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March  2021, 14(3): 1161-1180. doi: 10.3934/dcdss.2020227

Comparison of modern heuristics on solving the phase stability testing problem

Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13,120 00 Prague 2, Czech Republic

* Corresponding author: Tomáš Smejkal

Received  January 2019 Revised  November 2019 Published  December 2019

In this paper, we are concerned with the phase stability testing at constant volume, temperature, and moles ($ VTN $-specification) of a multicomponent mixture, which is an unconstrained minimization problem. We present and compare the performance of five chosen optimization algorithms: Differential Evolution, Cuckoo Search, Harmony Search, CMA-ES, and Elephant Herding Optimization. For the comparison of the evolution strategies, we use the Wilcoxon signed-rank test. In addition, we compare the evolution strategies with the classical Newton-Raphson method based on the computation times. Moreover, we present the expanded mirroring technique, which mirrors the computed solution into a given simplex.

Citation: Tomáš Smejkal, Jiří Mikyška, Jaromír Kukal. Comparison of modern heuristics on solving the phase stability testing problem. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1161-1180. doi: 10.3934/dcdss.2020227
References:
[1]

A. Auger and N. Hansen, Performance evaluation of an advanced local search evolutionary algorithm, 2005 IEEE Congress on Evolutionary Computation, 2 (2005), 1777-1784.   Google Scholar

[2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, N. J., 1957.   Google Scholar
[3]

J. BrestS. GreinerB. BoškovićM. Mernik and V. Žumer, Self-adapting control parameters in Differential Evolution: A comparative study on numerical benchmark problems, IEEE Transactions on Evolutionary Computation, 10 (2006), 646-657.  doi: 10.1109/TEVC.2006.872133.  Google Scholar

[4]

O. Dunn, Multiple comparisons among means, Journal of the American Statistical Association, 56 (1961), 52-64.  doi: 10.1080/01621459.1961.10482090.  Google Scholar

[5]

Z. W. Geem, Music-Inspired Harmony Search Algorithm: Theory and Applications, Springer, 2001. doi: 10.1007/978-3-642-00185-7.  Google Scholar

[6]

N. Hansen and A. Ostermeier, Completely derandomized self-adaptation in evolution strategies, Evolutionary Computation, 9 (2001), 159-195.  doi: 10.1162/106365601750190398.  Google Scholar

[7]

N. Hansen, The CMA evolution strategy: A tutorial, phCoRR, URL http://arXiv.org/abs/1604.00772. Google Scholar

[8]

T. Jindrová and J. Mikyška, General algorithm for multiphase equilibria calculation at given volume, temperature, and moles, Fluid Phase Equilibria, 393 (2015), 7-25.   Google Scholar

[9]

W. H. Kruskal, Historical notes on the Wilcoxon unpaired two-sample test, Journal of the American Statistical Association, 52 (1957), 356-360.  doi: 10.1080/01621459.1957.10501395.  Google Scholar

[10]

J. Kukal and M. Mojzeš, Quantile and mean value measures of search process complexity, Journal of Combinatorial Optimization, 35 (2018), 1261-1285.  doi: 10.1007/s10878-018-0251-4.  Google Scholar

[11]

M. L. Michelsen, The isothermal flash problem. Part I. Stability, Fluid Phase Equilibria, 9 (1982), 1-19.  doi: 10.1016/0378-3812(82)85001-2.  Google Scholar

[12]

M. L. Michelsen, The isothermal flash problem. Part 2. Phase-split computation, Fluid Phase Equilibria, 9 (1982), 21-40.   Google Scholar

[13]

J. Mikyška and A. Firoozabadi, Investigation of mixture stability at given volume, temperature, and moles, Fluid Phase Equilibria, 321 (2012), 1-9.   Google Scholar

[14]

A. W. MohamedH. Z. Sabry and M. Khorshid, An alternative differential evolution algorithm for global optimization, Journal of Advanced Research, 3 (2012), 149-165.  doi: 10.1016/j.jare.2011.06.004.  Google Scholar

[15]

