Development of A Novel Discharge Routing Method Based On the Large Discharge Dataset, Muskingum Model, Optimization Methods, and Multi-Criteria Decision Making

Journal of Soft Computing in Civil Engineering 8-4 (2024) 54-93
How to cite this article: Valikhan Anaraki M, Farzin S, Ahmadianfar I, Shams A. Development of a novel discharge routing
method based on the large discharge dataset, muskingum model, optimization methods, and multi-criteria decision making. J Soft
Comput Civ Eng 2024;8(4):54–93. https://doi.org/10.22115/scce.2023.400704.1664
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Development of A Novel Discharge Routing Method Based On the
Large Discharge Dataset, Muskingum Model, Optimization
Methods, and Multi-Criteria Decision Making
Mahdi Valikhan Anaraki 1
; Saeed Farzin 2*
; Iman Ahmadianfar 3
; Amin Shams 4
1. Ph.D., Faculty of Civil Engineering, Semnan University, Semnan, Iran
2. Associate Professor, Faculty of Civil Engineering, Semnan University, Semnan, Iran
3. Assistant Professor, Department of Civil Engineering, Behbahan Khatam Alanbia University of Technology, Behbahan,
Iran
4. Assistant Professor, Faculty of Civil Engineering, Semnan University, Semnan, Iran
Corresponding author: saeed.farzin@semnan.ac.ir
https://doi.org/10.22115/SCCE.2023.400704.1664
ARTICLE INFO ABSTRACT
Article history:
Received: 28 June 2023
Revised: 15 October 2023
Accepted: 20 November 2023
Discharge routing is a key method for predicting downstream
hydrograph in canals or rivers. Predicting the downstream hydrograph
using the upstream hydrograph can significantly decrease the flood
damages. A combination of a large dataset and Muskingum model can
enhance river discharge routing reliability. However, calibrating the
Muskingum model for large datasets requires powerful optimization
algorithms. Metaheuristic Optimization Algorithms (MOAs) can
accurately calibrate the Muskingum model, but their performance may
vary. Hence, in the present study, a new technique is introduced for
discharge routing based on the large discharge dataset: Muskingum
model, MOAs, and Multi-criteria Decision-Making (MCDM).
Different MOAs, including a Genetic Algorithm (GA), Particle Swarm
Optimization (PSO), Firefly Algorithm (FFA), Cuckoo Search (CS),
Bat Algorithm (BA), Shark Smell Optimization (SSO), Whale
Optimization Algorithm (WOA), Harris Hawk's Optimization (HHO),
and hybrid of WOA and CS (WOA_CS), were developed for
Muskingum calibration. The Mollasani-Ahvaz, Harmaleh-Bamdej
river reaches Karun basin, and Lighvan-Heravi in the Urmia basin are
considered case studies. Results of discharge routing based on the
evaluation criteria in the training period showed MOAs were trained
with high accuracy and reliability. While in the testing period, each
MOA achieves better results in some evaluation criteria. Considering
all evaluation criteria using the MCDM showed that WOA_CS, WOA,
and FFA were placed in the first, second, and third rankings,
respectively. The MCDM scores for WOA_CS, WOA, and FFA were
equal to 0.960, 0.913, and 0.907, respectively. The developed method
in this study has a good potential for discharge routing in various river
reaches.
Keywords:
Flood routing;
Large dataset flood;
Muskingum model;
Metaheuristic optimization
algorithms;
Multi criteria decision-making.

M. Valikhan Anaraki et al./ Journal of Soft Computing in Civil Engineering 8-4 (2024) 54-93 55
1. Introduction
Flood is one of the most important natural hazards that has caused humans to manage damages to
manufactured structures, human life, and countries' economies. Management of this phenomenon
has always been a solution to control and reduce its damages. Discharge forecasting is an
essential tool for managing it. In this regard, discharge and flood routing are two of the best
management methods. There are two crucial discharge or flood routing techniques: hydraulically
and hydrologically. Hydraulic methods have a complex structure and need many input
parameters, such as geometry and riverine characteristics. In contrast, hydrological methods have
a simple structure and are only executed by a few parameters. The Muskingum model is one of
the most famous hydrological discharge or flood routings researchers widely use because of its
simplicity and fewer computation times than similar methods [1,2].
So far, different Muskingum models have been introduced for the discharge or flood routings. In
following some of well-known Muskingum models are reviewed. Gavilan and Houck [3]
introduced two Muskingum models for flood routing. These models used three parameters and
were a linear relationship between storage and weighted inflow and outflow. Gill [4] reported
that there is much error if the relationship between storage and flow is linear. Therefore, Gill
modified the linear Muskingum model as a nonlinearity. This model was similar to the proposed
relationship with Gill [4], but it has a power parameter for inflow and outflow. The fourth
Muskingum model had four parameters and was presented by Easa [5]. This model is the
weighted sum of inflow and outflow in power form. In the fifth relationship, Haddad et al. [6]
defined seven parameters for the Muskingum equation. This model was more accurate than other
previously established Muskingum models. Zhang et al. [7] proposed a Muskingum model for
considering lateral flow. The mentioned Muskingum models require parameter determination
prior to execution. There are three methods to find the best parameters for the Muskingum
model: graphical, moment and cumulants, and optimization algorithms. The mentioned methods
work for earlier versions of Muskingum models, but they don't apply to models with over three
parameters. Therefore, using optimization algorithms is necessary to find the best parameters for
new Muskingum models. Although some optimization algorithms estimate the Muskingum
model's parameters with good accuracy, however, mathematical and numeric optimization
algorithms have the limitation in solving Muskingum models with numerous parameters and
constraints. Mathematical optimization algorithms only solve differentiable objective functions.
As the number of decision variables and constraints increases, the computational efforts of
numeric optimization algorithms grow exponentially. However, metaheuristic optimization
algorithms (MOAs) can efficiently solve any optimization problem. In solving a wide range of
optimization problems, MOAs are faster and require less computation cost than mathematical or
numeric optimization methods [8]. Hence, in recent years, MOAs have successfully solved flood
discharge routing problems. Farzin et al. [9] used an improved bat algorithm (IBA) and the three-
parameter Muskingum model in different Wilson, Karahan, and Myanmar River flood case
studies. Results showed more accuracy of IBA than bat algorithm (BA), particle swarm
optimization (PSO), and genetic algorithm (GA). In Wilson, Karahan, as well as Viessman and
Lewis case studies, Node Farahani et al. [9] Employed the kidney algorithm (KA) to find the

56 M. Valikhan Anaraki et al./ Journal of Soft Computing in Civil Engineering 8-4 (2024) 54-93
optimal parameter of the four-parameter Muskingum model for flood routing. Their results
indicated better results for KA compared with GA and PSO. Also, this study reported that the
four-parameter Muskingum model outperforms the three-parameter Muskingum. Farahani et al.
[10] applied the shark algorithm (SA) to optimize the four-parameter Muskingum in flood
routing of three benchmark case studies, and the results showed that the SA is more capable than
various optimizers such as the harmony search (HS) algorithm. Akbari et al. [11] employed the
hybrid of PSO and GA to find the optimal parameters of the four-parameter Muskingum model
in the four benchmark case studies. They indicated good application of this method compared to
other metaheuristic algorithms such as KA. Norouzi and Bazargan [12] used linear Muskingum
for four floods in the Karun River. In this study, the parameters of the Muskingum model were
calibrated for all four floods, and the results demonstrated improved food routing accuracy using
an investigated approach. Niazkar and Zakwan [13] applied the new four parameters Muskingum
method to solve Wilson, Viessman and Lewis, and Wye floods. This study employs the modified
honey bee Mating Optimization, the Generalized Reduced Gradient, and the hybrid. Results
indicate the better accuracy of the hybrid algorithm. Moradi et al. [14] introduced the
Muskingum model with twelve parameters. In this model, the lateral flow was considered and
the parameters of the model were obtained with the gorilla troop optimizer.
In the most previously conducted studies, only floods with one peak or a few peaks were used for
flood routing. The calibration period in the mentioned studies needed to be longer compared to
the actual conditions. The Muskingum model may only accurately predict downstream discharge
data in different conditions if calibrated with more than one flood event. This issue can be
addressed by discharge routing. In discharge routing, the downstream river's discharge is
predicted based on the upstream river's discharge. In the discharge data with long periods, there
are many floods and drought events. Therefore, predicting discharge data can help water resource
management with high reliability. To address this issue, some studies have been performed by
researchers. Perumal and price [15] introduced variable parameters McCarthy–Muskingum
model, one physically based variable parameter Muskingum method developed from the full
Saint–Venant equations. This method was used in different studies, such as Yadav et al. [16],
Barbetta et al. [17], and Yadav et al. [18], for discharge routing in various case studies. However,
this method has drawbacks, such as needing initial and boundary conditions and needing too
much physical data for execution. Integrating the Muskingum model with optimization
algorithms can address discharge routing with a large discharge dataset. Recently, new
metaheuristic algorithms such as grey wolf optimizer (GWO) [19], whale optimization algorithm
(WOA) [20], grasshopper optimization algorithm (GOA) [21], modified ideal gas molecular
movement algorithm [22], Harris hawk's optimization (HHO) [23], hybrid generalized reduced
gradient-based particle swarm optimizer [24], gradient-based optimizer (GBO) [25], improved
chaotic ideal gas molecular movement [26], hybrid grey wolf optimizer (GWO) combined with
elephant herding optimization (EHO) [27], and opposition‐based ideal gas molecular movement
algorithm [28], because of their high ability in finding optimal solution gained more attention of
researchers. These algorithms have successfully been applied to engineering problems such as
predicting soil electrical conductivity [29], optimal design of water distribution networks [30],

