<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.2 20190208//EN"
  "https://jats.nlm.nih.gov/publishing/1.2/JATS-journalpublishing1.dtd">
<article xmlns:xlink="http://www.w3.org/1999/xlink"
         xmlns:mml="http://www.w3.org/1998/Math/MathML"
         article-type="research-article"
         dtd-version="1.2">

  <!-- ============================================================ FRONT -->
  <front>
    <journal-meta>
      <journal-id journal-id-type="publisher-id">IJLTEMAS</journal-id>
      <journal-title-group>
        <journal-title>International Journal of Latest Technology in Engineering, Management &amp; Applied Science (IJLTEMAS)</journal-title>
        <abbrev-journal-title abbrev-type="publisher">IJLTEMAS</abbrev-journal-title>
      </journal-title-group>
      <issn pub-type="epub">2278-2540</issn>
      <publisher>
        <publisher-name>IJLTEMAS</publisher-name>
      </publisher>
    </journal-meta>

    <article-meta>
      <!-- IDs -->
      <article-id pub-id-type="publisher-id">219</article-id>
            <article-id pub-id-type="doi">10.51583/IJLTEMAS.2026.150800027</article-id>
      
      <!-- Categories -->
            <article-categories>
        <subj-group subj-group-type="heading">
          <subject>Hybrid Axial</subject>
        </subj-group>
      </article-categories>
      
      <!-- Title -->
      <title-group>
        <article-title>Hybrid Axial Distances in Spherical Region of Central Composite Designs (CCDs)</article-title>
      </title-group>

      <!-- Authors -->
      <contrib-group>
                <contrib contrib-type="author">
                    <name>
            <surname>Gozie Chukwunenye</surname>
            <given-names>Victor</given-names>
          </name>
                              <aff>
            Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria                        <country>Nigeria</country>
                      </aff>
                    
        </contrib>
                <contrib contrib-type="author">
                    <name>
            <surname>Iheanyi Onyeagu</surname>
            <given-names>Sidney</given-names>
          </name>
                              <aff>
            Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria                        <country>Nigeria</country>
                      </aff>
                    
        </contrib>
                <contrib contrib-type="author">
                    <name>
            <surname>Chukwuemeka Eze</surname>
            <given-names>Francis</given-names>
          </name>
                              <aff>
            Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria                        <country>Nigeria</country>
                      </aff>
                    
        </contrib>
                <contrib contrib-type="author">
                    <name>
            <surname>Ifeanyi Onyishi</surname>
            <given-names>Linus</given-names>
          </name>
                              <aff>
            Department of Mathematics and Statistics, Federal Polytechnic, Nasarawa State, Nigeria                        <country>Nigeria</country>
                      </aff>
                    
        </contrib>
              </contrib-group>

      <!-- Volume / Issue / Pages -->
            <volume>15</volume>
                  <issue>8</issue>
                        <fpage>387</fpage>
            <lpage>401</lpage>
            
      <!-- Dates -->
      <history>
                <date date-type="received">
          <day>19</day>
          <month>08</month>
          <year>2026</year>
        </date>
                        <date date-type="accepted">
          <day>24</day>
          <month>08</month>
          <year>2026</year>
        </date>
              </history>

            <pub-date pub-type="epub">
        <day>05</day>
        <month>09</month>
        <year>2026</year>
      </pub-date>
      
      <!-- DOI Self-URI -->
            <self-uri xlink:href="https://doi.org/10.51583/IJLTEMAS.2026.150800027"/>
      
      <!-- Keywords -->
            <kwd-group kwd-group-type="author">
                <kwd>Hybrid Axial</kwd>
                <kwd>Distances</kwd>
                <kwd>Spherical Region</kwd>
              </kwd-group>
      
    </article-meta>
  </front>

  <!-- ============================================================ BODY (Abstract) -->
  <body>
        <sec>
      <title>Abstract</title>
      <p>Response Surface Methodology (RSM) relies heavily on Central Composite Designs (CCDs) for modeling, analysis, and optimization of complex multivariate systems. However, the choice of a suitable axial distance (alpha) continues to pose a major challenge in research. Existing classical spherical and rotatable axial distances most of the time push the star points into extreme operational regions that are not feasible, as the number of factors (k) increases, while practical and cuboidal alphas are heavily affected by variance inflation. Mean–based alternative axial distances (arithmetic, harmonic, and geometric) address boundary constraints but are highly sensitive to extreme values, thereby leading to boundaries that are unconstrained as well as spikes in the scaled prediction variance (SPV). To mitigate this gap, this study proposed a new class of variance–based hybrid axial distances for the CCDs in spherical regions. Variability across classical alphas was accounted for by the proposed variance–based alphas, so as to control bias of extreme values; this was done by directly factoring in dispersion instead of raw averages. The new set of variance–based alphas was constructed using factor spaces ranging from k=2 to k=10 and evaluated against existing axial distances using D–optimality, which is a numeric/alphabetic criterion, as well as Variance Dispersion Graphs (VDGs) and Fraction of Design Space Graphs (FDSGs). The study found that for higher–dimensional design spaces (k≥8), the proposed variance–based alphas were effective in preventing boundary SPV inflation, maintaining stable, low prediction variance across more than 90 percent of the volume of the design space. Furthermore, the well–conditioned information matrices produced competitive D–efficiencies compared to the practical and rotatable alphas. In conclusion, the proposed variance–based alphas provide a mathematically sound, operationally feasible alternative for high–dimensional RSM experiments in the spherical regions.</p>
    </sec>
      </body>