A. OstermeierA. Gawelczyk and N. Hansen, A derandomized approach to self-adaptation of evolution strategies, Evolutionary Computation, 2 (1994), 369-380.  doi: 10.1162/evco.1994.2.4.369.  Google Scholar

[16]

I. Pavlyukevich, Cooling down Lévy flights, Journal of Physics: A Mathematical and Theoretical, 40 (2007), 12299-12313.  doi: 10.1088/1751-8113/40/41/003.  Google Scholar

[17]

I. Pavlyukevich, Lévy flights, non-local search and simulated annealing, Journal of Computational Physics, 226 (2007), 1830–1844, URL http://www.sciencedirect.com/science/article/pii/S002199910700263X. doi: 10.1016/j.jcp.2007.06.008.  Google Scholar

[18]

D.-Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial & Engineering Chemistry Fundamentals, 15 (1976), 59-64.  doi: 10.1021/i160057a011.  Google Scholar

[19]

O. Polívka and J. Mikyška, Compositional modeling in porous media using constant volume flash and flux computation without the need for phase identification, Journal of Computational Physics, 272 (2014), 149-179.  doi: 10.1016/j.jcp.2014.04.029.  Google Scholar

[20]

K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution. A Practical Approach to Global Optimization, With 1 CD-ROM (Windows, Macintosh and UNIX). Natural Computing Series. Springer-Verlag, Berlin, 2005.  Google Scholar

[21]

A. K. QinV. L. Huang and P. N. Suganthan, Differential Evolution algorithm with strategy adaptation for global numerical optimization, IEEE Transactions on Evolutionary Computation, 13 (2009), 398-417.  doi: 10.1109/TEVC.2008.927706.  Google Scholar

[22]

A. M. Reynolds and M. A. Frye, Free-flight odor tracking in drosophila is consistent with an optimal intermittent scale-free search, PloS ONE, 2 (2007), e354. doi: 10.1371/journal.pone.0000354.  Google Scholar

[23]

R. M. Rizk-Allah, R. A. El-Sehiemy and G.-G. Wang, A novel parallel hurricane optimization algorithm for secure emission/economic load dispatch solution, Applied Soft Computing, 63 (2018), 206–222, URL http://www.sciencedirect.com/science/article/pii/S1568494617307160. doi: 10.1016/j.asoc.2017.12.002.  Google Scholar

[24]

R. B. Schnabel and E. Eskow, A revised modified Cholesky factorization algorithm, SIAM Journal on Optimization, 9 (1999), 1135-1148.  doi: 10.1137/S105262349833266X.  Google Scholar

[25]

T. Smejkal and J. Mikyška, Phase stability testing and phase equilibrium calculation at specified internal energy, volume, and moles, Fluid Phase Equilibria, 431 (2017), 82-96.  doi: 10.1016/j.fluid.2016.09.025.  Google Scholar

[26]

T. Smejkal and J. Mikyška, Unified presentation and comparison of various formulations of the phase stability and phase equilibrium calculation problems, Fluid Phase Equilibria, 476 (2018), 61-88.  doi: 10.1016/j.fluid.2018.03.013.  Google Scholar

[27]

T. Smejkal and J. Mikyška, VTN-phase stability testing using the branch and bound strategy and the convex-concave splitting of the Helmholtz free energy density, Fluid Phase Equilibria, 504 (2020), 112323, URL http://www.sciencedirect.com/science/article/pii/S037838121930384X. doi: 10.1016/j.fluid.2019.112323.  Google Scholar

[28]

R. Storn and K. Price, Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[29]

G. Wang and Y. Tan, Improving metaheuristic algorithms with information feedback models, IEEE Transactions on Cybernetics, 49 (2019), 542-555.   Google Scholar

[30]

G.-G. Wang, S. Deb and L. dos S. Coelho, Elephant herding optimization, 3rd International Symposium on Computational and Business Intelligence, 2015. doi: 10.1109/ISCBI.2015.8.  Google Scholar

[31]

