# A Hesitant Fuzzy Combined Compromise Solution Framework-Based on Discrimination Measure for Ranking Sustainable Third-Party Reverse Logistic Providers

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## Abstract

**:**

## 1. Introduction

- A novel integrated HF-CoCoSo methodology is introduced for solving MCDM problems with HFSs.
- An HF-discrimination measure based framework is employed for evaluating the criteria weights.
- The proposed method is implemented to choose an optimal S3PRLP for an automobile manufacturing company within the HFS context.
- At last, comparison with extant approaches and sensitivity assessment are studied to confirm the reliability and practicability of the outcomes.

## 2. Preliminaries

#### 2.1. Hesitant Fuzzy Set (HFS)

#### 2.2. Combined Compromise Solution (CoCoSo) Method

#### 2.3. Sustainable Third-Party Reverse Logistics Provider (S3PRLP) Assessment

## 3. Preliminaries

**Definition**

**1**

**[7].**An HFS on the fixed universal set $\Omega $ is often expressed by the following mathematical form:

**Example**

**1.**

**Definition**

**2**

**[7,15].**Let $h,{h}_{1},{h}_{2}\in HFEs\left(Y\right)$. Then, the operations on HFEs are discussed as follows:

(a) | ${h}^{c}={\displaystyle \underset{\kappa \in h}{{\displaystyle \cup}}\left\{1-\kappa \right\}};$ |

(b) | ${h}_{1}\cup {h}_{2}={\displaystyle \underset{{\kappa}_{1}\in {h}_{1},{\kappa}_{2}\in {h}_{2}}{{\displaystyle \cup}}\mathrm{max}\left\{{\kappa}_{1},{\kappa}_{2}\right\}};$ |

(c) | ${h}_{1}\cap {h}_{2}={\displaystyle \underset{{\kappa}_{1}\in {h}_{1},{\kappa}_{2}\in {h}_{2}}{{\displaystyle \cup}}\mathrm{min}\left\{{\kappa}_{1},{\kappa}_{2}\right\}};$ |

(d) | $\lambda h={\displaystyle \underset{\kappa \in h}{{\displaystyle \cup}}\left\{1-{\left(1-\kappa \right)}^{\lambda}\right\}},\lambda >0;$ |

(e) | ${h}^{\lambda}={\displaystyle \underset{\kappa \in h}{{\displaystyle \cup}}\left\{{\kappa}^{\lambda}\right\}},\lambda >0;$ |

(f) | ${h}_{1}\oplus {h}_{2}={\displaystyle \underset{{\kappa}_{1}\in {h}_{1},{\kappa}_{2}\in {h}_{2}}{{\displaystyle \cup}}\left\{{\kappa}_{1}+{\kappa}_{2}-{\kappa}_{1}{\kappa}_{2}\right\}};$ |

(g) | ${h}_{1}\otimes {h}_{2}={\displaystyle \underset{{\kappa}_{1}\in {h}_{1},{\kappa}_{2}\in {h}_{2}}{{\displaystyle \cup}}\left\{{\kappa}_{1}{\kappa}_{2}\right\}}.$ |

**Definition**

**3**

**[21].**For a given HFE $h(v)$, the score and variance functions are given by the following expressions (2) and (3):

- (i)
- If $\wp \left({h}_{1}\right)>\wp \left({h}_{2}\right)$, then ${h}_{1}>{h}_{2}$,
- (ii)
- If $\wp \left({h}_{1}\right)=\wp \left({h}_{2}\right)$, then
- a)
- If $\Im \left({h}_{1}\right)>\Im \left({h}_{2}\right)$, then ${h}_{1}<{h}_{2}$,
- b)
- If $\Im \left({h}_{1}\right)=\Im \left({h}_{2}\right)$, then ${h}_{1}={h}_{2}$.

**Definition**

**4**

**[15,21].**Consider a set of HFEs $E=\left\{{h}_{1},{h}_{2},\dots ,{h}_{n}\right\}$, then, the HF weighted average (HFWA) and HF weighted geometric (HFWG) operators are as follows:

- -
- It is a nonnegative and symmetric mapping of the two compared sets.
- -
- It becomes zero when the two sets coincide.
- -
- It decreases when the two subsets become “more similar” in some sense.

