9. Example 9: Cantilever Stepped Beam

Example of solving the constrained engineering optimization problem “Cantilever Stepped Beam” as well as a simpler square case using NEORL with the PSO, DE, SSA, GWO, MFO, BAT, and PESA2 algorithms.

9.1. Summary

  • Algorithms: PSO, DE, SSA, GWO, MFO, BAT, PESA2

  • Type: Continuous, Single-objective, Constrained

  • Field: Structural Engineering

9.2. Problem Description

The cantilever stepped beam design is an engineering optimization problem with the objective to evaluate the optimal heights \(x_1 - x_5\) and widths \(x_6 - x_{10}\) of the five rectangular segments such that the volume of the statically loaded cantilever beam is minimized accounting for stress \((\sigma)\) constraints. The figure below shows the dimensions of the cantilever structure

alternate text

The equation for the volume of the cantilever beam is

\[\min_{\vec{x}} f (\vec{x}) = \sum_{i=1}^{5} x_i x_{i+5} l_i,\]

where \(l_i\) = 100 cm. The design is subject to 11 constraints:

\[ \begin{align}\begin{aligned} g_1 = \frac{P l_5}{x_5 x_{10}^2} - 14000 \leq 0,\\ g_2 = \frac{P (l_5 + l_4)}{x_4 x_9^2} - 14000 \leq 0,\\ g_3 = \frac{P (l_5 + l_4 + l_3)}{x_3 x_8^2} - 14000 \leq 0,\\g_4 = \frac{P (l_5 + l_4 + l_3 + l_2)}{x_2 x_7^2} - 14000 \leq 0,\\g_5 = \frac{P (l_5 + l_4 + l_3 + l_2 + l_1)}{x_1 x_6^2} - 14000 \leq 0,\\g_6 = \frac{x_{10}}{x_5} - 20 \leq 0,\\g_7 = \frac{x_9}{x_4} - 20 \leq 0,\\g_8 = \frac{x_8}{x_3} - 20 \leq 0,\\g_9 = \frac{x_7}{x_2} - 20 \leq 0,\\g_{10} = \frac{x_6}{x_1} - 20 \leq 0,\\g_{11} = \frac{P l^3}{3 E} \left( \frac{1}{I_5} +\frac{7}{I_4} + \frac{19}{I_3} + \frac{37}{I_2} + \frac{61}{I_1} \right)-2.7 \leq 0,\end{aligned}\end{align} \]

where \(1 \leq x_i \leq 5\) \((i = 1,2, \ldots, 5)\), \(30 \leq x_i \leq 65\) \((i = 6,7,\ldots,10)\), \(P = 50,000 N\), and \(E = 2 \times 10^7 N/cm^2\).

9.3. NEORL script

#************************************************************
#               Cantilever Stepped Beam
#************************************************************

#---------------------------------
# Import packages
#---------------------------------
import numpy as np
import matplotlib.pyplot as plt
from neorl import PSO, DE, SSA, GWO, MFO, BAT, PESA2

#---------------------------------
# Fitness function
#---------------------------------
def CSB(individual):
  """Cantilever Stepped Beam
  individual[i = 0 - 4] are beam widths
  individual[i = 5 - 9] are beam heights
  """

  check=all([item >= BOUNDS['x'+str(i+1)][1] for i,item in enumerate(individual)]) \
		and all([item <= BOUNDS['x'+str(i+1)][2] for i,item in enumerate(individual)])
  if not check:
	  raise Exception ('--error check fails')

  P = 50000
  E = 2 * 10**7
  l = 100
  g = np.zeros(11)
  g[0] = 600*P/(individual[4] * individual[9]**2) - 14000
  g[1] = 6*P*(2*l)/(individual[3] * individual[8]**2) - 14000
  g[2] = 6*P*(3*l)/(individual[2] * individual[7]**2) - 14000
  g[3] = 6*P*(4*l)/(individual[1] * individual[6]**2) - 14000
  g[4] = 6*P*(5*l)/(individual[0] * individual[5]**2) - 14000
  g[5] = 0
  g[6] = individual[9]/individual[4] - 20
  g[7] = individual[8]/individual[3] - 20
  g[8] = individual[7]/individual[2] - 20
  g[9] = individual[6]/individual[1] - 20
  g[10] = individual[5]/individual[0] - 20

  g_round=np.round(g,6)
  w1=1000
  w2=1000

  phi=sum(max(item,0) for item in g_round)
  viol=sum(float(num) > 0 for num in g_round)
  
