Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1732706938"> <meta name="title" content="Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods"> <meta name="description" content="This study compares screw joints made of different materials, including screws of different diameters. For that purpose, 8, 10, 12, 14, 16 mm diameter steel screws and various parts made of aluminum (Al), steel (Stl), bronze (Brz), cast iron (CI), copper (Cu) and brass (Br) are considered. Multiple criteria decision making (MCDM) methods such as evaluation based on distance from average solution (EDAS), simple additive weighting (SAW), technique for order of preference by similarity to ideal solution (TOPSIS) and complex proportional assessment (COPRAS) are utilized to assess reliability of screw joints also considering cost issues. The entropy, criterion impact loss (CILOS) and integrated determination of objective criteria weights (IDOCRIW) methods are utilized to assess weights of decision criteria and find the best design alternative. Numerical results confirm the validity of the proposed approach." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/metals-logo.png?e13532d60707dc57" > <meta name="dc.title" content="Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods"> <meta name="dc.creator" content="Audrius Čereška"> <meta name="dc.creator" content="Askoldas Podviezko"> <meta name="dc.creator" content="Edmundas Kazimieras Zavadskas"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Metals 2018, Vol. 8, Page 318"> <meta name="dc.date" content="2018-05-04"> <meta name ="dc.identifier" content="10.3390/met8050318"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="This study compares screw joints made of different materials, including screws of different diameters. For that purpose, 8, 10, 12, 14, 16 mm diameter steel screws and various parts made of aluminum (Al), steel (Stl), bronze (Brz), cast iron (CI), copper (Cu) and brass (Br) are considered. Multiple criteria decision making (MCDM) methods such as evaluation based on distance from average solution (EDAS), simple additive weighting (SAW), technique for order of preference by similarity to ideal solution (TOPSIS) and complex proportional assessment (COPRAS) are utilized to assess reliability of screw joints also considering cost issues. The entropy, criterion impact loss (CILOS) and integrated determination of objective criteria weights (IDOCRIW) methods are utilized to assess weights of decision criteria and find the best design alternative. Numerical results confirm the validity of the proposed approach." > <meta name="dc.subject" content="civil engineering" > <meta name="dc.subject" content="bolt joint" > <meta name="dc.subject" content="parameters" > <meta name="dc.subject" content="MCDM" > <meta name="dc.subject" content="entropy" > <meta name="dc.subject" content="TOPSIS" > <meta name="dc.subject" content="EDAS" > <meta name="dc.subject" content="CILOS" > <meta name="dc.subject" content="IDOCRIW" > <meta name ="prism.issn" content="2075-4701"> <meta name ="prism.publicationName" content="Metals"> <meta name ="prism.publicationDate" content="2018-05-04"> <meta name ="prism.volume" content="8"> <meta name ="prism.number" content="5"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="318" > <meta name="citation_issn" content="2075-4701"> <meta name="citation_journal_title" content="Metals"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods"> <meta name="citation_publication_date" content="2018/5"> <meta name="citation_online_date" content="2018/05/04"> <meta name="citation_volume" content="8"> <meta name="citation_issue" content="5"> <meta name="citation_firstpage" content="318"> <meta name="citation_author" content="Čereška, Audrius"> <meta name="citation_author" content="Podviezko, Askoldas"> <meta name="citation_author" content="Zavadskas, Edmundas Kazimieras"> <meta name="citation_doi" content="10.3390/met8050318"> <meta name="citation_id" content="mdpi-met8050318"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2075-4701/8/5/318"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2075-4701/8/5/318/pdf?version=1525432787"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2075-4701/8/5/318/pdf?version=1525432787"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2075-4701/8/5/318/pdf?version=1525432787"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2075-4701/8/5/318/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2075-4701/8/5/318/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2075-4701/8/5/318/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2075-4701/8/5/318/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2075-4701/8/5/318/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2075-4701/8/5/318/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/metals-logo-social.png?e13532d60707dc57" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2075-4701/8/5/318" /> <meta property="og:title" content="Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods" /> <meta property="og:description" content="This study compares screw joints made of different materials, including screws of different diameters. For that purpose, 8, 10, 12, 14, 16 mm diameter steel screws and various parts made of aluminum (Al), steel (Stl), bronze (Brz), cast iron (CI), copper (Cu) and brass (Br) are considered. Multiple criteria decision making (MCDM) methods such as evaluation based on distance from average solution (EDAS), simple additive weighting (SAW), technique for order of preference by similarity to ideal solution (TOPSIS) and complex proportional assessment (COPRAS) are utilized to assess reliability of screw joints also considering cost issues. The entropy, criterion impact loss (CILOS) and integrated determination of objective criteria weights (IDOCRIW) methods are utilized to assess weights of decision criteria and find the best design alternative. Numerical results confirm the validity of the proposed approach." /> <meta property="og:image" content="https://pub.mdpi-res.com/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001-550.jpg?1570076406" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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<div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Department of Mechanical and Material Engineering, Vilnius Gediminas Technical University, Basanavičiaus str. 28, LT-03324 Vilnius, Lithuania</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Economics Engineering, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>3</sup></div> <div class="affiliation-name ">Research Institute of Sustainable Construction, Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania</div> </div> <div class="affiliation"> <div class="affiliation-item"><sup>*</sup></div> <div class="affiliation-name ">Author to whom correspondence should be addressed. </div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Metals</em> <b>2018</b>, <em>8</em>(5), 318; <a href="https://doi.org/10.3390/met8050318">https://doi.org/10.3390/met8050318</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 13 March 2018</span> / <span style="display: inline-block">Revised: 30 April 2018</span> / <span style="display: inline-block">Accepted: 30 April 2018</span> / <span style="display: inline-block">Published: 4 May 2018</span> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-144647" aria-controls="drop-supplementary-144647" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-144647" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content 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<a href="https://pub.mdpi-res.com/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001.png?1570076406" title=" <strong>Figure 1</strong><br/> <p>Screw joint: 1—screw, 2, 3—joint element; 4—screw-nut, <span class="html-italic">h</span>—thickness of joint elements, <span class="html-italic">d</span>—screw diameter, <span class="html-italic">P</span>—clamping force, <span class="html-italic">T</span>—torque.</p> "> </a> </div> <a class="button button--color-inversed" href="/2075-4701/8/5/318/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">This study compares screw joints made of different materials, including screws of different diameters. For that purpose, 8, 10, 12, 14, 16 mm diameter steel screws and various parts made of aluminum (Al), steel (Stl), bronze (Brz), cast iron (CI), copper (Cu) and brass (Br) are considered. Multiple criteria decision making (MCDM) methods such as evaluation based on distance from average solution (EDAS), simple additive weighting (SAW), technique for order of preference by similarity to ideal solution (TOPSIS) and complex proportional assessment (COPRAS) are utilized to assess reliability of screw joints also considering cost issues. The entropy, criterion impact loss (CILOS) and integrated determination of objective criteria weights (IDOCRIW) methods are utilized to assess weights of decision criteria and find the best design alternative. Numerical results confirm the validity of the proposed approach.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=civil+engineering">civil engineering</a>; <a href="/search?q=bolt+joint">bolt joint</a>; <a href="/search?q=parameters">parameters</a>; <a href="/search?q=MCDM">MCDM</a>; <a href="/search?q=entropy">entropy</a>; <a href="/search?q=TOPSIS">TOPSIS</a>; <a href="/search?q=EDAS">EDAS</a>; <a href="/search?q=CILOS">CILOS</a>; <a href="/search?q=IDOCRIW">IDOCRIW</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-metals-08-00318' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>Screw joints are widely used [<a href="#B1-metals-08-00318" class="html-bibr">1</a>,<a href="#B2-metals-08-00318" class="html-bibr">2</a>,<a href="#B3-metals-08-00318" class="html-bibr">3</a>,<a href="#B4-metals-08-00318" class="html-bibr">4</a>] in cars, trucks, marine engines, compressors and generators, railway track construction, aviation, industrial fans, refrigeration equipment, machine tools, presses, instrumentation, conveyors, all types of mining machines, etc. [<a href="#B5-metals-08-00318" class="html-bibr">5</a>,<a href="#B6-metals-08-00318" class="html-bibr">6</a>,<a href="#B7-metals-08-00318" class="html-bibr">7</a>,<a href="#B8-metals-08-00318" class="html-bibr">8</a>,<a href="#B9-metals-08-00318" class="html-bibr">9</a>,<a href="#B10-metals-08-00318" class="html-bibr">10</a>]. The most important concerns for screw joints performance are (i) relaxation caused by vibration; and (ii) relaxation caused by dynamic loading, fatigue or corrosion [<a href="#B5-metals-08-00318" class="html-bibr">5</a>]. The mechanical behavior of screw joints subject to transverse loading was investigated in [<a href="#B6-metals-08-00318" class="html-bibr">6</a>,<a href="#B7-metals-08-00318" class="html-bibr">7</a>,<a href="#B8-metals-08-00318" class="html-bibr">8</a>]. Bhattacharya [<a href="#B9-metals-08-00318" class="html-bibr">9</a>] showed that transverse or shear loads are the main cause of vibration-induced loosening. In [<a href="#B10-metals-08-00318" class="html-bibr">10</a>], it is shown that, at high loads, there is an appropriate ratio between screw length and diameter to prevent sliding between threads. The theoretical model of screw joint behavior of [<a href="#B11-metals-08-00318" class="html-bibr">11</a>,<a href="#B12-metals-08-00318" class="html-bibr">12</a>] introduced a few more parameters besides reciprocating and rotational degrees of freedom and considered technical characteristics of connecting elements. The effect of contact pressure distribution was analyzed in [<a href="#B13-metals-08-00318" class="html-bibr">13</a>]. In [<a href="#B14-metals-08-00318" class="html-bibr">14</a>], it was found that a screw joint is relaxed at smaller loads than expected due to the sliding of contact area. Zapico-Valle [<a href="#B15-metals-08-00318" class="html-bibr">15</a>] presented an experimental validation of a new statistical process control feature for damage detection. In [<a href="#B16-metals-08-00318" class="html-bibr">16</a>,<a href="#B17-metals-08-00318" class="html-bibr">17</a>], screw joints excited by vibrations of different frequencies were analyzed. Laser technologies were used in [<a href="#B18-metals-08-00318" class="html-bibr">18</a>,<a href="#B19-metals-08-00318" class="html-bibr">19</a>,<a href="#B20-metals-08-00318" class="html-bibr">20</a>]. Izumi and Yokoyama et al. [<a href="#B21-metals-08-00318" class="html-bibr">21</a>,<a href="#B22-metals-08-00318" class="html-bibr">22</a>] analyzed relaxation of screw joints with spring washers using the finite element method. In [<a href="#B23-metals-08-00318" class="html-bibr">23</a>,<a href="#B24-metals-08-00318" class="html-bibr">24</a>,<a href="#B25-metals-08-00318" class="html-bibr">25</a>,<a href="#B26-metals-08-00318" class="html-bibr">26</a>,<a href="#B27-metals-08-00318" class="html-bibr">27</a>,<a href="#B28-metals-08-00318" class="html-bibr">28</a>,<a href="#B29-metals-08-00318" class="html-bibr">29</a>,<a href="#B30-metals-08-00318" class="html-bibr">30</a>,<a href="#B31-metals-08-00318" class="html-bibr">31</a>,<a href="#B32-metals-08-00318" class="html-bibr">32</a>], loosening of screws of different types, geometry, and materials, was investigated by considering various clamping forces and temperatures: relaxation rate was related to clamping force. Friction between joint parts were investigated in [<a href="#B31-metals-08-00318" class="html-bibr">31</a>,<a href="#B32-metals-08-00318" class="html-bibr">32</a>] and connection reliability was related to friction. In [<a href="#B33-metals-08-00318" class="html-bibr">33</a>,<a href="#B34-metals-08-00318" class="html-bibr">34</a>], continuous wavelet analysis of mode shapes was utilized for damage detection.