# Adaptive Latin Hypercube Sampling for a Surrogate-Based Optimization with Artificial Neural Network

^{*}

## Abstract

**:**

## 1. Introduction

_{2}hydrogenation, and (3) CO

_{2}capture using the Rectisol process to represent three different types of surrogate-based optimization problems. Section 5 offers results, discussion, recommendations, and future work. Finally, Section 6 presents the conclusion.

## 2. Sampling Technique and Surrogate Modeling

#### 2.1. Latin Hypercube Sampling

#### 2.2. Random Sampling

#### 2.3. Artificial Neural Networks (ANNs)

## 3. Methodology

#### 3.1. Proposed Adaptive LHS for Surrogate-Based Optimization Algorithm

_{kn}(with the desired number of sample points) was generated using Latin hypercube sampling where k = 1, 2, 3, …, K and n = 1, 2, 3, …, N

_{0}, (K is number of decision variables, and N is number of sample points). The initial number of sample points was set at seven times the number of factors. The corresponding output values, y

_{n}, were obtained from process simulation combined with an economic analysis. The sets of input and output were normalized in the range of −1 to 1 for ANNs training using MATLAB with a three-layer design (input layer, hidden layer, and output layer).

^{2}) value. The training loop for ANNs was terminated when the MSE value of the ANN model remained at a minimum for 5 consecutive iterations, and the R

^{2}value of the model was greater than or equal to 0.995. The weights and biases of the most recent ANN model were then used as the objective function for the optimization problem to determine the optimal solution (x

_{i,opt}and ${\widehat{y}}_{opt}$). The optimal operating conditions (x

_{i,opt}) were then input into the process simulation to obtain the corresponding output value (y

_{opt}). The accuracy of the obtained optimal solution is deemed acceptable if the percent error between the predicted output value (obtained from the ANN model) and the actual output value (obtained from process simulation), as shown in Equation (3), is within a preset value of 1 percent. If the percent error is less than one, the optimization algorithm is terminated, and the most recent optimal solution is reported as the optimal solution of the problem.

_{i,opt}, y

_{opt}), while the other two data points are generated using the adaptive Latin hypercube sampling method (as described in Section 3.2). The updated dataset was then normalized, and these steps were repeated until the percent error in the output falls below one, ultimately yielding the optimal solution.

#### 3.2. Adaptive Latin Hypercube Sampling: Addition of Sample Points

_{0}+ 3i) predicted outputs is calculated for each sample point in the dataset, where i represents the number of iterations. The sample point that exhibits the highest deviation, denoted as (x

_{n}, y

_{n,max deviation}), is selected. Subsequently, the intervals corresponding to the sample point with the highest deviation (x

_{n}, y

_{n,max deviation}) are evenly divided into two intervals. Two additional sample points are then randomly selected from each of these intervals.

_{1}and x

_{2}. The highest deviation sample point (x

_{n}, y

_{n,max deviation}) is represented as a red dot. The intervals corresponding to these sample points are within the ranges of 0.4 to 0.6 for factor 1 and 0.2 to 0.4 for factor 2. Both factors are evenly divided into two intervals: 0.4 to 0.5 and 0.5 to 0.6 for factor 1, and 0.2 to 0.3 and 0.3 to 0.4 for factor 2 (as indicated by dashed lines). Next, one sample point is randomly selected from each interval of the factors. A total of two additional sample points are obtained and represented as blue dots.

#### 3.3. Verification of the Optimal Solution Using Random Sampling Technique

## 4. Case Study

_{2}absorption by methanol using the Rectisol process, which involves five factors. The details of each case study are provided below.

#### 4.1. Process Simulation and Economic Evaluation

#### 4.1.1. Case Study 1: Ammonia Production from Syngas

_{2}removal, and methanation (Equations (14) and (15)). The reactions for each step are as follows.

