An extension of TOPSIS for group decision making in intuitionistic fuzzy environment

Naziya Parveen; Prakash N. Kamble

doi:10.3934/mfc.2021002

Article Contents

2021, Volume 4, Issue 1: 61-71. Doi: 10.3934/mfc.2021002

This issue Previous Article The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators Next Article

An extension of TOPSIS for group decision making in intuitionistic fuzzy environment

Naziya Parveen^1, , and
Prakash N. Kamble^1,

Department of Mathematics, Dr Babasaheb Ambedker Marathwada University, Aurangabad, Maharashtra, India 431001

^* Corresponding author: Naziya Parveen
^* Corresponding author: Naziya Parveen

Received: July 31, 2020

Revised: October 31, 2020

Early access: February 2021

Published: February 2021

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Abstract

In the present paper, notion of the distance between two intuitionistic fuzzy elements is presented. Using the new distance measure, we extend TOPSIS (a technique for order preference by similarity to ideal solution) to group decision making for the intuitionistic fuzzy set. Also, group preferences are aggregated within the procedure. Two numerical examples concerning supplier selection in a manufacturing company and nurse selection in a hospital are constructed to show the practicability and the usefulness of this extension for group decision making to reach an optimum solution.

Keywords:

Distance measure of intuitionistic fuzzy set,
intuitionistic fuzzy set,
MCDM methods,
TOPSIS method.

Mathematics Subject Classification: Primary:90C29, 90C70, 90C90.

Citation:

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Table 1. Linguistic variables for rating of alternatives and weight of criteria

Linguistic variables for weights and ratings of criteria	Intuitionistic fuzzy set
Extremily Low (EL)	(0.5, 0.3, 0.2)
Poor (PO)	(0.6, 0.2, 0.2)
Medium (MD)	(0.7, 0.2, 0.1)
Good (GO)	(0.8, 0.1, 0.1)
Excellent (EX)	(0.9, 0.1, 0.0)

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Table 2. Linguistic importance of weight of criteria from three Decision maker

criteria	$ DM_{1} $	$ DM_{2} $	$ DM_{3} $
$ C_{1} $	EX	GO	EX
$ C_{2} $	GO	EX	GO
$ C_{3} $	MD	EX	EX
$ C_{4} $	EX	MD	MD
$ C_{5} $	GO	MD	GO
$ C_{6} $	EX	EX	EX

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Table 3. Linguistic decision matrix for three Decision maker

Decision makers	Criteria	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $	$ C_{6} $
	suppliers
	$ S_{1} $	MD	EX	MD	GO	GO	EX
	$ S_{2} $	GO	EX	GO	MD	GO	EX
$ DM_{1} $	$ S_{3} $	EX	GO	MD	GO	MD	MD
	$ S_{4} $	EX	MD	GO	GO	LO	EX
	$ S_{5} $	EX	GO	MD	MD	GO	GO
	$ S_{1} $	GO	MD	MD	MD	GO	GO
	$ S_{2} $	MD	EX	MD	GO	GO	GO
$ DM_{2} $	$ S_{3} $	MD	GO	GO	EX	MD	EX
	$ S_{4} $	EX	GO	MD	GO	GO	EX
	$ S_{5} $	MD	GO	MD	GO	EX	EX
	$ S_{1} $	MD	GO	MD	GO	EX	EX
	$ S_{2} $	EX	GO	GO	MD	MD	MD
$ DM_{3} $	$ S_{3} $	MD	MD	EX	GO	GO	GO
	$ S_{4} $	GO	MD	GO	GO	LO	MD
	$ S_{5} $	EX	EX	GO	GO	MD	GO

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Table 4. Weights of the criteria from three Decision maker

Criteria	$ DM_{1} $	$ DM_{2} $	$ DM_{3} $
$ C_{1} $	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)
$ C_{2} $	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)
$ C_{3} $	(0.7, 0.2, 0.1)	(0.9, 0.1, 0.0)	(0.9, 0.1, 0.0)
$ C_{4} $	(0.9, 0.1, 0.0)	(0.7, 0.2, 0.1)	(0.7, 0.2, 0.1)
$ C_{5} $	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)
$ C_{6} $	(0.9, 0.1, 0.0)	(0.9, 0.1, 0.0)	(0.9, 0.1, 0.0)

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Table 5. Decision matrix of $ DM_{1} $

Supplier	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $	$ C_{6} $
$ S_{1} $	(0.7, 0.2, 0.1)	(0.9, 0.1, 0.0)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)
$ S_{2} $	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)
$ S_{3} $	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.7, 0.2, 0.1)
$ S_{4} $	(0.9, 0.1, 0.0)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.8, 0.1, 0.1)	(0.6, 0.2, 0.2)	(0.9, 0.1, 0.0)
$ S_{5} $	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.8, 0.1, 0.1)

