# Pythagorean Fuzzy SWARA–VIKOR Framework for Performance Evaluation of Solar Panel Selection

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

_{4}constantly secures its highest ranking despite how the parameter values vary. In addition, a comparative study is discussed to analyze the validity of the obtained result. The results show that the proposed approach is more efficient and applicable with previously developed methods in the PFS environment.

## 1. Introduction

- (a)
- An integrated Pythagorean fuzzy–SWARA–VIKOR (PF–SWARA–VIKOR) framework is proposed.
- (b)
- The PFS-based SWARA method is utilized to assess the criteria weights.
- (c)
- A problem regarding the selection of solar panels is presented and evaluated by utilizing the proposed PF–SWARA–VIKOR method, which reveals the applicability of the introduced approach.
- (d)
- A comparative study and sensitivity analysis are also discussed to show the usefulness of the introduced approach.

## 2. Preliminaries

**Definition 1**

**[30,31].**

**Definition 2**

**[33].**

**Definition**

**3.**

**Definition 4**

**[30,31].**

**Definition 5**

**[33].**

## 3. Proposed Pythagorean Fuzzy–SWARA–VIKOR Method

**Step I:**Construct a decision matrix.

_{i}concerning the criteria ${T}_{j};j=1(1)n$ for the k

^{th}DE.

**Step II:**Evaluate the DEs’ weights.

^{th}DE, let ${E}_{k}=Y\left({\mu}_{k},{\nu}_{k}\right)$ be the Pythagorean fuzzy number, then the weight computation formula for k

^{th}DE is presented as follows:

**Step III:**Construct the aggregated Pythagorean fuzzy decision (APF-D) matrix.

**Step IV:**Evaluate the normalized APF-D matrix.

_{b}and T

_{n}denote the beneficial and non-beneficial criterion sets, respectively.

**Step V:**Calculate the criteria weights.

**Step V-A:**Calculate the crisp values. Score values ${\mathbb{S}}^{\ast}\left({\tilde{z}}_{kj}\right)$ of PFNs are computed by Equation (3) given in Definition 3.

**Step V-B:**Preference order of the criteria. The criteria are arranged according to the DE’s preferences from the most to the least significant criterion.

**Step V-C:**Evaluate the comparative significance of score value. The comparative significance is determined from the criteria that are preferred in the second place, and successive comparative significance is evaluated by differencing criterion j and j − 1.

**Step V-D:**Compute the comparative coefficient. The coefficient ${k}_{j}$ is given by

_{j}presents the comparative significance of score value [41].

**Step V-E:**Estimate the weight. The recalculated weight p

_{j}is defined by

**Step V-F:**Evaluate the criteria weights. The criteria weights are defined by

**Step VI:**Find the best and worst values.

**Step VII:**Calculate the group utility, individual regret, and compromise measure.

_{i}are evaluated by employing the Hamming distance measure given in Equation (4). The group utility, individual regret, and compromise degree of the options are computed by using the following procedures:

**Step VIII:**Estimate the ranking of the options.

**Step IX:**Find the compromise solution.

**(C**Acceptable advantage:

_{1}):**(C**Adequate stability: The option ${R}^{(1)}$ must also be ranked by ${S}_{i}$ and ${I}_{i}.$ The compromise solution ${Q}_{i}$ is stable within an MCDM procedure, which can be selected with “voting by majority rule $(\tau >0.5)$”, “by consensus $(\tau \approx 0.5)$”, or “by veto $(\tau <0.5)$”.

_{2}):_{1}) is not fulfilled, then the extreme value should be inspected by the given relation:

_{2}) is not fulfilled.

**Step X**: End.

## 4. An Empirical Study: Performance Evaluation of Solar Panel Selection

_{1}, R

_{2}, R

_{3}, R

_{4}and R

_{5}. A team of three DEs is selected to process this solar panel selection problem. This problem associated with the performances of solar panels includes eight attributes or criteria. The facts of the criteria are given in Table 1.