</div><div class='html-p'>Theoretical analyses and experimental tests serve to assess reliability of screw joints, clarifying relationships between the cost of the junction and the technical parameters of the joint. For this reason, multiple criteria decision-making (MCDM) analysis was carried out in [<a href="#B35-metals-08-00318" class="html-bibr">35</a>,<a href="#B36-metals-08-00318" class="html-bibr">36</a>,<a href="#B37-metals-08-00318" class="html-bibr">37</a>,<a href="#B38-metals-08-00318" class="html-bibr">38</a>,<a href="#B39-metals-08-00318" class="html-bibr">39</a>,<a href="#B40-metals-08-00318" class="html-bibr">40</a>]. Various MCDM methods were successfully utilized in engineering problems. For example, the analytical hierarchy process (AHP) was used for steel bridges [<a href="#B41-metals-08-00318" class="html-bibr">41</a>,<a href="#B42-metals-08-00318" class="html-bibr">42</a>], energy absorption parts [<a href="#B43-metals-08-00318" class="html-bibr">43</a>] and cold-formed thin-walled steel structures [<a href="#B44-metals-08-00318" class="html-bibr">44</a>]. The TOPSIS (technique for order of preference by similarity to ideal solution) method was applied to steel constructions [<a href="#B45-metals-08-00318" class="html-bibr">45</a>].</div><div class='html-p'>Welding processes were evaluated with fuzzy logic [<a href="#B46-metals-08-00318" class="html-bibr">46</a>]. Laser cutting technological parameters were selected with AHP-TOPSIS [<a href="#B47-metals-08-00318" class="html-bibr">47</a>]. Thin-walled structures were evaluated with the COPRAS (complex proportional assessment) method [<a href="#B48-metals-08-00318" class="html-bibr">48</a>,<a href="#B49-metals-08-00318" class="html-bibr">49</a>,<a href="#B50-metals-08-00318" class="html-bibr">50</a>,<a href="#B51-metals-08-00318" class="html-bibr">51</a>,<a href="#B52-metals-08-00318" class="html-bibr">52</a>,<a href="#B53-metals-08-00318" class="html-bibr">53</a>,<a href="#B54-metals-08-00318" class="html-bibr">54</a>]. A hybrid method, AHP-COPRAS-G, was used for designing cold-formed steel structures [<a href="#B55-metals-08-00318" class="html-bibr">55</a>]. MCDM methods helped to select materials for a cutting tool [<a href="#B56-metals-08-00318" class="html-bibr">56</a>], structural materials [<a href="#B57-metals-08-00318" class="html-bibr">57</a>,<a href="#B58-metals-08-00318" class="html-bibr">58</a>,<a href="#B59-metals-08-00318" class="html-bibr">59</a>,<a href="#B60-metals-08-00318" class="html-bibr">60</a>], machine tools for changing production flow [<a href="#B61-metals-08-00318" class="html-bibr">61</a>].</div><div class='html-p'>In this study, classical MCDM methods such as EDAS (evaluation based on distance from average solution), SAW (simple additive weighting), TOPSIS and COPRAS will be used for evaluating performance of screw joints with respect to technical and economic parameters. Furthermore, recently developed methods for rating the importance of decision making criteria, such as CILOS (criterion impact loss) and IDOCRIW (integrated determination of objective criteria weights), will be used in the general context of a MCDM framework.</div></section><section id='sec2-metals-08-00318' type=''><h2 data-nested='1'> 2. Subject of the Study</h2><div class='html-p'>This study focusses on screw joints like that shown in <a href="#metals-08-00318-f001" class="html-fig">Figure 1</a> as they are the most popular elements used in construction. We investigate behavior of a single joint, leaving investigation of more complex structures for future studies. The single-joint case is relevant, however, because a rather common approach in structural design and verification is to consider the most loaded screw of the connection.</div><div class='html-p'>In this study, we attempted to evaluate widely used screws M8, M10, M12, M14, M16 in civil and mechanical engineering for connecting flat elements made of different materials. The connecting details are made of Stl, CI, Al, Brz, Br and Cu. Parameters of the compounds used for calculations are presented in <a href="#metals-08-00318-t001" class="html-table">Table 1</a>.</div></section><section id='sec3-metals-08-00318' type=''><h2 data-nested='1'> 3. Theoretical Analysis of a Screw Joint</h2><div class='html-p'>Different requirements for material selections and construction must be satisfied in screw joint technology. If the joint can be disassembled during operation, joint parts are usually connected by screws. The definition of a joint’s strength depends on the operating loads. Slipping between connecting parts is not possible if external load is perpendicular to the bolt’s axis. The maximum clamping force that can be transmitted by a screw joint is [<a href="#B62-metals-08-00318" class="html-bibr">62</a>] <div class='html-disp-formula-info' id='FD1-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>0.6</mn> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where 0.6 is the average value of the knockdown factor; <span class='html-italic'>P</span> is the clamping force (expressed in Newtons); <span class='html-italic'>f</span><sub>1</sub> is the coefficient of friction between the nut and the component; <span class='html-italic'>f</span><sub>2</sub> is the coefficient of friction between the joint components.</div><div class='html-p'>Another requirement for reliability of the threaded connection is to ensure a good contact between joint components. The possibility that joint dismantles depends on the flexibility of screws and joint components. The separation force is [<a href="#B62-metals-08-00318" class="html-bibr">62</a>] <div class='html-disp-formula-info' id='FD2-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mrow> <mi>at</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mi>P</mi> <mrow> <mi mathvariant="sans-serif">ν</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="sans-serif">χ</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">χ</mi> <mo>=</mo> <mfrac> <mrow> <msub> <mi mathvariant="sans-serif">λ</mi> <mi mathvariant="normal">d</mi> </msub> </mrow> <mrow> <msub> <mi mathvariant="sans-serif">λ</mi> <mi mathvariant="normal">d</mi> </msub> <mo>+</mo> <msub> <mi mathvariant="sans-serif">λ</mi> <mi mathvariant="sans-serif">ν</mi> </msub> </mrow> </mfrac> </mrow> </semantics></math>; λ<sub>d</sub> is flexibility of joint components, λ<sub>ν</sub> is flexibility of screw stretching.</div><div class='html-p'>If the thickness of joint components is close to the screw’s diameter, it holds, <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="sans-serif">λ</mi> <mi mathvariant="normal">d</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>0.82</mn> </mrow> <mrow> <msub> <mi>E</mi> <mi mathvariant="normal">d</mi> </msub> </mrow> </mfrac> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi mathvariant="sans-serif">λ</mi> <mi mathvariant="sans-serif">ν</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mn>1.27</mn> </mrow> <mrow> <msub> <mi>E</mi> <mi mathvariant="normal">p</mi> </msub> </mrow> </mfrac> </mrow> </semantics></math>, where <span class='html-italic'>E</span><sub>d</sub> and <span class='html-italic'>E</span><sub>p</sub>, respectively, are the values of elastic modulus of joint parts and screw (usually, steel for the latter).</div><div class='html-p'>When the external load is oriented perpendicular to the bolt’s axis, the rationale of the screw joint is the following: the equality between the bolt’s resistance to cutting and resistance of the joint components to crumpling. The details are crumpled when the transverse force exceeds the permissible shear force (1). Consequently, the minimum thickness of the joint components is [<a href="#B62-metals-08-00318" class="html-bibr">62</a>] <div class='html-disp-formula-info' id='FD3-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>h</mi> <mrow> <mi>min</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>F</mi> <mi mathvariant="normal">S</mi> </msub> </mrow> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>adm</mi> <mo>,</mo> <mi>ql</mi> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> where σ<sub>adm,ql</sub> is the adhesion strength of joint parts.</div><div class='html-p'>The joint can be preliminarily designed using the following equation [<a href="#B62-metals-08-00318" class="html-bibr">62</a>]:<div class='html-disp-formula-info' id='FD4-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>adm</mi> <mo>,</mo> <mi>ql</mi> </mrow> </msub> <mo>=</mo> <mn>0.45</mn> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>adm</mi> </mrow> </msub> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where σ<sub>adm</sub> is the allowable tensile limit of the material.</div><div class='html-p'>Screw joints are also used for joining components subject to bending. Due to the normal tensions in the joint parts directed in opposite directions, the joint components can push one another. The maximum bending stress in case of non-slipping of the parts is [<a href="#B62-metals-08-00318" class="html-bibr">62</a>] <div class='html-disp-formula-info' id='FD5-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="sans-serif">σ</mi> <mrow> <mi>max</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>6</mn> <msub> <mi>M</mi> <mi mathvariant="normal">l</mi> </msub> </mrow> <mrow> <mi>b</mi> <msup> <mi>h</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(5)</label> </div> </div> where <span class='html-italic'>M</span><sub>l</sub> is the bending moment working on the anchorage, <span class='html-italic'>b</span> is the width of the joint components, <span class='html-italic'>h</span> is the total thickness of the joint components.</div><div class='html-p'>When the joint components are of the same thickness (<span class='html-italic'>h</span>/2), the following relationship can be used for each detail [<a href="#B62-metals-08-00318" class="html-bibr">62</a>]:<div class='html-disp-formula-info' id='FD6-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mi mathvariant="normal">t</mi> </msub> <mo>≤</mo> <mfrac> <mn>6</mn> <mn>4</mn> </mfrac> <mfrac> <mrow> <msub> <mi>M</mi> <mi mathvariant="normal">l</mi> </msub> </mrow> <mi>h</mi> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(6)</label> </div> </div></div><div class='html-p'>The slipping of joint parts does not occur if (7) holds [<a href="#B62-metals-08-00318" class="html-bibr">62</a>]:<div class='html-disp-formula-info' id='FD7-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>F</mi> <mi mathvariant="normal">t</mi> </msub> <mo>≤</mo> <msub> <mi>F</mi> <mi mathvariant="normal">f</mi> </msub> <mo>=</mo> <mi>P</mi> <msub> <mi>f</mi> <mn>2</mn> </msub> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(7)</label> </div> </div> where <span class='html-italic'>P</span> and <span class='html-italic'>f</span><sub>2</sub> are those defined for Equation (1).</div><div class='html-p'>The condition of prevention of slipping of the components can be expressed by the relative internal bending stress divided by the integral thickness of the parts of the joint [<a href="#B62-metals-08-00318" class="html-bibr">62</a>]:<div class='html-disp-formula-info' id='FD8-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mfrac> <mrow> <msub> <mi>M</mi> <mi mathvariant="normal">l</mi> </msub> </mrow> <mi>h</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>P</mi> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mn>9</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>Nm</mi> </mrow> <mrow> <mi>mm</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(8)</label> </div> </div></div><div class='html-p'>The strength of the joint is ensured by the tightening force created by the screw in the nut. Tightening strength of the joint <span class='html-italic'>P</span> is determined by the allowable stress of the screw material. Then, the torque is defined as [<a href="#B62-metals-08-00318" class="html-bibr">62</a>] <div class='html-disp-formula-info' id='FD9-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.5</mn> <mo>×</mo> <msup> <mrow> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mi>P</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mfrac> <mn>1</mn> <mi mathvariant="sans-serif">π</mi> </mfrac> <mo>×</mo> <mfrac> <mi>s</mi> <mi>d</mi> </mfrac> <mo>+</mo> <mfrac> <mi>D</mi> <mi>d</mi> </mfrac> <msub> <mi>f</mi> <mn>1</mn> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>d</mi> <mi mathvariant="normal">m</mi> </msub> </mrow> <mi>d</mi> </mfrac> <msub> <mi>f</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(9)</label> </div> </div> where <span class='html-italic'>d</span> is the diameter of the thread; <span class='html-italic'>s</span> is the step of the thread; <span class='html-italic'>D</span> is the diameter of the nut of the supporting surface (<span class='html-italic'>D</span> = 1.3<span class='html-italic'>d</span>); <span class='html-italic'>d</span><sub>m</sub> is the average diameter of the thread; the coefficient of friction <span class='html-italic'>f</span><sub>1</sub> between the nut and the detail (when both are made of steel) was accepted to be 0.16; <math display='inline'><semantics> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mrow> <mi>cos</mi> <mn>30</mn> <mo>°</mo> </mrow> </mfrac> </mrow> </semantics></math> is the coefficient of friction between the screw and the nut of the threaded surfaces.