Developing Four Metaheuristic Algorithms for Multiple-Objective Management of Groundwater

Journal of Soft Computing in Civil Engineering 2-4 (2018) 01-22
How to cite this article: El-Ghandour HA, Elbeltagi E. Developing four metaheuristic algorithms for multiple-objective
management of groundwater. J Soft Comput Civ Eng 2018;2(4):01–22. https://doi.org/10.22115/scce.2018.128344.1057.
2588-2872/ © 2018 The Authors. Published by Pouyan Press.
This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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Journal of Soft Computing in Civil Engineering
Journal homepage: www.jsoftcivil.com
Developing Four Metaheuristic Algorithms for Multiple-Objective
Management of Groundwater
H.A. El-Ghandour1
*, E. Elbeltagi2
1. Associate Professor, Irrigation & Hydraulics Department, Faculty of Engineering Mansoura University, Mansoura
35516, Egypt
2. Professor, Structural Engineering Department, Faculty of Engineering Mansoura University, Mansoura 35516,
Egypt
Corresponding author: hamdy_a@mans.edu.eg
https://doi.org/10.22115/SCCE.2018.128344.1057
ARTICLE INFO ABSTRACT
Article history:
Received: 23 April 2018
Revised: 19 June 2018
Accepted: 20 June 2018
Groundwater is one of the important sources of freshwater
and accordingly, there is a need for optimizing its usage. In
this paper, four multi-objective metaheuristic algorithms
with new evolution strategy are introduced and compared for
the optimal management of groundwater namely: Multi-
objective genetic algorithms (MOGA), multi-objective
memetic algorithms (MOMA), multi-objective particle
swarm optimization (MOPSO), and multi-objective shuffled
frog leaping algorithm (MOSFLA). The suggested evolution
process is based on determining a unique solution of the
Pareto solutions called the Pareto-compromise (PC) solution.
The advantages of the current development stem from: 1)
The new multiple objectives evolution strategy is inspired
from the single objective optimization, where fitness
calculations depend on tracking the PC solution only through
the search history; 2) a comparison among the performance
of the four algorithms is introduced. The development of
each algorithm is briefly presented. A comparison study is
carried out among the formulation and the results of the four
algorithms. The developed four algorithms are tested on two
multiple-objective optimization benchmark problems. The
four algorithms are then used to optimize two-objective
groundwater management problem. The results prove the
ability of the developed algorithms to accurately find the
Pareto-optimal solutions and thus the potential application on
real-life groundwater management problems.
Keywords:
Genetic algorithms;
Memetic algorithms;
Particle swarm;
Shuffled frog leaping;
Compromise solution;
Multiple objectives
optimization.

2 H.A. El-Ghandour, E. Elbeltagi/ Journal of Soft Computing in Civil Engineering 2-4 (2018) 01-22
1. Introduction
Groundwater is considered as one of the freshwater sources needed to serve human activities,
such as drinking, industrial, domestic and irrigation. Due to irregular pumping of groundwater,
serious environmental hazards have been occurred including groundwater levels decline and
interference among wells.
To pump the maximum amount of groundwater without aquifer depletion and achieving
minimum operation cost, powerful simulation-optimization models have to be applied to obtain
the best strategy. These models are adopted to deal with both the design and operation problems
corresponding to remediation, water supply and controlling of groundwater flow [1].
Optimization models can be classified according to their objectives into single objective and
multiple objectives. In single objective optimization, a unique optimal solution is produced
related to the maximization or minimization of a single objective [1–19]. In multiple objectives
optimization, a group of non-dominated solutions, called Pareto-front (PF), are usually produced
[20–25]. Optimization models are classified into deterministic and metaheuristic. Deterministic
methods include linear, non-linear and dynamic programming [6–9,26–33]. While, metaheuristic
algorithms are stochastic optimization techniques that mimic nature perception, learning and
reasoning such as: Genetic Algorithms (GA), Memetic Algorithms (MA), Particle Swarm (PSO)
and Shuffled Frog Leaping (SFLA) [1–5,10–25].
Deterministic optimization methods tend to become overly complicated and require massive
computations when dealing with complex optimization problems with numerous constraints and
decision variables, such as real-life water related optimization problems. While, metaheuristic
optimization algorithms are usually used due to their direct getting solutions without the need for
gradients and initial solutions. Simulation models are coupled with optimization techniques to
evaluate proposed objective function(s). Simulation models apply either numerical or analytical
approaches to simulate groundwater flow and calculate the hydraulic heads of the studied
aquifer. Numerical approaches include finite element (FEM), finite difference and boundary
element methods while, analytical approaches use analytical element method and Fourier series
[16,17,19].
The development made in the current paper is different from previous works in many aspects.
The use of a new evolution strategy for generating the Pareto optimal solutions in four different
multi-objectives algorithms namely: MOGA, MOMA, MOPSO and MOSFLA. Using such
evolution strategy resembles solving single optimization problems. Performing a comparison
study among the four multi-objectives optimization algorithms for optimizing ground water
management problems. While, all previous comparisons have been performed among single
objective optimization algorithms [34,35].
2. New evolution strategy
Multiple objectives optimization problems are characterized by the existence of many optimal
solutions (non-dominated) called Pareto-Front (PF). Many different algorithms are used to solve

