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A neutrosophic enhanced best–worst method for considering decision-makers’ confidence in the best and worst criteria

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Abstract

The best–worst method (BWM) is a multiple criteria decision-making (MCDM) method for evaluating ≤a set of alternatives based on a set of decision criteria where two vectors of pairwise comparisons are used to calculate the importance weight of decision criteria. The BWM is an efficient and mathematically sound method used to solve a wide range of MCDM problems by reducing the number of pairwise comparisons and identifying the inconsistencies derived from the comparison process. In spite of its simplicity and efficiency, the BWM does not consider the decision-makers’ (DMs’) confidence in their pairwise comparisons. We propose a neutrosophic enhancement to the original BWM by introducing two new parameters as the DMs’ confidence in the best-to-others preferences and the DMs’ confidence in the others-to-worst preferences. We present two real-world cases to illustrate the applicability of the proposed neutrosophic enhanced BWM (NE-BWM) by considering confidence rating levels of the DMs.

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Notes

  1. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS).

  2. Vlsekriterijumska Optimizacija I Kompromisno Resenje (in Serbian) (VIKOR).

  3. Fuzzy-Delphi Method (FDM).

  4. Data Envelopment Analysis (DEA).

  5. Probabilistic hesitant fuzzy elements (PHFE).

  6. TOmada de Deciso Interativa e Multicritrio (in Portuguese) (TODIM) meaning interactive and multicriteria decision-making.

  7. COmplex PRoportional ASsessment (COPRAS).

  8. ELimination Et Choix Traduisant la REalit (in French) (ELECTRE) or elimination and choice expressing reality.

  9. Multi-objective optimization by ratio analysis plus the full MULTIplicative form (MULTIMOORA).

  10. Interval rough number (IRN).

  11. Mixed integer linear model (MILM).

  12. Non-linear model (NLM).

  13. Intuitionistic Fuzzy BWM (IF-BWM).

  14. Intuitionistic Fuzzy Multiplicative BWM (IFM-BWM).

  15. Test problem 1 has not been considered because it regards the weights in the original BWM.

  16. The reason for that is clear in Eq. (9) as interchanging \( \mathop \rho \nolimits^{ + } \) and \( \rho^{ - } \) would not produce a new solution. For instance, for \( \mathop a\nolimits_{BW} = 2 \) and \( \mathop \rho \nolimits^{ - } = 0.68 \) and \( \mathop \rho \nolimits^{ + } = 0.90 \) the \( CI \) would be \( CI = 0.274 \) which is the same \( CI \) value for \( \mathop a\nolimits_{BW} = 2 \) and \( \mathop \rho \nolimits^{ - } = 0.90 \) and \( \mathop \rho \nolimits^{ + } = 0.68 \).

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Acknowledgement

The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions. Amin Vafadarnikjoo would like to acknowledge the Post-Graduate Research Scholarship from the University of East Anglia, Social Science Faculty (SSF) and Research Training Support Grant (RTSG) which supported him in this research at Norwich Business School. Dr. Madjid Tavana is grateful for the partial support he received from the Czech Science Foundation (GAČR19-13946S) for this research.

Author information

Authors and Affiliations

  1. Norwich Business School, University of East Anglia, Norwich, UK

    Amin Vafadarnikjoo, Tiago Botelho & Konstantinos Chalvatzis

  2. Distinguished Chair of Business Systems and Analytics, Business Systems and Analytics Department, La Salle University, Philadelphia, PA, 19141, USA

    Madjid Tavana

  3. Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, 33098, Paderborn, Germany

    Madjid Tavana

  4. Tyndall Centre for Climate Change Research, University of East Anglia, Norwich, UK

    Konstantinos Chalvatzis

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  1. Amin Vafadarnikjoo
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  2. Madjid Tavana
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  4. Konstantinos Chalvatzis
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Appendices

Appendix A: Preliminaries on the neutrosophic set theory (NST)

Smarandache (1999) introduced the NST as a rigorous general framework for generalizing the concept of IFS. Smarandache (2005) elaborated on the differences between the NS and the IFS theories. The neutrosophic set can independently quantify truth-membership (or membership), indeterminacy-membership, and falsity-membership (or non-membership) functions (Vafadarnikjoo et al. 2018; Ye 2014). Govindan et al. (2015) have reviewed some fundamental definitions of the IFS theory, and intuitionistic fuzzy numbers. Several researchers have recently integrated the NST with decision-making techniques such as ANP and VIKOR (Abdel-Baset et al. 2019), TOPSIS (Nancy and Garg 2019; Biswas et al. 2016), AHP (Bolturk and Kahraman 2018), and fuzzy cognitive maps (Ferreira and Meidutė-Kavaliauskienė 2019). We present some basic definitions of the NST here.

