Pythagorean fuzzy combinative distance-based assessment with pure linguistic information and its application to financial strategies of multi-national companies

Figures & data

Table 1. Some fuzzy extensions and applications of C.O.D.A.S.

Reference	Information type	Case study	Methods
Ghorabaee et al., 2017	TFNs	Market segment selection	CODAS
Panchal et al., 2017	FLNs	Optimal maintenance strategy selection	AHP, CODAS
Bolturk, 2018	PFS	Supplier selection problem	CODAS
Ren, 2018	IFNs	Ranking of energy storage technologies	AHP, CODAS
Pamucar et al., 2018	LNFNs	Selection of power-generation technologies	PW, CODAS
Peng and Garg 2018	IVFNs	Evaluation and decision of emergency plans	WDBA, CODAS
Bolturk and Kahraman, 2018	IVIFNs	Selection among wave energy plant locations	CODAS
Peng and Li, 2019	IVPFNs	Emergency decision-making of mine explosions	Entropy, CODAS
Fatma and Gökhan, 2018	IVAIFNs	Selection problem for an engineering position	VIKOR, CODAS

Note: TFNs – trapezoidal fuzzy numbers; FLNs – fuzzy linguistic numbers; PFNs – Pythagorean fuzzy numbers;.

IFNs – Intuitionistic fuzzy numbers; LNFNs – linguistic neutrosophic fuzzy numbers;.

IVFNs – interval-valued fuzzy numbers; IVNFNs – interval-valued neutrosophic fuzzy numbers;.

IVIF – interval-valued intuitionistic fuzzy numbers; IVPF – interval-valued Pythagorean fuzzy numbers;.

IVAIF – interval-valued Atanassov intuitionistic fuzzy numbers.

Source: Designed by the authors.

Table 2. List of notations.

Notation	Explanation
$A=\{〈x,{\mu }_A(x),{\nu }_A(x)〉|x\in X\}$	an intuitionistic fuzzy set, $0\le {\mu }_A(x)+{\nu }_A(x)\le 1$
$S=\left\{s_h\right|h=0,1,\dots ,t\}$	a collection of linguistic terms
$A=\{〈x,{\mu }_A(x),{\nu }_A(x)〉|x\in X\}$	a Pythagorean fuzzy set, ${({\mu }_A(x))}^2+{({\nu }_A(x))}^2\le 1$
$S(\alpha )={\mu }_A^2-{\nu }_A^2,S(\alpha )\in [-1,1]$	score function
$H(\alpha )={\mu }_A^2+{\nu }_A^2,H(\alpha )\in [0,1]$	accuracy function
$T(·,·)$	t-norm function
$S(·,·)$	t-conorm function
$A\bigvee B$	the generalised intersection operators
$A\bigwedge B$	the generalised union operators
$A=\{(x,s_{\theta }(x),s_{\sigma }(x))|x\in X\}$	a linguistic Pythagorean fuzzy value, ${\theta }^2+{\sigma }^2\le t^2$
EDi	the Euclidean distance
HLDi	the Hamming-like distance
$NS={[n{s}_j]}_{1\times n}$	the negative ideal solution

Source: Designed by the authors.

Figure 1. The procedure of the proposed LPFN-CODAS method. Source: Designed by the authors.

Table 3. The pair-wise comparison of the importance of criteria by expert D₁.

Criteria	C1	C2	C3	C4
C1	(s4, s4)	(s6, s6)	(s5, s3)	(s6, s5)
C2	(s6, s6)	(s4, s4)	(s4, s4)	(s6, s2)
C3	(s3, s5)	(s4, s4)	(s4, s4)	(s3, s3)
C4	(s5, s6)	(s2, s6)	(s3, s3)	(s4, s4)

Source: Designed by the authors.

Table 4. The pair-wise comparison of the importance of criteria by expert D₂.

Criteria	C1	C2	C3	C4
C1	(s4, s4)	(s3, s5)	(s7, s3)	(s2, s6)
C2	(s6, s3)	(s4, s4)	(s3, s5)	(s3, s7)
C3	(s3, s7)	(s5, s3)	(s4, s4)	(s1, s5)
C4	(s6, s2)	(s7, s3)	(s5, s1)	(s4, s4)

Source: Designed by the authors.

Table 5. The pair-wise comparison of the importance of criteria by expert D₃.

Criteria	C1	C2	C3	C4
C1	(s4, s4)	(s3, s6)	(s2, s1)	(s3, s5)
C2	(s6, s3)	(s4, s4)	(s3, s5)	(s3, s2)
C3	(s1, s2)	(s5, s3)	(s4, s4)	(s6, s3)
C4	(s5, s3)	(s2, s3)	(s3, s6)	(s4, s4)

Source: Designed by the authors.

Table 6. Decision matrix $N^{(1)}$ provided by expert D₁.

	C1	C2	C3	C4
A1	(s6, s1)	(s3, s1)	(s3, s3)	(s1, s6)
A2	(s3, s4)	(s3, s4)	(s2, s5)	(s2, s4)
A3	(s1, s3)	(s2, s3)	(s3, s2)	(s6, s1)
A4	(s6, s2)	(s4, s3)	(s5, s1)	(s7, s1)

Source: Designed by the authors.

Table 7. Decision matrix $N^{(2)}$ provided by expert D₂.

	C1	C2	C3	C4
A1	(s3, s2)	(s4, s1)	(s3, s4)	(s2, s3)
A2	(s5, s2)	(s2, s1)	(s3, s4)	(s2, s5)
A3	(s2, s3)	(s3, s3)	(s1, s2)	(s3, s3)
A4	(s5, s2)	(s3, s3)	(s5, s2)	(s4, s1)

Source: Designed by the authors.