</div><div class='html-p'>Output quantities of Equations (1)–(9) provide entries for the decision matrix (10).</div></section><section id='sec4-metals-08-00318' type=''><h2 data-nested='1'> 4. The Multiple Criteria Evaluation</h2><div class='html-p'>Some multiple criteria methods were used for evaluating the combinations of technical and economic parameters of screw joints: EDAS [<a href="#B63-metals-08-00318" class="html-bibr">63</a>], SAW [<a href="#B64-metals-08-00318" class="html-bibr">64</a>], TOPSIS [<a href="#B64-metals-08-00318" class="html-bibr">64</a>] and COPRAS [<a href="#B65-metals-08-00318" class="html-bibr">65</a>]. The following methods for assessing importance weight of criteria were used: entropy, the criterion loss method CILOS for the determination of weighting criteria and the integrating IDOCRIW method [<a href="#B66-metals-08-00318" class="html-bibr">66</a>]. Combination of various methods with different logic for making quantitative evaluations allows smoothing over distortions introduced by intrinsic data transformations in each MCDM method [<a href="#B67-metals-08-00318" class="html-bibr">67</a>,<a href="#B68-metals-08-00318" class="html-bibr">68</a>,<a href="#B69-metals-08-00318" class="html-bibr">69</a>,<a href="#B70-metals-08-00318" class="html-bibr">70</a>].</div><section id='sec4dot1-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 4.1. The EDAS Method</h4><div class='html-p'>The EDAS method discerns alternatives based on the distance from the average solution [<a href="#B63-metals-08-00318" class="html-bibr">63</a>]. Two measures for calculating the cumulative criterion of the method are used. The first measure is the positive distance from the average solution (PD), and the second is the negative distance from the average solution (ND). Evaluation of the alternatives is made according to higher values of PD and lower values of ND. The steps for using the EDAS method are as follows:</div><div class='html-p'>Step 1: The decision matrix (<span class='html-italic'>R</span>) is constructed:<div class='html-disp-formula-info' id='FD10-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mrow> <mo>‖</mo> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>‖</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(10)</label> </div> </div></div><div class='html-p'>The matrix is presented in Table 2. This decision matrix will also be used in other MCDM methods.</div><div class='html-p'>Step 2: The vector of weights is created:<div class='html-disp-formula-info' id='FD11-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mo>Ω</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(11)</label> </div> </div> where <span class='html-italic'>i</span> = 1, 2, …, <span class='html-italic'>n</span>; <span class='html-italic'>j</span> = 1, 2, …, <span class='html-italic'>m</span>; <span class='html-italic'>m</span>—the number of criteria; <span class='html-italic'>n</span>—the number of options compared [<a href="#B63-metals-08-00318" class="html-bibr">63</a>].</div><div class='html-p'>Step 3: The average solution is calculated:<div class='html-disp-formula-info' id='FD12-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> <mo>=</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mrow> <mrow> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>/</mo> <mi>n</mi> </mrow> </mrow> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(12)</label> </div> </div></div><div class='html-p'>Step 4: Positive distances from average (PD) and the negative distances from the average solution (ND) are calculated for the maximizing criteria:<div class='html-disp-formula-info' id='FD13-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>PD</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>max</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">V</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(13)</label> </div> </div><div class='html-disp-formula-info' id='FD14-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>ND</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>max</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">V</mi> <mi>j</mi> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(14)</label> </div> </div> and for the minimizing criteria:<div class='html-disp-formula-info' id='FD15-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>PD</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>max</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(15)</label> </div> </div><div class='html-disp-formula-info' id='FD16-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>ND</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>max</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi mathvariant="normal">A</mi> <msub> <mi mathvariant="normal">V</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mrow> <mi>AV</mi> </mrow> <mi>j</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(16)</label> </div> </div> where PD<span class='html-italic'><sub>ij</sub></span> and ND<span class='html-italic'><sub>ij</sub></span> denote the positive and negative distance of the <span class='html-italic'>i</span>-th alternative from the average solution in terms of <span class='html-italic'>j</span>-th criterion, respectively.</div><div class='html-p'>Step 5: The weighted sum of PD and ND is calculated for all alternatives:<div class='html-disp-formula-info' id='FD17-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>SP</mi> </mrow> <mi>i</mi> </msub> <mo>=</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <mo>⋅</mo> <msub> <mrow> <mi>PD</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(17)</label> </div> </div><div class='html-disp-formula-info' id='FD18-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>SN</mi> </mrow> <mi>i</mi> </msub> <mo>=</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <mo>⋅</mo> <msub> <mrow> <mi>ND</mi> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(18)</label> </div> </div> where ω<span class='html-italic'><sub>j</sub></span> is the weight of <span class='html-italic'>j</span>-th criterion.</div><div class='html-p'>Step 6: Values of SP and SN are normalized for all alternatives:<div class='html-disp-formula-info' id='FD19-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>NSP</mi> </mrow> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>SP</mi> </mrow> <mi>i</mi> </msub> </mrow> <mrow> <msub> <mrow> <mi>max</mi> </mrow> <mi>i</mi> </msub> <msub> <mrow> <mi>SP</mi> </mrow> <mi>i</mi> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(19)</label> </div> </div><div class='html-disp-formula-info' id='FD20-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>NSN</mi> </mrow> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mfrac> <mrow> <msub> <mrow> <mi>SN</mi> </mrow> <mi>i</mi> </msub> </mrow> <mrow> <msub> <mrow> <mi>max</mi> </mrow> <mi>i</mi> </msub> <msub> <mrow> <mi>SN</mi> </mrow> <mi>i</mi> </msub> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(20)</label> </div> </div></div><div class='html-p'>Step 7: The cumulative criterion AS of the EDAS method is calculated for all alternatives:<div class='html-disp-formula-info' id='FD21-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>AS</mi> </mrow> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <msub> <mrow> <mi>NSP</mi> </mrow> <mi>i</mi> </msub> <mo>+</mo> <msub> <mrow> <mi>NSN</mi> </mrow> <mi>i</mi> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(21)</label> </div> </div> where 0 ≤ AS<span class='html-italic'><sub>i</sub></span> ≤ 1.</div></section><section id='sec4dot2-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 4.2. The SAW Method</h4><div class='html-p'>The basic idea behind the MCDM methods is to combine values of criteria and weights to obtain a single cumulative criterion for evaluation of the method. The most common example is the SAW method [<a href="#B64-metals-08-00318" class="html-bibr">64</a>], where the method’s cumulative criterion <span class='html-italic'>S<sub>i</sub></span> is calculated as <div class='html-disp-formula-info' id='FD22-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>S</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mrow> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>w</mi> <mi>j</mi> </msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> </mrow> </mstyle> </mrow> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(22)</label> </div> </div> where <span class='html-italic'>w<sub>j</sub></span> are weights of the <span class='html-italic'>j</span>-th criterion and <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> are normalized (dimensionless) values of the <span class='html-italic'>j</span>-th criterion for the <span class='html-italic'>i</span>-th alternative:<div class='html-disp-formula-info' id='FD23-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(23)</label> </div> </div></div></section><section id='sec4dot3-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 4.3. The TOPSIS Method</h4><div class='html-p'>The TOPSIS method is characterized by the use of vector normalization [<a href="#B64-metals-08-00318" class="html-bibr">64</a>] and by its specific features described in [<a href="#B71-metals-08-00318" class="html-bibr">71</a>] which allow using it for making an evaluation of a single alternative:<div class='html-disp-formula-info' id='FD24-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msqrt> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </mstyle> </mrow> </msqrt> </mrow> </mfrac> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mtext> </mtext> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(24)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> are normalized values of <span class='html-italic'>j</span>-th criterion for <span class='html-italic'>i</span>-th alternative.</div><div class='html-p'>The best alternative <math display='inline'><semantics> <mrow> <msup> <mi>V</mi> <mo>*</mo> </msup> </mrow> </semantics></math> and the worst alternative <math display='inline'><semantics> <mrow> <msup> <mi>V</mi> <mo>−</mo> </msup> </mrow> </semantics></math> are determined as <div class='html-disp-formula-info' id='FD25-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>V</mi> <mo>*</mo> </msup> <mo>=</mo> <mo stretchy="false">{</mo> <msubsup> <mi>V</mi> <mn>1</mn> <mo>*</mo> </msubsup> <mo>,</mo> <msubsup> <mi>V</mi> <mn>2</mn> <mo>*</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>V</mi> <mi>m</mi> <mo>*</mo> </msubsup> <mo stretchy="false">}</mo> <mo>=</mo> <mo stretchy="false">{</mo> <mo stretchy="false">(</mo> <mrow> <mrow> <munder> <mrow> <mi>max</mi> </mrow> <mi>i</mi> </munder> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>/</mo> <mi>j</mi> </mrow> <mo>∈</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow> <mrow> <munder> <mrow> <mi>min</mi> </mrow> <mi>i</mi> </munder> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>/</mo> <mi>j</mi> </mrow> <mo>∈</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">}</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(25)</label> </div> </div><div class='html-disp-formula-info' id='FD26-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msup> <mi>V</mi> <mo>−</mo> </msup> <mo>=</mo> <mo>{</mo> <msubsup> <mi>V</mi> <mn>1</mn> <mo>−</mo> </msubsup> <mo>,</mo> <msubsup> <mi>V</mi> <mn>2</mn> <mo>−</mo> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>V</mi> <mi>m</mi> <mo>−</mo> </msubsup> <mo>}</mo> <mo>=</mo> <mo>{</mo> <mo stretchy="false">(</mo> <mrow> <mrow> <munder> <mrow> <mi>min</mi> </mrow> <mi>i</mi> </munder> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>/</mo> <mi>j</mi> </mrow> <mo>∈</mo> <msub> <mi>J</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mrow> <mrow> <munder> <mrow> <mi>max</mi> </mrow> <mi>i</mi> </munder> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>/</mo> <mi>j</mi> </mrow> <mo>∈</mo> <msub> <mi>J</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>}</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(26)</label> </div> </div> where <span class='html-italic'>J</span><sub>1</sub> is a set of indices of the maximized criteria, <span class='html-italic'>J</span><sub>2</sub> is a set of indices of the minimized criteria.