There are two main issues with a simulation-optimization framework that couples a high-fidelity numerical model with an optimization algorithm. First, the numerical models are often computationally expensive and second, the optimization algorithm requires the numerical model to run several thousands of times which makes the whole process overly time-consuming. In addition, the computational complexity can increase further if the considered optimal management problem has multiple criteria and/or a high number of decision variables (see Table 1). This computational burden can be reduced significantly by using different surrogate modeling techniques (Rao *et al.* 2004; Bhattacharjya & Datta 2009; Dhar & Datta 2009; Papadopoulou *et al.* 2010; Sreekanth & Datta 2011, 2015; Ketabchi & Ataie-Ashtiani 2015). A surrogate model is an empirical or lower-fidelity alternative to either (A) the response of the high-fidelity numerical simulation model itself or (B) the objective function of the considered optimization problem or (C) the operating policy model in case of optimal control problems (Ratto *et al.* 2012). A surrogate model is simpler, considerably faster to run, and is generally derived through a training framework from the original simulation model (Razavi *et al.* 2012b).

Table 1

Reference . | Model^{a}
. | Problem^{b}
. | Objective function^{c}
. | #DV^{d}
. | SM^{e}
. | OA^{f}
. | ST^{g}
. | TD^{h}
. | CE^{i}
. |
---|---|---|---|---|---|---|---|---|---|

Bhattacharjya & Datta (2009) | 3D DA-GW | GW (q) | max(Q), min(_{out}C), min(Q) _{inj} | 33 | MLP | NSGA-II | HC | TR3 | 1,000 × 2,500 |

Bray & Yeh (2008) | FEMWATER | GW (q,x,y) | min(Q) _{inj} | 215 | – | GA | RC | TR5 | Parallel |

Dhar & Datta (2009) | FEMWATER | GW(q) | max(Q), min(_{out}Q) _{inj} | 33 | MLP | NSGA-II | HC | TR3 | 24 × 1,800 |

Haddad & Marino (2011) | 3D DA-GW | GW(q) | max(C) | 26 | – | HBMO | HC, RC | TI | 6,000 |

Kourakos & Mantoglou (2010) | SEAWAT (2D) | GW(q) | min(C), min(_{ök}C) _{umw} | 4 | – | NSGA-II | HC | TI | ? × 400 |

Papadopoulou et al. (2010) | PTC | GW(q) | max(Q) _{out} | 5 | RBF | DE | RC | TI | 20 × 200 |

Rao et al. (2004) | SEAWAT | GW(q) | max(Q), min(_{out}Q) _{inj} | 27 | MLP | SA | HC | TR3 | 10,000 |

Sreekanth & Datta (2011) | FEMWATER | GW(q) | max(Q), min(_{out}Q) _{inj} | 33 | GP | NSGA-II | HC | TR3 | 200 × 750 |

Reference . | Model^{a}
. | Problem^{b}
. | Objective function^{c}
. | #DV^{d}
. | SM^{e}
. | OA^{f}
. | ST^{g}
. | TD^{h}
. | CE^{i}
. |
---|---|---|---|---|---|---|---|---|---|

Bhattacharjya & Datta (2009) | 3D DA-GW | GW (q) | max(Q), min(_{out}C), min(Q) _{inj} | 33 | MLP | NSGA-II | HC | TR3 | 1,000 × 2,500 |

Bray & Yeh (2008) | FEMWATER | GW (q,x,y) | min(Q) _{inj} | 215 | – | GA | RC | TR5 | Parallel |

Dhar & Datta (2009) | FEMWATER | GW(q) | max(Q), min(_{out}Q) _{inj} | 33 | MLP | NSGA-II | HC | TR3 | 24 × 1,800 |

Haddad & Marino (2011) | 3D DA-GW | GW(q) | max(C) | 26 | – | HBMO | HC, RC | TI | 6,000 |

Kourakos & Mantoglou (2010) | SEAWAT (2D) | GW(q) | min(C), min(_{ök}C) _{umw} | 4 | – | NSGA-II | HC | TI | ? × 400 |

Papadopoulou et al. (2010) | PTC | GW(q) | max(Q) _{out} | 5 | RBF | DE | RC | TI | 20 × 200 |

Rao et al. (2004) | SEAWAT | GW(q) | max(Q), min(_{out}Q) _{inj} | 27 | MLP | SA | HC | TR3 | 10,000 |

Sreekanth & Datta (2011) | FEMWATER | GW(q) | max(Q), min(_{out}Q) _{inj} | 33 | GP | NSGA-II | HC | TR3 | 200 × 750 |

^{a}3D DA-GW = 3D density driven groundwater model, FEMWATER, SEAWAT, PTC = see reference.

^{b}GW = groundwater management, q = abstraction (injection) rate, x = x-coordinate, y = y-coordinate of the well.

^{c}*Q _{out}* = pump volume,

^{d}Number of decision variables.

^{e}Surrogate model: GP = genetic programming, MLP = multilayer perceptron, RBF = radial basis function network.

^{f}Optimization algorithm: GA = genetic algorithm, HBMO = honey-bee mating algorithm, NSGA-II = nondominated sorting genetic algorithm II, DE = differential evolution, SA = simulated annealing.

^{g}Type of study: HC = hypothetical case, RC = real case.

^{h}Time dependence of decision variables: TI = time-invariant, TR <x> transient over x periods.

^{i}Computational effort: <population size > x < nr. of generations > .

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