Optimization of heat transfer in a channel with stretching walls using a magnetized tetra-hybrid nanofluid

Abstract

Nanoparticles have numerous applications and are used frequently in different cooling, heating, treatment of cancer cells and manufacturing processes. The current investigation covers the utilization of tetra hybrid nanofluid (aluminium oxide, iron dioxide, titanium dioxide and copper) for Crossflow model over a vertical disk by considering the shape effects (bricks, cylindrical and platelet) of nanoparticles, electro-magneto-hydrodynamic effect and quadratic thermal radiation. The present study is devoted to present the mathematical formulation of the tetra-hybrid nanofluid flow in a porous channel with stretching/shrinking walls. The present inspection model is derived from given partial differential equations (PDEs) and then transformed into a system of ordinary differential equations (ODEs) by incorporating similarity variables. The transformed ODEs are solved using the bvp4c methodology, which yields numerical results. From the obtained results it is observed that the maximum amount of $C{f}_x$ happens when there is slightly increase in $M$ with stretched wall that is $\lambda \in [0,2]$ . A high radiation value indicates a considerable contribution from radiative heat transfer, whereas stretching and shrinking increase and reduce the size of the boundary. The combined impact caused an increase in the Nusselt number. The graphical and tabular representations are used to explain the physical behaviour of various model parameters. Previous outcomes are also contrasted with the current outcomes.

Introduction

Due to the numerous uses for non-Newtonian fluids in the engineering and industrial sectors, scientists and academicians are becoming more and more interested in this study. Casson developed the fluid flow model that includes non-Newtonian liquid flow in 1995. One essential nanofluid that is used in numerous situations is Casson fluid. Human existence’s use of the Casson fluid flow model has attracted a lot of interest late. Casson fluids include a wide range of materials, such as blood, jelly, chili sauce and honey. In modern research, the Casson fluid flow model is quite important. Yield tension features are present in the Casson fluid. A considerable yielding tension is the point at which a Casson fluid becomes a Newtonian fluid. Even though the strain rate is far less than the shear stress, the Casson fluid change. Eldabe et al. ¹ were the first to demonstrate the energy transmission of a consistent magnetohydrodynamic (MHD) non-Newtonian Casson fluid flow between two co-axial tubes. The investigation for this study took several years to develop. Prashu and Nandkeolyar ² introduced a mathematical model that specifically addresses the significant parameters of Casson fluid in practical scenarios. The model simulates the flow of electrically conducting magnetohydrodynamic fluid over a stretching layer, taking into account the combined influence of radiative heat transfer and Hall current. The objective of this model is to provide valuable insights into the power characteristics of unsteady 3D incompressible fluids. Nadeem et al. ³ conducted a study to examine the impact of an incident particle on the motion of Casson fluid in a two-dimensional flow via a porous and circular extended surface. Interesting results were obtained from modifying the Casson flow form in addition to other fluid flow properties. A number of important and relevant investigations have been carried out recently by a number of scholars, including those of Usman et al., ⁴ Soomro et al., ⁵ Mahmood et al., ⁶ Khan et al., ⁷ Waqas et al., ⁸ Pattnaik et al., ⁹ and Abbas et al. ¹⁰

A specific fluid can travel via a narrow or narrow channel with permeable walls, enabling the liquid to enter or exit, in order to mimic the fluid motions observed in living creatures. ¹¹ The suggested model can be used to study how hybrid nanofluids (HNFs) move in different physiological systems, such as the respiratory system, blood circulation, effervescent diaphragms, percolation, blood current, synthetic dialysis, dual fume dissemination, lymphatic system and blood flow. It also has application in the dispensing of drugs. The computational solutions of plasma flow via a stenosed vein were obtained by Omama et al. ¹² The authors also looked at how quickly plasma transfers heat through these veins. These are really important scenarios. As the primary biological filters in human bodies, the kidneys and respiratory system are vital for the movement of numerous chemicals and blood-related components. Substances exit the side end of the tunnel bedsteads during this process, and then they reappear at the intravenous end. Wan Azmi et al. ¹³ looked into the effects of a magnetic field and slip conditions on the passage of nanofluid through a stenotic artery. Given the respiratory system’s critical involvement in the movement of mass, oxygen and other crucial biological constituents, this modelling could prove to be a useful and thorough representation.^14,15 In their study, Fangfang et al. ¹⁶ employed a novel mass-based model to analyse the movement of blood HNF in the direction of an impermeable stretched slip. The numerical solution for blood flow via a slanted vein with a mild stricture was effectively obtained. They also looked at how a hybrid nanofluid affected the blood flow’s thermal performance. Abdelsalam et al. ¹⁷ investigated the impact of electroosmotic pressures on blood flow in sick tissues. Ijaz and Nadeem ¹⁸ have observed that pure blood forms the basis of our concept. Sharma et al. ¹⁹ conducted a study to examine the impact of tilted magnetic force on the flow of blood via a stenosed vein. Whereas thin blood vessels, which exhibit a greater pressure gradient and are more obviously non-Newtonian, broad blood arteries are different and behave as Newtonian fluids. The heat transfer impact of blood flow in a bell-shaped occluded artery was investigated by Gandhi et al., ²⁰ taking into account factors such as viscous dissipation and Joule heating. Shahzadi and Bilal ²¹ conducted a comprehensive analysis on a larger artery form, aiming to predict that blood behaviour will adhere to Newtonian rules.

Applications for ternary hybrid nanofluids (THNFs) in blood flow studies have recently been proposed. Karmakar and Das ²² examined the thermal properties of the cilia-driven transition of distributed blood-based THNFs while considering the impact of a slanted magneto field. In their study, Dolui et al. ²³ investigated the impact of current and rapidly moving slides on blood flow. They employed the THNFs model, which incorporated a combined magnetic field and thermal-based radiation. Sajid et al. ²⁴ investigated the effects of heat radiation and Cross non-Newtonianism liquid type on blood flow via a stenotic artery using a recently developed tetra hybrid nanoliquid prototype. The drug delivery aspect of a blood-based tri-hybridity fluid flowing via a perforated vessel was investigated by Alnahdi et al. ²⁵ The researchers observed that the heat transference rate increases due to an expansion in the thermal radiative effect. In their study, Rathore and Sandeep ²⁶ developed a computational framework to create a hybrid nanofluid that mimics the flow of blood via a narrowed artery. Riaz et al. ²⁷ examined the comprehension of the cilia signal of electrically guiding Cu-blood THNFs by analysing a consistent coiled frequency in the presence of significant entropy formation. Abidi et al. ²⁸ conducted a comprehensive analysis of the benefits of THNFs, specifically focusing on their impact on blood-base flow. Recent research publications^29–35 provide significant insights into the blood flow characteristics of hybrid nanofluids across different geometries.

Raza et al.^36,37 investigated the problem of fluid flow in a channel with stretching or shrinking walls. In Raza et al., ³⁶ they considered the problem of nanofluid flow in a semi-porous channel with stretching walls and the problem of casson fluid flow in a channel with shrinking and stationary walls is considered in Raza et al. ³⁷ On the other hand, the present study is dealing with tetra-hybrid nanofluid with stretching and porous walls. Moreover, in order to find the best combination of the physical parameters to enhance the heat transfer rate, we employ Fuzzy optimization called Fuzzy TOPSIS. Therefore, present study is the generalization of the previously published results and never been published before.

Since the non-Newtonian fluid model can faithfully depict the shear thickening and thinning events found in human blood, it is the model of choice. The unique features of the present study are:

• ➢  This study focuses on the mathematical model that describes the fluid flow containing four different kinds of nanoparticles in a porous channel with stretching/shrinking walls. The model incorporates the interaction between two parallel plates and a Tetra-hybrid nanofluid consisting of gold, silver, aluminium and titanium nanoparticles.

• ➢  This study also examines the influence of suction and injection walls.

• ➢  The highly nonlinear system of the derived model is solved numerically by use of the bvp4c method.

• ➢  The examination of fuzzy TOPSIS is conducted.

• ➢  The primary objective is to find the best combination of the physical parameter to enhance the heat transfer rate by Fuzzy TOPSIS.