</div><div class='html-p'>The distance <math display='inline'><semantics> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </semantics></math> of every considered alternative to the ideal (best) solution and its distance <math display='inline'><semantics> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>−</mo> </msubsup> </mrow> </semantics></math> from the worst solution are calculated (27):<div class='html-disp-formula-info' id='FD27-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>=</mo> <msqrt> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>V</mi> <mi>j</mi> <mo>*</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mstyle> </mrow> </msqrt> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(27)</label> </div> </div><div class='html-disp-formula-info' id='FD28-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>−</mo> </msubsup> <mo>=</mo> <msqrt> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>V</mi> <mi>j</mi> <mo>−</mo> </msubsup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mstyle> </mrow> </msqrt> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(28)</label> </div> </div></div><div class='html-p'>The criterion <math display='inline'><semantics> <mrow> <msubsup> <mi>C</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </semantics></math> of the TOPSIS method is determined:<div class='html-disp-formula-info' id='FD29-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msubsup> <mi>C</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>=</mo> <mfrac> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>−</mo> </msubsup> </mrow> <mrow> <msubsup> <mi>D</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>+</mo> <msubsup> <mi>D</mi> <mi>i</mi> <mo>−</mo> </msubsup> </mrow> </mfrac> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(29)</label> </div> </div><math display='inline'><semantics> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>≤</mo> <msubsup> <mi>C</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</div><div class='html-p'>The largest value of the criterion <math display='inline'><semantics> <mrow> <msubsup> <mi>C</mi> <mi>i</mi> <mo>*</mo> </msubsup> </mrow> </semantics></math> corresponds to the best alternative.</div></section><section id='sec4dot4-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 4.4. The COPRAS Method</h4><div class='html-p'>The <span class='html-italic'>Z<sub>i</sub></span> criterion of the COPRAS method is determined as [<a href="#B65-metals-08-00318" class="html-bibr">65</a>] <div class='html-disp-formula-info' id='FD30-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>Z</mi> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mo>+</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <msub> <mi>S</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mrow> <msub> <mi>S</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mrow> <mfrac> <mn>1</mn> <mrow> <msub> <mi>S</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> </mfrac> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(30)</label> </div> </div> where <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mo>+</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mo>+</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </semantics></math> is the sum of the weighted values of the maximized criteria <math display='inline'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mo>+</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mi>S</mi> <mrow> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mstyle displaystyle="true"> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </munderover> <mrow> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mo>−</mo> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </semantics></math>. The same applies for the minimizing criteria.</div><div class='html-p'>In order to compute values of the <span class='html-italic'>Z<sub>i</sub></span> criterion, input data are normalized using Equation (23).</div></section><section id='sec4dot5-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 4.5. Methods for Determining the Weights of the Criteria</h4><section id='sec4dot5dot1-metals-08-00318' type=''><h4 class='' data-nested='3'> 4.5.1. The Entropy Method</h4><div class='html-p'>The entropy method was proposed by Claude E. Shannon in [<a href="#B72-metals-08-00318" class="html-bibr">72</a>]. Entropy weights are defined as follows [<a href="#B64-metals-08-00318" class="html-bibr">64</a>,<a href="#B66-metals-08-00318" class="html-bibr">66</a>]:</div><div class='html-p'><ul class='html-order'><li><div class='html-p'>The values of criteria are normalized as <div class='html-disp-formula-info' id='FD31-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(31)</label> </div> </div></div></li><li><div class='html-p'>The entropy level of each criterion is calculated as follows:<div class='html-disp-formula-info' id='FD32-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>E</mi> <mi>j</mi> </msub> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>ln</mi> <mi>n</mi> </mrow> </mfrac> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mstyle> <mo>⋅</mo> <mi>ln</mi> <msub> <mover accent="true"> <mi>r</mi> <mo>˜</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mtext> </mtext> <mn>0</mn> <mo>≤</mo> <msub> <mi>E</mi> <mi>j</mi> </msub> <mo>≤</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(32)</label> </div> </div></div></li><li><div class='html-p'>The variation level of each criterion is calculated:<div class='html-disp-formula-info' id='FD33-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>d</mi> <mi>j</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>E</mi> <mi>j</mi> </msub> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(33)</label> </div> </div></div></li><li><div class='html-p'>Entropy weights are calculated by normalizing values <span class='html-italic'>d<sub>j</sub></span>:<div class='html-disp-formula-info' id='FD34-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>W</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>d</mi> <mi>j</mi> </msub> </mrow> <mrow> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>d</mi> <mi>j</mi> </msub> </mrow> </mstyle> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(34)</label> </div> </div></div></li></ul></div><div class='html-p'>Entropy weights reflect the structure of data revealing the degree of non-homogeneity of data. Weights of homogeneous data (when the values of the criteria do not differ considerably) for a certain criterion obtained by the entropy method [<a href="#B66-metals-08-00318" class="html-bibr">66</a>] should be close to zero and would have only a minor influence on the results of evaluation. The largest weight of a criterion obtained from entropy method corresponds to the criterion with the highest rate of data inhomogeneity.</div></section><section id='sec4dot5dot2-metals-08-00318' type=''><h4 class='' data-nested='3'> 4.5.2. Method of Criterion Impact Loss—CILOS</h4><div class='html-p'>In this method, impact loss is measured and objective weights are determined based on this measurement [<a href="#B60-metals-08-00318" class="html-bibr">60</a>,<a href="#B73-metals-08-00318" class="html-bibr">73</a>]. The method evaluates the loss of impact for each criterion until one of the remaining criteria acquires the optimum—the maximum or the minimum value. The algorithm of the method, formalization, description and application has been presented in [<a href="#B66-metals-08-00318" class="html-bibr">66</a>]. The logic of the method of criteria significance loss, its basic ideas, stages and implementation steps are as follows.</div><div class='html-p'>The method can be applied only to the maximizing criteria, therefore minimizing criteria are to be transformed into the maximizing ones. There could be different transformations for this purpose. The inverse transformation was applied:<div class='html-disp-formula-info' id='FD35-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mover accent="true"> <mi>r</mi> <mo>¯</mo> </mover> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mrow> <mi>min</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>r</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(35)</label> </div> </div></div><div class='html-p'>The new matrix with transformed minimizing values and the same maximizing values is denoted as <math display='inline'><semantics> <mrow> <mi>X</mi> <mo>=</mo> <mrow> <mo>‖</mo> <mrow> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>‖</mo> </mrow> </mrow> </semantics></math>. The maximum values for each column (i.e., each criterion) are calculated: <math display='inline'><semantics> <mrow> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mrow> <mi>max</mi> </mrow> <mi>i</mi> </msub> <msub> <mi>x</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>k</mi> <mi>j</mi> </msub> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math>, where <span class='html-italic'>k<sub>j</sub></span> is the number of the row of <span class='html-italic'>j</span>-th column hosting the largest value, is obtained.</div><div class='html-p'>A square matrix <math display='inline'><semantics> <mrow> <mi>A</mi> <mo>=</mo> <mrow> <mo>‖</mo> <mrow> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>‖</mo> </mrow> </mrow> </semantics></math> is formed from values of <span class='html-italic'>k<sub>j</sub></span> rows of the <span class='html-italic'>X</span> matrix. <math display='inline'><semantics> <mrow> <msub> <mi>x</mi> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> correspond to the maximum values of <span class='html-italic'>i</span>-th criterion: <math display='inline'><semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>j</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> (<span class='html-italic'>i</span>,<span class='html-italic'>j</span> = 1, 2, …, <span class='html-italic'>m</span>; <span class='html-italic'>m</span>—number of criteria).</div><div class='html-p'>A matrix <math display='inline'><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mrow> <mo>‖</mo> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mo>‖</mo> </mrow> </mrow> </semantics></math> of the relative losses is created:<div class='html-disp-formula-info' id='FD36-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mi>j</mi> </msub> </mrow> </mfrac> <mo>.</mo> <mtext> </mtext> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>;</mo> <mtext> </mtext> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(36)</label> </div> </div></div><div class='html-p'>Elements <math display='inline'><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </semantics></math> of <span class='html-italic'>P</span> matrix show the loss of alternative relatively to the <span class='html-italic'>j</span>-th criterion, if the <span class='html-italic'>i</span>-th criterion is selected as the best.</div><div class='html-p'>Weights <math display='inline'><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>q</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>q</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> can be determined by solving the linear system:<div class='html-disp-formula-info' id='FD37-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mi>F</mi> <mo>⋅</mo> <mi>q</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(37)</label> </div> </div> where, the matrix <span class='html-italic'>F</span> is as follows:<div class='html-disp-formula-info' id='FD38-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <mi>F</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable> <mtr> <mtd> <mrow> <mo>−</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mtd> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mn>12</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mn>1</mn> <mi>m</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mn>21</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <mo>−</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mrow> </mstyle> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>p</mi> <mrow> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mrow> <mo>−</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>p</mi> <mrow> <mi>i</mi> <mi>m</mi> </mrow> </msub> </mrow> </mstyle> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(38)</label> </div> </div></div><div class='html-p'>The method based on the criterion significance loss offsets the drawback of the entropy method. Thus, when the values of a criterion do not considerably differ, the element s<span class='html-italic'>p<sub>ij</sub></span> of the matrix <span class='html-italic'>P</span> of relative loss of criterion impact (36) becomes close to zero, while the respective criterion weight increases and has a strong impact on the evaluation. If data are homogeneous, when values of a criterion are similar for all alternatives, all relative losses of the criterion, as well as its total loss, become close to zero. Consequently, the linear system of Equation (37) loose of significance because one column of elements in matrix <span class='html-italic'>P</span> is close to zero.</div></section><section id='sec4dot5dot3-metals-08-00318' type=''><h4 class='' data-nested='3'> 4.5.3. Aggregate Objective Weights (IDOCRIW Method)</h4><div class='html-p'>Using the idea of comprising different weights of significance obtained by different methods [<a href="#B64-metals-08-00318" class="html-bibr">64</a>,<a href="#B74-metals-08-00318" class="html-bibr">74</a>,<a href="#B75-metals-08-00318" class="html-bibr">75</a>], the entropy weights <span class='html-italic'>W<sub>j</sub></span> [<a href="#B64-metals-08-00318" class="html-bibr">64</a>], and weights <span class='html-italic'>q<sub>j</sub></span> of criteria impact loss [<a href="#B74-metals-08-00318" class="html-bibr">74</a>] into a single overall weight, the ultimate weights ω<span class='html-italic'><sub>j</sub></span> for the evaluation are obtained:<div class='html-disp-formula-info' id='FD39-metals-08-00318'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mi mathvariant="sans-serif">ω</mi> <mi>j</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>q</mi> <mi>j</mi> </msub> <msub> <mi>W</mi> <mi>j</mi> </msub> </mrow> <mrow> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> <mrow> <msub> <mi>q</mi> <mi>j</mi> </msub> <msub> <mi>W</mi> <mi>j</mi> </msub> </mrow> </mstyle> </mrow> </mfrac> </mrow> </semantics></math> </div> <div class='l'> <label >(39)</label> </div> </div></div><div class='html-p'>The method of obtaining such ultimate weights is called Integrated determination of objective criteria weights (IDOCRIW) [<a href="#B66-metals-08-00318" class="html-bibr">66</a>].