Table 8. Decision matrix $N^{(3)}$ provided by expert D₃.

	C1	C2	C3	C4
A1	(s3, s3)	(s3, s5)	(s6, s1)	(s2, s6)
A2	(s3, s2)	(s2, s4)	(s2, s1)	(s3, s4)
A3	(s6, s1)	(s2, s5)	(s3, s4)	(s1, s3)
A4	(s5, s1)	(s4, s4)	(s6, s2)	(s5, s2)

Source: Designed by the authors.

Table 9. The weighted normalised aggregated decision matrix G.

Alternatives	C1	C2	C3	C4
A1	$(s_{1.9042},s_{5.6299})$	$(s_{1.7701},s_{6.2294})$	$(s_{2.1464},s_{5.9268})$	$(s_{0.6694},s_{7.4275})$
A2	$(s_{2.1112},s_{5.9796})$	$(s_{1.0966},s_{6.3075})$	$(s_{1.4612},s_{6.2659})$	$(s_{0.8804},s_{7.3388})$
A3	$(s_{1.1546},s_{5.9783})$	$(s_{1.2286},s_{6.6339})$	$(s_{1.0030},s_{5.5101})$	$(s_{1.0942},s_{6.7587})$
A4	$(s_{2.9671},s_{5.3904})$	$(s_{1.7547},s_{6.4634})$	$(s_{3.3244},s_{4.8245})$	$(s_{2.0810},s_{6.0627})$

Source: Designed by the authors.

Table 10. LPF Weighted Euclidean and Hamming-like distances.

	HLD					ED
	C 1	C2	C3	C4		C1	C2	C3	C4
A1	0.0625	0.0561	0.0953	0.0000	A1	0.0663	0.0599	0.1038	0.0000
A2	0.0797	0.0272	0.0382	0.0176	A2	0.0976	0.0333	0.0468	0.0187
A3	0.0001	0.0110	0.0630	0.0558	A3	0.0001	0.0135	0.0771	0.0599
A4	0.1510	0.0548	0.1934	0.1176	A4	0.1634	0.0604	0.2072	0.1417

Source: Designed by the authors.

Table 11. Relative assessment matrix (Q).

Alternatives	A1	A2	A3	A4	Si	Rank
A1	−0.1635	−0.0787	0.0000	−0.8093	−1.0515	4
A2	0.1291	0.0000	0.1245	0.0895	0.3430	2
A3	0.0000	0.0848	0.1635	−0.6458	−0.3975	3
A4	0.6458	0.7306	0.8093	0.0000	2.1857	1

Source: Designed by the authors.

Table 12. Comparison of ranking results with various fuzzy extensions of C.O.D.A.S.

Methodology	Ranking of alternatives	Best alternatives	Worst alternatives
FLNs-AHP	$A_4\succ A_3\succ A_2\succ A_1$	A4	A1
IFS-AHP	$A_4\succ A_3\succ A_2\succ A_1$	A4	A1
LNF-PW	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1
IVF-WDBA	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1
IVNF-TOPSIS	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1
IVPF-Entropy	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1
IVAIF-VIKOR	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1
Proposed approach	$A_4\succ A_2\succ A_3\succ A_1$	A4	A1

Source: Designed by the authors.

Figure 2. Ranking results with various values of parameter ε.

Note: Cases 1–15 correspond to the different values of $\epsilon \in [0.01,1.00]$ as shown in Table 13. Source: Designed by the authors.

Table 13. Ranking results with different values of threshold ε.

ε
Alter.	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.10	0.30	0.50	0.70	0.90	1.00
A1	4	4	4	4	4	4	4	4	4	4	4	3	3	3	3
A2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4
A3	3	3	3	3	3	3	3	3	3	3	3	2	2	2	2
A4	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1

Source: Designed by the authors.

Ghorabaee, M. K., Amiri, M., Zavadskas, E. K., Hooshmand, R., & Antucheviciene, J. (2017). Fuzzy extension of the CODAS method for multi-criteria market segment evaluation. Journal of Business Economics and Management, 18(1), 1–19. doi:10.3846/16111699.2016.1278559

 Web of Science ®Google Scholar

Panchal, D., Chatterjee, P., Shukla, R. K., Choudhury, T., Tamosaitiene, J., et al. (2017). Integrated fuzzy AHP-CODAS framework for maintenance decision in urea fertilizer industry. Economic Computation and Economic Cybernetics Studies and Research, 51(3), 179–196.

 Web of Science ®Google Scholar

Pamucar, D., Badi, I., Sanja, K., & Obradović, R. (2018). A novel approach for the selection of power-generation technology using a linguistic Neutrosophic CODAS method: a case study in Libya. Energies, 11(9), 2489–2514. doi:10.3390/en11092489

 Web of Science ®Google Scholar

Peng, X. D., & Garg, H. (2018). Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure. Computers and Industrial Engineering, 119, 439–452.

 PubMed Web of Science ®Google Scholar

Bolturk, E., & Kahraman, C. (2018). Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem. Journal of Intelligent and Fuzzy Systems, 35(4), 4865–4877. doi:10.3233/JIFS-18979

 Web of Science ®Google Scholar

Peng, X. D., & Li, W. (2019). Algorithms for interval-valued Pythagorean fuzzy sets in emergency decision making based on multi-parametric similarity measures and WDBA. IEEE Access., 7, 7419–7441. doi:10.1109/ACCESS.2018.2890097

 Web of Science ®Google Scholar

Fatma, B. Y., & Gökhan, Ö. (2018). Interval-valued Atanassov intuitionistic fuzzy CODAS method for multi-criteria group decision making problems. Group Decision and Negotiation, 28, 433–452.

 Web of Science ®Google Scholar