# Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Motivation and Contributions

#### 3.1. Motivation and Objective

#### 3.2. Contribution

- The well-known MCDM techniques with hesitant fuzzy environment: MOORA, TOPSIS and VIKOR are used to select the best water distribution system of drought-prone areas.
- The first proposed method is to weights the criteria via extended standard deviation weighting technique, called as HFSDV-MOORA.
- The second proposed method uses the extended standard deviation weighting technique to assign weights to the criteria under HFSDV-TOPSIS.
- The third proposed method uses the extended standard deviation weighting technique to assign weights to the criteria under hesitant fuzzy environment in order to give a compromise solution, because it provides a maximum “group utility” of the “majority” and a minimum of the individual regret of the “opponent” called as HFSDV-VIKOR.
- The ranking result concluded that the proposed SDV-MCDM methods have outperformed the existing WEM-MCDM in the literature review by giving compromise and best solution.
- The comparison study shows that the proposed three methods(SDV-MOORA, TOPSIS, VIKOR) with hesitant fuzzy environment have perfect ranking.
- The sensitivity analysis is to analyze the ranking order when changing the significance of objective weights by making two cases.
- Testing and analyzing the two cases of sensitivity analysis with our proposed MCDM methods and provide the ranking order of comparison study of this sensitive part. And also test the correlation co-efficient between the proposed HFSDV-MOORA, HFSDV-TOPSIS and HFSDV-VIKOR by use of Spearman’s rank correlation method.
- Our methods provide the best alternative water distribution system for the ground water irrigation scheme.The solution is rectifying the water problem in drought-prone areas of selected three district. Moreover, we compare our proposed SDV hesitant fuzzy MOORA method with some research articles Li, (2014); Xu, (2013); Liao, (2013) [50,51,52]. And also compared our proposed weight finding results with entropy weights.

## 4. Preliminaries

**Definition**

**1.**

**Example**

**1.**

**Definition**

**2.**

**Definition**

**3.**

- ${h}_{1}>{h}_{2}$, if $s({h}_{1})>s({h}_{2})$;
- ${h}_{1}<{h}_{2}$, if $s({h}_{1})<s({h}_{2})$;
- ${h}_{1}={h}_{2}$, if $s({h}_{1})=s({h}_{2})$.

## 5. The MCDM Methods in Hesitant Environment with Standard Deviation Weights

#### 5.1. HFSDV for Calculating Criteria Weights

**Step**

**1:**In first, construct the score matrix (SM) by using (Definition 3), we get score values of given hesitant fuzzy elements.

**Step**

**2:**

#### 5.2. HFSDV-MOORA

**Step 1:**

**Step 2:**

**Step 3:**

**Step 4:**

**Step 5:**

**Step 6:**

#### 5.3. HFSDV-TOPSIS

**Step 1:**Construct normalized score matrix $(\tilde{{C}_{ij}})$.

**Step 2:**Construct the weighted normalized matrix by multiplying each column of the normalized matrix by its associated weight.The weight values are find by using (4). Hence we get a new matrix

**Step 3:**Find the positive ideal solution (PIS)

**Step 4:**Find the negative ideal solution (NIS)

**Step 5:**Determine the separation from positive ideal solution ${S}_{i}^{+}$

**Step 6:**Determine the separation from negative ideal solution ${S}_{i}^{-}$

**Step 7:**Calculate the relative closeness to the ideal solution

**Step 8:**Based on the decrease order values of closeness to the ideal solution, alternatives are ranked from higher value to lower value.

#### 5.4. HFSDV-VIKOR

**Step 1:**Construct normalized score matrix $(\tilde{{X}_{ij}})$.

**Step 2:**Calculate the positive ideal solution (PIS)

**Step 3:**Calculate the negative ideal solution (NIS)

**Step 4:**Calculate utility measures$({S}_{ij})$. The weight values are find by using (4).

**Step 5:**Calculate utility measure(S) and regard measure(G) for each alternative

**Step 6:**Computation the index value $({Q}_{i})$ ith alternative as follows,

**Step 7:**These values are sorted by rank, it is based on S G and Q in decreasing order.

**Step 8:**Consider the alternative i, corresponding to ${Q}_{\left[i\right]}$ as a trade-off solution if the two conditions given below are satisfied:

**${C}_{1}$. Acceptable advantage:**

## 6. Study Area

#### Description of Problem

**(i). Closed concrete pipes laid along agricultural land $({P}_{1})$**

**(ii). Closed concrete pipes laid along roadsides of national highway $({P}_{2})$**

**(iii). Concrete canal through agricultural land $({P}_{3})$**

**(iv). Earthen canal through agricultural land $({P}_{4})$**

**(i) Effectiveness $({R}_{1})$**

**(ii) Economic $({R}_{2})$**

**(iii) Social $({R}_{3})$**

**(iv) Operational time $({R}_{4})$**

**(v) Environment production $({R}_{5})$**

## 7. Numerical Evaluation

**Step 1:**

**Score matrix:**

**Step 2:**

**Normalized score matrix:**

**Step 3:**

**Step 4:**

**Step 5:**Here, the HF- benefcial and HF-non benefcial criteria value are calculated by using (3) & (4). Here, Effectiveness $({R}_{1})$, Social $({R}_{4})$ and Environmental production $({R}_{5})$ are the benefcial criteria; Economic $({R}_{2})$ and Operational time $({R}_{3})$ are the non-benefcial criteria values are give as follows,

**Step 6:**By use of performance of alternatives, the selected alternatives are ranked in decreasing order. The resultant alternative values are shown in Table 4. Here, ${P}_{1}$ is considered as one of the best water distribution system. Also, the result is shown in the Figure 6. Thus, we conclude that one of the most suitable water distribution system is Closed concrete pipes laid along agricultural land.

**Step 1:**

**Step 2:**

**Normalized score matrix:**

**Step 3:**

**Step 4:**

**Step 5:**

**Step 6:**

**Step 7:**

**Step 1:**

**Step 2:**

**Step 3:**

**Step 4:**

**Step 5:**

**Step 6:**

**Step 7:**

## 8. Comparison Analysis

- In HFSDV-MOORA ${P}_{1}>{P}_{2}>{P}_{4}>{P}_{3}$
- In HFSDV-TOPSIS ${P}_{1}>{P}_{2}>{P}_{4}>{P}_{3}$
- In HFSDV-VIKOR ${P}_{1}>{P}_{2}>{P}_{4}>{P}_{3}$

- In HFWEM-MOORA ${P}_{1}>{P}_{2}>{P}_{4}>{P}_{3}$
- In HFWEM-TOPSIS ${P}_{1}>{P}_{2}>{P}_{3}>{P}_{4}$
- In HFWEM-VIKOR ${P}_{1}>{P}_{2}>{P}_{3}>{P}_{4}$

## 9. Sensitivity Analysis

**Case 1:**Equal importance for assigning weights: positive criteria = 0.5 and negative criteria = 0.5.

**Case 2:**Assigning weights when importance to positive criteria: positive criteria= 1 and negative criteria = 0.

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

DM | Decision Making |

MCDM | Multi-Criteria Decision Making |

MAGDM | Multi- Attribute Group Decision Making |

HMCDM | Hesitant Multi-Criteria Decision Making |

TMC | Thousand Million Cubic |

HFLTS | Hesitant Fuzzy Linguistic Term Set |

HFS | Hesitant Fuzzy Set |

HFE | Hesitant Fuzzy Element |

HFMCDM | Hesitant Fuzzy Multi-Criteria Decision Making |

MOORA | Multi objective optimization on the basis of ratio analysis |

TOPSIS | Technique for order preference by similarity to an ideal solution |

VIKOR | VIsekriterijumsko KOmpromisno Rangiranje |

HFSDV | Hesitant Fuzzy Standard Deviation |

SDV | Standard Deviation |

WEM | Weighted Entropy Measure |

SAW | Simple Additive Weighting |

SA | Similarity Analysis |

DA | Decision Analysis |

DA | Decision Analysis |

IVIFN | Interval Valued Intuitionistic Fuzzy Numbers |

FUDS | FUzzy Decision System |

AASP | Aringar Anna Sugar Project |

PIS | Positive Ideal Solution |

PIS | Positive Ideal Solution |

NIS | Negative Ideal Solution |

FPIS | Fuzzy Positive Ideal Solution |

FNIS | Fuzzy Negative Ideal Solution |

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**Figure 11.**Comparison results: (

