# Multi-Criteria Decision Analyses for the Selection of Hydrological Flood Routing Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. Routing Approach

#### 2.2. Flood Routing Optimization

#### 2.3. Flood Datasets

Flood Name | Label | Source | Hydrograph Category |
---|---|---|---|

Wilson’s 1974 flood | DS1 | [32] | Smooth Single Peak |

Brutsaert’s data | DS2 | [38] | Smooth Single Peak |

River Wye 1960 flood | DS3 | in [27] | Non-smooth Single Peak |

Karun river flood | DS4 | [36] | Non-smooth Single Peak |

Viessman and Lewis’s data | DS5 | [39] | Multi-Peak |

Sütçüler flood | DS6 | [11] | Multi-Peak |

River Wye 1982 flood | DS7 | in [11] | Irregular |

Chenggou and Lingqing river | DS8 | [37] | Irregular |

#### 2.4. Comparison of Muskingum Models

- Step 1:
- The entries of decision matrix Z are normalized as$${h}_{ij}=\frac{{z}_{ij}}{\sqrt{{\sum}_{i=1}^{p}{z}_{ij}^{2}}},\hspace{1em}i=1,2,\dots ,p,\hspace{1em}j=1,2,\dots ,r$$
- Step 2:
- Each criterion has a definite weight such that ${\sum}_{i=1}^{r}{w}_{i}=1$. The weighted normalized matrix is then computed as$${v}_{ij}={w}_{j}\times {h}_{ij},\hspace{1em}i=1,2,\dots ,p,\hspace{1em}j=1,2,\dots ,r$$
- Step 3:
- As stated earlier, some criteria can be maximized (a higher value is preferred) or minimized (a lower value is preferred). The ideal best ${V}^{+}$ and the ideal worst ${V}^{-}$ solutions are, respectively, determined as$${V}_{j}^{+}=\left\{\begin{array}{c}max\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{maximized}\\ min\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}\right.i=1,2,\dots ,pj=1,2,\dots ,r$$$${V}_{j}^{-}=\left\{\begin{array}{c}min\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{maximized}\\ max\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}\right.i=1,2,\dots ,pj=1,2,\dots ,r$$
- Step 4:
- The Euclidean distance for the ideal best and the ideal worst solutions will be$${\zeta}_{i}^{+}=\sqrt{\sum _{i=1}^{r}{\left({v}_{ij}-{V}_{j}^{+}\right)}^{2}}i=1,2,\dots ,pj=1,2,\dots ,r$$$${\zeta}_{i}^{-}=\sqrt{\sum _{i=1}^{r}{\left({v}_{ij}-{V}_{j}^{-}\right)}^{2}}i=1,2,\dots ,pj=1,2,\dots ,r$$
- Step 5:
- The closeness to the ideal solution for each alternative is then computed by$${\chi}_{i}=\frac{{\zeta}_{i}^{-}}{{\zeta}_{i}^{-}+{\zeta}_{i}^{+}}i=1,2,\dots ,p$$
- Step 6:
- Finally, ${\chi}_{i}$ values are sorted. A larger $\chi $ value implies a better alternative.

- Step 1:
- Step 1: The decision matrix Z in Equation (28) is normalized first according to the criteria whether some of them are maximized or minimized as$${z}_{ij}=\left\{\begin{array}{c}\frac{{c}_{j}\left({a}_{i}\right)-min\left({c}_{j}\right)}{max\left({c}_{j}\right)-min\left({c}_{j}\right)},if{c}_{j}\mathrm{is}\mathrm{maximized}\\ \frac{max\left({c}_{j}\right)-{c}_{i}\left({a}_{i}\right)}{max\left({c}_{j}\right)-min\left({c}_{j}\right)},if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}\right.i=1,2,\dots ,pj=1,2,\dots ,r$$
- Step 2:
- The pairwise distance is$${d}_{k}({a}_{i},{a}_{j})={c}_{k}\left({a}_{i}\right)-{c}_{k}\left({a}_{j}\right)i\ne j,i,j=1,2,\dots ,pk=1,2,\dots ,r$$
- Step 3:
- The preference degree is then determined by a preference function as$$\pi ({a}_{i},{a}_{j})=\sum _{k=1}^{r}{P}_{k}\left({d}_{k}\left({a}_{i},{a}_{j}\right)\right){w}_{k}$$$$P\left({d}_{k}\right)=\left\{\begin{array}{c}0,if{d}_{k}\le 0\\ 1,if{d}_{k}0\end{array}\right.$$Thus, $\pi $ values can be expressed by a $p\times p$ matrix whose diagonal entries are empty as$$\Gamma ={\left[\begin{array}{cccccc}-& \pi ({a}_{1},{a}_{2})& \pi ({a}_{1},{a}_{3})& \xb7& \xb7& \pi ({a}_{1},{a}_{p})\\ \pi ({a}_{2},{a}_{1})& -& \pi ({a}_{2},{a}_{3})& \xb7& \xb7& \pi ({a}_{2},{a}_{p})\\ \xb7& \xb7& \xb7& \xb7& \xb7& \xb7\\ \pi ({a}_{p},{a}_{1})& \pi ({a}_{p},{a}_{2})& \xb7& \xb7& \pi ({a}_{p-1},{a}_{p})& -\end{array}\right]}_{p\times p}$$
- Step 4:
- The ideal best (positive flow) and the ideal worst (negative flow) solutions are$${\xi}_{i}^{+}\left({a}_{i}\right)=\frac{1}{p-1}\sum _{i=1}^{p}\pi \left({a}_{i},x\right)$$$${\xi}_{i}^{-}\left({a}_{i}\right)=\frac{1}{p-1}\sum _{i=1}^{p}\pi \left(x,{a}_{i}\right)$$In other words, ${\xi}_{i}^{+}$ is the mean of the ith row of matrix $\Gamma $, whereas ${\xi}_{i}^{-}$ is the mean of the ith column of matrix $\Gamma $.
- Step 5:
- The ideal solution (net flow) is$${\xi}_{i}\left({a}_{i}\right)={\xi}_{i}^{+}\left({a}_{i}\right)-{\xi}_{i}^{-}\left({a}_{i}\right)$$
- Step 6:
- The PROMETHEE provides the complete ranking by ordering alternatives according to the decreasing values of the ideal solution $\xi $.

- 1.
- 2.
- 3.
- For each dataset, the estimation score of each model is noted and sorted. Since there is a total of 10 models, the model with the smallest N-RMSE is given a rank of 1 while the highest one is given a rank of 10. A smaller rank indicates a better performance.
- 4.
- Based on the N-RMSE values, the ranks for each dataset are found and then averaged.
- 5.
- Similarly, the $Ad-{R}^{2}$ values of each model for each dataset are sorted. The model with the highest $Ad-{R}^{2}$ value is given a rank of 1 while the smallest one is given a rank of 10. A higher rank score shows a better performance.
- 6.
- With $Ad-{R}^{2}$ values, the rank scores are averaged.
- 7.
- The decision matrix is formed by NP and MB together with the rank scores of N-RMSE and $Ad-{R}^{2}$. A binary code (1 for the models with physical parameters, 0 for the models with empirical parameters) is used to classify the MB.
- 8.
- The obtained decision matrix acquired from the previous steps is then studied by TOPSIS and PROMETHEE to elucidate the most efficient Muskingum model.