Definition A.1 [Neutrosophic set (NS) (Smarandache 1999)]

Let \( U \) be a universal discourse and let \( x \) signify a generic element in \( U \). The NS \( A \) in \( U \) is characterized by a truth-membership function \( \mathop T\nolimits_{A} \left( x \right) \), an indeterminacy-membership function \( \mathop I\nolimits_{A} \left( x \right) \) and a falsity-membership function \( \mathop F\nolimits_{A} \left( x \right) \). The \( \mathop T\nolimits_{A} \left( x \right) \), \( \mathop I\nolimits_{A} \left( x \right) \) and \( \mathop F\nolimits_{A} \left( x \right) \) are elements of \( \left] {\mathop 0\nolimits^{ - } ,\mathop 1\nolimits^{ + } } \right[ \), where \( \mathop 1\nolimits^{ + } = 1 + \varepsilon \) and \( \mathop 0\nolimits^{ - } = 0 - \varepsilon \) are non-standard finite numbers. The NS can be represented as Eq. (A.1). Note that \( \mathop 0\nolimits^{ - } \le \mathop T\nolimits_{A} \left( x \right) + \mathop I\nolimits_{A} \left( x \right) + \mathop F\nolimits_{A} \left( x \right) \le \mathop 3\nolimits^{ + } \)

$$ A = \left\{ {\left\langle {x,\left( {\mathop T\nolimits_{A} \left( x \right),\mathop I\nolimits_{A} \left( x \right),\mathop F\nolimits_{A} \left( x \right)} \right)} \right\rangle :x \in U,\mathop T\nolimits_{A} \left( x \right),\mathop I\nolimits_{A} \left( x \right),\mathop F\nolimits_{A} \left( x \right) \in \left] {\mathop 0\nolimits^{ - } ,\mathop 1\nolimits^{ + } } \right[} \right\} $$
(A.1)

Definition A.2 [Single-valued neutrosophic set (SVNS) (Wang et al. 2010)]

Let \( U \) be a universal discourse, and let \( x \) signify a generic element in \( U \). The SVNS \( A \) in \( U \) is characterized by a truth-membership function \( \mathop T\nolimits_{A} \left( x \right) \), an indeterminacy-membership function \( \mathop I\nolimits_{A} \left( x \right) \) and a falsity-membership function \( \mathop F\nolimits_{A} \left( x \right) \). The \( \mathop T\nolimits_{A} \left( x \right) \), \( \mathop I\nolimits_{A} \left( x \right) \) and \( \mathop F\nolimits_{A} \left( x \right) \) are real numbers of \( \left[ {0,1} \right] \). The SVNS can be represented as Eq. (A.2). Note that \( 0 \le \mathop T\nolimits_{A} \left( x \right) + \mathop I\nolimits_{A} \left( x \right) + \mathop F\nolimits_{A} \left( x \right) \le 3 \)

$$ A = \left\{ {\left\langle {x,\left( {\mathop T\nolimits_{A} \left( x \right),\mathop I\nolimits_{A} \left( x \right),\mathop F\nolimits_{A} \left( x \right)} \right)} \right\rangle :x \in U,\mathop T\nolimits_{A} \left( x \right),\mathop I\nolimits_{A} \left( x \right),\mathop F\nolimits_{A} \left( x \right) \in \left[ {0,1} \right]} \right\} $$
(A.2)

Definition A.3 [Single-valued trapezoidal neutrosophic number (SVTNN) (Deli and Subas 2014)]

A SVTNN \( \tilde{a} = \left\langle {\left( {\mathop a\nolimits_{1} ,\mathop b\nolimits_{1} ,\mathop c\nolimits_{1} ,\mathop d\nolimits_{1} } \right);\mathop w\nolimits_{{\tilde{a}}} ,\mathop u\nolimits_{{\tilde{a}}} ,\mathop y\nolimits_{{\tilde{a}}} } \right\rangle \) is a particular SVNN where \( w_{{\tilde{a}}} ,u_{{_{{\tilde{a}}} }} ,y_{{\tilde{a}}} \in \left[ {0,1} \right] \) and \( a_{1} ,b_{1} ,c_{1} ,d_{1} \in {\mathbb{R}} \) then \( \mathop T\nolimits_{{\tilde{a}}} \left( x \right) \), \( \mathop I\nolimits_{{\tilde{a}}} \left( x \right) \) and \( \mathop F\nolimits_{{\tilde{a}}} \left( x \right) \) are presented as Eqs. (A.3) to (A.5) respectively.