**a**)—Comparison results between HFSDV-MOORA and HFWEM-MOORA, (

**b**)—Comparison results between HFSDV-TOPSIS and HFWEM-TOPSIS, (

**c**)—Comparison results between HFSDV-VIKOR and HFWEM-VIKOR

**Figure 12.**Sensitivity analysis results: (

**a**)—When ${W}_{1}=0.20$, ${W}_{2}=0.25$, ${W}_{3}=0.15$, ${W}_{4}=0.25$, ${W}_{5}=0.15$, (

**b**)—When ${W}_{1}=0.5$, ${W}_{2}=0$, ${W}_{3}=0.25$, ${W}_{4}=0$, ${W}_{5}=0.25$.

Authors | Study of MCDM Methods | Applications |
---|---|---|

Arabsheybani et al. (2018) [1] | An integrated fuzzy MOORA and FMEA technique | Supplier selection |

Brauers and Zavadskas, (2006) [2] | MOORA method | Privatization in a transition economy |

Karande and Chakraborty, (2012) [3] | Fuzzy-MOORA method | ERP system selection |

Cevik Onar et al. (2014) [4] | HF-TOPSIS and Interval Type-2 Fuzzy AHP | Strategic decision selection |

Sun and Ouyang, (2015) [5] | HFMADM Based on TOPSIS With Entropy | An energy police selection |

Zhang et al. (2016) [6] | FMCGDM: IVDHF-TOPSIS and IVDHF-VIKOR | Community development |

Ozcelik et al. (2014) [7] | Hybrid MOORA-Fuzzy | Special education center selection |

Buyukozkan and Gocer, (2017) [8] | An extension of MOORA based on IVIFN | Digital supply chain |

Santawy and Ahmed, (2012) [9] | SDV-MOORA | Analysis of project selection |

Achebo and Odinikuku, (2015) [10] | SDV and MOORA | Gas metal arc welding process |

Myers and Sirois, (2006) [11] | Spearman correlation coefficients | A general view |

Beg and Rashid, (2013) [12] | HF-TOPSIS with linguistic | Investment company |

Tavana et al. (2018) [13] | An extended VIKOR | Decision maker’s risk |

Suh Park et al. (2019) [14] | Integrated weighting fuzzy VIKOR | Mobile services |

Zhang and Wei, (2013) [15] | Extension of VIKOR method | Enterprises projects |

Narayanamoorthy et al. (2019) [16] | IVIHF entropy based VIKOR | Industrial robots selection |

Narayanamoorthy et al. (2019) [17] | HF-MAUT with HF-CRITIC method | Water management |

Narayanamoorthy et al. (2019) [18] | Normal wiggly dual hesitant fuzzy sets | Energy management |

Geetha et al. (2019)[19] | HF-MULTIMOORA method | Healthcare waste management |

Narayanamoorthy et al. (2020) [20] | HF-MOOSRA with HF-SOWIA method | Bio-medical waste management |

Nadaban et al. (2016) [21] | Fuzzy TOPSIS | Team alternative selection |

Senvar et al. (2016) [22] | Hesitant Fuzzy TOPSIS | Hospital Site Selection |

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ⋯ | ${\mathit{R}}_{\mathit{n}}$ | |
---|---|---|---|---|

${P}_{1}$ | $\u2329x({h}_{11})\u232a$ | $\u2329x({h}_{12})\u232a$ | ⋯ | $\u2329x({h}_{1n})\u232a$ |

${P}_{2}$ | $\u2329x({h}_{21})\u232a$ | $\u2329x({h}_{22})\u232a$ | ⋯ | $\u2329x({h}_{2n})\u232a$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

${P}_{m}$ | $\u2329x({h}_{m1})\u232a$ | $\u2329x({h}_{m2})\u232a$ | ⋯ | $\u2329x({h}_{mn})\u232a$ |

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | |
---|---|---|---|---|---|