#### 4.1.2. Case Study 2: Methanol Production via Carbon Dioxide Hydrogenation

#### 4.1.3. Case Study 2: Methanol Production via Carbon Dioxide Hydrogenation

_{2}absorption by methanol via the Rectisol process. The process consists of three main sections: water removal, absorption, and CO

_{2}/methanol separation. The syngas feed contains a mole fraction of 0.2462 of CO

_{2}, 0.0002 of NH

_{3}, 0.0044 of CO, 0.0050 of Ar, 0.4148 of N

_{2}, 0.3186 of H

_{2}, 0.0035 of H

_{2}O, and 0.0073 of CH

_{4}at 17.24 bar and 18.30 °C [59]. A small amount of methanol was mixed with the syngas feed and sent to the first separator to separate water from the feed gas. Subsequently, the syngas was fed to an absorber column, where CO

_{2}gas was captured by chilled methanol added at the top of the column. The rich methanol then passes through a three-stage separator and is fed into a stripper column to separate CO

_{2}from methanol. The lean methanol leaving the stripper was mixed with 40 kmole per hour of makeup methanol and recycled to the absorber. The vapor product from the stripper, primarily containing CO

_{2}, was combined to other CO

_{2}product streams from the second and third separators. Details for replicating the process simulation are provided in Supplementary Information S1.3.

## 5. Results and Discussion

#### 5.1. The Results of Monte Carlo or Random Sampling

^{2}value of the model has to be greater than or equal to 0.995. The criterion for the optimal solution is that the percent error of the optimal cost has to be less than one (1).

_{2}absorption by methanol via the Rectisol process), the optimization problem required 400 sample points to meet the criteria. The minimum CO

_{2}capture cost was USD 43.40.

#### 5.2. The Convergence of the Proposed Adaptive LSH Optimization Algorithm

#### 5.3. Comparison of Optimal Solutions between Proposed Sampling and Random Sampling

_{3}) in the third replication. In the third replication, the steam’s temperature entering the separator (x

_{3}) is 76 °C, which was slightly different from the temperatures obtained in the other replications (80 °C). A slight decrease in the temperature of the stream entering the separator resulted in an increase in the methanol production cost. This change did not have a significant impact in terms of model prediction error. Furthermore, the third replication used only 40 sample points to represent the entire surface, while the other two replications used 46 sample points. This indicates that the model with 40 sample points found a local optimum. The optimal operating conditions for this case study were a pressure of 70 bar for the equilibrium reactor, a temperature of 190 °C for the reactor, a temperature of 80 °C for the steam entering a separator, and a recycling ratio of 1. The lowest cost to generate methanol was USD 942.45 per tonne of methanol. While random sampling required 100 sample points to find the optimal solution for this problem with four decision variables, the proposed adaptive LHS optimization approach only required 46 sample points.

_{1}). The results for the lean methanol temperature (x

_{1}) from three replications varied from −26.70 to −20.0 °C, which was slightly different from the values obtained from random sampling (−29.0 °C). The lowest CO

_{2}capture costs identified by the proposed algorithm for three replications ranged from USD 43.25 to USD 45.91 per ton of CO

_{2}capture, which is consistent with the lowest cost identified by random sampling (USD 43.40 per ton of CO

_{2}capture). The optimal operating parameters for this case study included a lean methanol temperature of −26.70 °C, a third-stage separator pressure of 1.28 bar, a stripper reflux ratio of 5, a stripper input temperature of 40 °C, and a distillation reflux ratio of 1. The minimum cost for CO

_{2}capture was USD 43.19 per tonne of CO

_{2}. The proposed technique required only 53 sample points for this problem with five decision variables, while random sampling needed 400 sample points to achieve the same level of ANN model accuracy and obtain the optimal solutions.

#### 5.4. Recommendation for Future Work

## 6. Conclusions

^{2}) values. The criterion for determining the accuracy of the optimal solution was that the percent error should be less than one percent. The results demonstrate that the proposed algorithm is capable of obtaining optimal solutions that are similar to the random sampling approach but require fewer sample points.

