# An Extended Shapley TODIM Approach Using Novel Exponential Fuzzy Divergence Measures for Multi-Criteria Service Quality in Vehicle Insurance Firms

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fuzzy Sets (FSs)

**Definition 1.**

**Definition 2.**

#### 2.2. Divergence Measure for FSs

**Definition 3.**

**Definition 4.**

## 3. New Divergence Measure for FSs

#### Jensen–Shannon Exponential Divergence Measures for FSs

**Definition 5.**

**Theorem 1.**

**Proof.**

**Proposition 1.**

**Proof.**

**Corollary 1.**

**Proof.**

**Proposition 2.**

**Proof.**

**Theorem 2.**

**Proof.**

**Corollary 2.**

**Proof.**

## 4. An Integrated TODIM Approach Using the Shapley Function and Divergence Measure

#### 4.1. Shapley Function

**Definition 6.**

**Property 1.**

**Property 2.**

#### 4.2. Models for Criteria Weight Based on the Optimal Additive Measure

_{i}concerning the criterion E

_{j}. In accordance with the entropy rule, if the entropy of an object is small, it would deliver valued information to the DEs. As a result, the criterion needs to be consigned with a greater weight; unless, the criterion would be considered insignificant by most DEs. In addition, this criterion needs to be estimated as a smaller weight. Hence, the optimal F-measure will form a greater inclusive value for each alternative that is preferable.

#### 4.3. Shapley Function-Based TODIM Technique for MCDM

**Step I:**Construct a decision matrix $D={\left({d}_{ij}\right)}_{m\times n},$ in which ${d}_{ij}$ presents an assessment value of an option S

_{i}concerning the criterion E

_{j}. Initially, the information should be normalized. A larger value shows a higher quality assessment of benefit criteria, but the same condition reveals the poorer quality performance of a cost criterion. As a result, for the purpose of guaranteeing all criteria to be with complete compatibility, the cost criteria were transformed into benefit one using the formula below: ${\ell}_{ij}=\{\begin{array}{l}{d}_{ij},\mathrm{for}\mathrm{benefit}\mathrm{criterion}{E}_{j}\\ {\left({d}_{ij}\right)}^{c},\mathrm{for}\mathrm{cos}\mathrm{t}\mathrm{criterion}{E}_{j}\end{array},$ where ${\left({d}_{ij}\right)}^{c}$ denotes the complement of ${d}_{ij},$ through this procedure, and it is possible to attain the normalized fuzzy decision matrix $L={\left({\ell}_{ij}\right)}_{m\times n}.$

**Step II:**For a better judgment, we need to identify the significance of the judgment of each DE. Therefore, each criterion weight needs to be determined. If the characteristics among the criteria are interdependent, the weight vector of the criteria is calculated in forms of Shapley values. Employ (19), and model (20) and (21) with respect to (11), to find the criteria weight.

**Step III:**Calculate the criteria’s relative weight vector using the formula ${\phi}_{jr}\left(g,V\right)=\frac{{\phi}_{j}\left(g,V\right)}{{\phi}_{r}\left(g,V\right)},$ where ${\phi}_{r}(g,V)=\mathrm{max}\{{\phi}_{j}(g,V)\}$ and ${\phi}_{j}\left(g,V\right)$ is the criteria weight E

_{j}.

**Step IV:**Determine the degree of dominance of an option ${S}_{i}$ over each option ${S}_{t}$ by:

**Step V:**Evaluate the overall dominance value of an option ${S}_{i}$ over each option ${S}_{t}$ by:

**Step VI:**Determine the overall value of each option S

_{i}using the expression:

**Step VII:**Determine the rank of options based on the overall values.

## 5. Case Study of the Proposed Method

_{1}), National Insurance (S

_{2}), Bajaj Insurance (S

_{3}), and New India Insurance (S

_{4}). The questionnaires indices contain four evaluation criteria, i.e., confidence (E

_{1}), responsiveness (E

_{2}), reliability (E

_{3}), and tangibles (E

_{4}).

**Step I:**Through the integration of the preference value results obtained from the four above-noted firms based on four evaluation criteria, the decision matrix of options is attained, as presented in Table 1. The importance of criteria is given by $\left[0.15,0.4\right],\left[0.25,0.5\right],\left[0.25,0.6\right]$, and $\left[0.2,0.3\right].$

**Step II:**By using (11), the entropy measure of the option ${S}_{i}(i=1,\dots ,4),$ considering the criteria ${E}_{j};j=1(1)4,$ is listed in Table 2.