  <!-- ============================================================ BACK (References) -->
    <back>
    <ref-list>
      <title>References</title>
            <ref id="ref1">
        <label>1</label>
        <mixed-citation>Alanazi, K., Georgiou, S. D. and Stylianou, S. (2023). On the robustness of central composite designs when missing two observations. Quality and Reliability Engineering International, 39(4), 1143-1171. https://doi.org/10.1002/qre.3281</mixed-citation>
      </ref>
            <ref id="ref2">
        <label>2</label>
        <mixed-citation>Anderson, M. J. and Whitcomb, P. J. (2016). RSM simplified: optimizing processes using response surface methods for design of experiments. Productivity press. https://doi.org/10.1201/9781315382326</mixed-citation>
      </ref>
            <ref id="ref3">
        <label>3</label>
        <mixed-citation>Bartholomew, D. C., Obite, C. P. and Joan, I. C. (2021). Scaled Prediction Variances of Equiradial Design under Changing Design Sizes, Axial Distances and Center Runs. Asian Journal of Probability and Statistics 11(1): 1-13.</mixed-citation>
      </ref>
            <ref id="ref4">
        <label>4</label>
        <mixed-citation>Bhattacharya, S. (2021). Central composite design for response surface methodology and its application in pharmacy. In Response surface methodology in engineering science. IntechOpen. DOI: 10.5772/intechopen.95835</mixed-citation>
      </ref>
            <ref id="ref5">
        <label>5</label>
        <mixed-citation>Box, G. E. and Wilson, K. B. (1992). On the experimental attainment of optimum conditions. In Breakthroughs in statistics: methodology and distribution (pp. 270-310). New York, NY: Springer New York.</mixed-citation>
      </ref>
            <ref id="ref6">
        <label>6</label>
        <mixed-citation>Lamidi, S., Olaleye, N., Bankole, Y., Obalola, A., Aribike, E. and Adigun, I. (2022). Applications of response surface methodology (RSM) in product design, development, and process optimization. IntechOpen.</mixed-citation>
      </ref>
            <ref id="ref7">
        <label>7</label>
        <mixed-citation>Li, J., Li, L., Borror, C. M., Anderson-Cook, C. and Montgomery, D. C. (2009). Graphical summaries to compare prediction variance performance for variations of the central composite design for 6 to 10 factors. Quality Technology &amp; Quantitative Management, 6(4), 433-449. https://doi.org/10.1080/16843703.2009.11673209</mixed-citation>
      </ref>
            <ref id="ref8">
        <label>8</label>
        <mixed-citation>Montgomery, D. C. (2017). Design and analysis of experiments. John wiley &amp; sons.</mixed-citation>
      </ref>
            <ref id="ref9">
        <label>9</label>
        <mixed-citation>Myers, R. H., Montgomery, D. C. and Anderson-Cook, C. M. (2016). Response surface methodology: process and product optimization using designed experiments. John Wiley &amp; Sons.</mixed-citation>
      </ref>
            <ref id="ref10">
        <label>10</label>
        <mixed-citation>Onwuamaeze, C. U. (2021). Optimal prediction variance properties of some central composite designs in the hypercube. Communications in Statistics-Theory and Methods, 50(8), 1911-1924. https://doi.org/10.1080/03610926.2019.1656746</mixed-citation>
      </ref>
            <ref id="ref11">
        <label>11</label>
        <mixed-citation>Onyishi, L. I. and Eze, F. C. (2020). Alternative Axial Distances for Spherical Regions of Central Composite Designs. Earthline Journal of Mathematical Sciences, 4(2), 273-285.</mixed-citation>
      </ref>
            <ref id="ref12">
        <label>12</label>
        <mixed-citation>Yakubu, Y., Chukwu, A. U., Adebayo, B. T. and Nwanzo, A. G. (2015). Effects of missing observations on predictive capability of central composite designs. arXiv preprint arXiv:1603.09716. https://doi.org/10.48550/arXiv.1603.09716</mixed-citation>
      </ref>
          </ref-list>
  </back>
  
</article>