${P}_{1}$ | $\{0.6,0.7,0.8,0.9\}$ | $\{0.5,0.6,0.7\}$ | $\{0.9,0.6,0.8\}$ | $\{0.7,0.8,0.2,0.9\}$ | $\{0.7,0.9,0.8\}$ |

${P}_{2}$ | $\{0.5,0.7,0.6\}$ | $\{0.3,0.4,0.6,0.5,\}$ | $\{0.4,0.5,0.3,0.7\}$ | $\{0.6,0.7,0.4\}$ | $\{0.7,0.5,0.8\}$ |

${P}_{3}$ | $\{0.3,0.5,0.6,0.7\}$ | $\{0.4,0.7,0.5,0.6,\}$ | $\{0.2,0.3,0.4,0.6\}$ | $\{0.3,0.4,0.5\}$ | $\{0.2,0.4,0.5,0.6\}$ |

${P}_{4}$ | $\{0.4,0.6,0.7,0.8\}$ | $\{0.8,0.5,0.7,0.9\}$ | $\{0.5,0.7,0.9,0.6\}$ | $\{0.2,0.3,0.1,0.4\}$ | $\{0.1,0.2,0.3\}$ |

Alternatives | Performance of Alternative Values | Rank |
---|---|---|

${P}_{1}$ | $0.299$ | 1 |

${P}_{2}$ | $0.117$ | 2 |

${P}_{3}$ | $-0.069$ | 4 |

${P}_{4}$ | $0.055$ | 3 |

${\mathit{R}}_{1}$ | ${\mathit{R}}_{2}$ | ${\mathit{R}}_{3}$ | ${\mathit{R}}_{4}$ | ${\mathit{R}}_{5}$ | |
---|---|---|---|---|---|

${P}_{1}$ | 0 | $0.087$ | 0 | 0 | 0 |

${P}_{2}$ | $0.127$ | $0.191$ | $0.157$ | $0.042$ | $0.045$ |

${P}_{3}$ | $0.191$ | $0.121$ | $0.21$ | $0.127$ | $0.128$ |

${P}_{4}$ | $0.106$ | 0 | $0.049$ | $0.203$ | $0.205$ |

${\mathit{G}}_{\mathit{i}}$ | ${\mathit{S}}_{\mathit{i}}$ | ${\mathit{Q}}_{\mathit{i}}$ | |
---|---|---|---|

${P}_{1}$ | 0.087 | 0.087 | 0 |

${P}_{2}$ | 0.191 | 0.562 | 0.785 |

${P}_{3}$ | 0.191 | 0.777 | 0.941 |

${P}_{4}$ | 0.106 | 0.563 | 0.845 |

Alternatives | HFWEM-TOPSIS | Rank | HFWEM-VIKOR | Rank | HFWEM-MOORA | Rank |
---|---|---|---|---|---|---|

${P}_{1}$ | 0.901 | 1 | 0 | 1 | 0.125 | 1 |

${P}_{2}$ | 0.177 | 2 | 0.6251 | 2 | 0.058 | 2 |

${P}_{3}$ | 0.193 | 3 | 0.8414 | 3 | 0.0101 | 4 |

${P}_{4}$ | 0.245 | 4 | 0.8763 | 4 | 0.0324 | 3 |

Alternatives | HFSDV-TOPSIS | Rank | HFSDV-VIKOR | Rank | HFSDV-MOORA | Rank |
---|---|---|---|---|---|---|

${P}_{1}$ | 0.893 | 1 | 0 | 1 | 0.299 | 1 |

${P}_{2}$ | 0.0783 | 2 | 0.785 | 2 | 0.117 | 2 |

${P}_{3}$ | 0.033 | 4 | 0.941 | 4 | -0.069 | 4 |

${P}_{4}$ | 0.037 | 3 | 0.845 | 3 | 0.055 | 3 |

Proposed Methods | HFSDV-MOORA | HFSDV-TOPSIS | HFSDV-VIKOR |
---|---|---|---|

HFSDV-MOORA | - | 1 | 1 |

HFSDV-TOPSIS | - | 1 | |

HFSDV-VIKOR | - |

Methods | HFWEM-MOORA | HFWEM-TOPSIS | HFWEM-VIKOR |
---|---|---|---|

HFWEM-MOORA | - | 0.8 | 0.8 |

HFWEM-TOPSIS | - | 0.8 | |

HFWEM-VIKOR | - |

Alternatives | Li, (2014) Weights | Rank of Alternatives | Proposed SDV Weights | Rank of Alternatives |
---|---|---|---|---|