This work presents novel applications in the areas of chemical processes, biological applications, human endoscopy, thermal systems, engineering processes, extrusion systems and heating phenomenon control.^38–42

Problem formulation

Let’s consider the following assumptions in order to develop the mathematical model.

• The fluid considered is a steady, incompressible Tetra-Hybrid nanofluid (TTHNF) between two parallel plates.

• The lower wall is located at $y=0$ , and the upper wall is located at $y=a$ .

• The lower wall is permeable, allowing fluid to be sucked or injected into it.

• The lower wall can be stretched at a constant rate $\mathrm{cx}$ in the direction of the flow ( $x$ -axis).

• The temperature at the lower wall is $T_o$ .

• The upper wall is shrinking, impermeable and has a temperature of $T_1$ .

• A magnetic field is applied at the lower wall in the direction of the $y$ -axis, opposite to the flow direction.

• Heat source/sink effects are considered, accounting for possible temperature increases or decreases.

• A comprehensive model of single-phase nanofluid is used for the special case of TTHNF.

After the consideration of above assumptions, the governing equations can be written as ³⁶ :

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{{\rho }_{\mathrm{ttnf}}}\frac{\partial p}{\partial x}+\frac{{\mu }_{\mathrm{ttnf}}}{{\rho }_{\mathrm{ttnf}}}(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2})-\frac{{\sigma }_{\mathrm{ttnf}}{B}_{°}^2}{{\rho }_{\mathrm{ttnf}}}u$$

(2)

$$u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{{\rho }_{\mathrm{ttnf}}}\frac{\partial p}{\partial y}+\frac{{\mu }_{\mathrm{ttnf}}}{{\rho }_{\mathrm{ttnf}}}(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2})$$

(3)

$$\begin{array}{c}\multicolumn{1}{c}{u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{\mathrm{ttnf}}}{{\left(\rho \mathrm{Cp}\right)}_{\mathrm{ttnf}}}(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2})-\frac{1}{{\left(\rho C_p\right)}_{\mathrm{ttnf}}}\frac{\partial q_r}{\partial y}}\\ +\frac{Q_{°}}{{\left(\rho C_p\right)}_{\mathrm{ttnf}}}(T-T_1)\hfill \end{array}$$

(4)

In the above equations, u, v are the components of the velocity vectors, T is the temperature, $q_r$ is the thermal radiation, $Q_{°}$ is the coefficient of heat produce/absorption, ${\mu }_{\mathrm{ttnf}}$ is dynamic viscosity, ${\rho }_{\mathrm{ttnf}}$ density, ${\sigma }_{\mathrm{ttnf}}$ electric conductivity, ${\left(\rho \mathrm{Cp}\right)}_{\mathrm{ttnf}}$ is heat capacitance, $k_{\mathrm{ttnf}}$ is the thermal conductivity of the tetra-hybrid nanoparticles (TTHNP).

Subject to the boundary conditions:

$$u\left(\eta \right)=\mathrm{cx},v\left(\eta \right)=-v_o,T\left(\eta \right)=T_1,(\eta =\frac{y}{a})\mathrm{at}y=0$$

(5)

$$u\left(\eta \right)=-\mathrm{cx},v\left(\eta \right)=0,T\left(\eta \right)=T_2,(\eta =\frac{y}{a})\mathrm{at}y=a$$

(6)

The thermos-physical correlations of TTHNF ⁴³ are given by:

$${\mu }_{\mathrm{ttnf}}=\frac{{\mu }_f}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}{(1-{\phi }_3)}^{2.5}{(1-{\phi }_4)}^{2.5}},$$

(7)

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_{\mathrm{ttnf}}=[(1-{\phi }_1)\{(1-{\phi }_2)(1-{\phi }_3)[(1-{\phi }_4){\phi }_f+{\rho }_4{\phi }_4]}\\ +{\rho }_3{\phi }_3+{\phi }_2{\rho }_2\}+{\rho }_1{\phi }_1]\hfill \end{array}$$

(8)

$$\begin{array}{c}\multicolumn{1}{c}{{\left(\rho C_p\right)}_{\mathrm{ttnf}}=[(1-{\phi }_1)\{(1-{\phi }_2)(1-{\phi }_3)[(1-{\phi }_4){\left(\rho C_p\right)}_f+{\left(\rho C_p\right)}_{s_4}{\phi }_4]}\\ +{\left(\rho C_p\right)}_{s_3}{\phi }_3+{\phi }_2{\left(\rho C_p\right)}_{s_2}\}+{\left(\rho C_p\right)}_{s_1}{\phi }_1]\hfill \end{array}$$

(9)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{k_{\mathrm{ttnf}}}{k_{\mathrm{tnf}}}}& =\frac{(k_4+2k_{\mathrm{tnf}}-2{\phi }_4(k_{\mathrm{tnf}}-k_4))}{k_4+2k_{\mathrm{tnf}}+{\phi }_4(k_{\mathrm{tnf}}-k_4)},\hfill \\ \multicolumn{1}{c}{\frac{k_{\mathrm{tnf}}}{k_{\mathrm{hnf}}}}& =\frac{(k_3+2k_{\mathrm{hnf}}-2{\phi }_3(k_{\mathrm{hnf}}-k_3))}{k_3+2k_{\mathrm{hnf}}+{\phi }_3(k_{\mathrm{hnf}}-k_3)},\hfill \\ \multicolumn{1}{c}{\frac{k_{\mathrm{hnf}}}{k_{\mathrm{nf}}}}& =\frac{(k_2+2k_{\mathrm{nf}}-2{\phi }_2(k_{\mathrm{nf}}-k_2))}{k_2+2k_{\mathrm{nf}}+{\phi }_2(k_{\mathrm{nf}}-k_2)},\hfill \\ \multicolumn{1}{c}{\frac{k_{\mathrm{nf}}}{k_f}}& =\frac{(k_1+2k_f-2{\phi }_1(k_f-k_1))}{k_1+2k_f+{\phi }_1(k_f-k_1)}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{ttnf}}}{{\sigma }_{\mathrm{tnf}}}}& =\frac{((1-2{\phi }_4){\sigma }_4+(1-2{\phi }_4){\sigma }_{\mathrm{tnf}})}{(1-{\phi }_4){\sigma }_4+(1+{\phi }_4){\sigma }_{\mathrm{tnf}}},\hfill \\ \multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{tnf}}}{{\sigma }_{\mathrm{hnf}}}}& =\frac{((1-2{\phi }_3){\sigma }_3+(1-2{\phi }_3){\sigma }_{\mathrm{hnf}})}{(1-{\phi }_3){\sigma }_3+(1+{\phi }_3){\sigma }_{\mathrm{hnf}}},\hfill \\ \multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{hnf}}}{{\sigma }_{\mathrm{nf}}}}& =\frac{((1-2{\phi }_2){\sigma }_2+(1-2{\phi }_2){\sigma }_{\mathrm{nf}})}{(1-{\phi }_2){\sigma }_2+(1+{\phi }_2){\sigma }_{\mathrm{nf}}},\hfill \\ \multicolumn{1}{c}{\frac{{\sigma }_{\mathrm{nf}}}{{\sigma }_f}}& =\frac{((1-2{\phi }_1){\sigma }_1+(1-2{\phi }_1){\sigma }_f)}{(1-{\phi }_1){\sigma }_1+(1+{\phi }_1){\sigma }_f}\hfill \end{array}$$

(11)

Thermophysical properties of gold, TiO2, Ag, Al2O3 and base fluid can be depicted from Table 1.

Table 1.

Thermophysical properties of TTHNP.

