# Unified Fuzzy Divergence Measures with Multi-Criteria Decision Making Problems for Sustainable Planning of an E-Waste Recycling Job Selection

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## Abstract

**:**

## 1. Introduction

#### Motivation and Novelty

- Some new divergence measures are introduced for FSs based on probabilistic divergence measures.
- Based on directed divergence measures and Jensen’s difference divergence measures, a class of unified divergence measures is developed for FSs.
- Based on proposed measures, an MCDM technique is presented to solve the MCDM problems over FSs.
- A decision-making problem of e-waste recycling partner selection is solved to illustrate the applicability and usefulness of the proposed method.
- A comparison with existing methods is discussed to reveal the validity of the developed method.

## 2. Preliminaries

**Definition**

**1**

**.**Let $X=\left\{{x}_{1},\hspace{0.17em}\hspace{0.17em}{x}_{2},\hspace{0.17em}\hspace{0.17em}\dots ,\hspace{0.17em}\hspace{0.17em}{x}_{n}\right\}$ be the finite discourse set. An FS $K$ defined on $X$ is given as

**Definition**

**2**

**.**Let $K=\left\{\left({x}_{i},\hspace{0.17em}{\mu}_{K}({x}_{i})\right):{x}_{i}\in X\right\}$ and $L=\left\{\left({x}_{i},\hspace{0.17em}{\mu}_{L}({x}_{i})\right):{x}_{i}\in X\right\}$ be two FSs in the finite discourse set $X.$ Then, the function ${D}_{m}:FS(X)\times FS(X)\to \mathbb{R}$ is called the divergence measure for FSs if it holds the following axioms:

- (P1).
- ${D}_{m}\left(K\left|\right|L\right)={D}_{m}\left(L\left|\right|K\right),$
- (P2).
- ${D}_{m}\left(K\left|\right|\hspace{0.17em}L\right)=0$ if $K=L,$
- (P3).
- ${D}_{m}(K\cap T||L\cap T)\le \hspace{0.17em}{D}_{m}(K||L)$ for every $T\in FS(X),$
- (P4).
- ${D}_{m}(K\cup T||L\cup T)\le {D}_{m}(K||L)$ for every $T\in FS(X).$

## 3. Proposed Method

#### 3.1. New Divergence for FSs

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

- (J1).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|L\right)={D}_{m}{}_{\gamma}\left(L\left|\right|K\right)$ and $0\le {D}_{m}{}_{\gamma}\left(K\left|\right|L\right)\le 1,$
- (J2).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|\hspace{0.17em}L\right)=0$ if $K=L,$
- (J3).
- ${D}_{m}{}_{\gamma}\left(K\cap T\left|\right|L\cap T\right)\le \hspace{0.17em}{D}_{m}{}_{\gamma}\left(K\left|\right|L\right)$ for every $T\in FS(X),$
- (J4).
- ${D}_{m}{}_{\gamma}\left(K\cup T\left|\right|L\cup T\right)\le {D}_{m}{}_{\gamma}\left(K\left|\right|L\right)$ for every $T\in FS(X),$
- (J5).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|L\right)={D}_{m}{}_{\gamma}\left({K}^{c}\left|\right|{L}^{c}\right),$
- (J6).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|{L}^{c}\right)={D}_{m}{}_{\gamma}({K}^{c}||L),$
- (J7).
- ${D}_{m}{}_{\gamma}(K||{K}^{c})=1,$ if $K$ is crisp set,
- (J8).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|K\cup L\right)={D}_{m}{}_{\gamma}\left(K\cap L\left|\right|L\right)\le {D}_{m}{}_{\gamma}\left(K\left|\right|L\right)$ for $K\subseteq L$ and $L\subseteq K,$
- (J9).
- ${D}_{m}{}_{\gamma}\left(K\cup L\left|\right|K\cap L\right)={D}_{m}{}_{\gamma}\left(K\left|\right|L\right),$
- (J10).
- ${D}_{m}{}_{\gamma}\left(K\left|\right|L\right)\le {D}_{m}{}_{\gamma}\left(K\left|\right|T\right)$ and ${D}_{m}{}_{\gamma}\left(L\left|\right|T\right)\le {D}_{m}{}_{\gamma}\left(K\left|\right|T\right)$ for $K\subseteq L\subseteq T.$