tuning machine learning methods [25−27], water resources demand forecasting [34],
optimization of hydrothermal scheduling [35] as well as open channel designing [36].
The other topic of debate for scientific society is selecting the best algorithm for optimization
engineering problems. Because there are many criteria for choosing the best optimization
algorithm, such as accuracy and time computing in finding the optimal solution, multi-criteria
decision-making (MCDM) can be one ideal solution to choosing the best algorithms by
considering all the mentioned criteria. Researchers widely used the MCDM methods to select the
best optimization algorithm in assessing the relationships between the water control
infrastructure and water governance [37], optimal operation of benchmark reservoirs [38] for
choosing the best machine learning [39,40], and hydrochemical assessment of groundwater
suitability for irrigation [41].
According to the best author's knowledge, the lack of using a large discharge dataset, discharge
routing with efficient methods, integrating the new Muskingum model with powerful MOAs, and
selecting the best MOAs for optimizing the Muskingum model are the main gaps in previous
research. Hence, this study aims to address these gaps using developing one new approach to
discharge routing. In this approach, the integration of the Muskingum model and different MOAs
are developed for discharge routing. In this approach, a large discharge river flow dataset is
employed for training and testing the Muskingum Model. Moreover, the MCDM is used to select
the best MOAs in the performance optimization of the Muskingum model. This approach can
apply to other case studies for discharge routing.
2. Research significance
Integrating MOA, the Muskingum method, and MCDM can increase the accuracy and efficiency
of large-scale discharge routing. This approach unlike other discharge simulation methods only
needs to upstream hydrograph of river discharge and can be used in water resources management
and flood warning systems. Some studies used the Muskingum method for flood routing;
however, they only consider large-scale river discharge, leading to unreliable models and
underestimating results. Some studies used physical methods such as variable parameter
McCarthy–Muskingum for large-scale river discharge. However, this method needs many
datasets, such as lateral flow, physical and geometric characteristics of rivers, and high-time
computing for solving physical equations in the problem domain. Hence, the present research has
focused on developing a new technique by integrating MOAs, the Muskingum method, and
MCDM. Moreover, in the present study, long-term daily runoff data containing many flood
events will calibrate a seven-parameter Muskingum model. The presented method can provide
more reliable results by considering the entire river flow (base flow + flood wave) in a long
statistical period. Unlike in previous studies, part of the data will be used to validate the obtained
model. It is worth mentioning that three different case studies, including the Mollasani-Ahvaz,
Harmaleh-Bamdej river reaches in the Karun basin, and Lighvan-Heravi in the Urmia basin in
Iran, are chosen to investigate the application of MOAs in optimizing the Muskingum model.
The results of MOAs are also compared, and the best algorithm is selected by employing

MCDM. Finally, the best parameters of the Muskingum model and their outcomes for flood
routing are reported. Figure 1 shows the flowchart of the present study.
Fig. 1. The flowchart of present study.
3. Methods
3.1. Case studies and data collection
The present study's three reaches are chosen as case studies, including Mollasani-Ahvaz and
Harmaleh-Bamdej in the Karun basin and Lighvan-Heravi in the Urmia basin. The Mollasani-
Ahvaz and Harmaleh-Bamdej reach the Karun basin and Karun River, one of Iran's perennial
basin rivers. In addition, Lighvan-Heravi's reach is placed in mountainous regions of the Urmia
basin, one of the most famous basins in Iran. Each of these reaches is placed in various
morphological and climate conditions. Therefore, using these case studies can reveal the
accuracy of the proposed method of discharge routing. Figure 2 shows the locations of the case
studies in Iran. The inflow and outflow data of the mentioned reaches from 2011/03/21 to
2017/03/20 are used for the discharge routing. Table 1 shows the present study's usage data and
its statistical criteria.
3.2. Muskingum model
The present study employs a new non-linear Muskingum model [7] with seven parameters,
called the NL5 (mean fifth equation for non-linear storage-discharge relationship in natural
stream), for the discharge routing. The NL5 considers the morphological changes between the
upstream and downstream sections in a river reach, while other methods consider the same
morphology characteristic along one river reach.

Table 1
Usage flow data and its statistical criteria.
Flow
Date
Mollasani
(m3
/s)
Ahvaz (m3
/s) Harmaleh (m3
/s) Bamdej (m3
/s) Lighvan (m3
/s) Heravi (m3
/s)
2011-03-21 257 268.00 62.20 18.00 0.91 0.02
2011-03-22 241 263.00 58.10 17.60 0.96 0.02
2011-03-23 243 267.00 56.10 18.00 0.75 0.03
2011-03-24 243 268.00 54.80 17.60 0.70 0.03
2011-03-25 246 270.00 55.50 17.50 0.86 0.02
2011-03-26 249 271.00 54.80 17.50 0.91 0.20
2011-03-27 252 270.00 52.80 17.90 0.68 0.02
2011-03-28 243 245.00 53.50 18.70 0.46 0.02
2011-03-29 223 232.00 52.80 18.30 0.23 0.01
. . . . . . .
. . . . . . .
. . . . . . .
2017-03-11 386 374.00 105.00 5.88 0.16 0.011
2017-03-12 400 403.00 105.00 6.28 0.16 0
2017-03-13 378 351.00 105.00 6.49 0.16 0
2017-03-14 364 346.00 94.50 6.92 0.25 0
2017-03-15 366 345.00 94.50 6.81 0.30 0.015
2017-03-16 362 354.00 93.90 6.39 0.35 0.031
2017-03-17 368 370.00 93.90 6.28 0.30 0.046
2017-03-18 348 319.00 93.30 6.39 0.25 0.062
2017-03-19 326 315.00 99.80 6.7 0.25 0.077
2017-03-20 342 336.00 101.00 6.7 0.16 0.112
Statistical criteria
Mean (m3
/s) 300.88 308.30 95.80 93.76 0.70 0.42
Max (m3
/s) 1357.00 1358.00 1360.00 1082.00 20.60 19.00
Min (m3
/s) 0.00 0.00 0.00 0.00 0.00 0.00
Std (m3
/s) 118.03 116.68 89.31 86.97 1.04 0.79
Fig. 2. The locations of the case study.

The main relationship in this method is expressed as follows:
    
1 2
1 2
1
S K X C I X C O

 
 
  
  (1)
In which, 𝐼 is inflow, 𝑂 is outflow, 𝑆 is storage, 𝐾 is time storage coefficient, 𝑋 is a weighted
factor that indicates the effect of inflow and outflow on storage, 𝐶1, 𝐶2, 𝛼1and 𝛼2 are parameters
for considering different morphology between the upstream and downstream sections, and 𝛽 is
one exponent.
3.3. Genetic algorithm (GA)
GA is an evolutionary optimization algorithm which had applications in different aspects, such as
optimizing parameters of machine learning in streamflow forecasting [42]. In the GA first, we
randomly generated the initial population. Then, the population evolves to achieve an optimal
objective function in the optimization process. In this process, the selection operator selects a
population of individuals with better objective functions. Afterward, the selected individuals are
considered parent individuals. These parents produce children using the crossover and mutation
operators. Each operator applies to parent individuals by a certain probability. The probability of
crossover is called crossover probability, and the probability for mutation is called mutation
probability. The relationship for the crossover operator is as follows:
 
1
* 1
t t t
i i j
Pop Pop Pop
 

   (2)
 
1
* 1
t t t
j j i
Pop Pop Pop
 

   (3)
In which, 𝑃𝑜𝑝𝑖
𝑡+1
is ith
children, 𝑃𝑜𝑝𝑖
𝑡
is ith
parent, 𝑃𝑜𝑝𝑗
𝑡+1
is jth
children, 𝑃𝑜𝑝𝑗
𝑡
is jth
parent, and
𝛼 is a random number from 0 to 1. The mutation is defined based on the following relationship:
 