Properties	$\mathrm{Gold}$	$\mathrm{Ti}{O}_2$	$\mathrm{Ag}$	$A{l}_2{O}_3$	$\mathrm{Blood}$
$\rho$	19,300	4250	10,500	3970	1050
$C_p$	129	690	235	765	3617
$k$	310	8953	429	40	0.52
$\sigma$	$4.1\times {10}^6$	$2.4\times {10}^6$	$6.3\times {10}^7$	$3.5\times {10}^7$	0.668

For radiative heat flux, we employ the Rosseland approximation as:

$$q_r=-\frac{4{\sigma }^{*}}{3k^{*}}\frac{\partial T^4}{\partial y},$$

(12)

Here, ${\sigma }^{*}$ and $k^{*}$ are Stefan Boltzmann constant and mean absorption coefficient. Expand $T^4$ by Taylor series approximation, we get:

$$T^4\approx -3T_{\infty }^4+4T_{\infty }^{3}T.$$

(13)

Similarity Solution:

Now, we introduce following similarity transformation ³⁶ on equations (1)–(6). We get:

$$u=\mathrm{bx}{f}^{\prime }\left(\eta \right),v=-\mathrm{abf}\left(\eta \right),\theta \left(\eta \right)=\frac{(T-T_2)}{(T_1-T_2)},\eta =\frac{y}{a}$$

(14)

Once, we incorporated the equation (12) into the equations (1)–(4), the law of conservation of mass (equation (1)) satisfies and eliminating the pressure terms, the momentum equation and heat equation can be written as:

$$\begin{array}{c}\multicolumn{1}{c}{f^{\mathrm{IV}}\left(\eta \right)-\left(R\stackrel{~}{\frac{A_2}{\widetilde{A_1}}}\right)(f^I\left(\eta \right)f^{\mathrm{II}}\left(\eta \right)-f\left(\eta \right)f^{\mathrm{III}}\left(\eta \right))}\\ -\left(\stackrel{~}{\frac{A_3}{\widetilde{A_1}}}\right)M^2{f}^{\mathrm{II}}\left(\eta \right)=0\hfill \end{array}$$

(15)

$$(\widetilde{A_4}+\frac{4}{3}\mathrm{Rd}){\theta }^{\mathrm{II}}\left(\eta \right)+\mathrm{Pr}\widetilde{A_5}f\left(\eta \right){\theta }^I\left(\eta \right)+Q^{*}\theta \left(\eta \right)=0$$

(16)

Here, stretching Reynolds number $R=\frac{a^{2}b}{{\upsilon }_f}$ , magnetic parameter $M=\frac{{\sigma }_f{B}_o^2}{{\mu }_f}$ , Prandtl number $\mathrm{Pr}=\left[\frac{a^{2}b}{k_f}{\left(\rho c_p\right)}_f\right]$ , heat source/sink parameter $Q^{*}=\frac{Q_o{a}^2}{k_f}$ and radiation parameter $\mathrm{Rd}=\left(\frac{4{\sigma }^{*}{T}_{\infty }^3}{k^{*}{k}_f}\right)$ . Moreover, ${\tilde{A}}_1,{\tilde{A}}_2,{\tilde{A}}_3,{\tilde{A}}_4,{\tilde{A}}_5$ are given below:

$$\begin{array}{c}\widetilde{A_1}=\frac{1}{{(1-{\phi }_1)}^{2.5}{(1-{\phi }_2)}^{2.5}{(1-{\phi }_3)}^{2.5}{(1-{\phi }_4)}^{2.5}}\hfill \\ \widetilde{A_2}=[(1-{\phi }_1)\left\{\begin{array}{c}(1-{\phi }_2)(1-{\phi }_3)[(1-{\phi }_4)+\frac{{\rho }_4{\phi }_4}{{\phi }_f}]\hfill \\ +\hfill \\ \frac{{\rho }_3{\phi }_3}{{\phi }_f}+\frac{{\phi }_2{\rho }_2}{{\phi }_f}\hfill \end{array}\right\}+\frac{{\rho }_1{\phi }_1}{{\phi }_f}]\hfill \\ \widetilde{A_3}=\frac{{\sigma }_{\mathrm{ttnf}}}{{\sigma }_f}\hfill \\ \widetilde{A_4}=\frac{k_{\mathrm{ttnf}}}{k_f}\hfill \\ \widetilde{A_5}=[(1-{\phi }_1)\left\{\begin{array}{c}(1-{\phi }_2)(1-{\phi }_3)[(1-{\phi }_4)+\frac{{\left(\rho C_p\right)}_{s_4}{\phi }_4}{{\left(\rho C_p\right)}_f}]\hfill \\ +\hfill \\ \frac{{\left(\rho C_p\right)}_{s_3}}{{\left(\rho C_p\right)}_f}{\phi }_3+\frac{{\phi }_2{\left(\rho C_p\right)}_{s_2}}{{\left(\rho C_p\right)}_f}\hfill \end{array}\right\}+\frac{{\left(\rho C_p\right)}_{s_1}}{{\left(\rho C_p\right)}_f}{\phi }_1]\hfill \end{array}\}$$

(17)

The boundary conditions become:

$$\begin{array}{c}f\left(0\right)=S,f^I\left(0\right)=\lambda ,\theta \left(0\right)=1\hfill \\ f\left(1\right)=0,f^I\left(1\right)=-\lambda ,\theta \left(1\right)=0\hfill \end{array}\}$$

(18)

where,

$S=\{\begin{array}{c}S>0:\mathrm{Suction}\hfill \\ S<0:\mathrm{Injection}\hfill \end{array}:\mathrm{is}\mathrm{the}\mathrm{suction}/\mathrm{injection}\mathrm{parameter}$ ,

$\lambda =\{\begin{array}{c}\lambda >0:\mathrm{Stretching}\mathrm{walls}\hfill \\ \lambda <0:\mathrm{Shrinking}\mathrm{walls}\hfill \end{array}$ :is the shrinking/stretching parameter.

Now, we have to solve the similarity equations (15) and (16) subject to the boundary conditions (18) numerically. Since, our governing similarity equations are fourth order nonlinear ODEs therefore we convert them into system of first order ODEs and then solve. For missing initial conditions, we will employ shooting method. For this, we use Matlab built-in routine called bvp4c.

Numerical procedure

We employed bvp4c routine in order to find the numerical solution of equations (15) and (16) subject to the boundary condition 18. The advantage of bvp4c lies in its high accuracy, which is attributed to its use of collocation methods, adaptive meshing, flexible boundary conditions, robustness and automatic error control. For this we reduce the governing equations (15) and (16) into initial value problem by considering:

$$\eta =\widetilde{{\xi }_o},f=\widetilde{{\xi }_1},f^I=\widetilde{{\xi }_2},f^{\mathrm{II}}=\widetilde{{\xi }_3},f^{\mathrm{III}}=\widetilde{{\xi }_4},\theta =\widetilde{{\xi }_5},{\theta }^I=\widetilde{{\xi }_6}$$

Then the following system is obtained:

$$\left[\begin{array}{c}{\tilde{\xi }}_o^{\prime }\\ {\tilde{\xi }}_1^{\prime }\\ {\tilde{\xi }}_2^{\prime }\\ {\tilde{\xi }}_3^{\prime }\\ {\tilde{\xi }}_4^{\prime }\\ \begin{array}{l}{\tilde{\xi }}_5^{\prime }\\ {\tilde{\xi }}_6^{\prime }\end{array}\end{array}\right]=\left[\begin{array}{l}\begin{array}{c}1\\ {\tilde{\xi }}_2\\ {\tilde{\xi }}_3\\ {\tilde{\xi }}_4\\ R\left(\frac{{\tilde{A}}_2}{\tilde{A}{A}_1}\right)\left({\tilde{\xi }}_2{\tilde{\xi }}_3-{\tilde{\xi }}_1{\tilde{\xi }}_4\right)+\left(\frac{{\tilde{A}}_3}{{\tilde{A}}_1}\right)M^2{\tilde{\xi }}_3\\ {\tilde{\xi }}_6\end{array}\\ -\frac{\left(Pr{\tilde{A}}_5{\tilde{\xi }}_1{\tilde{\xi }}_6+Q^{\ast }{\tilde{\xi }}_5\right)}{\left({\tilde{A}}_4+\frac{4}{3}Rd\right)}\end{array}\right]$$

Subject to the initial conditions:

$$\begin{array}{c}\multicolumn{1}{c}{\left[\begin{array}{c}\widetilde{{\xi }_1}\left(0\right)\hfill \\ \multicolumn{1}{c}{\widetilde{{\xi }_2}\left(0\right)}\\ \multicolumn{1}{c}{\widetilde{{\xi }_3}\left(0\right)}\\ \multicolumn{1}{c}{\widetilde{{\xi }_4}\left(0\right)}\\ \multicolumn{1}{c}{\widetilde{{\xi }_5}\left(0\right)}\\ \multicolumn{1}{c}{\widetilde{{\xi }_6}\left(0\right)}\end{array}\right]=\left[\begin{array}{c}S\hfill \\ \multicolumn{1}{c}{\lambda }\\ \multicolumn{1}{c}{{\alpha }_1^{*}}\\ \multicolumn{1}{c}{{\alpha }_2^{*}}\\ \multicolumn{1}{c}{1}\\ \multicolumn{1}{c}{{\alpha }_3^{*}}\end{array}\right]}\end{array}$$

Where $\{{\alpha }_i^{*}:1\le i\le 3\}$ are missing initial conditions and is determined by shooting technique.