**Definition**

**5**

**[74].**For all $R,S,T\in HFSs(\Omega )$, a function $D:HFS(\Omega )\times HFS(\Omega )\to \mathbb{R}$ is said to be an HF—Discrimination measure if it satisfies

(J_{1}). | $D\left(R,S\right)=D\left(S,R\right);$ |

(J_{2}). | $D\left(R,R\right)=0;$ |

(J_{3}). | $\mathrm{max}\left\{D\left(R\cup T,S\cup T\right),D\left(R\cup T,S\cap T\right)\right\}\le D\left(R,S\right).$ |

**Definition**

**6.**

## 4. New Discrimination Measure for HFSs

**Theorem**

**1.**

**Proof.**

**D**)

_{1}**.**For any two real numbers $\alpha ,\beta \in \mathbb{R}$, the following inequalities assure:

**D**)

_{2}**.**It is obvious from Definition 5.

**D**)

_{3}**.**Suppose that $R=S$, therefore, ${h}_{R}^{\sigma \left(j\right)}\left({v}_{i}\right)={h}_{S}^{\sigma \left(j\right)}\left({v}_{i}\right)$, for each $i=1,2,\dots ,n,j=1,2,\dots ,l\left(h({v}_{i})\right)$. Then, Equation (7) becomes

**Theorem**

**2.**

**(P1).**$D\left(R,S\right)\le D\left(R,T\right)$ and $D\left(S,T\right)\le D\left(R,T\right)$, for $R\subseteq S\subseteq T$;

**Proof.**

**(P2).**$D\left(R\cup S,R\cap S\right)=D\left(R,S\right)$;

**(P3).**$D\left(R\cup S,T\right)\le D\left(R,T\right)+D\left(S,T\right),\forall T\in HFS(\Omega )$;

**(P4).**$D\left(R\cap S,T\right)\le D\left(R,T\right)+D\left(S,T\right),\forall T\in HFS(\Omega )$;

**Proof.**

**(P5).**$D\left(R\cap T,S\cap T\right)\le D\left(R,S\right),\forall T\in HFS(\Omega )$;

**(P6).**$D\left(R\cup T,S\cup T\right)\le D\left(R,S\right),\forall T\in HFS(\Omega )$.

**Proof.**

## 5. Hesitant Fuzzy Combined Compromise Solution (HF-CoCoSo) Approach

**Step 1:**Problem description.

**Step 2:**Determine crisp DMEs’ weights.

**Step 3:**Create the aggregated HF-DM.

**Step 4:**Determine the criteria weights.

**Step 4.1:**To estimate the objective criteria weights, we execute the following process by employing the proposed HF-discrimination measure (7):

**Step 4.2:**Calculate the subjective criteria weights.

_{j}. For this, create the individual importance degree matrix $\left({\eta}_{j}^{k}\right)$ for kth expert by using the process

_{j}given by kth expert, $j=1,2,\dots ,n$, $k=1,2,\dots ,\ell $. To estimate the subjective weight, we get

**Step 4.3:**Evaluate the overall importance degree $\left({\eta}_{j}\right)$ of criteria H

_{j}.

**Step 4.4:**Determine the final weight of criteria.

**Step 5:**Construct the normalized aggregated HF-DM.

**Step 6:**Determine the weighted sum and power weight comparability sequences.

**Step 7:**Computation of relative weights or balanced compromise scores.

**Step 8:**Estimate the final aggregating compromise index.

**Step 9**: End.

## 6. Case Study of S3PRLP Selection

_{1}, P

_{2}, P

_{3}). After establishing the expert team, six candidate providers (M

_{1}, M

_{2}, M

_{3}, M

_{4}, M

_{5}, M

_{6}) are identified for a further evaluation process. These alternatives are evaluated based on the following 13 criteria: Green warehousing (H

_{1}), Pollution control cost (H

_{2}), Green product and eco-design cost (H

_{3}), RL cost (H

_{4}), Green R&D and innovation (H

_{5}), Air emissions (H

_{6}), Environmental management system (H

_{7}), Flexibility (H

_{8}), Quality (H

_{9}), Financial risk (H

_{10}), Health and safety practices (H

_{11}), Social responsibility (H

_{12}), and Employment Practices (H

_{13}). In this example, the criteria S

_{2}, S

_{3}, S

_{4}, S

_{6}, and S

_{10}are of cost type, and the rest all are of the benefit type. Next, the implementation procedures of the proposed HF-CoCoSo model for assessing the S3PRLPs are expressed as follows.

