# 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$${V}_{j}^{+}=\left(\right)open="\{"\; close>\begin{array}{c}max\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{maximized}\\ min\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}$$$${V}_{j}^{-}=\left(\right)open="\{"\; close>\begin{array}{c}min\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{maximized}\\ max\left({v}_{ij}\right),if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}$$
- Step 4:
- The Euclidean distance for the ideal best and the ideal worst solutions will be$${\zeta}_{i}^{+}=\sqrt{\sum _{i=1}^{r}{\left(\right)}^{{v}_{ij}}2}$$$${\zeta}_{i}^{-}=\sqrt{\sum _{i=1}^{r}{\left(\right)}^{{v}_{ij}}2}$$
- 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$${z}_{ij}=\left(\right)open="\{"\; close>\begin{array}{c}\frac{{c}_{j}\left({a}_{i}\right)-min\left({c}_{j}\right)}{max\left({c}_{j}\right)-min\left({c}_{j}\right)},if{c}_{j}\mathrm{is}\mathrm{maximized}\\ \frac{max\left({c}_{j}\right)-{c}_{i}\left({a}_{i}\right)}{max\left({c}_{j}\right)-min\left({c}_{j}\right)},if{c}_{j}\mathrm{is}\mathrm{minimized}\end{array}$$
- 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$$\pi ({a}_{i},{a}_{j})=\sum _{k=1}^{r}{P}_{k}\left(\right)open="("\; close=")">{d}_{k}\left(\right)open="("\; close=")">{a}_{i},{a}_{j}{w}_{k}$$$$P\left(\right)open="("\; close=")">{d}_{k}$$Thus, $\pi $ values can be expressed by a $p\times p$ matrix whose diagonal entries are empty as$$\Gamma ={\left(\right)}_{\begin{array}{cccccc}-& \pi ({a}_{1},{a}_{2})& \pi ({a}_{1},{a}_{3})& \xb7& \xb7& \pi ({a}_{1},{a}_{p})\\ \pi ({a}_{2},{a}_{1})& -& \pi ({a}_{2},{a}_{3})& \xb7& \xb7& \pi ({a}_{2},{a}_{p})\\ \xb7& \xb7& \xb7& \xb7& \xb7& \xb7\\ \pi ({a}_{p},{a}_{1})& \pi ({a}_{p},{a}_{2})& \xb7& \xb7& \pi ({a}_{p-1},{a}_{p})& -\end{array}}p\times p$$
- Step 4:
- The ideal best (positive flow) and the ideal worst (negative flow) solutions are$${\xi}_{i}^{+}\left(\right)open="("\; close=")">{a}_{i}$$$${\xi}_{i}^{-}\left(\right)open="("\; close=")">{a}_{i}$$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 |

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**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 |

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**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