## Supplementary Materials

_{2}absorption by methanol via Rectisol process, S2.1: Final dataset of Case Study I: ammonia production from syngas, S2.2: Final dataset of Case Study II: methanol production via carbon dioxide hydrogenation, S2.3: Final dataset of Case Study III: CO

_{2}absorption by methanol via Rectisol process, S3.1: Weights and biases of an artificial neural networks for training the dataset of random sampling, S3.1.1 Case study I: ammonia production from syngas, S3.1.2 Case study II: methanol production via carbon dioxide hydrogenation, S3.1.3 Case study III: CO

_{2}absorption by methanol via Rectisol process, S3.2: Weights and biases of an artificial neural networks for training the dataset of adaptive Latin hypercube sampling, S3.2.1 Case study I: ammonia production from syngas, S3.2.2 Case study II: methanol production via carbon dioxide hydrogenation, S3.2.3 Case study III: CO

_{2}absorption by methanol via Rectisol process.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Crombecq, K.; Laermans, E.; Dhaene, T. Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling. Eur. J. Oper. Res.
**2011**, 214, 683–696. [Google Scholar] [CrossRef] - Ahmadi, S.; Mesbah, M.; Igwegbe, C.A.; Ezeliora, C.D.; Osagie, C.; Khan, N.A.; Dotto, G.L.; Salari, M.; Dehghani, M.H. Sono electro-chemical synthesis of LaFeO3 nanoparticles for the removal of fluoride: Optimization and modeling using RSM, ANN and GA tools. J. Environ. Chem. Eng.
**2021**, 9, 105320. [Google Scholar] [CrossRef] - Bashir, M.J.; Amr, S.A.; Aziz, S.Q.; Ng, C.A.; Sethupathi, S. Wastewater treatment processes optimization using response surface methodology (RSM) compared with conventional methods: Review and comparative study. Middle-East J. Sci. Res.
**2015**, 23, 244–252. [Google Scholar] - Bezerra, M.A.; Santelli, R.E.; Oliveira, E.P.; Villar, L.S.; Escaleira, L.A. Response surface methodology (RSM) as a tool for optimization in analytical chemistry. Talanta
**2008**, 76, 965–977. [Google Scholar] [CrossRef] - Hoseiny, S.; Zare, Z.; Mirvakili, A.; Setoodeh, P.; Rahimpour, M. Simulation–based optimization of operating parameters for methanol synthesis process: Application of response surface methodology for statistical analysis. J. Nat. Gas Sci. Eng.
**2016**, 34, 439–448. [Google Scholar] [CrossRef] - Leonzio, G. Optimization through Response Surface Methodology of a Reactor Producing Methanol by the Hydrogenation of Carbon Dioxide. Processes
**2017**, 5, 62. [Google Scholar] [CrossRef] - Mohammad, N.K.; Ghaemi, A.; Tahvildari, K. Hydroxide modified activated alumina as an adsorbent for CO
_{2}adsorption: Experimental and modeling. Int. J. Greenh. Gas Control**2019**, 88, 24–37. [Google Scholar] [CrossRef] - Veljković, V.B.; Velickovic, A.V.; Avramović, J.M.; Stamenković, O.S. Modeling of biodiesel production: Performance comparison of Box–Behnken, face central composite and full factorial design. Chin. J. Chem. Eng.
**2018**, 27, 1690–1698. [Google Scholar] [CrossRef] - Simpson, T.W.; Poplinski, J.D.; Koch, P.N.; Allen, J.K. Metamodels for Computer-Based Engineering Design: Survey and Recommendations. Eng. Comput.
**2001**, 17, 129–150. [Google Scholar] [CrossRef] - Beers, W.C.M.v.; Kleijnen, J.P.C. Kriging interpolation in simulation: A survey. In Proceedings of the 2004 Winter Simulation Conference 2004, Washington, DC, USA, 5–8 December 2004. [Google Scholar]
- Eason, J.; Cremaschi, S. Adaptive sequential sampling for surrogate model generation with artificial neural networks. Comput. Chem. Eng.