1
, , , ,
*
t
i j i j i j i j
Pop Lb Ub Lb


   (4)
In which, 𝑃𝑜𝑝𝑖,𝑗
𝑡+1
denotes the ith
gene in jth
chromosome, 𝑈𝑏𝑖,𝑗 is the upper bound of ith
gene in
jth
chromosome, 𝐿𝑏𝑖,𝑗 is the lower bound of ith
gene in jth
chromosome, 𝛽 is a random number
between 0 to 1. For more information, please see [43].
3.4. Particle swarm optimization (PSO)
PSO is a swarm-based algorithm which was employed in different fields, such as modelling
streamflow [44]. The population in PSO is a group of particles. Each of these particles has one
position, velocity, and an objective function. PSO updates the position and velocity of particles
based on the best global experience of particles ( Gbest
X ) and the best personal experience of
particles ( ,
Pbest i
X ). The best global experience of particles is an optimal solution for all particles
found so far, and the best personal experience of particles is a better solution than each particle
experiences. In PSO, the search process begins by generating a random initial population. Then
the velocity and position of particles are updated based on the following relationship:
   
1
1 2 ,
* * * * *
t t t t
i i Gbest i Pbest i i
V V c rand X X c rand X X


     (5)

1 1
t t t
i i i
X X V
 
  (6)
In which, 𝑉𝑖
𝑡+1
is the updated velocity of ith
particle, 𝑉𝑖
𝑡
is the old velocity of ith
particle, 𝑋𝐺𝑏𝑒𝑠𝑡 is
the position of best global experience of all particles, 𝑋𝑃𝑏𝑒𝑠𝑡,𝑖 is the position of best personal
experience of ith
particles, 𝑋𝑖
𝑡+1
is the updated position of ith
particle, 𝑋𝑖
𝑡
is the old position of ith
particle, 𝜔 is inertia weight, 𝑐1 and 𝑐2 are acceleration coefficients, and 𝑟𝑎𝑛𝑑 is a random
number between 0 and 1. For more information, please refer to [45].
3.5. Firefly algorithm (FFA)
FFA [46] inspires the firefly's attraction to the flashing light. FFA was used by [47] for
streamflow forecasting. This algorithm works based on the three following concepts:
1. Each firefly is attracted to other fireflies, regardless of their sex.
2. The firefly's attractiveness is proportional to light intensity.
3. The objective function determines the light intensity of the firefly. In FFA, fireflies move
toward fireflies with more light intensity (objective function). For more information about
FFA, please see [48].
3.6. Cuckoo search (CS)
The reproduction strategy of cuckoo birds inspires CS [49]. In this strategy, the cuckoos' lay eggs
in the other birds' nests. CS is developed based on the following assumptions:
1. Each cuckoo lays one egg in each iteration, and the host nest is randomly selected.
2. The nests with better quality are transferred to the next iteration.
3. There is the probability of Pa (between 0 and 1) to find the egg laid by the cuckoo.
After that, the cuckoo lays an egg in a host's nest, and other birds may find the cuckoo's egg and
discard the egg or leave the nest. This process is down by the probability of Pa. Also, in CS,
cuckoos search according to levy flight distribution to find host nests. For more information
about CS, please refer to the study of [50].
3.7. Bat algorithm (BA)
The echolocation ability of microbat inspires the BA [51]. This algorithm was used in other
fields, such as drought forecasting [52]. In nature, microbats emit loud sounds to the surrounding
environment and receive their echo. Then, according to the time interval between the emitted
sound and received echo, microbats detect the prey and obstacles. The BA algorithm is designed
based on the following rules:
1. All bats use echolocation ability to find prey and the obstacle.
2. All bats randomly fly in the problem's search space by the velocity of 𝑉𝑖
𝑡
, the minimum
frequency of 𝑓𝑚𝑖𝑛, and loudness of 𝐴 in the position of 𝑋𝑖
𝑡
.

3. The loudness is varied between 𝐴𝑚𝑖𝑛 and 𝐴𝑚𝑎𝑥. For more details about BA, please refer to
[53].
3.8. Shark smell optimization (SSO)
SSO [54] is inspired by the hunting ability of sharks to move to an odor source based on their
sense of smell.
This algorithm is designed based on the following idea:
1. The prey is injured fish, and blood is regularly released into seawater. Also, the effect of
seawater flow in distorting the odor particles is not considered.
2. Only one blood source (injured fish or prey) exists.
In SSO, each shark has one velocity vector, one position vector, and one objective function, in
which the objective function shows the odor intensity. In this algorithm, first, the velocity of each
shark is updated based on the gradient of the objective function. Then, the position of the sharks
is updated according to the current velocity. Moreover, the SSO uses rotation movement to
achieve more accuracy. The position of sharks in this strategy is updated as follows:
1, 1 1
*
t m t t
i i i
Z X R X
  
  (7)
In which, 𝑍𝑖
𝑡+1,𝑚
is the updated position of the shark by rotation strategy, 𝑋𝑖
𝑡+1
is the current
position of the shark, 𝑅 is a random number between −1 and 1. For more information about SSO,
please see [43].
3.9. Whale optimization algorithm (WOA)
The WOA inspires of hunting behaviors of humpback whales. In this behavior, humpback whales
establish unique babbles as a circle or spiral path around school fish to hunt them. This algorithm
constructs from encircling prey and bubble-net attack sections. The encircling of prey is done by
moving whales toward prey as follows:
   
* best
D C X t X t
  (8)
   
1 *
best
X t X t A D
   (9)
In which, 𝑋𝑏𝑒𝑠𝑡 is the best position of all whales in iteration, 𝑡, 𝑋 position of the whale, 𝐴, and 𝐷
are coefficient vectors calculated by relations 8 and 9. Coefficient vectors are calculated as
follows:
1
2
A a r a
    (10)
2
2
C r
  (11)
In which, 𝑟1 and 𝑟2 are random vectors between 0 and 1, and 𝑎 is linearly reduced from 2 to 0 by
increasing iteration.
Also, bubble-net attacking is down as follows:
     
'
1 * *cos 2
bl
best
X t D e l X t

   (12)

In which, 𝐷′
is Euclidean distance between 𝑋𝑏𝑒𝑠𝑡 and 𝑋, 𝑏 is a parameter that relates to the shape
of the spiral, 𝑙 is a random number between −1 and 1 [20].
3.10. Harris hawks optimization (HHO)
The HHO [23] is another optimization algorithm inspired by the cooperative strategy of Harris
hawk's in hunting prey. HHO was applied in other works, such as groundwater level predicting
[55]. In HHO, hawks cooperatively besiege prey from different directions. In the hunting
strategy of Harris, hawks are seen in different patterns based on the escaping methods of prey
[56]. HHO simulates this strategy in four phases: soft besiege, hard besiege, soft besiege with
progressive rapid dives, and hard besiege with advanced rapid dives. In HHO, these four phases
are simulated according to two important parameters. These critical parameters are escaping
energy and one random number (𝑞). The escaping energy in HHO is calculated as follows:
0
2* * 1
t
E E
MaxIt
 
 
 
 
(13)
In which, 𝐸 is current escaping energy, 𝐸0 is initial escaping energy, 𝑡 is the current number of
iterations, 𝑀𝑎𝑥𝐼𝑡 is the maximum number of iterations. When 𝑐 and |𝐸| are both equal to or
greater than 0.5, the soft besiege phase is executed. If 𝑞 ≥ 0.5 and |𝐸| ≤ 0.5, the besiege phase
is done. If 𝑞 < 0.5, and |𝐸| ≥ 0.5, the hard besiege with progressive rapid dives step is done.
When 𝑞 is smaller than 0.5 and the absolute value of 𝐸 is less than 0.5, the approach of hard
besiege with progressive rapid dives is carried out. All phases have a unique formula shown in
[33].
3.11. The hybrid of WOA and CS (WOA_CS)
WOA and CS are well-known swarm-based algorithms which have unique abilities. WOA is
good in local search, while CS has the ability for global search. Therefore, a combination of them
can create one optimization algorithm with a good balance between local and global search. The
central concept of WOA_CS is based on the communication approach between WOA and CS. In
this approach, the worst solutions of each algorithm are replaced with the best solutions of
another algorithm. In the communication approach, the initial population is divided into two
subgroups that search independently in the problem's search space and then share information.
Afterward, each subgroup's worst and best solutions are introduced to other subgroups [43]. In
this method, the weaknesses of each algorithm are covered by the other algorithm. The pseudo-
code of WOA_CS is as follows (Algorithm1):
3.12. Evaluation criteria
In the present study, different evaluation criteria include sum squared deviation (SSQ), sum
absolute deviation (SAD), the error between calculated and observed peak outflow (EQp), the
error between the time of computed and observed peak outflow (ETp), mean absolute relative
error (MARE), variance index (VarexQ), agreement index (d) are used for evaluating the
accuracy of MOAs. The SSQ, SAD, EQp, ETp, MARE, VarexQ, and d are defined as follows:

1: Define search agent number (Searchagent_n), number of nests (n), 𝑃
𝑎, MaxIt, 2: Swap number
(Ri)
3: Generate the initial population for WOA
4: Calculate the objective function (SSQ) for each whale
5: Generate the initial population for CS
6: Calculate the objective function (SSQ) for each nest
7: Ri=1: R: MaxIt
8: For it=1:MaxIt
9: Run WOA
Sort solution: evaluations of WOA based on the objective function
11: Run CS
12: Sort solutions of CS based on the objective function
13: If it==Ri then
14: Replace k number of best solutions of WOA with k number of worst
solutions of CS
15: Replace k number of best solutions of CS with k number of the worst solutions of WOA
16: End
17: Update the best solution of two algorithms
18: End
19: Return the best solution (bet Muskingam parameters with minimum SSQ)
Algorithm 1. The pseudo-code of WOA_CS.
 