Results and discussions

In this section, we present the graphical and tabulation analysis of the numerical results of the current study (Figure 1). Figure 2 elucidates the effect of Reynolds number $R$ on velocity $f^{\prime }\left(\eta \right)$ and temperature profile $\theta \left(\eta \right)$ . From this plot, it is depicted that velocity of the nanofluid near the lower wall $\eta =0$ of the channel decreases as increase in the Reynolds number $R$ and increases near the centre of the channel. It implies that near the lower channel wall the momentum boundary layer thickness declines by the enhancement of the Reynolds number $R$ . Physically, we can say that lower wall is under the influence of suction and stretching effect therefore fluids’ particle offers great reduction and hence there is a great reduction in the velocity of the nanofluid. However, temperature profile $\theta \left(\eta \right)$ increases gradually as increase in the Reynolds number $R$ . The effect of stretching parameter $\lambda$ on velocity $f^{\prime }\left(\eta \right)$ and temperature profile $\theta \left(\eta \right)$ can be depicted from Figure 3. It came to know that velocity profile shifted upwards from lower wall to the centre of the channel and reverse trend is seen from centre to upper wall of the channel. Additionally, the trend of temperature profile $\theta \left(\eta \right)$ is decreasing as increase in the stretching parameter. This shows that the thermal boundary layer thickness reduces by increasing the stretching parameter. The minimal effect of magnetic parameter $M$ on velocity $f^{\prime }\left(\eta \right)$ and temperature profile $\theta \left(\eta \right)$ is seen from Figure 4. From Figure 5, it is noticed that the velocity $f^{\prime }\left(\eta \right)$ and temperature profile decline rapidly as the enhancement in the suction parameter. This means that drag forces and heat transfer rate increase in this case. The effect of radiation parameter and heat source/sink parameter is presented in Figure 6. The temperature profile $\theta \left(\eta \right)$ enhances by increasing the strength of radiation and heat source/sink parameters.

Figure 1.

Schematic configuration of the proposed problem.

Figure 2.

Effect of stretching Reynolds number on velocity and temperature profile.

Figure 3.

Effect of stretching parameter on velocity and temperature profile.

Figure 4.

Effect of magnetic parameter on velocity and temperature profile.

Figure 5.

Effect of suction/injection parameter on velocity and temperature profile.

Figure 6.

Effect of radiation and heat source/sink parameters on temperature profile.

The combine effect of $M,\lambda ,R,S$ on skin friction coefficient $(C{f}_x)$ can be depicted from Figure 7. Figure 7(a) shows combine effect of $M$ and $\lambda$ . Maximum amount of $C{f}_x$ happens when there is slightly increase in $M$ with stretched wall that is $\lambda \in [0,2]$ . Because when wall stretches in the direction of flow, it increases the velocity gradient near it. The increasing velocity gradient results in higher shear stresses and, thus, higher $C{f}_x$ . Figure 7(b) displays the manner in which $R$ and $\lambda$ determine that is to say due to stretching effect $C{f}_x$ increases by increase in $R$ . Figure 7(c) shows combine impact of R and $M$ on $C{f}_x$ . The presence of a magnetic field alters the behaviour of the flow, causing Lorentz forces to impact the velocity distribution and shear stress at the boundary (i.e. skin friction). ⁴⁴ Figure 7(d) dispatch the combined effect of $\lambda$ and $S$ . As $S>0$ and $\lambda >0$ , suction reduces boundary layer thickness, but stretching elongates fluid components, potentially resulting in thinner boundary layers and greater skin friction due to increasing velocity gradients. The combined impact relies on the relative intensities of suction and stretching. Figure 7(e) transmit combined effect of $M$ and $S$ . Whenever there is injection implies decrease in $C{f}_x$ while with the increase in $M$ there is increase in $C{f}_x$ . Figure 7(f) put out combined effect of $R$ and $S$ . High $R$ values enhance the stretching effects, resulting in a more stretched flow field. In this situation, the combination of suction and high stretching can reduce $C{f}_x$ by increasing velocity gradients and transferring momentum. ⁴⁵ Figure 8 demonstrates the way factors: heat source/sink parameter $(Q)$ , shrinking/stretching parameter $(\lambda )$ , radiation parameter $(\mathrm{Rd})$ and suction/injection parameter $(S)$ affect the Nusselt number $(N{u}_x)$ combined. Figure 7(a) shows combine effect of $Q$ and $\lambda .$ Shrinking parameter decreases the boundary dimensions, whilst the heat source heats the system, hence $N{u}_x$ increases. ⁴⁶ Figure 8(b) displays combine effect of $\lambda$ and $\mathrm{Rd}$ . High $\mathrm{Rd}$ indicates a considerable contribution from radiative heat transfer, which raises the heat transfer rate and $N{u}_x$ . Figure 8(c) shows combine impact of $\mathrm{Rd}$ and $Q$ on $N{u}_x.$ As heat source supplies heat to the system, whereas a high radiation value indicates a considerable contribution from radiative heat transfer. The Nusselt number increases depending on the flow arrangement and the relative intensity of these influences. Figure 8(d) transmit the combined effect of $\lambda$ and $S$ . At the same time as $\lambda \in [-2,2]$ and $S\in [-2,2],S>0$ thins the boundary layer by withdrawing fluid near the boundary, whereas $\lambda <0$ lowers the size of the boundary, hence the $N{u}_x$ increases. $S<0$ continues to build the boundary layer, whilst $\lambda >0$ increases its size. When $\lambda >0$ effects prevail, the combined impact causes a rise in the $N{u}_x$ , which promotes mixing and increases the heat transfer area in spite of $S<0$ . Figure 8(e) transmit combined effect of $Q$ and $S$ . The $Q$ provides heat to the system, while stretching and shrinking change the size of the boundary. The combined impact result in a rise in $N{u}_x$ . Figure 8(f) elucidates the combined effect of $\mathrm{Rd}$ and $S$ . A high radiation value indicates a considerable contribution from radiative heat transfer, whereas stretching and shrinking increase and reduce the size of the boundary. The combined impact caused an increase in the Nusselt number.

Figure 7.

(a-f) Combined effects of $M,\lambda ,R,S$ on Skin Friction coefficient.

Figure 8.

(a-f) Combined effects of $Q,\lambda ,\mathrm{Rd},S$ on Nusselt number.

Results of fuzzy TOPSIS

In this step we will demonstrate fuzzy TOPSIS (Table 2).

Table 2.

Validations of the results for velocity and temperature profiles with previous published results.

	$f^{″}(-1)$	${\theta }^{\prime }(-1)$
Raza et al. 36	−3.91075	−0.75900
Present study $(\mathrm{Rd}=0,\lambda =1,S=1,Q^{*}=0,{\phi }_2={\phi }_3=0,R=2)$	−3.91069	−0.75912

We will follow the following steps:

Step 1

According to the response of experiments Case 1, 2 and 3 from Table 4 and using linguistic variables from Table 3 we make Tables 5 to 7. |−θ(0)| remain same according to Experiment 1, Experiment 2 and Experiment 3.

Table 3.

Weight assigned to linguistic variables for each output response.