H.A. El-Ghandour, E. Elbeltagi/ Journal of Soft Computing in Civil Engineering 2-4 (2018) 01-22 3
multi-objectives optimization problems. Grierson [36] proposed a methodology to determine a
single Pareto-Compromise (PC) solution located in the PF which fairly satisfies all objectives.
This research adopts such PC solution for the fitness calculations and then driving the evolution
process in MOGA, MOMA, MOPSO, and MOSFLA algorithms.
The proposed new evolutionary strategy which is based on finding the PC solution is adopted
due to their proven effectiveness in terms of the processing time and number of Pareto solutions
obtained in the PF, as stated by Elbeltagi et al. [37]. Also, the proposed multi-objective
evolutionary strategy is reminiscent of single-objective optimization, where its fitness
assignment is based on tracking a single evolving solution over the search history. In addition,
knowing the PC solution helps decision makers to choose a single solution that uniquely
represents a mutually agreeable trade-off between all competing objectives for the problem.
The suggested methodology for developing each algorithm to handle multiple objective
problems depends mainly on the principle of non-dominant solutions located in more densely
populated regions in the objective space having lower probabilities to be selected. Consequently,
better distributions of optimal solutions can occur in the PF. The following steps illustrate the
suggested methodology:
1. All solutions in an evolutionary cycle are evaluated according to the values of their
objectives to determine a set of non-dominated solutions called PF. Each non-dominated
solution belongs to the PF can beat all other solutions in at least one objective.
2. For each obtained PF, the PC solution is determined mathematically as given by Grierson
[36].
3. The distance ( k
PC
d , ) between the PC and each Pareto optimal solution (k) is calculated
based on Eq. 1:
 




n
i
ki
PCi
k
PC f
f
d
1
2
, k = 1, 2, 3……N (1)
in which, fPCi: the objective function i value corresponding to the PC, fki: the objective function i
value corresponding to solution (k) located in the PF, N: number of Pareto optimal solutions
located in the PF, and n: number of objective functions.
4. The fitness of each Pareto-optimal solution (k), Eq. 2, equals its distance ( k
PC
d , ) from the
PC solution. This step ensures the disperse of the solutions along the solution space and
prevents solutions crowdies.
k
PC
k d
Fitness ,
 k = 1, 2, 3……N (2)
5. The relative fitness of each solution in the PF is then calculated as follows:

Relative Fitnessk =


N
k
k
K
Fitness
Fitness
1
(3)
6. To form the next generation, solutions from the current PF are randomly selected based on
their Relative Fitness. Accordingly, Pareto optimal solutions with larger values of
Euclidean distance have the higher probability to be selected.
3. Descriptions of the proposed metaheuristic algorithms
A brief description of the MOGA, MOMA, MOPSO, MOSFLA algorithms are presented in the
following subsections.
3.1. Multiple objectives genetic algorithms
GA simulates mechanisms of natural principle of the survival of the fittest. In GA, solutions are
represented using chromosomes containing a group of genes, having values for the variables of
an optimization problem. Each chromosome is, then, evaluated to determine its fitness using
specific objective function. Then, best chromosomes are exchanging their information through
different GA operation (i.e. selection, crossover and mutation), as follows, to produce offspring
chromosomes:
- Selection: Fit chromosomes are likely to be selected to pass to next generations.
- Crossover: Parent chromones are exchanging their genetic information to produce
offspring chromosomes.
- Mutation: It allows for random change of genetic information of individual genes. This
process introduces new genetic information to the current chromosomes, this may lead to
avoid stagnation and pre-mature convergence.
The GA operators are continued for large number of generations until an optimal or near-optimal
solution is obtained. The parameters that affect the performance of the GA are: number of
chromosomes, number of generations, crossover rate, mutation rate and crossover type. The
probability of obtaining a global optimum solution is achieved by increasing the numbers of both
population and generations but, this substantially increases processing time.
In the proposed MOGA, for each generation, the steps from (1) to (5) mentioned in the previous
section are carried out then, step (6) is continued to produce offspring chromosomes through
crossover and mutation operators until a new population of chromosomes is formed. Also, a
replacement strategy is carried out to replace the weakest chromosome in the current generation
with a randomly selected one located in the PF of the previous generation. These processes are
repeated for large number of evolutionary cycles until an optimum or near-optimal solution is
reached. Figure 1 shows a flow chart of the proposed MOGA.

Fig. 1. Flowchart describing both the MOGA and MOMA algorithms.
Applied only in MOMA
A
A
Read input data
Evaluation of all objective
functions and constraints
Start
Determination of non-
dominated solutions located
in the PF
Determination of the
compromise solution for the
obtained PF
Calculation of final PF from
the obtained ones in each
generation
Is the
termination
condition
satisfied?
No
Yes
Stop
Separation of PF in an
external repository
Write the final PF
Generation of initial
population of chromosomes
Calculation of fitness and relative
fitness, Eqs. 2 and 3, for each
chromosome in the PF
Application of replacement
strategy
Application of GA operators
(selection, crossover, and
mutation) to form new
population of chromosomes
Application of local search on
each chromosome
Re-evaluation of all objective
Is max.
number of
swaps
reached?
No
Yes
The change is kept if the
chromosome’s performance
improves

3.2. Multiple objectives memetic algorithm
Similar to the GA, MA chromosome’s elements are called memes, not genes. In MA,
chromosomes are allowed to perform local search first, before being involved in the global
evolutionary search [38]. Similar to GA, a population of chromosomes is, initially, generated
randomly. Then, each generated solution is subjected to a local search to improve its fitness.
Afterwards, the GA operators are applied, to produce new individuals. These new individuals are
then performing some local search to maintain local optimality. In this paper, the local search is
carried out through a pair-wise interchange (or in other words, swapping chromosome’s memes)
as shown in Fig. 2, [38].
Fig. 2. Applying local search using pair-wise interchange (Elbeltagi et al. 2005).
The maximum number of swaps is limited to 10 for each chromosome in order to maintain
reasonable processing time. After swapping, if the chromosome’s fitness improves, the change is
kept; otherwise, ignore the change. As previously mentioned, the MA parameters are the same as
the parameters of the GA, in addition to the local-search mechanism.
The proposed MOMA is the same as the MOGA in addition to the local search previously
mentioned. After performing each swap, a chromosome before change (chbefore) is compared with
its mate after (chafter) the swaps, as follows:
- If the chafter dominates the chbefore, then the change is kept.
- If the chbefore dominates the chafter, then the change is ignored and swaps are repeated for a
specified number of times (10 times).
- If no domination occurs, then one of the two chromosomes is randomly chosen as the new
chromosome.
Figure 1 shows the flow chart of the MOMA.
3.3. Multiple objectives particle swarm
The PSO was initially developed by Kennedy and Eberhart [39]. In single objective PSO, a set of
random particles (solutions) is initially generated to begin the evolution process. Through such
process, three values are monitored by each particle i: its current location [Xi(I)], the best
position it arrived [Pi(I)], and its velocity of flying [Vi(I)] where, I is the current evolution cycle.
In each evolutionary cycle, the position of best particle (Pg) is determined which is considered
the best fitness for all particles. Consequently, the velocity of each particle is updated as given by
Shi and Eberhart [40]:

       
     
 ,
1 2
2
1
1 I
X
I
P
r
C
I
X
I
P
r
C
I
V
I
V i
g
i
i
i
i 









  (4)
  ,
1 max
max V
I
V
V i 


 p
n
i ,.....,
1

Each particle position is updated as follows (Eq. 5):
     