2
1
n
t t
Observed Routed
t
SSQ Q Q

 
 (14)
1
n
t t
Observed Routed
t
SAD Q Q

 
 (15)
peak peak
Observed Routed
peak
Observed
Q Q
EQp
Q

 (16)
peak peak
Observed Routed
ETp T T
  (17)
 
*
t t
Observed Routed
t
Observed
Q Q
MARE
N Q

 (18)
 
 
2
1
2
1
1 *100
N
t t
Observed Routed
t
N
t mean
Observed Observed
t
Q Q
VarexQ
Q Q


 

 
 
 
 

 
 


(19)
 
2
1
2 2
1 1
1
N
t t
Observed Routed
t
N N
t mean t mean
Routed observed Observed Observed
t t
Q Q
d
Q Q Q Q

 

 
  

 
(20)

In which, 𝑄𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑡
is observed outflow, 𝑄𝑅𝑜𝑢𝑡𝑒𝑑
𝑡
is routed outflow, 𝑄𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑚𝑒𝑎𝑛
is the mean of the
observed outflow, 𝑄𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑝𝑒𝑎𝑘
is the peak of observed outflow, 𝑄𝑅𝑜𝑢𝑡𝑒𝑑
𝑝𝑒𝑎𝑘
is the peak of routed
outflow, 𝑇𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑
𝑝𝑒𝑎𝑘
is the time of observed peak outflow, 𝑇𝑅𝑜𝑢𝑡𝑒𝑑
𝑝𝑒𝑎𝑘
is the time of routed peak
outflow, and N is the number of data.
3.13. Multi criteria decision-making (MCDM)
The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) [57] is one of
the MCDM methods that select the best alternative based on choosing what is closest and most
far from the Positive Ideal Solution (PIS) and Negative Ideal Solution (NIS), respectively [58].
This method has been used in the present study because it is more straightforward than other
MCDM methods. [59]. TOPSIS chooses solutions from a set of alternatives. Specifically, the
selected alternative has the smallest distance to the positive ideal solution 𝐴+
(PIS) and the most
significant distance to the negative ideal solution 𝐴−
(NIS). TOPSIS is suggested for order
preservation for ranking the alternatives under natural experts' errors made during expert
estimation usability for many of the alternatives [60].
The TOPSIS method can be summarized as follows:
1. Create a decision matrix (𝐷), based on m alternative and n criteria as follows:
11 12 1
21 22 2
1 2
...
...
...
...
n
n
m m mn
d d d
d d d
D
d d d
 
 
 

 
 
 
(21)
In which, 𝑑𝑖𝑗 is element of decision matrix in ith
row and jth
column.
2. Normalize the decision matrix as follows:
1
ij
ij m
ij
i
d
r
d



(22)
Here, 𝑟𝑖𝑗 is a normalized decision matrix.
3. Calculate the weighted normalized decision matrix by defining the weight vector and
multiplying the normalized decision matrix:
 
* *
*
m n j ij m n
V W r
 (23)
In which, 𝑉
𝑚∗𝑛 is a weighted normalized decision matrix, and 𝑊
𝑗 is a weight of jth
criterion.
4. Estimate the distance of the ith
alternative from the 𝐴+
and 𝐴−
as follows:
 
 
1 2
1 2
, ,..., max | 1,2,...,
, ,..., min | 1,2,...,
n ij
n ij
A a a a v i m
A a a a v i m
   
   
  
  
  

 
  
  

(24)

5. Estimate the Euclidean distance for each solution from the positive ideal (𝑆+) and negative
ideal (𝑆−) as follows:
 
 
2
2
, 1,2,...,
, 1,2,...,
ij j
ij j
S v a i m
S v a i m
 
 

  


   

(25)
Estimate the relative closeness or score of each alternative as follows:
, 1,2,...,
i
i
i i
S
Score i m
S S


 
 

(26)
In which, 𝑆𝑐𝑜𝑟𝑒𝑖
+
is the score of the ith
alternative. The alternative with more scores is the better
solution. In this study, alternatives are MOAs and SSQ, SAD, EQp, ETp, MARE, VarexQ, and d
are evaluation criteria.
3.14. Presented technique for discharge routing
In the present study, nine MOAs, GA, PSO, BA, CS, FFA, SSO, WOA, HHO, and WOA_CS,
optimize one of the Muskingum's powerful methods to simulate the discharge in three reaches.
The optimization algorithms used in this study are well-known MOAs used in many previously
conducted studies. Each of these algorithms has unique operators and a specific structure for
optimizing problems. The GA is the evolutionary algorithm, PSO, FFA, CS, BA, SSO, WOA,
and HHO are swarm-based algorithms, and WOA_CS is the hybrid algorithm. GA works based
on the natural selection theory [61]. The PSO developed based on the swarm movement of
particles, birds or school fish [62]. The BA was inspired by the echolocation ability of bats [51].
CS was designed based on the lifestyle of a family of cuckoos [63]. The FFA works according to
the flashing behaviour of fireflies. The SSO was designed based on the hunting method of sharks
to use their smell sense [54]. The WOA was inspired by the hunting behaviour of humpback
whales [20]. HHO was developed based on the hunting approach of Haris hawk's [23]. The
WOA_CS worked based on the parallel hybridization of WOA and CS.
The simulation process is described as follows: Determine the parameter of MOAs.
1. The NL5 parameters include 𝐾, 𝑋, 𝐶1, 𝐶2, 𝛼1, 𝛼2 and 𝛽 are considered decision variables.
2. Generate the initial population for each optimization algorithm.
3. Calculate the initial storage 𝑆0 for each search agent and each algorithm, considering the
equality of inflow and outflow:
    
1 2
0 1 0 2 0
1
S K X C I X C O

 
 
  
  (27)
4. Calculate the change in storage for each search agent and each algorithm on time:

   
 
2
1
1
1
1
2 2
1 1
* * * *
* 1 * 1
t t
t t
S S
I X C I
t C X K C X



 
   
  
   
  
    
   
  
 
   
 
   
 
(28)
5. Calculate the storage for each search agent and algorithm at the time 𝑡:
1 *
t
t t
S
S S t
t


  

(29)
6. Calculate the outflow for each search agent and each algorithm at the time 𝑡 + 1:
   
 
2
1
1
1
1 1 1
2 2
1 1
* * * *
* 1 * 1
t
t t
S
O X C I
C X K C X



 
 
   
 
   
 
    
   
 
 
   
 
   
 
(30)
7. Calculate the objective function (SSQ) for each search agent and each algorithm.
8. Update the position of search agents based on the operators of each algorithm.
9. If the termination criterion is satisfied, go to stage 11. Else, go to stage 4.
10. Calculate the evaluation criterion.
11. Select the best optimization algorithm based on the evaluation criteria and the TOPSIS
method. Figure 3 shows the presented technique for discharge routing.
Fig. 3. Presented technique for discharge routing.