_{i}, I

_{i}, and Q

_{i}are calculated. The obtained results are given in Table 8. By employing the decreasing values of S

_{i}, I

_{i}, and Q

_{i}, the preference order of the solar panel alternatives is acquired in Table 8. The lowest value of Q

_{i}denotes the optimal solar panel, i.e., R

_{4}is the best solar panel alternative.

#### 4.1. Sensitivity Analysis

_{i}of R

_{1}decreases when the value of $\tau $ increases, while R

_{2}, R

_{3}, and R

_{5}increases when the value of $\tau $ increases. Meanwhile, the fourth alternative R

_{4}is stable in each set. Accordingly, despite the change of weights in the criterion set, the preference order of the five solar power alternatives remains the same. The final ranking of solar panel alternatives is presented with respect to following performance scores in Table 9, and it is observed that solar panel R

_{4}is best of all options.

#### 4.2. Comparative Study

#### PF-TOPSIS Method

**Steps I–VI:**Same as the previous method.

**Step VII:**Compute the degree of distances from PF-PIS and PF-NIS.

**Step VIII:**Evaluate the relative closeness coefficient (CC).

**Step IX:**Choose the highest value, $\mathbb{R}\left({R}_{k}\right),$ among the values $\mathbb{R}\left({R}_{i}\right),i=1\left(1\right)m.$ Hence, ${R}_{k}$ is the optimal choice.

- (a)
- The PF–SWARA–VIKOR method represents the Pythagorean fuzzy information, which can depict the MD, ND, and hesitation degree with an effortless mathematical description. Based on it, we can determine the significance degree of the DEs without any modification and, therefore, the developed method can successfully avoid the loss of information.
- (b)
- As some of the previous measures under the PFSs [33] have been incapable of providing the preference order of the alternatives accurately, thus, their consequent methods may not present relevant outcomes. Alternatively, the proposed approach has the capability to prevail over their limitations and is therefore able to order the alternatives appropriately, which makes it a more desirable approach to solving MCDM problems.
- (c)
- The SWARA approach is utilized to compute the subjective weights of criteria in the process of performance evaluation of solar panels, which makes the developed PF–SWARA–VIKOR approach more sensible, flexible, and efficient.
- (d)
- The developed framework has the following benefits when choosing solar panels:
- An innovative procedure is utilized to enumerate tangible sub-criteria successfully.
- The integrated approach eradicates the subjective estimation of indistinct sub-criteria.
- Pythagorean fuzzy SWARA is used to achieve appropriate harmonizing of criteria.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Comparison of preference order of the Solar Panel Selection (SPS) alternatives with various approaches.

Criteria | Descriptions | Type |
---|---|---|

Peak power rating (T_{1}) | Refers to the maximum output (in Watts) under standard test conditions | Benefit |

Peak efficiency (T_{2}) | Refers to the high peak efficiency | Benefit |

Maximum power current (T_{3}) | Refers to the high value of current | Benefit |

Maximum power voltage (T_{4}) | Refers to the high value of power current | Benefit |

Weight (T_{5}) | Prefers to the solar panel with less weight | Cost |

Price (T_{6}) | Considers the price of solar panels | |

Reliability (T_{7}) | Measures the reliability of the solar panel | Benefit |

Spare parts availability (T_{8}) | The availability of solar panel (SP) spare parts is one of the factors deciding customer fulfillment | Benefit |

R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | |
---|---|---|---|---|---|

T_{1} | E_{1}: (0.29, 0.75)E _{2}: (0.40, 0.70)E _{3}: (0.45, 0.65) | E_{1}: (0.70, 0.45)E _{2}: (0.72, 0.50)E _{3}: (0.65, 0.50) | E_{1}: (0.58, 0.55)E _{2}: (0.55, 0.60)E _{3}: (0.60, 0.55) | E_{1}: (0.55, 0.65)E _{2}: (0.52, 0.66)E _{3}: (0.60, 0.55) | E_{1}: (0.60, 0.55)E _{2}: (0.70, 0.45)E _{3}: (0.65, 0.50) |

