# Fermatean Fuzzy-Based Personalized Prioritization of Barriers to IoT Adoption within the Clean Energy Context

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## Abstract

**:**

## 1. Introduction

#### Motivation and Research Contributions

- Initial qualitative rating data from experts via questionnaires are converted to Fermatean fuzzy data (FFD) [13], which not only offers flexibility to experts in terms of opinion sharing from both preference and non-preference aspects but also helps to model uncertainty better by using three grades of uncertainty viz., membership, hesitancy, and non-membership as claimed in [13]. The inequality constraint ${a}^{3}+{b}^{3}\le 1$ ($a$ is the membership grade and $b$ is the non-membership grade) allows flexibility in the orthopair values, thereby providing a window for experts to express their opinions effectively compared to fuzzy set, intuitionistic, and Pythagorean fuzzy sets. Based on the discussion with experts regarding the quantification of the qualitative terms for the degree of preference and non-preference, the conversion is made by adhering to the constraint of FFD.
- Weights of criteria are determined methodically by presenting a regret measure, which not only captures the hesitation of experts but also considers the nature of criteria during weight assessment.
- A new ranking algorithm is developed by considering the formulation of “weighted aggregated sum product assessment” (WASPAS) and personal choices from experts to obtain personalized ordering of barriers, which not only provides a sense of personalization but also adds rationality to the decision process by considering the rating for each criterion and their overall opinion for a particular option (barrier in this case).
- Further, a case example of barriers to IoT adoption in the clean energy sector within India is demonstrated to understand the model’s usefulness.
- Finally, sensitivity analysis for weight values followed by a comparison of the proposed model with extant models from both the application and method perspectives is performed to understand the merits and limitations of the current work.

## 2. Literature Review

#### 2.1. Decision Models for Barriers Ranking

#### 2.2. Fermatean Fuzzy-Based Decision Models

## 3. Methodology

#### 3.1. Preliminaries

**Definition**

**1**

**Note**

**1:**

**Definition**

**3**

**Example**

**1:**

#### 3.2. Weight Etimation by Regret Measure

#### 3.3. Rank Algorithm for Ordering Barriers

## 4. Case Example of Barrier Ranking in the Clean Energy Sector

#### Sensitivity Analysis

## 5. Comparative Analysis of the Proposed Model vs. Other Models

- FFN is used as the preferred structure that could not only model uncertainty from three dimensions, viz., membership, non-membership, and hesitation but also allow flexible elicitation of preferences by providing a broader window for preference expression, which is lacking in the extant barrier ranking models considered for comparison. As evidence to the claim, readers can refer to the work in [13], which is the inception of FFNs, where the authors clarify the flexibility that FFNs offer to experts during the preference elicitation process by extending the window of expression, which is lacking in classical fuzzy sets and PFS.
- Furthermore, it can be observed that the weights are methodically determined by considering the nature of the criteria and experts’ hesitation. Unlike the extant models, in the proposed method, criteria type is considered that intuitively aids in the rationality of weight calculation. Precisely, when experts provide similar opinions or ratings to a particular criterion, at that time, the effect of the risk component measured via the Von Neumann measure becomes subtle owing to the criteria type factor of the Von Neumann measure in the utility function that suppresses the risk component. As a result, there is only a regret component, and the risk component is either negligible or zero, indicating that the particular criterion is less important than others and that experts exhibit a higher level of hesitation towards that criterion. On the other hand, if both the risk and regret components are involved, and their aversion values $a$ and $b$, respectively, are chosen close to the complete aversion that unity, the risk and regret values become negligible or close to zero, with high rejoice and as a result, the criterion gets high importance with less hesitation from the experts’ viewpoint. The hesitation of experts is mapped onto the consideration of risk and regret components. When a particular component is suppressed (either by considering less aversion of risk/regret or less variability in the preference distribution), the hesitancy level of experts is high, and the net utility value is small for the criterion indicating less importance. In other words, if risk and regret are high, hesitation is high, and utility value is low, eventually leading to less weight for the criterion.
- Further, the importance of experts is considered during the criteria weight determination, as the experts are crucial owing to their choice sharing for each criterion. Unlike other models, in the proposed model, consideration is given to the weights of experts that can be intuitively observed as potential information in the decision process. Moreover, the complexity of the proposed model is moderate. At the same time, some approaches have high complexity owing to their pairwise comparison formulation that adds overhead to the model and increases the computational complexity.
- Finally, the personal choices of experts on each barrier are collected in the form of a vector and utilized in the formulation for determining rank values of barriers with a sense of personalization intuitively; the process provides rationality in the rank estimation and gives a feel of the customizable ordering of barriers. Such a feature needs to be improved in the extant barrier ranking models compared to the proposed model.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Symbol | Meaning |
---|---|