4. Results
4.1. Sensitivity analysis of the optimization algorithm
The MOAs have random parameters, which are necessary to determine. One of the best methods
for choosing these parameters is sensitivity analysis. This method estimates the objective
function changes by varying the optimization algorithm parameters. Then the value of the
intended parameter that leads to the minimum value of the objective function (SSQ) is
considered the best parameter value of the optimization algorithm. The mentioned value of SSQ
is calculated based on the average of SSQ from 30 random runs of each MOA. Table 2 to Table 7
listed the best values of MOA parameters obtained by sensitivity analysis. These tables report
different values for each parameter and their corresponding objective functions. Also, the best
parameters of each algorithm with the minimum objective function are bold. For example, the
best population size for all MOAs except for FFA and WOA_CS was 125, while the best
population size for FFA and WOA_CS was 25 and 75, respectively. According to the other table
results, the best Pc and Pm for GA in Ahvaz-Mollasani were equal to 0.1 and 0.6, respectively. In
these tables, a dashed line in the cell corresponding to each algorithm and parameter indicates
that the corresponding parameter is unrelated to that algorithm.
Table 2
Sensitivity analysis of GA, PSO, FFA, and CS for Ahvaz-Mollasani.
Parameters Value GA PSO FFA CS
PopSize
50 7670000.00 7480000.00 9620000.00 7490000.00
70 7630000.00 7470000.00 8550000.00 7470000.00
100 7640000.00 7460000.00 7710000.00 7470000.00
125 7610000.00 7450000.00 7390000.00 7460000.00
Pc
0.1 7610000.00 - - -
0.3 7620000.00 - - -
0.6 7610000.00 - - -
0.9 7610000.00 - - -
Pm
0.1 7620000.00 - - -
0.3 7600000.00 - - -
0.6 7580000.00 - - -
0.9 7670000.00 - - -
c1=c2
1.8 - 7460000.00 - -
1.9 - 7450000.00 - -
2.0 - 7450000.00 - -
2.1 - 7440000.00 - -
Wdamp
0.7 - 7460000.00 - -
0.8 - 7460000.00 - -
0.9 - 7470000.00 - -
1.0 - 7460000.00 - -
α
0.1 - - 7450000.00 -
0.15 - - 7440000.00 -
0.20 - - 7430000.00 -
0.30 - - 7400000.00 -
β
0.01 - - 7640000.00 -
0.05 - - 7610000.00 -
0.10 - - 7560000.00 -
0.20 - - 7460000.00 -
Pa
0.1 - - - 7450000.00
0.15 - - - 7460000.00
0.20 - - - 7460000.00
0.25 - - - 7460000.00

Table 3
Sensitivity analysis of GA, PSO, FFA, and CS for Harmaleh-Bamdej.
PopSize
50 756000.00 740000.00 2000000000.00 739000.00
70 755000.00 739000.00 971000.00 738000.00
100 753000.00 739000.00 751000.00 738000.00
125 749000.00 739000.00 737000.00 738000.00
Pc
0.1 753000.00 - - -
0.3 749000.00 - - -
0.6 748000.00 - - -
0.9 748000.00 - - -
Pm
0.1 750000.00 - - -
0.3 747000.00 - - -
0.6 744000.00 - - -
0.9 743000.00 - - -
c1=c2
1.8 - 739000.00 - -
1.9 - 739000.00 - -
2.0 - 739000.00 - -
2.1 - 739000.00 - -
Wdamp
0.7 - 739000.00 - -
0.8 - 739000.00 - -
0.9 - 739000.00 - -
1.0 - 739000.00 - -
α
0.1 - - 737000.00 -
0.15 - - 737000.00 -
0.20 - - 737000.00 -
0.30 - - 737000.00 -
β
0.01 - - 750000.00 -
0.05 - - 746000.00 -
0.10 - - 742000.00 -
0.20 - - 737000.00 -
Pa
0.1 - - - 738000.00
0.15 - - - 738000.00
0.20 - - - 738000.00
0.25 - - - 738000.00
Table 4
Sensitivity analysis of GA, PSO, FFA, and CS for Lighvan-Heravi.
PopSize
50 667000000.00 1330000000.00 10000000000.00 256.00
70 264.00 1000000000.00 10000000000.00 256.00
100 263.00 667000000.00 9000000000.00 256.00
125 266.00 255.00 7330000000.00 256.00
Pc
0.1 265.00 - - -
0.3 286.00 - - -
0.6 273.00 - - -
0.9 263.00 - - -
Pm
0.1 263.00 - - -
0.3 259.00 - - -
0.6 260.00 - - -
0.9 259.00 - - -
c1=c2
1.8 - 1000000000.00 - -
1.9 - 255.00 - -
2.0 - 333000000.00 - -
2.1 - 667000000.00 - -

Table 4 (continuous)
Sensitivity analysis of GA, PSO, FFA, and CS for Lighvan-Heravi.
Wdamp
0.7 - 1000000000.00 - -
0.8 - 333000000.00 - -
0.9 - 257.00 - -
1.0 - 255.00 - -
α
0.1 - - 9000000000.00 -
0.15 - - 7330000000.00 -
0.20 - - 7000000000.00 -
0.30 - - 6670000000.00 -
β
0.01 - - 6670000000.00 -
0.05 - - 5000000000.00 -
0.10 - - 7330000000.00 -
0.20 - - 7330000000.00 -
Pa
0.1 - - - 254.00
0.15 - - - 256.00
0.20 - - - 256.00
0.25 - - - 256.00
Table 5
Sensitivity analysis of BA, SSO, WOA, HHO, and WOA_CS for Ahvaz-Mollasani.
Parameters Value BA SSO WOA HHO WOA_CS
PopSize
50 9870000.00 7690000.00 7620000.00 7520000.00 7520000.00
70 8180000.00 7660000.00 7530000.00 7530000.00 7460000.00
100 8220000.00 7660000.00 7500000.00 7500000.00 7430000.00
125 7940000.00 7650000.00 7500000.00 7490000.00 7420000.00
A
0.1 8240000.00 - - -
0.3 8150000.00 - - -
0.4 8220000.00 - - -
0.6 8180000.00 - - -
Qmim
0.0 8060000.00 - - - -
1.0 8120000.00 - - - -
3.0 8400000.00 - - - -
4.0 8370000.00 - - - -
Qmax
1 8200000.00 - - - -
2 8140000.00 - - - -
4 8330000.00 - - - -
5 8280000.00 - - - -
θ
0.7 - 7660000.00 - - -
0.8 - 7650000.00 - - -
0.9 - 7630000.00 - - -
1.0 - 7650000.00 - - -
Pa
0.1 - - - - 7410000.00
0.15 - - - - 7400000.00
0.20 - - - - 7440000.00
0.30 - - - - 7420000.00
Ri
0.01 - - - - 7410000.00
0.05 - - - - 7350000.00
0.10 - - - - 7390000.00
0.20 - - - - 7410000.00
N_Swap
0.1 - - - - 7440000.00
0.15 - - - - 7400000.00
0.20 - - - - 7410000.00
0.25 - - - - 7370000.00

Table 6
Sensitivity analysis of BA, SSO, WOA, HHO, and WOA_CS for Harmaleh-Bamdej.
PopSize
50 1040000.00 763000.00 749000.00 748000.00 739000.00
70 862000.00 761000.00 743000.00 746000.00 738000.00
100 817000.00 761000.00 743000.00 741000.00 737000.00
125 800000.00 754000.00 742000.00 740000.00 737000.00
A
0.1 825000.00 - - -
0.3 803000.00 - - -
0.4 884000.00 - - -
0.6 812000.00 - - -
Qmim
0.0 807000.00 - - - -
1.0 820000.00 - - - -
3.0 838000.00 - - - -
4.0 817000.00 - - - -
Qmax
1 826000.00 - - - -
2 832000.00 - - - -
4 833000.00 - - - -
5 817000.00 - - - -
θ
0.7 - 754000.00 - - -
0.8 - 756000.00 - - -
0.9 - 755000.00 - - -
1.0 - 758000.00 - - -
Pa
0.1 - - - - 737000.00
0.15 - - - - 737000.00
0.20 - - - - 737000.00
0.30 - - - - 737000.00
Ri
0.01 - - - - 739000.00
0.05 - - - - 737000.00
0.10 - - - - 737000.00
0.20 - - - - 737000.00
N_Swap
0.1 - - - - 738000.00
0.15 - - - - 737000.00
0.20 - - - - 737000.00
0.25 - - - - 737000.00
Table 7
Sensitivity analysis of BA, SSO, WOA, HHO, and WOA_CS for Lighvan-Heravi.
PopSize
50 1670000000.00 294.00 333000000.00 333000000.00 267.00
70 667000000.00 288.00 333000000.00 259.00 256.00
100 333000000.00 284.00 270.00 258.00 255.00
125 318.00 282.00 258.00 257.00 255.00
A
0.1 317.00 - - - -
0.3 451.00 - - - -
0.4 667000000.00 - - - -
0.6 333000000.00 - - - -
Qmim
0.0 667000000.00 - - - -
1.0 337.00 - - - -
3.0 417.00 - - - -
4.0 336.00 - - - -
Qmax
1 532.00 - - - -
2 319.00 - - - -
4 401.00 - - - -
5 339.00 - - - -