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Table 6. Normalization matrix of $ DM_{1} $

Supplier	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $	$ C_{6} $
$ S_{1} $	(0.78, 0.11, 0.11)	(1.0, 0.0, 0.0)	(0.87, 0.11, 0.02)	(1.0, 0.0, 0.0)	(1.0, 0.0, 0.0)	(0.78, 0.11, 0.11)
$ S_{2} $	(0.89, 0.0, 0.11)	(1.0, 0.0, 0.0)	(1.0, 0.0, 0.0)	(0.87, 0.11, 0.02)	(1.0, 0.0, 0.0)	(0.78, 0.11, 0.11)
$ S_{3} $	(1.0, 0.0, 0.0)	(0.89, 0.0, 0.11)	(0.87, 0.11, 0.02)	(1.0, 0.0, 0.0)	(0.87, 0.11, 0.02)	(1.0, 0.0, 0.0)
$ S_{4} $	(1.0, 0.0, 0.0)	(0.78, 0.11, 0.11)	(1.0, 0.0, 0.0)	(1.0, 0.0, 0.0)	(0.75, 0.11, 0.14)	(0.78, 0.11, 0.11)
$ S_{5} $	(1.0, 0.0, 0.0)	(0.89, 0.0, 0.11)	(0.87, 0.11, 0.02)	(0.87, 0.11, 0.02)	(1.0, 0.0, 0.0)	(0.87, 0.11, 0.02)

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Table 7. Weighted normalization matrix of $ DM_{1} $

Supplier	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $	$ C_{6} $
$ S_{1} $	(0.70, 0.20, 0.10)	(0.80, 0.10, 0.10)	(0.61, 0.29, 0.10)	(0.90, 0.10, 0.0)	(0.80, 0.10, 0.10)	(0.70, 0.20, 0.10)
$ S_{2} $	(0.80, 0.10, 0.10)	(0.80, 0.10, 0.10)	(0.70, 0.20, 0.10)	(0.78, 0.20, 0.02)	(0.80, 0.10, 0.01)	(0.70, 0.20, 0.10)
$ S_{3} $	(0.90, 0.10, 0.0)	(0.71, 0.10, 0.19)	(0.61, 0.29, 0.10)	(0.90, 0.10, 0.0)	(0.70, 0.20, 0.10)	(0.90, 0.10, 0.0)
$ S_{4} $	(0.90, 0.10, 0.0)	(0.62, 0.20, 0.18)	(0.70, 0.20, 0.10)	(0.90, 0.10, 0.0)	(0.60, 0.20, 0.20)	(0.70, 0.20, 0.10)
$ S_{5} $	(0.90, 0.10, 0.0)	(0.71, 0.10, 0.19)	(0.61, 0.29, 0.10)	(0.78, 0.20, 0.02)	(0.80, 0.10, 0.10)	(0.78, 0.20, 0.02)

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Table 8. Separation measures of all decision maker

Supplier	$ DM_{1} $		$ DM_{2} $		$ DM_{3} $
	$ s_{1}^{+} $	$ s_{1}^{-} $	$ s_{2}^{+} $	$ s_{2}^{-} $	$ s_{3}^{+} $	$ s_{3}^{-} $
$ S_{1} $	0.49	0.50	0.64	0.28	0.69	0.45
$ S_{2} $	0.42	0.57	0.46	0.48	0.46	0.68
$ S_{3} $	0.28	0.72	0.53	0.37	0.59	0.54
$ S_{4} $	0.58	0.41	0.48	0.46	0.64	0.49
$ S_{5} $	0.42	0.58	0.58	0.34	0.40	0.73

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Table 9. Evaluation Table

Supplier	$ S^{+} $	$ S^{-} $	Relative closeness	Rank
$ S_{1} $	0.61	0.41	0.5980	5
$ S_{2} $	0.45	0.58	0.4368	1
$ S_{3} $	0.47	0.54	0.4653	3
$ S_{4} $	0.57	0.45	0.5588	4
$ S_{5} $	0.47	0.55	0.4607	2

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Table 10. Linguistic importance of weights of criteria from three Decision maker

Criteria	$ DM_{1} $	$ DM_{2} $	$ DM_{3} $
$ C_{1} $	EX	MD	MD
$ C_{2} $	GO	LO	GO
$ C_{3} $	EX	GO	MD
$ C_{4} $	GO	GO	EX
$ C_{5} $	MD	EL	EL

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Table 11. Weights of the criteria from three Decision maker

