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Reliability and Optimization of the Manufacturing of Steel Flanges, by using Mechanical Method, Statistical Method, Firefly Algorithm
<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Archiving and Interchange DTD v1.0 20120330//EN" "JATS-archivearticle1.dtd"> <article article-type="research-article" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mml="http://www.w3.org/1998/Math/MathML"> <front> <journal-meta> <journal-id journal-id-type="publisher-id">IJMPERD</journal-id> <journal-title-group> <journal-title>International Journal of Mechanical and Production Engineering Research and Development</journal-title> </journal-title-group> <issn pub-type="ppub">2249-6890</issn> <issn pub-type="epub">2249-8001</issn> <publisher> <publisher-name>Trans Stellar Journal Publications Research Consultancy</publisher-name> <publisher-loc>India</publisher-loc> </publisher> </journal-meta> <article-meta> <article-id pub-id-type="publisher-id">IJMPERD-9-99</article-id> <article-id pub-id-type="paper-id">IJMPERDAPR201910</article-id> <article-categories> <subj-group subj-group-type="heading"> <subject>Original Article</subject> </subj-group> </article-categories> <title-group> <article-title>RELIABILITY AND OPTIMIZATION OF THE MANUFACTURING OF STEEL FLANGES, BY USING MECHANICAL METHOD, STATISTICAL METHOD, FIREFLY ALGORITHM</article-title> </title-group> <contrib-group content-type="article"> <contrib contrib-type="author"> <name> <surname>HAOUACHE</surname> <given-names>NOUR-EDDINE Mʼ</given-names> </name> <xref ref-type="aff" rid="aff1"><sup>1</sup></xref> </contrib> <contrib contrib-type="author"> <name> <surname>ELKHECHAFI</surname> <given-names>MARIAM</given-names> </name> <xref ref-type="aff" rid="aff2"><sup>2</sup></xref> </contrib> <contrib contrib-type="author"> <name> <surname>HANAAHACHIMI</surname> </name> <xref ref-type="aff" rid="aff2"><sup>2</sup></xref> </contrib> <contrib contrib-type="author"> <name> <surname>MOULAYTAIBBELGHITI</surname> </name> <xref ref-type="aff" rid="aff1"><sup>1</sup></xref> </contrib> </contrib-group> <aff id="aff1"> <label>1</label>Laboratory of EECOMAS, National School of Applied Sciences ENSA, IbnTofail University Kenitra, Morocco</aff> <aff id="aff2"> <label>2</label>Laboraroty of LGS, National School of Applied Sciences ENSA, IbnTofail University Kenitra, Morocco</aff> <pub-date pub-type="ppub"> <season>Apr</season> <year>2019</year> </pub-date> <volume>9</volume> <issue>2</issue> <fpage>99</fpage> <lpage>110</lpage> <history> <date date-type="received"> <day>01</day> <month>Dec</month> <year>2018</year> </date> <date date-type="accepted"> <day>20</day> <month>Dec</month> <year>2018</year> </date> <date date-type="published"> <day>06</day> <month>Feb</month> <year>2019</year> </date> </history> <permissions> <copyright-statement>© TJPRC Pvt. Ltd.</copyright-statement> <copyright-year>2018</copyright-year> <copyright-holder>www.tjprc.org</copyright-holder> </permissions> <abstract> <p>This paper presents the reliability calculation and the optimization ofthe manufacturing process of steel flanges, by using three methods: the first is the mechanical method based on feedback and the analysis of failure modes. The second one is the statistical method for determining the mathematical model of the reliability factors. The last one is the optimization method by the Firefly algorithm.</p> <p>On the mechanical method we used the feedback experience to define all the elements related to reliability, and the identification of failure modes, the target of the statistical analysis method is to value the results of the mechanical, we used the optimization method through the Firefly algorithm to calculate the values of the reliability factors for the two steel flanges, by concluding on the most reliable and optimal manufacturing process.</p> </abstract> <kwd-group> <kwd>Reliability</kwd> <kwd>Regression</kwd> <kwd>Algorithm</kwd> <kwd>Optimization</kwd> <kwd>Firefly</kwd> </kwd-group> <counts> <fig-count count="3"/> <table-count count="6"/> <ref-count count="13"/> <page-count count="12"/> </counts> </article-meta> </front> <body> <sec id="sec1-1" sec-type="intro"> <title>INTRODUCTION</title> <p>We have started this study with an objective, even the reclamation of our customer for the new structure, the Quadra-sector flange (<xref ref-type="fig" rid="fig2">figure 2</xref>),and aftermakingthe results of the stress marked by a good result ofthe stressof the new Quadra sector flange,we should communicatetoour client thereliabilityof this structure.</p> <p>With the three methods the mechanical,statistical and the firefly algorithm,we must justify that the new Quadra-sector flange is reliable as the ring flange (<xref ref-type="fig" rid="fig1">figure 1</xref>), which we have an experience of manufacturing and we have also the satisfaction of our client for this one.