T_{2} | E_{1}: (0.63, 0.40,)E _{2}: (0.55, 0.60)E _{3}: (0.68, 0.35) | E_{1}: (0.63, 0.45)E _{2}: (0.60, 0.50)E _{3}: (0.55, 0.60) | E_{1}: (0.60, 0.45)E _{2}: (0.65, 0.50)E _{3}: (0.58, 0.44) | E_{1}: (0.60, 0.57)E _{2}: (0.55, 0.60)E _{3}: (0.50, 0.60) | E_{1}: (0.60, 0.50)E _{2}: (0.55, 0.60)E _{3}: (0.55, 0.50) |

T_{3} | E_{1}: (0.55, 0.65)E _{2}: (0.60, 0.70)E _{3}: (0.50, 0.70) | E_{1}: (0.70, 0.45)E _{2}: (0.70, 0.50)E _{3}: (0.68, 0.45) | E_{1}: (0.64, 0.55)E _{2}: (0.55, 0.57)E _{3}: (0.60, 0.55) | E_{1}: (0.60, 0.55)E _{2}: (0.70, 0.50)E _{3}: (0.65, 0.55) | E_{1}: (0.70, 0.50)E _{2}: (0.65, 0.50)E _{3}: (0.68, 0.50) |

T_{4} | E_{1}: (0.55, 0.60)E _{2}: (0.59, 0.45)E _{3}: (0.60, 0.50) | E_{1}: (0.55, 0.65)E _{2}: (0.50, 0.65)E _{3}: (0.55, 0.60) | E_{1}: (0.50, 0.60)E _{2}: (0.55, 0.60)E _{3}: (0.45, 0.65) | E_{1}: (0.55, 0.65)E _{2}: (0.63, 0.42)E _{3}: (0.60, 0.50) | E_{1}: (0.55, 0.50)E _{2}: (0.60, 0.50)E _{3}: (0.45, 0.65) |

T_{5} | E_{1}: (0.51, 0.55)E _{2}: (0.60, 0.50)E _{3}: (0.60, 0.55) | E_{1}: (0.65, 0.48)E _{2}: (0.60, 0.55)E _{3}: (0.66, 0.47) | E_{1}: (0.60, 0.45)E _{2}: (0.65, 0.55)E _{3}: (0.60, 0.50) | E_{1}: (0.65, 0.48)E _{2}: (0.65, 0.50)E _{3}: (0.70, 0.45) | E_{1}: (0.65, 0.58)E _{2}: (0.50, 0.65)E _{3}: (0.65, 0.45) |

T_{6} | E_{1}: (0.65, 0.45)E _{2}: (0.60, 0.48)E _{3}: (0.55, 0.50) | E_{1}: (0.62, 0.50)E _{2}: (0.60, 0.52)E _{3}: (0.58, 0.65) | E_{1}: (0.58, 0.49)E _{2}: (0.55, 0.50)E _{3}: (0.68, 0.48) | E_{1}: (0.65, 0.45)E _{2}: (0.57, 0.48)E _{3}: (0.60, 0.50) | E_{1}: (0.62, 0.55)E _{2}: (0.60, 0.55)E _{3}: (0.58, 0.55) |

T_{7} | E_{1}: (0.50, 0.58)E _{2}: (0.55, 0.50)E _{3}: (0.52, 0.57) | E_{1}: (0.58, 0.60)E _{2}: (0.50, 0.60)E _{3}: (0.45, 0.60) | E_{1}: (0.55, 0.65)E _{2}: (0.53, 0.64)E _{3}: (0.50, 0.60) | E_{1}: (0.45, 0.55)E _{2}: (0.60, 0.50)E _{3}: (0.65, 0.53) | E_{1}: (0.48, 0.70)E _{2}: (0.50, 0.60)E _{3}: (0.55, 0.60) |