Table 7 (continuous)
Sensitivity analysis of BA, SSO, WOA, HHO, and WOA_CS for Lighvan-Heravi.
θ
0.7 - 282.00 - - -
0.8 - 283.00 - - -
0.9 - 283.00 - - -
1.0 - 279.00 - - -
Pa
0.1 - - - - 255.00
0.15 - - - - 255.00
0.20 - - - - 254.00
0.30 - - - - 255.00
Ri
0.01 - - - - 255.00
0.05 - - - - 255.00
0.10 - - - - 255.00
0.20 - - - - 255.00
N_Swap
0.1 - - - - 255.00
0.15 - - - - 255.00
0.20 - - - - 255.00
0.25 - - - - 255.00
4.2. Results of 30 random runs of MOAs for discharge routing
The results of 30 random runs of MOAs and statistical criteria for Ahvaz-Mollasani, Harmaleh-
Bamdej, and Lighvan-Heravi are presented in Table 8, Table 9, and Table 10. According to the
results of mentioned Tables, the minimum SSQ for Ahvaz-Mollasani and Lighavan-Heravi
reaches were for WOA_CS, while in Bamdej-Harmaleh reach, it is for FFA and WOA_CS with a
slight difference. The minimum average of SSQ in all reaches was also for WOA_CS.
Table 8
Statistical criteria run of MOAs for discharge routing in Ahvaz-Mollasani over 30 random runs.
Algorithm Mean Max Min Std CV Run time (sec)
GA 7582243.73 7641530.3 7401508.4 48823.76 0.01 36.64
PSO 7462058.7 7496591.27 7305166.65 31597.77 0 85.23
FFA 7461242.7 7685104.83 7305122.75 94766.55 0.01 13.89
CS 7463442.87 7509080.19 7315186.62 46570.52 0.01 26.99
BA 8180967.69 14049832.1 7476748.45 1211026.73 0.15 17.55
SSO 7656993.44 7799479.44 7449225.41 61484.03 0.01 115.1
WOA 7497059.49 7606087.6 7346273.79 46892.73 0.01 44.84
HHO 7490400.36 7624217.55 7328248.59 43290.6 0.01 46.09
WOA_CS 7352808.49 7450544.36 7305126.2 61849.65 0.01 20.72
The lower values of the difference between maximum and minimum, standard deviation (Std),
and coefficient of variation (CV) show the quality of the results of MOAs. Whatever the
mentioned statistical criteria for one MOA are lower, the quality of its results is higher. CV is one
of the best criteria for analyzing the quality of results. As seen, for the reach of Ahvaz-Mollasani,
the minimum 30 random runs of CV was related to the PSO algorithm. While the CV of other
MOAs was equal to 0.01. In Harmaleh-Bamdej's reach, the CV of PSO, FFA, CS, and WOA_CS
was equal to 0.00, and those for GA, SSO, WOA, and HHO were equal to 0.01. The maximum
values of CV (CV=0.18) in this reach were related to BA.

Table 9
Statistical criteria of MOAs for discharge routing in Harmaleh-Bamdej over 30 random runs.
GA 749012.82 775758.2 736123.39 8046.12 0.01 10.67
PSO 738874.88 739514.13 735886.41 750.01 0 22.77
FFA 736875.46 739864.29 735752.67 1258.45 0 3.93
CS 738508.48 740280.52 735918.14 1008.92 0 8.96
BA 862266.55 1376172.59 741222.83 152136.45 0.18 8.29
SSO 754167.51 778137.34 738655.57 9425.79 0.01 31.61
WOA 741774.79 781567.02 735934.19 7736.52 0.01 16.35
HHO 741019.44 761434.67 736943.89 4250.31 0.01 13.13
WOA_CS 736588.9 738682.96 735761.19 1053.87 0 6.62
The results of the Lighvan-Heravi reach revealed which PSO and HHO have a minimum CV
(CV=0.00), and BA has a maximum CV (CV=0.58). Moreover, in the mentioned reach, the CVs
for other MOAs ranged from 0.02 to 0.09. The difference between maximum and minimum and
standard deviation results confirms these results. Therefore, MOAs in Ahvaz-Mollasani
optimized the Muskingum model with the most qualified, and those in Lighvan-Heravi
optimized the Muskingum model with less quality. Furthermore, maximum time computing was
related to the reach of Ahvaz-Mollasani, and minimum time computing was associated with the
reach of Lighvan-Heravi.
Table 10
Statistical criteria of MOAs for discharge routing in Lighvan-Heravi over 30 random runs.
GA 260.46 291.48 255.37 6.44 0.02 8.69
PSO 255.37 255.51 255.28 0.1 0 23.81
FFA 258.56 310.73 255.28 10.84 0.04 0.86
CS 254.08 256.11 211.48 7.91 0.03 6.26
BA 400.53 1351.95 257.38 230.85 0.58 6.11
SSO 282.02 306.19 255.71 12.54 0.04 29.56
WOA 269.82 314.09 255.3 24.43 0.09 10.56
HHO 255.87 257.64 255.34 0.52 0 8.68
WOA_CS 253.54 255.74 200.77 9.8 0.04 4.62
Figure 4 shows the violin plot for the objective function results over 30 runs. In this chart, the
distribution, minimum, maximum and average of data are displayed. Moreover, the longer the
violin diagram, the greater the uncertainty of the results. According to this diagram, PSO, CS,
WOA, HHO, and WOA_CS algorithms were less uncertain than the other algorithms in the
Ahvaz-Mollasani reach. Additionally, in the other two cases, the PSO, FFA, CS, WOA, HHO,
and WOA_CS algorithms had less uncertainty than the other algorithms. Regarding the average
objective function, WOA_CS achieved better results than other algorithms. The results of this
plot are consistent with the results of Tables 8 to 10.

a)
b)
c)
Fig. 4. violin plot of MOAs for discharge routing over 30 random runs and in a) Ahvaz-Mollasani, b)
Harmaleh-Bamdej, and c) Lighvan-Heravi.

4.3. Convergence curve of MOAs in discharge routing
Figure 5 shows the convergence curve of MOAs for different reaches. This Figure shows the
minimum, mean, and maximum of 30 random runs. The lower difference between the minimum
and maximum results shows more quality of results. As seen, PSO, CS, HHO, and WOA_CS
have results with more quality than other MOAs. In addition, FFA converged to the optimal
solution in 150 iterations, while other algorithms converged to the optimal solution in 300
iterations.

Fig. 5. Convergence curve of MOAs in different reaches.

4.4. Accuracy of MOAs in discharge routing
In this section, the results of MOAs are compared. Table 11 tabulates the calculated evaluation
criteria of MOAs for the three investigated river reaches in the training period. As seen in the
training period, all investigated algorithms reasonably simulate river discharge. The results of
discharge routing in the testing period are presented in Table 12. In this table, the best outcome
for each criterion is bold. Although the results are close to each other, each of the competing
algorithms was more accurate than the others based on some evaluation criteria. In contrast,
other algorithms were more accurate considering other criteria. For example, in the Lighvan-
Heravi River reach, BA had better results on the Varex and d, while SSO had better results based
on the SSQ, EQp, ETp, and MARE criteria. Therefore, it cannot say which algorithm is superior
to other algorithms. Hence, the TOPSIS method chooses the best algorithm based on all the
criteria.
Table 11
Accuracy of optimization algorithm in the training period.
Algorithms Mollasani-Ahvaz
SSQ SAD EQP ETP MARE Varex d
GA 7401508.40 53866.43 0.21 2.00 0.09 47.20 0.81
PSO 7305166.65 53711.04 0.27 2.00 0.09 47.89 0.79
FFA 7305122.75 53699.64 0.27 2.00 0.09 47.89 0.79
CS 7315186.62 53772.16 0.29 2.00 0.09 47.82 0.79
BA 7591075.36 56965.11 0.19 2.00 0.10 45.85 0.79
SSO 7449225.41 55400.05 0.25 2.00 0.09 46.86 0.80
WOA 7346273.79 53594.59 0.25 2.00 0.09 47.59 0.80
HHO 7328248.59 53702.94 0.24 2.00 0.09 47.72 0.80
WOA_CS 7305126.20 53703.91 0.27 2.00 0.09 47.89 0.79
Algorithms Harmaleh-Bamdej
GA 736123.39 21607.70 0.12 22.00 0.19 75.53 0.93
PSO 735886.41 21600.12 0.10 22.00 0.19 75.54 0.93
FFA 735752.67 21589.72 0.09 22.00 0.19 75.54 0.93
CS 735918.14 21629.20 0.09 22.00 0.19 75.54 0.93
BA 747313.11 21743.55 0.19 22.00 0.20 75.16 0.93
SSO 738655.57 21716.26 0.11 22.00 0.19 75.44 0.93
WOA 735934.19 21620.14 0.10 22.00 0.19 75.53 0.93
HHO 736943.89 21610.83 0.14 22.00 0.19 75.50 0.93
WOA_CS 735761.19 21597.55 0.12 22.00 0.19 75.53 0.93
Algorithms Lighvan-Heravi
GA 255.37 517.00 0.07 1.00 3.61 22.63 0.80
PSO 255.28 516.55 0.05 1.00 3.60 22.66 0.80
FFA 255.28 516.55 0.05 1.00 3.60 22.66 0.80
CS 255.67 515.32 0.13 1.00 3.60 22.54 0.80
BA 264.84 535.57 0.08 1.00 3.84 19.76 0.78
SSO 255.71 514.41 0.08 1.00 3.61 22.53 0.80
WOA 255.30 516.98 0.05 1.00 3.60 22.65 0.80
HHO 255.34 516.96 0.02 1.00 3.60 22.64 0.80
WOA_CS 256.42 520.02 0.06 1.00 3.62 22.31 0.79