1
1 


 I
V
I
X
I
X i
i
i (5)
in which, Vi(I+1): particle i velocity in iteration (I+1); Vi(I): particle i velocity in iteration (I);
Pi(I): particle i best position in iteration (I); Pg(I): the global best particle position in iteration (I);
Xi(I): particle i position in iteration (I); Xi(I+1): particle i position in iteration (I+1);  : constant
called inertia weight; C1 and C2: learning factors (C1 = C2 = 2); r1 and r2: two random numbers
(ranging from 0 to 1); Vmax: the maximum value of velocity change.
The main PSO parameters are: the population size; number of cycles; Vmax; C1; C2; and ω.
In the suggested MOPSO, both Eqs. 4 and 5 are, also, adopted but, the two parameters Pi and Pg
need to be determined repetitively during each cycle [41]. Initially, Pi represents particle i initial
position which updated in the subsequent iterations as given by Baltar and Fontane [42] for each
particle i:
- If the current position Pi dominates the new position, then the new Pi is set as the current
Pi.
- If the current Pi is dominated by the new position, then the new position is set the new Pi.
- If no domination occurs, then the new Pi is selected randomly from the two positions.
In the MOPSO, there is no best particle position (Pg) for all particles, but there are several
equally good particles positions (PF) in each evolution cycle stored in the external repository.
For each evolution cycle, the steps from (1) to (6) mentioned in the previous section are carried
out to select one solution from the stored ones which is considered as Pg. The flow chart of the
MOPSO is shown in Fig. 3.
3.4. Multiple objectives shuffled frog leaping
SFLA combines the advantages of both MA and PSO algorithms. In the single objective SFLA,
the evolution process is initialized with a number of memeplexes, each contains a number of
frogs (individuals). Throughout each memeplex, two individuals (frogs) are determined based on
their fitness: the best individual (XB) and the worst individual (XW). Also, in each cycle of
evolution, the individual with the highest global fitness (XG) is determined. Then, the individual
with the worst fitness is improved in each cycle (not all frogs) through a process similar to the
PSO. Accordingly, the change in worst frog position (Di) and its new position (XWnew) are
calculated as follows [38,43]:
Di = C × r × (XB – XW) (6)
XWnew= XWcurrent + Di, Dmax ≥ Di ≥ - Dmax (7)

in which, C: the search acceleration factor; r: a random number in the range [0, 1]; and Dmax: the
maximum allowed frog’s position change.
Fig. 3. Flowchart describing the MOPSO algorithm.
Read input data
Randomly initialize all particles
positions
Start
Determination of the best
position vector of particle i
(Pi)
Determination of the best
particle position, Pg using
roulette wheel selection
Calculation of velocity and
updating position for each
particle, Eqs. 4 and 5
Is the
termination
condition
satisfied?
No
Yes
Stop
Determination of PF and
Separation in the external
repository
Write the final
PF
obtained PF
Calculation of fitness and
relative fitness, Eqs. 2 and 3,
for each particle in the PF

If a better individual (solution) is produced when applying Eqs. 6 and 7, the worst frog is
replaced by the new produced one. Otherwise, Eqs. 6 and 7 is repeatedly calculated but XB is
exchanged with the global best frog XG. If no improvement is occurred, a new frog (solution) is
created randomly replacing the worst one. This process continues for a given number of
iterations [38].
The SFLA parameters are: number of frogs; number of memeplexes; number of trials in each
memeplex; number of iterations; Dmax; and C.
In the proposed MOSFLA, there are several solutions for (XB) and several other solutions for
(XW) in each memeplex. Consequently, both the (XB) and (XW) are randomly selected from the
corresponding solutions. Also, there is no global best frog (XG) for all frogs, but there are several
equally good frogs’ positions (PF) in each evolution cycle stored in the external repository. In
this case, XG is determined by applying the steps from (1) to (6) mentioned in the previous
section in each evolution cycle. Figure 4 shows the flow diagram of the MOSFLA algorithm.
4. Models implementation and verification
To facilitate the implementation of the proposed four algorithms, each algorithm is coded by the
authors using the FORTRAN language. Accordingly, all parameters related to each algorithm are
studied. The most common parameters presented in the literature are studied here in the current
development. Then, the proposed algorithms’ performance is compared using two multiple
objectives benchmark problems (Appendix A) where the obtained PFs are compared with the
corresponding known true PFs of the two benchmark problems.
Selection of Parameters’ values for each algorithm is an essential part of this study. As such, a
large number of trials are performed to obtain the most suitable parameters’ values that suit the
two test problems and the example application (presented later). In this step, initial parameters’
values are set based on relevant literature. Then, parameters’ values are changed through many
experiments while the results are monitored. The final parameters’ values adopted for each
model are presented in Table 1. Also, different termination criteria have been experimented with
including the solution convergence and number of evolutionary cycles. Based on many trials and
experiments, the termination criterion has been set based on the number of evolutionary cycles
(iteration-oriented) as presented in Table 1.
For the purpose of comparing the proposed algorithms performance, different metrics are used,
namely: the generational distance (Gd); the distance between the PC solution of the obtained PF
and the true PF; the processing time; the number of Pareto-optimal solutions located on the PF
and the mean square error between the obtained compromise solution and the nearest best
alternative Pareto optimal solution. The values of all these metrics are calculated and presented
in Table 2.