T_{8} | E_{1}: (0.67, 0.46)E _{2}: (0.65, 0.45)E _{3}: (0.60, 0.50) | E_{1}: (0.57, 0.68)E _{2}: (0.52, 0.57)E _{3}: (0.50, 0.60) | E_{1}: (0.58, 0.65)E _{2}: (0.55, 0.60)E _{3}: (0.50, 0.62) | E_{1}: (0.60, 0.50)E _{2}: (0.65, 0.55)E _{3}: (0.68, 0.53) | E_{1}: (0.57, 0.58)E _{2}: (0.52, 0.60)E _{3}: (0.45, 0.65) |

R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | |
---|---|---|---|---|---|

T_{1} | Y (0.381,0.702, 0.601) | Y (0.691,0.480, 0.541) | Y (0.579,0.564, 0.589) | Y (0.559,0.618, 0.552) | Y (0.648,0.504, 0.572) |

T_{2} | Y (0.628,0.430, 0.649) | Y (0.598,0.509, 0.620) | Y (0.609,0.460, 0.646) | Y (0.556,0.588, 0.587) | Y (0.571,0.526, 0.630) |

T_{3} | Y (0.551,0.680, 0.484) | Y (0.680,0.483, 0.551) | Y (0.604,0.556, 0.571) | Y (0.648,0.535, 0.542) | Y (0.680,0.500, 0.536) |

T_{4} | Y (0.578,0.521, 0.649) | Y (0.537,0.633, 0.620) | Y (0.500,0.616, 0.646) | Y (0.591,0.528, 0.587) | Y (0.538,0.544, 0.630) |

T_{5} | Y (0.568,0.535, 0.625) | Y (0.640,0.495, 0.587) | Y (0.615,0.493, 0.615) | Y (0.667,0.476, 0.573) | Y (0.615,0.552, 0.563) |

T_{6} | Y (0.607,0.474, 0.638) | Y (0.602,0.550, 0.579) | Y (0.609,0.490, 0.624) | Y (0.613,0.474, 0.632) | Y (0.602,0.550, 0.579) |

T_{7} | Y (0.521,0.553, 0.650) | Y (0.520,0.600, 0.608) | Y (0.529,0.631, 0.568) | Y (0.570,0.529, 0.629) | Y (0.510,0.638, 0.578) |

T_{8} | Y (0.643,0.470, 0.605) | Y (0.535,0.621, 0.573) | Y (0.548,0.626, 0.556) | Y (0.642,0.523, 0.560) | Y (0.521,0.608, 0.599) |

R_{1} | R_{2} | R_{3} | R_{4} | R_{5} | |
---|---|---|---|---|---|

T_{1} | Y(0.381, 0.702, 0.601) | Y (0.691, 0.480, 0.541) | Y (0.579, 0.564, 0.589) | Y (0.559, 0.618, 0.552) | Y (0.648, 0.504, 0.572) |

T_{2} | Y (0.628, 0.430, 0.649) | Y (0.598, 0.509, 0.620) | Y (0.609, 0.460, 0.646) | Y (0.556, 0.588, 0.587) | Y (0.571, 0.526, 0.630) |

T_{3} | Y (0.551, 0.680, 0.484) | Y (0.680, 0.483, 0.551) | Y (0.604, 0.556, 0.571) | Y (0.648, 0.535, 0.542) | Y (0.680, 0.500, 0.536) |

T_{4} | Y (0.578, 0.521, 0.649) | Y (0.537, 0.633, 0.620) | Y (0.500, 0.616, 0.646) | Y (0.591, 0.528, 0.587) | Y (0.538, 0.544, 0.630) |

T_{5} | Y (0.535, 0.568, 0.625) | Y (0.495, 0.640, 0.587) | Y (0.493, 0.615, 0.615) | Y (0.476, 0.667, 0.573) | Y (0.552, 0.615, 0.563) |

T_{6} | Y (0.474, 0.607, 0.638) | Y (0.550, 0.602, 0.579) | Y (0.490, 0.609, 0.624) | Y (0.474, 0.613, 0.632) | Y (0.550, 0.602, 0.579) |