$$ \mathop T\nolimits_{{\tilde{a}}} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\left( {x - \mathop a\nolimits_{1} } \right)\mathop w\nolimits_{{\tilde{a}}} }}{{\left( {\mathop b\nolimits_{1} - \mathop a\nolimits_{1} } \right)}}} \hfill & {\mathop a\nolimits_{1} \le x < \mathop b\nolimits_{1} } \hfill \\ {w_{{\tilde{a}}} } \hfill & {\mathop b\nolimits_{1} \le x \le \mathop c\nolimits_{1} } \hfill \\ {\frac{{\left( {\mathop d\nolimits_{1} - x} \right)\mathop w\nolimits_{{\tilde{a}}} }}{{(\mathop d\nolimits_{1} - \mathop c\nolimits_{1} )}}} \hfill & {\mathop c\nolimits_{1} < x \le \mathop d\nolimits_{1} } \hfill \\ 0 \hfill & {otherwise} \hfill \\ \end{array} } \right. $$
(A.3)
$$ \mathop I\nolimits_{{\tilde{a}}} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\left( {\mathop b\nolimits_{1} - x + \mathop u\nolimits_{{\tilde{a}}} \left( {x - \mathop a\nolimits_{1} } \right)} \right)}}{{\left( {\mathop b\nolimits_{1} - \mathop a\nolimits_{1} } \right)}}} \hfill & {\mathop a\nolimits_{1} \le x < \mathop b\nolimits_{1} } \hfill \\ {\mathop u\nolimits_{{\tilde{a}}} } \hfill & {\mathop b\nolimits_{1} \le x \le \mathop c\nolimits_{1} } \hfill \\ {\frac{{\left( {x - \mathop c\nolimits_{1} + \mathop u\nolimits_{{\tilde{a}}} \left( {\mathop d\nolimits_{1} - x} \right)} \right)}}{{\left( {\mathop d\nolimits_{1} - \mathop c\nolimits_{1} } \right)}}} \hfill & {\mathop c\nolimits_{1} < x \le \mathop d\nolimits_{1} } \hfill \\ 1 \hfill & {otherwise} \hfill \\ \end{array} } \right. $$
(A.4)
$$ \mathop F\nolimits_{{\tilde{a}}} \left( x \right) = \left\{ {\begin{array}{*{20}l} {\frac{{\left( {\mathop b\nolimits_{1} - x + \mathop y\nolimits_{{\tilde{a}}} \left( {x - \mathop a\nolimits_{1} } \right)} \right)}}{{\left( {\mathop b\nolimits_{1} - \mathop a\nolimits_{1} } \right)}}} \hfill & {\mathop a\nolimits_{1} \le x < \mathop b\nolimits_{1} } \hfill \\ {\mathop y\nolimits_{{\tilde{a}}} } \hfill & {\mathop b\nolimits_{1} \le x \le \mathop c\nolimits_{1} } \hfill \\ {\frac{{\left( {x - \mathop c\nolimits_{1} + \mathop y\nolimits_{{\tilde{a}}} \left( {\mathop d\nolimits_{1} - x} \right)} \right)}}{{\left( {\mathop d\nolimits_{1} - \mathop c\nolimits_{1} } \right)}}} \hfill & {\mathop c\nolimits_{1} < x \le \mathop d\nolimits_{1} } \hfill \\ 1 \hfill & {otherwise} \hfill \\ \end{array} } \right. $$
(A.5)

Definition A.4 [Score function of a SVTNN (Wang and Zhong 2009; Ye 2017)]

Let \( \tilde{a} = \left\langle {\left( {a,b,c,d} \right);\mathop w\nolimits_{{\tilde{a}}} ,\mathop u\nolimits_{{\tilde{a}}} ,\mathop y\nolimits_{{\tilde{a}}} } \right\rangle \) be a SVTNN. Then the score function of \( \tilde{a} \) (i.e., \( S\left( {\tilde{a}} \right) \in \left[ {0,1} \right] \)) can be calculated according to Eq. (A.6).

$$ S\left( {\tilde{a}} \right) = \frac{1}{12}\left( {a + b + c + d} \right)\left( {2 + \mathop w\nolimits_{{\tilde{a}}} - \mathop u\nolimits_{{\tilde{a}}} - \mathop y\nolimits_{{\tilde{a}}} } \right). $$
(A.6)