Importance	Abbreviation	Fuzzy weight
Extremely low	EL	(0, 0, 0.1)
Very low	VL	(0, 0.1, 0.3)
Low	L	(0.1, 0.3, 0.5)
Medium	M	(0.3, 0.5, 0.7)
High	H	(0.5, 0.7, 0.9)
Very high	VH	(0.7, 0.9, 1.0)
Extremely high	EH	(0.9, 1.0, 1.0)

Table 4.

Feedback on output responses.

Number of experiment	1	2	3	Range
Experiment 1
Rd = 0.5	EL	EL	M
Rd = 1	M	H	H
Rd = 2	EH	EH	EH
Experiment 2
Q = 0.5	EL	VL	L	0 <|−θ(0)| <1 = VL
Q = 1	M	L	M	1 <|−θ(0)| <2 = L
Q = 1.5	H	M	H	2 <|−θ(0)| <3 = M
				3 <|−θ(0)| <4 = H
Experiment 3
S = −2	EL	VL	EL	4 <|−θ(0)| <5 = VH
S = 0	L	L	VL	5 <|−θ(0)| <6 = EH
S = 2	EH	VH	VH
Experiment 4
L = 0.5	VL	EL	L
L = 1	M	L	M
L = 1.5	EH	H	EH

Table 5.

Matrix (in the light of Experiment 1).

No.	Rd	Q	S	L	|−θ(0)|
1	(0.3, 0.5, 0.7)	(0.3, 0.5, 0.7)	(0.9, 1.0, 1.0)	(0.3, 0.5, 0.7)	(0.5, 0.7, 0.9)
2	(0, 0, 0.1)	(0.5, 0.7, 0.9)	(0, 0, 0.1)	(0, 0.1, 0.3)	(0, 0.1, 0.3)
3	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
4	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
5	$(0,0,0.1)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)
6	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0,0,0.1)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$
7	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$
8	$(0.9,1.0,1.0)$	$(0,0,0.1)$	$(0,0,0.1)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$
9	$(0.9,1.0,1.0)$	$(0.5,0.7,0.9)$	$(0,0,0.1)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$
10	$(0.9,1.0,1.0)$	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$
11	$(0.9,1.0,1.0)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$
12	$(0,0,0.1)$	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)
13	$(0,0,0.1)$	$(0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
14	$(0,0,0.1)$	$(0,0,0.1)$	$(0,0,0.1)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$
15	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$
16	$(0,0,0.1)$	$(0,0,0.1)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
17	$(0,0,0.1)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$
18	$(0.9,1.0,1.0)$	$(0,0,0.1)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
19	$(0.3,0.5,0.7)$	$(0,0,0.1)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
20	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
21	(0.9, 1.0, 1.0)	$(0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
22	(0.3, 0.5, 0.7)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
23	(0.3, 0.5, 0.7)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
24	(0.3, 0.5, 0.7)	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
25	(0, 0, 0.1)	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$
26	(0.3, 0.5, 0.7)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
27	(0.3, 0.5, 0.7)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
28	(0.3, 0.5, 0.7)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
29	(0, 0, 0.1)	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$

Table 6.

Matrix (in the light of Experiment 2).

No.	Rd	Q	S	L	|−θ(0)|
1	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)	$(0.1,0.3,0.5)$	$(0.5,0.7,0.9)$
2	$(0,0,0.1)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0,0,0.1)$	$(0,0.1,0.3)$
3	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
4	( $0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
5	$(0,0,0.1)$	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)	$(0,0,0.1)$	(0.7, 0.9, 1.0)
6	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
7	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0,0,0.1)$	$(0,0.1,0.3)$
8	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$	$(0,0,0.1)$	$(0,0.1,0.3)$
9	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0,0,0.1)$	$(0,0.1,0.3)$
10	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0,0,0.1)$	$(0.3,0.5,0.7)$
11	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$
12	$(0,0,0.1)$	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0,0,0.1)$	(0.7, 0.9, 1.0)
13	$(0,0,0.1)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.5,0.7,0.9)$	$(0,0.1,0.3)$
14	$(0,0,0.1)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$	$(0,0,0.1)$	$(0,0.1,0.3)$
15	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$
16	$(0,0,0.1)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$	$(0.5,0.7,0.9)$	$(0,0.1,0.3)$
17	$(0,0,0.1)$	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$
18	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$	$(0,0.1,0.3)$	$(0.5,0.7,0.9)$	$(0,0.1,0.3)$
19	$(0.5,0.7,0.9)$	$(0,0.1,0.3)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
20	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
21	(0.9,1.0,1.0)	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.5,0.7,0.9)$	$(0,0.1,0.3)$
22	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
23	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
24	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
25	(0, 0, 0.1)	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0.5,0.7,0.9)$	$(0.9,1.0,1.0)$
26	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
27	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
28	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
29	(0, 0, 0.1)	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)	$(0,0,0.1)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$
28	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
29	(0, 0, 0.1)	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0,0.1,0.3)$	(0.7, 0.9, 1.0)	$(0,0,0.1)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$

Table 7.

Matrix (in the light of Experiment 3).

No.	Rd	Q	S	L	|−θ(0)|
1	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	(0.7, 0.9, 1.0)	$(0.3,0.5,0.7)$	$(0.5,0.7,0.9)$
2	$(0.3,0.5,0.7)$	( $0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
3	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
4	( $0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
5	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)
6	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0,0.1)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$
7	$(0.5,0.7,0.9)$ ,	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
8	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$	$(0,0,0.1)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
9	$(0.9,1.0,1.0)$	( $0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
10	$(0.9,1.0,1.0)$	( $0.5,0.7,0.9)$	(0.7, 0.9, 1.0)	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$
11	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$
12	$(0.3,0.5,0.7)$	( $0.5,0.7,0.9)$	(0.7, 0.9, 1.0)	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)
13	$(0.3,0.5,0.7)$	( $0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
14	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0,0,0.1)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$
15	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$
16	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
17	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$
18	$(0.9,1.0,1.0)$	$(0.1,0.3,0.5)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
19	$(0.5,0.7,0.9)$	$(0.1,0.3,0.5)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
20	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
21	(0.9, 1.0, 1.0)	( $0.5,0.7,0.9)$	$(0,0,0.1)$	$(0.9,1.0,1.0)$	$(0,0.1,0.3)$
22	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
23	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
24	$(0.5,0.7,0.9)$	( $0.5,0.7,0.9)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
25	$(0.3,0.5,0.7)$	( $0.5,0.7,0.9)$	(0.7, 0.9, 1.0)	$(0.9,1.0,1.0)$	$(0.9,1.0,1.0)$
26	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
27	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
28	$(0.5,0.7,0.9)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
29	$(0.3,0.5,0.7)$	$(0.3,0.5,0.7)$	$(0,0.1,0.3)$	$(0.3,0.5,0.7)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0.1,0.3,0.5)$	(0.7, 0.9, 1.0)	$(0.1,0.3,0.5)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	( $0.5,0.7,0.9)$	(0.7, 0.9, 1.0)	$(0.9,1.0,1.0)$	$(0.3,0.5,0.7)$

Step 2

In this step, from Tables 5 to 7 we form Fuzzy comparison matrix as shown in Table 8 with the help of formula as given below.

$${\alpha }_{\mathrm{ij}}=({\mathrm{a}}_{\mathrm{ij}},{\mathrm{b}}_{\mathrm{ij}},{\mathrm{c}}_{\mathrm{ij}})$$

$${\mathrm{a}}_{\mathrm{ij}}=\underset{\mathrm{k}}{min}\left\{{\mathrm{a}}_{\mathrm{ij}}\right\},{\mathrm{b}}_{\mathrm{ij}}=\frac{1}{\mathrm{k}}\sum _{\mathrm{k}=1}{\mathrm{b}}_{\mathrm{ij}},{\mathrm{c}}_{\mathrm{ij}}=\underset{\mathrm{k}=1}{max}\left\{{\mathrm{c}}_{\mathrm{ij}}\right\}$$

Table 8.

Fuzzy comparison matrix.