## 3. Results and Discussion

#### 3.1. Estimation Performance of Models

^{3}/s)

^{2}) for this dataset were obtained as 39.8 by [44], 17.55 by [45], 9.82 by [11], 7.67 by [2], 5.124 by [46], 4.11 by [47], 4.04 by [25], 1.92 by [31], 1.092 by [48], 0.799 by [49], and 0.65 by [50]. In this study, the SSE value for DS1 with the WCOA was found to be 7.66. Moreover, the proposed model (M10) achieved an SSE value of 4.09. The range of SSE values reported by different researchers suggests that there may be considerable variability in flood modeling results for this dataset, depending on the specific optimization algorithm and routing approach used. For instance, the SSE value of model M7 in this study (with the four-parameter non-linear model proposed by [2]) was calculated as 7.67, while the same metric was found to be 9.82 for model M9 (with the four-parameter non-linear model by [11]). Although these two models have the same number of model parameters, the reported SSE values are different because the researchers used different routing approaches and optimization algorithms as well. Similar findings can also be drawn for other datasets used in this study. For the Viessman and Lewis flood dataset (DS5), the reported SSE values were 71,708 in [45], 65,324 in [44], 28,855 in [50], and 8449 in [49]. However, the implemented routing model with the WCOA in this study provided notably smaller SSE values using the examined models except for M4, as shown in Table A1. It is quite remarkable that the SSE value of model M1 (with the two-parameter linear model proposed by [5]) was 14.53, while a model proposed by [49] with 12 parameters yielded an SSE value of 8449. However, comparing the reported results in the literature directly (without taking into account the applied routing approaches and optimization methods) can be misleading in interpreting which model exhibited a better estimation performance.

#### 3.2. Multi-Criteria Decision Analysis (MCDA) Results

#### 3.3. Comparison of MCDA Tools

## 4. Summary and Conclusions

- (1)
- The WCOA used in this study showed a viable ability in parameter estimation when compared to other algorithms used in similar studies.
- (2)
- The criteria, $Ad-{R}^{2}$ and N-RMSE, revealed that the non-linear models M9 and M10 (with four and six parameters) showed better predictive performance than the rest of the models on all datasets. However, the temporal distribution of the relative error showed diversity in defining the best routing models overall. Such findings indicate the importance of the criteria in the selection of the proper model and parameters.
- (3)
- The model performances were also assessed by clustering the hydrograph types, smooth single peak (SSP), non-smooth single peak (NSP), multi-peak (MP), and irregular. According to the average of the ranked criteria overall, models M9 and M10 showed the best performance according to the error metric criteria, including $Ad-{R}^{2}$ and N-RMSE. For hydrograph types, the definition of the best model showed some variability, the proposed model M10 showed better performance for the majority of the datasets by considering these criteria.
- (4)
- To obtain a more rigid conclusion about defining the best model, two different multi-criteria decision analysis (MCDA) tools were applied by using two additional criteria: the number of model parameters (NP), which indicates the model complexity; and the model background (MB), which characterizes the model parameters as either physical-based or empirical, with the purpose of improving the fitting.
- (5)
- The effects of the relative importance of each criterion were investigated by assigning three different weight scenarios in both tools, TOPSIS and PROMETHEE. The most outstanding remark is the compatibility of both tools in all scenarios, giving similar model results in the rankings. However, it was seen that the governing factor in defining the best model depends on the scenario selection. In scenario 1, where additional criterion weights are balanced when compared to the other scenarios, physical-based models were more successful. However, there is a variety of ways to define the best model for hydrograph types. Model M5, a physically sound model, showed the best results for MP and irregular hydrograph types, whereas models M8 and M10, empirical models with a high number of parameters, showed better results compared to the rest of the models for the SSP and NSP types. Such outcomes may indicate that the routing application in complex hydrographs that have more than one peak with irregular limps can be assessed better using physical-based routing models that have fewer parameters.
- (6)
- The MCDA in a single pool ignoring the clustering of hydrograph types showed that model M5 had the greatest score in the rankings. Moreover, the cumulative score evaluation of 10 Muskingum models that cover the cumulative percentages of the model ranking place shows that models M5, M9, and M10 have the leading results with scores of 85% in the top four ranking places for both tools, TOPSIS and PROMETHEE. Such a result can be beneficial for researchers in choosing the proper routing models in their area of interest.
- (7)
- There are a variety of Muskingum models applied in the literature. Within this study, the success of the models was assessed concerning the model parameters (complexity), model constructions (empirical or physical), error metrics, and hydrograph types. The most suitable model, model M5, showed that the use of physical-based models with fewer parameters among the possible model sets studied here, may be beneficial.
- (8)
- Any MCDA method can inherently generate a different ranking for the problem of interest. Although the tools used in this study have different capabilities in terms of relating the criteria and being sensitive to the weights of the criteria, their results are compatible with each other. This outcome reveals the independence of the criteria and stability of the weights in application. This finding was also testified by the consistency rates (CR) that involve the proximity of model rankings for each scenario and hydrograph types performed for both tools.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AR | Averaged rank |

MB | Model background |

MCDA | Multi-criteria decision analysis |

MP | Multi-peak |

NP | Number of parameters |

N-RMSE | Normalized root mean squared error |

NSP | Non-smooth single peak |

PROMETHEE | The preference ranking organization method for enrichment evaluation |

TOPSIS | The technique for order of preference by similarity to ideal solution |

SSE | Sum of square errors |

SSP | Smooth single peak |

WCOA | Water cycle optimization algorithm |

## Appendix A. Results of Parameter Estimation for Models

**Table A1.**Results for the parameter estimation for each model and each dataset ($\alpha $ is the flow exponent, m is the general model exponent, $\gamma $ is the lateral flow, $\beta $ is the flow exponent, and $\eta $ is the coefficient of the explicit/implicit scheme).

Dataset | Model | K | w | $\mathit{\gamma}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | m | $\mathit{\eta}$ | SSE | $\mathit{Ad}-{\mathit{R}}^{2}$ | N-RMSE | MAE |
---|---|---|---|---|---|---|---|---|---|---|---|---|