**2014**, 68, 220–232. [Google Scholar] [CrossRef] - Bliek, L. A Survey on Sustainable Surrogate-Based Optimisation. Sustainability
**2022**, 14, 3867. [Google Scholar] [CrossRef] - Panerati, J.; Schnellmann, M.A.; Patience, C.; Beltrame, G.; Patience, G.S. Experimental methods in chemical engineering: Artificial neural networks–ANNs. Can. J. Chem. Eng.
**2019**, 97, 2372–2382. [Google Scholar] [CrossRef] - Jin, Y.; Li, J.; Du, W.; Qian, F. Adaptive Sampling for Surrogate Modelling with Artificial Neural Network and Its Application in an Industrial Cracking Furnace. Can. J. Chem. Eng.
**2016**, 94, 262–272. [Google Scholar] [CrossRef] - Vaklieva-Bancheva, N.G.; Vladova, R.K.; Kirilova, E.G. Simulation of heat-integrated autothermal thermophilic aerobic digestion system operating under uncertainties through artificial neural network. Chem. Eng. Trans.
**2019**, 76, 325–330. [Google Scholar] - Hoseinian, F.S.; Rezai, B.; Kowsari, E.; Safari, M. A hybrid neural network/genetic algorithm to predict Zn(II) removal by ion flotation. Sep. Sci. Technol.
**2020**, 55, 1197–1206. [Google Scholar] [CrossRef] - Bahrami, S.; Ardejani, F.D.; Baafi, E. Application of artificial neural network coupled with genetic algorithm and simulated annealing to solve groundwater inflow problem to an advancing open pit mine. J. Hydrol.
**2016**, 536, 471–484. [Google Scholar] [CrossRef] - Mousavi, S.M.; Mostafavi, E.S.; Jiao, P. Next generation prediction model for daily solar radiation on horizontal surface using a hybrid neural network and simulated annealing method. Energy Convers. Manag.
**2017**, 153, 671–682. [Google Scholar] [CrossRef] - Zendehboudi, S.; Rezaei, N.; Lohi, A. Applications of hybrid models in chemical, petroleum, and energy systems: A systematic review. Appl. Energy
**2018**, 228, 2539–2566. [Google Scholar] [CrossRef] - Eslamimanesh, A.; Gharagheizi, F.; Mohammadi, A.H.; Richon, D. Artificial neural network modeling of solubility of supercritical carbon dioxide in 24 commonly used ionic liquids. Chem. Eng. Sci.
**2011**, 66, 3039–3044. [Google Scholar] [CrossRef] - Mohammadpoor, M.; Torabi, F. A new soft computing-based approach to predict oil production rate for vapour extraction (VAPEX) process in heavy oil reservoirs. Can. J. Chem. Eng.
**2018**, 96, 1273–1283. [Google Scholar] [CrossRef] - Bhutani, N.; Rangaiah, G.; Ray, A. First-principles, data-based, and hybrid modeling and optimization of an industrial hydrocracking unit. Ind. Eng. Chem. Res.
**2006**, 45, 7807–7816. [Google Scholar] [CrossRef] - Zorzetto, L.F.M.; Filho, R.M.; Wolf-Maciel, M. Processing modelling development through artificial neural networks and hybrid models. Comput. Chem. Eng.
**2000**, 24, 1355–1360. [Google Scholar] [CrossRef] - Tahkola, M.; Keranen, J.; Sedov, D.; Far, M.F.; Kortelainen, J. Surrogate Modeling of Electrical Machine Torque Using Artificial Neural Networks. IEEE Access
**2020**, 8, 220027–220045. [Google Scholar] [CrossRef] - Adeli, H. Neural Networks in Civil Engineering: 1989–2000. Comput.-Aided Civ. Infrastruct. Eng.
**2001**, 16, 126–142. [Google Scholar] [CrossRef] - Arora, S.; Shen, W.; Kapoor, A. Neural network based computational model for estimation of heat generation in LiFePO
_{4}pouch cells of different nominal capacities. Comput. Chem. Eng.**2017**, 101, 81–94. [Google Scholar] [CrossRef] - Himmelblau, D.M. Applications of artificial neural networks in chemical engineering. Korean J. Chem. Eng.