  V = 0
  for i in range(5):
	  V += individual[i] * individual[i+5] * l
	  
  return V + w1*phi + w2*viol

#---------------------------------
# Parameter space
#---------------------------------
nx=10
BOUNDS={}
for i in range(1, 6):
  BOUNDS['x'+str(i)]=['float', 1, 5]
for i in range(6, 11):
  BOUNDS['x'+str(i)]=['float', 30, 65]

#---------------------------------
# PSO
#---------------------------------
pso=PSO(mode='min', bounds=BOUNDS, fit=CSB, c1=2.05, c2=2.1, npar=50, speed_mech='constric', ncores=1, seed=1)
pso_x, pso_y, pso_hist=pso.evolute(ngen=300, verbose=0)

#---------------------------------
# DE
#---------------------------------
de=DE(mode='min', bounds=BOUNDS, fit=CSB, npop=50, F=0.5, CR=0.7, ncores=1, seed=1)
de_x, de_y, de_hist=de.evolute(ngen=300, verbose=0)

#---------------------------------
# SSA
#---------------------------------
ssa=SSA(mode='min', bounds=BOUNDS, fit=CSB, nsalps=50, ncores=1, seed=1)
ssa_x, ssa_y, ssa_hist=ssa.evolute(ngen=300, verbose=0)

#---------------------------------
# BAT
#---------------------------------
bat=BAT(mode='min', bounds=BOUNDS, fit=CSB, nbats=50, fmin = 0 , fmax = 1, A=0.5, r0=0.5, levy = True, seed = 1, ncores=1)
bat_x, bat_y, bat_hist=bat.evolute(ngen=300, verbose=0)

#---------------------------------
# GWO
#---------------------------------
gwo=GWO(mode='min', fit=CSB, bounds=BOUNDS, nwolves=50, ncores=1, seed=1)
gwo_x, gwo_y, gwo_hist=gwo.evolute(ngen=300, verbose=0)

#---------------------------------
# MFO
#---------------------------------
mfo=MFO(mode='min', bounds=BOUNDS, fit=CSB, nmoths=50, b = 0.2, ncores=1, seed=1)
mfo_x, mfo_y, mfo_hist=mfo.evolute(ngen=300, verbose=0)

#---------------------------------
# PESA2
#---------------------------------
pesa2=PESA2(mode='min', bounds=BOUNDS, fit=CSB, npop=50, nwolves=5, ncores=1, seed=1)
pesa2_x, pesa2_y, pesa2_hist=pesa2.evolute(ngen=600, replay_every=2, verbose=0)

#---------------------------------
# Plot
#---------------------------------
plt.figure()
plt.plot(pso_hist['global_fitness'], label = 'PSO')
plt.plot(de_hist['global_fitness'], label = 'DE')
plt.plot(ssa_hist['global_fitness'], label = 'SSA')
plt.plot(bat_hist['global_fitness'], label = 'BAT')
plt.plot(gwo_hist['fitness'], label = 'GWO')
plt.plot(mfo_hist['global_fitness'], label = 'MFO')
plt.plot(pesa2_hist, label = 'PESA2')
plt.legend()
plt.xlabel('Generation')
plt.ylabel('Fitness')
plt.ylim(0,150000)
plt.savefig('CSB_fitness.png',format='png', dpi=300, bbox_inches="tight")
plt.close()

#************************************************************
#              Square Cantilever Stepped Beam
#************************************************************

#---------------------------------
# Import packages
#---------------------------------
import numpy as np
from math import cos, pi, exp, e, sqrt
import matplotlib.pyplot as plt
from neorl import PSO, DE, SSA, GWO, MFO, BAT, PESA2

#---------------------------------
# Fitness function
#---------------------------------
def CSB_square(individual):
  """Square Cantilever Stepped Beam
  individual[i = 0 - 4] are beam heights and widths
  """

  check=all([item >= BOUNDS['x'+str(i+1)][1] for i,item in enumerate(individual)]) \
		and all([item <= BOUNDS['x'+str(i+1)][2] for i,item in enumerate(individual)])
  if not check:
	  raise Exception ('--error check fails')

  g = 61/individual[0]**3 + 37/individual[1]**3 + 19/individual[2]**3 + 7/individual[3]**3 + 1/individual[4]**3 - 1

  g_round=np.round(g,6)
  w1=1000
  #phi=max(g_round,0)
  if g_round > 0:
	  phi = 1
  else:
	  phi = 0