Fig. 4. Flowchart describing the MOSFLA algorithm.
Read input data
Randomly initialize all frogs’
positions
Start
Determination of PF and it is
separated in the external
repository
Local search A
Shuffle the memeplexes
Is the
termination
condition
satisfied?
No
Yes
Stop
obtained PF
Write the final
PF
Calculation of fitness and
relative fitness, Eqs. 2 and 3,
for each frog in the PF
Determination of the global
best frog (XG) using roulette
wheel selection
For each memeplex,
determine XB and XW
Apply Eqs. (6) and (7)
Is the new
frog better
than the
worst?
No
Apply Eqs. (6) and (7) with
replacing XB by XG
Is the new
frog better
than the
worst?
No
Yes
Yes
Generate a new frog randomly
Replace the worst frog
A Local search
Partition all frogs into
memeplexes

Table 1
Parameters’ values used in the two test problems and the example application.
Algorithm Parameter names Test problem (1):
Kita function
Test problem (2):
Kursawe function
Example
application
MOGA - Number of chromosomes:
- Number of generations:
- Crossover ratio:
- Mutation ratio:
- Crossover type:
200
1000
0.9
0.2
Uniform
100
500
0.9
0.2
Uniform
100
300
0.8
0.05
Uniform
MOMA - Number of chromosomes:
- Number of generations:
- Crossover rate:
- Mutation rate:
- Crossover type:
200
1000
0.9
0.2
Uniform
100
500
0.9
0.1
Uniform
100
300
0.8
0.05
Uniform
MOPSO - Population size:
- Number of cycles:
- The maximum velocity
Vmax:
- Values of learning factors
C1 and C2:
- Value of inertia weight ω:
100
500
0.35
2
Linear [1.4-0.9]
100
500
0.6
2
Linear [1.4-0.9]
100
300
5500
2
Linear [1.3-
0.9]
MOSFLA - Number of frogs:
- Number of memeplexes:
- Number of trials before
shuffling:
- Number of iterations after
shuffling:
- The maximum change in
frog position Dmax:
- The search acceleration
factor C:
200
20
30
100
7
2.4
200
20
30
100
10
1.7
200
20
20
300
7000
4.2
The Gd metric [44] is considered to test the performance of the developed four models for coping
with multiple objectives problems. The Gd metric is used to measure the distance between the
obtained and the true PFs:
N
d
G
N
i
i
d


 1
2
(8)
in which, di: the Euclidean distance between the non-dominated solution i located in the obtained
PF and the nearest one in the true PF and N is the number of non-dominated solutions located in
the obtained PF. A very small value for Gd metric means that all the generated non-dominated
solution nearly coincides with the true PF.

