Article Contents
Article Contents

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

• * Corresponding author: Naziya Parveen
• 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.

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

 Citation:

• 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)

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

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

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)

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)

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)

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)

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

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

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

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)

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

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)

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)

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)

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

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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