DS 1 | M1 | 29.8106 | 0.2388 | 1 | 415.55 | 0.9624 | 0.0659 | 3.77 | ||||

M2 | 2.7637 | 0.2287 | 1.5012 | 1 | 245.58 | 0.9766 | 0.0506 | 2.76 | ||||

M3 | 0.5175 | 0.2869 | 1.8681 | 1 | 36.77 | 0.9965 | 0.0196 | 1.07 | ||||

M4 | 235.8340 | 0.1661 | 0.6000 | 1 | 752.00 | 0.9320 | 0.0886 | 4.96 | ||||

M5 | 29.7496 | 0.2477 | −0.0091 | 1 | 411.69 | 0.9607 | 0.0655 | 3.74 | ||||

M6 | 1.6247 | 0.0003 | 3.0418 | 1.5676 | 1 | 184.32 | 0.9814 | 0.0439 | 2.45 | |||

M7 | 0.8338 | 0.2956 | 0.4331 | 4.0790 | 1 | 7.67 | 0.9992 | 0.0089 | 0.47 | |||

M8 | 0.8646 | 0.0395 | 0.7438 | 0.3679 | 4.3500 | 0.9307 | 4.96 | 0.9995 | 0.0072 | 0.29 | ||

M9 | 0.5342 | 0.3005 | −0.0216 | 1.8642 | 1.0 | 9.82 | 0.9990 | 0.0101 | 0.57 | |||

M10 | 0.6663 | 0.0246 | −0.0077 | 0.9850 | 0.4953 | 3.3866 | 0.8657 | 4.09 | 0.9995 | 0.0065 | 0.31 | |

DS 2 | M1 | 2.0839 | 0.1102 | 1 | 15,895.60 | 0.9987 | 0.0109 | 16.73 | ||||

M2 | 1.0809 | 0.1332 | 1.0839 | 0.9602 | 11,846.22 | 0.9990 | 0.0094 | 14.73 | ||||

M3 | 0.8189 | 0.0388 | 1.1107 | 0 | 12,144.80 | 0.9990 | 0.0095 | 14.95 | ||||

M4 | 51.9261 | 0.0000 | 0.6000 | 0.5320 | 155,580.39 | 0.9874 | 0.0341 | 53.38 | ||||

M5 | 2.0077 | 0.0251 | 0.0129 | 1 | 11,913.60 | 0.9990 | 0.0094 | 15.04 | ||||

M6 | 586.1200 | 0.9992 | 0.2314 | 1.1572 | 0.9554 | 8631.99 | 0.9992 | 0.0080 | 12.78 | |||

M7 | 1.5962 | 0.1616 | 1.7865 | 0.5823 | 1.0000 | 8244.86 | 0.9993 | 0.0078 | 12.31 | |||

M8 | 0.9137 | 0.0001 | 5.0000 | 4.0957 | 0.2687 | 0.5103 | 7711.48 | 0.9993 | 0.0076 | 11.77 | ||

M9 | 1.1235 | 0.0560 | 0.0117 | 1.0746 | 1.0000 | 9043.47 | 0.9992 | 0.0082 | 12.48 | |||

M10 | 13.6508 | 0.9675 | 0.0041 | 0.8243 | 1.4851 | 0.7180 | 0.8723 | 8510.25 | 0.9992 | 0.0080 | 12.45 | |

DS 3 | M1 | 26.6573 | 0.4274 | 1 | 88,387.01 | 0.9431 | 0.0565 | 36.16 | ||||

M2 | 1.6370 | 0.4024 | 1.4050 | 1 | 44,478.76 | 0.9704 | 0.0401 | 25.01 | ||||

M3 | 0.4624 | 0.4091 | 1.5856 | 1 | 35,194.51 | 0.9766 | 0.0356 | 22.65 | ||||

M4 | 491.3309 | 0.3475 | 0.6000 | 1 | 200,258.51 | 0.8711 | 0.0850 | 58.54 | ||||

M5 | 26.4413 | 0.3992 | 0.0465 | 1 | 82,057.20 | 0.9454 | 0.0544 | 35.47 | ||||

M6 | 0.6773 | 0.0022 | 2.2497 | 1.4641 | 1 | 37,352.54 | 0.9743 | 0.0367 | 20.56 | |||

M7 | 0.4502 | 0.4074 | 1.2022 | 1.3229 | 1 | 29,474.96 | 0.9797 | 0.0326 | 20.19 | |||

M8 | 0.9540 | 0.7331 | 1.0038 | 1.1888 | 1.3742 | 1 | 29,132.20 | 0.9792 | 0.0324 | 20.41 | ||

M9 | 0.3691 | 0.3830 | 0.0547 | 1.6141 | 1 | 25,915.27 | 0.9822 | 0.0306 | 17.09 | |||

M10 | 0.7500 | 0.0452 | 0.1323 | 0.5429 | 0.2676 | 5.0000 | 1 | 20,048.64 | 0.9852 | 0.0269 | 14.51 | |

DS 4 | M1 | 12.5606 | 0.2424 | 1 | 89,417.76 | 0.9735 | 0.0544 | 31.20 | ||||

M2 | 1000 | 0.1846 | 0.4563 | 1 | 65,726.23 | 0.9801 | 0.0466 | 27.22 | ||||

M3 | 1000 | 0.1749 | 0.4570 | 1 | 64,593.54 | 0.9804 | 0.0462 | 27.07 | ||||

M4 | 297.3967 | 0.1989 | 0.6000 | 1 | 68,814.88 | 0.9796 | 0.0477 | 28.12 | ||||

M5 | 12.3521 | 0.3608 | −0.0314 | 1 | 63,325.14 | 0.9808 | 0.0458 | 27.55 | ||||

M6 | 1000 | 0.1101 | 0.5159 | 0.4462 | 1 | 65,666.31 | 0.9796 | 0.0466 | 27.19 | |||

M7 | 1000 | 0.1146 | 2.8707 | 0.1600 | 1 | 62,607.58 | 0.9806 | 0.0455 | 27.00 | |||

M8 | 1000 | 0.0000 | 5.0000 | 3.4967 | 0.1301 | 1 | 60,243.11 | 0.9808 | 0.0446 | 26.12 | ||

M9 | 980.2646 | 0.2431 | −0.0198 | 0.4564 | 1 | 54,947.64 | 0.9829 | 0.0426 | 25.64 | |||

M10 | 1000 | 0.0000 | −0.0249 | 5.0000 | 2.8688 | 0.1551 | 1 | 47,663.45 | 0.9845 | 0.0397 | 22.80 | |

DS 5 | M1 | 3.8158 | 0.4489 | 1 | 14.53 | 1.0000 | 0.0006 | 0.62 | ||||

M2 | 3.8376 | 0.4490 | 0.9993 | 1 | 13.98 | 1.0000 | 0.0006 | 0.60 | ||||

M3 | 3.8003 | 0.4489 | 1.0005 | 1 | 14.33 | 1.0000 | 0.0006 | 0.62 | ||||

M4 | 83.3925 | 0.3243 | 0.6000 | 1 | 118,587.65 | 0.9725 | 0.0513 | 57.10 | ||||

M5 | 3.8174 | 0.4497 | −0.0007 | 1 | 7.57 | 1.0000 | 0.0004 | 0.21 | ||||

M6 | 3.7546 | 0.4300 | 1.0074 | 0.9978 | 1 | 10.56 | 1.0000 | 0.0005 | 0.42 | |||

M7 | 3.7960 | 0.4490 | 0.9952 | 1.0055 | 1 | 8.96 | 1.0000 | 0.0004 | 0.36 | |||

M8 | 3.7918 | 0.4476 | 0.9960 | 0.9953 | 1.0052 | 1 | 8.95 | 1.0000 | 0.0004 | 0.36 | ||

M9 | 3.7977 | 0.4496 | −0.0007 | 1.0007 | 1 | 7.24 | 1.0000 | 0.0004 | 0.22 | |||

M10 | 3.7721 | 0.4411 | −0.0007 | 1.0053 | 1.0010 | 0.9986 | 1 | 6.98 | 1.0000 | 0.0004 | 0.22 | |

DS 6 | M1 | 1.0478 | 0.0000 | 0 | 561.30 | 0.9903 | 0.0217 | 3.12 | ||||

M2 | 1000 | 0.6980 | 5.0000 | 0.9836 | 327.35 | 0.9941 | 0.0166 | 1.86 | ||||

M3 | 0.9867 | 0.0000 | 1.0106 | 0.9995 | 560.62 | 0.9899 | 0.0217 | 3.14 | ||||

M4 | 11.4763 | 0.0000 | 0.6000 | 0.9895 | 1979.65 | 0.9657 | 0.0408 | 6.19 | ||||