No.	Rd	Q	S	L	|−θ(0)|
1	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	(0.7, 0.93, 1.0)	$(0.1,0.43,0.7)$	$(0.5,0.7,0.9)$
2	$(0,0.16,0.7)$	( $0.3,0.63,0.9)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
3	$(0.9,1.0,1.0)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
4	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
5	$(0,0.16,0.7)$	$(0.1,0.13,0.5)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	(0.7, 0.9, 1.0)
6	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.03,0.3)$	$(0.1,0.43,0.7)$	$(0,0.1,0.3)$
7	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
8	$(0.9,1.0,1.0)$	$(0,0.13,0.5)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
9	$(0.9,1.0,1.0)$	( $0.3,0.63,0.9)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
10	$(0.9,1.0,1.0)$	( $0.3,0.63,0.9)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	$(0.3,0.5,0.7)$
11	$(0.9,1.0,1.0)$	$(0,0.13,0.5)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.3,0.5,0.7)$
12	$(0,0.16,0.7)$	( $0.3,0.63,0.9)$	(0.7, 0.93, 1.0)	$(0,0.13,0.5)$	(0.7, 0.9, 1.0)
13	$(0,0.16,0.7)$	( $0.3,0.63,0.9)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
14	$(0,0.16,0.7)$	$(0,0.13,0.5)$	$(0,0.03,0.3)$	$(0,0.13,0.5)$	$(0,0.1,0.3)$
15	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.5,0.9,1.0)$	$(0.1,0.3,0.5)$
16	$(0,0.16,0.7)$	$(0,0.13,0.5)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
17	$(0,0.16,0.7)$	$(0,0.13,0.5)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.9,1.0,1.0)$
18	$(0.9,1.0,1.0)$	$(0,0.13,0.5)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
19	$(0.5,0.63,0.9)$	$(0,0.13,0.5)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
20	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
21	(0.9, 1.0, 1.0)	( $0.3,0.63,0.9)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
22	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
23	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
24	$(0.5,0.63,0.9)$	( $0.3,0.63,0.9)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
25	$(0,0.16,0.7)$	( $0.3,0.63,0.9)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.9,1.0,1.0)$
26	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
27	$(0.5,0.63,0.9)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
28	$(0.5,0.63,0.9),$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
29	$(0,0.16,0.7)$	$(0.1,0.43,0.7)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0,0.13,0.5)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	$(0.1,0.13,0.5)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.3,0.5,0.7)$

Step 3

In third step we find the Fuzzy geometric mean, weights, Fuzzy weights and Normalized weights as shown in Table 9 for different parameters.

Table 9.

Fuzzy weight.

	Rd	Q	S	L	|−θ (0)|	Fuzzy geometric mean value	Fuzzy weights	Weights	Normalized weights
Rd	(0, 0.1, 0.3)	(0.1, 0.3, 0.5)	(0.3, 0.5, 0.7)	(0.5, 0.7, 0.9)	(0.7, 0.9, 1.0)	(0, 0.39, 0.62)	(0, 0.08, 0.23)	0.1033	0.065
Q	(2, 3.3, 10)	(0, 0.1, 0.3)	(0.5, 0.7, 0.9)	(0.5, 0.7, 0.9)	(0.3, 0.5, 0.7)	(0, 0.6, 1.1)	(0, 0.13, 0.418)	0.548	0.347
S	(1.4, 2, 3.3)	(1.1, 1.4, 2)	(0.3, 0.5, 0.7)	(0.5, 0.7, 0.9)	(0.3, 0.5, 0.7)	(0.5, 0.8, 1.23)	(0.08, 0.176, 0.4)	0.2186	0.138
L	(1.1, 1.4, 2)	(1.1, 1.4, 2)	(1.1, 1.4, 2)	(0.9, 1.0, 1.0)	(0.9, 1.0, 1.0)	(1.01, 1.22, 1.5)	(0.16, 0.26, 0.57)	0.33	0.209
|−θ (0)|	(1, 1.1, 1.4)	(1.4, 2, 3.3)	(1.4, 2, 3.3)	(1, 1, 1.1)	(0.9, 1.0, 1.0)	(1.12, 1.34, 1.75)	(0.179, 0.29, 0.66),	0.376	0.238
						=(2.63, 4.35, 6.2)		=1.575	=0.997

We find Fuzzy geometric mean with the help of formula given

$$A_1\times A_2=(l_1,m_1,u_1)\times (l_2,m_2,u_2)$$

$$=[\frac{({\mathrm{l}}_1*{\mathrm{l}}_2*{\mathrm{l}}_3*{\mathrm{l}}_4)}{4},\frac{({\mathrm{m}}_1*{\mathrm{m}}_2*{\mathrm{m}}_3*{\mathrm{m}}_4)}{4},\frac{({\mathrm{u}}_1*{\mathrm{u}}_2*{\mathrm{u}}_3*{\mathrm{u}}_4)}{4}]$$

Weights =  $\frac{(\mathrm{l}+\mathrm{m}+\mathrm{u})}{3}$

Step 4

In fourth step, Table 10 Normalized fuzzy decision matrix are form with the help of the following formula by using Table 9 as shown below.

$${\mathrm{r}}_{\mathrm{ij}}=(\frac{{\mathrm{a}}_{\mathrm{ij}}}{{\mathrm{c}}_{\mathrm{j}}^{*}},\frac{{\mathrm{b}}_{\mathrm{ij}}}{{\mathrm{c}}_{\mathrm{j}}^{*}},\frac{{\mathrm{c}}_{\mathrm{ij}}}{{\mathrm{c}}_{\mathrm{j}}^{*}})\mathrm{and}$$

$${\mathrm{c}}_{\mathrm{j}}^{*}=\underset{\mathrm{i}}{max}\left\{{\mathrm{c}}_{\mathrm{ij}}\right\}$$

Table 10.

Normalized fuzzy decision matrix.

No.	Rd	Q	S	L	|−θ(0)|
1	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	(0.7, 0.93, 1.0)	$(0.1,0.43,0.7)$	$(0.5,0.7,0.9)$
2	$(0,0.16,0.7)$	( $0.33,0.7,1)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
3	$(0.9,1.0,1.0)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
4	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
5	$(0,0.16,0.7)$	$(0,0.14,0.55)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	(0.7, 0.9, 1.0)
6	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.03,0.3)$	$(0.1,0.43,0.7)$	$(0,0.1,0.3)$
7	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
8	$(0.9,1.0,1.0)$	$(0,0.14,0.5)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
9	$(0.9,1.0,1.0)$	( $0.33,0.7,1)$	$(0,0.03,0.3)$	$(0,0.1,0.5)$	$(0,0.1,0.3)$
10	$(0.9,1.0,1.0)$	( $0.33,0.7,1)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	$(0.3,0.5,0.7)$
11	$(0.9,1.0,1.0)$	$(0,0.14,0.55)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.3,0.5,0.7)$
12	$(0,0.16,0.7)$	( $0.33,0.7,1)$	(0.7, 0.93, 1.0)	$(0,0.13,0.5)$	(0.7, 0.9, 1.0)
13	$(0,0.16,0.7)$	( $0.33,0.7,1)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
14	$(0,0.16,0.7)$	$(0,0.14,0.55)$	$(0,0.03,0.3)$	$(0,0.13,0.5)$	$(0,0.1,0.3)$
15	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.5,0.9,1.0)$	$(0.1,0.3,0.5)$
16	$(0,0.16,0.7)$	$(0,0.14,0.55)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
17	$(0,0.16,0.7)$	$(0,0.14,0.55)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.9,1.0,1.0)$
18	$(0.9,1.0,1.0)$	$(0,0.14,0.55)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
19	$(0.5,0.63,0.9)$	$(0,0.14,0.55)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
20	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
21	(0.9, 1.0, 1.0)	( $0.33,0.7,1)$	$(0,0.03,0.3)$	$(0.5,0.9,1.0)$	$(0,0.1,0.3)$
22	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
23	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
24	$(0.5,0.63,0.9)$	( $0.33,0.7,1)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
25	$(0,0.16,0.7)$	( $0.33,0.7,1)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.9,1.0,1.0)$
26	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
27	$(0.5,0.63,0.9)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
28	$(0.5,0.63,0.9),$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
29	$(0,0.16,0.7)$	$(0.11,0.47,0.77)$	$(0,0.23,0.5)$	$(0.1,0.43,0.7)$	$(0.1,0.3,0.5)$
30	(0.9, 1.0, 1.0)	$(0,0.14,0.55)$	(0.7, 0.93, 1.0)	$(0,0.1,0.5)$	$(0.3,0.5,0.7)$
31	(0.9, 1.0, 1.0)	( $0.33,0.7,1)$	(0.7, 0.93, 1.0)	$(0.5,0.9,1.0)$	$(0.3,0.5,0.7)$