</p> <p>The new Quadra sector flange and the ring flangerealizedin the same workstations, but theprocess is different between the two structures, on welding and machining, because we shouldweldthe four sectors for the new Quadra-sector flange, but for the ring flange we had one element without welding, the welding of sectorprovidesus a benefit in raw materials, it gives us a benefit of purchasing of this new Quadra sector <sup><bold>5</bold></sup>.</p> <p>Our objective is to assume that the new Quadr-sector flange is reliable and the optimal structure, according to the difference in the process of manufacturing and the benefit of raw materials, and to prove that our new Quadra-sector flange is also reliable and the new optimal structure.</p> <p>we support our study of the two flanges using the three methods, mechanical, statistical and the fire fly algorithm, in goal to communicate the reliability and optimization of those structures, especially the new Quadras-sector flange.</p> <fig id="fig1"> <label>Figure 1</label> <caption> <p><bold>Ring Flange Structure</bold></p> </caption> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_1.jpg"/> </fig> <fig id="fig2"> <label>Figure 2</label> <caption> <p><bold>Quadr-Sector Flange Structure</bold></p> </caption> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_2.jpg"/> </fig> </sec> <sec id="sec1-2" sec-type="materials|methods"> <title>MATERIALS AND METHODS</title> <p>The first step we followed is the identification of all the reliability factors, those are parameters regarding the operating environment of the structure, which can be factors related to reliability, while starting with an identification process, we used the mechanical method of experience feedback for line of production of flange structures, over a period of one year.</p> <p>The second step is the analysis of the results factors of experience feedback, we analysed the failure modes, their effects; as a target of this method, we evaluated the factors related to the reliability of this flange structure, for both models; at the same time, we got the failure modes that we can have to avoid in order to have a more reliable structure.</p> <p>In objective to determinate the mathematical model for each type of metallic flange, we benefit from the mechanical method and the results of the mechanical method, we decided to use the results of the mechanical method of the statistical method.</p> <p>Both the statistical and mechanical methods, can only give the mathematical parameter and the analysis of factors related to reliability, hence, we decided with our laboratory of mathematics applied to use the algorithm firefly, by using the firefly we can have the values of reliability of the two flange structures, then the most optimized flange structure.</p> <p>Before moving to use methods, we introduce below the structure’s environment. The structure is metallic, in contact with the connected pipe environment, the fluid is the element that flows through the pipes (<xref ref-type="fig" rid="fig3">figure 3</xref>), and the two environments are as follows:</p> <p>We present in the next table the elements of the external environment:</p> <table-wrap id="tab1"> <label>Table 1</label> <caption> <p><bold>Elements of External Environment</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle">Environment</th> <th align="center" valign="middle">Nbr</th> <th align="center" valign="middle">Designation Elements</th> </tr> </thead> <tbody> <tr> <td align="center" valign="middle" rowspan="6">External</td> <td align="center" valign="middle">1</td> <td align="left" valign="middle">Ms: Mechanical forces, which are the results of the external environment efforts (N)</td> </tr> <tr> <td align="center" valign="middle">2</td> <td align="left" valign="middle">At: Operating temperature, it’s the temperature of the external environment (°C)</td> </tr> <tr> <td align="center" valign="middle">3</td> <td align="left" valign="middle">Grm: The grade of structural materials manufacturing, it is the grade of the structural metal raw materials.</td> </tr> <tr> <td align="center" valign="middle">4</td> <td align="left" valign="middle">Cs: Conditions of storage and transportation of metallic structures.</td> </tr> <tr> <td align="center" valign="middle">5</td> <td align="left" valign="middle">F: The fluid, which is in most cases steam or water, which flows through the piping.</td> </tr> <tr> <td align="center" valign="middle">6</td> <td align="left" valign="middle">mt: The melting temperature of the filler metal, using during welding. (°C)</td> </tr> </tbody> </table> </table-wrap> <p>We present in the next table the elements of the internal environment:</p> <table-wrap id="tab2"> <label>Table 2</label> <caption> <p><bold>Elements of Internal Environment</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle">Environment</th> <th align="center" valign="middle">Nbr</th> <th align="center" valign="middle">Designation Elements</th> </tr> </thead> <tbody> <tr> <td align="center" valign="middle" rowspan="9">Internal</td> <td align="center" valign="middle">1</td> <td align="left" valign="middle">OP: Operating pressure.