M5 | 0.5197 | 0.9999 | −1.0001 | 0.9802 | 283.39 | 0.9949 | 0.0154 | 1.68 | ||||

M6 | 1000 | 0.7460 | 5.0000 | 5.0535 | 0.9877 | 322.01 | 0.9940 | 0.0165 | 1.92 | |||

M7 | 0.0100 | 0.6980 | 5.0000 | 4.7868 | 0.9836 | 327.35 | 0.9939 | 0.0166 | 1.86 | |||

M8 | 1.2950 | 0.0091 | 0.6912 | 0.1798 | 5.0000 | 1 | 319.37 | 0.9938 | 0.0164 | 1.94 | ||

M9 | 126.2182 | −1.0000 | 0.8970 | 5.0000 | 0.9945 | 292.35 | 0.9945 | 0.0157 | 1.82 | |||

M10 | 1000 | 0.3908 | −2.6209 | 1.6807 | 1.7052 | 1.9928 | 0.9996 | 280.56 | 0.9943 | 0.0154 | 1.81 | |

DS 7 | M1 | 131.5774 | 0.9917 | 0.9561 | 36,896.67 | −0.3563 | 0.3752 | 26.36 | ||||

M2 | 307.8855 | 0.7268 | 0.4827 | 0.2951 | 5272.04 | 0.7993 | 0.1418 | 10.04 | ||||

M3 | 1000 | 0.9622 | 0.2656 | 0.4545 | 24,252.22 | 0.0766 | 0.3042 | 16.28 | ||||

M4 | 325.5926 | 0.9808 | 0.6000 | 0.8240 | 25,624.60 | 0.0580 | 0.3127 | 17.65 | ||||

M5 | 5.1523 | 0.2293 | 2.5276 | 1 | 54.86 | 0.9979 | 0.0145 | 0.95 | ||||

M6 | 341.8120 | 0.9086 | 0.4249 | 0.6222 | 0.7361 | 3222.91 | 0.8727 | 0.1109 | 8.69 | |||

M7 | 31.6292 | 0.7387 | 0.5061 | 4.0949 | 0.4961 | 3296.05 | 0.8699 | 0.1121 | 7.43 | |||

M8 | 125.8095 | 0.8845 | 0.4640 | 0.6121 | 2.4301 | 1 | 3244.81 | 0.8670 | 0.1113 | 7.88 | ||

M9 | 5.6765 | 0.2271 | 2.5298 | 0.9800 | 1 | 53.66 | 0.9979 | 0.0143 | 0.90 | |||

M10 | 6.8449 | 0.3347 | 2.5003 | 1.1999 | 1.2497 | 0.7737 | 1 | 44.52 | 0.9981 | 0.0130 | 0.84 | |

DS 8 | M1 | 116.3420 | 0.9914 | 1 | 4582.58 | 0.9903 | 0.0302 | 10.18 | ||||

M2 | 0.1268 | 0.0000 | 1.3211 | 1 | 4542.59 | 0.9900 | 0.0301 | 9.80 | ||||

M3 | 0.1268 | 0.0000 | 1.3211 | 1 | 4542.59 | 0.9900 | 0.0301 | 9.80 | ||||

M4 | 19.4275 | 0.0000 | 0.6000 | 0.8915 | 5898.41 | 0.9875 | 0.0343 | 11.24 | ||||

M5 | 0.0321 | 0.9994 | −1.0003 | 0.1399 | 992.93 | 0.9978 | 0.0141 | 4.70 | ||||

M6 | 0.1268 | 0.0000 | 0.4908 | 1.3211 | 1 | 4542.59 | 0.9895 | 0.0301 | 9.80 | |||

M7 | 0.1268 | 0.0000 | 0.2642 | 4.9999 | 0.5749 | 4542.59 | 0.9895 | 0.0301 | 9.80 | |||

M8 | 2.5489 | 0.8092 | 0.1000 | 0.5000 | 3.0526 | 1 | 4534.57 | 0.9891 | 0.0301 | 9.81 | ||

M9 | 0.2358 | −1.0000 | −0.0027 | 1.2134 | 0.1516 | 918.24 | 0.9979 | 0.0135 | 4.51 | |||

M10 | 1000 | 0.9450 | −1.1817 | 5.0 | 5.1915 | 0.1637 | 0.9390 | 3160.90 | 0.9921 | 0.0251 | 8.48 |

## References

- Afzali, S.H. Variable-parameter Muskingum model. Iran. J. Sci. Technol. Trans. Civ. Eng.
**2016**, 40, 59–68. [Google Scholar] [CrossRef] - Easa, S.M. New and Improved Four-Parameter Non-Linear Muskingum Model. Proc. Inst. Civ. Eng. Water Manag.
**2014**, 167, 288–298. [Google Scholar] [CrossRef] - Gąsiorowski, D.; Romuald, S. Dimensionally Consistent Nonlinear Muskingum Equation. J. Hydrol. Eng.
**2018**, 23, 04018039. [Google Scholar] [CrossRef] - Bai, T.; Wei, J.; Yang, W.; Huang, Q. Multi-objective parameter estimation of improved Muskingum model by wolf pack algorithm and its application in Upper Hanjiang River, China. Water
**2018**, 10, 1415. [Google Scholar] [CrossRef] [Green Version] - McCarthy, G.T. The unit hydrograph and flood routing. In Conference of North Atlantic Division; US Army Corps of Engineers: Washington, DC, USA, 1938. [Google Scholar]
- Chow, V.T. Open-Channel Hydraulics; McGraw-Hill: New York, NY, USA, 1959. [Google Scholar]
- Gill, M.A. Flood routing by the Muskingum method. J. Hydrol.
**1978**, 36, 353–363. [Google Scholar] [CrossRef] - O’donnell, T. A Direct Three-Parameter Muskingum Procedure Incorporating Lateral Inflow. Hydrol. Sci. J.
**1985**, 30, 479–496. [Google Scholar] [CrossRef] [Green Version] - Gavilan, G.; Houck, M.H. Optimal Muskingum river routing. In Proceedings of the ASCE Computer Applications in Water Resources, Buffalo, NY, USA, 10–12 June 1985; American Society of Civil Engineers: New York, NY, USA, 1985; pp. 1294–1302. [Google Scholar]
- Vatankhah, A.R. Discussion of parameter estimation of the nonlinear Muskingum flood routing model using a hybrid harmony search algorithm by Halil Karahan, Gurhan Gurarslan, and Zong Woo Geem. J. Hydrol. Eng.