After analysis of comparison matrix we have come to know that

$$\begin{array}{c}{\mathrm{for}\mathrm{Rd}\mathrm{c}}_{\mathrm{j}}^{*}=1,{\mathrm{for}\mathrm{Q}\mathrm{c}}_{\mathrm{j}}^{*}=0.9,{\mathrm{for}\mathrm{S}\mathrm{c}}_{\mathrm{j}}^{*}=1,\hfill \\ {\mathrm{for}\mathrm{L}\mathrm{c}}_{\mathrm{j}}^{*}=1,|\mathrm{\theta }\left(0\right)|{\mathrm{c}}_{\mathrm{j}}^{*}=1\hfill \end{array}$$

Step 5

In fifth step we find out the Weighted Normalized Fuzzy Decision Matrix in Table 11.

$${\mathrm{v}}_{\mathrm{ij}}={\mathrm{r}}_{\mathrm{ij}}\times {\mathrm{w}}_{\mathrm{j}}$$

We know the fuzzy Weight of Rd = (0, 0.08, 0.23) and Q = (0, 0.13, 0.418), S = (0.08, 0.175, 0.4), L = (0.16, 0.26, 0.57), |θ(0)| = (0.179, 0.29, 0.66) which we already find in Table 9.

Table 11.

Weighted normalized decision matrix.

No.	Rd	Q	S	L	|−θ(0)|
1	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	(0.056, 0.16368, 0.4)	$(0.016,0.33318,0.399)$	$(0.0895,0.203,0.5)$
2	$(0,0.0128,0.161)$	( $0,0.091,0.418)$	$(0,0.005,0.12)$	$(0,0.026,0.285)$	$(0,0.029,0.198)$
3	$(0,0.08,0.23)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
4	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$		$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
5	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0,0.026,0.285)$ .	(0.1253, 0.261, 0.66)
6	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.005,0.12)$	$016,0.33318,0.399$	$(0,0.029,0.198)$
7	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048.2)$	$(0,0.026,0.285)$	$(0,0.029,0.198)$
8	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$ .	$(0,0.005,0.12)$	$(0,0.026,0.285)$	$(0,0.029,0.198)$
9	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0,0.026,0.285)$	$(0,0.029,0.198)$
10	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0,0.026,0.285)$	$(0.0537,0.145,0.462)$
11	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0.08,0.234,0.57)$	$(0.0537,0.145,0.462)$
12	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0,0.0338,0.285)$	(0.1253, 0.261, 0.66)
13	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0.08,0.234,0.57)$	$(0,0.029,0.198)$
14	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0,0.0338,0.285)$	$(0,0.029,0.198)$
15	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$(0.08,0.234,0.57)$	$(0.0179,0.087,0.33)$
16	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0.08,0.234,0.57)$	$(0,0.029,0.198)$
17	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0.08,0.234,0.57)$	(0.1253, 0.261, 0.66)
18	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0.08,0.234,0.57)$	$(0,0.029,0.198)$
19	$(0,0.0504,0.207)$	$(0,0.0182,0.2299)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
20	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
21	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0.08,0.234,0.57)$	$(0,0.029,0.198)$
22	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
23	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
24	$(0,0.0504,0.207)$	$(0,0.0182,0.2299)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
25	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0.08,0.234,0.57)$	(0.1253, 0.261, 0.66)
26	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
27	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
28	$(0,0.0504,0.207)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
29	$(0,0.0128,0.161)$	$(0,0.0611,0.32186)$	$(0,0.048,0.2)$	$016,0.33318,0.399$	$(0.0179,0.087,0.33)$
30	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0,0.0338,0.285)$	$(0.0537,0.145,0.462)$
31	$(0,0.08,0.23)$	$(0,0.0182,0.2299)$	(0.056, 0.16368, 0.4)	$(0.08,0.234,0.57)$	$(0.0537,0.145,0.462)$
$A^{*}$	$(0,0.0504,0.207)$	$(0,0.091,0.418)$	$(0.056,0.16368,0.4)$	$(0.08,0.234,0.57)$	$(0.1253,0.261,0.66)$
$A^{-}$	$(0,0.0128,0.161)$	$(0,0.0182,0.2299)$	$(0,0.005,0.12)$	$(0,0.026,0.285)$	$(0,0.029,0.198)$

The value of (Fuzzy positive ideal solution) $A^{*}$ and $A^{-}$ (Fuzzy negative ideal solution) of every attribute were calculated in the next step, shown in Table 11. ${\mathrm{A}}^{*}=({\mathrm{v}}_1^{*},{\mathrm{v}}_2^{*},{\mathrm{v}}_3^{*},\dots {\mathrm{v}}_{\mathrm{n}}^{*}){\mathrm{where}\mathrm{v}}_1^{*}=\underset{\mathrm{i}}{max}\{{\mathrm{v}}_{\mathrm{ij}3}\}$

$${\mathrm{A}}^{-}=({\mathrm{v}}_1^{-},{\mathrm{v}}_2^{-},,\dots {\mathrm{v}}_{\mathrm{n}}^{-}){\mathrm{where}\mathrm{v}}_1^{-}=\underset{\mathrm{i}}{min}\{{\mathrm{v}}_{\mathrm{ij}1}\}$$

Step 6

Distance of Fuzzy positive ideal solution (FPIS) and Fuzzy Negative ideal solution (FNIS) for each cell is calculated with the help of formula, as shown in Tables 12 and 13.

$$\mathrm{d}(\check{\mathrm{x}},\check{\mathrm{y}})=\sqrt{\frac{1}{3}}\left[{\left({\mathrm{a}}_1-{\mathrm{a}}_2\right)}^2+{\left({\mathrm{b}}_1-{\mathrm{b}}_2\right)}^2+{\left({\mathrm{c}}_1-{\mathrm{c}}_2\right)}^2\right]$$

Table 12.

Distance of FPSIS.

No.	Rd	Q	S	L	|−θ(0)|
1	0	0.0537	0	0.1267	0.0547
2	0.284	0	0.187	0.1715	0.3071
3	0.2446	0.0537	0.1836	0.1267	0.2231
4	0	0.0537	0.1836	0.1267	0.2231
5	0.284	0.0228	0	0.1715	0
6	0	0.0537	0.187	0.1267	0.3071
7	0	0.0537	0.1836	0.1715	0.3071
8	0.2446	0.0228	0.187	0.1715	0.3071
9	0.2446	0	0.187	0.1715	0.3071
10	0.2446	0	0	0.1715	0.1387
11	0.2446	0.0228	0	0	0.1387
12	0.284	0	0	0.2062	0
13	0.284	0	0.187	0	0.3071
14	0.284	0.0228	0.187	0.2062	0.3071
15	0	0.0537	0.1836	0	0.2231
16	0.284	0.0228	0.187	0	0.3071
17	0.284	0.0228	0	0	0.0265
18	0.2446	0.0228	0.187	0	0.3071
19	0	0.0228	0.1836	0.1267	0.2231
20	0	0.0537	0.1836	0.1267	0.2231
21	0.2446	0	0.187	0	0.3071
22	0	0.0537	0.1836	0.1267	0.2231
23	0	0.0537	0.1836	0.1267	0.2231
24	0	0	0.1836	0.1267	0.2231
25	0.284	0	0	0	0.0265
26	0	0.0537	0.1836	0.1267	0.2231
27	0	0.0537	0.1836	0.1267	0.2231
28	0	0.0537	0.1836	0.1267	0.2231
29	0.284	0.0537	0.1836	0.1267	0.2231
30	0.2446	0.0228	0	0.2062	0.1387
31	0.2446	0	0	0	0.1387

Table 13.

