures of pairs of alternatives in NHFPRs ${H}^{*q},$$q=1,2,3,4$, are:$${T}_{L}CI\left({h}_{ij}^{*1}\right)=\left[\begin{array}{cccc}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1\\ 1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1\\ 1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1\\ 1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}1\end{array}\right],\phantom{\rule{4pt}{0ex}}{T}_{L}CI\left({h}_{ij}^{*2}\right)=\left[\begin{array}{cccc}1& 0.9500& 0.9667& 0.9667\\ 0.9500& 1& 0.9667& 0.9500\\ 0.9667& 0.9667& 1& 0.9667\\ 0.9667& 0.9500& 0.9667& 1\end{array}\right],$$$${T}_{L}CI\left({h}_{ij}^{*3}\right)=\left[\begin{array}{cccc}1& 0.9667& 0.9667& 1\\ 0.9667& 1& 0.9667& 1\\ 0.9667& 0.9667& 1& 1\\ 1& 1& 1& 1\end{array}\right],\phantom{\rule{4pt}{0ex}}{T}_{L}CI\left({h}_{ij}^{*4}\right)=\left[\begin{array}{cccc}1& 0.9500& 0.9667& 0.9667\\ 0.9500& 1& 0.9667& 0.9500\\ 0.9667& 0.9667& 1& 0.9667\\ 0.9667& 0.9500& 0.9667& 1\end{array}\right].$$
- (ii).
- The consistency measures of alternatives ${a}_{1},{a}_{2},{a}_{3}$ and ${a}_{4}$ are:$$\begin{array}{cccc}\hfill {T}_{L}CI\left({a}_{1}\right)& =(1,\phantom{\rule{4.pt}{0ex}}0.9611,\phantom{\rule{4.pt}{0ex}}0.9778,\phantom{\rule{4.pt}{0ex}}1),\hfill & \hfill \phantom{\rule{1.em}{0ex}}& {T}_{L}CI\left({a}_{2}\right)=(1,\phantom{\rule{4.pt}{0ex}}0.9556,\phantom{\rule{4.pt}{0ex}}0.9778,\phantom{\rule{4.pt}{0ex}}1),\hfill \\ \hfill {T}_{L}CI\left({a}_{3}\right)& =(1,\phantom{\rule{4.pt}{0ex}}0.9667,\phantom{\rule{4.pt}{0ex}}0.9778,\phantom{\rule{4.pt}{0ex}}1),\hfill & \hfill \phantom{\rule{1.em}{0ex}}& {T}_{L}CI\left({a}_{4}\right)=(1,\phantom{\rule{4.pt}{0ex}}0.9611,\phantom{\rule{4.pt}{0ex}}1,\phantom{\rule{4.pt}{0ex}}1).\hfill \end{array}$$
- (iii).
- The consistency measures of NHFPRs are:$$\begin{array}{cc}\hfill {T}_{L}CI\left({H}^{*1}\right)& =1,\phantom{\rule{3.33333pt}{0ex}}{T}_{L}CI\left({H}^{*2}\right)=0.961125,\hfill \\ \hfill {T}_{L}CI\left({H}^{*3}\right)& =0.98335,\phantom{\rule{3.33333pt}{0ex}}{T}_{L}CI\left({H}^{*4}\right)=1.\hfill \end{array}$$The global consistency index under the use of (11) is obtained as:$$CI=0.9861.$$Now, the consistency weights of the decision makers ${D}_{1},{D}_{2},{D}_{3}$ and ${D}_{4}$ are estimated using (12) as:$$\begin{array}{cc}\hfill Cw\left({D}_{1}\right)& =0.2535,\phantom{\rule{3.33333pt}{0ex}}Cw\left({D}_{2}\right)=0.2437,\hfill \\ \hfill Cw\left({D}_{3}\right)& =0.2493,\phantom{\rule{3.33333pt}{0ex}}Cw\left({D}_{4}\right)=0.2535.\hfill \end{array}$$

- (i).
- The consensus measures on each pair of alternatives are shown in the following, collectively aggregated, similarity matrix using (13):$$S=\left[\begin{array}{cccc}1& 0.9000& 0.7500& 0.8361\\ 0.9000& 1& 0.6486& 0.7736\\ 0.7500& 0.6486& 1& 0.8083\\ 0.8361& 0.7736& 0.8083& 1\end{array}\right].$$
- (ii).
- Based on similarity matrix S, the consensus measures on the alternatives ${a}_{1},{a}_{2},{a}_{3}$ and ${a}_{4},$ applying (14) are:$$\begin{array}{cc}\hfill C{D}_{1}& =0.8287,\phantom{\rule{3.33333pt}{0ex}}C{D}_{2}=0.7708,\hfill \\ \hfill C{D}_{3}& =0.7356,\phantom{\rule{3.33333pt}{0ex}}C{D}_{4}=0.8060.\hfill \end{array}$$
- (iii).
- The consensus measure on the information provided by the decision makers is:$$CR=0.7853.$$

## 5. Comparison

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Methods | Consistency Levels of | Consensus Level | Ranking | |||
---|---|---|---|---|---|---|

${\mathit{H}}^{1}$ | ${\mathit{H}}^{2}$ | ${\mathit{H}}^{3}$ | ${\mathit{H}}^{4}$ | $\left(\mathbf{CR}\right)$ | Order | |

Xu et al. [31] | $0.9750$ | $0.8833$ | $0.9389$ | $0.9847$ | $0.7653$ | ${a}_{3}\succ {a}_{2}\succ {a}_{1}\succ {a}_{4}$ |

Proposed (Round 1) | 1 | $0.9611$ | $0.9834$ | 1 | $0.7853$ | ${a}_{3}\succ {a}_{2}\succ {a}_{1}\succ {a}_{4}$ |

Proposed (Round 2) | 1 | 1 | 1 | 1 | $0.9009$ | ${a}_{3}\succ {a}_{2}\succ {a}_{1}\succ {a}_{4}$ |

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**MDPI and ACS Style**

Rehman, A.-u.; Wątróbski, J.; Faizi, S.; Rashid, T.; Tarczyńska-Łuniewska, M.
Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study. *Symmetry* **2020**, *12*, 1957.
https://doi.org/10.3390/sym12121957

**AMA Style**

Rehman A-u, Wątróbski J, Faizi S, Rashid T, Tarczyńska-Łuniewska M.
Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study. *Symmetry*. 2020; 12(12):1957.
https://doi.org/10.3390/sym12121957

**Chicago/Turabian Style**

Rehman, Atiq-ur, Jarosław Wątróbski, Shahzad Faizi, Tabasam Rashid, and Małgorzata Tarczyńska-Łuniewska.
2020. "Sustainable Decision Making Using a Consensus Model for Consistent Hesitant Fuzzy Preference Relations—Water Allocation Management Case Study" *Symmetry* 12, no. 12: 1957.
https://doi.org/10.3390/sym12121957