</td> </tr> <tr> <td align="center" valign="middle">2</td> <td align="left" valign="middle">Gw: Geometry of the weld bead (m<sup>2</sup>)</td> </tr> <tr> <td align="center" valign="middle">3</td> <td align="left" valign="middle">Sts: Surface treatment against corrosion. (g/m<sup>2</sup>)</td> </tr> <tr> <td align="center" valign="middle">4</td> <td align="left" valign="middle">Sc: Stress concentration. (N/m<sup>2</sup>)</td> </tr> <tr> <td align="center" valign="middle">5</td> <td align="left" valign="middle">Sc: Security Constraint (N/m<sup>2</sup>)</td> </tr> <tr> <td align="center" valign="middle">6</td> <td align="left" valign="middle">Tmp: Thickness of the manufacturing plate. (mm)</td> </tr> <tr> <td align="center" valign="middle">7</td> <td align="left" valign="middle">Gs: Geometry of the structure. (m<sup>2</sup>)</td> </tr> <tr> <td align="center" valign="middle">8</td> <td align="left" valign="middle">Ws: Weight of the structure. (kg)</td> </tr> <tr> <td align="center" valign="middle">9</td> <td align="left" valign="middle">Qwe: Quality of the welding electrode for the elements of the structure</td> </tr> </tbody> </table> </table-wrap> <fig id="fig3"> <label>Figure 3</label> <caption> <p><bold>Piping with Steel Flanges</bold></p> </caption> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/fig_3.jpg"/> </fig> <p>The reliability of the structures depends on many factors, because the structure is on the interaction medium between many elements, the list of these elements is not limited, and we focused on the elements that had direct contact with steel flanges.</p> <p>We will address the reliability analysis mentioned above, to identify all the factors related to the reliability using each method.</p> <sec id="sec2-1"> <title>Mechanical Method</title> <p>We present in the following method, the identification of factors related to the reliability of the metal flanges, through our experience feedback based on one-year of manufacturing of flanges in our factories, we analysed each element defined in the environment, the importance to our metallic structure, thereafter, we had all important elements evaluated as factors related to the reliability analysis of the flange structure.</p> </sec> <sec id="sec2-2"> <title>Factors Related to the Reliability Analysis</title> <p>The analysis through our experience feedback method provide us the elements that we characterized as factors related to the reliability of the two metallic flanges, the ring and the Quadr-sector flanges, we can see the analysis results in the following table:</p> <table-wrap id="tab3"> <label>Table 3</label> <caption> <p><bold>Reliability Factors of the Flange Structure</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle">nbr</th> <th align="center" valign="middle">sym</th> <th align="center" valign="middle">Designation</th> <th align="center" valign="middle">Environment</th> </tr> </thead> <tbody> <tr> <td align="center" valign="middle">6</td> <td align="center" valign="middle">mt</td> <td align="left" valign="middle">The melting temperature of the filler metal.(°C)</td> <td align="center" valign="middle">External</td> </tr> <tr> <td align="center" valign="middle">1</td> <td align="center" valign="middle">Op</td> <td align="left" valign="middle">Operating pressure.(br)</td> <td align="center" valign="middle">Internal</td> </tr> <tr> <td align="center" valign="middle">2</td> <td align="center" valign="middle">Gw</td> <td align="left" valign="middle">Geometry of the weld bead. (m<sup>2</sup>)</td> <td align="center" valign="middle">Internal</td> </tr> <tr> <td align="center" valign="middle">6</td> <td align="center" valign="middle">Tmp</td> <td align="left" valign="middle">Thickness of the manufacturing plate. (mm)</td> <td align="center" valign="middle">Internal</td> </tr> <tr> <td align="center" valign="middle">7</td> <td align="center" valign="middle">Gs</td> <td align="left" valign="middle">Geometry of the structure.(m<sup>2</sup>)</td> <td align="center" valign="middle">Internal</td> </tr> <tr> <td align="center" valign="middle">6</td> <td align="center" valign="middle">Ws</td> <td align="left" valign="middle">Weight of the structure.(Kg)</td> <td align="center" valign="middle">Internal</td> </tr> </tbody> </table> </table-wrap> <p>As we can see in the table that 99% of the elements characterized as factors related to the analysis of reliability, using the experience feedback method, are the internal environment elements.</p> </sec> <sec id="sec2-3"> <title>Analysis of the Failure Modes</title> <p>The analysis of the failure mode method gives us the determination of the effects of the failure modes.</p> <p>After analysis of all the failure modes, their effects, we have the occasion to present the most critical and degrading failure modes; the list of these failure modes is as follows:</p> <sec id="sec3-1"> <title>Failure in Erection of Piping with Flanges</title> <list list-type="bullet"> <list-item><p>Bad state of the pipes and wires.</p></list-item> <list-item><p>The propagation of surface cracks on the flange structure.</p></list-item> <list-item><p>Wrong adjustment of the joint welding equipment of the weld bead.