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Neutrosophic Fuzzy Decision-Making Using TOPSIS and Autocratic Methodology for Machine Selection in an Industrial Factory

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Abstract

Multi-attribute decision-making (MADM) is a crucial component of modern decision theory, with extensive application in several practical problems. In this particular setting, the pervasive problem of uncertainty has prompted the development of many theoretical frameworks, most notably the fuzzy set (FS) and its subsequent expansions, to address imprecise and ambiguous information. Among the many frameworks, the neutrosophic set (NS) has become a significant paradigm due to its ability to clearly define truth, falsehood, and indeterminacy. This unique characteristic allows for a more nuanced approach to dealing with ambiguity within decision-making scenarios. However, both the FS and the NS demonstrate constraints when faced with ambiguous and inconsistent circumstances. To address these constraints, this research explores the conceptual domain of neutrosophic fuzzy set (NFS), which represents a fusion of fuzzy FS and NS theories. This study presents a novel scoring mechanism that enables a comprehensive evaluation of NFSs and a customized distance measure designed to accurately identify inconsistencies within the dataset in both forward and backward directions. Following that, a decision-making framework is presented, derived from the extensions of the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) and Autocratic approaches for group MADM using NFSs. The amalgamation of these techniques is further augmented using frequency as a criterion to get a definitive ranking for alternatives. Simultaneously, using the distinctive benefits of the TOPSIS and the Autocratic technique enables a thorough assessment, enhancing the strength and knowledge of decision-making in situations that include group decision-makers and NFS. The suggested strategy is applied to deal with industrial machine selection problems, where the decision values of the attributes for alternatives and the weights of the attributes are given by decision-makers based on NF values. These values facilitate assessing determination levels of alternative characteristics and attribute weights. This study contributes significantly to the broader discussion on decision-making in situations involving uncertainty and its direct relevance to the selection of machines. One notable characteristic of these approaches to reduce computation complexity is integrating decision and weight matrices from various management viewpoints into a unified aggregated assessment matrix. Despite strategies aimed at addressing decision-making challenges involving group suggestions and the use of neutrosophic fuzzy numbers, the analytical results indicate that the proposed approaches demonstrate a desirable level of computational simplicity.

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Author information

Authors and Affiliations

  1. Department of Industrial Engineering and Management, National Taipei University of Technology, Taipei, 10608, Taiwan

    Amirhossein Nafei, Chien-Yi Huang & Shu-Chuan Chen

  2. Department of Computer Science and Technology (Cyberspace Security), Harbin Institute of Technology, Shenzhen, China

    Amir Javadpour

  3. ADiT-Lab, Electrotechnics and Telecommunications Department, Instituto Politécnico de Viana do Castelo, Porto, 4900-347, Portugal

    Amir Javadpour

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

    Harish Garg

  5. Department of Electrical Engineering, National Taiwan Ocean University, Keelung, 202, Taiwan

    S. Pourmohammad Azizi

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Appendix 1

Appendix 1

Proof of Theorem 1

Based on Definition 1, we have:

$$S({M}_{1})=\frac{\left((2-{B}_{1})\left({A}_{1}-\frac{{B}_{1}}{2}-{C}_{1}+3\right)(2-{C}_{1})\right)+4{\mu }_{1}}{4},$$

and

$$S\left({M}_{2}\right)=\frac{\left((2-{B}_{2})\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)(2-{C}_{2})\right)+4{\mu }_{2}}{4}.$$