T_{7} | Y (0.521, 0.553, 0.650) | Y (0.520, 0.600, 0.608) | Y (0.529, 0.631, 0.568) | Y (0.570, 0.529, 0.629) | Y (0.510, 0.638, 0.578) |

T_{8} | Y (0.643, 0.470, 0.605) | Y (0.535, 0.621, 0.573) | Y (0.548, 0.626, 0.556) | Y (0.642, 0.523, 0.560) | Y (0.521, 0.608, 0.599) |

Linguistic Values | PFNs |
---|---|

Extremely Low (EL) | Y(0.1500, 0.9500) |

Very Low (VL) | Y(0.2500, 0.9000) |

Low (L) | Y(0.3000, 0.8500) |

Medium Low (ML) | Y(0.3500, 0.7500) |

Medium (M) | Y(0.4500, 0.6500) |

Medium High (MH) | Y(0.6000, 0.5000) |

High (H) | Y(0.7000, 0.3500) |

Very High (VH) | Y(0.8000, 0.3000) |

**Table 6.**Criteria weights given by the decision experts (DEs) in terms of LVs for solar panel evaluation.

Criteria | E_{1} | E_{2} | E_{3} | Aggregated PFNs | Score Values |
---|---|---|---|---|---|

T_{1} | H | VH | VVH | Y(0.788, 0.300, 0.538) | 0.765 |

T_{2} | MH | ML | H | Y(0.592, 0.500, 0.633) | 0.550 |

T_{3} | M | M | MH | Y(0.507, 0.597, 0.621) | 0.450 |

T_{4} | ML | ML | MH | Y(0.456, 0.658, 0.633) | 0.388 |

T_{5} | MH | H | ML | Y(0.579, 0.515, 0.632) | 0.535 |

T_{6} | H | M | MH | Y(0.614, 0.468, 0.635) | 0.579 |

T_{7} | VH | H | VH | Y(0.776, 0.313, 0.547) | 0.752 |

T_{8} | VH | H | H | Y(0.745, 0.329, 0.580) | 0.723 |

**Table 7.**Results obtained by the Stepwise Weight Assessment Ratio Analysis (SWARA) method for solar panel selection.

Criteria | Crisp Values | $\mathbf{Comparative}\mathbf{Significance}\mathbf{of}\mathbf{Criteria}\mathbf{Value}\left({\mathit{s}}_{\mathit{j}}\right)$ | $\mathbf{Coefficient}\left({\mathit{k}}_{\mathit{j}}\right)$ | $\mathbf{Recalculated}\mathbf{Weight}\left({\mathit{p}}_{\mathit{j}}\right)$ | $\mathbf{Criteria}\mathbf{Weight}\left({\mathit{w}}_{\mathit{j}}\right)$ |
---|---|---|---|---|---|

T_{1} | 0.765 | - | 1.000 | 1.000 | 0.1463 |

T_{7} | 0.752 | 0.013 | 1.013 | 0.987 | 0.1444 |

T_{8} | 0.723 | 0.029 | 1.029 | 0.959 | 0.1403 |

T_{6} | 0.579 | 0.144 | 1.144 | 0.838 | 0.1226 |

T_{2} | 0.550 | 0.029 | 1.029 | 0.814 | 0.1191 |

T_{5} | 0.535 | 0.015 | 1.015 | 0.802 | 0.1173 |

T_{3} | 0.450 | 0.085 | 1.085 | 0.739 | 0.1081 |

T_{4} | 0.388 | 0.062 | 1.062 | 0.696 | 0.1019 |

S_{i} | I_{i} | Q_{i} | |
---|---|---|---|

R_{1} | 0.547 | 0.183 | 0.857 |

R_{2} | 0.545 | 0.140 | 0.519 |

R_{3} | 0.634 | 0.145 | 0.669 |

R_{4} | 0.262 | 0.119 | 0.000 |

R_{5} | 0.661 | 0.144 | 0.695 |

Ranking order | ${S}_{4}\succ {S}_{2}\succ {S}_{1}\succ {S}_{3}\succ {S}_{5}$ | ${I}_{4}\succ {I}_{2}\succ {I}_{5}\succ {I}_{3}\succ {I}_{1}$ | ${Q}_{4}\succ {Q}_{2}\succ {Q}_{3}\succ {Q}_{5}\succ {Q}_{1}$ |

