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  <doi_batch_id>aspg-21-672-1791472081</doi_batch_id>
  <timestamp>20261008150801</timestamp>
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  <journal>
   <journal_metadata language="en">
    <full_title>International Journal of Neutrosophic Science</full_title>
    <abbrev_title>IJNS</abbrev_title>
    <issn media_type="print">2692-6148</issn>
    <issn media_type="electronic">2690-6805</issn>
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   <journal_issue>
    <publication_date media_type="online">
     <year>2021</year>
    </publication_date>
    <journal_volume>
     <volume>14</volume>
    </journal_volume>
    <issue>1</issue>
   </journal_issue>
   <journal_article publication_type="full_text">
    <titles>
     <title>Linear and Non-Linear Decagonal Neutrosophic numbers: Alpha Cuts, Representation, and solution of large MCDM problems</title>
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      <given_name>Sara</given_name>
      <surname>Farooq</surname>
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        <institution_name>Department of Mathematics and Statistics, The University of Lahore,Pakistan</institution_name>
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      <given_name>Ali</given_name>
      <surname>Hamza</surname>
      <affiliations>
       <institution>
        <institution_name>Department of Mathematics and Statistics, The University of Lahore,Pakistan</institution_name>
       </institution>
      </affiliations>
     </person_name>
     <person_name sequence="additional" contributor_role="author">
      <given_name>Florentin</given_name>
      <surname>Smarandache</surname>
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        <institution_name>Dept. Math and Sciences, University of New Mexico, Gallup, NM, USA</institution_name>
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     <jats:p>The postulation of neutrosophic numbers has been analyzed from different angles in this paper. In this current era, our main purpose is to mention Decagonal Neutrosophic numbers. The types of linear and non-linear generalized decagonal neutrosophic numbers play a very critical role in the theory related to uncertainty This approach is helpful in getting a crisp number from a neutrosophic number. The definitions regarding Linear, Non-Linear, symmetry, Asymmetry, alpha cuts have been introduced and large decision-making problems using fuzzy TOPSIS have been solved.</jats:p>
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     <jats:p>Name=&quot;index 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;index 7&quot;/&gt;</jats:p>
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     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 2&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 3&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 4&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 5&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 6&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 7&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 8&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;toc 9&quot;/&gt;</jats:p>
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     <jats:p>Name=&quot;footnote text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;annotation text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;header&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;footer&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;index heading&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; QFormat=&quot;true&quot; Name=&quot;caption&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;table of figures&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;envelope address&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;envelope return&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;footnote reference&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;annotation reference&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;line number&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;page number&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;endnote reference&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;endnote text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;table of authorities&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;macro&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;toa heading&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Bullet&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Number&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Bullet 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Bullet 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Bullet 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Bullet 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Number 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Number 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Number 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Number 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Closing&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Signature&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;Default Paragraph Font&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text Indent&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Continue&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Continue 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Continue 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Continue 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Continue 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Message Header&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Salutation&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Date&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text First Indent&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text First Indent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Note Heading&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text Indent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Body Text Indent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Block Text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Hyperlink&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;FollowedHyperlink&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Document Map&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Plain Text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;E-mail Signature&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Top of Form&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Bottom of Form&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Normal (Web)&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Acronym&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Address&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Cite&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Code&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Definition&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Keyboard&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Preformatted&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Sample&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Typewriter&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;HTML Variable&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Normal Table&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;annotation subject&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;No List&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Outline List 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Outline List 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Outline List 3&quot;/&gt;</jats:p>
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     <jats:p>Name=&quot;Table Simple 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Simple 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Classic 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Classic 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Classic 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Classic 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Colorful 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Colorful 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Colorful 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Columns 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Columns 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Columns 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Columns 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Columns 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 7&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Grid 8&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 7&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table List 8&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table 3D effects 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table 3D effects 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table 3D effects 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Contemporary&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Elegant&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Professional&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Subtle 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Subtle 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Web 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Web 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Web 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Balloon Text&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Table Theme&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Paragraph&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Intense Quote&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Subtle Emphasis&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Intense Emphasis&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Subtle Reference&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Intense Reference&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; Name=&quot;Bibliography&quot;/&gt;</jats:p>
     <jats:p>UnhideWhenUsed=&quot;true&quot; QFormat=&quot;true&quot; Name=&quot;TOC Heading&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 1 Light Accent 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 6 Colorful Accent 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;Grid Table 7 Colorful Accent 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 1&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 2&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 3&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 4&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 5&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 1 Light Accent 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 6 Colorful Accent 6&quot;/&gt;</jats:p>
     <jats:p>Name=&quot;List Table 7 Colorful Accent 6&quot;/&gt;</jats:p>
     <jats:p>/* Style Definitions */</jats:p>
     <jats:p>table.MsoNormalTable</jats:p>
     <jats:p>{mso-style-name:&quot;Tableau Normal&quot;;</jats:p>
     <jats:p>mso-tstyle-rowband-size:0;</jats:p>
     <jats:p>mso-tstyle-colband-size:0;</jats:p>
     <jats:p>mso-style-noshow:yes;</jats:p>
     <jats:p>mso-style-priority:99;</jats:p>
     <jats:p>mso-style-parent:&quot;&quot;;</jats:p>
     <jats:p>mso-padding-alt:0cm 5.4pt 0cm 5.4pt;</jats:p>
     <jats:p>mso-para-margin-top:0cm;</jats:p>
     <jats:p>mso-para-margin-right:0cm;</jats:p>
     <jats:p>mso-para-margin-bottom:10.0pt;</jats:p>
     <jats:p>mso-para-margin-left:0cm;</jats:p>
     <jats:p>line-height:115%;</jats:p>
     <jats:p>mso-pagination:widow-orphan;</jats:p>
     <jats:p>font-size:11.0pt;</jats:p>
     <jats:p>font-family:&quot;Calibri&quot;,&quot;sans-serif&quot;;</jats:p>
     <jats:p>mso-ascii-font-family:Calibri;</jats:p>
     <jats:p>mso-ascii-theme-font:minor-latin;</jats:p>
     <jats:p>mso-hansi-font-family:Calibri;</jats:p>
     <jats:p>mso-hansi-theme-font:minor-latin;</jats:p>
     <jats:p>mso-ansi-language:EN-US;</jats:p>
     <jats:p>mso-fareast-language:EN-US;}</jats:p>
    </jats:abstract>
    <publication_date media_type="online">
     <year>2021</year>
    </publication_date>
    <pages>
     <first_page>24</first_page>
     <last_page>41</last_page>
    </pages>
    <publisher_item>
     <item_number item_number_type="article-number">672</item_number>
    </publisher_item>
    <ai:program name="AccessIndicators">
     <ai:license_ref applies_to="vor">https://creativecommons.org/licenses/by/4.0/</ai:license_ref>
    </ai:program>
    <doi_data>
     <doi>10.54216/IJNS.140102</doi>
     <resource>https://www.americaspg.com/journal/21/article/672</resource>
     <collection property="text-mining">
      <item>
       <resource mime_type="application/pdf">https://www.americaspg.com/storage/5133.pdf</resource>
      </item>
     </collection>
    </doi_data>
   </journal_article>
  </journal>
 </body>
</doi_batch>