${A}_{1}$ | w1 = 0.3 | 2 | w1 = 0.2470 | 2 |

${A}_{2}$ | w2 = 0.25 | 1 | w2 = 0.2430 | 4 |

${A}_{3}$ | w3 = 0.25 | 3 | w3 = 0.2429 | 3 |

${A}_{4}$ | w4 = 0.2 | 4 | w4 = 0.2698 | 1 |

Alternatives | Xu and Zhang, (2013) Weights | Rank of Alternatives | Proposed SDV Weights | Rank of Alternatives |
---|---|---|---|---|

${A}_{1}$ | w1 = 0.2341 | 5 | w1 = 0.25 | 5 |

${A}_{2}$ | w2 = 0.2474 | 3 | w2 = 0.24 | 3 |

${A}_{3}$ | w3 = 0.3181 | 2 | w3 = 0.30 | 2 |

${A}_{4}$ | w4 = 0.2004 | 1 | w4 = 0.20 | 4 |

${A}_{5}$ | - | 4 | - | 4 |

**Table 13.**The Comparison results with Liao and Xu, (2013) [52].

Alternatives | Liao and Xu, (2013) Weights | Rank of Alternatives | Proposed SDV Weights | Rank of Alternatives |
---|---|---|---|---|

${A}_{1}$ | w1 = 0.1 | 4 | w1 = 0.229 | 2 |

${A}_{2}$ | w2 = 0.2 | 1 | w2 = 0.23 | 1 |

${A}_{3}$ | w3 = 0.4 | 3 | w3 = 0.268 | 3 |

${A}_{4}$ | w4 = 0.3 | 2 | w4 = 0.273 | 4 |

Alternatives | HFSDV-MOORA | Rank | HFSDV-TOPSIS | Rank | HFSDV-VIKOR | Rank |
---|---|---|---|---|---|---|

${P}_{1}$ | 0.2137 | 2 | 0.8461 | 1 | 0 | 1 |

${P}_{2}$ | 0.3684 | 1 | 0.5174 | 2 | 0.1391 | 2 |

${P}_{3}$ | −0.1283 | 4 | 0.3485 | 4 | 0.1844 | 3 |

${P}_{4}$ | 0.0460 | 3 | 0.3505 | 3 | 0.2429 | 4 |

Alternatives | HFSDV-MOORA | Rank | HFSDV-TOPSIS | Rank | HFSDV-VIKOR | Rank |
---|---|---|---|---|---|---|

${P}_{1}$ | 1 | 1 | 1 | 1 | 0 | 1 |

${P}_{2}$ | 0.427 | 2 | 0.0.560 | 2 | 0.654 | 2 |

${P}_{3}$ | 0.093 | 4 | 0.258 | 4 | 0.950 | 4 |

${P}_{4}$ | 0.411 | 3 | 0.337 | 3 | 0.820 | 3 |

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**MDPI and ACS Style**

Narayanamoorthy, S.; Annapoorani, V.; Kalaiselvan, S.; Kang, D.
Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System. *Symmetry* **2020**, *12*, 2096.
https://doi.org/10.3390/sym12122096

**AMA Style**

Narayanamoorthy S, Annapoorani V, Kalaiselvan S, Kang D.
Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System. *Symmetry*. 2020; 12(12):2096.
https://doi.org/10.3390/sym12122096

**Chicago/Turabian Style**

Narayanamoorthy, Samayan, Veerappan Annapoorani, Samayan Kalaiselvan, and Daekook Kang.
2020. "Hybrid Hesitant Fuzzy Multi-Criteria Decision Making Method: A Symmetric Analysis of the Selection of the Best Water Distribution System" *Symmetry* 12, no. 12: 2096.
https://doi.org/10.3390/sym12122096