</p></list-item> <list-item><p>Welding in an inadequate area, presence of dust…</p></list-item> <list-item><p>Bad tightening of bolts, poor application of a tightening torque.</p></list-item> <list-item><p>Abnormal pressure.</p></list-item> </list></sec> <sec id="sec3-2"> <title>Failure in Manufacturing and Process of Control</title> <list list-type="bullet"> <list-item><p>Lack of dimensional control of the flange structure.</p></list-item> <list-item><p>Lack of grade and plate control of the raw material of the flange structure.</p></list-item> <list-item><p>Lack of aspect and plate quality control of the raw material of the flange structure.</p></list-item> <list-item><p>Lack of dimensional geometry control of the flange structure.</p></list-item> <list-item><p>Lack of hole geometry control of the flange structure.</p></list-item> <list-item><p>The machining of inconvenient thickness flange.</p></list-item> </list> <p>The different steps are going to allow the manufacturers of these types of structures to avoid these failure modes, with implementation of the prevention programs to eliminate a maximum of these failure modes, need to reinforce their improvement process of manufacturing, in order to have a good quality control during manufacturing of the metal flange structure and a qualified team in the erection of piping with flanges.</p> <p>The mechanical method allowed us to group the factors related to reliability, we also evaluated their failure modes, we analyse in the following method done statistically, to quantify and identify, what are the elements that related to the reliability of the flange structure, by using the statistic method?</p> </sec></sec></sec> <sec id="sec1-3"> <title>STATISTICAL METHOD</title> <sec id="sec2-4"> <title>Objective of the Statistical Method</title> <p>The objective of using the statistical method is to select statistical the factors related to the reliability of the structure, quantify analytic the mathematical model of the reliability of the flange structure, this method done for the first time on these metallic structures, then we can evaluate and compare the results of statistical method with the results of the mechanical method.</p> <p>We started with the methodology of the statistical method, step by step to find the mathematical model for the two flanges.</p> </sec> <sec id="sec2-5"> <title>Database of the Statistical Method</title> <p>To answer this question of the statistical analysis of the factors related to the reliability of the structures, we analysed our manufacturing of the metallic flange structures in our production plant, process of manufacturing realised on the machine shops and metal workshops, in which the flange structure followed a many production’s step.</p> <p>We collected all the parameters related to geometry, the strength of the structure, the fluidity, to the welding of the structure, the manufacturing time of the flange structure, and other elements for the 36 flanges manufactured on different diameter, with a pressure from 10 bar to 16 bar.</p> <p>The collect of information done through the observation of production workstations, and the timing of production workstations, we also used the standards and security norms related to structure, namely the pressure, the security constraints, the Young module.</p> </sec> <sec id="sec2-6"> <title>Alimentation of Database</title> <p>We used in the statistical method all the data collected, through the method of observation of the workstations established in one year of manufacturing of the metallic flanges structure for 36 flanges, we joined our data bases of the two metallic structure to qualify the parameters that have a significant weight, in the reliability analysis and the reliability calculation of these two metallic flange structure.</p> <p>The use of the statistical method performed us to write the mathematical model for the two different flange structures, this model is, the better results of the statistical analysis, and this mathematical model used in the analysis of the algorithmic optimization method.</p> </sec> <sec id="sec2-7"> <title>Algorithmic Optimization Method</title> <sec id="sec3-3"> <title>The Necessity of the Algorithmic Method</title> <p>Our analysis of important factors and parameters related to the reliability is an essential method, but it is preliminary in the evaluation of the most reliable metallic flange model and the most optimized manufacturing process.</p> <p>The mechanical method and the statistical method allowed us to have mathematical model, by using the linear function regression, which groups all the reliability factors of the two metallic structures.</p> <p>Our objective is the evaluation of the reliability of the two metallic structures (the ring and the quadr-sector flanges) and also the evaluation of the most optimized manufacturing process.</p> <p>The analytical calculation of the regression function will be too long, according to all the parameters that we have as reliability factors, and in regards to the complexity of the regression function. So to find the reliability factors, we used our mathematical model of the two structures by using the algorithmic method<sup><bold>1</bold>.