</div><div class='html-p'>These weights incorporate both deviation of particular values of criteria (entropy characteristic) and with relative losses of impact (36). Therefore, extreme possible values obtained by the entropy method are compensated by the CILOS method.</div><div class='html-p'>Calculated weights of the entropy and criteria loss of impact are combined into aggregated weights and then are used in multiple criteria assessment for ranking options and for selecting the best alternative.</div></section></section></section><section id='sec5-metals-08-00318' type='results'><h2 data-nested='1'> 5. Results and Discussion</h2><section id='sec5dot1-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 5.1. Results of Theoretical Analysis of Screw Joints</h4><div class='html-p'>Results of theoretical analysis carried out on screw joints are presented in <a href="#metals-08-00318-t002" class="html-table">Table 2</a>. The values listed in the table are obtained from Equations (1)–(9).</div></section><section id='sec5dot2-metals-08-00318' type=''><h4 class='html-italic' data-nested='2'> 5.2. Assessment of Results and Analysis</h4><div class='html-p'>The results of theoretical evaluation of screw joints (<a href="#metals-08-00318-t002" class="html-table">Table 2</a>) are analyzed using multiple criteria analytical methods: EDAS, TOPSIS, COPRAS and SAW. All variants are evaluated twice: using seven criteria (with price) and using six criteria (without price). All calculations are performed with the aggregate IDOCRIW weights.</div><div class='html-p'>The results obtained using all seven criteria are presented in <a href="#metals-08-00318-t003" class="html-table">Table 3</a>, <a href="#metals-08-00318-t004" class="html-table">Table 4</a>, <a href="#metals-08-00318-t005" class="html-table">Table 5</a>, <a href="#metals-08-00318-t006" class="html-table">Table 6</a> and <a href="#metals-08-00318-t007" class="html-table">Table 7</a>.</div><div class='html-p'>The weights of decision criteria resulting from the entropy method are computed with Equations (31)–(34). They are presented in <a href="#metals-08-00318-t003" class="html-table">Table 3</a>.</div><div class='html-p'>The weights of decision criteria resulting from the CILOS method are computed with Equations (35)–(38). They are presented in <a href="#metals-08-00318-t004" class="html-table">Table 4</a>.</div><div class='html-p'>The weights of decision making criteria resulting from the IDOCRIW method are computed with Equation (39) and are presented in <a href="#metals-08-00318-t005" class="html-table">Table 5</a>.</div><div class='html-p'>Joints including parts made of different materials and screws of different diameters are evaluated with the EDAS, TOPSIS, COPRAS and SAW multiple criteria methods including generalized IDOCRIW weights. For example, <a href="#metals-08-00318-t006" class="html-table">Table 6</a> shows the results of the evaluation carried out for the joints with 8 mm diameter screws.</div><div class='html-p'>Comparing the results obtained by the different methods of multiple criteria analyses, it can be observed that, depending on the screw diameter, the joints are ranked in a different way. Generally, the first rank is attained by the Stl joint, although the EDAS method ranked the brass joint as first.</div><div class='html-p'>Evaluation results for the other screw diameters are presented in <a href="#metals-08-00318-t007" class="html-table">Table 7</a>.</div><div class='html-p'>The results show that the best variant, after incorporating the price criterion into the evaluation, appears to be Stl irrespective of the diameters of the screws. Furthermore, the worst variant is a joint made of Brz.</div><div class='html-p'>The average price of the material of the joint (the seventh evaluation criterion of screw) has significant influence on the evaluation. All calculations were repeated without considering the cost of the joint material criterion. Weights determined with the entropy, CILOS and IDOCRIW methods were calculated using only 6 criteria, Equations (31)–(39). The weights obtained by the IDOCRIW method are provided in <a href="#metals-08-00318-t008" class="html-table">Table 8</a>.</div><div class='html-p'>Now, the EDAS, TOPSIS, COPRAS and SAW multiple criteria methods with generalized IDOCRIW weights will be applied to joints of all diameters, but omitting the price criterion. The results of such evaluation of the joints with screws of <span class='html-italic'>d</span> = 8 mm diameter are presented in <a href="#metals-08-00318-t009" class="html-table">Table 9</a>.</div><div class='html-p'>Differences between the results with the price and without the price are observed. If cost is included in the analysis, the steel joint is the best design concept, followed by brass and cast-iron joints; the bronze joint is the worst design. Conversely, if cost is not considered, the brass joint is the best design concept, followed by copper and steel joints; bronze is again the worst design. Consequently, it can be observed that a single parameter such as the price makes a considerable influence on distribution ranks of screw joints in the multiple criteria analysis. More details on triangulation tools in MCDM are given in [<a href="#B76-metals-08-00318" class="html-bibr">76</a>].</div><div class='html-p'>Similarly, applying the EDAS, TOPSIS, COPRAS and SAW multiple criteria methods with generalized IDOCRIW weights joints with screws of <span class='html-italic'>d</span> = 10, 12, 14, 16 mm, omitting the price criterion, are compared.</div><div class='html-p'><a href="#metals-08-00318-t010" class="html-table">Table 10</a> presents the evaluation results for the other screw diameters of 10, 12, 14 and 16 mm, yet without considering cost.</div><div class='html-p'>Multiple criteria evaluation based only on technical parameters (i.e., regardless of cost) indicated that the best choice is to use brass joints, while the worst design is the bronze joint. If cost is included in the analysis, steel joints are the best design concept while bronze joints take again the last rank. The same conclusion was obtained by all EDAS, COPRAS and SAW methods used in the evaluation.</div><div class='html-p'>The results show that the price has a significant influence on the choice of the variant. A decision-maker may decide to account for cost effects or even avoid selecting joint material.</div><div class='html-p'>The cost of screw joints may greatly affect the global cost of a designed component. This study proved that it is possible to obtain a satisfactory solution using multiple criteria analysis of screw joints that encompasses both technical and economic factors. This would lead to reducing cost of the final product. </div><div class='html-p'>Since screw joints are used in almost all modern machines, the results obtained in this study may be very useful for the design stage of screw joints in civil and mechanical engineering.</div></section></section><section id='sec6-metals-08-00318' type='conclusions'><h2 data-nested='1'> 6. Conclusions</h2><div class='html-p'>Various screw joints were evaluated by MCDM methods using two sets of criteria. The first set included only technical parameters of screw joints, while in the second set, the price together with technical criteria was used.</div><div class='html-p'>The multiple criteria evaluation indicated that cost should have a significant impact on the selection of best screw joint.</div><div class='html-p'>Evaluation based on technical parameters only was carried out using four multiple criteria methods: SAW, TOPSIS, COPRAS and EDAS. It was found that the brass joint is the best choice while the worst option is the bronze joint.</div><div class='html-p'>If cost is included in the analysis, steel joints are the best design concept regardless of the diameter of the screw, while bronze joints are again the worst design.</div><div class='html-p'>The present results confirm that MCDM methods can be applied to engineering design problems including reliability and rationality aspects.</div><div class='html-p'>Since it was not possible to assign weights using only expert opinions, the entropy method and the more recently developed CILOS IDOCRIW methods were utilized in order to objectively assign weights to different criterions.</div><div class='html-p'>The proposed approach appears to be general and suited for a wide range of technical and technological problems.</div></section> </div> <div class="html-back"> <section class='html-notes'><h2 >Author Contributions</h2><div class='html-p'>The individual contribution and responsibilities of the authors were as follows: A.Č. provided the research concept and the purpose and designed the research; E.K.Z. provided extensive advice throughout the study, regarding the research design, methodology, findings and revised the manuscript; A.P. collected and analyzed the data. All the authors have read and approved the final manuscript.</div></section><section class='html-notes'><h2 >Funding</h2><div class='html-p'>This research received no external funding.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>We appreciate anonymous referees and the editor for their remarkable comments, deep editing suggestions, and manuscript processing.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>The authors declare no conflict of interest.</div></section><section id='html-references_list'><h2>References</h2><ol class='html-xx'><li id='B1-metals-08-00318' class='html-x' data-content='1.'>Longqi, W.L.; Tjhen, S.; Yao, Z. Detection of loose fasteners in a third rail support assembly using a wavelet analysis. <span class='html-italic'>IES J. Part A Civ. Struct. Eng.</span> <b>2014</b>, <span class='html-italic'>7</span>, 73–82. 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data-lsrc="/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#fig_body_display_metals-08-00318-f001"></a> </div> </div> <div class="html-fig_description"> <b>Figure 1.</b> Screw joint: 1—screw, 2, 3—joint element; 4—screw-nut, <span class='html-italic'>h</span>—thickness of joint elements, <span class='html-italic'>d</span>—screw diameter, <span class='html-italic'>P</span>—clamping force, <span class='html-italic'>T</span>—torque.  </div> </div> <div class="html-fig_show mfp-hide" id ="fig_body_display_metals-08-00318-f001" > <div class="html-caption" > <b>Figure 1.</b> Screw joint: 1—screw, 2, 3—joint element; 4—screw-nut, <span class='html-italic'>h</span>—thickness of joint elements, <span class='html-italic'>d</span>—screw diameter, <span class='html-italic'>P</span>—clamping force, <span class='html-italic'>T</span>—torque.</div> <div class="html-img"><img alt="Metals 08 00318 g001" data-large="/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001.png" data-original="/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001.png" data-lsrc="/metals/metals-08-00318/article_deploy/html/images/metals-08-00318-g001.png" /></div> </div><div class="html-table-wrap" id="metals-08-00318-t001"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t001'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Parameters of screw joints. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t001" > <div class="html-caption" ><b>Table 1.</b> Parameters of screw joints.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Row No.</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Material Elastic Coefficient <span class='html-italic'>E</span>, MPa</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Friction Coefficient between Screw and Element, <span class='html-italic'>f</span><sub>1</sub></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Friction Coefficient between Elements, <span class='html-italic'>f</span><sub>2</sub></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Clamping Force, <span class='html-italic'>P</span>, N</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Torque, <span class='html-italic'>T</span>, N·m</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Average Price, €</th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >8–16</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4629–19,123</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7.83–63.33</td><td align='center' valign='middle' class='html-align-center' >1.40–5.43</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.25–0.97</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >4.82–18.72</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >0.27–1.07</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >6.25–24.25</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.05–14.42</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t002"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t002'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t002"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 2.</b> Results of evaluation of screw joints. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t002" > <div class="html-caption" ><b>Table 2.</b> Results of evaluation of screw joints.