Criteria	$ DM_{1} $	$ DM_{2} $	$ DM_{3} $
$ C_{1} $	(0.9, 0.1, 0.0)	(0.7, 0.2, 0.1)	(0.7, 0.2, 0.1)
$ C_{2} $	(0.8, 0.1, 0.1)	(0.6, 0.2, 0.2)	(0.8, 0.1, 0.1)
$ C_{3} $	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)
$ C_{4} $	(0.8, 0.1, 0.1)	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)
$ C_{5} $	(0.7, 0.2, 0.1)	(0.5, 0.3, 0.2)	(0.5, 0.3, 0.2)

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Table 12. Linguistic decision matrix for three Decision maker

Criteria	Decision	$ D_{1} $	$ D_{2} $	$ D_{3} $
	makers
	$ A_{1} $	GO	EX	EX
$ C_{1} $	$ A_{2} $	MD	MD	GO
	$ A_{3} $	GO	LO	MD
	$ A_{1} $	LO	GO	GO
$ C_{2} $	$ A_{2} $	GO	GO	MD
	$ A_{3} $	EX	MD	EX
	$ A_{1} $	GO	MD	GO
$ C_{3} $	$ A_{2} $	MD	GO	GO
	$ A_{3} $	MD	MD	GO
	$ A_{1} $	GO	MD	MD
$ C_{4} $	$ A_{2} $	EX	GO	EX
	$ A_{3} $	MD	GO	MD
	$ A_{1} $	LO	MD	LO
$ C_{5} $	$ A_{2} $	GO	MD	GO
	$ A_{3} $	GO	EX	MD

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Table 13. Decision matrix of $ DM_{1} $

Alternatives	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $
$ A_{1} $	(0.8, 0.1, 0.1)	(0.6, 0.2, 0.2)	(0.8, 0.1, 0.1)	(0.8, 0.1, 0.1)	(0.6, 0.2, 0.2)
$ A_{2} $	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)	(0.9, 0.1, 0.0)	(0.8, 0.1, 0.1)
$ A_{3} $	(0.8, 0.1, 0.1)	(0.9, 0.1, 0.0)	(0.7, 0.2, 0.1)	(0.7, 0.2, 0.1)	(0.8, 0.1, 0.1)

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Table 14. Normalization matrix of $ DM_{1} $

Alternatives	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $
$ A_{1} $	(0.875, 0.11, 0.14)	(0.67, 0.11, 0.22)	(1.0, 0.0, 0.0)	(0.89, 0.0, 0.11)	(0.75, 0.11, 0.14)
$ A_{2} $	(1.0, 0.0, 0.0)	(0.89, 0.0, 0.11)	(0.875, 0.11, 0.14)	(1.0, 0.0, 0.0)	(1.0, 0.0, 0.0)
$ A_{3} $	(0.875, 0.11, 0.14)	(1.0, 0.0, 0.0)	(0.875, 0.11, 0.14)	(0.78, 0.11, 0.11)	(1.0, 0.0, 0.0)

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Table 15. Weighted normalization matrix of $ DM_{1} $

Alternatives	$ C_{1} $	$ C_{2} $	$ C_{3} $	$ C_{4} $	$ C_{5} $
$ A_{1} $	(0.79, 0.20, 0.01)	(0.54, 0.20, 0.26)	(0.9, 0.1, 0.0)	(0.71, 0.1, 0.18)	(0.53, 0.29, 0.18)
$ A_{2} $	(0.9, 0.1, 0.0)	(0.71, 0.1, 0.18)	(0.79, 0.20, 0.01)	(0.8, 0.1, 0.1)	(0.7, 0.2, 0.1)
$ A_{3} $	(0.79, 0.20, 0.01)	(0.8, 0.1, 0.1)	(0.79, 0.20, 0.01)	(0.624, 0.20, 0.18)	(0.7, 0.2, 0.1)

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Table 16. Separation measure of each decision maker

Alternatives	$ DM_{1} $		$ DM_{2} $		$ DM_{3} $
	$ s_{1}^{+} $	$ s_{1}^{-} $	$ s_{2}^{+} $	$ s_{2}^{-} $	$ s_{3}^{+} $	$ s_{3}^{-} $
$ A_{1} $	0.63	0.20	0.62	0.07	0.56	0.09
$ A_{2} $	0.13	0.63	0.31	0.39	0.26	0.44
$ A_{3} $	0.30	0.43	0.17	0.52	0.26	0.39

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Table 17. Evaluation Table

Alternatives	$ S^{+} $	$ S^{-} $	Relative closeness	Rank
$ A_{1} $	0.60	0.12	0.83	3
$ A_{2} $	0.23	0.49	0.32	1
$ A_{3} $	0.24	0.45	0.35	2

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