G.-G. WangA. GandomiX.-S. Yang and A. Alavi, A new hybrid method based on Krill Herd and Cuckoo Search for global optimisation tasks, International Journal of Bio-Inspired Computation, 8 (2016), 286-299.  doi: 10.1504/IJBIC.2016.079569.  Google Scholar

[32]

G.-G. Wang, A. H. Gandomi and A. H. Alavi, An effective krill herd algorithm with migration operator in biogeography-based optimization, Applied Mathematical Modelling, 38 (2014), 2454–2462, URL http://www.sciencedirect.com/science/article/pii/S0307904X13006756. doi: 10.1016/j.apm.2013.10.052.  Google Scholar

[33]

G.-G. WangA. H. GandomiA. H. Alavi and G.-S. Hao, Hybrid krill herd algorithm with differential evolution for global numerical optimization, Neural Computing and Applications, 25 (2014), 297-308.  doi: 10.1007/s00521-013-1485-9.  Google Scholar

[34]

G.-G. Wang, L. H. Guo, A. H. Gandomi, G.-S. Hao and H. Q. Wang, Chaotic krill herd algorithm, Information Sciences, 274 (2014), 17-24. doi: 10.1016/j.ins.2014.02.123.  Google Scholar

[35]

G. Wang, L. Guo, H. Wang, H. Duan, L. Liu and J. Li, Incorporating mutation scheme into Krill Herd algorithm for global numerical optimization, Neural Computing and Applications, 24 (2014), 853–871, URL https://doi.org/10.1007/s00521-012-1304-8. Google Scholar

[36]

H. Q. Wang and J.-H. Yi, An improved optimization method based on krill herd and artificial bee colony with information exchange, Memetic Computing, 10 (2018), 177-198.  doi: 10.1007/s12293-017-0241-6.  Google Scholar

[37]

Y. Q. Wu and Z. M. Chen, The application of high-dimensional sparse grids in flash calculations: From theory to realisation, Fluid Phase Equilibria, 464 (2018), 22-31.  doi: 10.1016/j.fluid.2018.02.013.  Google Scholar

[38]

Y. Q. WuC. KowitzS. Y. Sun and A. Salama, Speeding up the flash calculations in two-phase compositional flow simulations - The application of sparse grids, Journal of Computational Physics, 285 (2015), 88-99.  doi: 10.1016/j.jcp.2015.01.012.  Google Scholar

[39]

Y. Q. Wu and M. Q. Ye, A parallel sparse grid construction algorithm based on the shared memory architecture and its application to flash calculations, phComputers & Mathematics with Applications, 77 (2019), 2114–2129, URL http://www.sciencedirect.com/science/article/pii/S0898122118307004. doi: 10.1016/j.camwa.2018.12.008.  Google Scholar

[40]

X.-S. Yang, Cuckoo search via Levy flights, World Congress on Nature Biologically Inspired Computing, (2009), 210–214. Google Scholar

[41]

J.-H. Yi, S. Deb, J. Y. Dong, A. H. Alavi and G.-G. Wang, An improved NSGA-Ⅲ algorithm with adaptive mutation operator for big data optimization problems, Future Generation Computer Systems, 88 (2018), 571–585, URL http://www.sciencedirect.com/science/article/pii/S0167739X18306137. doi: 10.1016/j.future.2018.06.008.  Google Scholar

[42]

J.-H. Yi, L.-N. Xing, G.-G. Wang, J. Dong, A. V. Vasilakos, A. H. Alavi and L. Wang, Behavior of crossover operators in NSGA-Ⅲ for large-scale optimization problems, Information Sciences, 509 (2020), 470–487, URL http://www.sciencedirect.com/science/article/pii/S0020025518308016. doi: 10.1016/j.ins.2018.10.005.  Google Scholar

[43]

J. Zhang and A. C. Sanderson, JADE: Adaptive differential evolution with optional external archive, IEEE Transactions on Evolutionary Computation, 13 (2009), 945-958.   Google Scholar

show all references

References:
[1]

A. Auger and N. Hansen, Performance evaluation of an advanced local search evolutionary algorithm, 2005 IEEE Congress on Evolutionary Computation, 2 (2005), 1777-1784.   Google Scholar