**2000**, 17, 373–392. [Google Scholar] [CrossRef] - Shahin, M.A.; Jaksa, M.B.; Maier, H.R. State of the art of artificial neural networks in geotechnical engineering. Electron. J. Geotech. Eng.
**2008**, 8, 1–26. [Google Scholar] - Nayak, R.; Jain, L.C.; Ting, B.K.H. Artificial Neural Networks in Biomedical Engineering: A Review. In Computational Mechanics–New Frontiers for the New Millennium; Valliappan, S., Khalili, N., Eds.; Elsevier: Oxford, UK, 2001; pp. 887–892. [Google Scholar]
- Wang, C.; Luo, Z. A Review of the Optimal Design of Neural Networks Based on FPGA. Appl. Sci.
**2022**, 12, 771. [Google Scholar] [CrossRef] - Garud, S.S.; Karimi, I.A.; Kraft, M. Design of computer experiments: A review. Comput. Chem. Eng.
**2017**, 106, 71–95. [Google Scholar] [CrossRef] - Morshed, M.N.; Pervez, N.; Behary, N.; Bouazizi, N.; Guan, J.; Nierstrasz, V.A. Statistical Modeling and Optimization of Heterogeneous Fenton-like Removal of Organic Pollutant Using Fibrous Catalysts: A Full Factorial Design. Sci. Rep.
**2020**, 10, 16133. [Google Scholar] [CrossRef] - Villa Montoya, A.C.; Mazareli, R.C.d.S.; Delforno, T.P.; Centurion, V.B.; de Oliveira, V.M.; Silva, E.L.; Varesche, M.B.A. Optimization of key factors affecting hydrogen production from coffee waste using factorial design and metagenomic analysis of the microbial community. Int. J. Hydrogen Energy
**2020**, 45, 4205–4222. [Google Scholar] [CrossRef] - McKay, M.D.; Beckman, R.J.; Conover, W.J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code. Technometrics
**1979**, 21, 239–245. [Google Scholar] - Sheikholeslami, R.; Razavi, S. Progressive Latin Hypercube Sampling: An Efficient Approach for Robust Sampling-Based Analysis of Environmental Models. Environ. Model. Softw.
**2017**, 93, 109–126. [Google Scholar] [CrossRef] - Grosso, A.; Jamali, A.R.M.J.U.; Locatelli, M. Finding Maximin Latin Hypercube Designs by Iterated Local Search heuristics. Eur. J. Oper. Res.
**2009**, 197, 541–547. [Google Scholar] [CrossRef] - Morris, M.D.; Mitchell, T.J. Exploratory designs for computational experiments. J. Stat. Plan. Inference
**1995**, 43, 381–402. [Google Scholar] [CrossRef] - Pholdee, N.; Bureerat, S. An efficient optimum Latin hypercube sampling technique based on sequencing optimisation using simulated annealing. Int. J. Syst. Sci.
**2015**, 46, 1780–1789. [Google Scholar] [CrossRef] - Park, J.-S. Optimal Latin-hypercube designs for computer experiments. J. Stat. Plan. Inference
**1994**, 39, 95–111. [Google Scholar] [CrossRef] - Ye, K.Q.; Li, W.; Sudjianto, A. Algorithmic construction of optimal symmetric Latin hypercube designs. J. Stat. Plan. Inference
**2000**, 90, 145–159. [Google Scholar] [CrossRef] - Fang, K.-T. Theory, Method and Applications of the Uniform Design. Int. J. Reliab. Qual. Saf. Eng.
**2002**, 9, 305–315. [Google Scholar] [CrossRef] - Bates, S.; Sienz, J.; Toropov, V. Formulation of the Optimal Latin Hypercube Design of Experiments Using a Permutation Genetic Algorithm. In Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, CA, USA, 19–22 April 2004; American Institute of Aeronautics and Astronautics: Reston VA, USA, 2004. [Google Scholar]
- Jin, R.; Chen, W.; Sudjianto, A. An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Inference