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Trial</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Young’s Modulus <span class='html-italic'>E</span>, MPa</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Friction Coefficient between Screw and Elements, <span class='html-italic'>f</span><sub>1</sub></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Friction Coefficient between Elements, <span class='html-italic'>f</span><sub>2</sub></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Clamping Force <span class='html-italic'>P</span>, N</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Torque, <span class='html-italic'>T</span>, N·m</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Limit Shear Force <span class='html-italic'>F</span><sub>s</sub>, N</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Limit Abruption Force <span class='html-italic'>F</span><sub>at</sub>, N</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Minimal Thickness of Connected Elements, <span class='html-italic'>h</span>, mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Limit Relative Bending Stress, Nm/mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Average Price, €</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>1</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>2</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' > </th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>3</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>4</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>5</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>6</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>7</sub></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >8</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4629</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7834</td><td align='center' valign='middle' class='html-align-center' >3055</td><td align='center' valign='middle' class='html-align-center' >6827</td><td align='center' valign='middle' class='html-align-center' >9.93</td><td align='center' valign='middle' class='html-align-center' >1.851</td><td align='center' valign='middle' class='html-align-center' >1.40</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >889</td><td align='center' valign='middle' class='html-align-center' >5904</td><td align='center' valign='middle' class='html-align-center' >0.398</td><td align='center' valign='middle' class='html-align-center' >0.329</td><td align='center' valign='middle' class='html-align-center' >0.25</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >807</td><td align='center' valign='middle' class='html-align-center' >6341</td><td align='center' valign='middle' class='html-align-center' >1.64</td><td align='center' valign='middle' class='html-align-center' >0.329</td><td align='center' valign='middle' class='html-align-center' >4.82</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >2221</td><td align='center' valign='middle' class='html-align-center' >5904</td><td align='center' valign='middle' class='html-align-center' >4.11</td><td align='center' valign='middle' class='html-align-center' >1.027</td><td align='center' valign='middle' class='html-align-center' >0.27</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >3471</td><td align='center' valign='middle' class='html-align-center' >7035</td><td align='center' valign='middle' class='html-align-center' >4.38</td><td align='center' valign='middle' class='html-align-center' >1.643</td><td align='center' valign='middle' class='html-align-center' >6.25</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3523</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4903</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.99</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1.687</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.05</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7423</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >15,663</td><td align='center' valign='middle' class='html-align-center' >4899</td><td align='center' valign='middle' class='html-align-center' >10,492</td><td align='center' valign='middle' class='html-align-center' >12.73</td><td align='center' valign='middle' class='html-align-center' >2.968</td><td align='center' valign='middle' class='html-align-center' >2.46</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >1425</td><td align='center' valign='middle' class='html-align-center' >9427</td><td align='center' valign='middle' class='html-align-center' >0.51</td><td align='center' valign='middle' class='html-align-center' >0.527</td><td align='center' valign='middle' class='html-align-center' >0.44</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >1291</td><td align='center' valign='middle' class='html-align-center' >9427</td><td align='center' valign='middle' class='html-align-center' >2.10</td><td align='center' valign='middle' class='html-align-center' >0.527</td><td align='center' valign='middle' class='html-align-center' >8.49</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >3563</td><td align='center' valign='middle' class='html-align-center' >9427</td><td align='center' valign='middle' class='html-align-center' >5.27</td><td align='center' valign='middle' class='html-align-center' >1.647</td><td align='center' valign='middle' class='html-align-center' >0.48</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >5567</td><td align='center' valign='middle' class='html-align-center' >11,283</td><td align='center' valign='middle' class='html-align-center' >5.62</td><td align='center' valign='middle' class='html-align-center' >2.635</td><td align='center' valign='middle' class='html-align-center' >11.0</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5611</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7861</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2.55</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2.710</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7.13</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10,008</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >25,220</td><td align='center' valign='middle' class='html-align-center' >6605</td><td align='center' valign='middle' class='html-align-center' >14,752</td><td align='center' valign='middle' class='html-align-center' >14.30</td><td align='center' valign='middle' class='html-align-center' >3.113</td><td align='center' valign='middle' class='html-align-center' >3.75</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >1921</td><td align='center' valign='middle' class='html-align-center' >12,710</td><td align='center' valign='middle' class='html-align-center' >0.576</td><td align='center' valign='middle' class='html-align-center' >0.71</td><td align='center' valign='middle' class='html-align-center' >0.67</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >1741</td><td align='center' valign='middle' class='html-align-center' >12,710</td><td align='center' valign='middle' class='html-align-center' >2.36</td><td align='center' valign='middle' class='html-align-center' >0.710</td><td align='center' valign='middle' class='html-align-center' >12.93</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >4804</td><td align='center' valign='middle' class='html-align-center' >12,710</td><td align='center' valign='middle' class='html-align-center' >5.92</td><td align='center' valign='middle' class='html-align-center' >2.22</td><td align='center' valign='middle' class='html-align-center' >0.73</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >7506</td><td align='center' valign='middle' class='html-align-center' >15,212</td><td align='center' valign='middle' class='html-align-center' >6.31</td><td align='center' valign='middle' class='html-align-center' >3.552</td><td align='center' valign='middle' class='html-align-center' >16.75</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7026</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10,060</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2.87</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3.643</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10.85</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14,362</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >42,124</td><td align='center' valign='middle' class='html-align-center' >9479</td><td align='center' valign='middle' class='html-align-center' >21,169</td><td align='center' valign='middle' class='html-align-center' >17.60</td><td align='center' valign='middle' class='html-align-center' >4.446</td><td align='center' valign='middle' class='html-align-center' >4.98</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >2757</td><td align='center' valign='middle' class='html-align-center' >18,240</td><td align='center' valign='middle' class='html-align-center' >0.709</td><td align='center' valign='middle' class='html-align-center' >1.019</td><td align='center' valign='middle' class='html-align-center' >0.89</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >2498</td><td align='center' valign='middle' class='html-align-center' >18,239</td><td align='center' valign='middle' class='html-align-center' >2.91</td><td align='center' valign='middle' class='html-align-center' >1.0197</td><td align='center' valign='middle' class='html-align-center' >17.18</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >6862</td><td align='center' valign='middle' class='html-align-center' >18,240</td><td align='center' valign='middle' class='html-align-center' >7.29</td><td align='center' valign='middle' class='html-align-center' >3.188</td><td align='center' valign='middle' class='html-align-center' >0.98</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >10,771</td><td align='center' valign='middle' class='html-align-center' >21,830</td><td align='center' valign='middle' class='html-align-center' >7.77</td><td align='center' valign='middle' class='html-align-center' >5.098</td><td align='center' valign='middle' class='html-align-center' >22.25</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10,082</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >15,209</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3.53</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5.228</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14.42</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >Al, 7 × 10<sup>4</sup></td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' class='html-align-center' >0.4</td><td align='center' valign='middle' class='html-align-center' >0.7</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19,123</td><td rowspan='6' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >63,335</td><td align='center' valign='middle' class='html-align-center' >12,621</td><td align='center' valign='middle' class='html-align-center' >28,187</td><td align='center' valign='middle' class='html-align-center' >20.51</td><td align='center' valign='middle' class='html-align-center' >5.947</td><td align='center' valign='middle' class='html-align-center' >5.43</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >Stl, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >0.16</td><td align='center' valign='middle' class='html-align-center' >3672</td><td align='center' valign='middle' class='html-align-center' >24,286</td><td align='center' valign='middle' class='html-align-center' >0.826</td><td align='center' valign='middle' class='html-align-center' >1.358</td><td align='center' valign='middle' class='html-align-center' >0.97</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >Brz, 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.13</td><td align='center' valign='middle' class='html-align-center' >0.19</td><td align='center' valign='middle' class='html-align-center' >3327</td><td align='center' valign='middle' class='html-align-center' >26,007</td><td align='center' valign='middle' class='html-align-center' >3.39</td><td align='center' valign='middle' class='html-align-center' >1.357</td><td align='center' valign='middle' class='html-align-center' >18.72</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >CI, 2.1 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.3</td><td align='center' valign='middle' class='html-align-center' >0.5</td><td align='center' valign='middle' class='html-align-center' >9179</td><td align='center' valign='middle' class='html-align-center' >24,286</td><td align='center' valign='middle' class='html-align-center' >8.49</td><td align='center' valign='middle' class='html-align-center' >4.245</td><td align='center' valign='middle' class='html-align-center' >1.07</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >Cu, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' class='html-align-center' >0.45</td><td align='center' valign='middle' class='html-align-center' >0.8</td><td align='center' valign='middle' class='html-align-center' >13,842</td><td align='center' valign='middle' class='html-align-center' >39,641</td><td align='center' valign='middle' class='html-align-center' >9.05</td><td align='center' valign='middle' class='html-align-center' >6.788</td><td align='center' valign='middle' class='html-align-center' >24.25</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Br, 1.2 × 10<sup>5</sup></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.35</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.82</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14,024</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >20,251</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4.11</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6.960</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >14.42</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t003"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t003'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t003"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 3.