</sup></p></sec> <sec id="sec3-4"> <title>Firefly Algorithm Method</title> <p>The firefly is one of the most impressive performances swarm intelligence algorithms in solving optimization problems which were based on the flashing patterns and behaviour of fireflies.</p> <p>Fireflies are insects characterized by their amazing flashing light produced by a process of bioluminescence. The main role of such flashes can be reduced to the following two tasks: Firstly, to attract mating partners and secondly to remind potential predators of the bitter taste of fireflies. The movement of a firefly I is attracted to another more attractive (brighter) firefly J<sup><bold>9,10,11</bold></sup>.</p> </sec></sec> <sec id="sec2-8"> <title>Method of Calculation Reliability Factors</title> <p>To find the reliability factors, we worked in collaboration with our numerical laboratory (LGS), we provide them our data bases, the parameters of reliability and the mathematical models of each flange.</p> <p>Our target from the algorithmic method is the evaluation of the reliability factors, by using the analysis program of the function, and defining the constraints of the parameters related to this function.</p> <p>We have chosen the Firefly algorithm, as a new one characterized quickly compilation, the ability of analysis for these types of functions<sup><bold>6,7,8</bold></sup>.</p> </sec></sec> <sec id="sec1-4" sec-type="results|discussion"> <title>RESULTS AND DISCUSSIONS</title> <sec id="sec2-9"> <title>Results of the Statistical and Mechanical Methods</title> <sec id="sec3-5"> <title>Coefficients of the Ring Flange</title> <p>After using the data bases, in the statistical method, by the alimentation of the data base elements for the ring flange, we present the following table.</p> <p>In the table we see the factors related to the reliability of the ring flange structure:</p> <table-wrap id="tab4"> <label>Table 4</label> <caption> <p><bold>Reliability Coefficients of the Ring Flange</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle" rowspan="2">Model</th> <th align="center" valign="middle" colspan="2">Non Standardized Coefficients</th> <th align="center" valign="middle">Standardized Coefficients</th> </tr> <tr> <th align="center" valign="middle">A</th> <th align="center" valign="middle">Standard Error(Ɛ)</th> <th align="center" valign="middle">Bêta</th> </tr> </thead> <tbody> <tr> <td align="left" valign="middle">Constant</td> <td align="center" valign="middle">-1,530</td> <td align="center" valign="middle">0,732</td> <td align="center" valign="middle"> </td> </tr> <tr> <td align="left" valign="middle">Pressure</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">0,002</td> <td align="center" valign="middle">0,061</td> </tr> <tr> <td align="left" valign="middle">Flow temperature</td> <td align="center" valign="middle">0,004</td> <td align="center" valign="middle">0,004</td> <td align="center" valign="middle">0,186</td> </tr> <tr> <td align="left" valign="middle">Inside diameter</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">-3,144</td> </tr> <tr> <td align="left" valign="middle">Outside diameter</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">-4,817</td> </tr> <tr> <td align="left" valign="middle">Thickness</td> <td align="center" valign="middle">-0,002</td> <td align="center" valign="middle">0,001</td> <td align="center" valign="middle">-2,195</td> </tr> <tr> <td align="left" valign="middle">Number of holes</td> <td align="center" valign="middle">0,001</td> <td align="center" valign="middle">0,001</td> <td align="center" valign="middle">0,483</td> </tr> <tr> <td align="left" valign="middle">Diameter of the holes</td> <td align="center" valign="middle">-0,002</td> <td align="center" valign="middle">0,002</td> <td align="center" valign="middle">-1,109</td> </tr> <tr> <td align="left" valign="middle">Weight</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">4,336</td> </tr> <tr> <td align="left" valign="middle">Security Constraint.</td> <td align="center" valign="middle">0,011</td> <td align="center" valign="middle">0,006</td> <td align="center" valign="middle">6,383</td> </tr> </tbody> </table> </table-wrap> <p>In the table of reliability coefficients, we have the important parameters of modelling, which are designated by the element <italic>A</italic>. The elimination of this element <italic>A</italic> is interpreted by the no-consideration as a reliability factor, a real modelling takes into consideration the standard Error, both elements <italic>A</italic> and standard Error are sufficient for the writing of mathematical model of multi-variable linear regression.</p> <p>Our analysis of the table of coefficients focalises on the interpretation below, which allowed us to identify the factors related to the reliability are appointed reliability coefficients of the metallic flange structure which are:</p> <list list-type="bullet"> <list-item><p><bold>Ft</bold>: Flow temperature.