$\mathit{\tau}$ | R_{1} | R_{2} | R_{3} | R_{4} | R_{5} |
---|---|---|---|---|---|

0.0 | 1.000 | 0.328 | 0.406 | 0.000 | 0.391 |

0.1 | 0.971 | 0.366 | 0.459 | 0.000 | 0.452 |

0.2 | 0.943 | 0.404 | 0.511 | 0.000 | 0.512 |

0.3 | 0.914 | 0.442 | 0.564 | 0.000 | 0.573 |

0.4 | 0.886 | 0.481 | 0.617 | 0.000 | 0.634 |

0.5 | 0.857 | 0.519 | 0.669 | 0.000 | 0.695 |

0.6 | 0.829 | 0.557 | 0.722 | 0.000 | 0.756 |

0.7 | 0.800 | 0.595 | 0.775 | 0.000 | 0.817 |

0.8 | 0.771 | 0.633 | 0.827 | 0.000 | 0.878 |

0.9 | 0.743 | 0.671 | 0.880 | 0.000 | 0.939 |

1.0 | 0.714 | 0.709 | 0.932 | 0.000 | 1.000 |

**Table 10.**Computational results of the PF–Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method for solar panel selection.

Alternative | ${\mathit{D}}_{\mathit{h}}\left({\tilde{\mathit{z}}}_{\mathit{i}\mathit{j}},{\mathit{\sigma}}_{\mathit{j}}^{+}\right)$ | ${\mathit{D}}_{\mathit{h}}\left({\tilde{\mathit{z}}}_{\mathit{i}\mathit{j}},{\mathit{\sigma}}_{\mathit{j}}^{-}\right)$ | $\mathit{\u2102}\left({\mathit{R}}_{\mathit{i}}\right)$ | Ranking | $\mathit{\mathbb{R}}\left({\mathit{R}}_{\mathit{i}}\right)$ | Ranking |
---|---|---|---|---|---|---|

R_{1} | 0.101 | 0.085 | 0.457 | 4 | −1.0446 | 4 |

R_{2} | 0.068 | 0.102 | 0.598 | 2 | −0.3036 | 2 |

R_{3} | 0.099 | 0.081 | 0.449 | 5 | −1.0446 | 4 |

R_{4} | 0.056 | 0.112 | 0.665 | 1 | 0.0000 | 1 |

R_{5} | 0.086 | 0.083 | 0.491 | 3 | −0.7946 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Rani, P.; Mishra, A.R.; Mardani, A.; Cavallaro, F.; Štreimikienė, D.; Khan, S.A.R.
Pythagorean Fuzzy SWARA–VIKOR Framework for Performance Evaluation of Solar Panel Selection. *Sustainability* **2020**, *12*, 4278.
https://doi.org/10.3390/su12104278

**AMA Style**

Rani P, Mishra AR, Mardani A, Cavallaro F, Štreimikienė D, Khan SAR.
Pythagorean Fuzzy SWARA–VIKOR Framework for Performance Evaluation of Solar Panel Selection. *Sustainability*. 2020; 12(10):4278.
https://doi.org/10.3390/su12104278

**Chicago/Turabian Style**

Rani, Pratibha, Arunodaya Raj Mishra, Abbas Mardani, Fausto Cavallaro, Dalia Štreimikienė, and Syed Abdul Rehman Khan.
2020. "Pythagorean Fuzzy SWARA–VIKOR Framework for Performance Evaluation of Solar Panel Selection" *Sustainability* 12, no. 10: 4278.
https://doi.org/10.3390/su12104278