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Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets pp 587–602Cite as

VIKOR Method for Decision Making Problems in Interval Valued Neutrosophic Environment

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 369))

Abstract

In this chapter, we discuss the VIKOR method for solving MCDM problem with interval valued neutrosophic number . First we define interval-valued neutrosophic number . The different properties of that type of numbers are also discussed. For solving MCDM problem we propose Interval-valued neutrosophic weighted arithmetic average (INNWAA) operator and Interval-valued neutrosophic weighted geometric average (INNWGA) operator in neutrosophic environment.

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Acknowledgements

The second author of the article wishes to convey his heartiest thanks to Miss. Gullu for inspiring him to write the chapter.

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Authors and Affiliations

  1. Department of Production Engineering, National Institute of Technology, Agartala, Jirania, 799046, India

    Syed Abou Iltaf Hussain & Uttam Kumar Mandal

  2. Department of Mathematics, Midnapore College (Autonomous), Midnapore, West Midnapore, 721101, West Bengal, India

    Sankar Prasad Mondal

Authors
  1. Syed Abou Iltaf Hussain
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  2. Sankar Prasad Mondal
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  3. Uttam Kumar Mandal
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Corresponding author

Correspondence to Sankar Prasad Mondal .

Editor information

Editors and Affiliations

  1. Department of Industrial Engineering, Istanbul Technical University, Macka, Istanbul, Turkey

    Cengiz Kahraman

  2. Department of Industrial Engineering, Istanbul Okan University, Akfirat-Tuzla, Istanbul, Turkey

    İrem Otay

Appendices

Appendix 1

$$ \begin{aligned} \Delta_{1} & = \left\langle {\left[ {0.174^{{p_{1} }} 0.242^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.134^{{p_{2} }} 0.313^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.193^{{p_{3} }} 0.268^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \Delta_{2} & = \left\langle {\left[ {0.169^{{p_{1} }} 0.228^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.091^{{p_{2} }} 0.234^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.168^{{p_{3} }} 0.240^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \Delta_{3} & = \left\langle {\left[ {0.178^{{p_{1} }} 0.237^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.157^{{p_{2} }} 0.334^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.210^{{p_{3} }} 0.276^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \Delta_{4} & = \left\langle {\left[ {0.159^{{p_{1} }} 0.219^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.110^{{p_{2} }} 0.256^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.122^{{p_{3} }} 0.174^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \Delta_{5} & = \left\langle {\left[ {0.179^{{p_{1} }} 0.238^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.191^{{p_{2} }} 0.366^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.153^{{p_{3} }} 0.226^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \end{aligned} $$

Appendix 2

$$ \begin{aligned} \sigma_{1} & = \left\langle {\left[ {0.039^{{p_{1} }} 0.066^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.042^{{p_{2} }} 0.091^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.058^{{p_{3} }} 0.072^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \sigma_{2} & = \left\langle {\left[ {0.049^{{p_{1} }} 0.065^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.030^{{p_{2} }} 0.074^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.055^{{p_{3} }} 0.069^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \sigma_{3} & = \left\langle {\left[ {0.049^{{p_{1} }} 0.066^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.052^{{p_{2} }} 0.103^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.065^{{p_{3} }} 0.081^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \sigma_{4} & = \left\langle {\left[ {0.053^{{p_{1} }} 0.073^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.025^{{p_{2} }} 0.066^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.027^{{p_{3} }} 0.037^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \sigma_{5} & = \left\langle {\left[ {0.049^{{p_{1} }} 0.065^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {0.103^{{p_{2} }} 0.148^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {0.035^{{p_{3} }} 0.046^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \end{aligned} $$

Appendix 3

$$ \begin{aligned} Pr_{1} & = \left\langle {\left[ {\left( { - 2.365} \right)^{{p_{1} }} 1.235^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {\left( { - 1.483} \right)^{{p_{2} }} 1.443^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {\left( { - 9.340} \right)^{{p_{3} }} 3.393^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ Pr_{2} & = \left\langle {\left[ {\left( { - 1.812} \right)^{{p_{1} }} 1.135^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {\left( { - 1.130} \right)^{{p_{2} }} 1.524^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {\left( { - 9.004} \right)^{{p_{3} }} 2.878^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ Pr_{3} & = \left\langle {\left[ {\left( { - 1.958} \right)^{{p_{1} }} 1.041^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {\left( { - 1.490} \right)^{{p_{2} }} 1.438^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {\left( { - 11.16} \right)^{{p_{3} }} 3.730^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ Pr_{4} & = \left\langle {\left[ {\left( { - 1.553} \right)^{{p_{1} }} 1.553^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {\left( { - 1.332} \right)^{{p_{2} }} 1.322^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {\left( { - 2.937} \right)^{{p_{3} }} 2.937^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ Pr_{5} & = \left\langle {\left[ {\left( { - 1.974} \right)^{{p_{1} }} 1.005^{{\left( {1 - p_{1} } \right)}} } \right],\left[ {\left( { - 1.423} \right)^{{p_{2} }} 2.010^{{\left( {1 - p_{2} } \right)}} } \right],\left[ {\left( { - 4.428} \right)^{{p_{3} }} 2.169^{{\left( {1 - p_{3} } \right)}} } \right]} \right\rangle \\ \end{aligned} $$

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Hussain, S.A.I., Mondal, S.P., Mandal, U.K. (2019). VIKOR Method for Decision Making Problems in Interval Valued Neutrosophic Environment. In: Kahraman, C., Otay, İ. (eds) Fuzzy Multi-criteria Decision-Making Using Neutrosophic Sets. Studies in Fuzziness and Soft Computing, vol 369. Springer, Cham. https://doi.org/10.1007/978-3-030-00045-5_22

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