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P. 59
bounds of the design variables are: 0.0625 ≤ x1, x2 ≤ average value for function evaluations found in literature.
99 ∗ 0.0625, and 10 ≤ x3, x4 ≤ 200. The parameters of HSA-CS were set according to the authors
in [8], while the population size was varied as: N p = 30 for
4.3 Spring design welded beam, spring design, and speed reducer, while for
pressure vessel N p = 50. Each experiment was replicated
The spring design optimization problem deals with an opti- 50 times, thus the results reported here are the average of
mal design of a tension spring. The problem consists of three those runs.
variables, which are the number of spring coils, winding di-
ameter, and wire diameter. The objective is to minimize the Table 1 holds the results of the experiments. For each prob-
weight of the spring, subject to minimum deflection, surge lem the minimum (min), maximum (max), mean, median
frequency, shear stress, and limits on the outside diameter. (md), and standard deviation (std) values are reported.
Mathematically it can be formulated as:
f (x) = (x3 + 2) ∗ x12 ∗ x2, The results in Table 1 indicate that the HSA-CS was able
subject to the following constraints: to find the same solution for the welded beam and speed
(8) reducer problems in all 50 runs of the algorithm. On the
contrary, the HSA-CS had trouble in converging towards a
single solution.
g0 : 1− x23x3 ≤ 0, g1 : 4x22 − x2x3 + 1 −1 ≤ 0, 6. DISCUSSION
71785x41 12566(x2x33 − x34) 5108x32
In this section we analyze the results from our experiments
g2 :1− 140.45x1 ≤ 0, g3 : (x1 + x2) − 1.5 ≤0 (9) and compare them to those found in the literature. For this
x22x3 purpose a Table 2 is provided, where results from literature
are gathered. It can be seen, that the HSA-CS achieved com-
The search space of design variables are limited as: 0.05 ≤ petitive results if not better results on all test optimization
x1 ≤ 2, 0.25 ≤ x2 ≤ 1.3, and 2 ≤ x3 ≤ 15. problems. For the welded beam problem our method and
the method in [2] converged to a single solution, whereas
4.4 Speed reducer other methods were not as successful. It is also hard to
say, which method performed the best, since the findings
This problem deals with a minimum weight design of a speed in other papers are reported only to 6 digits. On the pres-
reducer, subject to bending stress of the gear teeth, surface sure vessel problem, the HSA-CS achieved similar results as
stress, stresses in the shafts, and transverse deflections of for the welded beam problem. Based on the mean value
the shafts. This problem is formulated with seven design the only competitive method was again the one proposed
variables, using the following mathematical definition: in [2]. HSA-CS acheived the best results for the speed re-
ducer. Again, like for the welded beam, the results were
f (x) = 0.7854x1x22(3.3333x32 + 14.9334x3 − 43.0934)− unanimous, converging to a single solution, which was also
the smallest based on mean value. Good results were also
1.508x1(x62 + x27) + 7.477(x63 + x37) + 0.7854(x4x26 + x5x27), obtained on the spring design problem, were the HSA-CS
(10) had the smallest std value over the 50 runs, while obtain-
ing good results based on the mean value. We can conclude
subject to: the HSA-CS would be suitable for use in real-life constraint
optimization problems.
g0 : 27.0 − 1.0 ≤ 0, g1 : 397.5 − 1.0 ≤ 0,
x1x22x3 x1 x22 x32
g2 : 1.93x34 − 1.0 ≤ 0, g3 : 1.93x35 − 1.0 ≤ 0,
x2x3x46 x2x3x74
( 745x4 )2 + 16.9 ∗ 10e6) 7. CONCLUSION
g4 : x2 x3 − 1.0 ≤ 0, This paper investigated the recently proposed HSA-CS al-
gorithm, on four well known engineering design optimiza-
(110x36) tion problems with constraints. The problems at hand were:
welded beam, pressure vessel design, speed reducer design,
( 745x5 )2 + 157.5 ∗ 10e6) and spring design. The obtained results were compared to
(85x37) the some state-of-the-art methods, were the HSA-CS per-
g5 : x2 x3 − 1.0 ≤ 0, formed very well, thus we can conclude it would be suitable
for use in real-life engineering applications.
g6 : x2x3 − 1 ≤ 0, g7 : 5x2 − 1 ≤ 0, g8 : x1 − 1 ≤ 0,
40 x1 12x2
g9 : 1.5x6 + 1.9 − 1 ≤ 0, g10 : 1.1x7 + 1.9 − 1.0 ≤ 0. (11)
x4 x5
The search of the design variables is defined as: 8. REFERENCES
(2.6, 0.7, 17.0, 7.3, 7.8, 2.9, 5.0)T ≤ x ≤ (3.6, 0.8, 28.0, 8.3, 8.3, 3.9, 5.5)T [1] Bahriye Akay and Dervis Karaboga. Artificial bee
5. RESULTS colony algorithm for large-scale problems and
engineering design optimization. Journal of Intelligent
The HCS-SA was applied to solve the design optimization Manufacturing, 23(4):1001–1014, 2012.
problems, which were described in the previous section. To [2] Adil Baykaso˘glu and Fehmi Burcin Ozsoydan.
provide for a fair comparison with the literature, the number Adaptive firefly algorithm with chaos for mechanical
of function evaluations was set to 50000, as advised in [2], design optimization problems. Applied Soft
where the authors determined, that such a number is an Computing, 36:152–164, 2015.