#### 3.2. Unified $\left(\alpha ,\beta \right)-$ Divergence Measure for FSs

#### 3.2.1. First Generalization of the Unified Expression

#### 3.2.2. Second Generalization of Unified Expression

## 4. Fuzzy MCDM Method for E-Waste Recycling Job Selection

**Definition**

**3.**

**Definition**

**4**

**.**For TFN${\zeta}_{ij}=\left({f}_{ij},{g}_{ij},{h}_{ij}\right)$ the graded mean integration representation of TFN ${\zeta}_{ij}$ is defined by

- Step 1:
- Construct the fuzzy decision matrix $\mathbb{F}={\left({\zeta}_{ij}\right)}_{r\times s}$.

- Step 2:
- Compute ideal solution (IS) and anti-ideal solution (A-IS).

- Step 3:
- Compute the criteria weights

- Step 4:
- Compute the closeness degree of the alternative(s).

- Step 5:
- Rank the alternatives.

## 5. Investigating the Sustainable Planning of an E-Waste Recycling Job Selection

- Step 1:
- Fuzzy IS and A-IS are calculated by using (49) and (50) are as follows:$$\epsilon {\hspace{0.17em}}^{+}=\left\{0.757,\hspace{0.17em}0.76,\hspace{0.17em}0.231,\hspace{0.17em}0.317,\hspace{0.17em}0.244\right\},$$$$\epsilon {\hspace{0.17em}}^{-}=\left\{0.24,\hspace{0.17em}0.25,\hspace{0.17em}0.767,\hspace{0.17em}0.229,\hspace{0.17em}0.77\right\}.$$
- Step 2:
- Corresponding to (51) and (52), the divergence measure of ${\zeta}_{ij}$ form ${\epsilon}^{+}$ and ${\zeta}_{ij}\hspace{0.17em}$ form ${\epsilon}^{-}$ are evaluated as follows:$${D}_{m}{}_{11}^{+}=0.3021,\hspace{0.17em}{D}_{m}{}_{12}^{+}=0.0000,\hspace{0.17em}{D}_{m}{}_{13}^{+}=0.0000,\hspace{0.17em}{D}_{m}{}_{14}^{+}=0.0236,\hspace{0.17em}{D}_{m}{}_{15}^{+}=0.0000,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{21}^{+}=0.2211,\hspace{0.17em}{D}_{m}{}_{22}^{+}=0.3913,\hspace{0.17em}{D}_{m}{}_{23}^{+}=0.5253,\hspace{0.17em}{D}_{m}{}_{24}^{+}=0.0000,\hspace{0.17em}{D}_{m}{}_{25}^{+}=0.4674,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{31}^{+}=0.0000,\hspace{0.17em}{D}_{m}{}_{32}^{+}=0.0000,\hspace{0.17em}{D}_{m}{}_{33}^{+}=0.000566,\hspace{0.17em}{D}_{m}{}_{34}^{+}=0.1138,\hspace{0.17em}{D}_{m}{}_{35}^{+}=0.0000315,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{41}^{+}=0.4910,\hspace{0.17em}{D}_{m}{}_{42}^{+}=0.058,\hspace{0.17em}{D}_{m}{}_{43}^{+}=0.0089,\hspace{0.17em}{D}_{m}{}_{44}^{+}=0.0015,\hspace{0.17em}{D}_{m}{}_{45}^{+}=0.0053.$$And$${D}_{m}{}_{11}^{-}=0.0000525,\hspace{0.17em}{D}_{m}{}_{12}^{-}=0.3913,\hspace{0.17em}{D}_{m}{}_{13}^{-}=0.5253,\hspace{0.17em}{D}_{m}{}_{14}^{-}=0.008647,\hspace{0.17em}{D}_{m}{}_{15}^{-}=0.4674,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{21}^{-}=0.0004378,\hspace{0.17em}{D}_{m}{}_{22}^{-}=0.0000,\hspace{0.17em}{D}_{m}{}_{23}^{-}=0.0000,\hspace{0.17em}{D}_{m}{}_{24}^{-}=0.1138,\hspace{0.17em}{D}_{m}{}_{25}^{-}=0.0000,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{31}^{-}=0.4583,\hspace{0.17em}{D}_{m}{}_{32}^{-}=0.3913,\hspace{0.17em}{D}_{m}{}_{33}^{-}=0.2478,\hspace{0.17em}{D}_{m}{}_{34}^{-}=0.0000,\hspace{0.17em}{D}_{m}{}_{35}^{-}=0.3250,\hspace{0.17em}\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{41}^{-}=0.0000,\hspace{0.17em}{D}_{m}{}_{42}^{-}=0.0101,\hspace{0.17em}{D}_{m}{}_{43}^{-}=0.1009,\hspace{0.17em}{D}_{m}{}_{44}^{-}=0.0172,\hspace{0.17em}{D}_{m}{}_{45}^{-}=0.1081.