**2005**, 134, 268–287. [Google Scholar] [CrossRef] - Van Dam, E.R.; Husslage, B.; Hertog, D.D.; Melissen, H. Maximin Latin Hypercube Designs in Two Dimensions. Oper. Res.
**2007**, 55, 158–169. [Google Scholar] [CrossRef] - Viana, F.A.C.; Venter, G.; Balabanov, V. An algorithm for fast optimal Latin hypercube design of experiments. Int. J. Numer. Methods Eng.
**2010**, 82, 135–156. [Google Scholar] [CrossRef] - Aziz, M.; Tayarani-N, M.-H. An adaptive memetic Particle Swarm Optimization algorithm for finding large-scale Latin hypercube designs. Eng. Appl. Artif. Intell.
**2014**, 36, 222–237. [Google Scholar] [CrossRef] - Chen, R.-B.; Hsieh, D.-N.; Hung, Y.; Wang, W. Optimizing Latin hypercube designs by particle swarm. Stat. Comput.
**2013**, 23, 663–676. [Google Scholar] [CrossRef] - Pan, G.; Ye, P.; Wang, P. A Novel Latin hypercube algorithm via translational propagation. Sci. World J.
**2014**, 2014, 163949. [Google Scholar] [CrossRef] [PubMed] - Husslage, B.G.M.; Rennen, G.; van Dam, E.R.; Hertog, D.D. Space-filling Latin hypercube designs for computer experiments. Optim. Eng.
**2011**, 12, 611–630. [Google Scholar] [CrossRef] - Wang, G.G. Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points. J. Mech. Des.
**2003**, 125, 210–220. [Google Scholar] [CrossRef] - Chang, Y.; Sun, Z.; Sun, W.; Song, Y. A New Adaptive Response Surface Model for Reliability Analysis of 2.5D C/SiC Composite Turbine Blade. Appl. Compos. Mater.
**2018**, 25, 1075–1091. [Google Scholar] [CrossRef] - Roussouly, N.; Petitjean, F.; Salaun, M. A new adaptive response surface method for reliability analysis. Probabilistic Eng. Mech.
**2013**, 32, 103–115. [Google Scholar] [CrossRef] - Liu, Z.-z.; Li, W.; Yang, M. Two General Extension Algorithms of Latin Hypercube Sampling. Math. Probl. Eng.
**2015**, 2015, 450492. [Google Scholar] [CrossRef] - Liu, Z.; Yang, M.; Li, W. A Sequential Latin Hypercube Sampling Method for Metamodeling. In Theory, Methodology, Tools and Applications for Modeling and Simulation of Complex Systems; Springer: Singapore, 2016. [Google Scholar]
- Metropolis, N.; Ulam, S. The Monte Carlo Method. J. Am. Stat. Assoc.
**1949**, 44, 335–341. [Google Scholar] [CrossRef] [PubMed] - Anawkar, S.; Panchwadkar, A. Comparison between Levenberg Marquart, Bayesian and scaled conjugate algorithm for prediction of cutting forces in face milling operation. AIP Conf. Proc.
**2022**, 2653, 030006. [Google Scholar] - Borisut, P.; Nuchitprasittichai, A. Process Configuration Studies of Methanol Production via Carbon Dioxide Hydrogenation: Process Simulation-Based Optimization Using Artificial Neural Networks. Energies
**2020**, 13, 6608. [Google Scholar] [CrossRef] - Janosovský, J.; Danko, M.; Labovský, J.; Jelemenský, L. Software approach to simulation-based hazard identification of complex industrial processes. Comput. Chem. Eng.
**2019**, 122, 66–79. [Google Scholar] [CrossRef] - Adams, T.A.; Salkuyeh, Y.K.; Nease, J. Chapter 6—Processes and simulations for solvent-based CO
_{2}capture and syngas cleanup. In Reactor and Process Design in Sustainable Energy Technology; Shi, F., Ed.; Elsevier: Amsterdam, The Netherlands, 2014; pp. 163–231. [Google Scholar] - Nuchitprasittichai, A.; Cremaschi, S. An algorithm to determine sample sizes for optimization with artificial neural networks. AIChE J.
**2013**, 59, 805–812. [Google Scholar] [CrossRef]