$\mu $ | Degree of membership or membership grade |

$\upsilon $ | Degree of non-membership or non-membership grade |

$\pi $ | Degree of hesitancy or hesitancy grade |

${FF}_{i}$ | Fermatean fuzzy number |

$\rho $ | Any scalar value greater than 0 |

$R$ | Number of experts |

$G$ | Number of criteria |

${NU}_{lj}$ | von-Neumann value of expert $l$ rating criterion $j$ |

${TY}_{lj}$ | Regret value of expert $l$ rating criterion $j$ |

$l$ | Index of expert |

$j$ | Index of criterion |

$a$ | Parameter reflecting risk aversion coefficients |

$b$ | Parameter reflecting regret aversion coefficient |

${TI}_{j}$ | The utility value of criterion $j$ |

${W}_{j}$ | Weight of criterion $j$ |

$H$ | Number of barriers |

${h}_{i}$ | The choice value associated with barrier $i$ |

$i$ | Index of barrier |

${WA}_{ij}$ | Weighted accuracy value associated with barrier $i$ rated over criterion $j$ |

${SM}_{i}$ | The weighted sum of barrier $i$ |

${PT}_{I}$ | Weighted product of barrier $i$ |

${\gamma}_{i}$ | The final rank value of barrier $i$ |

$\beta $ | Strategy value |

$S\left(*\right)$ | The score value of * |

$A\left(*\right)$ | Accuracy value of * |

## Appendix B

Correlation | Proposed | No Choice Mode | [31] | [37] | [44] | Choice with Equal Weights |
---|---|---|---|---|---|---|

Proposed | 1 | 0.01 | 0.04 | 0.04 | 0.055 | 0.2 |

No choice model | 0.01 | 1 | 0.5 | 0.5 | 0.6 | 0.03 |

[31] | 0.04 | 0.5 | 1 | 0.5 | 0.4 | 0.04 |

[37] | 0.04 | 0.5 | 0.5 | 1 | 0.4 | 0.04 |

[44] | 0.055 | 0.6 | 0.4 | 0.4 | 1 | 0.05 |

Choice with equal weights | 0.2 | 0.03 | 0.04 | 0.04 | 0.05 | 1 |

Barriers | Set 1 | Set 2 | Set 3 | Set 4 | Set 5 | Set 6 | Set 7 | Set 8 | Set 9 |
---|---|---|---|---|---|---|---|---|---|

${X}_{1}$ | 8 | 10 | 7 | 7 | 6 | 5 | 6 | 4 | 5 |

${X}_{2}$ | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 |

${X}_{3}$ | 7 | 5 | 8 | 4 | 10 | 7 | 9 | 8 | 7 |

${X}_{4}$ | 2 | 2 | 2 | 1 | 3 | 2 | 2 | 2 | 3 |

${X}_{5}$ | 4 | 3 | 3 | 3 | 2 | 3 | 3 | 3 | 1 |

${X}_{6}$ | 6 | 7 | 5 | 5 | 5 | 8 | 4 | 7 | 9 |

${X}_{7}$ | 3 | 9 | 6 | 9 | 4 | 9 | 5 | 0 | 4 |

${X}_{8}$ | 15 | 14 | 15 | 14 | 13 | 14 | 15 | 15 | 14 |

${X}_{9}$ | 14 | 15 | 14 | 15 | 12 | 15 | 14 | 13 | 15 |

${X}_{10}$ | 12 | 11 | 10 | 12 | 11 | 13 | 11 | 14 | 12 |

${X}_{11}$ | 10 | 13 | 13 | 13 | 15 | 12 | 12 | 11 | 13 |

${X}_{12}$ | 13 | 12 | 12 | 11 | 13 | 10 | 13 | 12 | 11 |

${X}_{13}$ | 5 | 6 | 4 | 6 | 7 | 6 | 7 | 6 | 6 |

${X}_{14}$ | 9 | 8 | 9 | 10 | 8 | 11 | 8 | 10 | 8 |

${X}_{15}$ | 11 | 4 | 11 | 8 | 9 | 4 | 10 | 5 | 10 |

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**Figure 2.**Sensitivity analysis (

**a**–

**i**) denotes nine sets of criteria weight vectors obtained via rotation.

Source | Application | Methods Proposed | No of Alternatives | No of Criteria | Whether Sensitivity Analysis Is Done or Not | Whether Comparative The Analysis Is Done or Not | Fuzzy Set Used |
---|---|---|---|---|---|---|---|

[11] | IoT barriers | CoCoSo SWARA | 4 | 25 | Yes | No | PFS |

[12] | Sustainability barriers | AHP ELECTRE | 15 | 9 | No | Yes | Fuzzy set |

[16] | Offshore outsourcing barriers evaluation | AHP | 3 | 10 | Yes | No | Interval fuzzy set |