_{1}= 0.3333, λ

_{2}= 0.2689, λ

_{3}= 0.3978}. The linguistic decision matrix is presented in Table 3.

_{j}= (0.0624, 0.1082, 0.0630, 0.0794, 0.0812, 0.0437, 0.1074, 0.0675, 0.0996, 0.0597, 0.0904, 0.0559, 0.0815).

_{2}, H

_{3}, H

_{4}, H

_{6}, and H

_{10}are of non-benefit, and the rest all are of the benefit-type criteria; therefore, by using Table 4 and Equation (23), the normalized aggregated HF-DM is depicted in Table 6.

#### 6.1. Comparative Study

#### 6.1.1. HF-TOPSIS Method

**Step 1–4:**Same as HF-CoCoSo methodology.

**Step 5:**Determine the HF-positive ideal solution (HF-PIS) and HF-negative ideal solution (HF-NIS).

**Step 6:**Calculation of discrimination measures from HF-PIS and HF-NIS

**Step 7:**Estimation off relative closeness coefficient (CF).

**Step 8:**Selection of optimal alternative

**Step 9:**End.

_{2}.

#### 6.1.2. HF-COPRAS Method

**Step 1–4:**Same as HF-CoCoSo framework.

**Step 5:**Sum the values of criteria for benefit and cost.

**Step 6:**Determine the relative weight.

**Step 7:**Compute the priority order.

**Step 8:**Evaluate the utility degree.

**Step 9:**End.

_{2}is the best alternative. Figure 3 presents the comparison of score values of S3PRLP alternatives with various methods.

- ⮚
- The proposed approach utilizes a comparability sequence, and then, the weights are aggregated through two manners. One of them follows the usual multiplication rule, and the second one narrates the weighted power of the distance from the comparability sequence. To validate the preference order, we have defined three different measures (aggregation strategy) for a given alternative. At the ultimate, a cumulative equation reports a ranking. There is not any algorithm among MCDM tools supporting this kind of aggregation. Each strategy would offer a ranking score, which would be further improved by a complete ranking index. This procedure is based on a combination of compromise attitudes.
- ⮚
- The CoCoSo methodology has the potential to entail the DMEs’ estimations into decision-making processes, whilst the COPRAS technique is incapable of doing so.
- ⮚
- The hesitancy degree is considered independently significant in the complete implementation process, and preference ordering of the candidates is obtained using the transaction degrees of all three parameters.
- ⮚
- Assessment of criteria weights is one of the main challenges in the MCDM process. In the developed method, the objective weights, which are determined by new discrimination measure-based formula and the subjective weights, which are expressed by DMEs, are combined, and the aggregated weights are employed in the extended CoCoSo framework.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic diagram of proposed hesitant fuzzy Combined Compromise Solution (HF-CoCoSo) method.

LVs | HFEs | DMEs Risk Preferences | ||
---|---|---|---|---|

Pessimist | Moderate | Optimist | ||

Very high (VH) | [0.85, 1.00] | 0.85 | 0.925 | 1.00 |

High (H) | [0.70, 0.85] | 0.70 | 0.775 | 0.85 |

Medium (M) | [0.55, 0.70] | 0.55 | 0.625 | 0.70 |

Low (L) | [0.40, 0.55] | 0.40 | 0.475 | 0.55 |

Very low (VL) | [0.25, 0.40] | 0.25 | 0.325 | 0.40 |

LVs | HFEs | DMEs Risk Preferences | ||
---|---|---|---|---|

Pessimist | Moderate | Optimist | ||

Extremely preferable (EP) | [0.90, 1.00] | 0.90 | 0.95 | 1.00 |

Strong preferable (SP) | [0.75, 0.90] | 0.75 | 0.825 | 0.90 |

Preferable (P) | [0.60, 0.75] | 0.60 | 0.675 | 0.75 |

Moderately preferable (MP) | [0.50, 0.60] | 0.50 | 0.55 | 0.60 |

Moderate (M) | [0.40, 0.50] | 0.40 | 0.45 | 0.50 |

Moderately undesirable (MU) | [0.30, 0.40] | 0.30 | 0.35 | 0.40 |

Undesirable (U) | [0.20, 0.30] | 0.20 | 0.25 | 0.30 |

Strong undesirable (SU) | [0.10, 0.20] | 0.10 | 0.15 | 0.20 |

Extremely undesirable (EU) | [0.00, 0.10] | 0.00 | 0.05 | 0.10 |

**Table 3.**Linguistic decision matrix given by different experts for sustainable third-party reverse logistics provider (S3PRLP) evaluation.