Table 2
Comparison among algorithms’ results for the two test problems.
Algorithm metric Test problem (1): Kita function Test problem (2): Kursawe function
MOGA
Gd 0.00076 0.00159
Dist. 0.1843 0.1196
Processing time (second) 2.917 1.451
Compromise solution (2.79, 8.11) (-16.7, -4.72)
Best alternative (2.63, 8.12) (-16.69, -4.64)
MSE 0.001733 0.000163
No. of PF solutions 187 419
MOMA
Gd 0.00086 0.00051
Dist. 0.2951 0.0150
MSE 0.007058 0.000089
MOPSO
Gd 0.00064 0.00326
Dist. 0.061 0.0114
MSE 0.000018 0.000002
MOSFLA
Gd 0.00102 0.00290
Dist. 0.0051 0.0301
Compromise solution (2.78, 8.11) (16.71, -4.79)
MSE 0.000061 0.000004
The graphical comparison between the obtained PFs, associated with each proposed model, and
the true PF is presented in Fig. 5. It can be seen that, the obtained PFs seem regular and
distributed throughout the solution space. The Gd value is found to be 0.00076, 0.00086,
0.00064, and 0.00102 for MOGA, MOMA, MOPSO, and MOSFLA respectively. Consequently,
the obtained PFs are nearly similar to the true PF as per the very small values obtained of the Gd.
Also, the minimum Gd value obtained is associated with the MOPSO algorithm.
Figure 6, on the other hand, shows the obtained PFs for the proposed four models along with
their corresponding true PF for the second benchmark problem. As noticed in the previous test
problem, the obtained PFs seem distributed regularly all over the solution space. The Gd values
are calculated as: 0.00159, 0.00051, 0.00326, and 0.00290 for MOGA, MOMA, MOPSO, and
MOSFLA models respectively. Then, the obtained PFs are nearly similar to the true PF as per the
very small distances of Gd metric, with the best value associated with the MOMA.

Fig. 5. PFs produced by the four proposed algorithms and the true PF for the Kita function.
The distance between the compromise solution of the obtained PF and the corresponding one of
the true PF is calculated according to Eq. 9 and listed in Table 2 for each studied algorithm. The
minimum distance reported is for the MOPSO with the second test problem and for the
MOSFLA with the first test problem.
   2
2
. obt
true
obt
true y
y
x
x
Dist 


 (9)
in which, xtrue and ytrue: the coordinates of the compromise solution corresponding to the true PF
and xobt and yobt: the coordinates of compromise solution corresponding to the obtained PF.
Also, the Mean Square Error (MSE) is calculated based on the values of the criteria 0
j
f for the
PC solution and the values of the corresponding criteria *
ij
f for each of the N Pareto-optimal
solutions (Eq. 10) [36]:
  )
,......,
1
(
;
1
)
1
(
2
0
*
n
i
f
f
n
MSE i
i 


  (10)
True
MOGA
True
MOMA
True
MOPSO
True
MOSFLA

where, n: number of criteria (i.e. objective functions). The solution that has the smallest MSE
value is considered as the best alternative solution of the PC solution.
Fig. 6. PFs produced by the four proposed algorithms and the true PF for the Kursawe function.
All codes are running on a personal computer of Intel (R) Core (TM) I3 CPU and having 4 GB
Ram. The least processing time reported with the MOPSO for both test problems. The MOMA
outperforms all other algorithms for the obtained number of solutions located on the PF for the
second test problem, while the MOPSO outperforms other algorithms for the same metric in the
first test problem. Generally, the results show that the MOPSO outperforms other algorithms in
most of the comparison criteria as presented in Table 2.
The results of the previous two test problems show that, the obtained PFs corresponding to the
four models are very close to the true PFs. Also, both the rapid convergence and good results are
obtained in the two cases. Accordingly, the proposed four models having the ability for dealing
with problems of multiple-objective functions.
5. Example application
In this section, the four proposed models with the new evolution strategy are applied on a
popular groundwater management problem in order to maximize the pumping rates and
True
MOGA
True
MOMA
True
MOPSO
True
MOSFLA

minimize the operation costs [24]. Also, a comparative study among the algorithms’ results is
carried out. Both the plan of the studied aquifer and its cross-sectional elevation along with the
required data are shown in Fig. 7. The aquifer is formed of sand and gravel with homogeneous
and isotropic porous medium. The hydraulic conductivity (K) equals 50 m/day and it is assumed
that the aquifer is subjected to uniform rainfall (W) of 0.001 m/day. The objective of this problem
is to determine the pumping rates from the pre-specified system of the ten wells, achieving
hydraulic constraints. The mathematical formulation of the optimization problem is as follows
[15,24]:







 

W
N
i
i h
P
Q
f
1
1 )
(
.
max (11)
           
 












 

W
N
i
i
i
i
b
i
i
b
i
b
i Q
P
h
P
h
d
Q
a
h
d
Q
a
d
y
a
f
1
3
2
1
2
3
2
1
.
min (12)
Subjected to:



W
N
i
req
i Q
Q
1
. (13)
hi ≥ hi, min, i = 1, 2, 3,…….NW (14)
Qi min ≤ Qi ≤ Qi max, i = 1, 2, 3,….... NW (15)






0
0
0
1
i
i
Q
if
Q
if
k i = 1, 2, 3,…....NW (16)







min
min
min
0
)
(
i
i
i
i
i
i
h
h
if
h
h
if
h
h
h
P i = 1, 2, 3,…… NW (17)












.
.
.
0
)
(
req
req
req
Q
Q
if
Q
Q
if
Q
Q
Q
P i = 1, 2, 3,…… NW (18)
where, Qi: well i pumping rate; NW: number of wells; )
(h
P : head violation penalty; hi, min:
minimum head allowable at well i (= 0); Qi min and Qi max: well i minimum and maximum
allowable pumping rates (equals 0 and 7000 m3
/day respectively); a1, a2, and a3: cost coefficients
for drilling, pumping equipment and operation of well respectively (a1 = 4221 $/m, a2 = 0.12
$/m4
, a3 = 0.03 $/m4
); y: the penalty term; b1, b2, and b3: constants indicating economies of scale
(b1 = 0.299 and b2 = b3 =1); Qreq: the water demand required (= 30000 m3
/day); di: the depth of
well i; and )
(Q
P : penalty term for the minimum pumping violation.

Fig. 7. The plan view and sectional elevation of the studied aquifer [24].
In this application, a simulation model, based on FEM, is linked with each proposed optimization
algorithm to predict the hydraulic head at each well location shown in the plan, Fig. 7. FEM is
used to discretize the studied plan to number of linear rectangular elements and the two-
dimensional flow equation, Eq. 19 is solved over each element [25].
   
i
N
i
i
i y
y
x
x
Q
W
y
K
x
K
W















1
2
2
2
2
(19)
in which, K: the hydraulic conductivity of aquifer;  : the potential (= h2
/2); h: the hydraulic
head; W: the uniform rainfall or uniform evaporation; Qi: the rate of injection or pumping for the
well i; δ(z): the Dirac delta function which equals 1 if z equal zero otherwise equals zero; and
NW: number of field wells.
In the simulation model, Eq. 19 is written as follows:
[𝐴] × [φ] = [𝐹] (20)
(a)
Z
Z
4500 m
x
y
(1) (2) (3)
(4) (5) (6)
(10)
(9)
(8)
(7)
River, h=20 m
Swamp, h=20 m
(b)
980
1000
1020
Elevation
(m)
Slate Bedrock
Swamp
River
Areal Recharge
W = 0.001 m/d
Sand & Gravel
K = 50 m/d
N

in which, A: the conductance/stiffness matrix, φ: the unknown vector of potentials and F: the
load vector which contains the external source or sink of water and flux concentration.
In order to make a comparison among the obtained PFs of the developed four models, in addition
to the metrics described before, the Spacing (SP) metric [44] is considered, Eq. 21. The SP is used
to measure the spread of the Pareto optimal solutions over the whole solution space.
 





N
i
i
P d
d
N
S
1
2
1
1
(21)
in which, di: the minimum distance between Pareto-solution i and all other solutions located in
the PF, d : the arithmetic mean of di, and N: the number of solutions located in the PF.
Fig. 8. PFs for the example application produced by the developed four algorithms.
A small value of SP means that solutions located in the PF are equidistantly spaced. Figure 8
illustrates the obtained PFs associated with each developed model. As presented in Table 3, the
SP metric is computed for each algorithm as 231.6, 2842.6, 129.3, and 479.9 for MOGA,
MOMA, MOPSO, and MOSFLA respectively. The SP value for MOPSO model is the minimum
then, the performance of this model is the best compared with the other models. Accordingly, the
obtained PF from MOPSO model is more regular and distributed throughout the solution space.
MOPSO
Compromise solution
MOSFLA
Compromise solution
(45625.6 & 189477.3) (44655.1 & 184663.4)
MOGA
Compromise solution
MOMA
Compromise solution
(45077.4 & 186944.6) (45868.7 &
189665.6)