Table 12
Accuracy of optimization algorithm in the testing period.
Algorithms Mollasani-Ahvaz
GA 5750457.90 28926.76 0.14 1.00 0.10 60.51 0.87
PSO 5899562.65 29324.37 0.23 1.00 0.09 59.49 0.85
FFA 5900942.00 29321.90 0.23 1.00 0.09 59.48 0.85
CS 5977784.03 29463.27 0.25 1.00 0.09 58.95 0.84
BA 5835758.33 30315.48 0.11 1.00 0.10 59.92 0.87
SSO 5900762.44 30381.30 0.19 1.00 0.10 59.48 0.86
WOA 5813904.44 28989.04 0.19 1.00 0.09 60.07 0.86
HHO 5786612.47 28888.49 0.19 1.00 0.09 60.26 0.86
WOA_CS 5902464.18 29327.04 0.23 1.00 0.09 59.47 0.85
Algorithms Harmaleh-Bamdej
GA 3062702.70 19129.99 0.01 2.00 0.21 74.08 0.91
PSO 3112585.15 19564.90 0.05 2.00 0.21 73.65 0.91
FFA 3116244.06 19547.68 0.05 2.00 0.21 73.62 0.91
CS 3112828.61 19463.93 0.05 2.00 0.21 73.65 0.91
BA 3040121.99 18798.16 0.09 2.00 0.19 74.27 0.92
SSO 3081733.89 19015.86 0.03 2.00 0.19 73.92 0.91
WOA 3088148.35 19190.95 0.03 2.00 0.20 73.86 0.91
HHO 3038832.86 18846.19 0.02 2.00 0.20 74.28 0.92
WOA_CS 3124220.37 19687.93 0.05 2.00 0.21 73.56 0.91
Algorithms Lighvan-Heravi
GA 628.39 337.41 0.54 1.00 5.40 39.44 0.76
PSO 628.67 338.46 0.56 1.00 5.43 39.42 0.75
FFA 628.68 338.46 0.56 1.00 5.43 39.42 0.75
CS 614.63 332.05 0.48 1.00 5.29 40.77 0.77
BA 648.77 349.79 0.53 1.00 5.76 37.48 0.74
SSO 622.01 337.71 0.52 1.00 5.26 40.06 0.76
WOA 630.63 337.89 0.55 1.00 5.45 39.23 0.75
HHO 633.40 340.35 0.58 1.00 5.47 38.96 0.75
WOA_CS 614.57 336.31 0.52 1.00 5.52 40.77 0.77
In the following, Taylor's diagram (Figure 6) is employed to evaluate MOAs based on three
criteria: standard deviation, root mean square of deviations and correlation coefficient. As can be
seen, all the MOAs have a reasonable performance in terms of accuracy. Also, the results of the
MOAs were similar to each other. By comparing the Taylor diagrams of the three reaches, the
results show that the modeling was done more accurately in the Harmaleh-Bamdej reach and less
accurately in the Lighvan-Heravi reach.
Figure 7 shows the relative results of the Nemenyi test for all MOAs based on the absolute
difference between the observed and the routed outflow. Methods with higher rankings are to the
right and those with lower rankings are to the left. MOAs within a horizontal line of critical
distance or shorter are statistically identical. It was noted that on the first and second reach,
HHO, and third reach, CS outperforms other MOAs.

a)
b)
c)
Fig. 6. Taylor diagram of MOAs in a) Ahvaz-Mollasani, b) Harmaleh-Bamdej, and c) Lighvan-Heravi.

a)
b)
c)
Fig. 7. Graphical representation of the Nemenyi test of MOAs in a) Ahvaz-Mollasani, b) Harmaleh-
Bamdej, and c) Lighvan-Heravi.
4.5. MCDA results in discharge routing
Because choosing the best MOAS according to different criteria such as average results and best
results leads to choosing different MOAs. The TOPSIS method is used for the selection of the
best algorithm. In this method, various criteria such as error evaluation metrics and Nemenyi test
ranks for the best results in 30 runs, the average value of the objective function in 30 runs, the
value of the coefficient of variation in 30 runs and execution time have been employed to select

the best algorithm. Table 13 presents the scores and rankings of MOAs in the discharge routing
obtained by TOPSIS. According to the author's experience, the weights of these criteria were
equal to 0.0313 for error evaluation metrics and Nemenyi test, and 0.25 for the average value of
the objective function in 30 runs, the value of the coefficient of variation in 30 runs, and
execution time. WOA_CS (rank=1), FFA (rank=1), and WOA_CS (rank=1) have more scores in
Mollasani-Ahvaz, Harmaleh-Bamdej, and Lighvan-Heravi River reaches, respectively.
Table 13
The optimization algorithm scores and ranks.
Algorithms Mollasani-Ahvaz Harmaleh-Bamdej Lighvan-Heravi
GA 0.73 (Rank=6) 0.89 (Rank=6) 0.90 (Rank=4)
PSO 0.85 (Rank=3) 0.91 (Rank=5) 0.85 (Rank=5)
FFA 0.83 (Rank=4) 0.97 (Rank=1) 0.92 (Rank=3)
CS 0.62 (Rank=7) 0.83 (Rank=7) 0.68 (Rank=7)
BA 0.02 (Rank=9) 0.08 (Rank=9) 0.13 (Rank=9)
SSO 0.82 (Rank=5) 0.92 (Rank=4) 0.84 (Rank=6)
WOA 0.85 (Rank=2) 0.93 (Rank=2) 0.96 (Rank=2)
HHO 0.02 (Rank=8) 0.08 (Rank=8) 0.13 (Rank=8)
WOA_CS 0.98 (Rank=1) 0.92 (Rank=3) 0.97 (Rank=1)
The average results of Table 13 (Figure 8) demonstrates that WOA_CS (score = 0.960), WOA
(score = 0.913) and FFA (score = 0.907) were ranked first, second and third.
Fig.8. Average TOPSIS ranking MOAs in discharge routing.
4.6. Optimal Muskingum parameters
The optimal solutions (𝐾, 𝑋, 𝐶1, 𝐶2, 𝛼1, 𝛼2 and 𝛽) by different MOAs are listed in Table 14.
According to this table, the 𝐾, 𝑋, 𝐶1, 𝐶2, 𝛼1, 𝛼2, and 𝛽 values are varied from 0.74 to 100.00,
−0.80 to −0.29, 0.27 to 1, 0.00 to 0.86, 0.51 to 9.10, 0.40 to 10.00, and 0.12 to 10, respectively.
Also, the range of optimal solutions was varied versus MOAs and river reaches. This issue can
be found in different assumptions of algorithms and different statistical characteristics of inflow
and outflow time series in river reach.