</p></list-item> <list-item><p><bold>T</bold>: Thickness.</p></list-item> <list-item><p><bold>Nh</bold>: Number of holes.</p></list-item> <list-item><p><bold>Dh</bold>: Diameter of holes.</p></list-item> <list-item><p><bold>Sc</bold>: Security Constraint.</p></list-item> </list> </sec></sec> <sec id="sec2-10"> <title>Mathematical Model of the Reliability of the Ring Flange</title> <p>The mathematical model designed in the analysis of reliability, through the objective function, to find the optimum of the functions under the effect of well-defined constraints.</p> <p>The mathematical model is a function grouped all important factors results of the reliability, and this model helps us to find after as a result the optimum of a function.</p> <p>We have written the mathematical model with the reliability function f(X<sub>i</sub>), by using the following linear regression function:</p> <p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn1.png"/></p> <p>Where:</p> <p>f(X<sub>i</sub>)<italic>Rf</italic>: Reliability function of the structure ring flange.</p> <p>a<sub>o</sub>: The original constant of the reliability function</p> <p>X<sub>i</sub>: Reliability coefficient factor</p> <p>a<sub>i</sub>: The A value of the reliability coefficient factor</p> <p>Ɛ: The standard error of the reliability coefficient factor</p> <p>By using the values of the table of the reliability coefficient factor, the reliability function f(X<sub>i</sub>) of the ring flange is written as follows:</p> <p><bold>f(X<sub>i</sub>)<italic>Rf</italic></bold> = -1.530 + 4x10<sup>-3</sup><bold>Ft</bold> -2x10<sup>-3</sup> <bold>T</bold>+10<sup>-3</sup><bold>Nh</bold> - 2x10<sup>-3</sup><bold>Dh</bold>+11x10<sup>-2</sup> <bold>Sc</bold> +14x10<sup>-2</sup>.</p> <p>Finally, we have only the following coefficients:</p> <p>Ft: the fluid, which is in most cases steam or water, which flows on the pipe.</p> <p>T: Thickness of the manufacturing plate</p> <p>Nh and Dh: the number of holes and the Diameter of hols are two parameters of Geometry surface(Gs),</p> <p>Sc: Security Constraint depend of the material of flanges.</p> </sec> <sec id="sec2-11"> <title>Coefficients of the Quadra-Sector Flange</title> <p>After using the database, in the statistical method, by the alimentation of the data base elements for the Quadra-sector flange, we present the following table.</p> <p>In this table we present the factors related to the reliability of the Quadra-sector flange structure:</p> <table-wrap id="tab5"> <label>Table 5</label> <caption> <p><bold>Reliability Coefficients of the Quadra-Sector Flange</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle">Model</th> <th align="center" valign="middle" colspan="2">Non Standardized Coefficients</th> <th align="center" valign="middle">Standardized Coefficients</th> </tr> <tr> <td align="center" valign="middle"> </td> <th align="center" valign="middle">A</th> <th align="center" valign="middle">Standard Error(Ɛ)</th> <th align="center" valign="middle">Bêta</th> </tr> </thead> <tbody> <tr> <td align="left" valign="middle">Constant</td> <td align="center" valign="middle">-20,097</td> <td align="center" valign="middle">10,510</td> <td align="center" valign="middle"> </td> </tr> <tr> <td align="left" valign="middle">Pressure</td> <td align="center" valign="middle">0,001</td> <td align="center" valign="middle">0,030</td> <td align="center" valign="middle">0,017</td> </tr> <tr> <td align="left" valign="middle">Flow temperature</td> <td align="center" valign="middle">0,116</td> <td align="center" valign="middle">0,054</td> <td align="center" valign="middle">0,369</td> </tr> <tr> <td align="left" valign="middle">Inside diameter</td> <td align="center" valign="middle">-0,002</td> <td align="center" valign="middle">0,005</td> <td align="center" valign="middle">-4,392</td> </tr> <tr> <td align="left" valign="middle">Outside diameter</td> <td align="center" valign="middle">-0,001</td> <td align="center" valign="middle">0,005</td> <td align="center" valign="middle">-1,166</td> </tr> <tr> <td align="left" valign="middle">Thickness</td> <td align="center" valign="middle">-0,030</td> <td align="center" valign="middle">0,017</td> <td align="center" valign="middle">-2,107</td> </tr> <tr> <td align="left" valign="middle">Number of holes</td> <td align="center" valign="middle">0,000</td> <td align="center" valign="middle">0,011</td> <td align="center" valign="middle">-0,006</td> </tr> <tr> <td align="left" valign="middle">Diameter of the holes</td> <td align="center" valign="middle">-0,007</td> <td align="center" valign="middle">0,028</td> <td align="center" valign="middle">-0,274</td> </tr> <tr> <td align="left" valign="middle">Weight</td> <td align="center" valign="middle">0,003</td> <td align="center" valign="middle">0,002</td> <td align="center" valign="middle">2,730</td> </tr> <tr> <td align="left" valign="middle">Security Constraint</td> <td align="center" valign="middle">0,128</td> <td align="center" valign="middle">0,081</td> <td align="center" valign="middle">5,194</td> </tr> </tbody> </table> </table-wrap> <p>The factors related to the reliability are appointed reliability coefficients of the metallic Quadra-sector flange, which are:</p> <list list-type="bullet"> <list-item><p><bold>P</bold>: Pressure</p></list-item> <list-item><p><bold>Ft</bold>: flow time</p></list-item> <list-item><p><bold>Id</bold>: Internal diameter</p></list-item> <list-item><p><bold>Od</bold>: Outside diameter</p></list-item> <list-item><p><bold>T</bold>: Thickness</p></list-item> <list-item><p><bold>Dh</bold>: Diameter of the holes</p></list-item> <list-item><p><bold>W</bold>: Weight</p></list-item> <list-item><p><bold>Sc</bold>: Security Constraint.