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Another View on Intuitionistic Fuzzy Preference Relation-Based Aggregation Operators and Their Applications

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Abstract

Multi-attribute group decision making (\(\mathrm {MAGDM}\)) can be considered as the process of ranking alternatives or selecting an optimal alternative by many decision makers based on multiple criteria. By comparing any two alternatives pairwise, preference relations provide efficient ways to represent the preference degrees of decision makers. The aim of this paper is to introduce the notions of upward intuitionistic fuzzy preference relations from fuzzy information system and to study some properties of these relations. Further the series of upward intuitionistic fuzzy preference aggregation operators including the upward intuitionistic fuzzy preference weighted averaging \(\left( \mathrm {UIFPWA}\right) \) operator, upward intuitionistic fuzzy preference ordered weighted averaging \( \left( \mathrm {UIFPOWA}\right) \) operator and upward intuitionistic fuzzy preference hybrid averaging \(\left( \mathrm {UIFPHA}\right) \) operator and their related results are also investigated. We developed a \(\mathrm {MAGDM}\) method based on the proposed operators under the fuzzy environment and illustrated with a numerical example to study the applicability of the new approach on a candidate selection decision-making problem.

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Acknowledgements

This research is supported by National Natural Science Foundation of China (Nos. 71771140 and 71471172), (Project of cultural masters and “the four kinds of a batch” talents), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045).

Author information

Authors and Affiliations

  1. School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan, 250014, Shandong, China

    Peide Liu

  2. Department of Mathematics & Statistics, Riphah International University, Hajj Complex I-14, Islamabad, Pakistan

    Abbas Ali

  3. Department of Mathematics & Statistics, Bacha Khan University, Charsadda, Khyber Pakhtunkhwa, Pakistan

    Noor Rehman

  4. Department of Mathematics, Islamia College University, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Syed Inayat Ali Shah

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  1. Peide Liu
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  2. Abbas Ali
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  3. Noor Rehman
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  4. Syed Inayat Ali Shah
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Corresponding author

Correspondence to Noor Rehman.

Appendices

Appendices

1.1 Appendix 1

$$\begin{aligned} \mathcal {R}_{C\left( \mathcal {K}\right) }^{\uparrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.4167,0.2500\right) &{} \left( 0.0417,0.0417\right) &{} \left( 0.0000,0.0000\right) \\ \left( 0.2500,0.4167\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.1250,0.2917\right) &{} \left( 0.0833,0.2500\right) \\ \left( 0.0417,0.0417\right) &{} \left( 0.2917,0.1250\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.4583,0.4583\right) \\ \left( 0.0000,0.0000\right) &{} \left( 0.2500,0.0833\right) &{} \left( 0.4583,0.4583\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) . \end{aligned}$$

1.2 Appendix 2

$$\begin{aligned} {}_{A_{1}}\mathcal {R}^{1\downarrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.0000,0.9000\right) &{} \left( 0.4000,0.6000\right) &{} \left( 0.3000,0.4000\right) \\ \left( 0.9000,0.0000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.8000,0.1000\right) &{} \left( 0.8000,0.0000\right) \\ \left( 0.6000,0.4000\right) &{} \left( 0.1000,0.8000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.4000,0.3000\right) \\ \left( 0.4000,0.3000\right) &{} \left( 0.0000,0.8000\right) &{} \left( 0.3000,0.4000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.3 Appendix 3

$$\begin{aligned} {}_{A_{2}}\mathcal {R}^{1\uparrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.1000,0.7000\right) &{} \left( 0.4000,0.5000\right) &{} \left( 0.1000,0.6000\right) \\ \left( 0.7000,0.1000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.7000,0.2000\right) &{} \left( 0.5000,0.4000\right) \\ \left( 0.5000,0.4000\right) &{} \left( 0.2000,0.7000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.2000,0.4000\right) \\ \left( 0.6000,0.1000\right) &{} \left( 0.4000,0.5000\right) &{} \left( 0.6000,0.2000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.4 Appendix 4

