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January  2020, 16(1): 445-467. doi: 10.3934/jimo.2018162

## Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment

 School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University) Patiala-147004, Punjab, India

Received  October 2017 Revised  July 2018 Published  October 2018

Fund Project: The author would like to thank the Editor-in-Chief and referees for providing very helpful comments and suggestions.

In the present article, we extended the idea of the linguistic intuitionistic fuzzy set to linguistic interval-valued intuitionistic fuzzy (LIVIF) set to represent the data by the interval-valued linguistic terms of membership and non-membership degrees. Some of the desirable properties of the proposed set are studied. Also, we propose a new ranking method named as possibility degree measures to compare the two or more different LIVIF numbers. During the aggregation process, some LIVIF weighted and ordered weighted aggregation operators are proposed to aggregate the collections of the LIVIF numbers. Finally, based on these proposed operators and possibility degree measure, a new group decision making approach is presented to rank the different alternatives. A real-life case has been studied to manifest the practicability and feasibility of the proposed group decision making method.

Citation: Harish Garg, Kamal Kumar. Group decision making approach based on possibility degree measure under linguistic interval-valued intuitionistic fuzzy set environment. Journal of Industrial & Management Optimization, 2020, 16 (1) : 445-467. doi: 10.3934/jimo.2018162
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LIVIFNs decision matrix $\widetilde{R}^{(1)}$ of decision maker $D^{(1)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_2 ])$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_3 , s_4 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_2 , s_4 ], [ s_1, s_2 ])$ $( [s_2 , s_4 ], [ s_3 , s_4 ])$ $( [s_1 , s_3 ], [ s_2 , s_3])$ $A_{3}$ $( [s_4 , s_6 ], [ s_1 , s_2 ])$ $( [s_5 , s_6 ], [ s_1 , s_1 ])$ $( [s_3 , s_4 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_1 , s_3 ], [ s_3 , s_4 ])$ $( [s_3 , s_5 ], [ s_1 , s_3 ])$ $( [s_6 , s_7 ], [ s_1 , s_1])$ Weights 0.40 0.25 0.20 0.15
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_2 ])$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_3 , s_4 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_2 , s_3 ])$ $( [s_2 , s_4 ], [ s_1, s_2 ])$ $( [s_2 , s_4 ], [ s_3 , s_4 ])$ $( [s_1 , s_3 ], [ s_2 , s_3])$ $A_{3}$ $( [s_4 , s_6 ], [ s_1 , s_2 ])$ $( [s_5 , s_6 ], [ s_1 , s_1 ])$ $( [s_3 , s_4 ], [ s_2 , s_3 ])$ $( [s_4 , s_5 ], [ s_1 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_2 , s_3 ])$ $( [s_1 , s_3 ], [ s_3 , s_4 ])$ $( [s_3 , s_5 ], [ s_1 , s_3 ])$ $( [s_6 , s_7 ], [ s_1 , s_1])$ Weights 0.40 0.25 0.20 0.15