Criteria | DMEs | M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | M_{6} |
---|---|---|---|---|---|---|---|

H_{1} | P_{1} | MP | M | MU | MP | M | MP |

P_{2} | M | MU | P | MU | P | SU | |

P_{3} | P | MP | M | P | MU | MU | |

H_{2} | P_{1} | MP | MU | MP | MP | P | U |

P_{2} | SP | MP | P | M | MU | MP | |

P_{3} | P | U | MU | P | P | U | |

H_{3} | P_{1} | U | M | M | P | P | M |

P_{2} | MP | SP | MU | M | MU | P | |

P_{3} | M | P | M | P | M | MU | |

H_{4} | P_{1} | M | MU | U | M | P | M |

P_{2} | M | U | P | MU | U | P | |

P_{3} | MP | MU | U | U | MP | M | |

H_{5} | P_{1} | M | M | M | MP | MP | M |

P_{2} | SP | M | MU | M | P | MP | |

P_{3} | M | SU | M | SU | MU | M | |

H_{6} | P_{1} | MP | P | M | MP | SP | SP |

P_{2} | M | MP | MU | U | MU | U | |

P_{3} | P | M | SP | P | U | MP | |

H_{7} | P_{1} | MU | P | MP | M | SU | P |

P_{2} | MP | M | SP | SU | MP | M | |

P_{3} | P | SP | P | M | M | SU | |

H_{8} | P_{1} | SP | EP | MP | P | M | M |

P_{2} | M | P | P | MP | P | P | |

P_{3} | P | MP | P | M | SP | P | |

H_{9} | P_{1} | MP | EP | SP | M | MP | SP |

P_{2} | SP | MP | MP | MU | M | M | |

P_{3} | P | MU | U | M | M | M | |

H_{10} | P_{1} | MU | MP | P | MU | P | SP |

P_{2} | MP | M | M | MP | U | M | |

P_{3} | M | P | MU | P | MU | MU | |

H_{11} | P_{1} | MP | P | M | SP | MU | P |

P_{2} | P | MP | MP | MP | P | MP | |

P_{3} | SP | P | MU | M | MU | SP | |

H_{12} | P_{1} | U | P | M | P | P | M |

P_{2} | SP | SP | SP | M | MP | U | |

P_{3} | M | MU | P | MP | M | MP | |

H_{13} | P_{1} | MU | SP | MU | P | MP | MP |

P_{2} | SP | MP | P | MU | P | MU | |

P_{3} | P | M | SP | SU | M | M |

M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | M_{6} | |
---|---|---|---|---|---|---|