</b> Weights of decision criteria determined with the entropy method. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t003" > <div class="html-caption" ><b>Table 3.</b> Weights of decision criteria determined with the entropy method.</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th colspan='7' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Weights</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>1</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>2</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>3</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>4</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>5</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>6</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>7</sub></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >8</td><td align='center' valign='middle' class='html-align-center' >0.067</td><td align='center' valign='middle' class='html-align-center' >0.119</td><td align='center' valign='middle' class='html-align-center' >0.105</td><td align='center' valign='middle' class='html-align-center' >0.005</td><td align='center' valign='middle' class='html-align-center' >0.256</td><td align='center' valign='middle' class='html-align-center' >0.139</td><td align='center' valign='middle' class='html-align-center' >0.310</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >0.067</td><td align='center' valign='middle' class='html-align-center' >0.119</td><td align='center' valign='middle' class='html-align-center' >0.105</td><td align='center' valign='middle' class='html-align-center' >0.005</td><td align='center' valign='middle' class='html-align-center' >0.256</td><td align='center' valign='middle' class='html-align-center' >0.139</td><td align='center' valign='middle' class='html-align-center' >0.309</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >0.068</td><td align='center' valign='middle' class='html-align-center' >0.120</td><td align='center' valign='middle' class='html-align-center' >0.103</td><td align='center' valign='middle' class='html-align-center' >0.007</td><td align='center' valign='middle' class='html-align-center' >0.259</td><td align='center' valign='middle' class='html-align-center' >0.130</td><td align='center' valign='middle' class='html-align-center' >0.314</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >0.068</td><td align='center' valign='middle' class='html-align-center' >0.120</td><td align='center' valign='middle' class='html-align-center' >0.103</td><td align='center' valign='middle' class='html-align-center' >0.005</td><td align='center' valign='middle' class='html-align-center' >0.259</td><td align='center' valign='middle' class='html-align-center' >0.130</td><td align='center' valign='middle' class='html-align-center' >0.314</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.067</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.119</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.102</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.019</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.256</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.128</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.310</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t004"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t004'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t004"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 4.</b> Weights of decision criteria determined with the criterion impact loss (CILOS) method. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t004" > <div class="html-caption" ><b>Table 4.</b> Weights of decision criteria determined with the criterion impact loss (CILOS) method.</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th colspan='7' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Weights</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>1</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>2</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>3</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>4</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>5</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>6</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>7</sub></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >8</td><td align='center' valign='middle' class='html-align-center' >0.116</td><td align='center' valign='middle' class='html-align-center' >0.158</td><td align='center' valign='middle' class='html-align-center' >0.172</td><td align='center' valign='middle' class='html-align-center' >0.224</td><td align='center' valign='middle' class='html-align-center' >0.107</td><td align='center' valign='middle' class='html-align-center' >0.123</td><td align='center' valign='middle' class='html-align-center' >0.101</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >0.117</td><td align='center' valign='middle' class='html-align-center' >0.157</td><td align='center' valign='middle' class='html-align-center' >0.174</td><td align='center' valign='middle' class='html-align-center' >0.214</td><td align='center' valign='middle' class='html-align-center' >0.108</td><td align='center' valign='middle' class='html-align-center' >0.127</td><td align='center' valign='middle' class='html-align-center' >0.102</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >0.118</td><td align='center' valign='middle' class='html-align-center' >0.170</td><td align='center' valign='middle' class='html-align-center' >1.127</td><td align='center' valign='middle' class='html-align-center' >0.204</td><td align='center' valign='middle' class='html-align-center' >0.110</td><td align='center' valign='middle' class='html-align-center' >0.170</td><td align='center' valign='middle' class='html-align-center' >0.100</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >0.117</td><td align='center' valign='middle' class='html-align-center' >0.166</td><td align='center' valign='middle' class='html-align-center' >0.125</td><td align='center' valign='middle' class='html-align-center' >0.217</td><td align='center' valign='middle' class='html-align-center' >0.109</td><td align='center' valign='middle' class='html-align-center' >0.166</td><td align='center' valign='middle' class='html-align-center' >0.099</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.120</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.172</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.190</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.105</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.127</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.172</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.113</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t005"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t005'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t005"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 5.</b> Weights of decision criteria determined with the integrated determination of objective criteria weights (IDOCRIW) method. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t005" > <div class="html-caption" ><b>Table 5.</b> Weights of decision criteria determined with the integrated determination of objective criteria weights (IDOCRIW) method.</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th colspan='7' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Weights</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>1</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>2</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>3</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>4</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>5</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>6</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>7</sub></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >8</td><td align='center' valign='middle' class='html-align-center' >0.064</td><td align='center' valign='middle' class='html-align-center' >0.154</td><td align='center' valign='middle' class='html-align-center' >0.149</td><td align='center' valign='middle' class='html-align-center' >0.010</td><td align='center' valign='middle' class='html-align-center' >0.225</td><td align='center' valign='middle' class='html-align-center' >0.140</td><td align='center' valign='middle' class='html-align-center' >0.258</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >0.064</td><td align='center' valign='middle' class='html-align-center' >0.152</td><td align='center' valign='middle' class='html-align-center' >0.149</td><td align='center' valign='middle' class='html-align-center' >0.008</td><td align='center' valign='middle' class='html-align-center' >0.225</td><td align='center' valign='middle' class='html-align-center' >0.144</td><td align='center' valign='middle' class='html-align-center' >0.258</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >0.064</td><td align='center' valign='middle' class='html-align-center' >0.163</td><td align='center' valign='middle' class='html-align-center' >0.104</td><td align='center' valign='middle' class='html-align-center' >0.011</td><td align='center' valign='middle' class='html-align-center' >0.229</td><td align='center' valign='middle' class='html-align-center' >0.176</td><td align='center' valign='middle' class='html-align-center' >0.252</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >0.065</td><td align='center' valign='middle' class='html-align-center' >0.163</td><td align='center' valign='middle' class='html-align-center' >0.105</td><td align='center' valign='middle' class='html-align-center' >0.010</td><td align='center' valign='middle' class='html-align-center' >0.230</td><td align='center' valign='middle' class='html-align-center' >0.175</td><td align='center' valign='middle' class='html-align-center' >0.253</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.058</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.146</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.139</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.014</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.233</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.158</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.252</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t006"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t006'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t006"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 6.</b> Comparison of screwing joints made of different materials by multiple criteria methods (case <span class='html-italic'>d</span> = 8 mm). EDAS: evaluation based on distance from average solution; TOPSIS: technique for order of preference by similarity to ideal solution; COPRAS: complex proportional assessment; SAW: simple additive weighting. </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t006" > <div class="html-caption" ><b>Table 6.</b> Comparison of screwing joints made of different materials by multiple criteria methods (case <span class='html-italic'>d</span> = 8 mm). EDAS: evaluation based on distance from average solution; TOPSIS: technique for order of preference by similarity to ideal solution; COPRAS: complex proportional assessment; SAW: simple additive weighting.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Ranks</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Al), <span class='html-italic'>E</span>, 7 × 10<sup>4</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Stl), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Brz), <span class='html-italic'>E</span>, 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(CI), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Cu), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Br), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th></tr></thead><tbody ><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >EDAS</td><td align='center' valign='middle' class='html-align-center' >0.524</td><td align='center' valign='middle' class='html-align-center' >0.678</td><td align='center' valign='middle' class='html-align-center' >0.145</td><td align='center' valign='middle' class='html-align-center' >0.717</td><td align='center' valign='middle' class='html-align-center' >0.444</td><td align='center' valign='middle' class='html-align-center' >0.786</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >TOPSIS</td><td align='center' valign='middle' class='html-align-center' >0.4862</td><td align='center' valign='middle' class='html-align-center' >0.6805</td><td align='center' valign='middle' class='html-align-center' >0.4778</td><td align='center' valign='middle' class='html-align-center' >0.6977</td><td align='center' valign='middle' class='html-align-center' >0.4490</td><td align='center' valign='middle' class='html-align-center' >0.6357</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >COPRAS</td><td align='center' valign='middle' class='html-align-center' >0.1416</td><td align='center' valign='middle' class='html-align-center' >0.3588</td><td align='center' valign='middle' class='html-align-center' >0.0594</td><td align='center' valign='middle' class='html-align-center' >0.1375</td><td align='center' valign='middle' class='html-align-center' >0.1458</td><td align='center' valign='middle' class='html-align-center' >0.1570</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >SAW</td><td align='center' valign='middle' class='html-align-center' >0.1463</td><td align='center' valign='middle' class='html-align-center' >0.2802</td><td align='center' valign='middle' class='html-align-center' >0.0695</td><td align='center' valign='middle' class='html-align-center' >0.2001</td><td align='center' valign='middle' class='html-align-center' >0.1442</td><td align='center' valign='middle' class='html-align-center' >0.1597</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Sum rank</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >7</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >23</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >9</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Total rank</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t007"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t007'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t007"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 7.