LIVIFNs decision matrix $\widetilde{R}^{(2)}$ of decision maker $D^{(2)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $([ s_ 4 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_1 , s_3] )$ $( [s_1 , s_2 ], [ s_1 , s_4] )$ $( [s_2 , s_3 ], [ s_3 , s_4 ] )$ $( [ s_3 , s_5 ], [ s_1 , s_3])$ $A_{3}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2 ] )$ $([ s_3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_3 , s_4 ], [ s_2 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $( [s_1 , s_2 ], [ s_3 , s_5 ] )$ $( [s_3 , s_3 ], [ s_2 , s_3 ] )$ $( [ s_2 , s_3 ], [ s_1 , s_2])$ Weights 0.30 0.35 0.25 0.10
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $([ s_ 4 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_3 , s_5 ], [ s_1 , s_3] )$ $( [s_1 , s_2 ], [ s_1 , s_4] )$ $( [s_2 , s_3 ], [ s_3 , s_4 ] )$ $( [ s_3 , s_5 ], [ s_1 , s_3])$ $A_{3}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2 ] )$ $([ s_3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_3 , s_4 ], [ s_2 , s_3])$ $A_{4}$ $( [s_4 , s_5 ], [ s_1 , s_2 ] )$ $( [s_1 , s_2 ], [ s_3 , s_5 ] )$ $( [s_3 , s_3 ], [ s_2 , s_3 ] )$ $( [ s_2 , s_3 ], [ s_1 , s_2])$ Weights 0.30 0.35 0.25 0.10
LIVIFNs decision matrix $\widetilde{R}^{(3)}$ of decision maker $D^{(3)}$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_2 ] )$ $( [s_2 , s_3 ], [ s_2 , s_4 ] )$ $([ s_ 3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_1 , s_4 ], [ s_2 , s_3] )$ $( [s_4 , s_5 ], [ s_1 , s_2] )$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [ s_3 , s_4 ], [ s_2 , s_4])$ $A_{3}$ $( [s_2 , s_3 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3 ] )$ $([ s_3 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3])$ $A_{4}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_1 , s_2 ], [ s_3 , s_4 ] )$ $( [s_3 , s_5 ], [ s_1 , s_{2}])$ $([ s_{5} , s_6 ], [ s_1 , s_2])$ Weights 0.35 0.40 0.15 0.10
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $A_{1}$ $( [s_2 , s_4 ], [ s_1 , s_2 ] )$ $( [s_2 , s_3 ], [ s_2 , s_4 ] )$ $([ s_ 3 , s_5 ], [ s_2 , s_3 ] )$ $( [s_5 , s_6 ], [ s_1 , s_2])$ $A_{2}$ $( [s_1 , s_4 ], [ s_2 , s_3] )$ $( [s_4 , s_5 ], [ s_1 , s_2] )$ $( [s_2 , s_4 ], [ s_1 , s_3 ] )$ $( [ s_3 , s_4 ], [ s_2 , s_4])$ $A_{3}$ $( [s_2 , s_3 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3 ] )$ $([ s_3 , s_5 ], [ s_1 , s_3 ] )$ $( [s_3 , s_5 ], [ s_2 , s_3])$ $A_{4}$ $( [s_3 , s_4 ], [ s_2 , s_3 ] )$ $( [s_1 , s_2 ], [ s_3 , s_4 ] )$ $( [s_3 , s_5 ], [ s_1 , s_{2}])$ $([ s_{5} , s_6 ], [ s_1 , s_2])$ Weights 0.35 0.40 0.15 0.10
Collective individual performance of each decision maker
 $D^{(1)}$ $D^{(2)}$ $D^{(3)}$ $A_{1}$ $( [ s_{3.4146 }, s_{ 4.8354 }], [s_{ 1.6184 }, s_{ 2.6217 }])$ $([ s_{ 3.1569 }, s_{ 4.7623 }], [s_{ 1.0000 }, s_{ 2.5725 }])$ $([ s_{2.3294 }, s_{ 3.8391 }], [s_{ 1.5690 }, s_{ 3.0359}])$ $A_{2}$ $( [s_{ 2.1199 }, s_{ 4.1887 }], [s_{ 1.9875 }, s_{ 2.9952 }])$ $([ s_{ 1.8455 }, s_{ 3.1932 }], [s_{ 1.5647 }, s_{ 3.6266 }])$ $([ s_{ 2.1562 }, s_{ 4.3734 }], [s_{ 1.4691 }, s_{ 2.7404}])$ $A_{3}$ $( [s_{ 3.9930 }, s_{ 5.3834 }], [s_{ 1.2125 }, s_{ 2.1497 }])$ $([ s_{ 3.5873 }, s_{ 4.8744 }], [s_{ 1.6674 }, s_{ 2.6705 }])$ $([ s_{ 2.6031 }, s_{ 4.1814 }], [s_{ 1.5193 }, s_{ 3.0000}])$ $A_{4}$ $( [s_{ 2.8377 }, s_{ 4.6284 }], [s_{ 1.9496 }, s_{ 3.0265 }])$ $([ s_{ 2.1380 }, s_{ 3.0342 }], [s_{ 2.0129 }, s_{ 3.5022 }])$ $([ s_{ 2.0345 }, s_{ 3.2643 }], [s_{ 2.2028 }, s_{ 3.2137}])$