StuCoSReC Proceedings of the 2016 3rd Student Computer Science Research Conference 59
Ljubljana, Slovenia, 12 October
99 ∗ 0.0625, and 10 ≤ x3, x4 ≤ 200. The parameters of HSA-CS were set according to the authors
in [8], while the population size was varied as: N p = 30 for
4.3 Spring design welded beam, spring design, and speed reducer, while for
pressure vessel N p = 50. Each experiment was replicated
The spring design optimization problem deals with an opti- 50 times, thus the results reported here are the average of
mal design of a tension spring. The problem consists of three those runs.
variables, which are the number of spring coils, winding di-
ameter, and wire diameter. The objective is to minimize the Table 1 holds the results of the experiments. For each prob-
weight of the spring, subject to minimum deflection, surge lem the minimum (min), maximum (max), mean, median
frequency, shear stress, and limits on the outside diameter. (md), and standard deviation (std) values are reported.
Mathematically it can be formulated as:
f (x) = (x3 + 2) ∗ x12 ∗ x2, The results in Table 1 indicate that the HSA-CS was able
subject to the following constraints: to find the same solution for the welded beam and speed
(8) reducer problems in all 50 runs of the algorithm. On the
contrary, the HSA-CS had trouble in converging towards a
single solution.
g0 : 1− x23x3 ≤ 0, g1 : 4x22 − x2x3 + 1 −1 ≤ 0, 6. DISCUSSION
71785x41 12566(x2x33 − x34) 5108x32
In this section we analyze the results from our experiments
g2 :1− 140.45x1 ≤ 0, g3 : (x1 + x2) − 1.5 ≤0 (9) and compare them to those found in the literature. For this
x22x3 purpose a Table 2 is provided, where results from literature
are gathered. It can be seen, that the HSA-CS achieved com-
The search space of design variables are limited as: 0.05 ≤ petitive results if not better results on all test optimization
x1 ≤ 2, 0.25 ≤ x2 ≤ 1.3, and 2 ≤ x3 ≤ 15. problems. For the welded beam problem our method and
the method in [2] converged to a single solution, whereas
4.4 Speed reducer other methods were not as successful. It is also hard to
say, which method performed the best, since the findings
This problem deals with a minimum weight design of a speed in other papers are reported only to 6 digits. On the pres-
reducer, subject to bending stress of the gear teeth, surface sure vessel problem, the HSA-CS achieved similar results as
stress, stresses in the shafts, and transverse deflections of for the welded beam problem. Based on the mean value
the shafts. This problem is formulated with seven design the only competitive method was again the one proposed
variables, using the following mathematical definition: in [2]. HSA-CS acheived the best results for the speed re-
ducer. Again, like for the welded beam, the results were
f (x) = 0.7854x1x22(3.3333x32 + 14.9334x3 − 43.0934)− unanimous, converging to a single solution, which was also
the smallest based on mean value. Good results were also
1.508x1(x62 + x27) + 7.477(x63 + x37) + 0.7854(x4x26 + x5x27), obtained on the spring design problem, were the HSA-CS
(10) had the smallest std value over the 50 runs, while obtain-
ing good results based on the mean value. We can conclude
subject to: the HSA-CS would be suitable for use in real-life constraint
optimization problems.
g0 : 27.0 − 1.0 ≤ 0, g1 : 397.5 − 1.0 ≤ 0,
x1x22x3 x1 x22 x32
g2 : 1.93x34 − 1.0 ≤ 0, g3 : 1.93x35 − 1.0 ≤ 0,
x2x3x46 x2x3x74
( 745x4 )2 + 16.9 ∗ 10e6) 7. CONCLUSION
g4 : x2 x3 − 1.0 ≤ 0, This paper investigated the recently proposed HSA-CS al-
gorithm, on four well known engineering design optimiza-
(110x36) tion problems with constraints. The problems at hand were:
welded beam, pressure vessel design, speed reducer design,
( 745x5 )2 + 157.5 ∗ 10e6) and spring design. The obtained results were compared to
(85x37) the some state-of-the-art methods, were the HSA-CS per-
g5 : x2 x3 − 1.0 ≤ 0, formed very well, thus we can conclude it would be suitable
for use in real-life engineering applications.
g6 : x2x3 − 1 ≤ 0, g7 : 5x2 − 1 ≤ 0, g8 : x1 − 1 ≤ 0,
40 x1 12x2
g9 : 1.5x6 + 1.9 − 1 ≤ 0, g10 : 1.1x7 + 1.9 − 1.0 ≤ 0. (11)
x4 x5
The search of the design variables is defined as: 8. REFERENCES
(2.6, 0.7, 17.0, 7.3, 7.8, 2.9, 5.0)T ≤ x ≤ (3.6, 0.8, 28.0, 8.3, 8.3, 3.9, 5.5)T [1] Bahriye Akay and Dervis Karaboga. Artificial bee
5. RESULTS colony algorithm for large-scale problems and
engineering design optimization. Journal of Intelligent
The HCS-SA was applied to solve the design optimization Manufacturing, 23(4):1001–1014, 2012.
problems, which were described in the previous section. To [2] Adil Baykaso˘glu and Fehmi Burcin Ozsoydan.
provide for a fair comparison with the literature, the number Adaptive firefly algorithm with chaos for mechanical
of function evaluations was set to 50000, as advised in [2], design optimization problems. Applied Soft
where the authors determined, that such a number is an Computing, 36:152–164, 2015.
StuCoSReC Proceedings of the 2016 3rd Student Computer Science Research Conference 59
Ljubljana, Slovenia, 12 October