[17] | Supply chain management barriers | ELECTRE | 6 | 9 | No | Yes | IFS |

[18] | IoT barriers in manufacturing industry | AHP TOPSIS | 10 | 13 | No | Yes | Fuzzy set |

[19] | Sustainable consumption barriers | ANP | 20 | 4 | No | Yes | Fuzzy set |

[20] | Industry 4.0 implementation barriers | SWARA WASPAS | 6 | 5 | Yes | No | Fuzzy set |

[21] | Waste management barriers | COCOSO | 15 | 4 | Yes | Yes | FFS |

[22] | Hydrogen up-site barriers | WASPAS COPRAS | 14 | 4 | No | Yes | IFS |

[23] | Blockchain in CE adoption barriers | ANP | 14 | 7 | Yes | Yes | HFLTS |

[24] | Prioritization of barriers in Indian manufacturing industries | AHP ANP | 15 | 4 | Yes | No | Fuzzy set |

[25] | Spray painting robot barriers | SWARA COCOSO | 6 | 7 | No | Yes | IFS |

[26] | Tourism barrier evaluation | DEMATEL ISM | 17 | 4 | No | Yes | IFS |

[27] | CE barriers | CoCoSo | 15 | 5 | Yes | No | Fuzzy set |

[10] | Digital technology barrier selection | SWARA WASPAS | 4 | 24 | Yes | Yes | HFS |

Sources | Expressing Both Preference and Non-Preference | Level of Broadness for Choice Expression | Experts’ Hesitation during Weight Calculation | Criteria Type during Weight Calculation | Personalized Ranking |
---|---|---|---|---|---|

[11] | Yes | Moderate | Not considered | Not considered | No |

[12] | No | No | Not considered | Not considered | No |

[16] | No | No | Not considered | Not considered | No |

[17] | Yes | Low | Not considered | Not considered | No |

[18] | No | No | Not considered | Not considered | No |

[19] | No | No | Not considered | Not considered | No |

[20] | No | No | Not considered | Not considered | No |

[21] | Yes | High | Not considered | Not considered | No |

[22] | Yes | Low | Not considered | Not considered | No |

[23] | No | No | Not considered | Not considered | No |

[24] | No | No | Not considered | Not considered | No |

[25] | Yes | Low | Not considered | Not considered | No |

[26] | Yes | Low | Yes | Not considered | No |

[27] | No | No | Not considered | Not considered | No |

[10] | No | No | Not considered | Not considered | No |

Proposed | Yes | High | Yes | Yes | Yes |

Likert Scale | FIN | Likert Scale | FIN |
---|---|---|---|

Extremely low (EL) | (0.10,0.95) | Very highly preferred (VHP) | (0.95, 0.10) |

Very low (VL) | (0.60,0.90) | Highly preferred (HP) | (0.80, 0.60) |

Moderately low (ML) | (0.70,0.80) | Moderately preferred (MP) | (0.80, 0.65) |

Low (L) | (0.60,0.70) | Preferred (P) | (0.75, 0.60) |

Moderate (M) | (0.50,0.50) | Neutral (N) | (0.50, 0.50) |

High (H) | (0.75,0.60) | Slightly preferred (SP) | (0.60, 0.70) |

Moderately high (MH) | (0.80,0.65) | Less preferred (LP) | (0.70, 0.80) |

Very high (VH) | (0.80,0.60) | Very less preferred (VLP) | (0.60, 0.90) |

Extremely high (EH) | (0.95,0.10) | Not preferred (NP) | (0.10, 0.95) |

$\mathit{X}$ | CE Criteria | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | ${\mathit{Z}}_{5}$ | ${\mathit{Z}}_{6}$ | ${\mathit{Z}}_{7}$ | ${\mathit{Z}}_{8}$ | ${\mathit{Z}}_{9}$ | |

${X}_{1}$ | M | M | VH | ML | L | L | L | MH | ML |

${X}_{2}$ | MH | MH | VH | ML | H | MH | VL | MH | MH |

${X}_{3}$ | MH | MH | H | VH | L | L | M | VL | H |

${X}_{4}$ | MH | MH | VH | H | L | L | H | ML | VH |

${X}_{5}$ | VL | M | MH | MH | L | MH | ML | ML | L |

${X}_{6}$ | VH | MH | L | L | MH | H | VL | MH | H |

${X}_{7}$ | H | H | ML | MH | VL | M | MH | L | MH |

${X}_{8}$ | L | M | H | ML | M | M | MH | H | VH |

${X}_{9}$ | MH | M | M | VL | MH | M | L | M | VL |

${X}_{10}$ | MH | VH | H | L | H | M | H | MH | VH |

${X}_{11}$ | M | ML | H | ML | H | M | M | MH | H |

${X}_{12}$ | VH | M | L | L | M | VH | M | VH | VH |

${X}_{13}$ | H | L | MH | MH | L | H | H | MH | MH |

${X}_{14}$ | ML | VL | ML | L | L | M | L | M | M |

${X}_{15}$ | ML | L | VH | VL | M | VL | VL | VH | M |

$\mathit{Y}$ | CE Criteria | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{Z}}_{1}$ | ${\mathit{Z}}_{2}$ | ${\mathit{Z}}_{3}$ | ${\mathit{Z}}_{4}$ | ${\mathit{Z}}_{5}$ | ${\mathit{Z}}_{6}$ | ${\mathit{Z}}_{7}$ | ${\mathit{Z}}_{8}$ | ${\mathit{Z}}_{9}$ | |