 $D^{(1)}$ $D^{(2)}$ $D^{(3)}$ $A_{1}$ $( [ s_{3.4146 }, s_{ 4.8354 }], [s_{ 1.6184 }, s_{ 2.6217 }])$ $([ s_{ 3.1569 }, s_{ 4.7623 }], [s_{ 1.0000 }, s_{ 2.5725 }])$ $([ s_{2.3294 }, s_{ 3.8391 }], [s_{ 1.5690 }, s_{ 3.0359}])$ $A_{2}$ $( [s_{ 2.1199 }, s_{ 4.1887 }], [s_{ 1.9875 }, s_{ 2.9952 }])$ $([ s_{ 1.8455 }, s_{ 3.1932 }], [s_{ 1.5647 }, s_{ 3.6266 }])$ $([ s_{ 2.1562 }, s_{ 4.3734 }], [s_{ 1.4691 }, s_{ 2.7404}])$ $A_{3}$ $( [s_{ 3.9930 }, s_{ 5.3834 }], [s_{ 1.2125 }, s_{ 2.1497 }])$ $([ s_{ 3.5873 }, s_{ 4.8744 }], [s_{ 1.6674 }, s_{ 2.6705 }])$ $([ s_{ 2.6031 }, s_{ 4.1814 }], [s_{ 1.5193 }, s_{ 3.0000}])$ $A_{4}$ $( [s_{ 2.8377 }, s_{ 4.6284 }], [s_{ 1.9496 }, s_{ 3.0265 }])$ $([ s_{ 2.1380 }, s_{ 3.0342 }], [s_{ 2.0129 }, s_{ 3.5022 }])$ $([ s_{ 2.0345 }, s_{ 3.2643 }], [s_{ 2.2028 }, s_{ 3.2137}])$
Worse alternative $A_{4}^{\prime}$ for each decision maker
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $D^{(1)}$ $( [s_{2}, s_{ 3 }], [s_{ 3 }, s_{ 4}])$ $( [s_{ 0 }, s_{ 3 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 2 }, s_{ 4 }])$ $( [s_{ 3 }, s_{ 4 }], [s_{ 2 }, s_{3}])$ $D^{(2)}$ $( [s_{ 2 }, s_{ 3 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 0 }, s_{ 1 }], [s_{ 4 }, s_{ 6 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4}])$ $D^{(3)}$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 5 }])$ $( [s_{ 1 }, s_{ 1 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 2 }, s_{ 3 }], [s_{ 2 }, s_{3}])$
 $G_{1}$ $G_{2}$ $G_{3}$ $G_{4}$ $D^{(1)}$ $( [s_{2}, s_{ 3 }], [s_{ 3 }, s_{ 4}])$ $( [s_{ 0 }, s_{ 3 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 2 }, s_{ 4 }])$ $( [s_{ 3 }, s_{ 4 }], [s_{ 2 }, s_{3}])$ $D^{(2)}$ $( [s_{ 2 }, s_{ 3 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 0 }, s_{ 1 }], [s_{ 4 }, s_{ 6 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 4}])$ $D^{(3)}$ $( [s_{ 1 }, s_{ 2 }], [s_{ 3 }, s_{ 5 }])$ $( [s_{ 1 }, s_{ 1 }], [s_{ 4 }, s_{ 5 }])$ $( [s_{ 2 }, s_{ 4 }], [s_{ 3 }, s_{ 4 }])$ $( [s_{ 2 }, s_{ 3 }], [s_{ 2 }, s_{3}])$
The characteristic comparisons of different methods
 Whether flexibly to express a wider range of information Whether describe information using linguistic features Whether describe information by interval-valued numbers Whether have the characteristic of generalization Xu and Yager [48] no no no no Xu [44] yes no yes no Zhang [51] no yes no yes The proposed method yes yes yes yes
 Whether flexibly to express a wider range of information Whether describe information using linguistic features Whether describe information by interval-valued numbers Whether have the characteristic of generalization Xu and Yager [48] no no no no Xu [44] yes no yes no Zhang [51] no yes no yes The proposed method yes yes yes yes
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