**2014**, 19, 839–842. [Google Scholar] [CrossRef] - Karahan, H.; Gurarslan, G.; Geem, Z.W. A New Nonlinear Muskingum Flood Routing Model Incorporating Lateral Flow. Eng. Optim.
**2015**, 47, 737–749. [Google Scholar] [CrossRef] - Yoon, J.W.; Padmanabhan, G. Parameter estimation of linear and nonlinear Muskingum models. ASCE J. Water Resour. Plan. Manag.
**1993**, 119, 600–610. [Google Scholar] [CrossRef] - Das, A. Parameter estimation for Muskingum models. ASCE J. Irrig. Drain. Eng.
**2004**, 130, 140–147. [Google Scholar] [CrossRef] - Geem, Z.W. Parameter estimation for the nonlinear Muskingum model using the BFGS technique. ASCE J. Irrig. Drain. Eng.
**2006**, 132, 474–478. [Google Scholar] [CrossRef] - Barati, R. Parameter estimation of nonlinear Muskingum models using Nelder–Mead simplex algorithm. ASCE J. Hydrol. Eng.
**2011**, 16, 946–954. [Google Scholar] [CrossRef] - Mohan, S. Parameter estimation of nonlinear Muskingum models using genetic algorithm. ASCE J. Hydraul. Eng.
**1997**, 123, 137–142. [Google Scholar] [CrossRef] - Zhang, S.; Ling, K.; Liwei, Z.; Xiaoming, G. A New Modified Nonlinear Muskingum Model and Its Parameter Estimation Using the Adaptive Genetic Algorithm. Hydrol. Res.
**2017**, 48, 17–27. [Google Scholar] [CrossRef] - Kim, J.H.; Geem, Z.W.; Kim, E.S. Parameter estimation of the nonlinear Muskingum model using harmony search. J. Am. Water Resour. Assoc.
**2001**, 37, 1131–1138. [Google Scholar] [CrossRef] - Chu, H.J.; Chang, L.C. Applying particle swarm optimization to parameter estimation of the nonlinear Muskingum model. ASCE J. Hydrol. Eng.
**2009**, 14, 1024–1027. [Google Scholar] [CrossRef] - Xu, D.; Qiu, L.; Chen, S. Estimation of nonlinear Muskingum model parameters using differential evolution. ASCE J. Hydrol. Eng.
**2015**, 16, 348–353. [Google Scholar] [CrossRef] - Luo, J.; Xie, J. Parameter estimation for nonlinear Muskingum model based on immune clonal selection algorithm. ASCE J. Hydrol. Eng.
**2010**, 15, 844–851. [Google Scholar] [CrossRef] - Ehteram, M.; Binti Othman, F.; Mundher Yaseen, Z.; Abdulmohsin Afan, H.; Falah Allawi, M.; Najah Ahmed, A.; Shahid, S.; Singh, V.P.; El-Shafie, A. Improving the Muskingum flood routing method using a hybrid of particle swarm optimization and bat algorithm. Water
**2018**, 10, 807. [Google Scholar] [CrossRef] [Green Version] - Karahan, H.; Gurarslan, G.; Geem, Z.W. Parameter estimation of the nonlinear Muskingum flood-routing model using a hybrid harmony search algorithm. J. Hydrol. Eng.
**2013**, 18, 352–360. [Google Scholar] [CrossRef] - Bozorg-Haddad, O.; Hamedi, F.; Orouji, H.; Pazoki, M.; Loáiciga, H.A. A re-parameterized and improved nonlinear Muskingum model for flood routing. Water Resour. Manag.
**2015**, 29, 3419–3440. [Google Scholar] [CrossRef] - Niazkar, M.; Afzali, S.H. Parameter estimation of an improved nonlinear Muskingum model using a new hybrid method. Hydrol. Res.
**2017**, 48, 1253–1267. [Google Scholar] [CrossRef] - Farzin, S.; Singh, V.P.; Karami, H.; Farahani, N.; Ehteram, M.; Kisi, O.; Allawi, M.F.; Mohd, N.S.; El-Shafie, A. Flood routing in river reaches using a three-parameter Muskingum model coupled with an improved bat algorithm. Water
**2018**, 10, 1130. [Google Scholar] [CrossRef] [Green Version] - Geem, Z.W. Issues in Optimal Parameter Estimation for the Nonlinear Muskingum Flood Routing Model. Eng. Optim.
**2014**, 46, 328–339. [Google Scholar] [CrossRef] - Gąsiorowski, D.; Romuald, S. Identification of Parameters Influencing the Accuracy of the Solution of the Nonlinear Muskingum Equation. Water Resour. Manag.
**2020**, 34, 3147–3164. [Google Scholar] [CrossRef] - Easa, S.M. Improved nonlinear Muskingum model with variable exponent parameter. ASCE J. Hydrol. Eng.
**2013**, 18, 1790–1794. [Google Scholar] [CrossRef] - Karahan, H. Discussion of ‘‘Improved nonlinear Muskingum model with variable exponent parameter” by Said M Easa. J. Hydrol. Eng.
**2014**, 19, 1–9. [Google Scholar] [CrossRef] - Ayvaz, M.T.; Gurarslan, G. A new partitioning approach for nonlinear Muskingum flood routing models with lateral flow contribution. J. Hydrol.
**2017**, 553, 142–159. [Google Scholar] [CrossRef] - Wilson, E.M. Engineering Hydrology; Macmillan Education: Basingstoke, UK, 1974. [Google Scholar]
- Eskandar, H.; Sadollah, A.; Bahreininejad, A.; Hamdi, M. Water cycle algorithm-A novel metaheuristic optimization method for solving constrained engineering optimization problems. Comput. Struct.
**2012**, 110–111, 151–166. [Google Scholar] [CrossRef] - Sahin, A.U. Automatic Shifting Method for the Identification of Generalized Radial Flow Parameters by Water Cycle Optimization. Water Resour. Manag.
**2021**, 35, 5205–5223. [Google Scholar] [CrossRef] - Nasir, M.; Sadollah, A.; Choi, Y.H.; Kim, J.H. A comprehensive review on water cycle algorithm and its applications. Neural Comput. Appl.
**2020**, 32, 17433–17488. [Google Scholar] [CrossRef] - Orouji, H.; Bozorg, H.O.; Fallah-Mehdipour, E.; Marino, M.A. Estimation of muskingum parameter by meta-heuristic algorithms. Proc. Inst. Civ. Eng. Water Manag.
**2013**, 166, 315–324. [Google Scholar] [CrossRef] - Wang, W.; Xu, Z.; Qiu, L.; Xu, D. Hybrid chaotic genetic algorithms for optimal parameter estimation of Muskingum flood routing model. In Proceedings of the International Joint Conference on Computational Sciences and Optimization, Sanya, China, 24–26 April 2009; pp. 215–218. [Google Scholar]
- Brutsaert, W. Hydrology: An Introduction; Cambridge University Press: New York, NY, USA, 2005. [Google Scholar]