</p></list-item> </list> </sec> <sec id="sec2-12"> <title>Mathematical Model of the Reliability of the Quadra-Sector Flange</title> <p>We written the mathematical model of the reliability function f(X<sub>i</sub>), by using the following linear regression function:</p> <p><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="images/eqn2.png"/></p> <p>Where:</p> <p><bold>f(X<sub>i</sub>)Qsf:</bold> Reliability function of the structure Quadra-sector flange.</p> <p><bold>a<sub>o</sub>:</bold> Original constant of the reliability function</p> <p><bold>X<sub>i</sub>:</bold> Reliability coefficient factor</p> <p><bold>a<sub>i</sub>:</bold> Value A of the reliability coefficient factor</p> <p><bold>Ɛ:</bold> Standard error of the reliability coefficient factor</p> <p>By using the values of the table of the reliability coefficient factor, the reliability function Y2 of the Quadra-sector flange is written as follows:</p> <p>f(X<sub>i</sub>)Qsf = -20.097+10<sup>-3</sup> <bold>P</bold> +116x10<sup>-3</sup><bold>Ft</bold>-2x10<sup>-3</sup><bold>Id</bold> -10-3 <bold>Od</bold> -30x10<sup>-2</sup> <bold>T</bold>-7x10<sup>-3</sup><bold>Dh+</bold>3x10<sup>-3</sup><bold>W+</bold>128x10<sup>-3</sup><bold>Sc+</bold>222 x 10<sup>-3</sup></p> <p>Finally, we have only the following coefficients:</p> <p><bold>P:</bold> Pressure (operating pressure).</p> <p><bold>Ft:</bold> the fluid, which is in most cases steam or water, which flows on pipe.</p> <p><bold>T:</bold> Thickness of the manufacturing plate</p> <p><bold>Id and Ed, Dh:</bold> Internal and external diameter, Dimeter of hols, all are parameters of the Geometry surface (Gs)</p> <p><bold>W:</bold> Weight of the structure</p> <p><bold>Sc:</bold> Security Constraint depend of the material of flanges.</p> </sec> <sec id="sec2-13"> <title>Results the Firefly Algorithmic Optimization</title> <p>The writing, the programming and the compilation of the algorithm Firefly, was based on the results of the two methods, the mechanical and the statistical methods, we used the mathematical model for each flange structure 5.</p> <p>The numerical analysis laboratory used the factors and parameters of our reliability analysis of the two flanges, the objective is to compile the algorithm and to visualize the results after many iterations.</p> <p>After 1000 iterations, we got the result of the ring flange, the homogenous flange, without welding, cut and processed according to the normal manufacturing process, the result is the reliability factor value: 1.850.</p> <p>With the same method we got the result of the Quadra-sector flange, the flange welded into four sectors, cut and processed according to a new manufacturing process, the result is the reliability factor value: 1.789</p> <p>The following table presents the results of the algorithmic optimization method<sup><bold>2, 3,4</bold>.</sup></p> <table-wrap id="tab6"> <label>Table 6</label> <caption> <p><bold>Reliability Factor</bold></p> </caption> <table frame="box" rules="all"> <thead> <tr> <th align="center" valign="middle">Type of Flange</th> <th align="center" valign="middle">Reliability Factor</th> <th align="center" valign="middle">Manufacturing Process</th> </tr> </thead> <tbody> <tr> <td align="left" valign="middle">Ring flange</td> <td align="center" valign="middle">1.850</td> <td align="left" valign="middle">Standard Process</td> </tr> <tr> <td align="left" valign="middle">Quadra-sector flange</td> <td align="center" valign="middle">1.789</td> <td align="left" valign="middle">New Process</td> </tr> </tbody> </table> </table-wrap> <p>After the analysis and the algorithmic method firefly of the two flanges models, we observe that the value of the reliability factor of the two flanges had the approximate value with a negligible value of 0.061 between two factors.</p> </sec></sec> <sec id="sec1-5" sec-type="conclusions"> <title>CONCLUSIONS</title> <p>The using of the mechanical, statistical and algorithmic optimization methods, help us to prove that reliability of the Quadra-sector flange is approximately the same of the reliability of the ring flange structure, then the manufacturing process of the new Quadra-sector flange is the optimal one, because we use a less material to manufacture the flange [<xref ref-type="bibr" rid="ref5">5</xref>], we have ensured the satisfaction of the customer by the reliability factor and the comparison of the results of the two metallic flanges.