Distance of FNIS.

No.	Rd	Q	S	L	|−θ(0)|
1	0.0342	0.0590	0.18859	0.08237	0.2550
2	0	0.0136	0	0	0
3	0.0556	0.0590	0.2348	0.08237	0.0838
4	0.0342	0.0590	0.2348	0.08237	0.0838
5	0	0	0.18859	0	0.3071
6	0.0342	0.0590	0	0.08237	0
7	0.0342	0.0590	0.2348	0	0
8	0.0556	0	0	0	0
9	0.0556	0.0136	0	0	0
10	0.0556	0.0136	0.18859	0	0.1693
11	0.0556	0	0.18859	0.211	0.1693
12	0	0.0136	0.18859	0.0045	0.3071
13	0	0.0136	0	0.211	0
14	0	0	0	0.0045	0
15	0.0342	0.0590	0.2348	0.211	0.0838
16	0	0	0	0.211	0
17	0	0	0.18859	0.211	0.3071
18	0.0556	0	0	0.211	0
19	0.0342	0	0.2348	0.08237	0.0838
20	0.0342	0.0590	0.2348	0.08237	0.0838
21	0.0556	0.0136	0	0.211	0
22	0.0342	0.0590	0.2348	0.08237	0.0838
23	0.0342	0.0590	0.2348	0.08237	0.0838
24	0.0342	0.0136	0.2348	0.08237	0.0838
25	0	0.0136	0.18859	0.211	0.3071
26	0.0342	0.0590	0.2348	0.08237	0.0838
27	0.0342	0.0590	0.2348	0.08237	0.0838
28	0.0342	0.0590	0.2348	0.08237	0.0838
29	0	0.0590	0.2348	0.08237	0.0838
30	0.0556	0	0.18859	0.0045	0.16934
31	0.0556	0.0136	0.18859	0.211	0.16934

Step 7

In next step, the distance each alternative of the FPIS and FNIS is calculated with the help of formula. Next the closeness coefficient were determined seen in Table 14.

$${\mathrm{d}}_{\mathrm{i}}^{*}=\sum _{\mathrm{j}=1}^{\mathrm{n}}\mathrm{d}({\tilde{\mathrm{v}}}_{\mathrm{ij}},{{\tilde{\mathrm{v}}}_{\mathrm{j}}}^{*})$$

$${\mathrm{d}}_{\mathrm{i}}^{-}=\sum _{\mathrm{j}=1}^{\mathrm{n}}\mathrm{d}({{\tilde{\mathrm{v}}}_{\mathrm{j}}}^{-},{\tilde{\mathrm{v}}}_{\mathrm{ij}})$$

Table 14.

Ranking of alternatives.

No.	$d_i^{+}$	$d_i^{-}$	$c{c}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}}$	Rank
1	0.2351	0.61916	0.724	1
2	0.9456	0.0136	0.0141	30
3	0.8331	0.51557	0.3822	20
4	0.5871	0.49417	0.4570	14
5	0.4783	0.49569	0.5089	8
6	0.6745	0.17557	0.2065	27
7	0.7159	0.328	0.3142	22
8	0.933	0.0556	0.056	29
9	0.9102	0.0692	0.0706	28
10	0.5528	0.42709	0.4358	18
11	0.4061	0.62453	0.6059	5
12	0.4902	0.51379	0.51174	7
13	0.7781	0.2246	0.2196	25
14	1.0071	0.0045	0.00448	31
15	0.4604	0.6228	0.5749	6
16	0.8009	0.211	0.20851	25
17	0.333	0.71969	0.6836	3
18	0.7615	0.2666	0.2593	24
19	0.5562	0.43517	0.4389	17
20	0.5871	0.49417	0.4570	14
21	0.7387	0.2802	0.27500	23
22	0.5871	0.49417	0.45702	9
23	0.5871	0.49417	0.45702	9
24	0.5334	0.44877	0.70252	16
25	0.3105	0.73329	0.70252	2
26	0.5871	0.49417	0.45702	2
27	0.5871	0.49417	0.45702	9
28	0.5871	0.49417	0.45702	9
29	0.8711	0.45997	0.34556	21
30	0.6123	0.41803	0.40572	19
31	0.3833	0.63813	0.62474	4

Closeness coefficient C $C_i$ for each alternative

$$\mathrm{C}{\mathrm{C}}_{\mathrm{i}}=\frac{{\mathrm{d}}_{\mathrm{i}}^{-}}{{\mathrm{d}}_{\mathrm{i}}^{*}+{\mathrm{d}}_{\mathrm{i}}^{-}}$$

Finally, when all the findings have been obtained, the ranking of the alternative were completed in order to obtain the best possible collection of choices, given in Table 14. The alternative with the highest C $C_i$ value chosen best option. The A1 with Rd = (0.3, 0.633, 0.9), Q = (0.1, 0.43, 0.7) and S = (0.7, 0.93, 1.0), L = (0.1, 0.43, 0.7) and |−θ(0)| = (0.5, 0.7, 0.9) have highest value 1. The A14 has minimum value and ranked at position 25. The ranking of alternatives in descending order were based on the value of C $C_i$ as shown in Table 14.

The ranking of alternatives in the descending order were based on the value of C $C_i$ given as A1 > A25 = A26 > A17 > A31 > A11 > A15 > A12 > A5> A22 = A23 = A27 = A28 > A4 = A20 > A24 > A19 > A10 > A30 > A3 > A29 > A7 > A21> A18 > A16 = A13 > A6 > A9 > A8 > A2 > A14.

Graphical results of fuzzy TOPSIS

In Figure 9 we can see the highest weightage of the parameters Q which is 35%. The lowest weightage of Rd is 6%. |−θ(0)| has fuzzy weight 24%. L has 21% weightage and S has 14% weightage.

Figure 9.

Fuzzy weightage of parameters.

In Figure 10 the Ranking of the Alternatives are displaying. The A1 with Rd =  $(0.5,0.63,0.9)$ and Q = (0.1, 0.43, 0.7), .S = (0.7, 0.93, 1.0) and L =  $(0.1,0.43,0.7)$ θ(0)| =  $(0.5,0.7,0.9)$ (have highest value 0.724). The A14 has minimum value and ranked at position 31.

Figure 10.

Ranking of alternatives for optimization of HTR.

Conclusions

The goal of this work is to optimize the heat transfer rate of a magnetized tetra-hybrid Tiwari and Dass nanofluid through a channel in stretching walls. The heat transfer rate of magnetized tetra-hybrid nanofluid is taken into account in order to optimize it. Using the bvp4c approach in MATLAB software to solve ODEs and offer precise results and implications for multiple parameters. The results were successfully explained through graphs and tables. After the necessary model formulation was finished and numerical analysis was carried out while taking into account certain physical factors, the following outcomes were obtained.

1. Velocity of the nanofluid near the lower wall $\eta =0$ of the channel decreases as increase in the Reynolds number $R$ and increases near the centre of the channel.

2. It came to know that velocity profile shifted upwards from lower wall to the centre of the channel and reverse trend is seen from centre to upper wall of the channel. Additionally, the trend of temperature profile $\theta (\eta )$ is decreasing as increase in the stretching parameter.

3. The velocity $f\prime (\eta )$ and temperature profile decline rapidly as the enhancement in the suction parameter.

4. Maximum amount of $C{f}_x$ happens when there is slightly increase in $M$ with stretched wall that is $\lambda \in [0,2]$ .

5. Whenever there is injection implies decrease in $C{f}_x$ while with the increase in $C{f}_x$ there is increase in $C{f}_x$ .

Limitation and future recommendations

In the present study, we considered the solid volume fraction of nanoparticles is up to 1%. We considered two-dimensional flow in a channel. In future, this can be generalized for three-dimensional rotating flow for the variation of solid volume fraction. Moreover, in this study we employed fuzzy TOPSIS to find the best combination of the physical parameters to enhance the heat transfer rate. In future, some other optimization techniques can be applied.