$$Next, the overall performances, by using (53), of alternative are calculated as follows:$${D}_{m}{}_{11}=0.0001737,\hspace{0.17em}{D}_{m}{}_{12}=1.0000,\hspace{0.17em}{D}_{m}{}_{13}=1.0000,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{14}=0.2681,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{15}=1.0000,\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{21}=0.0020,\hspace{0.17em}{D}_{m}{}_{22}=0.0000,\hspace{0.17em}{D}_{m}{}_{23}=0.0000,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{24}=1.0000,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{25}=0.0000,\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{31}=1.0000,\hspace{0.17em}{D}_{m}{}_{32}=1.0000,\hspace{0.17em}{D}_{m}{}_{33}=0.9977,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{34}=0.0000,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{35}=0.9999,\phantom{\rule{0ex}{0ex}}{D}_{m}{}_{41}=0.0000,\hspace{0.17em}{D}_{m}{}_{42}=0.1483,\hspace{0.17em}{D}_{m}{}_{43}=0.9189,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{44}=0.9198,\hspace{0.17em}\hspace{0.17em}{D}_{m}{}_{45}=0.9533.$$
- Step 3:
- To compute the weight vector, construct the model$$\mathrm{max}\hspace{0.17em}J=1.0022{\omega}_{1}+2.1483{\omega}_{2}+2.9166{\omega}_{3}+2.1879{\omega}_{4}+2.9532{\omega}_{5}\phantom{\rule{0ex}{0ex}}s.\hspace{0.17em}\hspace{0.17em}t.\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\{\begin{array}{l}0.25\le {\omega}_{1}\le 0.4,\hspace{0.17em}0.16\le {\omega}_{2}\le 0.27,\hspace{0.17em}\hspace{0.17em}\\ 0.1\le {\omega}_{4}\le 0.18,\hspace{0.17em}0.2\le {\omega}_{5}\le 0.35,\hspace{0.17em}\hspace{0.17em}\\ 0.15\le {\omega}_{3}\le 0.25,\hspace{0.17em}\hspace{0.17em}{\omega}_{1}\ge 0.2{\omega}_{4},\hspace{0.17em}\hspace{0.17em}{\omega}_{5}-\hspace{0.17em}{\omega}_{2}\le {\omega}_{3}\\ {\omega}_{j}={\left({\omega}_{1},{\omega}_{2},\dots ,{\omega}_{s}\right)}^{T},\hspace{0.17em}{\omega}_{j}\ge 0,\hspace{0.17em}\hspace{0.17em}{\displaystyle \sum _{j=1}^{s}{\omega}_{j}=1}.\end{array}$$Using MATHEMATICA, model (57) is computed and the criteria’s weight vector is computed by$${\left({\omega}_{j}\right)}^{T}={\left(0.25,0.16,0.165,0.1,0.325\right)}^{T}.$$
- Step 4:
- The calculated closeness degrees of the alternatives are given as$$J\left({Y}_{1}\right)=0.6769,\hspace{0.17em}\hspace{0.17em}J\left({Y}_{2}\right)=0.1005,\hspace{0.17em}\hspace{0.17em}J\left({Y}_{3}\right)=0.8996,\hspace{0.17em}J\left({Y}_{4}\right)=0.5771.$$
- Step 5:
- Based on calculated closeness degrees of the alternatives, the ranking of the associations is ${Y}_{3}\succ {Y}_{1}\succ {Y}_{4}\succ {Y}_{2}.$