  V = 0.0624*(np.sum(individual)) 

  return V + w1*phi

#---------------------------------
# Parameter space
#---------------------------------
nx=5
BOUNDS={}
for i in range(1, 6):
  BOUNDS['x'+str(i)]=['float', 0.01, 100]
  
#---------------------------------
# PSO
#---------------------------------
pso=PSO(mode='min', bounds=BOUNDS, fit=CSB_square, c1=2.05, c2=2.1, npar=50, speed_mech='constric', ncores=1, seed=1)
pso_x, pso_y, pso_hist=pso.evolute(ngen=200, verbose=0)

#---------------------------------
# DE
#---------------------------------
de=DE(mode='min', bounds=BOUNDS, fit=CSB_square, npop=50, F=0.5, CR=0.7, ncores=1, seed=1)
de_x, de_y, de_hist=de.evolute(ngen=200, verbose=0)

#---------------------------------
# SSA
#---------------------------------
ssa=SSA(mode='min', bounds=BOUNDS, fit=CSB_square, nsalps=50, ncores=1, seed=1)
ssa_x, ssa_y, ssa_hist=ssa.evolute(ngen=200, verbose=0)

#---------------------------------
# BAT
#---------------------------------
bat=BAT(mode='min', bounds=BOUNDS, fit=CSB_square, nbats=50, fmin = 0 , fmax = 1, A=0.5, r0=0.5, levy = True, seed = 1, ncores=1)
bat_x, bat_y, bat_hist=bat.evolute(ngen=200, verbose=0)

#---------------------------------
# GWO
#---------------------------------
gwo=GWO(mode='min', bounds=BOUNDS, fit=CSB_square, nwolves=50, ncores=1, seed=1)
gwo_x, gwo_y, gwo_hist=gwo.evolute(ngen=200, verbose=0)

#---------------------------------
# MFO
#---------------------------------
mfo=MFO(mode='min', bounds=BOUNDS, fit=CSB_square, nmoths=50, b = 0.2, ncores=1, seed=1)
mfo_x, mfo_y, mfo_hist=mfo.evolute(ngen=200, verbose=0)

#---------------------------------
# PESA2
#---------------------------------
pesa2=PESA2(mode='min', bounds=BOUNDS, fit=CSB_square, npop=50, nwolves=5, ncores=1, seed=1)
pesa2_x, pesa2_y, pesa2_hist=pesa2.evolute(ngen=400, replay_every=2, verbose=0)

#---------------------------------
# Plot
#---------------------------------
plt.figure()
plt.plot(pso_hist['global_fitness'], label = 'PSO')
plt.plot(de_hist['global_fitness'], label = 'DE')
plt.plot(ssa_hist['global_fitness'], label = 'SSA')
plt.plot(bat_hist['global_fitness'], label = 'BAT')
plt.plot(gwo_hist['fitness'], label = 'GWO')
plt.plot(mfo_hist['global_fitness'], label = 'MFO')
plt.plot(pesa2_hist, label = 'PESA2')
plt.legend()
plt.xlabel('Generation')
plt.ylabel('Fitness')
plt.savefig('CSB_square_fitness.png',format='png', dpi=300, bbox_inches="tight")
plt.close()

9.4. Results

A summary of the results for the different methods is shown below with the best \(\vec{x}\) and \(y=f(x)\) (minimum volume). DE and PESA2 show the best results