- Viessman, W.; Lewis, G.L. Introduction to Hydrology; Pearson Education, Inc.: Upper Saddle River, NJ, USA, 2003. [Google Scholar]
- Hwang, C.L.; Yoon, K. Multiple Attribute Decision Making: Methods and Applications; Springer: New York, NY, USA, 1981. [Google Scholar]
- Yoon, K. A reconciliation among discrete compromise situations. J. Oper. Res. Soc.
**1987**, 38, 277–286. [Google Scholar] [CrossRef] - Hwang, C.L.; Lai, Y.J.; Liu, T.Y. A new approach for multiple objective decision making. Comput. Oper. Res.
**1993**, 20, 889–899. [Google Scholar] [CrossRef] - Brans, J.P. L’ingénierie de la décision: Elaboration d’instruments d’aide à la décision. La méthode PROMETHEE. In L’aide à la Décision: Nature, Instruments et Perspectives d’Avenir; Presses de l’Université Laval: Quebec, QC, Canada, 1982. [Google Scholar]
- Vatankhah, A.R. The lumped muskingum flood routing model revisited: The storage relationship. Hydrol. Sci. J.
**2021**, 66, 1625–1637. [Google Scholar] [CrossRef] - Lu, C.; Ji, K.; Wang, W.; Zhang, Y.; Ealotswe, T.K.; Qin, W.; Lu, J.; Liu, B.; Shu, L. Estimation of the interaction between groundwater and surface water based on flow routing using an improved nonlinear muskingum-cunge method. Water Resour. Manag.
**2021**, 35, 2649–2666. [Google Scholar] [CrossRef] - Farahani, N.; Karami, H.; Farzin, S.; Ehteram, M.; Kisi, O.; El Shafie, A. A new method for flood routing utilizing four-parameter nonlinear muskingum and shark algorithm. Water Resour. Manag.
**2019**, 33, 4879–4893. [Google Scholar] [CrossRef] - Lee, E.H. Development of a new 8-parameter muskingum flood routing model with modified inflows. Water
**2021**, 13, 3170. [Google Scholar] [CrossRef] - Akbari, R.; Hessami-Kermani, M.-R.; Shojaee, S. Flood routing: Improving outflow using a new non-linear muskingum model with four variable parameters coupled with PSO-GA algorithm. Water Resour. Manag.
**2020**, 34, 3291–3316. [Google Scholar] [CrossRef] - Moradi, E.; Yaghoubi, B.; Shabanlou, S. A new technique for flood routing by nonlinear Muskingum model and artificial gorilla troops algorithm. Appl. Water Sci.
**2023**, 13, 49. [Google Scholar] [CrossRef] - Bozorg-Haddad, O.; Mohammad-Azari, S.; Hamedi, F.; Pazoki, M.; Loáiciga, H.A. Application of a new hybrid non-linear Muskingum modelto flood routing. Proc. Inst. Civ. Eng. Water Manag.
**2020**, 173, 109–120. [Google Scholar] [CrossRef] - Zlaugotne, B.; Zihare, L.; Balode, L.; Kalnbalkite, A.; Khabdullin, A.; Blumberga, D. Multi-Criteria Decision Analysis Methods Comparison. Environ. Clim. Technol.
**2020**, 24, 454–471. [Google Scholar] [CrossRef] - Kaya, G.K.; Ozturk, F. A Comparison of the Multi-criteria Decision-Making Methods for the Selection of Researchers. In Industrial Engineering in the Internet-of-Things World: Selected Papers from the Virtual Global Joint Conference on Industrial Engineering and Its Application Areas, GJCIE 2020, Virtually, 14–15 August 2020; Calisir, F., Ed.; Lecture Notes in Management and Industrial Engineering Series; Springer: Cham, Switzerland, 2020. [Google Scholar] [CrossRef]
- Vassoney, E.; Mammoliti Mochet, A.; Desiderio, E.; Negro, G.; Pilloni, M.G.; Comoglio, C. Comparing Multi-Criteria Decision-Making Methods for the Assessment of FlowRelease Scenarios From Small Hydropower Plants in the Alpine Area. Front. Environ. Sci.
**2021**, 9, 635100. [Google Scholar] [CrossRef] - Khan, I.; Pintelon, L.; Martin, H. The Application of Multicriteria Decision Analysis Methods in Health Care: A Literature Review. Med. Decis. Mak.
**2022**, 42, 262–274. [Google Scholar] [CrossRef] [PubMed] - Cinelli, M.; Burgherr, P.; Kadziński, M.; Słowiński, R. Proper and improper uses of MCDA methods in energy systems analysis. Decis. Support Syst.
**2022**, 163, 113848. [Google Scholar] [CrossRef] - Zavadskas, E.K.; Kaklauskas, A. The new method of multicriteria evaluation of projects. In Deutsch-Litauisch-Polnisches Kolloquim zum Baubetriebswesen; 3 Jahrgang, Sonderheft; Hochschule fur Technik, Wirtschaft und Kultur: Leipzig, Germany, 1996; pp. 3–8. [Google Scholar]
- Zavadskas, E.K.; Kaklauskas, A.; Vilutiene, T. Multicriteria evaluation of apartment blocks maintenance contractors: Lithuanian case study. Int. J. Strateg. Prop. Manag.
**2009**, 13, 319–338. [Google Scholar] [CrossRef]