Appendix B: Acquiring DMs’ confidence in the best-to-others and the others-to-worst preferences

Q1. Reflecting on your chosen best criterion and your provided preferences, to what degree do you have confidence in your provided best-to-others preferences? Please choose one of the following choices:

□ No Confidence

□ Low Confidence

□ Fairly Low confidence

□ Moderate confidence

□ Fairly high confidence

□ High confidence

□ Absolute confidence

Q2. Reflecting on your chosen worst criterion and your provided preferences, to what degree do you have confidence in your provided others-to-worst preferences? Please choose one of the following choices:

□ No confidence

□ Low confidence

□ Fairly low confidence

□ Moderate confidence

□ Fairly high confidence

□ High confidence

□ Absolute confidence

Appendix C: The CI values

In this appendix, \( CI \) values corresponding to various \( \mathop a\nolimits_{BW} \), \( \mathop \rho \nolimits^{ + } \) and \( \rho^{ - } \) values have been shown. Note that by swapping values for \( \mathop \rho \nolimits^{ + } \) and \( \rho^{ - } \) the \( CI \) values will not change.Footnote 16 Thus, for convenience, we have shown those \( \rho^{ - } \) and \( \mathop \rho \nolimits^{ + } \) values that produce unique \( CI \) values. The \( CI \) values for \( \mathop a\nolimits_{BW} = 1 \) are not shown because the best and worst criteria cannot be equally important.

\( \mathop \rho \nolimits^{ + } \)

\( \rho^{ - } \)

\( \mathop a\nolimits_{BW} = 2 \)

\( \mathop a\nolimits_{BW} = 3 \)

\( \mathop a\nolimits_{BW} = 4 \)

\( \mathop a\nolimits_{BW} = 5 \)

\( \mathop a\nolimits_{BW} = 6 \)

\( \mathop a\nolimits_{BW} = 7 \)

\( \mathop a\nolimits_{BW} = 8 \)

\( \mathop a\nolimits_{BW} = 9 \)

0.26

0.26

0.092

0.218

0.363

0.520

0.687

0.860

1.040

1.224

0.26

0.38

0.109

0.257

0.428

0.612

0.807

1.010

1.218

1.432

0.26

0.50

0.120

0.283

0.468

0.668

0.878

1.095

1.318

1.546

0.26

0.68

0.132

0.307

0.506

0.718

0.941

1.169

1.403

1.641

0.26

0.90

0.140

0.325

0.533

0.754

0.984

1.219

1.459

1.702

0.26

1.00

0.143

0.331

0.542

0.765

0.997

1.235

1.476

1.721

0.38

0.38

0.135

0.318

0.530

0.760

1.004

1.258

1.520

1.789

0.38

0.50

0.153

0.361

0.600

0.860

1.134

1.420

1.715

2.017

0.38

0.68

0.172

0.404

0.670

0.956

1.258

1.571

1.892

2.220

0.38

0.90

0.187

0.438

0.723

1.028

1.348

1.678

2.015

2.358

0.38

1.00

0.193

0.450

0.740

1.051

1.376

1.711

2.053

2.400

0.50

0.50

0.177

0.419

0.697

1.000

1.321

1.655

2.000

2.354

0.50

0.68

0.204

0.481

0.800

1.146

1.511

1.892

2.284

2.686

0.50

0.90

0.227

0.533

0.883

1.261

1.658

2.070

2.493

2.926

0.50

1.00

0.234

0.551

0.911

1.298

1.706

2.127

2.559

3.000

0.68

0.68

0.241

0.570

0.948

1.360

1.796

2.251

2.720

3.202

0.68

0.90

0.274

0.647

1.076

1.542

2.034

2.547

3.075

3.617

0.68

1.00

0.286

0.675

1.121

1.605

2.115

2.646

3.193

3.752

0.90

0.90

0.319

0.754

1.255

1.800

2.377

2.979

3.600

4.238

0.90

1.00

0.336

0.793

1.320

1.893

2.500

3.132

3.785

4.455

1.00

1.00

0.354

0.838

1.394

2.000

2.641

3.310

4.000

4.708

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Vafadarnikjoo, A., Tavana, M., Botelho, T. et al. A neutrosophic enhanced best–worst method for considering decision-makers’ confidence in the best and worst criteria. Ann Oper Res 289, 391–418 (2020). https://doi.org/10.1007/s10479-020-03603-x

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