**Figure 1.**Illustration of Latin hypercube sampling for N = 10 of (

**a**) good filling design and (

**b**) poor filling design.

**Figure 5.**Additional sample points generation using maximum deviation. (

**a**) original Latin hypercube sampling and (

**b**) adaptive Latin hypercube sampling.

**Figure 10.**The convergence of the proposed algorithm for all three case studies: (

**a**) ammonia production from syngas, (

**b**) the production of methanol from carbon dioxide hydrogenation, (

**c**) carbon dioxide absorption by methanol via Rectisol process.

Algorithm | Criteria | Pros. | Cons. |
---|---|---|---|

Simulated annealing [36,37,38] | Maximize the inter-point distance | Effective for small-size problems | Converge very slowly |

Exchange type and Newton type [39] | Maximize entropy | Fast to find optimal design for large-size problems | |

Columnwise–pairwise [40] | Maximize the inter-point distance and Maximize entropy | Retain some orthogonality and high efficiency for small designs | Does not significantly reduce the searching time |

Threshold accepting [41] | Minimize L_{2} discrepancy | Can be applied to both factorial and computer experiment | Cannot give a good design for small dimension |

Genetic algorithm [42] | Maximize the inter-point distance | Requires a small amount of computational time | |

Enhanced stochastic evolutionary algorithm [43] | Maximize the inter-point distance, maximize entropy and minimize L_{2} discrepancy | Needs a small number of exchanges; effective for large-size problems | |

Branch-and-bound [44] | Maximize the inter-point distance | Can be used for non-collapsing designs | Obtain the optimum design for N < 70 |

Translational propagation [45] | Maximize the inter-point distance | Obtain near-optimum LHDs up to medium dimensions | High computational cost for large number of sample points |

Particle swarm optimization [46,47] | Maximize the inter-point distance | Fast accessibility to reach solutions | Local search algorithms become trapped in local optima |

Translational propagation [48] | Maximize the inter-point distance and minimize L_{2} discrepancy | Effective in terms of the computation time and space-filling and projective properties | Not good in terms of performance of sampling points |

Enhanced stochastic evolutionary algorithm [49] | Maximize the inter-point distance | Effective for large-size problems |

Decision Variables | Range |
---|---|

Reformer temperature (°C) | 900 to 1200 |

Combuster temperature (°C) | 1400 to 1700 |

Low-temperature conversion reactor temperature (°C) | 160 to 290 |

Decision Variables | Range |
---|---|

Pressure of the equilibrium reactor (bar) | 50 to 70 |

Temperature of the equilibrium reactor (°C) | 190 to 210 |

Temperature of the steam entering a separator (°C) | 60 to 80 |

Recycle ratio | 0 to 1 |

Decision Variables | Range |
---|---|

Lean methanol temperature (°C) | −55 to −20 |

The 3rd stage separator pressure (bar) | 1.2 to 2 |

Stripper reflux ration | 5 to 20 |

Stripper inlet temperature (°C) | 10 to 40 |

Distillation reflux ratio | 1 to 10 |

Case Studies | Case Study I | Case Study II | Case Study III | |||||
---|---|---|---|---|---|---|---|---|

Sample points | 50 | 100 | 50 | 100 | 50 | 100 | 200 | 400 |

R-squared | 0.9998 | 0.9999 | 1.0000 | 1.0000 | 0.9951 | 0.9999 | 0.9912 | 0.9998 |

Minimum cost | 495.87 | 495.87 | 942.45 | 942.45 | 49.70 | 43.66 | 43.20 | 43.40 |

Predicted cost | 505.89 | 496.73 | 926.29 | 948.23 | 39.37 | 42.83 | 45.88 | 43.66 |