#### Comparison and Discussion for the Sustainable Planning of an E-Waste Recycling Job Selection

_{3}, i.e., desirable e-waste recycling job. In general, the advantages of the extended approach over the existing methods are presented by

- The portrayal of the relative significance of various criteria is made simple with the help of linguistic evaluations enabling the attainment of the desirable stability between parameter performance and desirable e-waste recycling job in various circumstances.
- The aggregation of various criteria (e.g., health and safety at workplace, public acceptability, and green technology innovation) is performed efficiently with the proposed method whereas, the preference order abnormality problem is evaded with the help of objective utility functions.
- The developed method utilizes a conventional concept of the synchronized satisfaction of the given objectives that comprises the compromise doctrine of TOPSIS, that is, to be as closer as likely to an IS and as farther as likely from an A-IS.
- The aggregation of various criteria is made with FSR TOPSIS Chamodrakas, et al. [57], a proposed method to evade possible inconsistency of the ranking outcomes. Furthermore, the utilization of parameterized utility functions for evaluating the normalized decision matrix in FSR TOPSIS reduces the order abnormality concern.
- As the significance of DEs is considered, we have discussed a method based graded mean integration representation (GMIR) of TFN, which provides more precise outcomes for MCDM problems.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Linguistic Terms | Fuzzy Score |
---|---|

Very Strong (VS) | (0.7, 0.9, 1.0) |

Fairly Strong (FS) | (0.5, 0.7, 0.9) |

Equal (E) | (0.3, 0.5, 0.7) |

Fairly Weak (FW) | (0.1, 0.3, 0.5) |

Very Weak (VW) | (0.0, 0.1, 0.3) |

Sustainability Criteria under Each Dimension | Firm’s E-Waste Products | Alternatives of the E-Waste Recycling Job | |||
---|---|---|---|---|---|

Sustainability Dimension | Sustainability Criteria | Description | E-Waste Product | Description | |

Social $\left(S\right)$ | Health and safety at the workplace $\left({Z}_{1}\right)$ | The number of decreased workers’ compensation claimed | Computer | Personal computers, CRT monitors, notebook computers, PC keyboards, LCD monitors, modem, cables associated with PC system, mouse, etc. | ${Y}_{1}$ |

Public acceptability $\left({Z}_{2}\right)$ | General attitudes/ public perceptions regard to the firms’ e-recycling services | Communication equipment | Server, telephone handsets, hub, rack mount cabinets, routers, switch, assorted network gear, PABX controller units, modems/print servers, uninterruptible power supplies, etc. | ||

Economic $\left({E}_{c}\right)$ | Direct/Indirect cost $\left({Z}_{3}\right)$ | The expenditure is given/The expenses for exploring business opportunities | Battery | Lead acid batteries, lithium ion, lithium batteries, NiCad batteries (vented/sealed), NiMH batteries, Alkaline batteries, etc. | ${Y}_{2}$ |

Environmental $\left({E}_{n}\right)$ | Green technology Innovation $\left({Z}_{4}\right)$ | The new technology innovations Made to decrease the negative environmental Effects | Cell phone | Cell phones, battery, charger, accessories, etc. | ${Y}_{3}$ |

The problem decreased the volume of trash/waste within the landfill | Office electrical equipment | Desktop printers, enterprise printer, photocopy machines, fax machines, desktop scanners, desktop multifunction printers/scanners, etc. | |||

Landfill reduction $\left({Z}_{5}\right)$ | Consumer electrical equipment | CRT televisions, LCD televisions, plasma televisions, VCR/DVD/set top box, speaker devices, Hi-Fi stereo, domestic vacuum cleaners, microwave ovens, cordless phones, digital still cameras, video cameras, etc. | ${Y}_{4}$ |