Also, the processing time of MOPSO model (138.97 sec) is found to be the minimum compared
with the processing time of the other models. Also, the highest number of Pareto optimal
solutions obtained was with the MOPSO. From the visual inspection of Fig. 8, the PFs obtained
using MOGA and MOMA models are more uniformly distributed compared with the other two
models. The Pareto compromise solution for each obtained PF is, also, determined as shown in
Fig. 8. It is noticed that, the deviation among the values of compromise solutions is small. In
general, the results showed the effectiveness of using the suggested evolution strategy in each
model to drive the final PF for the example application. Consequently, the proposed four models
can be used to deal with groundwater real life applications having multiple objectives.
Table 3
Comparison among algorithms results for the example application.
metric
Algorithm
Sp Processing
time (second)
Compromise
solution
Best
alternative
MSE No. of PF
solutions
MOGA
231.6 174.408 (45094.6,
186891,0)
(45077.5,
186944.6)
0.000000114 1221
MOMA
2842.6 745.166 (45792.4,
189431.5)
(45868.7,
189665.6)
0.00000215 1551
MOPSO
129.3 138.965 (45582.2,
189013.0)
(45625.6,
189477.3)
0.00000347 3968
MOSFLA
479.9 593.794 (44672.6,
184592.2)
(44655.1,
184663.4)
0.000000151 186
In general, the MOPSO outperforms all other algorithms in most of the comparison criteria,
which confirm with some previous studies. Also, identifying a unique PC solution helps the
decision makers to determine a unique solution that satisfies all objectives fairly.
6. Limitations and suggested improvements
The four developed metaheuristic models have been applied on a ground water example
application and worked effectively. Also, the models were experimented with two bench mark
problems with different shapes of PFs and performed well. The developed models used a unique
solution that satisfies fairly all competing criteria of the problem "PC solution" to drive the
fitness. As such, tracking a single point resembles optimizing a single-objective. Also, it helps
the decision maker by presenting a unique PC solution especially in complex objective spaces.
Despite their perceived benefits, the developed model still has some limitations with a number of
possible improvements, including:
- For the groundwater management problem, considering other objectives in addition to the
total pumping and total cost;
- studying different aquifers shapes other than rectangular or near rectangular shapes.; and
- Compare the performance of the proposed multi-objective optimization strategy with other
strategies used in the literature

7. Conclusions
This research applied new evolution strategy of tracking a single compromise solution to modify
the GA, MA, PSO, and SFLA algorithms to tackle multiple objectives problems. Two multiple
objectives standard test problems with known PFs are used to check the ability of the proposed
models with the proposed evolution strategy to arrive at the correct PFs. The proposed four
algorithms are then applied on a popular groundwater management example for maximizing
pumping rates and minimizing operation costs.
The suggested four models proved their ability to obtain the close true PFs for the two
benchmark problems in addition to the groundwater application. Accordingly, they can
effectively be used to deal with real-life groundwater management application. The performance
of the different multi-objective algorithms is studied and compared. The MOPSO outperformed
all other algorithms based on the used performance metrics in terms of regularity and distribution
of PF throughout the solution space and smallest processing time when compared with other
multi-objective optimization algorithms. In this application, a finite element-based simulation
model, with a new idea to reduce the run time, is linked with each proposed model for evaluating
the potential solution.
Appendix A
A.1. Benchmark problem (1)
The first benchmark problem is called the “Kita” problem with two objectives (Eq. A.1), two
unknowns, and three constraints (Eq. 9) [44]:
7
,
0
,
1
5
.
0
)
,
(
,
)
,
( 2
2
1 







 y
x
p
y
x
y
x
f
p
x
y
y
x
f (A.1)
Subjected to:
  0
30
5
,
0
2
15
2
1
,
0
2
13
6
1
3
2
1 






















 y
x
p
y
x
p
y
x
p (A.2)


 



Otherwise
p
p
p
if
p
p
p
P
0
0
,
,
12 3
2
1
3
2
1
(A.3)
where, P is the penalty term for constraints violations. In this study, different penalty functions
are experimented with (linear, exponential, etc.) and the best results were associated with the
proposed linear penalty function (Eq. A.3).
A.2. Benchmark problem (2)
The second benchmark problem is called the “Kursawe” problem with two objectives and three
unknowns [44].

   
 








1
1
2
1
2
1 2
.
0
exp
10
n
i
i
i
i x
x
x
f (A.4)
   
 




n
i
i
i
i x
x
x
f
1
3
8
.
0
2 sin
5 (A.5)
where, the three unknowns are in the range [-5, 5].
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