February  2021, 4(1): 61-71. doi: 10.3934/mfc.2021002

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

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

* Corresponding author: Naziya Parveen

Received  August 2020 Revised  November 2020 Published  February 2021

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.

Citation: Naziya Parveen, Prakash N. Kamble. An extension of TOPSIS for group decision making in intuitionistic fuzzy environment. Mathematical Foundations of Computing, 2021, 4 (1) : 61-71. doi: 10.3934/mfc.2021002
References:
[1]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.  Google Scholar

[2]

K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Application, Studies in Fuzziness and Soft Computing, 35, Physica-Verlag, Heidelberg, 1999. doi: 10.1007/978-3-7908-1870-3.  Google Scholar

[3]

P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications, Word Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/2326.  Google Scholar

[4]

P. A. EjegwaS. O. AkoweP. M. Otene and J. M. Ikyule, An overview on intuitionistic fuzzy sets, Internat. J. Sci. Tech. Res., 3 (2014), 142-145.   Google Scholar

[5]

C. L. Hwang and K. Yoon, Multiple Attribute Decision Making. Methods and Applications, Lecture Notes in Economics and Mathematical Systems, 186, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/978-3-642-48318-9.  Google Scholar

[6]

P. N. Kamble and Na ziya Parveen, An application of integrated fuzzy AHP and fuzzy TOPSIS method for staff selection, J. Comput. Math. Sci., 9 (2018), 1161-1169.  doi: 10.29055/jcms/855.  Google Scholar

[7]

K. Palczewski and W. Sałabun, The fuzzy TOPSIS applications in the last decade, Procedia Comput. Sci., 159 (2019), 2294-2303.  doi: 10.1016/j.procs.2019.09.404.  Google Scholar

[8]

N. Parveen and P. N. Kamble, Decision making problem using fuzzy TOPSIS method with hexagonal fuzzy number, in Computing in Engineering and Technology, Advances in Intelligent Systems and Computing, 1025, Springer, Singapore, 2020, 421-430. doi: 10.1007/978-981-32-9515-5_40.  Google Scholar

[9]

C. SamantraS. Data and S. S. Mahapatra, Application of fuzzy based VIKOR approach for multi-attribute group decision making (MAGDM): A case study in supplier selection, Decision Making in Manufacturing and Services, 6 (2012), 25-39.  doi: 10.7494/dmms.2012.6.1.25.  Google Scholar

[10]

F. ShenX. MaZ. LiZ. Xu and D. Cai, An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation, Inform. Sci., 428 (2018), 105-119.  doi: 10.1016/j.ins.2017.10.045.  Google Scholar

[11]

H. S. ShihH. J. Shyur and E. S. Lee, An extension of TOPSIS for group decision making, Math. Comput. Modelling, 45 (2007), 801-813.  doi: 10.1016/j.mcm.2006.03.023.  Google Scholar

[12]

E. B. Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G.-W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, J. Cleaner Production, 250 (2020). doi: 10.1016/j.jclepro.2019.119517.  Google Scholar

[13]

B. VahdaniS. M. Mousavi and R. Tavakkoli-Moghaddam, Group decision making based on novel fuzzy modified TOPSIS method, Appl. Math. Model., 35 (2011), 4257-4269.  doi: 10.1016/j.apm.2011.02.040.  Google Scholar

[14]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

show all references

References:
[1]

K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.  doi: 10.1016/S0165-0114(86)80034-3.  Google Scholar

[2]

K. T. Atanassov, Intuitionistic Fuzzy Sets. Theory and Application, Studies in Fuzziness and Soft Computing, 35, Physica-Verlag, Heidelberg, 1999. doi: 10.1007/978-3-7908-1870-3.  Google Scholar

[3]

P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets. Theory and Applications, Word Scientific Publishing Co., Inc., River Edge, NJ, 1994. doi: 10.1142/2326.  Google Scholar

[4]

P. A. EjegwaS. O. AkoweP. M. Otene and J. M. Ikyule, An overview on intuitionistic fuzzy sets, Internat. J. Sci. Tech. Res., 3 (2014), 142-145.   Google Scholar

[5]

C. L. Hwang and K. Yoon, Multiple Attribute Decision Making. Methods and Applications, Lecture Notes in Economics and Mathematical Systems, 186, Springer-Verlag, Berlin-New York, 1981. doi: 10.1007/978-3-642-48318-9.  Google Scholar

[6]

P. N. Kamble and Na ziya Parveen, An application of integrated fuzzy AHP and fuzzy TOPSIS method for staff selection, J. Comput. Math. Sci., 9 (2018), 1161-1169.  doi: 10.29055/jcms/855.  Google Scholar

[7]

K. Palczewski and W. Sałabun, The fuzzy TOPSIS applications in the last decade, Procedia Comput. Sci., 159 (2019), 2294-2303.  doi: 10.1016/j.procs.2019.09.404.  Google Scholar

[8]

N. Parveen and P. N. Kamble, Decision making problem using fuzzy TOPSIS method with hexagonal fuzzy number, in Computing in Engineering and Technology, Advances in Intelligent Systems and Computing, 1025, Springer, Singapore, 2020, 421-430. doi: 10.1007/978-981-32-9515-5_40.  Google Scholar

[9]

C. SamantraS. Data and S. S. Mahapatra, Application of fuzzy based VIKOR approach for multi-attribute group decision making (MAGDM): A case study in supplier selection, Decision Making in Manufacturing and Services, 6 (2012), 25-39.  doi: 10.7494/dmms.2012.6.1.25.  Google Scholar

[10]

F. ShenX. MaZ. LiZ. Xu and D. Cai, An extended intuitionistic fuzzy TOPSIS method based on a new distance measure with an application to credit risk evaluation, Inform. Sci., 428 (2018), 105-119.  doi: 10.1016/j.ins.2017.10.045.  Google Scholar

[11]

H. S. ShihH. J. Shyur and E. S. Lee, An extension of TOPSIS for group decision making, Math. Comput. Modelling, 45 (2007), 801-813.  doi: 10.1016/j.mcm.2006.03.023.  Google Scholar

[12]

E. B. Tirkolaee, A. Mardani, Z. Dashtian, M. Soltani and G.-W. Weber, A novel hybrid method using fuzzy decision making and multi-objective programming for sustainable-reliable supplier selection in two-echelon supply chain design, J. Cleaner Production, 250 (2020). doi: 10.1016/j.jclepro.2019.119517.  Google Scholar

[13]

B. VahdaniS. M. Mousavi and R. Tavakkoli-Moghaddam, Group decision making based on novel fuzzy modified TOPSIS method, Appl. Math. Model., 35 (2011), 4257-4269.  doi: 10.1016/j.apm.2011.02.040.  Google Scholar

[14]

L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

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)
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)
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
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
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
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
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)
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)
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)
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)
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)
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)
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)
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)
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
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
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
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
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
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
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)
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)
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
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
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)
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)
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)
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)
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)
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)
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
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
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
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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