Algorithm 3: SHM Analysis using MOHGPSO, deploying GRA and FDM

Input: structure information from the FEM analysis, number of sensors $m.$

Output: optimal locations of the sensors.

(1)Get the random binary matrix for the sensor’s placement $X_{iϵ\left[\mathrm{0,1}\right]}^t, t=\mathrm{1,2}\dots maxiter$ (2)$\mathrm{Calculate} \mathrm{the} \mathrm{multi}-\mathrm{objective} \mathrm{functions} F^t$ form the structural analysis (3)$\mathrm{Store} \mathrm{the} X_{iϵ\left[\mathrm{0,1}\right]}^t$ and $F^t$ in the external archive when $t=1$ (4) Update the particle’s position using HGPSO Input: epoch size, swarm size $m$, $\omega$, $a$, $a_2$, $a_3$ Initialize: initial position of the swarm (a) Calculate $q,$ $k$, $s$, $f\left(x_i\right)$ for the initial positions (b) do (i) for each sequence x in the swarm do (ii) Update the velocity using $v_{i,t}=V_{i,t+1}(x, q, k, s, \omega , a_1, a_2, a_3)$ (iii) Calculate new position ${\overline{x}}_{i,t}$ to update particle’s position (iv) If $f\left(\overline{x}\right)$ Is better than $f\left(x\right)$ then (1) $f\left(x\right)=f\left(\overline{x}\right)$ (2) $x=\overline{x}$ (v) end_if (vi) If $f\left(\overline{x}\right)$ Is better than $f\left(k\right)$ then (1) $f\left(k\right)=f\left(\overline{x}\right)$ (vii) End_for (viii) Calculate $\overline{s}$ for current epoch and particle positions (ix) If ($\overline{s}$ Is better than $s$) (1) $s=\overline{s}$ (x) end_if (xi) while (number of epochs are not satisfied) (5)Repeat the steps 2 and 3 for $t=2$ (6) Use the Grey relation analysis (GRA) on the archived particles to select the non-dominated solution (a) Input: Calculated Fitness value matrix $f$$\mathrm{using} F^t$ (b) Normalize the matrix $f$using Equation (15) (c) For $i = 1 : n$ (i) $\mathrm{For} j = 1 : k$ (1) Calculate the grey relational coefficient ${\gamma }_{ij}$ using Equation (16) (ii) end for (d) end for (e) Generate a graph object from ${\gamma }_{ij}$ (f) $\mathrm{Calculate} \mathrm{the} \mathrm{betweenness} g\left(v\right)$ for each element (g) $OS{P}_{indx}=\max g\left(v\right)$ (7)Update the archive with the obtained value, if needed (8)If iterations are finished (a)Stop (9) Else (a) Repeat steps 2 and 3 (10)End (11)Select the single solution from the final archive using Fuzzy Decision modelling (FDM) as described through Equations (38) and (39)