</p> <p>The manufacturing process of these two rings and Quadra-sector flanges is different, this different located on the steps of their cutting and welding. The manufacturing of the new Quadr-sector flange gives us a better benefit on raw material, also the optimal time and process of manufacturing, but the production of the ring flange provides scrap material and a long process of manufacturing<sup><bold>5</bold></sup>.</p> <p>This study of the ring and Quadra-sector flanges, was very important to prove the difference and the benefit to our customer, and the use of the three methods are appropriate with the two models of flanges, the three methods also are complementary to achieve the results.</p> <p>For this new manufacturing process, we have a new Quadra-sector flange, optimized physically in raw materials, proved by the using the mechanical and statistical methods, the firefly algorithmic optimization method.</p> </sec> </body> <back> <ref-list> <title>REFERENCES</title> <ref id="ref1"> <label>1.</label> <element-citation publication-type="journal"> <person-group person-group-type="author"> <name> <surname>Hachimi</surname> <given-names>H.</given-names> </name> <name> <surname>Assif</surname> <given-names>S.</given-names> </name> <name> <surname>Aoues</surname> <given-names>Y.</given-names> </name> <name> <surname>El hami</surname> <given-names>A.</given-names> </name> <name> <surname>Ellaia</surname> <given-names>R.</given-names> </name> <name> <surname>Agouzoul</surname> <given-names>M.</given-names> </name> </person-group> <article-title>Optimization of the Solder Joints of an Electronic Card Using Heuristic Algorithm</article-title> <source>International Journal of Engineering Research in Africa ISSN: 1663-4144</source> <volume>30</volume> <fpage>39</fpage> <lpage>48</lpage> </element-citation> </ref> <ref id="ref2"> <label>2.</label> <element-citation publication-type="conf-proc"> <person-group person-group-type="author"> <name> <surname>Hachimi</surname> <given-names>H.</given-names> </name> <name> <surname>Makhloufi A</surname> <given-names>A.</given-names> </name> <name> <surname>Ellaia</surname> <given-names>R</given-names> </name> <name> <surname>Elhami</surname> <given-names>A.</given-names> </name> </person-group> <article-title>Optimization of engineering structures by heuristic algorithms</article-title> <conf-name>5 th IESM Conference</conf-name> <conf-date>October 2013</conf-date> <conf-loc>Rabat, Morocco</conf-loc> <comment> I 4 e 2 2013.</comment> </element-citation> </ref> <ref id="ref3"> <label>3.</label> <element-citation publication-type="journal"> <person-group person-group-type="author"> <name> <surname>Assif</surname> <given-names>S.</given-names> </name> <name> <surname>Hachimi</surname> <given-names>H.</given-names> </name> <name> <surname>Agouzoul</surname> <given-names>M.</given-names> </name> <name> <surname>Ellaia R</surname> <given-names>R.</given-names> </name> <name> <surname>El Hami</surname> <given-names>A.</given-names> </name> <name> <surname>Aoues Y</surname> <given-names>Y.</given-names> </name> </person-group> <article-title>Optimization for an electronic cardwith a new hybrid method</article-title> <source>Advanced Materials Research</source> <volume>682</volume> <year>2013</year> <fpage>143</fpage> <lpage>151</lpage> </element-citation> </ref> <ref id="ref4"> <label>4.</label> <element-citation publication-type="journal"> <person-group person-group-type="author"> <name> <surname>Elkhechafi</surname> <given-names>M.</given-names> </name> <name> <surname>Hachimi</surname> <given-names>H.</given-names> </name> <name> <surname>Amine, A</surname> <given-names>H.</given-names> </name> <name> <surname>ElKettani</surname> <given-names>Y.</given-names> </name> </person-group> <year>2017</year> <article-title>Construction of a Novel Water Quality Classifier: A case study in the Sebou region.</article-title> <source>International Journal of Applied Mathematics and Statistics ™</source> <volume>56</volume> <issue>5</issue> <fpage>141</fpage> <lpage>158</lpage> </element-citation> </ref> <ref id="ref5"> <label>5.</label> <element-citation publication-type="journal"> <person-group person-group-type="author"> <name> <surname>M’haouache,</surname> <given-names>N,</given-names> </name> <name> <surname>Hachimi</surname> <given-names>H.</given-names> </name> <name> <surname>Assif</surname> <given-names>S.</given-names> </name> <name> <surname>Belghiti</surname> <given-names>MT.</given-names> </name> </person-group> <article-title>Modeling and Evaluating the Stress of Two Mechanical Structures in Piping Industry -The Ring Flange and the new Quadra-Sector Flange</article-title> <source>International Journal of Engineering and future technology</source> <year>2017</year> <volume>14</volume> <issue>2</issue> </element-citation> </ref> <ref id="ref6"> <label>6.</label> <element-citation publication-type="book"> <person-group person-group-type="author"> <name> <surname>Yang</surname> <given-names>Xin-She</given-names> </name> </person-group> <article-title>Cuckoo search and firefly algorithm. 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