alternate text
      ------------------------ PSO Summary --------------------------
Best fitness (y) found: 67335.91306205148
Best individual (x) found: [3.5688543506518657, 2.7791479019993583, 2.525097748486896, 2.5931709214818746, 3.0184400437701497, 54.792949608125554, 55.55687861890972, 50.45798494270985, 40.66285990348347, 30]
--------------------------------------------------------------
      ------------------------ DE Summary --------------------------
Best fitness (y) found: 62253.08088965673
Best individual (x) found: [3.0024484908957634, 2.805060577916484, 2.5489794459150485, 2.2104364777790244, 1.7812440232731417, 59.83601779613085, 55.29672939889619, 50.28650239860514, 44.1239666136573, 34.83693680366301]
--------------------------------------------------------------
      ------------------------ SSA Summary --------------------------
Best fitness (y) found: 64883.21386055779
Best individual (x) found: [ 3.01217612  3.12691389  2.66584391  2.56789564  2.26413108 59.64053204
 52.35626719 49.10660541 40.85291814 30.76422498]
--------------------------------------------------------------
------------------------ BAT Summary --------------------------
Best fitness (y) found: 85509.99706478164
Best individual (x) found: [ 3.24944962  3.15901556  3.71034211  2.88044491  2.72539593 57.42189701
 63.18019888 41.62463495 57.60433819 54.50783013]
--------------------------------------------------------------
------------------------ GWO Summary --------------------------
Best fitness (y) found: 64217.56500668205
Best individual (x) found: [ 3.0490241   2.82805632  2.54882906  2.60647295  1.91892183 59.32877115
 55.25800387 50.22171706 40.6816295  36.98217938]
--------------------------------------------------------------
------------------------ MFO Summary --------------------------
Best fitness (y) found: 68284.66539072228
Best individual (x) found: [ 3.00200798  3.04903017  2.52470544  4.76307633  2.38149254 59.7430926
 53.04777538 50.46763269 30.00000001 30.00007271]
--------------------------------------------------------------
------------------------PESA2 Summary-------------------------
Best fitness (y) found: 62491.80715494685
Best individual (x) found: [3.0160894636962894, 2.7812536032709683, 2.5788148288206765, 2.22105253054037, 1.9267356857359532, 59.60349505892154, 55.515872598414006, 49.94170599870354, 43.93631599428571, 33.408629968533766]
--------------------------------------------------------------

9.5. Simple Square Case

A simpler case of the cantilever stepped beam design is shown in the figure below where the heights and widths of each section of the beam are equal \((x_i = x_{i+5})\) and the cantilever is hollow with constant thickness.

alternate text

The equation for the weight of the square cantilever beam to be minimized is:

\[\min_{\vec{x}} f (\vec{x}) = 0.0624 \sum_{i=1}^{5} x_i,\]

with a single constraint

\[g_1 = \frac{61}{x_1^3} + \frac{37}{x_2^3} + \frac{19}{x_3^3} + \frac{7}{x_4^3} + \frac{1}{x_5^3} - 1 \leq 0,\]

where \(0.001 \leq x_i \leq 100\)

#---------------------------------
# Import packages
#---------------------------------
import numpy as np
from math import cos, pi, exp, e, sqrt
import matplotlib.pyplot as plt
from neorl import PSO, DE, SSA, GWO, MFO, BAT, PESA2

#---------------------------------
# Fitness function
#---------------------------------
def CSB_square(individual):
  """Square Cantilever Stepped Beam
  individual[i = 0 - 4] are beam heights and widths
  """

  check=all([item >= BOUNDS['x'+str(i+1)][1] for i,item in enumerate(individual)]) \
                and all([item <= BOUNDS['x'+str(i+1)][2] for i,item in enumerate(individual)])
  if not check:
          raise Exception ('--error check fails')

  g = 61/individual[0]**3 + 37/individual[1]**3 + 19/individual[2]**3 + 7/individual[3]**3 + 1/individual[4]**3 - 1

  g_round=np.round(g,6)
  w1=1000
  #phi=max(g_round,0)
  if g_round > 0:
          phi = 1
  else:
          phi = 0

  V = 0.0624*(np.sum(individual))

  return V + w1*phi

#---------------------------------
# Parameter space
#---------------------------------
nx=5
BOUNDS={}
for i in range(1, 6):
  BOUNDS['x'+str(i)]=['float', 0.01, 100]

#---------------------------------
# PSO
#---------------------------------
pso=PSO(mode='min', bounds=BOUNDS, fit=CSB_square, c1=2.05, c2=2.1, npar=50, speed_mech='constric', ncores=1, seed=1)
pso_x, pso_y, pso_hist=pso.evolute(ngen=200, verbose=0)

#---------------------------------
# DE
#---------------------------------
de=DE(mode='min', bounds=BOUNDS, fit=CSB_square, npop=50, F=0.5, CR=0.7, ncores=1, seed=1)
de_x, de_y, de_hist=de.evolute(ngen=200, verbose=0)