</b> Ranking of joints with parts made of different materials connected by 10 to 16 mm diameter screws (cost included) (cases <span class='html-italic'>d</span> = 10, 12, 14, 16 mm). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t007" > <div class="html-caption" ><b>Table 7.</b> Ranking of joints with parts made of different materials connected by 10 to 16 mm diameter screws (cost included) (cases <span class='html-italic'>d</span> = 10, 12, 14, 16 mm).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Al), <span class='html-italic'>E</span>, 7 × 10<sup>4</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Stl), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Brz), <span class='html-italic'>E</span>, 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(CI), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Cu), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Br), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >2</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >2–3</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >2–3</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >1</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >2–3</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >2–3</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2–3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2–3</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t008"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t008'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t008"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 8.</b> Weights of decision criteria determined with the IDOCRIW method (cost criterion omitted). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t008" > <div class="html-caption" ><b>Table 8.</b> Weights of decision criteria determined with the IDOCRIW method (cost criterion omitted).</div> <table > <thead ><tr ><th rowspan='2' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th colspan='6' align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Weights</th></tr><tr ><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>1</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>2</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>3</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>4</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>5</sub></th><th align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >ω<sub>6</sub></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >8</td><td align='center' valign='middle' class='html-align-center' >0.072</td><td align='center' valign='middle' class='html-align-center' >0.212</td><td align='center' valign='middle' class='html-align-center' >0.207</td><td align='center' valign='middle' class='html-align-center' >0.009</td><td align='center' valign='middle' class='html-align-center' >0.296</td><td align='center' valign='middle' class='html-align-center' >0.204</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >0.072</td><td align='center' valign='middle' class='html-align-center' >0.208</td><td align='center' valign='middle' class='html-align-center' >0.207</td><td align='center' valign='middle' class='html-align-center' >0.007</td><td align='center' valign='middle' class='html-align-center' >0.296</td><td align='center' valign='middle' class='html-align-center' >0.210</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >0.072</td><td align='center' valign='middle' class='html-align-center' >0.233</td><td align='center' valign='middle' class='html-align-center' >0.135</td><td align='center' valign='middle' class='html-align-center' >0.009</td><td align='center' valign='middle' class='html-align-center' >0299</td><td align='center' valign='middle' class='html-align-center' >0.251</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >0.072</td><td align='center' valign='middle' class='html-align-center' >0.233</td><td align='center' valign='middle' class='html-align-center' >0.136</td><td align='center' valign='middle' class='html-align-center' >0.008</td><td align='center' valign='middle' class='html-align-center' >0.301</td><td align='center' valign='middle' class='html-align-center' >0.251</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.061</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.205</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.196</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.012</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.305</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >0.221</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t009"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t009'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t009"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 9.</b> Multiple criteria method-based evaluation of 8 mm diameter screw joints made of different materials (price omitted) (<span class='html-italic'>d</span> = 8 mm). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t009" > <div class="html-caption" ><b>Table 9.</b> Multiple criteria method-based evaluation of 8 mm diameter screw joints made of different materials (price omitted) (<span class='html-italic'>d</span> = 8 mm).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Ranks</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Al), <span class='html-italic'>E</span>, 7 × 10<sup>4</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Stl), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Brz), <span class='html-italic'>E</span>, 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(CI), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Cu), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Br), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th></tr></thead><tbody ><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >EDAS</td><td align='center' valign='middle' class='html-align-center' >0.303</td><td align='center' valign='middle' class='html-align-center' >0.316</td><td align='center' valign='middle' class='html-align-center' >0.210</td><td align='center' valign='middle' class='html-align-center' >0.428</td><td align='center' valign='middle' class='html-align-center' >0.807</td><td align='center' valign='middle' class='html-align-center' >0.998</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >TOPSIS</td><td align='center' valign='middle' class='html-align-center' >0.3815</td><td align='center' valign='middle' class='html-align-center' >0.5918</td><td align='center' valign='middle' class='html-align-center' >0.5543</td><td align='center' valign='middle' class='html-align-center' >0.5727</td><td align='center' valign='middle' class='html-align-center' >0.6752</td><td align='center' valign='middle' class='html-align-center' >0.8577</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >COPRAS</td><td align='center' valign='middle' class='html-align-center' >0.1719</td><td align='center' valign='middle' class='html-align-center' >0.2187</td><td align='center' valign='middle' class='html-align-center' >0.0842</td><td align='center' valign='middle' class='html-align-center' >0.1275</td><td align='center' valign='middle' class='html-align-center' >0.1897</td><td align='center' valign='middle' class='html-align-center' >0.2079</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td></tr><tr ><td rowspan='2' align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >SAW</td><td align='center' valign='middle' class='html-align-center' >0.1719</td><td align='center' valign='middle' class='html-align-center' >0.2187</td><td align='center' valign='middle' class='html-align-center' >0.0842</td><td align='center' valign='middle' class='html-align-center' >0.1275</td><td align='center' valign='middle' class='html-align-center' >0.1897</td><td align='center' valign='middle' class='html-align-center' >0.2079</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Sum rank</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >19</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >23</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >10</td><td align='center' valign='top' style='border-bottom:solid thin' class='html-align-center' >6</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >Total rank</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td></tr></tbody> </table> </div><div class="html-table-wrap" id="metals-08-00318-t010"> <div class="html-table_wrap_td" > <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href='#table_body_display_metals-08-00318-t010'> <img alt="Table" data-lsrc="https://www.mdpi.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2075-4701/8/5/318/display" href="#table_body_display_metals-08-00318-t010"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 10.</b> Ranking of joints with parts made of different materials connected by 10 to 16 mm diameter screws (cost not included). </div> </div> <div class="html-table_show mfp-hide " id ="table_body_display_metals-08-00318-t010" > <div class="html-caption" ><b>Table 10.</b> Ranking of joints with parts made of different materials connected by 10 to 16 mm diameter screws (cost not included).</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Screw Diameter, <span class='html-italic'>d</span>, mm</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Al), <span class='html-italic'>E</span>, 7 × 10<sup>4</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Stl), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Brz), <span class='html-italic'>E</span>, 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(CI), <span class='html-italic'>E</span>, 2.1 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Cu), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >(Br), <span class='html-italic'>E</span>, 1.2 × 10<sup>5</sup></th></tr></thead><tbody ><tr ><td align='center' valign='middle' class='html-align-center' >10</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >1</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >12</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >1</td></tr><tr ><td align='center' valign='middle' class='html-align-center' >14</td><td align='center' valign='middle' class='html-align-center' >5</td><td align='center' valign='middle' class='html-align-center' >3</td><td align='center' valign='middle' class='html-align-center' >6</td><td align='center' valign='middle' class='html-align-center' >4</td><td align='center' valign='middle' class='html-align-center' >2</td><td align='center' valign='middle' class='html-align-center' >1</td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >16</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4–5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >6</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4–5</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td></tr></tbody> </table> </div></section> <section id="html-copyright"><br>© 2018 by the authors. 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Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods. <em>Metals</em> <b>2018</b>, <em>8</em>, 318. https://doi.org/10.3390/met8050318 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Čereška A, Podviezko A, Zavadskas EK. Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods. <em>Metals</em>. 2018; 8(5):318. https://doi.org/10.3390/met8050318 </p> <b>Chicago/Turabian Style</b><br> <p> Čereška, Audrius, Askoldas Podviezko, and Edmundas Kazimieras Zavadskas. 2018. "Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods" <em>Metals</em> 8, no. 5: 318. https://doi.org/10.3390/met8050318 </p> <b>APA Style</b><br> <p> Čereška, A., Podviezko, A., & Zavadskas, E. K. (2018). 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Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods. <em>Metals</em> <b>2018</b>, <em>8</em>, 318. https://doi.org/10.3390/met8050318 </p> <div style="display: block"> <b>AMA Style</b><br> <p> Čereška A, Podviezko A, Zavadskas EK. Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods. <em>Metals</em>. 2018; 8(5):318. https://doi.org/10.3390/met8050318 </p> <b>Chicago/Turabian Style</b><br> <p> Čereška, Audrius, Askoldas Podviezko, and Edmundas Kazimieras Zavadskas. 2018. "Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods" <em>Metals</em> 8, no. 5: 318. https://doi.org/10.3390/met8050318 </p> <b>APA Style</b><br> <p> Čereška, A., Podviezko, A., & Zavadskas, E. K. (2018). 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Assessment of Different Metal Screw Joint Parameters by Using Multiple Criteria Analysis Methods