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TOPSIS Techniques on q-Rung Orthopair Fuzzy Sets and Its Extensions

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q-Rung Orthopair Fuzzy Sets

Abstract

An effective technique for dealing with multiple-criteria decision-making (MCDM) problems of real world is the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). q-Rung Orthopair Fuzzy Sets (q-ROFs) were introduced by Yager as a generalization of intuitionistic fuzzy sets, in which the sum of the qth powers of the (membership and non-membership) degrees is restricted to one. As the value of q increases, the feasible region for orthopair also increases. This results in more orthopairs satisfying the limitations and hence broadening the scope of representation of fuzzy information. This chapter makes an attempt to integrate the TOPSIS technique with q-ROF sets with some of its generalizations and extensions. Initially, we explore the concept of TOPSIS technique to solve MCDM problems under q-Rung Orthopair Fuzzy (q-ROF) environment and illustrate it with its application in solving a transport policy problem. Then we consider the TOPSIS technique to solve MCDM problems under q-Rung Orthopair Hesitant Fuzzy (q-ROHF) settings. To explain it we have mentioned an illustration of military aircraft overhaul effectiveness evaluation. In addition to the above-mentioned methods, we present the TOPSIS technique for solving decision-making problems under the newly introduced q-rung orthopair fuzzy soft set (q-ROFS\(_f\)S). Here, we tackle a problem of selection of a medical clinic utilizing the q-ROFS\(_f\) TOPSIS method.

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Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Calicut, Calicut, 673601, India

    V. Salsabeela, Aparna Sivadas, Bibin Mathew & Sunil Jacob John

  2. Department of Basic Science, Amal Jyothi College of Engineering, Kanjirappally, Kottayam, Kerala, 686518, India

    Bibin Mathew

Authors
  1. V. Salsabeela
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  2. Aparna Sivadas
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  3. Bibin Mathew
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  4. Sunil Jacob John
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Corresponding author

Correspondence to Sunil Jacob John .

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Editors and Affiliations

  1. Thapar Institute of Engineering and Technology, Patiala, Punjab, India

    Harish Garg

Appendices

Appendix 1

$$\begin{aligned} A&= {{Z_{1}} + {Z_{2}} + {Z_{3}} + {Z_{4}}} \\&= \left( \begin{array}{lllllllllllll} (0.83, 0.53)&{} (0.84, 0.21)&{} (0.32, 0.16)&{} (0.93, 0.14)&{} (0.81, 0.23)&{} (0.73, 0.21)\\ &{} (0.36, 0.79)&{} (0.26, 0.16)&{} (0.43, 0.71)&{} (0.16, 0.63)&{} (0.36, 0.71)\\ &{} (0.96, 0.12)&{} (0.86, 0.32)\\ (0.72, 0.63)&{} (0.56, 0.72)&{} (0.32, 0.63)&{} (0.51, 0.71)&{} (0.26, 0.71)&{} (0.36, 0.36)\\ &{} (0.24, 0.84)&{} (0.69, 0.34)&{} (0.49, 0.63)&{} (0.26, 0.61)&{} (0.61, 0.36)\\ &{} (0.71, 0.36)&{} (0.26, 0.37)\\ (0.39, 0.79)&{} (0.61, 0.72)&{} (0.23, 0.26)&{} (0.56, 0.36)&{} (0.26, 0.71)&{} (0.31, 0.76)\\ &{} (0.63, 0.49)&{} (0.39, 0.71)&{} (0.82, 0.26)&{} (0.71, 0.30)&{} (0.71, 0.21)\\ &{} (0.38, 0.81)&{} (0.18, 0.47) \end{array} \right) \\&= \left[ {{\dot{z}}_{j k}}\right] _{3 \times 13}. \end{aligned}$$

Appendix 2

$$\begin{aligned} J&= \ddot{z}_{{j k}_{3\times 13}}\\&= \left( \begin{array}{lllllllllllll} (0.04, 0.07)&{} (0.07, 0.02)&{} (0.01, 0.02)&{} (0.01, 0.07)&{} (0.02, 0.06)&{} (0.02, 0.06)\\ &{} (0.06, 0.03)&{} (0.01, 0.02)&{} (0.03, 0.06)&{} (0.01, 0.04)&{} (0.03, 0.05)\\ &{} (0.08, 0.01)&{} (0.07, 0.03)\\ (0.05, 0.02)&{} (0.04, 0.06)&{} (0.04, 0.02)&{} (0.06, 0.04)&{} (0.06, 0.02)&{} (0.03, 0.03)\\ &{} (0.07, 0.02)&{} (0.03, 0.06)&{} (0.04, 0.05)&{} (0.01, 0.04)&{} (0.03, 0.05)\\ &{} (0.08, 0.01)&{} (0.07, 0.03)\\ (0.06, 0.03)&{} (0.05, 0.06)&{} (0.02, 0.02)&{} (0.03, 0.04)&{} (0.06, 0.02)&{} (0.06, 0.02)\\ &{} (0.05, 0.04)&{} (0.06, 0.03)&{} (0.07, 0.02)&{} (0.05, 0.02)&{} (0.04, 0.01)\\ &{} (0.03, 0.06)&{} (0.01, 0.04) \end{array} \right) \end{aligned}$$

where \( \ddot{z}_{j k}=\mathfrak {y}_{k} \times \dot{z}_{j k} .\)

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Salsabeela, V., Sivadas, A., Mathew, B., John, S.J. (2022). TOPSIS Techniques on q-Rung Orthopair Fuzzy Sets and Its Extensions. In: Garg, H. (eds) q-Rung Orthopair Fuzzy Sets. Springer, Singapore. https://doi.org/10.1007/978-981-19-1449-2_5

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