**Figure 1.**Schematic summary of the entire computational process (blue dotted lines show the implementation and output of WCOA, red dotted lines show the details of the WCOA).

**Figure 2.**Flowrate datasets. (

**a**) Wilson’s 1974 flood (SSP), (

**b**) Brutsaert’s data (SSP), (

**c**) River Wye 1960 flood (NSP), (

**d**) Karun river flood (NSP), (

**e**) Viessman and Lewis’s data (MP), (

**f**) Sütçüler flood (MP), (

**g**) River Wye 1982 flood (irregular), and (

**h**) Chenggou and Lingqing river (irregular).

**Figure 4.**Percentage of ranking for both methods ($\left|\right|$ represents OR function; in the overall evaluation of PROMETHEE, model M9 takes 1st or 2nd place with 80%).

**Figure 6.**Consistency in ranking for each hydrograph type and scenario between TOPSIS and PROMETHEE (CR is consistency rate, CR(0) shows the exact coherency, CR(1) shows the coherency in the gray zone. For scenario 1 in SSP, 50% for CR(0) shows that 5 out of 10 models obtain the same rankings from both methods, whereas 90% for CR(1) shows that 9 out of 10 models obtain close rankings, located in the gray zone).

Model No. | Storage Equation * | # of Parameters | Studied by |
---|---|---|---|

M1 | $S=K[wI+(1-w\left)Q\right]$ | 2 | [5] |

M2 | $S=K[w{I}^{\alpha}+(1-w){Q}^{\alpha}]$ | 3 | [6] |

M3 | $S=K{[wI+(1-w)Q]}^{m}$ | 3 | [7] |

M4 | $S=K{[wI+(1-w)Q]}^{3/5}$ | 2 | [3] |

M5 | $S=K\left[w\right(1+\gamma )I+(1-w\left)Q\right]$ | 3 | [8] |

M6 | $S=K[w{I}^{\alpha}+(1-w){Q}^{\beta}]$ | 4 | [9] |

M7 | $S=K{[w{I}^{\alpha}+(1-w){Q}^{\alpha}]}^{m}$ | 4 | [2] |

M8 | $S=K{[w{I}^{\alpha}+(1-w){Q}^{\beta}]}^{m}$ | 5 | [10] |

M9 | $S=K{[w(1+\gamma )I+(1-w)Q]}^{m}$ | 4 | [11] |

M10 | $S=K{[w(1+\gamma ){I}^{\alpha}+(1-w){Q}^{\beta}]}^{m}$ | 6 | Proposed |

$\mathit{A}\mathit{d}-{\mathit{R}}^{2}$ | |||||||||||||

SSP | NSP | MP | Irregular | Overall | |||||||||

Model | DS1 | DS2 | AR | DS3 | DS4 | AR | DS5 | DS6 | AR | DS7 | DS8 | AR | AR |

M1 | 8 | 9 | 8.5 | 9 | 10 | 9.5 | 7 | 8 | 7.5 | 10 | 4 | 7 | 8.125 |

M2 | 7 | 6 | 6.5 | 7 | 7 | 7 | 8 | 4 | 6 | 7 | 5 | 6 | 6.375 |

M3 | 5 | 8 | 6.5 | 5 | 6 | 5.5 | 9 | 9 | 9 | 8 | 6 | 7 | 7 |

M4 | 10 | 10 | 10 | 10 | 9 | 9.5 | 10 | 10 | 10 | 9 | 10 | 9.5 | 9.75 |

M5 | 9 | 7 | 8 | 8 | 4 | 6 | 1 | 1 | 1 | 2 | 2 | 2 | 4.25 |

M6 | 6 | 3 | 4.5 | 6 | 8 | 7 | 6 | 5 | 5.5 | 4 | 7 | 5.5 | 5.625 |

M7 | 3 | 2 | 2.5 | 3 | 5 | 4 | 4 | 6 | 5 | 5 | 8 | 6.5 | 4.5 |

M8 | 2 | 1 | 1.5 | 4 | 3 | 3.5 | 5 | 7 | 6 | 6 | 9 | 7.5 | 4.625 |

M9 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 1 | 2 | 2.5 |

M10 | 1 | 5 | 3 | 1 | 1 | 1 | 3 | 3 | 3 | 1 | 3 | 2 | 2.25 |

N-RMSE | |||||||||||||

M1 | 9 | 9 | 9 | 9 | 10 | 9.5 | 9 | 9 | 9 | 10 | 9 | 9.5 | 9.25 |

M2 | 7 | 6 | 6.5 | 7 | 8 | 7.5 | 7 | 7 | 7 | 7 | 6 | 6.5 | 6.875 |

M3 | 5 | 8 | 6.5 | 5 | 6 | 5.5 | 8 | 8 | 8 | 8 | 7 | 7.5 | 6.875 |

M4 | 10 | 10 | 10 | 10 | 9 | 9.5 | 10 | 10 | 10 | 9 | 10 | 9.5 | 9.75 |

M5 | 8 | 7 | 7.5 | 8 | 5 | 6.5 | 3 | 2 | 2.5 | 3 | 2 | 2.5 | 4.75 |

M6 | 6 | 4 | 5 | 6 | 7 | 6.5 | 6 | 5 | 5.5 | 4 | 8 | 6 | 5.75 |

M7 | 3 | 2 | 2.5 | 4 | 4 | 4 | 5 | 6 | 5.5 | 6 | 5 | 5.5 | 4.375 |

M8 | 2 | 1 | 1.5 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 4 | 4.5 | 3.25 |

M9 | 4 | 5 | 4.5 | 2 | 2 | 2 | 2 | 3 | 2.5 | 2 | 1 | 1.5 | 2.625 |

M10 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 1.5 |

SSP | NSP | ||||||||

Model | $Ad-{R}^{2}$ | N-RMSE | NP | MB | Model | $Ad-{R}^{2}$ | N-RMSE | NP | MB |

M1 | 8.5 | 9 | 2 | 1 | M1 | 9.5 | 9 | 2 | 1 |

M2 | 6.5 | 6.5 | 3 | 0 | M2 | 7 | 7 | 3 | 0 |

M3 | 6.5 | 6.5 | 3 | 0 | M3 | 5.5 | 5 | 3 | 0 |

M4 | 10 | 10 | 2 | 1 | M4 | 9.5 | 9 | 2 | 1 |

M5 | 8 | 7.5 | 3 | 1 | M5 | 6 | 6 | 3 | 1 |

M6 | 4.5 | 5 | 4 | 0 | M6 | 7 | 6 | 4 | 0 |

M7 | 2.5 | 2.5 | 4 | 0 | M7 | 4 | 4 | 4 | 0 |

M8 | 1.5 | 1.5 | 5 | 0 | M8 | 3.5 | 3 | 5 | 0 |

M9 | 4 | 4.5 | 4 | 0 | M9 | 2 | 2 | 4 | 0 |

M10 | 3 | 2 | 6 | 0 | M10 | 1 | 1 | 6 | 0 |

MP | Irregular | ||||||||

Model | $Ad-{R}^{2}$ | N-RMSE | NP | MB | Model | $Ad-{R}^{2}$ | N-RMSE | NP | MB |

M1 | 7.5 | 9 | 2 | 1 | M1 | 7 | 9.5 | 2 | 1 |

M2 | 6 | 7 | 3 | 0 | M2 | 6 | 6.5 | 3 | 0 |

M3 | 9 | 8 | 3 | 0 | M3 | 7 | 7.5 | 3 | 0 |

M4 | 10 | 10 | 2 | 1 | M4 | 9.5 | 9.5 | 2 | 1 |

M5 | 1 | 2.5 | 3 | 1 | M5 | 2 | 2.5 | 3 | 1 |

M6 | 5.5 | 5.5 | 4 | 0 | M6 | 5.5 | 6 | 4 | 0 |

M7 | 5 | 5.5 | 4 | 0 | M7 | 6.5 | 5.5 | 4 | 0 |

M8 | 6 | 4 | 5 | 0 | M8 | 7.5 | 4.5 | 5 | 0 |

M9 | 2 | 2.5 | 4 | 0 | M9 | 2 | 1.5 | 4 | 0 |

M10 | 3 | 1 | 6 | 0 | M10 | 2 | 2 | 6 | 0 |

**Table 5.**Ranking results (starting from the best model) and scores of TOPSIS and PROMETHEE for three scenarios and four hydrograph types.

Dataset Type | Scenario 1 | Scenario 2 | Scenario 3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

TOPSIS | PROMETHEE | TOPSIS | PROMETHEE | TOPSIS | PROMETHEE | |||||||

Model | $\mathit{\chi}$ | Model | $\mathit{\xi}$ | Model | $\mathit{\chi}$ | Model | $\mathit{\xi}$ | Model | $\mathit{\chi}$ | Model | $\mathit{\xi}$ | |

SSP | 8 | 0.5475 | 8 | 0.1251 | 8 | 0.5971 | 8 | 0.1693 | 8 | 0.6970 | 8 | 0.2577 |

5 | 0.5398 | 7 | 0.1157 | 7 | 0.5779 | 7 | 0.1464 | 7 | 0.6730 | 7 | 0.2078 | |

7 | 0.5311 | 5 | 0.0507 | 10 | 0.5477 | 10 | 0.0353 | 10 | 0.6481 | 10 | 0.1245 | |

1 | 0.5116 | 1 | 0.0413 | 5 | 0.4954 | 5 | 0.0124 | 9 | 0.5656 | 9 | 0.0477 | |

10 | 0.4981 | 10 | −0.0093 | 9 | 0.4918 | 9 | 0.0092 | 6 | 0.5266 | 6 | 0.0020 | |

4 | 0.4684 | 9 | −0.0101 | 1 | 0.4622 | 1 | −0.0105 | 5 | 0.4087 | 5 | −0.0642 | |

9 | 0.4536 | 6 | −0.0461 | 6 | 0.4613 | 6 | −0.0301 | 2 | 0.3850 | 1 | −0.1141 | |

6 | 0.4268 | 4 | −0.0485 | 4 | 0.4179 | 4 | −0.1085 | 3 | 0.3850 | 2 | −0.1165 | |

2 | 0.3525 | 2 | −0.1094 | 2 | 0.3637 | 2 | −0.1118 | 1 | 0.3626 | 3 | −0.1165 | |

3 | 0.3525 | 3 | −0.1094 | 3 | 0.3637 | 3 | −0.1118 | 4 | 0.3158 | 4 | −0.2284 | |

NSP | 5 | 0.5939 | 5 | 0.1586 | 10 | 0.5812 | 9 | 0.1856 | 10 | 0.6834 | 10 | 0.2618 |

10 | 0.5307 | 9 | 0.1516 | 9 | 0.5771 | 10 | 0.1529 | 9 | 0.6723 | 9 | 0.2536 | |

9 | 0.5303 | 10 | 0.0985 | 5 | 0.5555 | 5 | 0.1301 | 8 | 0.5904 | 8 | 0.0975 | |

1 | 0.4693 | 7 | 0.0078 | 8 | 0.5037 | 8 | 0.0320 | 7 | 0.5459 | 5 | 0.0730 | |

4 | 0.4693 | 8 | −0.0007 | 7 | 0.4761 | 7 | 0.0288 | 5 | 0.4836 | 7 | 0.0706 | |

8 | 0.4593 | 1 | −0.0126 | 1 | 0.4188 | 3 | −0.0333 | 3 | 0.4293 | 3 | −0.0250 | |

7 | 0.4396 | 4 | −0.0126 | 4 | 0.4188 | 1 | −0.0693 | 1 | 0.3166 | 6 | −0.1810 | |

3 | 0.3804 | 3 | −0.0375 | 3 | 0.3973 | 4 | −0.0693 | 4 | 0.3166 | 1 | −0.1827 | |