Therefore,

$$\begin{aligned} S\left({M}_{2}\right)-S\left({M}_{1}\right)&=\frac{\left(\left(2-{B}_{2}\right)\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)\left(2-{C}_{2}\right)\right)+4{\mu }_{2}}{4} \\ & \quad -\frac{\left(\left(2-{B}_{1}\right)\left({A}_{1}-\frac{{B}_{1}}{2}-{C}_{1}+3\right)\left(2-{C}_{1}\right)\right)+4{\mu }_{1}}{4} \\ &=\frac{\left(\left(2-{B}_{2}-{B}_{1}+{B}_{1}\right)\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)\left(2-{C}_{2}+{C}_{1}-{C}_{1}\right)\right)+4{\mu }_{2}}{4} \\ & \quad -\frac{\left(\left(2-{B}_{1}\right)\left({A}_{1}-\frac{{B}_{1}}{2}-{C}_{1}+3\right)\left(2-{C}_{1}\right)\right)+4{\mu }_{1}}{4} =\frac{\left(\left(\left(2-{B}_{1}\right)+\left({B}_{1}-{B}_{2}\right)\right)\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)\left(\left(2-{C}_{1}\right)+\left({C}_{1}-{C}_{2}\right)\right)\right)+4{\mu }_{2}}{4} \\ & \quad -\frac{\left(\left(2-{B}_{1}\right)\left({A}_{1}-\frac{{B}_{1}}{2}-{C}_{1}+3\right)\left(2-{C}_{1}\right)\right)+4{\mu }_{1}}{4}\\ &=\frac{\left(\left({A}_{2}-{A}_{1}\right)+\frac{{B}_{1}-{B}_{2}}{2}+\left({C}_{1}-{C}_{2}\right)\right)\left(2-{C}_{1}\right)\left(2-{B}_{1}\right)}{4}\\ & \quad +\frac{\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)(2-{C}_{1})(2-{B}_{1})}{4} \\ & \quad +\frac{\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)(({B}_{1}-{B}_{2})(2-{C}_{1}))}{4}+\frac{\left({A}_{2}-\frac{{B}_{2}}{2}-{C}_{2}+3\right)(({B}_{1}-{B}_{2})({C}_{1}-{C}_{2}))}{4}+({\mu }_{2}-{\mu }_{1}).\end{aligned}$$

Since \({M}_{1}\subseteq {M}_{2}\) then, it follows that \({\mu }_{1}\le {\mu }_{2},{A}_{1}\le {A}_{2},{B}_{2}\le {B}_{1},\;text{and}\;{C}_{2}\le {C}_{1}.\) Based on this information, we can easily verify that all the values in the existing brackets of the above equation are greater or equal to zero. Therefore, \(S({M}_{2})-S({M}_{1})\ge 0,\) and subsequently \(S({M}_{2})\ge S({M}_{1}).\square \)

Proof of Proposition 2

The properties I, II, and III are straightforward. To prove property IV, assume that \(M=\{{x}_{i},{\mu _{\rm M}}({x}_{i}),{A_{\rm M}}({x}_{i},\mu ),{B_{\rm M}}({x}_{i},\mu ),{C_{\rm M}}({x}_{i},\mu )\}\). Since \(K\subseteq L\subseteq M,\)

$${\mu }_{K}\left({x}_{i}\right)\le {\mu }_{L}\left({x}_{i}\right)\le {\mu _{\rm M}}\left({x}_{i}\right),$$
$${A}_{K}\left({x}_{i},\mu \right)\le {A}_{L}\left({x}_{i},\mu \right)\le {A_{\rm M}}\left({x}_{i},\mu \right),$$
$${B}_{K}({x}_{i},\mu )\ge {B}_{L}({x}_{i},\mu )\ge {B_{\rm M}}({x}_{i},\mu ),$$

And,\({C}_{K}({x}_{i},\mu )\ge {C}_{L}({x}_{i},\mu )\ge {C_{\rm M}}({x}_{i},\mu ).\) Therefore,

$$\begin{aligned}\left|{\mu }_{K}\left({x}_{i}\right)-{\mu }_{L}\left({x}_{i}\right)\right| & \le \left|{\mu }_{K}\left({x}_{i}\right)-{\mu _{\rm M}}\left({x}_{i}\right)\right|, \\ \left|{\mu }_{L}\left({x}_{i}\right)-{\mu _{\rm M}}\left({x}_{i}\right)\right| & \le \left|{\mu }_{K}\left({x}_{i}\right)-{\mu _{\rm M}}\left({x}_{i}\right)\right|, \\ \left|{A}_{K}\left({x}_{i},\mu \right)-{A}_{L}\left({x}_{i},\mu \right)\right| & \le \left|{A}_{K}\left({x}_{i},\mu \right)-{A_{\rm M}}\left({x}_{i},\mu \right)\right|, \\ \left|{A}_{L}\left({x}_{i},\mu \right)-{A_{\rm M}}\left({x}_{i},\mu \right)\right| & \le \left|{A}_{K}\left({x}_{i},\mu \right)-{A_{\rm M}}\left({x}_{i},\mu \right)\right|, \\ \left|{B}_{K}\left({x}_{i},\mu \right)-{B}_{L}\left({x}_{i},\mu \right)\right| & \le \left|{B}_{K}\left({x}_{i},\mu \right)-{B_{\rm M}}\left({x}_{i},\mu \right)\right|, \\ \left|{B}_{L}\left({x}_{i},\mu \right)-{B_{\rm M}}\left({x}_{i},\mu \right)\right| & \le \left|{B}_{K}\left({x}_{i},\mu \right)-{B_{\rm M}}\left({x}_{i},\mu \right)\right|,\end{aligned}$$

