Manual Finite Neutrosophic Complex Numbers

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Further the new notion of finite complex modulo integers is defined. are constructed using these neutrosophic complex modulo integers.
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This book provides several open problems to the semigroup theorists. Account Options Sign in. Top Charts. New Arrivals. Dual Numbers. Vasantha Kandasamy, Florentin Smarandache. Vasantha Kandasamy. For the first time, the social problem of untouchability, which is peculiar to India, is being studied mathematically. Ramasamy This thought-provoking book will be of great interest to human rights activists, socio-scientists, historians, and above all, mathematicians.

Smarandache Notions, Vol. Book Most of the papers are in English, a few of them are in Spanish, Portuguese, or German. Book 4.


For the involvement of uncertainty of varying degrees, when the total of the membership degree exceeds one or less than one, then the newer mathematical paradigm shift, Fuzzy Theory proves appropriate. For the past two or three decades, Fuzzy Theory has become the potent tool to study and analyze uncertainty involved in all problems. But, many real world problems also abound with the concept of indeterminacy.

In this book, the new, powerful tool of neutrosophy that deals with indeterminacy is utilized. Innovative neutrosophic models are described. The theory of neutrosophic graphs is introduced and applied to fuzzy and neutrosophic models. Fuzzy Linguistic Topological Spaces. Innovative Uses of Matrices. The authors have used the concept of finite complex modulo integers to construct non associative algebraic structures like groupoids, loops and quasi-loops.

Using these structures we built non associative complex matrix groupoids and complex polynomial groupoids. Their distance measure is defined as in 6 :. This technique assesses alternatives and selects the best one based on the distance from the negative ideal solution NIS to the positive ideal solution PIS. The most suitable choice s should meet conditions of having the least distance to the PIS as well as the greatest distance to the NIS.

ISBN 13: 9781599731582

In practice, both benefit attributes and cost attributes may exist in MADM problems. Let B and S be the collections of benefit attributes and cost attributes, respectively. Then, we have the conversion rules in 8 :. The weighted SVNL decision matrix, Y , is formed as shown in 11 , using the operational laws given in Definition 2 above:. Step 4. Step 5. Step 6. Order the alternatives and identify the optimal one s in accordance with the decreasing values, C i , obtained from Step 5.

The ordered weighted averaging OWA operator developed by Yager [ 49 ] is important in most aggregation techniques and has been analyzed, applied and generalized by a number of researchers [ 38 , 50 , 51 , 52 , 53 , 54 , 55 ]. The most distinctive advantage of the OWA operator is its ordering mechanism for the argument under consideration, which enables the integration of the complex attitudinal character of the experts in the decision-making process. The attitudinal character formula is expressed in 16 :. Therefore, DMs have more choice in selecting cases according to their particular interests or the specific problem being considered.

For example,. This method includes the following steps:. Steps 1—3. These are the same as Steps 1—3 in the description in Section 3. However, some authors have pointed out that, in some cases, the relative closeness cannot reach the goal that the desired solution should simultaneously have the least distance from the PIS and the greatest distance from the NIS [ 20 , 52 ]. To present a complete picture of the aggregation results for a decision analysis, we can consider the different cases of the SVNLOWAD that are proposed in Section 4 to calculate distance measures in Step 3.

In recent years, global economic development and public health have been threatened by increased carbon emissions, which has prompted companies and governments around the world to attempt to stimulate and increase investments in low-carbon and green economics. In this decision-making process, a key stage is to choose a suitable green supplier in the low-carbon supply chain.

More importantly, the process involves various factors involving uncertain information, all of which have to be considered and evaluated simultaneously. Therefore, green supplier selection is a highly ambiguous and complex decision process. To date, several authors have investigated methods for low-carbon selection in such uncertain situations. Liu et al. Considering the available information, we employ the proposed approach to select the most suitable choice s. The main steps are summarized as follows:. Step 1. Step 2. Calculate the collective opinion and get the collective SVNL decision matrix, which is shown in Table 7 :.

Step 3. The result is shown in Table 8 :. Utilize Equation 19 to compute the revised relative coefficients:. This yields. Therefore, the most desirable alternative is A 4 , which coincides with the result obtained by the application of the method proposed by Ye [ 34 ]. The results are given in Table 10 and Table This model is very flexible, as it can provide more specific cases for DMs to choose from using the different families of the operators.

The main contributions of this paper can be summarized as follows: 1 Great progress has been achieved in single-valued neutrosophic linguistic distance measure methodology. The outstanding merit of the method is its ability to fuse the optimism or pessimism degrees of DMs. Moreover, it allows for thorough viewing of the decision process, as the decision-makers can choose different values of interest and thereby render many different scenarios. A revised relative coefficient has been proposed to rank the potential alternatives. Therefore, this method provides a more general and more accurate result in comparison with existing methods.

A comparison with existing methods illustrates that the proposed model can be expected to achieve more realistic and accurate results in single-valued neutrosophic linguistic situations.

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Thus, this paper offers a significant contribution in regard to the development of MADM frameworks for green supplier selection problems. Other empirical applications of this method will also be considered in future works.

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National Center for Biotechnology Information , U. Published online Jul 8. Find articles by Ji Chen. Find articles by Chonghui Zhang. Author information Article notes Copyright and License information Disclaimer.

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Received May 31; Accepted Jul 6. Introduction Due to growing awareness of the need to protect the environment and address global warming, low-carbon supply chains have become a leading research topic in previous decades [ 1 , 2 , 3 ]. Normalize individual decision matrices: In practice, both benefit attributes and cost attributes may exist in MADM problems.

Complex Numbers as Matrices

An Illustrative Example In recent years, global economic development and public health have been threatened by increased carbon emissions, which has prompted companies and governments around the world to attempt to stimulate and increase investments in low-carbon and green economics. Open in a separate window. Author Contributions J. Conflicts of Interest The authors declare no conflict of interest. References 1. Govindan K. Green supplier selection and order allocation in a low-carbon paper industry: Integrated multi-criteria heterogeneous decision-making and multi-objective linear programming approaches.

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Rao C. Low carbon supplier selection under multi-source and multi-attribute procurement. Fuzzy Syst. Krishnendu S. Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain. Expert Syst. Public Health. Wan S. Qin J. An extended TODIM multi-criteria group decision making method for green supplier selection in interval type-2 fuzzy environment.

Keshavarz G. Song W. Developing sustainable supplier selection criteria for solar air-conditioner manufacturer: An integrated approach. Energy Rev. Kou G. Enhancing data consistency in decision matrix: Adapting Hadamard model to mitigate judgment contradiction. Jain V. Universal supplier selection via multi-dimensional auction mechanisms for two-way competition in oligopoly market of supply chain. Davood M. A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs.

Pairwise comparison matrix in multiple criteria decision making.

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  7. Asadabadi M. A customer based supplier selection process that combines quality function deployment, the analytic network process and a Markov chain. Theisen S. Smarandache F. Neutrosophy, Neutrosophic Probability, Set, and Logic. Wang H. Single valued neutrosophic sets. Multispace Multistruct. Atanassov K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. Yager R. Pythagorean membership grades in multi-criteria decision making.

    IEEE Trans. Zhang X. Multicriteria decision-making method using the correlation coefficient under single-valued neutrosophic environment. Single valued neutrosophic cross entropy for multicriteria decision making problems. Biswas P. Entropy based grey relational analysis method for multi-attribute decision-making under single valued neutrosophic assessments.