<p>Nanofluids were proposed in the early 1990s and have gone through enormous growth from theoretical ideas to realistic technologies in numerous fields, including cooling systems and renewable energies, automobiles, and electronics, where they are complemented by improved thermal conductivity. This research investigates the three-dimensional boundary layer flow of a nanofluid over a stretching/shrinking sheet, aiming to enhance heat transfer efficiency. We have developed a mathematical model of the proposed problem dealing with continuity, momentum, heat, and concentration. A suitable similarity transformation has changed the system of partial differential equations (PDEs) into the system of ordinary differential equations (ODEs). The nonlinear ODEs are solved by one of the most powerful solvers in MATLAB known as bvp4c. Bvp4c proved to be effective when dealing with the ODEs. Along with bvp4c, we used grey relational analysis (GRA), for the optimization of heat transfer. The utilization of GRA for heat enhancement in this specific context is rather peculiar as it provides a detailed analysis of the enhancement of heat transfer with biaxial stretching sheets. The effects of stretching/shrinking parameter <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\((\lambda )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, suction/injection parameter <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\((S)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, Brownian <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\({N}_{b}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>b</mi> </msub> </math></EquationSource> </InlineEquation>, Thermophoresis (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\({N}_{t}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> and nusselt number <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">\((Nux(0))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi>u</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are produced. The present study revealed that the maximum nessult number emerges when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">\(0.5 \le \lambda \le 1.0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0.5</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1.0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq7.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(-2 \le S \le 2.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>2</mn> <mo>≤</mo> <mi>S</mi> <mo>≤</mo> <mn>2</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> For increased suction parameter S ∊ {0, 0.3, 0.6, 0.9}; there is a decrease in temperature and concentration fields. GRA identifies that the suction parameter is the most critical factor when it comes to heat transfer efficiency, contributing <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="41939_2024_712_Article_IEq8.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(82.4\%\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>82.4</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> in changing the multi-performance characteristics. To conclude, this work emphasizes how the suction parameter, optimized using bvp4c and Grey Relational Analysis, has a crucial impact on the efficiency of heat transfer in nanofluid boundary layer flows.</p>