_{j}are as follows:

**Step III:**The relative weights of the given criteria ${E}_{j};j=1(1)4$ are ${\phi}_{1r}(g,V)=1,{\phi}_{2r}(g,V)=0.1698,{\phi}_{3r}(g,V)=0.1698$, and ${\phi}_{4r}(g,V)=0.1698.$

**Step IV:**The dominance degree matrices of the options over the criteria ${E}_{j};j=1(1)4$ are as follows:

**Step V:**The total dominance degree matrix of option ${S}_{i}$ over each option ${S}_{t}$ is evaluated as:

**Step VI:**Now, the calculated overall values are as follows:

**Step VII:**Finally, the ranking of the alternatives is ${S}_{2}\succ {S}_{4}\succ {S}_{1}\succ {S}_{3}$ and thus, ${S}_{2}$ is the optimal choice.

_{1}and S

_{4}. Additionally, the ranking obtained via the proposed technique is totally different from the Krohling, and de Souza [69] and Fan et al. [52] techniques (see Table 3).

#### Comparative Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 1

#### Appendix A.2. Proof of Proposition 1

#### Appendix A.3. Proof of Proposition 2

_{1}and V

_{2}as:

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Option | E_{1} | E_{2} | E_{3} | E_{4} |
---|---|---|---|---|

S_{1} | 0.5395 | 0.7655 | 0.6770 | 0.5915 |

S_{2} | 0.7015 | 0.6990 | 0.7725 | 0.7650 |

S_{3} | 0.7365 | 0.5350 | 0.6595 | 0.6340 |

S_{4} | 0.7800 | 0.6985 | 0.5995 | 0.7455 |

Entropy | E_{1} | E_{2} | E_{3} | E_{4} |
---|---|---|---|---|

H(S_{1}) | 0.6029 | 0.4400 | 0.5329 | 0.5869 |

H(S_{2}) | 0.5111 | 0.5134 | 0.4310 | 0.4406 |

H(S_{3}) | 0.4747 | 0.6037 | 0.5468 | 0.5644 |

H(S_{4}) | 0.4211 | 0.5139 | 0.5834 | 0.4644 |

Techniques | Ranking | Optimal Choice |
---|---|---|

Krohling and de Souza [69] technique | ${S}_{1}\succ {S}_{2}\succ {S}_{4}\succ {S}_{3}$ | S_{1} |

Fan et al. [52] technique | ${S}_{1}\succ {S}_{4}\succ {S}_{2}\succ {S}_{3}$ | S_{1} |

Mishra et al. [68] technique | ${S}_{2}\succ {S}_{4}\succ {S}_{1}\succ {S}_{3}$ | S_{2} |

Mishra [67] technique | ${S}_{2}\succ {S}_{1}\succ {S}_{4}\succ {S}_{3}$ | S_{2} |

Proposed technique | ${S}_{2}\succ {S}_{4}\succ {S}_{1}\succ {S}_{3}$ | S_{2} |

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## Share and Cite

**MDPI and ACS Style**

Mishra, A.R.; Rani, P.; Mardani, A.; Kumari, R.; Zavadskas, E.K.; Kumar Sharma, D.
An Extended Shapley TODIM Approach Using Novel Exponential Fuzzy Divergence Measures for Multi-Criteria Service Quality in Vehicle Insurance Firms. *Symmetry* **2020**, *12*, 1452.
https://doi.org/10.3390/sym12091452

**AMA Style**

Mishra AR, Rani P, Mardani A, Kumari R, Zavadskas EK, Kumar Sharma D.
An Extended Shapley TODIM Approach Using Novel Exponential Fuzzy Divergence Measures for Multi-Criteria Service Quality in Vehicle Insurance Firms. *Symmetry*. 2020; 12(9):1452.
https://doi.org/10.3390/sym12091452

**Chicago/Turabian Style**

Mishra, Arunodaya Raj, Pratibha Rani, Abbas Mardani, Reetu Kumari, Edmundas Kazimieras Zavadskas, and Dilip Kumar Sharma.
2020. "An Extended Shapley TODIM Approach Using Novel Exponential Fuzzy Divergence Measures for Multi-Criteria Service Quality in Vehicle Insurance Firms" *Symmetry* 12, no. 9: 1452.
https://doi.org/10.3390/sym12091452