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

${Y}_{1}$ | (E,VW,FW,VW) | (VS,E,VS,VS) | (FW,FW,VW,VW) | (FW,VW,FW,VW) | (FW,FW,E,VW) |

${Y}_{2}$ | (FW,FW,E,VW) | (FW,FW,FW,VW) | (VS,VS,VS,E) | (VW,VW,VW,VW) | (VS,VS,FS,FS) |

${Y}_{3}$ | (FS,VS,FS,FS) | (VS,FS,FS,FS) | (FS,FW,VW,VW) | (FS,VW,FW,VW) | (VW,VW,FW,E) |

${Y}_{4}$ | (E,FW,FW,VW) | (FS,E,FW,FW) | (FS,FW,E,VW) | (VW,VW,VW,FW) | (VS,FW,FW,VW) |

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

${Y}_{1}$ | (0.1,0.25,0.45) | (0.6,0.8,0.93) | (0.05,0.2,0.4) | (0.05,0.2,0.35) | (0.13,0.3,0.5) |

${Y}_{2}$ | (0.15,0.35,0.5) | (0.08,0.25,0.45) | (0.6,0.8,0.93) | (0.0,0.1,0.3) | (0.65,0.8,0.95) |

${Y}_{3}$ | (0.55,0.75,0.93) | (0.5,0.75,0.9) | (0.15,0.3,0.5) | (0.1,0.3,0.5) | (0.1,0.2,0.45) |

${Y}_{4}$ | (0.12,0.3,0.5) | (0.25,0.45,0.65) | (0.23,0.4,0.6) | (0.03,0.15,0.35) | (0.22,0.4,0.55) |

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

${Y}_{1}$ | 0.2844 | 0.76 | 0.231 | 0.229 | 0.244 |

${Y}_{2}$ | 0.315 | 0.25 | 0.767 | 0.095 | 0.77 |

${Y}_{3}$ | 0.757 | 0.751 | 0.319 | 0.317 | 0.275 |

${Y}_{4}$ | 0.24 | 0.435 | 0.411 | 0.157 | 0.391 |

Methods | Benchmark | Ranking | Optimal Alternative |
---|---|---|---|

TOPSIS Tzeng and Huang [53] method | Crisp Sets | ${Y}_{3}\succ {Y}_{4}\succ {Y}_{1}\succ {Y}_{2}$ | ${Y}_{3}$ |

F-TOPSIS Chen [54] method | Fuzzy sets and distance measure | ${Y}_{3}\succ {Y}_{1}\succ {Y}_{4}\succ {Y}_{2}$ | ${Y}_{3}$ |

IF-TOPSIS Joshi and Kumar [55] method | Intuitionistic fuzzy sets and distance measure | ${Y}_{3}\succ {Y}_{1}\succ {Y}_{4}\succ {Y}_{2}$ | ${Y}_{3}$ |

IF-TOPSIS Mishra, et al. [56] method | Intuitionistic fuzzy sets and similarity measure | ${Y}_{3}\succ {Y}_{1}\succ {Y}_{4}\succ {Y}_{2}$ | ${Y}_{3}$ |

Proposed method | Fuzzy sets and divergence measure based linear programming model | ${Y}_{3}\succ {Y}_{1}\succ {Y}_{4}\succ {Y}_{2}$ | ${Y}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Rani, P.; Govindan, K.; Mishra, A.R.; Mardani, A.; Alrasheedi, M.; Hooda, D.S.
Unified Fuzzy Divergence Measures with Multi-Criteria Decision Making Problems for Sustainable Planning of an E-Waste Recycling Job Selection. *Symmetry* **2020**, *12*, 90.
https://doi.org/10.3390/sym12010090

**AMA Style**

Rani P, Govindan K, Mishra AR, Mardani A, Alrasheedi M, Hooda DS.
Unified Fuzzy Divergence Measures with Multi-Criteria Decision Making Problems for Sustainable Planning of an E-Waste Recycling Job Selection. *Symmetry*. 2020; 12(1):90.
https://doi.org/10.3390/sym12010090

**Chicago/Turabian Style**

Rani, Pratibha, Kannan Govindan, Arunodaya Raj Mishra, Abbas Mardani, Melfi Alrasheedi, and D. S. Hooda.
2020. "Unified Fuzzy Divergence Measures with Multi-Criteria Decision Making Problems for Sustainable Planning of an E-Waste Recycling Job Selection" *Symmetry* 12, no. 1: 90.
https://doi.org/10.3390/sym12010090