and

$$\left|{C}_{K}({x}_{i},\mu )-{C}_{L}({x}_{i},\mu )\right|\le \left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|,\left|{C}_{L}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\le \left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|.$$

So,

$$\left[\begin{array}{c}\frac{\left(\begin{array}{c}\left|{\mu }_{K}({x}_{i})-{\mu }_{L}({x}_{i})\right|+\left|{A}_{K}({x}_{i},\mu )-{A}_{L}({x}_{i},\mu )\right|+\\ \left|{B}_{K}({x}_{i},\mu )-{B}_{L}({x}_{i},\mu )\right|+\left|{C}_{K}({x}_{i},\mu )-{C}_{L}({x}_{i},\mu )\right|\end{array}\right)}{8}+\\ \frac{\text{max}\left(\begin{array}{c}\left|{A}_{K}({x}_{i},\mu )-{A}_{L}({x}_{i},\mu )\right|,\left|{B}_{K}({x}_{i},\mu )-{B}_{L}({x}_{i},\mu )\right|,\\ \left|{C}_{K}({x}_{i},\mu )-{C}_{L}({x}_{i},\mu )\right|\end{array}\right)}{3}\end{array}\right]\le \left[\begin{array}{c}\frac{\left(\begin{array}{c}\left|{\mu }_{K}({x}_{i})-{\mu _{\rm M}}({x}_{i})\right|+\left|{A}_{K}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|+\\ \left|{B}_{K}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|+\left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{8}+\\ \frac{\text{max}\left(\begin{array}{c}\left|{A}_{K}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|,\left|{B}_{K}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|,\\ \left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{3}\end{array}\right],$$

and

$$\left[\begin{array}{c}\frac{\left(\begin{array}{c}\left|{\mu }_{L}({x}_{i})-{\mu _{\rm M}}({x}_{i})\right|+\left|{A}_{L}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|+\\ \left|{B}_{L}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|+\left|{C}_{L}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{8}+\\ \frac{\text{max}\left(\begin{array}{c}\left|{A}_{L}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|,\left|{B}_{L}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|\\ ,\left|{C}_{L}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{3}\end{array}\right]\le \left[\begin{array}{c}\frac{\left(\begin{array}{c}\left|{\mu }_{K}({x}_{i})-{\mu _{\rm M}}({x}_{i})\right|+\left|{A}_{K}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|+\\ \left|{B}_{K}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|+\left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{8}+\\ \frac{\text{max}\left(\begin{array}{c}\left|{A}_{K}({x}_{i},\mu )-{A_{\rm M}}({x}_{i},\mu )\right|,\left|{B}_{K}({x}_{i},\mu )-{B_{\rm M}}({x}_{i},\mu )\right|,\\ \left|{C}_{K}({x}_{i},\mu )-{C_{\rm M}}({x}_{i},\mu )\right|\end{array}\right)}{3}\end{array}\right].$$

Hence,

$$d(K,M)\ge d(K,L),\;text{and}\;d(K,M)\ge d(L,M).\square $$

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Nafei, A., Huang, CY., Javadpour, A. et al. Neutrosophic Fuzzy Decision-Making Using TOPSIS and Autocratic Methodology for Machine Selection in an Industrial Factory. Int. J. Fuzzy Syst. 26, 860–886 (2024). https://doi.org/10.1007/s40815-023-01640-9

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