H_{1} | 0.584 | 0.461 | 0.497 | 0.499 | 0.476 | 0.401 |

H_{2} | 0.680 | 0.424 | 0.543 | 0.564 | 0.557 | 0.396 |

H_{3} | 0.472 | 0.641 | 0.461 | 0.581 | 0.484 | 0.476 |

H_{4} | 0.527 | 0.372 | 0.470 | 0.384 | 0.524 | 0.524 |

H_{5} | 0.581 | 0.368 | 0.455 | 0.425 | 0.539 | 0.523 |

H_{6} | 0.548 | 0.571 | 0.579 | 0.527 | 0.526 | 0.575 |

H_{7} | 0.527 | 0.652 | 0.665 | 0.423 | 0.439 | 0.472 |

H_{8} | 0.660 | 0.740 | 0.663 | 0.555 | 0.652 | 0.581 |

H_{9} | 0.665 | 0.699 | 0.581 | 0.424 | 0.505 | 0.579 |

H_{10} | 0.444 | 0.603 | 0.507 | 0.539 | 0.449 | 0.552 |

H_{11} | 0.686 | 0.621 | 0.488 | 0.619 | 0.466 | 0.675 |

H_{12} | 0.559 | 0.609 | 0.653 | 0.561 | 0.539 | 0.470 |

H_{13} | 0.651 | 0.606 | 0.632 | 0.416 | 0.555 | 0.477 |

Criteria | LVs Given by DMEs | HFNs Given by DMEs | $\mathit{s}\left({\mathit{\kappa}}_{\mathit{j}}\right)$ | ${\mathit{\eta}}_{\mathit{j}}$ | ||||
---|---|---|---|---|---|---|---|---|

P_{1} | P_{2} | P_{3} | P_{1} | P_{2} | P_{3} | |||

H_{1} | MU | MP | P | 0.35 | 0.55 | 0.675 | 0.575 | 0.0769 |

H_{2} | MP | M | P | 0.50 | 0.45 | 0.75 | 0.625 | 0.0835 |

H_{3} | U | MP | M | 0.25 | 0.55 | 0.50 | 0.472 | 0.0631 |

H_{4} | MP | M | P | 0.50 | 0.45 | 0.675 | 0.584 | 0.0781 |

H_{5} | M | MP | M | 0.40 | 0.60 | 0.50 | 0.528 | 0.0706 |

H_{6} | MU | P | P | 0.35 | 0.60 | 0.675 | 0.592 | 0.0791 |

H_{7} | U | P | SP | 0.30 | 0.60 | 0.75 | 0.623 | 0.0833 |

H_{8} | SP | M | SP | 0.75 | 0.40 | 0.825 | 0.734 | 0.0981 |

H_{9} | P | MP | P | 0.60 | 0.55 | 0.675 | 0.639 | 0.0854 |

H_{10} | U | P | MP | 0.30 | 0.60 | 0.55 | 0.524 | 0.0700 |

H_{11} | MP | M | SP | 0.50 | 0.45 | 0.75 | 0.625 | 0.0835 |

H_{12} | MU | MP | M | 0.40 | 0.55 | 0.50 | 0.509 | 0.0680 |

H_{13} | MP | M | MU | 0.50 | 0.45 | 0.35 | 0.452 | 0.0604 |

M_{1} | M_{2} | M_{3} | M_{4} | M_{5} | M_{6} | |
---|---|---|---|---|---|---|

H_{1} | 0.584 | 0.461 | 0.497 | 0.499 | 0.476 | 0.401 |

H_{2} | 0.320 | 0.576 | 0.457 | 0.436 | 0.443 | 0.604 |

H_{3} | 0.528 | 0.359 | 0.539 | 0.419 | 0.516 | 0.524 |

H_{4} | 0.473 | 0.628 | 0.530 | 0.616 | 0.476 | 0.476 |

H_{5} | 0.581 | 0.368 | 0.455 | 0.425 | 0.539 | 0.523 |

H_{6} | 0.452 | 0.429 | 0.421 | 0.473 | 0.474 | 0.425 |

H_{7} | 0.527 | 0.652 | 0.665 | 0.423 | 0.439 | 0.472 |

H_{8} | 0.660 | 0.740 | 0.663 | 0.555 | 0.652 | 0.581 |

H_{9} | 0.665 | 0.699 | 0.581 | 0.424 | 0.505 | 0.579 |

H_{10} | 0.556 | 0.397 | 0.493 | 0.461 | 0.551 | 0.448 |

H_{11} | 0.686 | 0.621 | 0.488 | 0.619 | 0.466 | 0.675 |

H_{12} | 0.559 | 0.609 | 0.653 | 0.561 | 0.539 | 0.470 |

H_{13} | 0.651 | 0.606 | 0.632 | 0.416 | 0.555 | 0.477 |

Option | ${\mathbf{\Lambda}}_{\mathit{i}}^{(1)}$ | ${\mathbf{\Lambda}}_{\mathit{i}}^{(2)}$ | ${\mathit{C}}_{\mathit{i}}^{\left(1\right)}$ | ${\mathit{C}}_{\mathit{i}}^{\left(2\right)}$ | ${\mathit{C}}_{\mathit{i}}^{\left(3\right)}$ | ${\mathit{C}}_{\mathit{i}}$ | Ranking |
---|---|---|---|---|---|---|---|