[2] R. Bellman, Dynamic Programming, Princeton University Press, Princeton, N. J., 1957.   Google Scholar
[3]

J. BrestS. GreinerB. BoškovićM. Mernik and V. Žumer, Self-adapting control parameters in Differential Evolution: A comparative study on numerical benchmark problems, IEEE Transactions on Evolutionary Computation, 10 (2006), 646-657.  doi: 10.1109/TEVC.2006.872133.  Google Scholar

[4]

O. Dunn, Multiple comparisons among means, Journal of the American Statistical Association, 56 (1961), 52-64.  doi: 10.1080/01621459.1961.10482090.  Google Scholar

[5]

Z. W. Geem, Music-Inspired Harmony Search Algorithm: Theory and Applications, Springer, 2001. doi: 10.1007/978-3-642-00185-7.  Google Scholar

[6]

N. Hansen and A. Ostermeier, Completely derandomized self-adaptation in evolution strategies, Evolutionary Computation, 9 (2001), 159-195.  doi: 10.1162/106365601750190398.  Google Scholar

[7]

N. Hansen, The CMA evolution strategy: A tutorial, phCoRR, URL http://arXiv.org/abs/1604.00772. Google Scholar

[8]

T. Jindrová and J. Mikyška, General algorithm for multiphase equilibria calculation at given volume, temperature, and moles, Fluid Phase Equilibria, 393 (2015), 7-25.   Google Scholar

[9]

W. H. Kruskal, Historical notes on the Wilcoxon unpaired two-sample test, Journal of the American Statistical Association, 52 (1957), 356-360.  doi: 10.1080/01621459.1957.10501395.  Google Scholar

[10]

J. Kukal and M. Mojzeš, Quantile and mean value measures of search process complexity, Journal of Combinatorial Optimization, 35 (2018), 1261-1285.  doi: 10.1007/s10878-018-0251-4.  Google Scholar

[11]

M. L. Michelsen, The isothermal flash problem. Part I. Stability, Fluid Phase Equilibria, 9 (1982), 1-19.  doi: 10.1016/0378-3812(82)85001-2.  Google Scholar

[12]

M. L. Michelsen, The isothermal flash problem. Part 2. Phase-split computation, Fluid Phase Equilibria, 9 (1982), 21-40.   Google Scholar

[13]

J. Mikyška and A. Firoozabadi, Investigation of mixture stability at given volume, temperature, and moles, Fluid Phase Equilibria, 321 (2012), 1-9.   Google Scholar

[14]

A. W. MohamedH. Z. Sabry and M. Khorshid, An alternative differential evolution algorithm for global optimization, Journal of Advanced Research, 3 (2012), 149-165.  doi: 10.1016/j.jare.2011.06.004.  Google Scholar

[15]

A. OstermeierA. Gawelczyk and N. Hansen, A derandomized approach to self-adaptation of evolution strategies, Evolutionary Computation, 2 (1994), 369-380.  doi: 10.1162/evco.1994.2.4.369.  Google Scholar

[16]

I. Pavlyukevich, Cooling down Lévy flights, Journal of Physics: A Mathematical and Theoretical, 40 (2007), 12299-12313.  doi: 10.1088/1751-8113/40/41/003.  Google Scholar

[17]

I. Pavlyukevich, Lévy flights, non-local search and simulated annealing, Journal of Computational Physics, 226 (2007), 1830–1844, URL http://www.sciencedirect.com/science/article/pii/S002199910700263X. doi: 10.1016/j.jcp.2007.06.008.  Google Scholar

[18]

D.-Y. Peng and D. B. Robinson, A new two-constant equation of state, Industrial & Engineering Chemistry Fundamentals, 15 (1976), 59-64.  doi: 10.1021/i160057a011.  Google Scholar

[19]

O. Polívka and J. Mikyška, Compositional modeling in porous media using constant volume flash and flux computation without the need for phase identification, Journal of Computational Physics, 272 (2014), 149-179.  doi: 10.1016/j.jcp.2014.04.029.  Google Scholar

[20]