$$\begin{aligned} {}_{A_{3}}\mathcal {R}^{1\downarrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.3000,0.0000\right) &{} \left( 0.1000,0.0714\right) &{} \left( 0.3000,0.2143\right) \\ \left( 0.0000,0.8000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.3000,0.5714\right) &{} \left( 0.2857,0.5000\right) \\ \left( 0.0714,0.6000\right) &{} \left( 0.5714,0.3000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.3571,0.3000\right) \\ \left( 0.2000,0.7857\right) &{} \left( 0.5000,0.2857\right) &{} \left( 0.3000,0.3571\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.5 Appendix 5

$$\begin{aligned} {}_{A_{4}}\mathcal {R}^{1\uparrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.5000,0.4000\right) &{} \left( 0.4000,0.4000\right) &{} \left( 0.5000,0.2000\right) \\ \left( 0.4000,0.5000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.3000,0.4000\right) &{} \left( 0.4000,0.2000\right) \\ \left( 0.4000,0.4000\right) &{} \left( 0.4000,0.3000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.6000,0.3000\right) \\ \left( 0.2000,0.5000\right) &{} \left( 0.2000,0.4000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.6 Appendix 6

$$\begin{aligned} {}_{A_{1}}\mathcal {R}^{2\downarrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.8000,0.2000\right) &{} \left( 0.6000,0.3000\right) &{} \left( 0.8000,0.1000\right) \\ \left( 0.2000,0.8000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.5000,0.4000\right) \\ \left( 0.3000,0.6000\right) &{} \left( 0.6000,0.3000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.7000,0.3000\right) \\ \left( 0.1000,0.2000\right) &{} \left( 0.4000,0.5000\right) &{} \left( 0.3000,0.3000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.7 Appendix 7

$$\begin{aligned} {}_{A_{2}}\mathcal {R}^{2\uparrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.6000,0.3000\right) &{} \left( 0.4000,0.5000\right) &{} \left( 0.6000,0.3000\right) \\ \left( 0.3000,0.6000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.2000,0.6000\right) &{} \left( 0.4000,0.4000\right) \\ \left( 0.5000,0.4000\right) &{} \left( 0.6000,0.2000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.7000,0.3000\right) \\ \left( 0.3000,0.6000\right) &{} \left( 0.4000,0.4000\right) &{} \left( 0.3000,0.7000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.8 Appendix 8

$$\begin{aligned} {}_{A_{3}}\mathcal {R}^{2\downarrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.0000,0.9000\right) &{} \left( 0.2000,0.8000\right) &{} \left( 0.3000,0.6000\right) \\ \left( 0.9000,0.0000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.6000,0.3000\right) &{} \left( 0.8000,0.2000\right) \\ \left( 0.8000,0.2000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.6000,0.3000\right) \\ \left( 0.6000,0.3000\right) &{} \left( 0.2000,0.6000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) \end{aligned}$$

1.9 Appendix 9

$$\begin{aligned} {}_{A_{4}}\mathcal {R}^{2\uparrow }= & {} \left( \begin{array}{cccc} \left( 0.5000,0.5000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.7000,0.1000\right) &{} \left( 0.4000,0.4000\right) \\ \left( 0.6000,0.3000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.9000,0.0000\right) &{} \left( 0.6000,0.3000\right) \\ \left( 0.1000,0.7000\right) &{} \left( 0.0000,0.9000\right) &{} \left( 0.5000,0.5000\right) &{} \left( 0.2000,0.8000\right) \\ \left( 0.4000,0.4000\right) &{} \left( 0.3000,0.6000\right) &{} \left( 0.8000,0.2000\right) &{} \left( 0.5000,0.5000\right) \end{array} \right) . \end{aligned}$$

1.10 Appendix 10

See Table 8.

Table 8 Different score functions
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Liu, P., Ali, A., Rehman, N. et al. Another View on Intuitionistic Fuzzy Preference Relation-Based Aggregation Operators and Their Applications. Int. J. Fuzzy Syst. 22, 1786–1800 (2020). https://doi.org/10.1007/s40815-020-00882-1

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