2 | 0.2849 | 2 | −0.1633 | 6 | 0.2835 | 2 | −0.1706 | 6 | 0.3023 | 4 | −0.1827 | |

6 | 0.2730 | 6 | −0.1899 | 2 | 0.2814 | 6 | −0.1869 | 2 | 0.2747 | 2 | −0.1851 | |

MP | 5 | 0.8853 | 5 | 0.4671 | 5 | 0.8856 | 5 | 0.4667 | 5 | 0.8863 | 5 | 0.4657 |

9 | 0.5361 | 9 | 0.1207 | 9 | 0.5821 | 9 | 0.1519 | 9 | 0.6743 | 9 | 0.2142 | |

1 | 0.5161 | 1 | 0.0883 | 10 | 0.5612 | 10 | 0.0593 | 10 | 0.6577 | 10 | 0.1525 | |

10 | 0.5125 | 10 | 0.0127 | 1 | 0.4687 | 1 | 0.0407 | 8 | 0.4792 | 7 | −0.0451 | |

4 | 0.4556 | 4 | −0.0306 | 8 | 0.4199 | 7 | −0.0704 | 7 | 0.4670 | 1 | −0.0543 | |

7 | 0.3898 | 7 | −0.0830 | 7 | 0.4174 | 4 | −0.0889 | 6 | 0.4454 | 8 | −0.0651 | |

8 | 0.3865 | 6 | −0.1000 | 4 | 0.4054 | 6 | −0.0889 | 1 | 0.3757 | 6 | −0.0667 | |

6 | 0.3746 | 2 | −0.1054 | 6 | 0.4000 | 8 | −0.1074 | 2 | 0.3708 | 2 | −0.1114 | |

2 | 0.3450 | 8 | −0.1285 | 2 | 0.3541 | 2 | −0.1074 | 4 | 0.3047 | 4 | −0.2056 | |

3 | 0.2444 | 3 | −0.2412 | 3 | 0.2319 | 3 | −0.2556 | 3 | 0.2072 | 3 | −0.2843 | |

Irregular | 5 | 0.8966 | 5 | 0.4697 | 5 | 0.8995 | 5 | 0.4694 | 5 | 0.9051 | 5 | 0.4690 |

9 | 0.5433 | 9 | 0.1954 | 9 | 0.5912 | 9 | 0.2333 | 9 | 0.6900 | 9 | 0.3093 | |

1 | 0.5349 | 1 | 0.0611 | 10 | 0.5528 | 10 | 0.1014 | 10 | 0.6574 | 10 | 0.2016 | |

10 | 0.5020 | 10 | 0.0513 | 1 | 0.4872 | 1 | 0.0111 | 6 | 0.4213 | 1 | −0.0889 | |

4 | 0.4896 | 4 | −0.0407 | 4 | 0.4387 | 2 | −0.0972 | 7 | 0.3986 | 6 | −0.0910 | |

2 | 0.3515 | 2 | −0.0961 | 6 | 0.3743 | 4 | −0.1000 | 8 | 0.3979 | 2 | −0.0995 | |

6 | 0.3494 | 6 | −0.1191 | 2 | 0.3634 | 6 | −0.1097 | 1 | 0.3919 | 7 | −0.1185 | |

7 | 0.3342 | 7 | −0.1407 | 7 | 0.3566 | 7 | −0.1333 | 2 | 0.3872 | 8 | −0.1634 | |

8 | 0.3193 | 3 | −0.1750 | 8 | 0.3475 | 3 | −0.1833 | 4 | 0.3344 | 3 | −0.2000 | |

3 | 0.2965 | 8 | −0.2058 | 3 | 0.2956 | 8 | −0.1917 | 3 | 0.2939 | 4 | −0.2185 |

**Table 6.**MCDA results of all flood events. (a) Decision matrix obtained from 8 flood events and (b) corresponding scores of TOPSIS and PROMETHEE for three scenarios.

(a) | |||||||||||

Model | $Ad-{R}^{2}$ | N-RMSE | NP | MB | |||||||

M1 | 8.125 | 9.25 | 2 | 1 | |||||||

M2 | 6.375 | 6.875 | 3 | 0 | |||||||

M3 | 7 | 6.875 | 3 | 0 | |||||||

M4 | 9.75 | 9.75 | 2 | 1 | |||||||

M5 | 4.25 | 4.75 | 3 | 1 | |||||||

M6 | 5.625 | 5.75 | 4 | 0 | |||||||

M7 | 4.5 | 4.375 | 4 | 0 | |||||||

M8 | 4.625 | 3.25 | 5 | 0 | |||||||

M9 | 2.5 | 2.625 | 4 | 0 | |||||||

M10 | 2.25 | 1.5 | 6 | 0 | |||||||

(b) | |||||||||||

Scenario 1 | Scenario 2 | Scenario 3 | |||||||||

TOPSIS | PROMETHEE | TOPSIS | PROMETHEE | TOPSIS | PROMETHEE | ||||||

Model | $\chi $ | Model | $\xi $ | Model | $\chi $ | Model | $\xi $ | Model | $\chi $ | Model | $\xi $ |

5 | 0.7539 | 5 | 0.2912 | 5 | 0.7346 | 5 | 0.2747 | 5 | 0.7008 | 5 | 0.2418 |

9 | 0.5271 | 9 | 0.1287 | 9 | 0.5742 | 9 | 0.1606 | 10 | 0.6722 | 9 | 0.2244 |

1 | 0.5201 | 10 | 0.0556 | 10 | 0.5687 | 10 | 0.1061 | 9 | 0.6711 | 10 | 0.2071 |

10 | 0.5179 | 1 | 0.0292 | 8 | 0.4946 | 7 | 0.0010 | 8 | 0.5833 | 8 | 0.0431 |

4 | 0.4821 | 7 | −0.0176 | 7 | 0.4804 | 8 | −0.0146 | 7 | 0.5558 | 7 | 0.0382 |

8 | 0.4498 | 8 | −0.0435 | 1 | 0.4707 | 1 | −0.0237 | 6 | 0.4473 | 6 | −0.0849 |

7 | 0.4420 | 4 | −0.0556 | 4 | 0.4313 | 6 | −0.1045 | 2 | 0.3717 | 1 | −0.1295 |

6 | 0.3686 | 6 | −0.1144 | 6 | 0.3959 | 4 | −0.1162 | 1 | 0.3701 | 2 | −0.1352 |

2 | 0.3429 | 2 | −0.1241 | 2 | 0.3527 | 2 | −0.1278 | 3 | 0.3420 | 3 | −0.1676 |

3 | 0.3248 | 3 | −0.1495 | 3 | 0.3306 | 3 | −0.1556 | 4 | 0.3278 | 4 | −0.2374 |

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**

Şahin, A.U.; Özkaya, A.
Multi-Criteria Decision Analyses for the Selection of Hydrological Flood Routing Models. *Water* **2023**, *15*, 2588.
https://doi.org/10.3390/w15142588

**AMA Style**

Şahin AU, Özkaya A.
Multi-Criteria Decision Analyses for the Selection of Hydrological Flood Routing Models. *Water*. 2023; 15(14):2588.
https://doi.org/10.3390/w15142588

**Chicago/Turabian Style**

Şahin, Abdurrahman Ufuk, and Arzu Özkaya.
2023. "Multi-Criteria Decision Analyses for the Selection of Hydrological Flood Routing Models" *Water* 15, no. 14: 2588.
https://doi.org/10.3390/w15142588