K. V. Price, R. M. Storn and J. A. Lampinen, Differential Evolution. A Practical Approach to Global Optimization, With 1 CD-ROM (Windows, Macintosh and UNIX). Natural Computing Series. Springer-Verlag, Berlin, 2005.  Google Scholar

[21]

A. K. QinV. L. Huang and P. N. Suganthan, Differential Evolution algorithm with strategy adaptation for global numerical optimization, IEEE Transactions on Evolutionary Computation, 13 (2009), 398-417.  doi: 10.1109/TEVC.2008.927706.  Google Scholar

[22]

A. M. Reynolds and M. A. Frye, Free-flight odor tracking in drosophila is consistent with an optimal intermittent scale-free search, PloS ONE, 2 (2007), e354. doi: 10.1371/journal.pone.0000354.  Google Scholar

[23]

R. M. Rizk-Allah, R. A. El-Sehiemy and G.-G. Wang, A novel parallel hurricane optimization algorithm for secure emission/economic load dispatch solution, Applied Soft Computing, 63 (2018), 206–222, URL http://www.sciencedirect.com/science/article/pii/S1568494617307160. doi: 10.1016/j.asoc.2017.12.002.  Google Scholar

[24]

R. B. Schnabel and E. Eskow, A revised modified Cholesky factorization algorithm, SIAM Journal on Optimization, 9 (1999), 1135-1148.  doi: 10.1137/S105262349833266X.  Google Scholar

[25]

T. Smejkal and J. Mikyška, Phase stability testing and phase equilibrium calculation at specified internal energy, volume, and moles, Fluid Phase Equilibria, 431 (2017), 82-96.  doi: 10.1016/j.fluid.2016.09.025.  Google Scholar

[26]

T. Smejkal and J. Mikyška, Unified presentation and comparison of various formulations of the phase stability and phase equilibrium calculation problems, Fluid Phase Equilibria, 476 (2018), 61-88.  doi: 10.1016/j.fluid.2018.03.013.  Google Scholar

[27]

T. Smejkal and J. Mikyška, VTN-phase stability testing using the branch and bound strategy and the convex-concave splitting of the Helmholtz free energy density, Fluid Phase Equilibria, 504 (2020), 112323, URL http://www.sciencedirect.com/science/article/pii/S037838121930384X. doi: 10.1016/j.fluid.2019.112323.  Google Scholar

[28]

R. Storn and K. Price, Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[29]

G. Wang and Y. Tan, Improving metaheuristic algorithms with information feedback models, IEEE Transactions on Cybernetics, 49 (2019), 542-555.   Google Scholar

[30]

G.-G. Wang, S. Deb and L. dos S. Coelho, Elephant herding optimization, 3rd International Symposium on Computational and Business Intelligence, 2015. doi: 10.1109/ISCBI.2015.8.  Google Scholar

[31]

G.-G. WangA. GandomiX.-S. Yang and A. Alavi, A new hybrid method based on Krill Herd and Cuckoo Search for global optimisation tasks, International Journal of Bio-Inspired Computation, 8 (2016), 286-299.  doi: 10.1504/IJBIC.2016.079569.  Google Scholar

[32]

G.-G. Wang, A. H. Gandomi and A. H. Alavi, An effective krill herd algorithm with migration operator in biogeography-based optimization, Applied Mathematical Modelling, 38 (2014), 2454–2462, URL http://www.sciencedirect.com/science/article/pii/S0307904X13006756. doi: 10.1016/j.apm.2013.10.052.  Google Scholar

[33]

G.-G. WangA. H. GandomiA. H. Alavi and G.-S. Hao, Hybrid krill herd algorithm with differential evolution for global numerical optimization, Neural Computing and Applications, 25 (2014), 297-308.  doi: 10.1007/s00521-013-1485-9.  Google Scholar

[34]

G.-G. Wang, L. H. Guo, A. H. Gandomi, G.-S. Hao and H. Q. Wang, Chaotic krill herd algorithm, Information Sciences, 274 (2014), 17-24. doi: 10.1016/j.ins.2014.02.123.  Google Scholar