M_{1} | 0.5676 | 0.5437 | 0.1747 | 2.3004 | 0.9818 | 1.8857 | 2 |

M_{2} | 0.5811 | 0.5508 | 0.1779 | 2.3428 | 1.0000 | 1.9206 | 1 |

M_{3} | 0.5551 | 0.5414 | 0.1724 | 2.2701 | 0.9688 | 1.8608 | 3 |

M_{4} | 0.4888 | 0.4773 | 0.1519 | 2.0000 | 0.8535 | 1.6394 | 6 |

M_{5} | 0.5079 | 0.5015 | 0.1587 | 2.0899 | 0.8918 | 1.7130 | 5 |

M_{6} | 0.5292 | 0.5172 | 0.1645 | 2.1663 | 0.9245 | 1.7758 | 4 |

**Table 8.**Overall outcomes of HF-Technique for Order of Preference by Similarity to Ideal Solution (HF-TOPSIS) framework.

Option | $\mathit{D}\left({\mathit{y}}_{\mathit{i}\mathit{j}},{\mathit{\xi}}^{+}\right)$ | $\mathit{D}\left({\mathit{y}}_{\mathit{i}\mathit{j}},{\mathit{\xi}}^{-}\right)$ | $\mathit{C}\mathit{F}\left({\mathit{M}}_{\mathit{i}}\right)$ | Ranking |
---|---|---|---|---|

M_{1} | 0.132 | 0.133 | 0.502 | 3 |

M_{2} | 0.093 | 0.145 | 0.610 | 1 |

M_{3} | 0.119 | 0.104 | 0.465 | 4 |

M_{4} | 0.153 | 0.083 | 0.352 | 6 |

M_{5} | 0.142 | 0.090 | 0.387 | 5 |

M_{6} | 0.101 | 0.137 | 0.577 | 2 |

**Table 9.**The overall computational results of the HF-Complex Proportional Assessment (HF-COPRAS) method.

Option | ${\mathit{\delta}}_{\mathit{i}}^{\left(1\right)}$ | ${\mathit{\delta}}_{\mathit{i}}^{\left(2\right)}$ | ${\mathbf{\u019b}}_{\mathit{i}}$ | ${\mathit{\eta}}_{\mathit{i}}$ | Ranking |
---|---|---|---|---|---|

M_{1} | 0.464 | 0.254 | 0.670 | 96.63 | 2 |

M_{2} | 0.460 | 0.224 | 0.693 | 100.00 | 1 |

M_{3} | 0.435 | 0.224 | 0.667 | 96.30 | 3 |

M_{4} | 0.352 | 0.230 | 0.578 | 83.45 | 6 |

M_{5} | 0.377 | 0.227 | 0.607 | 87.59 | 5 |

M_{6} | 0.391 | 0.213 | 0.635 | 91.70 | 4 |

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**MDPI and ACS Style**

Mishra, A.R.; Rani, P.; Krishankumar, R.; Zavadskas, E.K.; Cavallaro, F.; Ravichandran, K.S.
A Hesitant Fuzzy Combined Compromise Solution Framework-Based on Discrimination Measure for Ranking Sustainable Third-Party Reverse Logistic Providers. *Sustainability* **2021**, *13*, 2064.
https://doi.org/10.3390/su13042064

**AMA Style**

Mishra AR, Rani P, Krishankumar R, Zavadskas EK, Cavallaro F, Ravichandran KS.
A Hesitant Fuzzy Combined Compromise Solution Framework-Based on Discrimination Measure for Ranking Sustainable Third-Party Reverse Logistic Providers. *Sustainability*. 2021; 13(4):2064.
https://doi.org/10.3390/su13042064

**Chicago/Turabian Style**

Mishra, Arunodaya Raj, Pratibha Rani, Raghunathan Krishankumar, Edmundas Kazimieras Zavadskas, Fausto Cavallaro, and Kattur S. Ravichandran.
2021. "A Hesitant Fuzzy Combined Compromise Solution Framework-Based on Discrimination Measure for Ranking Sustainable Third-Party Reverse Logistic Providers" *Sustainability* 13, no. 4: 2064.
https://doi.org/10.3390/su13042064