#---------------------------------
# SSA
#---------------------------------
ssa=SSA(mode='min', bounds=BOUNDS, fit=CSB_square, nsalps=50, c1=0.05, ncores=1, seed=1)
ssa_x, ssa_y, ssa_hist=ssa.evolute(ngen=200, verbose=0)

#---------------------------------
# BAT
#---------------------------------
bat=BAT(mode='min', bounds=BOUNDS, fit=CSB_square, nbats=50, fmin = 0 , fmax = 1, A=0.5, r0=0.5, levy = True, seed = 1, ncores=1)
bat_x, bat_y, bat_hist=bat.evolute(ngen=200, verbose=0)

#---------------------------------
# GWO
#---------------------------------
gwo=GWO(mode='min', bounds=BOUNDS, fit=CSB_square, nwolves=50, ncores=1, seed=1)
gwo_x, gwo_y, gwo_hist=gwo.evolute(ngen=200, verbose=0)

#---------------------------------
# MFO
#---------------------------------
mfo=MFO(mode='min', bounds=BOUNDS, fit=CSB_square, nmoths=50, b = 0.2, ncores=1, seed=1)
mfo_x, mfo_y, mfo_hist=mfo.evolute(ngen=200, verbose=0)

#---------------------------------
# PESA2
#---------------------------------
pesa2=PESA2(mode='min', bounds=BOUNDS, fit=CSB_square, npop=50, nwolves=5, ncores=1, seed=1)
pesa2_x, pesa2_y, pesa2_hist=pesa2.evolute(ngen=400, replay_every=2, verbose=0)

#---------------------------------
# Plot
#---------------------------------
plt.figure()
plt.plot(pso_hist, label = 'PSO')
plt.plot(de_hist, label = 'DE')
plt.plot(ssa_hist['global_fitness'], label = 'SSA')
plt.plot(bat_hist['global_fitness'], label = 'BAT')
plt.plot(gwo_hist['fitness'], label = 'GWO')
plt.plot(mfo_hist['global_fitness'], label = 'MFO')
plt.plot(pesa2_hist, label = 'PESA2')
plt.legend()
plt.xlabel('Generation')
plt.ylabel('Fitness')
plt.savefig('CSB_square_fitness.png',format='png', dpi=300, bbox_inches="tight")

9.6. Results

A summary of the results for the different methods is shown below with the best \(\vec{x}\) and \(y=f(x)\) (minimum volume). All methods seem to provide competitive performances.

alternate text
------------------------ PSO Summary --------------------------
Best fitness (y) found: 1.3399752313729163
Best individual (x) found: [6.009251848152787, 5.30657072479825, 4.487735398237302, 3.5223666630459225, 2.148037406998374]
--------------------------------------------------------------
------------------------ DE Summary --------------------------
Best fitness (y) found: 1.3400333309297923
Best individual (x) found: [6.009134740254158, 5.313241105010421, 4.465292783694765, 3.529049287512202, 2.1581752074033305]
--------------------------------------------------------------
------------------------ SSA Summary --------------------------
Best fitness (y) found: 1.3588143704307478
Best individual (x) found: [5.78402889 5.66053404 4.28387098 3.33467882 2.71275859]
--------------------------------------------------------------
------------------------ BAT Summary --------------------------
Best fitness (y) found: 1.3399823246990532
Best individual (x) found: [6.00500481 5.29566351 4.51093598 3.49613637 2.16633504]
--------------------------------------------------------------
------------------------ GWO Summary --------------------------
Best fitness (y) found: 1.3401094136761165
Best individual (x) found: [6.03840936 5.27194855 4.46957895 3.52048465 2.17569089]
--------------------------------------------------------------
------------------------ MFO Summary --------------------------
Best fitness (y) found: 1.340072721543648
Best individual (x) found: [5.98884701 5.35435014 4.45583399 3.51864409 2.15784915]
--------------------------------------------------------------
------------------------ PESA2 Summary --------------------------
Best fitness (y) found: 1.3399951046267742
Best individual (x) found: [6.0302855076764095, 5.324816195983351, 4.498012579038754, 3.474272116432333, 2.1468941237341275]
--------------------------------------------------------------