${Y}_{1}$ | N | P | LP | MP | P | SP | P | MP | N |

${Y}_{2}$ | N | P | N | SP | MP | N | N | SP | MP |

${Y}_{3}$ | VLP | P | P | SP | N | P | P | N | MP |

${Y}_{4}$ | N | P | VLP | SP | P | VLP | LP | P | P |

$\mathit{X}$ | ${\mathit{S}\mathit{M}}_{\mathit{i}}$ | ${\mathit{P}\mathit{T}}_{\mathit{I}}$ | ${\mathit{\gamma}}_{\mathit{i}}$ |
---|---|---|---|

${X}_{1}$ | 0.749 | 0.732 | 0.741 |

${X}_{2}$ | 0.849 | 0.847 | 0.848 |

${X}_{3}$ | 0.754 | 0.742 | 0.748 |

${X}_{4}$ | 0.837 | 0.835 | 0.836 |

${X}_{5}$ | 0.780 | 0.764 | 0.772 |

${X}_{6}$ | 0.751 | 0.746 | 0.749 |

${X}_{7}$ | 0.769 | 0.754 | 0.761 |

${X}_{8}$ | 0.609 | 0.574 | 0.591 |

${X}_{9}$ | 0.627 | 0.573 | 0.600 |

${X}_{10}$ | 0.680 | 0.669 | 0.674 |

${X}_{11}$ | 0.717 | 0.692 | 0.705 |

${X}_{12}$ | 0.626 | 0.607 | 0.616 |

${X}_{13}$ | 0.749 | 0.747 | 0.748 |

${X}_{14}$ | 0.750 | 0.725 | 0.738 |

${X}_{15}$ | 0.711 | 0.671 | 0.691 |

**Table 7.**Summarized view of different characteristics of proposed and extant barrier prioritization models.

Characteristics | Proposed | Mardani et al. (2021) [10] | Cui et al. (2021) [11] | Rahman et al. (2021) [16] | Kumar et al. (2021) [12] |
---|---|---|---|---|---|

Data | FFN | HFS | PFS | Fuzzy set | Fuzzy set |

Criteria weights | Considered | Considered | Considered | Considered | Considered |

Flexibility | High | Moderate | Moderate | Low | Low |

Uncertainty | Modeled in three ways | Modeled in one way | Modeled in three ways | Modeled in one way | Modeled in one way |

Hesitation | Considered | Not considered | Not considered | Not considered | Not considered |

Criteria nature (weight calculation) | Considered | Not considered | Not considered | Not considered | Not considered |

Experts’ importance | Considered | Not considered | Not considered | Not considered | Not considered |

Complexity | Moderate | Moderate | Moderate | High | High |

Personal choices | Considered | Not considered | Not considered | Not considered | Not considered |

**Note:**FFN—Fermatean fuzzy number, HFS—hesitant fuzzy set, and PFS—Pythagorean fuzzy set.

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## Share and Cite

**MDPI and ACS Style**

Reddy, N.S.S.; Krishankumar, R.; Priya, S.S.; Cavallaro, F.; Mardani, A.; Ravichandran, K.S.
Fermatean Fuzzy-Based Personalized Prioritization of Barriers to IoT Adoption within the Clean Energy Context. *Information* **2023**, *14*, 309.
https://doi.org/10.3390/info14060309

**AMA Style**

Reddy NSS, Krishankumar R, Priya SS, Cavallaro F, Mardani A, Ravichandran KS.
Fermatean Fuzzy-Based Personalized Prioritization of Barriers to IoT Adoption within the Clean Energy Context. *Information*. 2023; 14(6):309.
https://doi.org/10.3390/info14060309

**Chicago/Turabian Style**

Reddy, N Sai Snehitha, Raghunathan Krishankumar, S Shanmugha Priya, Fausto Cavallaro, Abbas Mardani, and Kattur Soundarapandian Ravichandran.
2023. "Fermatean Fuzzy-Based Personalized Prioritization of Barriers to IoT Adoption within the Clean Energy Context" *Information* 14, no. 6: 309.
https://doi.org/10.3390/info14060309