[35]

G. Wang, L. Guo, H. Wang, H. Duan, L. Liu and J. Li, Incorporating mutation scheme into Krill Herd algorithm for global numerical optimization, Neural Computing and Applications, 24 (2014), 853–871, URL https://doi.org/10.1007/s00521-012-1304-8. Google Scholar

[36]

H. Q. Wang and J.-H. Yi, An improved optimization method based on krill herd and artificial bee colony with information exchange, Memetic Computing, 10 (2018), 177-198.  doi: 10.1007/s12293-017-0241-6.  Google Scholar

[37]

Y. Q. Wu and Z. M. Chen, The application of high-dimensional sparse grids in flash calculations: From theory to realisation, Fluid Phase Equilibria, 464 (2018), 22-31.  doi: 10.1016/j.fluid.2018.02.013.  Google Scholar

[38]

Y. Q. WuC. KowitzS. Y. Sun and A. Salama, Speeding up the flash calculations in two-phase compositional flow simulations - The application of sparse grids, Journal of Computational Physics, 285 (2015), 88-99.  doi: 10.1016/j.jcp.2015.01.012.  Google Scholar

[39]

Y. Q. Wu and M. Q. Ye, A parallel sparse grid construction algorithm based on the shared memory architecture and its application to flash calculations, phComputers & Mathematics with Applications, 77 (2019), 2114–2129, URL http://www.sciencedirect.com/science/article/pii/S0898122118307004. doi: 10.1016/j.camwa.2018.12.008.  Google Scholar

[40]

X.-S. Yang, Cuckoo search via Levy flights, World Congress on Nature Biologically Inspired Computing, (2009), 210–214. Google Scholar

[41]

J.-H. Yi, S. Deb, J. Y. Dong, A. H. Alavi and G.-G. Wang, An improved NSGA-Ⅲ algorithm with adaptive mutation operator for big data optimization problems, Future Generation Computer Systems, 88 (2018), 571–585, URL http://www.sciencedirect.com/science/article/pii/S0167739X18306137. doi: 10.1016/j.future.2018.06.008.  Google Scholar

[42]

J.-H. Yi, L.-N. Xing, G.-G. Wang, J. Dong, A. V. Vasilakos, A. H. Alavi and L. Wang, Behavior of crossover operators in NSGA-Ⅲ for large-scale optimization problems, Information Sciences, 509 (2020), 470–487, URL http://www.sciencedirect.com/science/article/pii/S0020025518308016. doi: 10.1016/j.ins.2018.10.005.  Google Scholar

[43]

J. Zhang and A. C. Sanderson, JADE: Adaptive differential evolution with optional external archive, IEEE Transactions on Evolutionary Computation, 13 (2009), 945-958.   Google Scholar