Table 14
The optimal Muskingum parameters.
Mollasani-Ahvaz
K X C1 C2 α1 α2 β
GA 42.155 -0.781 0.903 0.002 7.884 8.777 10.000
PSO 62.268 -0.547 0.884 0.000 8.335 10.000 10.000
BA 49.655 -0.799 1.000 0.000 8.212 9.858 10.000
CS 37.044 -0.800 0.838 0.000 6.696 8.207 10.000
FFA 60.974 -0.615 0.453 0.000 7.445 8.661 10.000
SSO 12.050 -0.548 0.905 0.001 8.853 10.000 8.756
WOA 42.127 -0.800 0.323 0.000 6.316 7.364 10.000
HHO 60.974 -0.615 0.453 0.000 7.445 8.661 10.000
WOA_CS 0.749 -0.460 1.000 0.000 8.223 9.874 10.000
Harmaleh-Bamdej
GA 42.656 -0.543 0.964 0.075 5.656 6.063 0.223
PSO 50.119 -0.800 0.938 0.018 8.021 8.791 0.150
BA 28.068 -0.297 0.543 0.003 9.099 10.000 0.151
CS 100.000 -0.531 0.678 0.006 9.038 9.927 0.123
FFA 6.516 -0.799 0.763 0.132 4.943 5.225 0.342
SSO 83.803 -0.288 0.838 0.012 7.705 8.380 8.380
WOA 8.884 -0.678 0.269 0.013 6.183 6.717 0.272
HHO 6.516 -0.799 0.763 0.132 4.943 5.225 0.342
WOA_CS 99.410 -0.457 0.997 0.008 9.092 9.995 0.118
Lighvan-Heravi
GA 66.347 -0.571 1.000 0.658 0.950 1.079 1.481
PSO 69.355 -0.800 0.805 0.644 0.814 0.922 1.899
BA 47.327 -0.800 0.984 0.787 0.815 0.922 1.897
CS 49.186 -0.800 0.855 0.724 0.932 0.982 1.797
FFA 65.475 -0.770 0.913 0.711 0.743 0.850 2.102
SSO 40.235 -0.762 0.952 0.803 0.903 0.969 2.195
WOA 65.850 -0.649 0.953 0.667 0.874 1.002 1.584
HHO 65.475 -0.770 0.913 0.711 0.743 0.850 2.102
WOA_CS 77.557 -0.493 0.988 0.864 0.510 0.403 6.368
4.7. Outcomes of discharge routing by best algorithms
The observed inflow, observed outflow, and simulated outflow in Mollasani-Ahvaz, Harmaleh-
Bamdej, and Lighvan-Heravi are demonstrated in Figures 9 to 11. The simulated outflow had
good agreement with the observed outflow in all reaches. However, Mollasani-Ahvaz has the
best accuracy, and Lighvan-Heravi has the weakest. As clear in Figure 10 and Figure 11, the peak
of the outflow in Mollasani-Ahvaz and Lighvan-Heravi are simulated with more precision than
Lighvan-Heravi, despite the excellent agreement simulated peak of outflow in all reaches.
The scatter plots of discharge routing by the best algorithms are shown in Figure 12. In this
figure, the more the distribution of data around the 45-degree line, the better the performance of
the algorithm. As seen, the discharge routing in Mollasani-Ahvaz and Harmaleh-Bamdej has
better performance than Lighvan-Heravi. The violin plots in Figure 13 demonstrate the
agreement between observed and routed discharge in Mollasani-Ahvaz and Harmaleh-Bamdej.
However, the distribution of routed discharge in Lighvan-Heravi shows underestimated results.
The results obtained are in agreement with the results shown in Figure 9 to Figure 11.

Fig. 9. The discharge routing by the best algorithm in Mollasani-Ahvaz.
Fig 10. The discharge routing by the best algorithm in Harmaleh-Bamdej.
Fig. 11. The discharge routing by the best algorithm in Lighvan-Heravi.
Train Test
0
200
400
600
800
1000
1200
1400
0 250 500 750 1000 1250 1500 1750 2000 2250
Flow
(cms)
Month
Obs inflow Mollasani Obs outflow Ahvaz
HHO Outflow Ahvaz WOA_CS Outflow Ahvaz
Train Test
0
200
400
600
800
1000
1200
1400
0 250 500 750 1000 1250 1500 1750 2000 2250
Flow
(cms)
Month
Obs inflow Harmaleh Obs outflow Bamdej
HHO outflow Bamdej WOA_CS Outflow Bamdej
Train Test
0
5
10
15
20
0 250 500 750 1000 1250 1500 1750 2000 2250
Flow
(cms)
Month
Obs inflow Lighvan Obs Outflow Heravi
CS Outflow Heravi WOA_CS Outflow Heravi

a)
b)
c)
Fig. 12. The scatter plot of best algorithm in a) Ahvaz-Mollasani, b) Harmaleh-Bamdej, c) Lighvan-
Heravi.
y = 0.5846x + 128.07
R² = 0.6926
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
1400.00
0 200 400 600 800 1000 1200 1400 1600
WOA_CS
Outflow
Ahvaz
Obs Outflow Ahvaz
y = 0.7073x + 26.816
R² = 0.7783
0.00
200.00
400.00
600.00
800.00
1000.00
1200.00
0 200 400 600 800 1000 1200
WOA_CS
Outflow
Bamdej
Obs Outflow Bamdej
y = 0.5202x + 0.4895
R² = 0.5061
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
0 5 10 15 20
WOA_CS
Outflow
Heravi
Obs Outflow Heravi

a)
b)
c)
Fig. 14. The violin plot of best algorithms in a) Ahvaz-Mollasani, b) Harmaleh-Bamdej, c) Lighvan-
Heravi.

5. Discussion
In this study, the results of the algorithms were different in investigated stations, and in each
station, one algorithm had more accurate results than other methods. This is because the river
was analyzed with different fluvial geomorphology and climate conditions, and these caused
changes in the statistical characteristics of the inflow and outflow hydrographs. These differences
also cause discharge routing in each river to become a specific optimization problem. According
to the "No free lunch theorems for optimization" [64], each optimization algorithm performs well
in a specific range of problems. For this reason, the accuracy of optimization algorithms is
different in determining the optimal parameters of the Muskingum method.
Furthermore, more correlation between the inflow and outflow time series can explain the better
accuracy of discharge routing in Mollasani-Ahvaz and Bamdej-Harmaleh. The method presented
in the current research can be used in non-structural flood warning systems. This method can be
very efficient in flood warnings when a flood occurs upstream. For this purpose, the amount of
flood discharge should be measured at the upstream hydrometric station and transferred to a
computer using an online system. Then, the downstream flow rate should be calculated using the
current research method. Since the algorithms are pre-calibrated, this is done quickly. Finally, a
flood warning is issued downstream if the calculated hydrograph peak value exceeds the allowed
peak flow.
In previous studies, other methods such as SWAT [65], ANN [66], and hybrid machine learning
algorithm [67] simulated river discharge with determination coefficients between 0.58-0.90 and
0.52-0.61 and 0.74-0.85, respectively. The results of the present study are close to previous
studies. The method developed in this study can compete with physical and other AI algorithms.
Also, the number of input features in the developed method is less than other methods such as
SWAT, ANN and hybrid machine learning algorithms.
6. Conclusions
In this study, a new technique is introduced for discharge routing. The presented approach was
investigated in the Mollasani-Ahvaz, Harmaleh-Bamdej River, which reaches the Karun basin,
and the Lighvan-Heravi River in the Urmia basin. This study employed a range of MOAs,
including GA, PSO, FFA, CS, BA, SSO, WOA, and the hybrid of WOA and CS (WOA_CS), to
optimize the parameters of a highly effective Muskingum discharge routing method over an
extensive dataset period. Moreover, the TOPSIS as MCDA was used for choosing the best
optimization algorithm for each river reach. Based on all results of the presented technique, the
following points can be concluded:
1. The accuracy of discharge routing by different MOAs was almost the same. However, each
of the MOAs had good results in specific assessment criteria. Thus, the TOPSIS method
was employed.
2. Results of discharge routing based on the TOPSIS showed that WOA_CS were better with
considering all reach, respectively.

3. The uncertainty of WOA_CS was comparable with other MOAs. It was proved by using a
coefficient of variation of 30 random runs and violin plots.
4. The execution time of WOA_CS was less than most investigated MOAs.
5. The discharge routing in all the stations was accurate, but the discharge routing in
Mollasani-Ahavas had more accuracy than the other two investigated river reaches. In
addition, the outflow peaks in the reaches of Ahvaz-Mollasani and Harmaleh-Bamdej were
simulated with better accuracy than Lighavan-Heravi.
The presented method in this research is suitable for small and large amounts of data. But if a
large dataset is used, powerful optimization algorithms are needed to determine the parameters of
the Muskingum model. It should be noted that the parameter values of the optimization
algorithms need sensitivity analysis before implementation. Discharge data should be divided
into two periods of calibration and validation.
According to mentioned results, the introduced approach has good potential for discharge routing
in other river reaches. Furthermore, this model can be improved by using other multi-criteria
decision-making methods, such as integrated qualitative group decision-making.
Acknowledgments
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Authors contribution statement
MVA, SF, IA: Conceptualization; MVA, SF: Data curation; MVA, SF, IA: Formal analysis; SF,
IA: Investigation; MVA, SF, IA: Methodology; SF, IA: Project administration; MVA, FA, IA:
Resources; MVA, SF, IA: Software; SF: Supervision; MVA, SF, IA: Validation; MVA, SF, IA:
Visualization; MVA, SF, IA: Roles/Writing – original draft; MVA, SF, IA: Writing – review &
editing.
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