Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making
<p>Comparison of grades of IFNs, PFNs, and <span class="html-italic">q</span>-ROFNs.</p> "> Figure 2
<p>Scores of alternative <span class="html-italic">A</span><sub>1</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 3
<p>Scores of alternative <span class="html-italic">A</span><sub>2</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 4
<p>Scores of alternative <span class="html-italic">A</span><sub>3</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 5
<p>Scores of alternative <span class="html-italic">A</span><sub>4</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 6
<p>Scores of alternatives <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when <span class="html-italic">t</span> = 1 and <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 7
<p>Scores of alternative <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when <span class="html-italic">s</span> = 1 and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 8
<p>Scores of alternative <span class="html-italic">A</span><sub>1</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-rung dual hesitant fuzzy weighted geometric Heronian mean (<span class="html-italic">q</span>-RDHFWGHM) operator (<span class="html-italic">q</span> = 3).</p> "> Figure 9
<p>Scores of alternative <span class="html-italic">A</span><sub>2</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 10
<p>Scores of alternative <span class="html-italic">A</span><sub>3</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 11
<p>Scores of alternative <span class="html-italic">A</span><sub>4</sub> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 12
<p>Scores of alternative <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when <span class="html-italic">t</span> = 1 and <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 13
<p>Scores of alternative <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when <span class="html-italic">s</span> = 1 and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator (<span class="html-italic">q</span> = 3).</p> "> Figure 14
<p>Scores of alternative <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when s = <span class="html-italic">t</span> = 1 and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWHM operator.</p> "> Figure 15
<p>Scores of alternative <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <mn>2</mn> <mo>,</mo> <mtext> </mtext> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mn>4</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> when s = <span class="html-italic">t</span> = 1 and <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>∈</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>10</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> based on the <span class="html-italic">q</span>-RDHFWGHM operator.</p> ">
Abstract
:1. Introduction
2. Basic Concepts
2.1. q-Rung Orthopair Fuzzy Set
- 1.
- .
- 2.
- .
- 3.
- .
- 4.
- .
- 1.
- If , then ;
- 2.
- If , thenif , then ;if , then .
2.2. q-Rung Dual Hesitant Fuzzy Set
- 1.
- If, thenis superior to, denoted by;
- 2.
- If, thenif , then is equivalent to , denoted by ;if , then is superior to , denoted by .In the following, we define some operations of the q-RDHFEs.
- 1.
- ;
- 2.
- ;
- 3.
- ;
- 4.
- .
2.3. Heronian Mean
3. The q-Rung Dual Hesitant Fuzzy Heronian Mean Operators
3.1. The q-Rung Dual Hesitant Fuzzy Heronian Mean Operator
- 1.
- If , then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy generalized linear descending weighted mean operator, and we can obtain
- 2.
- If , then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy generalized liner ascending weighted mean operator, and we can obtain
- 3.
- If , then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy basic Heronian mean operator, and we can obtain
- 4.
- If , then the q-RDHFHM reduces to a q-rung dual hesitant fuzzy line Heronian mean operator. It follows that
- 5.
- If , then the q-RDHFHM reduces to a dual hesitant Pythagorean fuzzy Heronian mean operator. So, we can obtain
- 6.
- If , then the q-RDHFHM reduces to the dual hesitant fuzzy Heronian mean operator proposed by Yu et al. [47]. It follows that
3.2. The q-Rung Dual Hesitant Fuzzy Weighted Heronian Mean (q-RDHFWHM) Operator
3.3. The q-Rung Dual Hesitant Fuzzy Geometric Heronian Mean Operator
- If , then the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy generalized geometric linear descending weighted mean operator, and we can obtain
- If , the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy generalized geometric liner ascending weighted mean operator, and we can obtain
- If , the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy basic geometric Heronian mean operator, and we can obtain
- If , the q-RDHFGHM reduces to a q-rung dual hesitant fuzzy line Heronian mean operator, and it follows that
- If , then the q-RDHFGHM reduces to the dual hesitant Pythagorean fuzzy Heronian mean operator, and can we can obtain
- If , then the q-RDHFGHM reduces to the dual hesitant fuzzy Heronian mean operator proposed by Yu et al. [47], and it follows that
3.4. The q-Rung Dual Hesitant Fuzzy Weighted Geometric Heronian Mean Operator
4. A Novel Approach to MAGDM with q-Rung Dual Hesitant Fuzzy Information
4.1. Description of a Typical MAGDM Problem with q-Rung Dual Hesitant Fuzzy Information
4.2. An Algorithm for q-Rung Dual Hesitant Fuzzy MAGDM Problems
5. Numerical Example
5.1. The Decision-Making Process
5.2. The Influence of the Parameters on the Results
5.3. Compared with Exiting MAGDM Methods
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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G1 | G2 | G3 | G4 | |
---|---|---|---|---|
A1 | {{0.3, 0.4}, {0.6}} | {{0.7, 0.9}, {0.1}} | {{0.4}, {0.2,0.3}} | {{0.5, 0.6}, {0.2}} |
A2 | {{0.2, 0.3}, {0.5}} | {{0.6, 0.7}, {0.2}} | {{0.7, 0.8}, {0.2}} | {{0.6}, {0.1, 0.2, 0.3}} |
A3 | {{0.4}, {0.2,0.3}} | {{0.2,0.3,0.4}, {0.6}} | {{0.7,0.8}, {0.1}} | {{0.7}, {0.2,0.3}} |
A4 | {{0.6,0.7}, {0.3}} | {{0.5}, {0.4}} | {{0.3,0.4}, {0.5}} | {{0.4, 0.6}, {0.1,0.2}} |
Parameters | Score Function | Ranking Results |
---|---|---|
s = t = 1/2 | ||
s = t = 1 | ||
s = t = 2 | ||
s = t = 5 | ||
s = 1, t = 2 | ||
s = 2, t = 1 | ||
s = 1, t = 5 | ||
s = 5, t = 1 |
Parameters | Score Function | Ranking Results |
---|---|---|
s = t = 1/2 | ||
s = t = 1 | ||
s = t = 2 | ||
s = t = 5 | ||
s = 1, t = 2 | ||
s = 2, t = 1 | ||
s = 1, t = 5 | ||
s = 5, t = 1 |
Methods | Score Function | Ranking Results |
---|---|---|
Wang et al.’ [48] method based on the DHFWA operator | ||
Yu et al.’ s [47] method based on the DHFWHM operator (s = t = 2) | ||
Tu et al.’s [49] method based on the DHFWBM operator | ||
Wei and Lu’s [42] method based on the DHPFHWA operator | ||
The proposed method in this paper |
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Xu, Y.; Shang, X.; Wang, J.; Wu, W.; Huang, H. Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making. Symmetry 2018, 10, 472. https://doi.org/10.3390/sym10100472
Xu Y, Shang X, Wang J, Wu W, Huang H. Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making. Symmetry. 2018; 10(10):472. https://doi.org/10.3390/sym10100472
Chicago/Turabian StyleXu, Yuan, Xiaopu Shang, Jun Wang, Wen Wu, and Huiqun Huang. 2018. "Some q-Rung Dual Hesitant Fuzzy Heronian Mean Operators with Their Application to Multiple Attribute Group Decision-Making" Symmetry 10, no. 10: 472. https://doi.org/10.3390/sym10100472