Error | 2.02% | 0.17% | 1.71% | 0.61% | 20.80% | 1.91% | 6.21% | 0.59% |

Case Study I: Ammonia Production from Syngas | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Sampling Techniques | Total number of sample points | R^{2} | x_{1} (°C) | x_{2} (°C) | x_{3} (°C) | x_{4} | x_{5} | y_{predicted}($\hat{y}$) (USD per ton) | y_{actual}(y) (USD per ton) | % error |

Random Sampling | 100 | 0.9999 | 900 | 1400 | 160 | N/A | N/A | 496.73 | 495.87 | 0.17 |

Proposed algorithm | ||||||||||

Replication 1 | 38 | 0.9928 | 900 | 1400 | 160 | N/A | N/A | 498.40 | 495.87 | 0.51 |

Replication 2 | 30 | 0.9995 | 900 | 1400 | 160 | N/A | N/A | 495.22 | 495.87 | 0.13 |

Replication 3 | 30 | 0.9987 | 900 | 1400 | 160 | N/A | N/A | 492.14 | 495.83 | 0.74 |

Case Study II: Methanol Production via Carbon Dioxide Hydrogenation | ||||||||||

Sampling Techniques | Total number of sample points | R^{2} | x_{1} (bar) | x_{2} (°C) | x_{3} (°C) | x_{4} (-) | x_{5} | y_{predicted}($\hat{y}$) (USD per ton) | y_{actual}(y) (USD per ton) | % error |

Random Sampling | 100 | 1.0000 | 70 | 190 | 80 | 1 | N/A | 942.80 | 942.45 | 0.61 |

Proposed algorithm | ||||||||||

Replication 1 | 46 | 1.0000 | 70 | 190 | 80 | 1 | N/A | 942.37 | 942.45 | 0.01 |

Replication 2 | 46 | 0.9993 | 70 | 190 | 80 | 1 | N/A | 943.85 | 942.45 | 0.15 |

Replication 3 | 40 | 0.9991 | 70 | 190 | 76 | 1 | N/A | 944.31 | 945.94 | 0.17 |

Case Study III: Carbon Dioxide Absorption by Methanol via Rectisol Process | ||||||||||

Sampling Techniques | Total number of sample points | R^{2} | x_{1} (°C) | x_{2} (bar) | x_{3} (-) | x_{4} (°C) | x_{5} (-) | y_{predicted}($\hat{y}$) (USD per ton) | y_{actual}(y) (USD per ton) | % error |

Random Sampling | 400 | 0.9998 | −29.0 | 1.20 | 5 | 40 | 1 | 43.66 | 43.40 | 0.59 |

Proposed algorithm | ||||||||||

Replication 1 | 53 | 0.9958 | −20.0 | 1.20 | 5 | 40 | 1 | 45.74 | 45.47 | 0.60 |

Replication 2 | 50 | 0.9987 | −20.0 | 1.27 | 5 | 40 | 1 | 45.91 | 45.67 | 0.51 |

Replication 3 | 53 | 0.9980 | −26.7 | 1.28 | 5 | 40 | 1 | 43.25 | 43.19 | 0.14 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Borisut, P.; Nuchitprasittichai, A.
Adaptive Latin Hypercube Sampling for a Surrogate-Based Optimization with Artificial Neural Network. *Processes* **2023**, *11*, 3232.
https://doi.org/10.3390/pr11113232

**AMA Style**

Borisut P, Nuchitprasittichai A.
Adaptive Latin Hypercube Sampling for a Surrogate-Based Optimization with Artificial Neural Network. *Processes*. 2023; 11(11):3232.
https://doi.org/10.3390/pr11113232

**Chicago/Turabian Style**

Borisut, Prapatsorn, and Aroonsri Nuchitprasittichai.
2023. "Adaptive Latin Hypercube Sampling for a Surrogate-Based Optimization with Artificial Neural Network" *Processes* 11, no. 11: 3232.
https://doi.org/10.3390/pr11113232