Figure 1.  Geometric interpretation of mirroring into the feasible simplex
Figure 2.  Global minimum of the $ \mathrm{TPD} $ function in the $ cT $-space and the number of successful runs of each evolution heuristic. The red line represents the phase boundary (above the line the global minimum is zero). Example 1: mixture C$ _1 $-C$ _3 $
Figure 3.  Global minimum of the $ \mathrm{TPD} $ function in the $ cT $-space and the number of successful runs of each evolution heuristic. The red line represents the phase boundary (above the line the global minimum is zero). Example 2: mixture N$ _2 $-CO$ _2 $-C$ _1 $-PC$ _i $
Table 1.  The advantages and disadvantages of the chosen evolution algorithms
advantages disadvantages
DE ● good convergence properties ● parameter tuning is necessary
● strong theoretical analysis ● easy to stuck
in a local minimum
CS ● supports local and global search ● small precision
● easy to hybridize ● no theoretical analysis
HS ● simple implementation ● slow convergence
● small population ● small precision
CMA-ES ● no curse of dimensionality [2] ● harder implementation
● in-variance properties ● a lot of parameters
EHO ● hard to stuck in a local minimum ● slow convergence
● fewer parameters ● fixed parameters
advantages disadvantages
DE ● good convergence properties ● parameter tuning is necessary
● strong theoretical analysis ● easy to stuck
in a local minimum
CS ● supports local and global search ● small precision
● easy to hybridize ● no theoretical analysis
HS ● simple implementation ● slow convergence
● small population ● small precision
CMA-ES ● no curse of dimensionality [2] ● harder implementation
● in-variance properties ● a lot of parameters
EHO ● hard to stuck in a local minimum ● slow convergence
● fewer parameters ● fixed parameters
Table 2.  Parameters of the Peng-Robinson equation of state used in Examples 1–2
Component $ T_{\mathrm{crit}} $ [K] $ P_{\mathrm{crit}} $ [MPa] $ \omega $ [-]
C$ _1 $ 190.40 4.60 0.0110
C$ _3 $ 369.80 4.25 0.1530
CO$ _2 $ 304.14 7.375 0.2390
N$ _2 $ 126.21 3.390 0.0390
PC$ _1 $ 333.91 5.329 0.1113
PC$ _2 $ 456.25 3.445 0.2344
PC$ _3 $ 590.76 2.376 0.4470
C$ _{12+} $ 742.58 1.341 0.9125
Component $ T_{\mathrm{crit}} $ [K] $ P_{\mathrm{crit}} $ [MPa] $ \omega $ [-]
C$ _1 $ 190.40 4.60 0.0110
C$ _3 $ 369.80 4.25 0.1530
CO$ _2 $ 304.14 7.375 0.2390
N$ _2 $ 126.21 3.390 0.0390
PC$ _1 $ 333.91 5.329 0.1113
PC$ _2 $ 456.25 3.445 0.2344
PC$ _3 $ 590.76 2.376 0.4470
C$ _{12+} $ 742.58 1.341 0.9125
Table 3.  The binary interaction coefficients between all components in Example 2
Component N$ _2 $ CO$ _2 $ C$ _1 $ PC$ _1 $ PC$ _2 $ PC$ _3 $ C$ _{12+} $
N$ _2 $ 0.000 0.000 0.100 0.100 0.100 0.100 0.100
CO$ _2 $ 0.000 0.000 0.150 0.150 0.150 0.150 0.150
C$ _1 $ 0.100 0.150 0.000 0.035 0.040 0.049 0.069
PC$ _1 $ 0.100 0.150 0.035 0.000 0.000 0.000 0.000
PC$ _2 $ 0.100 0.150 0.040 0.000 0.000 0.000 0.000
PC$ _3 $ 0.100 0.150 0.049 0.000 0.000 0.000 0.000
C$ _{12+} $ 0.100 0.150 0.069 0.000 0.000 0.000 0.000
Component N$ _2 $ CO$ _2 $ C$ _1 $ PC$ _1 $ PC$ _2 $ PC$ _3 $ C$ _{12+} $
N$ _2 $ 0.000 0.000 0.100 0.100 0.100 0.100 0.100
CO$ _2 $ 0.000 0.000 0.150 0.150 0.150 0.150 0.150
C$ _1 $ 0.100 0.150 0.000 0.035 0.040 0.049 0.069
PC$ _1 $ 0.100 0.150 0.035 0.000 0.000 0.000 0.000
PC$ _2 $ 0.100 0.150 0.040 0.000 0.000 0.000 0.000
PC$ _3 $ 0.100 0.150 0.049 0.000 0.000 0.000 0.000
C$ _{12+} $ 0.100 0.150 0.069 0.000 0.000 0.000 0.000
Table 4.  Computation times in seconds for Examples 1–2
Example 1 Example 2
Newton-Raphson 0.99 11.55
Differential Evolution 35.87 995.03
Cuckoo Search 78.55 394.63
Harmony Search 210.72 862.48
CMA-ES 26.48 408.91
Elephant Herding Optimization 500.30 1777.72
Example 1 Example 2
Newton-Raphson 0.99 11.55
Differential Evolution 35.87 995.03
Cuckoo Search 78.55 394.63
Harmony Search 210.72 862.48
CMA-ES 26.48 408.91
Elephant Herding Optimization 500.30 1777.72
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